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{subparagraph}{Acknowledgments}{12}{section*.2}\protected@file@percent } +\bibcite{albert_rmp02}{{1}{2002}{{Albert and Barab{\'a}si}}{{}}} +\bibcite{allen2017evolutionary}{{2}{2017}{{Allen et~al}}{{Allen, Lippner, Chen, Fotouhi, Nowak, and Yau}}} +\bibcite{BA}{{3}{1999}{{Barab\'{a}si and Albert}}{{}}} +\bibcite{barabasi_s99}{{4}{1999}{{Barab{\'a}si and Albert}}{{}}} +\bibcite{EHB34164}{{5}{2013}{{Barclay}}{{}}} +\bibcite{bowles_11}{{6}{2011}{{Bowles and Gintis}}{{}}} +\bibcite{boyd2005solving}{{7}{2005}{{Boyd and Richerson}}{{}}} +\bibcite{PCB14e1006347}{{8}{2018}{{Chen and Szolnok}}{{}}} +\bibcite{cressman_03}{{9}{2003}{{Cressman}}{{}}} +\bibcite{evans2019cooperation}{{10}{2019}{{Evans and Rand}}{{}}} +\bibcite{evans2015fast}{{11}{2015}{{Evans et~al}}{{Evans, Dillon, and Rand}}} +\bibcite{fehl_el11}{{12}{2011}{{Fehl et~al}}{{Fehl, van~der Post, and Semmann}}} +\bibcite{fehr_n02}{{13}{2002}{{Fehr and G{\"a}chter}}{{}}} +\bibcite{fotouhi_rsif19}{{14}{2019}{{Fotouhi et~al}}{{Fotouhi, 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[2][]{subsection.2.3}{Mathematical modeling the available time of individuals}{section.2}% 5 +\BOOKMARK [1][]{section.3}{Results}{}% 6 +\BOOKMARK [1][]{section.4}{Discussion}{}% 7 +\BOOKMARK [1][]{section.5}{Materials and Methods}{}% 8 +\BOOKMARK [2][]{subsection.5.1}{Experimental design}{section.5}% 9 +\BOOKMARK [2][]{subsection.5.2}{Experimental setup and game rules}{section.5}% 10 +\BOOKMARK [2][]{subsection.5.3}{Simulation on the social networks}{section.5}% 11 +\BOOKMARK [3][]{section*.2}{Acknowledgments}{subsection.5.3}% 12 diff --git a/Main/2nd ect.pdf b/Main/2nd ect.pdf new file mode 100644 index 0000000..3819631 Binary files /dev/null and b/Main/2nd ect.pdf differ diff --git a/Main/2nd ect.synctex.gz b/Main/2nd ect.synctex.gz new file mode 100644 index 0000000..42378fc Binary files /dev/null and b/Main/2nd ect.synctex.gz differ diff --git a/Main/2nd ect.tex b/Main/2nd ect.tex new file mode 100644 index 0000000..56a9604 --- /dev/null +++ b/Main/2nd ect.tex @@ -0,0 +1,510 @@ +\documentclass[default]{sn-jnl} +% Use the lineno option to display guide line numbers if required. +% Note that the use of elements such as single-column equations +% may affect the guide line number alignment. +\usepackage{graphicx}% +\usepackage{multirow}% +\usepackage{amsmath,amssymb,amsfonts}% +\usepackage{amsthm}% +\usepackage{mathrsfs}% +\usepackage[title]{appendix}% +\usepackage{xcolor}% +\usepackage{textcomp}% +\usepackage{manyfoot}% +\usepackage{booktabs}% +\usepackage{algorithm}% +\usepackage{algorithmicx}% +\usepackage{algpseudocode}% +\usepackage{listings}% +\usepackage{flushend} +\usepackage{float} +\usepackage{dsfont} +\usepackage{setspace} +\setstretch{0.95} +% \usepackage{widetext} + +%% as per the requirement new theorem styles can be included as shown below +\theoremstyle{thmstyleone}% +\newtheorem{theorem}{Theorem}% meant for continuous numbers +%%\newtheorem{theorem}{Theorem}[section]% meant for sectionwise numbers +%% optional argument [theorem] produces theorem numbering sequence instead of independent numbers for Proposition +\newtheorem{proposition}[theorem]{Proposition}% +%%\newtheorem{proposition}{Proposition}% to get separate numbers for theorem and proposition etc. + +\theoremstyle{thmstyletwo}% +\newtheorem{example}{Example}% +\newtheorem{remark}{Remark}% + +\theoremstyle{thmstylethree}% +\newtheorem{definition}{Definition}% + +\raggedbottom +%%\unnumbered% uncomment this for unnumbered level heads + +\begin{document} + +\title{Limitation of time promotes cooperation in temporal network games} +% Use letters for affiliations, numbers to show equal authorship (if applicable) and to indicate the corresponding author + +\author[1,2]{Jiasheng Wang} +%\email{1510478@tongji.edu.cn} +\affil*[1]{Department of Computer Science and Technology, Tongji University, 4800 Cao'an Road, Shanghai 201804, China} +\affil[2]{Key Laboratory of Embedded System and Service Computing (Tongji University), Ministry of Education, Shanghai 200092, China} + +\author*[1,2]{Yichao Zhang} +%\affil[1]{Department of Computer Science and Technology, Tongji University, 4800 Cao'an Road, Shanghai 201804, China \\ Key Laboratory of Embedded System and Service Computing (Tongji University), Ministry of Education, Shanghai 200092, China} + +\author*[3,4]{Guanghui Wen} +\affil[3]{Department of Systems Science, School of Mathematics, Southeast University, Nanjing 210016, China} +\affil[4]{School of Engineering, RMIT University, Melbourne VIC 3000, Australia} + +\author*[1,2]{Jihong Guan} +%\email{jhguan@tongji.edu.cn} + +\author[5,6]{Shuigeng Zhou} +%\email{sgzhou@fudan.edu.cn} +\affil[5]{Shanghai Key Laboratory of Intelligent Information Processing, Shanghai 200433, China} +\affil[6]{School of Computer Science, Fudan University, 220 Handan Road, Shanghai 200433, China} + +\author[7]{Guanrong Chen} +%\email{gchen@ee.cityu.edu.hk} +\affil[7]{Department of Electrical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Hong Kong SAR, China} + +\author[8]{Krishnendu Chatterjee} +\affil[8]{Institute for Science and Technology, A-3400 Klosterneuburg, Austria} + +\author[9,10,11]{Matja{\v z} Perc} +\affil[9]{Faculty of Natural Sciences and Mathematics, University of Maribor, Koro{\v s}ka cesta 160, 2000 Maribor, Slovenia} +\affil[10]{Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan} +\affil[11]{Complexity Science Hub Vienna, Josefst{\"a}dterstra{\ss}e 39, 1080 Vienna, Austria} + + +\abstract{Temporal networks are obtained from time-dependent interactions among individuals, whereas the interactions can be emails, phone calls, face-to-face meetings, or work collaboration. In this article, a temporal game framework is established, in which interactions among rational individuals are embedded into two-player games in a time-dependent manner. This allows studying the time-dependent complexity and variability of interactions, and the way they affect prosocial behaviors. Based on this simple mathematical model, it is found that the level of cooperation is promoted when the time of collaboration is equally limited for every individual. This observation is confirmed with a series of systematic human experiments that forms a foundation for comprehensively describing human temporal interactions in collaboration. The research results reveal an important incentive for human cooperation, leading to a better understanding of a fascinating aspect of human nature in society.} + +\keywords{cooperation, non-cooperative game, temporal network, time limitation} + +\maketitle + +\section{Introduction}\label{sec1} + +Many complex collaborative systems in nature, society, and engineering can be modeled through networks based on graph theory. In a network, nodes represent collaborating individuals, and links represent their friendships~\cite{albert_rmp02}. In simple or simplified network modeling, links are weightless, undirected, and static. In order to improve the ability to depict real systems, weighted~\cite{shen_rsos18}, directed~\cite{Pagan2019game}, and dynamic~\cite{melamed_pnas18} network models are established. The application of these network models in social science proved that the closer the framework is to reality, the stronger its ability to explain behaviors. As one of such social behavior in human interactive systems, cooperation is of particular importance which has attracted broad attention for more than half a century~\cite{boyd2005solving, gachter2009reciprocity, rand_tcs13, perc_pr17}. Although humans are not exempted from selfishness, and they obey the fundamental principles of Darwinian competition-based evolution, cooperation is ubiquitous in and across societies~\cite{nowak_11}. While the impetus for the human strong cooperative drive has been linked to the difficulties of the genus \textit{Homo} in rearing offspring that survived and to the emergence of alloparental care~\cite{hrdy_11}, and also to the formation of alliances in times of conflicts~\cite{bowles_11}, it is still puzzling as why they have achieved such high levels of cooperation in general. Human altruistic behavior distinguishes them remarkably from other mammals, forming the bedrock for their astonishing evolutionary success in history. + +The studies of human cooperation in $n$-person games begin with population games, also known as mean-field games~\cite{maynard_n73,hofbauer_98,cressman_03}. In a well-mixed population, cooperation can hardly prevail with imitative update rules when individuals play non-cooperative games such as the Prisoner's Dilemma (PD) game~\cite{szabo_pr07}. If the population exhibits a relatively stable social structure, the consequence may be different~\cite{santos_prl05, ohtsuki_n06, santos_pnas06, santos_n08, tanimoto_pre07, fu_pre09, lee_s_prl11, rand_pnas14, fu2017leveraging, allen2017evolutionary, fotouhi_rsif19} -- a finding rooted in the seminal paper by Nowak and May~\cite{nowak_n92b}, observing clusters of cooperators on a square lattice that protected them from invading defectors. Nevertheless, social networks are seldom static. People disconnect and then reconnect to form connections with new partners from time to time. This reality has revealed new mechanisms for cooperation that may sustain even under extremely adverse conditions, when the temptation to defect is high and where on static network cooperation is perishing~\cite{perc_bs10}. Moreover, an individual usually does not interact with all his friends all the time but likely does so only occasionally. + +To account for the above-observed phenomena, some researchers considered dynamic networks. Implications of dynamic interactions on human cooperation are profound. Recent human experiments as well as theoretical analysis both have confirmed this to the fullest~\cite{rand_pnas11, fehl_el11, wang_j_pnas12, szolnoki_epl14b, wang_z_njp14, shen_rsos18}. It is argued, for example, that these observations demonstrate the effects of reputation~\cite{melamed_pnas18}. Individuals may connect with unfamiliar individuals after browsing their gaming records but cut some existing connections with unsatisfactory partners. Some may take breaking ties, instead of performing defection, as a way to penalize defectors~\cite{rand_pnas11}. Interestingly, the implication of dynamic reconnection fades out as individuals are taking more specific moves to play games with their partners~\cite{melamed_pnas18}. In light of this, an interesting question is whether dynamic reconnection is relevant to the level of cooperation in a human collaborative system if there is a time limit on the duration of a game. From the perspective of biological markets~\cite{EHB34164}, the dynamic reconnection in such a system is a reallocation of collaboration time in a time-limited collaborative condition. Will too much emphasis put on the structure of our social networks result in neglecting the temporal aspects of our interactions? In this article, this critical question will be addressed. + +Due to the complexity of temporal systems, using evolutionary game theory to model collaboration behavior is quite challenging. First, the evolution mechanism of a temporal system itself is complicated and hard to describe by a simple mathematical model. Secondly, in a temporal game, the individual strategy involves not only the moves but also the allocation of time in each round of the game. Furthermore, this openness allows individual strategy and network topology to co-evolve in a more flexible way than the existing dynamical gaming networks~\cite{zhang2018gaming, miritello2013time}, which brings up the difficulty in modeling coupled systems. + +In this paper, a temporal gaming framework is proposed based on the structure of temporal networks~\cite{GTN,holme_sr12}. The main objective is to test the impact of limited time on the level of cooperation in two-player collaborative systems. Such systems are common in reality. For instance, it usually takes a team to accomplish a project when applying for funding. The project leader would typically collaborate with a member to accomplish a specific part of it. Meanwhile, the member or the leader may also be involved in more than one project. Simultaneously, the total number of working time, such as months, for each participant is limited, which is identical for everyone. In such a scenario, a temporal gaming network is naturally laid out. Here, the collaboration between two team members is closer to the stag hunt game than the PD game. Since cooperation normally dominates the collaboration system playing the stag hunt game, it is not easy to differentiate the impacts from various mechanisms. For this reason, the PD game is adopted in this paper. + +%We need to draw another illustration to explain the temporal games not that on networks. + +One of the main contributions of this paper is a detailed online experiment for demonstrating the proposed theoretical framework. First, a gaming platform is established to implement a temporal game. Then, the level of cooperation is tested on the platform in a divide and conquer (D\&C) mode~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18}, where the difference from the present setting with those of the existing works~\cite{fehl_el11,rand_pnas11,gracia-lazaro_pnas12,rand_pnas14} is in the targeted decisions. Finally, the level of cooperation is tested on the platform in a time-dependent mode, where both the time limitation for individuals and the targeted decisions are considered. The reasons for adopting these mechanisms are explained in Section~\emph{Experimental design}. The objective is to find whether the limitation on time resources governs human cooperation in the games, which is the focus of this study. + +%Demonstration of the assumption +To understand the impact of the limited time, we invited 183 human subjects and carried out a set of comparative online experiments. In a match of the game, the participants are allocated to the nodes of some pre-generated networks. Two classes of networks are tested, namely the Barab\'{a}si-Albert scale-free network~\cite{BA} and the Watts-Strogatz small-world network, for they are the most popular social network models. It will be shown that the limitation to the individuals' time resources indeed promotes the participants' level of cooperation, which aligns with the theoretical prediction, as further discussed below. + +\section{Theoretical framework of temporal games}\label{FTG} +\subsection{Temporal game model}\label{Tgm} + +In a two-strategy (i.e., only two moves are allowed) game, define $i$'s strategy as +$ +\Omega_i=\left( + \begin{array}{cc} + X_i \\ + 1-X_i \\ + \end{array} +\right),%\label{eqn:ti} +$ +where $X_i$ can only take $1$ or $0$ in each game and each round of the play. If $X_i=1$, $i$ is a cooperator denoted by $C$; if $X_i=0$, $i$ is a defector denoted by $D$. Take the PD game~\cite{maynard_82} for example. In the PD game, the payoff table is a $2\times2$ matrix. Given $i$'s strategy, $i$'s payoff in the game playing with all his neighbors (denoted by $N_i$) can be written as +$ + G_i={\Omega_i^T}\left( + \begin{array}{cc} + \mathcal{R} & \mathcal{S}\\ + \mathcal{T} & \mathcal{P}\\ + \end{array} +\right)\sum_{j \in N_i}{\Omega_j}.%\label{eqn:G_i} +$ +In this PD model, a player gains $\mathcal{T}$ (the temptation to defect) for defecting a cooperator, $\mathcal{R}$ (reward for mutual cooperation) for cooperating with a cooperator, $\mathcal{P}$ (punishment for mutual defection) for defecting a defector, and $\mathcal{S}$ (sucker's payoff) for cooperating with a defector. Normally, the four payoff values satisfy the following inequalities: $\mathcal{T}>\mathcal{R}>\mathcal{P}>\mathcal{S}$ and $2\mathcal{R}>\mathcal{T}+\mathcal{S}$. Here, $2\mathcal{R}>\mathcal{T}+\mathcal{S}$ makes mutual cooperation the best outcome from the perspective of a collective decision. + +The temporal game model proposed in the present paper is based on the game model used in~\cite{wang_js_sr17,zhang_yc_sr15}, taking into account the time of interactions. As the model is time-dependent, each interaction is assigned a specific duration. The total game time for each individual in one round is set to be a constant, which is the same for all individuals to be realistic in real-life scenarios. An individual's interactions with different partners are assumed to be independent. The payoff of the game between individuals $i$ and $j$ can be written as +$ + s_{i,j}=\Omega_{i,j}^T\left( + \begin{array}{cc} + \mathcal{R} & \mathcal{S}\\ + \mathcal{T} & \mathcal{P}\\ + \end{array} +\right)\Omega_{j,i}\,. %\label{eqn:sij} +$ +In temporal games, the payoff of each interaction is proportional to the time it spends. In one round of the game, the accumulated payoff of the individual $i$ is defined as +\begin{equation} +\Lambda_i = \sum\limits_{j \in {N_i}} {\frac{\tau_{i,j}}{\mathfrak{T}} \times s_{i,j}},\label{LambdaiN} +\end{equation} +where $N_i$ is the set of $i$'s neighbors and $\tau_{i,j}$ is the duration of the interaction between individuals $i$ and $j$. This is shown in Fig.~\ref{fig:ITG}A, where individual $i$ and $j$ are colored red and blue, $N_i=4$ and $\tau_{i,j}=8$. Note that $\tau_{i,j}$ should satisfy the constraints of $\tau_{i,j} \in [0, \mathfrak{T}]$ and $\sum\limits_{j \in {N_i}}{\tau_{i,j}} \leqslant \mathfrak{T}$. Here, $\mathfrak{T}$ is the total time resource an individual has in each round, which is a constant for every individual in the proposed model. In Fig.~\ref{fig:ITG}A, $\mathfrak{T}=24$. If individual $i$ does not want to collaborate with $j$, then $i$ will not play the game with $j$ any longer. Simultaneously, $i$ will reject $j$'s gaming request. In this case, $\tau_{i,j}$ will be $0$ as indicated by the relation between the red and the green in Fig.~\ref{fig:ITG}A. + +\begin{figure}[ht]%[tbhp] +\centering + \includegraphics[height=2 in,clip,keepaspectratio,]{Illustration_of_temporal_game.eps} + \caption{Illustration of the temporal game. Panel A shows one round of the temporal game among five individuals. The individual colored red has four friends, in which the individuals colored orange and blue are his gaming partners. If the game between two individuals lasts for 24 hours, the payoff of a cooperator is 3 and 0, gaining from a cooperator and a defector, respectively. The payoff of a defector is 5 and 1, gaining from a cooperator and a defector, respectively.} + \label{fig:ITG} +\end{figure} + +Let $P_i$ be the set of partners who are interacting with $i$ in this round. Then, Eq.~\ref{LambdaiN} can be written as +\begin{equation} + \Lambda_i = \sum\limits_{j \in {P_i}} {\frac{\tau_{i,j}}{\mathfrak{T}} \times s_{i,j}}, \label{LambdaiP} +\end{equation} +where $\tau_{i,j}$ is greater than 0. For the red individual in Fig.~\ref{fig:ITG}A, the orange and the blue ones are his partners in this round. Based on Eq.~\ref{LambdaiP}, the payoffs of the five individuals are listed in Fig.~\ref{fig:ITG}B. +% + +From a mean-field view, Eq.~\ref{LambdaiP} can be written as +\begin{equation} + \Lambda_{k_i} = \sum\limits_{k_j} {\frac{\tau_{k_i,k_j}}{\mathfrak{T}} P\left(k_i,k_j\right) s_{k_i,k_j}}, \label{LambdaiPkikj} +\end{equation} +where $P(k_i,k_j)$ is the probability that a link exists between $i$ and $j$, depending on the topology of the collaborative network. For a heterogeneous network as the Barab\'{a}si-Albert (BA) networks~\cite{barabasi_s99}, $P(k_i,k_j)\sim\frac{k_jP(k_j)}{\langle k\rangle}$. For homogeneous networks such as the Watts-Strogatz (WS) networks~\cite{Watts98Nature}, $P(k_i,k_j)\sim P(k_j)$. + +An illustration of such a collaborative network is shown in Fig.~\ref{fig:tdnc}A. To clarify the generating procedure of the network, the communication log among the individuals in this round is shown in Fig.~\ref{fig:tdnc}B. In the log, Alice tries to collaborate with Tom for $\mathfrak{T}$, while Tom had agreed to work with Jerry and Frank when he received Alice's request. Thus, Alice turns to Frank and Jerry, but it is a bit late to make appointments with them as they are partially engaged. As a result, Alice takes $0.8\mathfrak{T}$ to play with Frank and Jerry but wastes $0.2\mathfrak{T}$ in this round. +% +\begin{figure*}[!htbp]%[tbhp] + \centering + \includegraphics[width=\textwidth]{vis.eps} + \caption{Illustration of temporal games in a two-player collaborative system. (A) One round of the temporal game on a social network. The blue circle is Jerry's neighborhood. Alice, Bob, and Tom are Jerry's partners in this round. The color of a time slot represents a partner; for instance, yellow represents Frank. $C$ or $D$ in the time slot denotes the move from the individual at the tail of a directed dashed line to the indicated specific partner. (B) The generating procedure of the circumstance is presented in (A). In the communication log, the records are sorted by their sequence numbers in ascending order. Only if both players agree to collaborate (the response to a request is OK) will their colors appear in each other's collaboration schedule, i.e., a time slot in (A).} + \label{fig:tdnc} +\end{figure*} +% + +%To model the correlation between the level of cooperation and the available time resource in the neighborhood, shown in our experimental results, a factor $\theta$, called the backup to defect, is now introduced. +%%\begin{equation} +%% \theta_{k_i,r} = \hat \tau_{f_{k_i,r}} - \sigma_{k_i,r-1}\tau_{u_{k_i,r-1}} P_l, \label{eqn:ttd} +%%\end{equation} +%We define $\theta_{k_i,r}$ as the total available time resource of the neighbors of an individual $i$ with degree $k_i$ at round $r$, namely, +%\begin{equation} +% \theta_{k_i,r} = min\left(\sum\limits_{k_j} P\left(k_i,k_j\right)S_{k_j}\left(r\right),\mathfrak{T}\right), \label{eqn:tf} +%\end{equation} +%where $S_{k_j}\left(r\right)$ denotes the average available time of a neighbor with degree $k_j$ at round $r$. When $\theta > 0$, trading partners may bring $i$ more payoff, since $\mathcal{T}>\mathcal{R}$ in the PD game. Interestingly, one can see that $\theta_{k_i,r} = S\left(r\right)$ in the WS networks. Therefore, one can use $S\left(r\right)$ to measure the available time resource in the neighborhood. Although $\theta_{k_i,r} \neq S\left(r\right)$ in the BA networks, $S\left(r\right)$ is applicable to depict the average available time resource in the neighborhood. Therefore, we will adopt $S\left(r\right)$ to represent the available time resource in the system. + +\subsection{Proportion of cooperation in the temporal game} + +In the temporal game, each game between two players is performed for a duration of time. Thus, the level of cooperation is measured by the duration and their moves. Define the proportion of cooperation as ${P_c} = \frac{{{T_C}}}{T_G}$, where $T_G$ is the total duration of the moves and $T_C$ is the total duration of cooperation in the game. + +%\subsection{Dissipative system in the temporal game} +% +%In our experiment, two types of the temporal social dilemma, dissipative scenario and classical scenario, are tested. In the dissipative scenario, individuals are provided with an initial resource. In each round, individuals play the game with some of their friends to earn more payoffs. Specifically, the aggregated payoff of an individual $i$ in round $r$ ($r\in\mathds{N}$) can be written as +%\begin{equation} +%{\phi_{i,r}} = \left\{ {\begin{array}{*{20}{cl}} +%{{\phi_{i,r - 1}} + {S_{i,r}} - \varepsilon }, & {r\in\mathds{N^+}},\\ +%{{\phi_0}}, & {r = 0}, +%\end{array}} \right. \label{eqn:f} +%\end{equation} +%where $\varepsilon$ denotes the cost in each round, $\phi_0$ is the initial resource, and $S_{i,r}$ is the total payoff of individual $i$ in round $r$ defined in Eqn.~(\ref{LambdaiP}). In the dissipative scenario, we set $\phi_0=5$ and $\varepsilon=3$. In the classical scenario, we set $\varepsilon=\phi_0=0$, which naturally leads to a monotonic ascent of $\phi_{i,r}$ for all $i$ in a match. + +%Caveats + +Note that current studies on decision time~\cite{evans2019cooperation, evans2015fast} in experimental psychology and response time in experimental economics~\cite{yamagishi2017response, spiliopoulos2018bcd} focus on the time instant for making a decision rather than the duration of time spent by moves. As such, the object of those studies is different from that of temporal games. + +%In the following, we will propose a model to reproduce the statistical results in our empirical experiments. As modeling the level of cooperation is a rather challenging task, we adopt a mean-field method and make some necessary assumptions. Note that the assumptions are not the components of the temporal games but only the tools to modeling the level of cooperation. + +\subsection{Mathematical modeling the available time of individuals}\label{MMATI} + +As is well known, in a game between two players, each player has to practice one of the four possible actions, namely, cooperating with a cooperator (CC), cooperating with a defector (CD), defecting a cooperator (DC), and defecting a defector (DD). Here, define a state vector $\mathbf{\Phi}$ by $(\Phi_{CC},\Phi_{CD},\Phi_{DC},\Phi_{DD})$, in which each entry corresponds to the probability of the indicated action. Generally, a memory-one strategy can be written as $\mathbf{p}=(p_{CC},p_{CD},p_{DC},p_{DD})$, corresponding to the probabilities of cooperating under each of the previous action outcomes. Since players update their moves with the memory-one strategies in each time step, the update can be considered a Markov process. A Markov transition matrix $M_i$ can be used to realize the update. For two players, $i$ and $j$, one has +\begin{equation} +\small{ +\!\!M_i\!\!=\!\!\left(\! +\begin{array}{cccc} +\!p_{CC}s_{CC} &\!\! p_{CC}(1-s_{CC}) &\!\! (1-p_{CC})s_{CC} &\!\! (1-p_{CC})(1-s_{CC})\! \\ +\!p_{CD}s_{DC} &\!\! p_{CD}(1-s_{DC}) &\!\! (1-p_{CD})s_{DC} &\!\! (1-p_{CD})(1-s_{DC})\! \\ +\!p_{DC}s_{CD} &\!\! p_{DC}(1-s_{CD}) &\!\! (1-p_{DC})s_{CD} &\!\! (1-p_{DC})(1-s_{CD})\! \\ +\!p_{DD}s_{DD} &\!\! p_{DD}(1-s_{DD}) &\!\! (1-p_{DD})s_{DD} &\!\! (1-p_{DD})(1-s_{DD})\! \\ +\end{array} +\!\right), +} +\end{equation} +where the vectors $\mathbf{p}=(p_{CC},p_{CD},p_{DC},p_{DD})$ and $\mathbf{s}=(s_{CC},s_{CD},s_{DC},s_{DD})$ denote players $i$ and $j$'s probabilities of cooperation in the next round after experiencing $CC$, $CD$, $DC$, and $DD$, respectively. Thus, the evolution of $i$'s state vector $\mathbf{\Phi}_i(t)$ is given by +\begin{equation} +\mathbf{\Phi}_i(r)=\mathbf{\Phi}_i(r-1)M_i. +\end{equation} + +To model the total available time of individuals in the temporal games, assume that no players at round $r-1$ reject the requests from an individual $i$ if they are available. The time left for him to spend in round $r$ is denoted by $S_{i}\left(r\right)=\mathfrak{T}-\sum_{j\in{P_i}}\tau_{u_{ij} \left(r-1\right)}$, where $\mu_{ij}\left(r-1\right)$ is the random portion of time within the request from $i$ in round $r-1$. If $i$ applies for playing with $j$ from $S_{i}\left(r\right)\mu_{ij}\left(r\right)$, the successful probability of the request is +\begin{equation} +{\omega_{i,j}\left(r,\mu_{ij}\left(r\right)\right)} = \left\{ {\begin{array}{*{20}{cl}} +{1}, & {S_{j}\left(r\right)\geq S_{i}\left(r\right)\mu_{ij}\left(r\right)},\\ +{0}, & {S_{j}\left(r\right)< S_{i}\left(r\right)\mu_{ij}\left(r\right)}, +\end{array}} \right. \label{eqn:f} +\end{equation} +assuming that $j$ wishes to play. Therefore, the expectation of difference during individual $i$'s available time from round $r$ to $r+1$ is +\begin{eqnarray} +&\varrho_{i}\left(r\right)=-\sum_{j\in {N_i-P_i\left(r-1\right)}}\omega_{i,j}\left(r,\mu_{ij}\left(r\right)\right)\left(S_{i}\left(r\right)\right. \\\nonumber +&\left.+\sum_{l\in P_i\left(r-1\right)}\alpha_{il}\left(r-1\right) \left( \mathbf{\Phi}_{il}\left(r\right) \cdot \begin{bmatrix}\chi_{i,CC} \\ \chi_{i,CD} \\ \chi_{i,DC} \\ \chi_{i,DD}\end{bmatrix}\right)\right)\mu_{ij}\left(r\right), +\end{eqnarray} +where $\mathbf{\chi_i}$ denotes $i$'s probabilities of reassigning time after experiencing the four outcomes, and $\alpha_{il}\left(r\right)$ denotes the timeshare which $i$ assigned to $l$ at round $r$. Note that +\begin{equation} +\sum_{l\in {P_i}}\alpha_{il}\left(r\right)+S_{i}\left(r\right)=1. +\end{equation} + +%Then, the change in the total available time for the system will be +%\begin{equation} +%\varrho\left(r\right)=N\sum_{k_i}P(k_i)\left(S_{k_i}(r+1)-S_{k_i}\left(r\right)\right),\label{Gammar} +%\end{equation} +%where $N$ denotes the number of users. +% +%To simplify the fitting procedure of the maximum likelihood, we assume that individual $i$ for all $i$ evenly allocates his time to the neighbors, that is, $\mu=\frac{1}{k_i}$. Eqn.~\ref{Gammar} is reduced to +%\begin{eqnarray} +%&\varrho\left(r\right)=-2N\sum_{k_i}P(k_i)\left(\alpha\left(1-S_{k_i}\left(r\right)\right)\left(1-P_c\left(r\right)^2\right)\right.\\\nonumber +%&\left.+\omega_{k_i,k_j}\left(r,\frac{1}{k_i}\right)P(k_i,k_j)S_{k_i}\left(r\right)\right). +%\end{eqnarray} +%With $\tau_{i,r}$ for all $i$, one can then estimate an individual's neighbors' total available time resources and the backup to defect in round $r+1$. + +Since $S_i\left(r\right)\geq0$ for all $r$, the iterative formula of $S_i\left(r\right)$ can be written as +\begin{equation} +S_{i}\left(r+1\right)=Relu\left(\varrho_{i}\left(r\right)+S_{i}\left(r\right)\right), +\end{equation} +where $Relu\left(x\right)= \left\{ {\begin{array}{*{20}{cl}} +{x}, & x \geq 0,\\ +{0}, & x < 0. +\end{array}} \right.$ +As the evolution process of $S_i\left(r\right)$ in the system cannot be modeled in a mean-field manner, one can hardly find and present its analytical solution. Therefore, the following presents only simulation results and empirical results from human online experiments. In the simulations, all agents uniformly adopt the same strategy; therefore, the results are reproducible. Let the number of agents be $N_A$. It will be shown that the average available time $S\left(r\right)=\frac{\sum_i S_i\left(r\right)}{N_A}$ falls to a low level at the first round. It is stabilized thereafter, indicating that finding new partners is problematic from the beginning of a match. + +%Note that a lower $\theta$ value means a higher risk of being isolated after losing the current partners. If the maximum potential benefit of reconnection cannot bring any extra available time resources, individuals are likely to cooperate in order to maintain the existing partnerships. As a result, $\theta$ should be inversely proportional to the frequency of cooperation. Let $\theta^M$ be the increment of available time resource after an individual loses all his current partners in round $r$, i.e., $\sigma=1$. The measurement looks somewhat useless as most players seem unlikely to defect all their partners simultaneously. Indeed, it is a natural choice when the players only have one partner, or their resources are extremely limited. We observe that the move is quite common after the 12th round in our experiments. Therefore, we set $\theta_{i,r}$ to $\theta_{r}^M$ to reduce the parameters of the model. +% +%Let $\phi_r$ be an individual's aggregated payoff in round $r$. If he has a partner in round $r-1$, he will cooperate with the partner with the probability +%\begin{equation} +%{P_{c,r}} = \left\{ {\begin{array}{*{20}{c}} +% {0.5}&{{\phi_{r-1}} \leqslant {\phi_{alive}}} \\ +% {\alpha + \beta\cdot\theta_{r}^M}&{{\phi_{r-1}} > {\phi_{alive}}} +%\end{array}} \right., \label{eqn:pc} +%\end{equation} +%where $\phi_{alive}=\varepsilon-\mathcal{R}$, and $\varepsilon$ is the compulsory consumption each round. $\varepsilon>0$ in the dissipative mode, and $\varepsilon=0$ in the classical mode. We set $P_{c,r}=0.5$ for ${\phi_{r-1}} \leqslant {\phi_{alive}}$, since only if two players adopt different moves one of them can survive one round more when they are both in this condition. $\alpha\geq0$ and $\beta<0$ are two parameters of the model subject to $0 \leq \alpha + \beta\cdot\theta_{r}^M \leq 1$. To mimic the decay of $\theta$ with the number of rounds, we set +%\begin{equation} +%\theta_{r}^M=max\{-\mathfrak{T},\gamma \cdot r\},\label{eqn:theta} +%\end{equation} +%where $\gamma$ is the third parameter controlling the decreasing rate of $\theta_{r}^M$. +%\begin{tiny} +%\begin{equation} +% \alpha^*, \beta^*, \gamma^*, \xi^* = \mathop{\arg\min}_{ 0 \leq \alpha + \beta \cdot max\{-\mathfrak{T},\gamma \cdot r\} \leq 1, \gamma \in [-\mathfrak{T}, 0] ,\xi \in [0, 1]} \sum_{r=1}^{M_r}(f_{c,r}-P_{c,r})^2,\label{eqn:alpha} +%\end{equation} +%\end{tiny} +%where $f_{c,r}$ and $P_{c,r}$ are the frequency of cooperation obtained by the empirical experiments and the expected probability of cooperation derived from Eqn.~\ref{eqn:pc} in round $r$, respectively. $M_r$ is the maximum number of rounds. $\xi$ is the fourth parameter of the model, which will be introduced later in this section. After maximizing the likelihood of $P_{c,r}$, namely, minimizing the variance between $f_{c,r}$ and $P_{c,r}$, one can derive $\alpha^*$, $\beta^*$, $\gamma^*$, and $\xi^*$. +% +%Given that whether two individuals play again depends on the gaming outcome of the previous round, which may be one of $CC$, $CD$, $DC$, and $DD$. From the mean-field perspective, if all the individuals have a unified probability of cooperation $P_c$, which is defined in Eqn.~\ref{eqn:pc}. Following this assumption, the probabilities of the four outcomes in round $r$ are +%\begin{equation} +%\left\{ {\begin{array}{*{20}{l}} +% {P_{CC,r}} = P_{c,r}^2 \\ +% {P_{CD,r}} = P_{c,r}(1-P_{c,r}) \\ +% {P_{DC,r}} = (1-P_{c,r})P_{c,r} \\ +% {P_{DD,r}} = (1-P_{c,r})^2 +%\end{array}} \right.. \label{eqn:probabilities} +%\end{equation} +%We assume that all the individuals make full use of their time resources. The expected aggregated payoff for each individual in round $r$ ($r \geqslant 1$) can be written as +%\begin{equation} +%\phi_r = \phi_{r-1} + E\left(\Lambda_r\right) - \varepsilon \label{eqn:phir}, +%\end{equation} +%where $\Lambda_r$ is an individual's accumulated payoff in round $r$. Its expectation is defined as +%\begin{tiny} +%\begin{equation} +%\! E\left(\Lambda_r\right)\! = \! \frac{{P_{CC,r}}\omega(\mathcal{R},r)\mathcal{R}\!+\!{P_{CD,r}}\omega(\mathcal{S},r)\mathcal{S}\!+\!{P_{DC,r}}\omega(\mathcal{T},r)\mathcal{T}\!+\!{P_{DD,r}}\omega(\mathcal{P},r)\mathcal{P}}{{P_{CC,r}}\omega(\mathcal{R},r)\!+\!{P_{CD,r}}\omega(\mathcal{S},r)\!+\!{P_{DC,r}}\omega(\mathcal{T},r)\!+\!{P_{DD,r}}\omega(\mathcal{P},r)}. +%\end{equation}\label{Elambda_r} +%\end{tiny} +%% +%If the payoff $G$ can not keep the individual from elimination, the payoff in round $r$ will not be accumulated anymore for the individual. Given this fact, define $\omega(G, r)$ and $Q_k(G,r+1)$ as the effective selection factor and the probability of collaboration in round $r+1$, respectively. The formal definition of $\omega(G, r)$ is +%\begin{equation} +% \omega(G, r) = \left\{ {\begin{array}{*{20}{cc}} +% {0} & {\phi_{r-1} + G - \varepsilon \leqslant 0} \\ +% {Q_k(G,r+1)} & {otherwise} +% \end{array}} \right., \label{eqn:omega} +%\end{equation} +%where +%\begin{equation} +% Q_k(G,r+1) = \left\{ {\begin{array}{*{20}{cc}} +% {1} & {G = \mathcal{R}} \\ +% {\xi} & {otherwise} +% \end{array}} \right.. \label{eqn:pb} +%\end{equation} +%Here $\xi$ denotes the probability of collaboration after at least one of the two players defects, the range of which is $[0,1]$. +% +%Summarizing the iterative procedure, the backup to defect $\theta$ in Eqn.~\ref{eqn:theta} is substituted into Eqn.~\ref{eqn:pc} to derive $P_{c,r}$ first; second, $P_{c,r}$ is used to derive $P_{CC,r}$, $P_{CD,r}$, $P_{DC,r}$, and $P_{DD,r}$ in Eqn.~\ref{eqn:probabilities}; with the probabilities of the four outcomes, one has $E\left(\Lambda_r\right)$ in Eqn.~\ref{Elambda_r} and $\phi_r$ in Eqn.~\ref{eqn:phir}; $\phi_r$ is then used to derive $\omega(G, r+1)$ in Eqn.~\ref{eqn:omega} and $P_{c,r+1}$ in Eqn.~\ref{eqn:pc}. The initial payoff $\phi_0$s of both modes are given at the beginning of each match. Therefore, one can iterate the procedure to derive $P_{c,r}$ and optimize $\alpha$, $\beta$, $\gamma$, and $\xi$ in Eqn.~\ref{eqn:alpha} finally. Notably, the mathematical model is based on the statistical results and simulation results obtained by our experiments. We adopt a series of approximations to model the decision-making procedure of humans only to provide a better understanding of the procedure. The accuracy of prediction is not our focus. + + +\section{Results}\label{Er} + +To show the impact of time redistribution, first, the evolution of moves is simulated when agents play a traditional PD game with their neighbors in the BA and WS networks. In any network, a player starts a game with a gaming request to a neighbor. In simulations, all the agents in the network are selected one by one, following a random sequence. For a selected agent, it evenly allocates the time left to its requests to the uncoordinated neighbors. If the requested neighbor has enough time to accept a gaming request, he will accept it. After one round of the game, agents will uniformly update their moves with the Zero-Determinant Extortionate strategy proposed in \cite{stewart_pnas12}. The strategy will wipe the cooperators out in 100 rounds. If one agent defects in a round, the gaming pair will be taken apart with a certain probability. The separation means that the time assigned to the pair will be redistributed in the next round. More details on the simulations will be provided later in Section~\emph{Simulation on the social networks}. + +In Fig.~\ref{fig:APC}(a) and \ref{fig:APC}(b), the results show that the level of cooperation decays as the rounds increase for agents playing the `divide-and-conquer' (D\&C) games~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18} in both BA and WS networks. After being affected by the temporal mechanisms, the rates of decay slow down, as shown in Fig.~\ref{fig:APC}(c) and \ref{fig:APC}(d). The differences in the level of cooperation between the temporal games and the D\&C games~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18} are shown in Fig.~\ref{fig:APC}(e) and \ref{fig:APC}(f), which will be amplified when human subjects play the games. The amplification may originate from the $S\left(r\right)$ shown in Fig.~\ref{fig:APC}(g) and \ref{fig:APC}(h), which will be much lower when humans play the temporal games. + +%Assume an individual assigns his time to two opponents. Cooperating with one to expect long-term mutual cooperation and defecting the other to pursue a higher payoff in this round is a common choice. Especially after the mutual selection in the partner-seeking procedure, a high frequency of cooperation can be expected since they have contacted each other before selection. In the dissipative scenario, $f_c$ increases at first and reaches 100\% in the $8^{th}$ round. Then, it sharply decays and forms a valley from the $10^{th}$ to the $15^{th}$ round. Considering a pair of mutual cooperators without any other partners, the aggregated payoff of one of them at the $r_{th}$ round is $\phi_{r}=\phi_{0}-r\left(\varepsilon-\mathcal{R}\right)$, where $\phi_{0}$ denotes the initial resource and $\varepsilon$ is the compulsory consumption. In our experiment, $\phi_{0}=5$ and $\varepsilon=3$. One can see that mutual cooperation can afford an individual's ongoing cost for at most 12 rounds based on the designed payoff matrix, where $R=2.6$. The aggregated payoff left for the pure mutual cooperators is 0.2 at the $13^{th}$ round. In this case, only defectors have a certain chance to survive for more than one round, while the cooperators are doomed to be eliminated in that round. Accordingly, $f_c$ drops to approximately $0.5$ as expected. The recovery of $f_c$ in the $14^{th}$ round results from the number of survivors after the $13^{th}$ round is small. If an individual loses a partner at this moment, it will be unlikely to find another one again. Therefore, mutual cooperation is the best choice to survive. To model the procedure, we set all the agents to follow a simple strategy, which will be extended in Section \emph{Simulation on the temporal game model}. The comparison between the simulation result and the statistical result of the real data is shown in Fig.~\ref{fig:fc}A, in which each red circle denotes the average of $800$ simulation runs. + +\begin{figure}[ht] + \centering + \includegraphics[height=2.2in,clip,keepaspectratio,trim=35 35 50 50]{Merged-Simulation_BA_WS.eps} + + \caption{Evolution of the average proportion of cooperation $\langle P_c\left(r\right)\rangle$ in the `divide-and-conquer' (D\&C) and temporal gaming networks. (a) and (c) show $\langle P_c\left(r\right)\rangle$ of the D\&C games and temporal games in the BA networks, respectively. (b) and (d) show $\langle P_c\left(r\right)\rangle$ of the two types of games in the WS networks, respectively. (e) and (f) show the differences of $\langle P_c\left(r\right)\rangle$ between the D\&C games and the temporal games in the BA and WS networks, respectively. Each plot denotes the average of $10$ simulation runs. As the system evolves dramatically at the beginning of the experiments, the results are shown in semi-log coordinates.}\label{fig:APC} +\end{figure} + +To verify the above theoretical results, we invited 183 volunteers to attend eight online experiments. For clarity, the basic information of each match is summarized in Table~\ref{PRGNF}. + +%For the D\&C games, the numbers of participants and rounds of the experiments are 39 and 13 rounds for the first BA network (G1224), 17 and 16 rounds for the second BA network (G1230) shown in Fig.~\ref{fig:HMBAWS}(a); 34 and 13 rounds for the first WS network (G1228), 21 and 15 rounds for the second WS network (G1234) shown in Fig.~\ref{fig:HMBAWS}(c). +% +%For the temporal games, the numbers of participants and rounds of the experiments are 50 and 11 rounds for the first BA network (G646), 44 and 28 rounds for the second BA network (G903) shown in Fig.~\ref{fig:HMBAWS}(b); 22 and 24 rounds for the first WS network (G936), 22 and 28 rounds for the second WS network (G933) shown in Fig.~\ref{fig:HMBAWS}(d). +% +\begin{table*}[tbhp] + \caption{The basic information of matches.} + \centering + \resizebox{\textwidth}{!}{ + \begin{tabular}{|c|c|c|c||c||c|} + \hline + Game Number & Game Type & Type of Network & Number of Participants & Number of Rounds & Corresponding Panel in Fig.~\ref{fig:HMBAWS} \\ \hline + G1224 & D\&C & BA & 39 & 13 & Fig.~\ref{fig:HMBAWS}(a) \\ \hline + G1230 & D\&C & BA & 17 & 16 & Fig.~\ref{fig:HMBAWS}(a) \\ \hline + G646 & Temporal Games & BA & 50 & 11 & Fig.~\ref{fig:HMBAWS}(b) and Fig.~\ref{fig:HMBAWS}(e) \\ \hline + G903 & Temporal Games & BA & 44 & 28 & Fig.~\ref{fig:HMBAWS}(b) and Fig.~\ref{fig:HMBAWS}(f) \\ \hline + G1228 & D\&C & WS & 34 & 13 & Fig.~\ref{fig:HMBAWS}(c) \\ \hline + G1234 & D\&C & WS & 21 & 15 & Fig.~\ref{fig:HMBAWS}(c) \\ \hline + G936 & Temporal Games & WS & 22 & 24 & Fig.~\ref{fig:HMBAWS}(d) and Fig.~\ref{fig:HMBAWS}(g)\\ \hline + G933 & Temporal Games & WS & 22 & 28 & Fig.~\ref{fig:HMBAWS}(d) and Fig.~\ref{fig:HMBAWS}(h)\\ \hline + \end{tabular} + } + \label{PRGNF} +\end{table*} + +By comparing Fig.~\ref{fig:HMBAWS}(a) with Fig.~\ref{fig:HMBAWS}(b) and Fig.~\ref{fig:HMBAWS}(c) with Fig.~\ref{fig:HMBAWS}(d), one can see that the decay of $P_c\left(r\right)$ in the temporal games is slower than that in the D\&C games. The result confirms the theoretical prediction, indicating that the limitation on time promotes the level of cooperation in gaming over a real social network. + +To explain the observed behavior, the average available time $S\left(r\right)$ is measured for four time-involved matches. The evolution of $S\left(r\right)$ for the two BA networks and two WS networks are shown in Fig.~\ref{fig:HMBAWS}(e)-Fig.~\ref{fig:HMBAWS}(h), respectively. For clarity, the basic information of matches is listed in Table~\ref{PRGNF}. One can see that $S\left(r\right)$ fluctuates around a small positive value in the four panels, revealing the difficulty of finding new partners when humans play the temporal games is more significant than the theoretical prediction. The difference in $P_c\left(r\right)$ between the theoretical prediction and the human behavior suggests that the rising of the difficulty of finding new partners may lead to the promotion of $P_c\left(r\right)$, which to some extent explains why the limited time promotes the level of cooperation in a real social network. + +The other behavior that should be noted is that the level of cooperation generally decays with the increasing rounds in Fig.~\ref{fig:HMBAWS}. +This behavior is caused by the number of rounds for each match being limited, although it is random. This limitation mainly comes from the time of the subjects, since it is complicated to ask about 100 volunteers to play online for more than one hour simultaneously. Even reasonable participation fees and attractive rewards were paid to the winners of each match. Some of the winners' strategies will be shown in Section \textbf{Top Voted Strategies} of the \textbf{Supplementary Information} (\textbf{SI}), where one can see that the level of cooperation decays when the participants guess that the match is ended. + +\begin{figure}[ht] + \centering + \includegraphics[height=4in,clip,keepaspectratio,trim=0 0 0 0]{Human_Merged_BA_WS.eps} + \caption{Evolution of the proportion of cooperation $P_c\left(r\right)$ and the average available time $S\left(r\right)$ in the temporal games played by human subjects. (a) and (c) show the results of the D\&C games on the BA networks and WS networks, respectively. (b) and (d) show the results of the temporal games on the BA networks and WS networks, respectively. Horizontal coordinates denote the number of rounds. (e) and (f) show the results of two temporal games on the BA networks. (g) and (h) show the results of two temporal games on the WS networks.}\label{fig:HMBAWS} +\end{figure} + +%\begin{figure}[ht] +% \centering +% \includegraphics[]{fc.eps} +% \caption{Evolution of the average frequency of cooperation (blue circles) in (A) the dissipative scenario and (B) classical scenario. The grey crosses are the actual . of cooperation for each round in the match. The red squares denote the simulation result, the simulation procedure of which will be shown in Section~\emph{Simulation on the evolution of cooperation}. The green triangles represent the analytical result, the mathematical procedure of which will be shown in Section~\emph{Mathematical modeling the level of cooperation}. As a comparison, the two baseline results are shown in purple crosses and yellow stars, where individuals update their strategies following the rules proposed by Santos \& Pacheco~\cite{santos_prl05} and Nowak \& May~\cite{nowak_n92b}, respectively.} +% \label{fig:fc} +%\end{figure} + +%More surprisingly, the numerical solution of $P_{c,r}$ in Eqn.~\ref{eqn:pc} is almost exactly the same as $f_{c,r}$ as shown in Fig.~\ref{fig:fc}A. One can see that the simulation result and the numerical solution are remarkably consistent with the entire evolution procedure of $f_c$ for $\alpha^*=0.8$, $\beta^*=-\frac{1}{7200}$, $\gamma^*= -209.437$, and $\xi^*=0.1$. +%%The details on mathematical modeling and analysis will be shown in section \emph{Analytical solution to the model}. +%% +%To have a baseline comparison, we force the agents in the simulation to follow the updating rules proposed by Santos \& Pacheco~\cite{santos_prl05} and Nowak \& May~\cite{nowak_n92b}. The update is targeted, where an agent only changes the strategy for who the agent concerns. In the baseline scenario, the agents can't relink others. One can see that the results of the two baselines are nearly the same, i.e., $f_c$ drops at the $2^{th}$ round, and no agent can survive longer than 13 rounds. The evolution of the average number of survivals in our experiments is shown in Table~\ref{tb:survival}, indicating that the strategies of the human subjects are different from the baseline strategies. +%\begin{table*}[htb] +% \centering +% \caption{Average number of survivals for each round in the dissipative scenario.} +% \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} +% \hline +% Round & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline +% Survival & 15.4 & 12.5 & 11.6 & 10.8 & 10.3 & 9.8 & 9.4 & 9.3 & 9.2 & 9.0 & 6.3 & 4.6 & 3.4 & 1.9 & 0.7 \\ \hline +% \end{tabular} +% \label{tb:survival} +%\end{table*} +% +%Conversely, in the classical scenario, as shown in Fig.~\ref{fig:fc}B, $f_c$ is stable over time, which confirms the reported results on dynamic networks~\cite{rand_pnas11,fehl_el11,wang_j_pnas12}. Now, the question is how to explain the smooth growth of $f_c$, which even reaches near 100\% in Fig.~\ref{fig:fc}A. In this scenario, the baseline results show a low frequency of cooperation, since there is no punishment for the defectors. +% +%Intuitively, the aggregated payoff is the optimization objective for each individual. In replicator dynamics, the deviation from the average payoff will determine the contribution of a specific strategy and probably take one step forward to change individuals' moves in the next round. To clarify the response of each move, the distributions of the payoff loss after cooperation and defection in both scenarios are shown in Fig.~\ref{fig:fl}. In the dissipative scenario, both the average and the median of the payoff loss after cooperation become lower than that after defection, indicating that most defectors will lose more payoff than the cooperators. To avoid losing payoff, an intuitive choice is to cooperate. In the classical scenario, since there is no cost, the payoff loss is smaller or equal to 0. The medians of the payoff loss are the same after both moves, while the average payoff loss after cooperation is higher than that after defection. Although the average is slightly lower, the payoff loss after defection is highly polarized, implying that the risk of no reward is significant. This indicates that some defective moves are rather risky~\cite{zhang_yc_sr15}. Thus, most individuals tend to cooperate for a steady payoff after a long or short learning process. As a result, cooperation is prevailing in both scenarios, match by match. +% +% +%\begin{figure}[ht]%[tbhp] +% \centering +% \includegraphics[]{FoodLoss.eps} +% \caption{Distribution of the payoff loss in the following rounds for C and D in the dissipative (blue), and classical (orange) scenarios. The black dotted lines denote the average and the dashed lines denote the medians. The area represents the probability density, and the width indicates the frequency.} +% \label{fig:fl} +%\end{figure} +% +%However, the payoff loss cannot directly explain the smooth growth and sharp decay of $f_c$ in Fig.~\ref{fig:fc}A. To better understand the reasons for the decay of the defectors' aggregated payoff and unexpected evolution of $f_c$, we investigate the responses of moves from the defectors' standpoint. Firstly, the defectors are likely to suffer from a broken relationship after defection. The actions in the previous round significantly affect the choice of maintaining or breaking the partnership in both scenarios ($\chi^2=1189.5306$, $P<0.0001$ for the dissipative scenario and $\chi^2=600.7032$, $P<0.0001$ for the classical scenario). Specifically, $86.26\%$ and $42.09\%$ of the partnerships are broken apart after defection in the dissipative and the classical scenarios, while only $13.72\%$ and $8.33\%$ of them are broken after cooperation in the respective scenarios. +% +% +%After losing partners, a defector has to find new partners to make good use of the time, while the success rate of a gaming request from a friend without interaction is 16.12\% (364 out of 2,258) and 22.89\% (672 out of 2,936) in the dissipative scenario and classical scenario, respectively. One can see that the social risk of defection is significant, which can be gradually detected by the participants after a couple of matches. The result indicates that the majority of partnerships are not only unlikely to resist a defection but also difficult to be reestablished. Note that the partnerships are established by the interactions based on social connections. Confusingly, why is reestablishing the partnership so difficult? Evidence suggests that reconnection failure is closely related to time resources. In the dissipative scenario, $87.77\%$ of the requests are rejected due to the insufficient remaining time resource of the target that is not enough to afford the requested duration. In the classical scenario, the percentages are $77.52\%$. One can see that the limited time resource is not a barrier to make more friends but a critical obstruction in seeking new partners. +% +%After all, an apparent cause of the fast decay of the aggregated payoff after defection is that the defectors are losing partners. Notably, the requested friends are unable to observe the moving records of those who are not their partners as this is the users' privacy. But, they can learn the senders' reputation from the public chat room. Interestingly, we observe that the corresponding discussions in the chat room are somewhat limited after the match starts. The observation implies that the requested friends reject the requests generally because of the lack of time resources. In other words, the social risk of defection originates from limited time resources. This point was never reported in previous studies, since the majority of existing interaction frameworks are not time-involving. +% +%Fig.~\ref{fig:ttdc} shows the evolutions of the frequency of cooperation $f_c$ and backup to defect $\theta$ in both scenarios. In the dissipative scenario, as shown in Fig.~\ref{fig:ttdc}A, the value of $\theta$ is close to $-1,440$ from the $8^{th}$ to the $12^{th}$ round, which indicates that the available time resource is scarce. One can observe that $f_c$ in Fig.~\ref{fig:ttdc}A is approximately 100\% in the region. The observation confirms our previous inference. Conversely, in the classical scenario, the frequency of cooperation is almost irrelevant to $\theta$. Fig.~\ref{fig:ttdc}B shows that $\theta$ in the classical scenario is significantly higher than that in the dissipative scenario in each round, indicating that the correlation between $\theta$ and $f_c$ emerges only when $\theta$ is low, that is, the available time resources of friends are scarce. One explanation is that the cost in each round stimulates the individuals to move cautiously. Especially after detecting the risk of defection, they prefer to maintain the existing partners rather than seeking new partners. As a result, the frequency of cooperation generally grows with the number of rounds in the dissipative scenario while mildly fluctuates in the classical scenario. +% +%\begin{figure}[htb]%[tbhp] +% \centering +% \includegraphics[]{TtDCo.eps} +% \caption{Evolutions of $f_c$ (blue) and $\theta$ (red) in (A) the dissipative scenario and (B) the classical scenario. The length of the match is stochastic, ranging from 10 to 15 rounds inclusively.} +% \label{fig:ttdc} +%\end{figure} +% +%To confirm the negative correlation between $\theta$ and $f_c$, the result of linear regression analysis is shown in Fig.~\ref{fig:t2}A. For the dissipative scenario, the red dashed line in Fig.~\ref{fig:t2}A shows the linear regression result, where $slope=-3.5474\times10^{-4}$, $intercept=0.4419$ and $residual=0.0608$. The Pearson correlation coefficient of them is $-0.8379$ with P-value $1.83\times10^{-4}$, demonstrating the strong correlation between $\theta$ and $f_c$. For the classical scenario, Fig.~\ref{fig:t2}B shows that the correlation is absent, where the Pearson correlation coefficient is $0.3845$ with P-value $0.1746$. For an individual $i$ to understand the risk of defection when $\tau_{f_{i,r}}$ is low is not difficult, while to estimate whether $\tau_{f_{i,r}}$ is low takes some training. In our experiment, the procedure would last for a couple of matches. Therefore, the negative correlation is expected to grow with the number of matches, since the judgment of an individual is approaching the statistical result after sufficient training. As shown in Fig.~\ref{fig:t2}A, when $\theta$ is limited to a very low level, for instance, lower than $-1,250$, the frequency of cooperation is approaching $100\%$. On the contrary, when the individual's estimation on $\tau_{f_{i,r}}$ is not so pessimistic, Fig.~\ref{fig:t2}B shows that the level of cooperation is basically not affected by the backup to defect $\theta$. +% +% +%\begin{figure}[htb]%[tbhp] +% \centering +% \includegraphics[]{t2.eps} +% \caption{Correlation between the frequency of cooperation $f_c$ and the backup to defect $\theta$. Correlations in the dissipative scenario and classical scenario are shown in (A) and (B). The red dashed line in (A) is the linear regression line of the data. In (B), $f_c$ is clearly irrelevant to $\theta$. Note that the cluster of plots in (A) is located in the range lower than $-1,250$, indicating the strong correlation between the level of cooperation and the scarcity of the available time in the neighborhood. In (B), the cluster of plots is located in the range higher than $-200$, suggesting that the available time in the neighborhood usually is high in the classical mode.} +% \label{fig:t2} +%\end{figure} +% +%Although in our experiment, each individual has more than eight social contacts on average, the number of frequently interacting partners is normally less than 3 (see~\emph{Supplementary materials} for details). Establishing connections on a social network is nearly costless, while choosing the next move towards a partner may take some time if an individual has a delicate strategy to make decisions. Naturally, the number of decisions that can be made in one round is limited by the individual decision time, which is set to 45 seconds in our experiment. An individual's maximal number of partners can be roughly estimated from this individual's average decision time for each partner in a round, which can be calculated by $min({\mathfrak{T}}/{decision\_time},N_{i,r})$. Interestingly, the observed numbers of partners are much less than the estimation, as shown in Fig.~\ref{fig:dt}. For example, if the average decision time of an individual in a round is 1 second and the individual has eight neighbors, the maximal number of partners will be 8. But the individual's actual number of partners is only 1. Our observation indicates that humans tend to cut the number of social interactions (not to cut the number of friends) if more interactions cannot bring them apparent benefits, for instance, a higher payoff in the current round. +% +%\begin{figure}[ht]%[tbhp] +% \centering +% \includegraphics[]{DecisionTime.eps} +% \caption{Correlation between the number of partners and the decision time. The color shows the number of occurrences. Most individuals' average decision time is less than 5 seconds, while their numbers of partners are only one, much less than their maximal possible number of partners.} +% \label{fig:dt} +%\end{figure} +% +%If the individuals frequently interacting with each other are taken to be a cluster, our result shows that the lifespan of the cluster grows with its size (see SM for details). Intuitively, participants are able to detect the rules after a number of matches, which will lead to a significant rise in the sizes of the clusters after a number of matches, whereas the maximum size of the clusters is 3 in our experiments. Here, the cluster is defined as a connected graph where each pair of connected individuals play at least three games in a match. The observation again confirms the behavior mentioned at the end of the preceding paragraph. +% +%Finally, people are making targeted choices. In our experiment, an individual is allowed to choose their neighbors to play with and to move for a specific reason. Our result shows that 7.71\% of the individuals (96 out of 1,244) are identified as `divide-and-conquer' ($D\&C$) individuals~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18}, who may cooperate with some partners to stabilize their partnerships and defect the other partners so as to pursue higher payoffs in the current round. The rest individuals are uncertain, since they have only one partner or their partners' moves are identical. + +\section{Discussion}\label{Dis} + +As a theoretical framework closer to realistic scenarios, the proposed temporal game has demonstrated its ability to illuminate complex behaviors in the real social experiment presented. The human behaviors revealed from the human temporal games were not or rarely reported in the literature. When the available time resources of individuals in the gaming network are scarce, the individuals are more likely to maintain the currently existing relationships through cooperation. The underlying mechanism is that interactions are not obligated but spontaneous. If an individual's time resource cannot afford the requested time duration of the interaction, he will have no choice but abandon it, which actually makes him much harder to find new partners. The accordance of empirical and simulation results confirms the effectiveness of the mechanism. The new finding reveals a fundamental reason for lasting altruistic behaviors in real human interactions, providing a new perspective in understanding the prevailing human cooperative behaviors in temporal collaboration systems. + +%Our work can inspire several interesting future directions for theoretical research. The classification of successful strategies has received massive attention in the game-theoretic frameworks, such as Tit-for-Tat~\cite{axelrod_jcr80}, Generous Tit-for-Tat~\cite{axelrod_jcr80b}, Win-stay-lose-shift~\cite{nowak_n93}, partner-rivals and extortionate strategies~\cite{hilbe_nhb18,press_pnas12}, as well as strategies with various memories~\cite{milinski_pnas98}. Studying the classification of strategies in the temporal game framework is a challenging but exciting direction of future work. Besides direct reciprocity, the game-theoretic frameworks to study indirect reciprocity have also received tremendous attention~\cite{nowak_n98,nowak_n05}. The study of such frameworks with temporal aspects is another interesting direction for future work. + +%Please check some classical experimental paper in either static or dynamic networks. If there is some possible future extension for temporal game framework we can mention. + +It should be noted that the limitation of time is ubiquitous in human collaboration systems, which is essentially different from the incentives, such as global reputation~\cite{fu_pre08b,gallo_pnas15} and anonymity~\cite{wang2017onymity}, associated with human psychology. In a sense, the behavior observed in the performed experiments is more deterministic than random. Introducing some other mechanisms like rewarding~\cite{sefton_ei07} and costly punishment~\cite{fehr_n02,PCB14e1006347} to the temporal systems will be a natural extension of study in this direction. Apart from the mechanisms, the impact from different types of games, for instance, the snow-drift game~\cite{hauert_n04} and the public goods game~\cite{santos_n08}, is also of interest and significance. +% +%On the other hand, the random allocation of groups in our experiment is a way to constrain group cheat, limiting the social knowledge of subjects. Therefore, a systematic experimental study on the role of global social knowledge~\cite{gallo_pnas15} in the formation of cooperative communities is another promising area of future studies. + +This work considers the temporal game framework and presents some rather surprising new results. There are several interesting future directions for investigation in terms of both theoretical and experimental studies. However, the basic theoretical model and the key experimental results presented in this paper for temporal games are the first steps to modeling realistic networks with time-dependent interactions. Such realistic modeling will allow better analysis, prediction, and design for the emergence of cooperation from network models, profoundly impacting disciplines on preserving natural resources to designing institutional policies. + +\section{Materials and Methods} +\subsection{Experimental design}\label{ED} + +In order to build an experimental environment as close as possible to realistic temporal two-player collaborative systems, two issues are considered in the performed empirical study. First, the interactive time is determined by negotiation. The setting resembles the temporal properties of a real game in society. A dynamic reconnection is implemented in the network by rejecting a friend's request and then proposing a game with another friend~\cite{rand_pnas11,wang_j_pnas12}. Second, a $D\&C$ framework, also referred to as targeted decision, is adopted, in which the individuals who propose a game or accept a gaming request have to decide whether to cooperate or to defect in each round of the game~\cite{zhang_yc_sr15,wang_js_sr17,melamed_pnas18}. Most existing research on gaming networks is performed under a framework where individuals choose the same move to interact with all their neighbors~\cite{rand_pnas11,fehl_el11,wang_j_pnas12}. On the contrary, in real-world scenarios, people do not normally defect their long-term partners after being defected by some other partners. In a realistic social network, they would choose a specific move to play with another partner, referred to as the D\&C game in the literature~\cite{zhang_yc_sr15,wang_js_sr17}. When the diffusive decision scheme is replaced by the D\&C or a targeted decision scheme, the impact of dynamic reconnection on promoting cooperation will become negligible~\cite{melamed_pnas18}. + +The coupling between temporal interaction and rational decision-making can be seen everywhere in real life. Nevertheless, the existing theoretical frameworks seem insufficient to explain the widespread cooperation in temporal social games. Under the framework of temporal games, a series of online game experiments were performed. The experimental data reveal a surprising finding: limitation of time promotes cooperation in temporal games. This finding, on the one hand, urges us to reconsider how much the dynamic nature of networks can impact human cooperation; on the other hand, it demonstrates the potential of the temporal game framework to explain various collective behaviors in real two-player collaborative systems. + +%The results reported in this paper have a profound impact on the study of pro-social behavior. By accounting for the time-dependent aspect to model a realistic network, we report an interesting finding which can improve our understanding of widespread cooperation in time-dependent collaborations. + +\subsection{Experimental setup and game rules}\label{ESGR} + +A series of online human subject experiments were designed to build a two-player collaborative system of rational individuals. A total of 183 human subjects participated in 8 matches in the experiment. The majority of subjects are students from Tongji University and Southeast University in China. To implement the designed framework, a novel online gaming platform was developed, called the \emph{War of Strategies} (http://strategywar.net, see~(Section \textbf{Experimental Platform and Interface} of \textbf{SI} for the details of the platform). + +In the online experiments, participants played a traditional PD game, where $C$ and $D$ were the only available actions. Each participant interacted with the individuals who had agreements with him in one round, after which the agreements needed to be redrafted. + +Each match on the platform comprises two stages. In the first stage, the system generates a network with a social network model. The subjects are then allocated to the nodes of the network. In this setting, the connections among the subjects are randomly predetermined. The second stage is an $n$-round iterated PD game, where $10\le n \le 30$ is unknown to individuals so as to avoid the ending-game effects. + +In each round of the game, individuals can make requests to interact with their friends. In a request, the duration of the interaction is suggested by the sender and shown to the target. The request can be accepted, denied, ignored, or canceled. Once an individual accepts it, this individual has to choose a move as his response. The payoff of the game is proportional to the time duration suggested in the request, which is a part or all of the sender's time resource. Once the request is sent out, this part of the resource will be occupied before receiving a response, which cannot be used again in any other interaction. If the request is accepted, the time resource will be consumed. If the request is denied, ignored, or canceled, the time resource will be returned to the sender. The total time resource assigned to each individual is $1,440$ units in each round, mimicking one day in real life. The experiment adopts $1,440$ to help the participants to understand its meaning, the value of which is irrelevant to the final results. For all the individuals, each round lasts for 60 seconds. The initial aggregated payoff for each individual is 0. The payoff matrix is the same as that shown in Fig.~\ref{fig:ITG}. + +During a match, the individual IDs are randomly generated. The individuals can only see their own game records, where each record includes the moves of both sides and the time durations. The topological structures beyond their immediate neighbors are invisible to them. Besides, individuals are shown their aggregated payoff, time resources, number of rounds played, and their remaining decision time. + +\subsection{Simulation on the social networks}\label{SSN} + +Here is the process of the simulation. + +Step 1: Generate a structured population such as the BA network~\cite{BA} with degree $m_0=m=3$ or WS small-world network with $P_{rewire}=0.1$ and $K=6$. Randomly assign the agents to be cooperators with a probability of 0.5. The size of the population is set to 1,024. + +Step 2: Shuffle the agent list and iteratively ask an agent to broadcast gaming requests to its neighbors. In each request, the agent uniformly allocates his rest time to those uncoordinated neighbors, i.e., $\mu_{ij}\left(r\right)=\frac{1}{\left | N_i-P_i\left(r-1\right) \right | }$, where $j\in N_i-P_i\left(r-1\right)$. If a neighbor has enough time to accept the request, he will accept it. + +Step 3: Each pair of the matched agents play the game for one round and then updates their moves, following the Zero-Determinant Extortionate strategy proposed in \cite{stewart_pnas12}. + +Step 4: If an agent defects in a round, the pair will be taken apart with a probability of 0.5, that is, $\mathbf{\chi}=[0,0.5,0.5,0.5]$. + +Step 5: Repeat Steps 2, 3, and 4 until reaching the preset number of rounds. + +\bmhead{Acknowledgments} + +Y. Z. was supported by the National Natural Science Foundation of China (Grant No. 61503285) and the Municipal Natural Science Foundation of Shanghai (Grant No. 17ZR1446000). J. G. was supported by the National Natural Science Foundation of China (Grant No. 61772367) and the Program of Shanghai Science and Technology Committee (Grant No. 16511105200). S. Z. was supported by the Program of Science and Technology Innovation Action of the Science and Technology Commission of Shanghai Municipality (STCSM) (Grant No. 17511105204). G. C. was supported by the Hong Kong Research Grants Council (Grant No. CityU-11206320). K. C. was supported by ERC Consolidator Grant 863818 (FoRM-SMArt). M. P. was supported by the Slovenian Research Agency (Grant Nos. 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8388607) + 214 compressed objects within 3 object streams + 80 named destinations out of 1000 (max. 500000) + 53 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/SI/2nd SI.out b/SI/2nd SI.out new file mode 100644 index 0000000..e69de29 diff --git a/SI/2nd SI.pdf b/SI/2nd SI.pdf new file mode 100644 index 0000000..a96f553 Binary files /dev/null and b/SI/2nd SI.pdf differ diff --git a/SI/2nd SI.synctex.gz b/SI/2nd SI.synctex.gz new file mode 100644 index 0000000..095a620 Binary files /dev/null and b/SI/2nd SI.synctex.gz differ diff --git a/SI/2nd SI.tex b/SI/2nd SI.tex new file mode 100644 index 0000000..9fa9c72 --- /dev/null +++ b/SI/2nd SI.tex @@ -0,0 +1,1108 @@ +\documentclass[default]{sn-jnl} +% Use the lineno option to display guide line numbers if required. +% Note that the use of elements such as single-column equations +% may affect the guide line number alignment. +\usepackage{graphicx}% +\usepackage{multirow}% +\usepackage{amsmath,amssymb,amsfonts}% +\usepackage{amsthm}% +\usepackage{mathrsfs}% +\usepackage[title]{appendix}% +\usepackage{xcolor}% +\usepackage{textcomp}% +\usepackage{manyfoot}% +\usepackage{booktabs}% +\usepackage{algorithm}% +\usepackage{algorithmicx}% +\usepackage{algpseudocode}% +\usepackage{listings}% +\usepackage{dsfont} +\usepackage{times} +%%% better text circle +%%% see https://tex.stackexchange.com/questions/7032/good-way-to-make-textcircled-numbers +\usepackage{tikz} +\newcommand*\circled[1]{\tikz[baseline=(char.base)]{ + \node[shape=circle,draw,inner sep=2pt] (char) {#1};}} +\usepackage{enumitem} +\usepackage{stfloats} +% \usepackage[justification=centering]{caption} + + +\begin{document} + + +\title{Supplementary Information for Limitation of time promotes cooperation in temporal games} + +\author{Jiasheng Wang} +\author*{Yichao Zhang}\email{yichaozhang@tongji.edu.cn} +\author*{Guanghui Wen}\email{wenguanghui@gmail.com} +\author*{Jihong Guan}\email{jhguan@tongji.edu.cn} +\author{Shuigeng Zhou} +\author{Guanrong Chen} +\author{Krishnendu Chatterjee} +\author{Matja{\v z} Perc} + +% \instructionspage + +\maketitle + +%% Adds the main heading for the SI text. Comment out this line if you do not have any supporting information text. +\begin{appendices} + + +\section*{Volunteers Recruitment and Experimental Setup} + +The experiment was carried out with a total of 183 volunteers. The participants are students mainly from Tongji University and Southeast University in China. All the participants of the experiment are required to register an account on the experimental platform in advance (see below). After logging in, they are asked to check the experiment schedule on the landing page, where the type of scenario is shown to the participants who are going to enroll in the match. Meanwhile, they can choose whether to enroll in the next match or not. When the match begins, only the participants who have enrolled can play. After kicking off a game, their accounts will automatically be redirected to the gaming page. For beginners, there is a casual mode for their training. + +In the online experiments, participants played a traditional Prisoner's dilemma (PD) game, where C (cooperative) and D (defective) were the only available actions. Each participant interacted with the individuals who had agreements with him in one round, after which the agreements needed to be redrafted. + +Each match on the platform comprises two stages. In the first stage, the system generates a network with a social network model. In the experiments, two types of networks are generated. One is Barab\'{a}si and Albert (BA) scale-free network with degree $m_0=m=3$; the other is Watts and Strogatz (WS) small-world network with $P_{rewire}=0.1$ and $K=6$. There were 150 players participating in the experiment with the BA networks (56 for the `divide-and-conquer' (D\&C) games and 94 for the temporal games) and 99 players playing with the WS networks (55 for the D\&C games and 44 for the temporal games). The subjects are then allocated to the nodes of the network. Thus, the connections among the subjects are randomly predetermined. The second stage is an $n$-round iterated PD game, where $10\le n \le 30$ is unknown to individuals so as to avoid the ending-game effects. + +In the match, participants are shown their identities, which are in-game-generated participant IDs. They are allowed to see their own gaming histories, where each record includes the actions (cooperation or defection) of both sides and the gaming time duration. Some necessary information about the game progress is visible to them, including food, time resources, neighbors, number of rounds, and time left for consideration. + +After each match, the food resources of the players, namely payoffs, is the base for the reward. Each player gets 1 RMB for 1 unit of food resources as a basic reward. The top 3 players with the most food resources per round are the winners of the match. All the interaction logs of winners are opened to the participants, so they can vote for their favorite strategies. The winners who receive more votes can get more extra bonuses. The bonus pool is 1,000 RMB. + +\section*{Experimental Platform and Interface} + +Today, there are many platforms designed for empirical experiments~\cite{RN254,RN255,RN256,RN257,RN258,RN259,RN260,RN261,RN262}. The most widely used is the z-Tree toolkits~\cite{RN254}, proposed in 2007. It is used to perform social or economic experiments. But the questionnaire-like user interface cannot support complicated interactions, such as reconnection, chatting, etc. On the other hand, the system cannot support real-time interactions. Modern empirical platforms are quite different. One good example is nodeGame~\cite{RN255}, which provides online service based on the browser/server (B/S) architecture. To recruit more participants, they normally connect with Amazon Mechanical Turk~\cite{mturk} (AMT). These platforms support real-time interactions to make the environment closer to real scenarios. Unfortunately, none of them can support the divide-and-conquer (D\&C) gaming environment, let alone the temporal social dilemma process. + +\subsection*{Overview} + +To implement the experimental scenarios of the temporal divide-and-conquer games, a novel online gaming experimental platform was developed in this work, called \emph{War of Strategies\footnote{http://strategywar.net}} (\emph{WoS}). The features of the platform are listed below: + +\begin{enumerate}[label=(\arabic*)] + \item Supporting D\&C games. The platform provides an easy way to configure and conduct a D\&C game experiment. + \item Having built-in bots for training. Beginners can be familiarized with the platform by playing with the training bots. The current strategy of the bots is uniformly set to a random selection; that is, they will accept the gaming request with a probability 50\% and cooperate with a probability 50\%. + \item Performing real-time interactions. The user interface is similar to a browser-based online game. The interaction between participants is real-time and stressful, stimulating the participants to make fast and cautious decisions. + \item Having scalability. All modules of the experimental platform can be deployed on standalone servers or distributed machines. Docker containers are also supported. + \item Having customizability. The gaming settings are easy to adjust to fit the models. +\end{enumerate} + +\subsection*{Architecture} + +The platform is developed based on several open-source software, composed of three components: Portal, Distributor, and Worker. The architecture is shown in Fig.~\ref{fig:arch}. + +\begin{figure}[!htbp]%[tbhp] + \centering + \includegraphics[width=.7\linewidth]{architecture} + \caption{Platform architecture of the WoS.} + \label{fig:arch} +\end{figure} + +In the WoS, PostgreSQL~\cite{psql} and Redis~\cite{redis} are used to store platform data. PostgreSQL is responsible for data persistence, which manages the rarely updated data, such as configurations, user profiles, archived logs, and archived gaming results. Redis is used as an in-memory cache, storing the data which are read and written frequently, such as match data, participant data, in-game requests, and runtime logs. The Redis Pub/Sub message system supports the communications between the Worker and the Distributor module. + +The Portal module deals with HTTP requests and web pages, such as the landing page. The module is developed on Sinatra~\cite{sinatra} with Ruby~\cite{ruby}, which is a lightweight web platform. Thin~\cite{thin} is adopted as a web server. The module provides an authentication service and management interface. The module can be customized to provide a specific user interface for the participants and researchers. The default theme of the WoS is shown in Fig.~\ref{fig:landpage}. + +\begin{figure}[htbp] + \centering + \includegraphics[width=\linewidth]{landing-page} + \caption{Landing page of the WoS.\label{fig:landpage}} + \vspace*{\floatsep} + \includegraphics[width=\linewidth]{user-login} + \caption{Login page of the WoS.\label{fig:login}} + +\end{figure} + +During the experiment, the communication between the webpage and the server follows the WebSocket~\cite{rfc6455} protocol. In the Distributor module, the server communicates with a webpage by Faye-WebSocket~\cite{fws}. Unlike traditional web-based applications, WoS adopts WebSocket to process real-time requests of the participants, including friend requests, gaming requests, chat messages, match processing data, etc. The Distributor module listens on the Redis Pub/Sub channel to process the request from the Worker module, for instance, broadcasting the match progress information. + +The Worker module processes the delayed jobs, including starting the match according to the schedule, match process management, match result processing, etc. The module sends messages to the Redis Pub/Sub channel to notify the Distributor module. Since the match is conducted in the Worker module, one can modify this module to customize its functionality. + +\subsection*{User-Interface} + +Some screenshots of the default theme ``the Lost Island'' are provided, which are used in the experiment and shown to the participants. The volunteers first register or log in to participate in the experiment. The login interface is shown in Fig.~\ref{fig:login}. To register an account, a user requires an email address, a nickname, and a password. The section ``Privacy Policies'' covers the privacy issues related to the experiments. +\begin{figure}[htbp] + \centering + \includegraphics[width=\linewidth]{main-page} + \caption{The main part of the experiment.\label{fig:main}} + \vspace*{\floatsep} + \includegraphics[width=\linewidth]{response-1} + \begin{center} + The left panel shows:\\ + Here is the request from [nickname] with time resource to cost: 1440 \\ + Would you accept this request?\\ + Deny | Accept\\ + The right panel shows: Day No. | Time resource used | My move | Partner's move + \end{center} + \caption{The modal dialog box showing the request sent from a partner.\label{fig:resp1}} +\end{figure} + +Once successfully logged in, a participant will reach the landing page, as shown in Fig.~\ref{fig:landpage}. The left top of the screen displays the user's nickname and accumulated food resource, which is used to calculate rewards. The left panel shows the top 20 participants who won a match, ordered by their average payoffs per round in the match. The right panel is the main panel, where the information from top to bottom is: +``There is no pending match now. Please wait'', +``Casual mode'', +``Story'', +``Help page'', +``In order to have the best gaming experience, modern browsers including Chrome, Firefox, and Safari are suggested.'' +If there exists a pending match, the match schedule will be shown in the first line, followed by an enroll button. + +In the experiment, the network topology is generated by a network model. After the generation of the social network, as shown in Fig.~\ref{fig:main}, the main process of the experiment begins. The status of the current match is shown on the top of the page, which is ``Day 1, 33s left in the daytime''. On the left-hand side of the page, the upper panel shows the participant's personal information, including the nickname, food resource, and the remaining time source. These properties are only visible to the participant himself; no one else can see them. The bottom-left panel is the operation panel. The participant can send gaming requests to their friends. The nickname is randomly generated, and the participant's actual ID is hidden to clear the memory generated in the previous matches. The request can be canceled, accepted, denied, or ignored. The first line shows that a social request has been sent to a friend. The label on the button is ``cancel''. The button can be used for withdrawing the request. The second line shows that a request has been accepted. As shown in the figure, the participant's move is cooperation, and the assigned time resource is 720. The third line shows that the participant just received a request from a friend, where the label on the button is ``request received''. The button can be used to trigger a modal dialog box for further operation, shown in Fig.~\ref{fig:resp1}. The fourth line shows that there is no interaction yet. Therefore the label on the button is ``take action''. The button can be used to trigger a modal dialog box, as shown in Fig.~\ref{fig:request}. The buttons on its right-hand side show ``check gaming history'', which can trigger a modal dialog box to review the gaming records. + +Fig.~\ref{fig:resp1} shows the modal dialog box triggered by request. The left panel is the operation panel and the right panel shows the gaming history. For integrity, a complete translation of the modal dialog box is provided. +% \newpage + +Fig.~\ref{fig:request} shows the modal dialog box of the drafting request. The left panel is the operation panel, and the right panel shows the gaming history. For integrity, a complete translation of the modal dialog box is provided. + +\begin{figure}[tbhp] + \centering + \includegraphics[width=0.9\linewidth]{request} + + \begin{center} + The left panel shows:\\ + Choose a strategy to play with [nickname]\\ + The move to take (Cooperate or Compete)\\ + Assign the time resource. The more time resource you use, the more food you will gain in the same condition.\\ + Cancel | OK\\ + The right panel is the same as shown in Fig.~\ref{fig:resp1}. + \end{center} + \caption{The modal dialog box of request composing.\label{fig:request}} + \vspace*{\floatsep} + \includegraphics[width=0.9\linewidth]{response-2} + \caption{The modal dialog box shows that the participant accepts the request.\label{fig:resp2}} +\end{figure} + +Fig.~\ref{fig:resp2} shows the modal dialog box when the participant accepts a request. Then, the participant should choose his move as a response. Note that the opponent's move is not shown to the others. + +%Fig.~\ref{fig:over} shows the interface when the game is over and no one survives. The sentence on the screen says that ``The rescue team is finally here, but it is too late as all people died and no one survived.'' + +\begin{figure}[tbhp] + \includegraphics[width=\linewidth]{configure} + \caption{The new configuration page of the management interface.\label{fig:config}} +\end{figure} + +For researchers, the WoS provides a user-friendly interface to manage experiments. The management interface provides the services of checking, creating, and editing experimental configurations, schedules, and exporting data. Fig.~\ref{fig:config} shows the page for creating an experimental configuration. The bar on the top of the page includes two drop-down lists, which are ``configuration'' and ``matches''. The drop-down items are related operations, such as checking, creating, and editing. The WoS uses JSON~\cite{json} to store the configuration. The figure shows a sample of the temporal social dilemma experiment. In the configuration, researchers can specify the payoff matrix, the duration of a round, the resource consumption, etc. The properties can be modified and created if the corresponding implementation is developed to conduct a customized experiment. + +\subsection*{Privacy Policy} +The privacy policy of the platform is shown at (http://strategywar.net/privacy), the details of which are listed in the following (in italics). + +{\fontfamily{lmss}\selectfont\itshape + +\small The data management and bioinformatics (DMB) laboratory of Tongji University is responsible for running and maintaining the http://strategy +war.net website (the ``Service''). This page informs you of our policies regarding data collection, usage, privacy protection of personal data, and corresponding options. + +We use your data to provide and improve the Service. By using the Service, you agree to the collection and use of information in accordance with this policy. Unless otherwise specified in this Privacy Policy, the terms used in this Privacy Policy have the same meanings as those in our Terms and Conditions shown on http://strategywar.net. + +\Large \textbf{Information Collection and Use} + +\small Several different types of information are collected, including personal data, usage data, and tracking \& cookies data (the details are listed below) to provide and improve the Service to the user. + +\large \textbf{Types of Data Collected} + +\normalsize \textbf{Personal Data} + +\small When using the Service, the user may be asked to provide us with personally identifiable information that can be used to contact or identify the user (``Personal Data''). The personally identifiable information may include, but is not limited to: + +\begin{itemize} + \item Email address + \item Cookies + \item Usage data +\end{itemize} + +\normalsize \textbf{Usage Data} + +\small The information concerning how the Service is accessed and used (``Usage Data'') will be collected. The Usage Data may include the user's computer Internet Protocol address (e.g., IP address), browser type, browser version, the pages of the Service that the user accessed, the time and date of the user's visit, the time spent on those pages, unique device identifiers, and other diagnostic data. + +\normalsize \textbf{Tracking \& Cookies Data} + +\small Cookies and similar tracking technologies are used to track the activity on our Service and hold certain information. + +A cookie is a small file containing a string of characters that is sent to your computer when a user visits a website. When the user visits the site again, the cookie allows it to recognize the user's browser. Tracking technologies like beacons, tags, and scripts to collect and track information are applied to improve and analyze the Service. + +The user can configure his browser to refuse all cookies or to set when a cookie is being sent. However, if the user does not accept cookies, he may not be able to use some functions of the Service. + +Examples of Cookies used: + +\begin{itemize} + \item \textbf{Session Cookies.} Using Session Cookies to provide Service. + \item \textbf{Preference Cookies.} Using Preference Cookies to record user preferences and settings. + \item \textbf{Security Cookies.} Using Security Cookies for security purposes. +\end{itemize} + +\Large \textbf{Use of Data} + +\small War of Strategies uses the collected data for the following purposes: +\begin{itemize} + \item To provide and maintain the Service + \item To notify the user of changes to the Service + \item To allow the user to participate in the interactive features of the Service when the user chooses to do so + \item To provide customers with technical support + \item To analyze the performance of the system for improving the Service + \item To monitor the usage of the Service + \item To detect, prevent, and address technical issues +\end{itemize} + +\Large \textbf{Transfer of Data} + +\small The user's information, including personal data, may be transferred to and maintained on computers outside his state, province, country, or other governmental jurisdiction where the data protection laws may differ from those of his jurisdiction. + +If the user is located outside China and chooses to provide information for the Service, it should be noted that the data, including Personal Data, will be transferred to China and processed there. + +The user's consent to this Privacy Policy, followed by his submission of such information, represents his agreement to that transfer. + +War of Strategies will take all steps reasonably necessary to ensure that personal data is treated securely and in accordance with this Privacy Policy. No transfer of Personal Data will take place to an organization or a country unless there are adequate controls in place, including the security of the user's data and other personal information. + +\Large \textbf{Disclosure of Data} + +\small \textbf{Legal Requirements} + +War of Strategies may disclose the user's data only in the case that such action is necessary to: + +\begin{itemize} + \item Comply with a legal obligation + \item Protect and defend the rights or property of War of Strategies + \item Prevent or investigate possible wrongdoing in connection with the Service +\end{itemize} + +\Large \textbf{Security of Data} + +\small The security of the user's data is important, but remember that no method of transmission over the Internet or method of electronic storage is 100\% secure. Although the means of data protection adopted in the Service is commercially acceptable, there is no guarantee of its absolute security. + +\Large \textbf{Service Providers} + +\small A third-party companies and individuals may be employed to facilitate the Service (``Service Providers''), to perform Service-related services, or to assist in the runtime analysis of the system. + +For the users' data, the third parties are only allowed to accomplish the tasks specified by the host and are obligated not to disclose or use it for any other purpose. + +\Large \textbf{Links to Other Sites} + +\small This Service may contain links to other sites that are not operated by the host. If the user clicks on a third-party link, he will be directed to its site. The user is strongly advised to review the Privacy Policy of every site that he visits. + +The content and privacy policies of the third parties are not under control. Accordingly, the host is not responsible for them. + +\Large \textbf{Changes to This Privacy Policy} + +\small This Privacy Policy may update aperiodically. The user will be notified of any changes by seeing the new Privacy Policy on this page. The user is advised to review this Privacy Policy periodically for any changes. Changes to this Privacy Policy are effective when they are posted on this page. +\normalsize +} + +\section*{Questionnaire for Volunteers} + +Volunteers were required to read the privacy policy and answer the questions. +They would only participate in the experiment if they accepted the privacy policy. +In the experiments, 40 valid answers were received, where 37 volunteers ($92.5\%$) accepted the policy and became participants. + +%Question 1. Q: We will collect some personal information of the volunteers. To take part in the experiment, you need to agree the privacy policy http://strategywar.net/privacy. +\begin{table}[ht] + \caption{The answer of acceptance of the privacy policy.} + \centering + \begin{tabular}{|l|l|l|} + \hline + Answer & Count & Ratio \\ \hline + Yes, I agree & 37 & 92.5\% \\ \hline + No, I disagree & 3 & 7.5\% \\ \hline + \end{tabular} +\end{table} + +Among the participants, 25 of them are male, and the male-to-female ratio is $2.08:1$. + +\begin{table}[ht] + \caption{The distribution of the sex.} + \centering + \begin{tabular}{|l|l|l|} + \hline + Sex & Count & Ratio \\ \hline + Male & 25 & 67.57\% \\ \hline + Female & 12 & 32.43\% \\ \hline + \end{tabular} +\end{table} + +The average age of the participants was 24.41 ($standard variation= 3.77$). +The detail of the age distribution is shown in Table~\ref{tbl:age}. + +\begin{table}[ht] + \caption{The ages of participants.\label{tbl:age}} + \centering + \begin{tabular}{|l|l|l|} + \hline + Age range & Count & Ratio \\ \hline + < 20 & 1 & 2.70\% \\ \hline + 20 - 24 & 23 & 62.16\% \\ \hline + 25 - 29 & 10 & 27.03\% \\ \hline + $ \geqslant $ 30 & 3 & 8.11\% \\ \hline + \end{tabular} + +\end{table} + +The institutions of participants are listed in Table~\ref{tbl:inst}. + +\begin{table}[ht] + \caption{The institutions of participants.} + \centering + \begin{tabular}{|l|l|l|} + \hline + Institution & Count & Ratio \\ \hline + Southeast University & 21 & 56.76\% \\ \hline + Beijing Institute of Technology & 6 & 16.22\% \\ \hline + Tongji University & 5 & 13.51\% \\ \hline + Others & 5 & 13.51\% \\ \hline + \end{tabular} + \label{tbl:inst} +\end{table} + +%\section*{Instructions for Participants} +% +% +%Fig.~\ref{fig:help} is a part of the poster enclosed in the CCCN 2017 conference brochures, which shows a brief instruction for participants to be familiar with WoS and the experiment. Here, a complete translation of the poster is provided. +% +%{\fontfamily{lmss}\selectfont +%When you are forced to land on a lost island, you want nothing but to survive, live until the rescue team comes. You have some food, which is found in the wreckage. But the food cannot keep you from starvation for a long time. You have to hunt for food with your friends, since the island is full of danger. When you and your friend get some food, the food needs to be assigned. At the moment, everyone tries to gain more food. When your food cannot compensate for your daily consumption, death is approaching. Therefore, in order to survive, you have to try your best to hunt for food. +% +% +%\begin{figure}[htbp] +% \centering +% \includegraphics[width=\linewidth]{help} +% \caption{Instructions for participants.\label{fig:help}} +%\end{figure} +% +%Game Procedures +% +%Prepare +% +%Before the game starts, there is a preparation period. In the period, you need to make friends in order to hunt for food together. +% +%\circled{\small{1}} The current participant panel shows all the opponents you can see. You can choose some of them to chat or to make friends with. +% +%\circled{\small{2}} The chat panel shows private and public messages. +% +%\circled{\small{3}} The friend-request panel shows the friend requests sent by other participants. +% +%Survive +% +%When the game starts, you can play the game with your friends. +% +%\circled{\small{1}} Displays the current day and remaining operation time in the day. Interaction is merely allowed in the daytime. You can check the payoff of the game at night. +% +%\circled{\small{2}} The interface displays all the friends whom you can send requests to. +% +%\circled{\small{3}} You can send a request to your friend. See the next section. +% +%\circled{\small{4}} If your friend doesn't accept your request for a long time, you can cancel the request or wait. +% +%\circled{\small{5}} If your friend accepts your request, you both will become partners and your interaction will be settled for today, which cannot be changed anymore. +% +%\circled{\small{6}} If you receive a request from your friend, the button will appear. +% +%Request +% +%If you want to interact with your friend, you have to operate the request interface. +% +%\circled{\small{1}} You can choose to cooperate or to compete, which directly affects your food payoff. +% +%\circled{\small{2}} The time resource is restricted to 1,440 units. Arrange your time reasonably. +% +%\circled{\small{3}} When you have decided, press OK. +% +%\circled{\small{4}} This panel shows your gaming history with that friend, FYI. +% +%Response +% +%You can handle the requests from your friends in this modal dialog box. +% +%\circled{\small{1}} The duration of the request. If your remaining time resource is less than the requirement, you cannot accept the request. +% +%\circled{\small{2}} Your decision. +% +%\circled{\small{3}} Your gaming history with the requester. +% +%If you accept the request, then +% +%\circled{\small{4}} Choose your move. +% +%\circled{\small{5}} Your decision. +% +%Casual Mode +% +%You can select the ``Casual Mode'' in the landing page to be familiar with the experiment process and test your strategies. The result of the casual mode will not be accumulated in your account. +% +%Strategy +% +%If you have no idea of how to design your own strategy, a sample ``Tit-for-Tat'' strategy is provided. +% +%\begin{enumerate} +% \item Make more friends. +% \item Cooperate first. +% \item If the opponent cooperates, cooperate. +% \item If the opponent competes, compete or seek for new friends. +%\end{enumerate} +% +%} + + +\newpage +\section*{Experiment Procedures} + +In a round of the match, a player's operation flow is listed below: + \begin{enumerate} + \item Refill the time resource + \item Choose a friend to interact with + \begin{enumerate} + \item Choose your move and assign proper time resources to the game + \item Send request + \item Wait for response + \item Cancel the request if necessary + \end{enumerate} + \item Handle the requests from friends + \begin{enumerate} + \item Deny the request if the remaining time resource is not sufficient + \item Accept the request and choose your move + \item Reject the request + \item Ignore the request + \end{enumerate} + \item Repeat the above two steps until the decision-making time is up + \item Review the strategy and prepare for the next round. + \item Go to step (a) + \end{enumerate} + + +\section*{Top Voted Strategies} + +The top three winning strategies in the temporal games are listed below. + +\textbf{Strategy 1} + +Round 1: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 20039. + +Round 2: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 20039. + +4. He/She denied the request from player 20024. + +Round 3: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +Round 4: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 20039. + +4. He/She denied the request from player 19982. + +Round 5: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +Round 6: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 20024. + +4. He/She denied the request from player 19971. + +Round 7: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +Round 8: + +1. He/She played `C,' and player 20047 played `C' with time resource 1440. + +2. He/She denied the request from player 20039. + +3. He/She denied the request from player 19982. + +Round 9: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 19971. + +Round 10: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 19982. + +4. He/She denied the request from player 19971. + +Round 11: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 20039. + +4. He/She denied the request from player 20039. + +Round 12: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 19982. + +Round 13: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 20024. + +4. He/She denied the request from player 19971. + +Round 14: + +1. He/She sent request `C' to 20047 with time resource 1440. + +2. He/She played `C,' and player 20047 played `C' with time resource 1440. + +Round 15: + +1. He/She sent request `D' to 20047 with time resource 1440. + +2. He/She played `D,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 20039. + +4. He/She denied the request from player 20024. + +5. He/She denied the request from player 20039. + +Round 16: + +1. He/She sent request `D' to 20047 with time resource 1440. + +2. He/She played `D,' and player 20047 played `C' with time resource 1440. + +3. He/She denied the request from player 20039. + +\textbf{Strategy 2} + + +Round 1: + +1. He/She sent request `C' to 19844 with time resource 720. + +2. He/She played `C,' and player 19902 played `C' with time resource 120. + +3. He/She sent request `C' to 19890 with time resource 600. + +4. He/She played `C,' and player 19890 played `C' with time resource 600. + +5. He/She played `C,' and player 19863 played `C' with time resource 360. + +6. He/She sent request `C' to 19868 with time resource 360. + +7. He/She sent request `C' to 19873 with time resource 360. + +8. He/She sent request `C' to 19873 with time resource 360. + +9. He/She played `C,' and player 19868 played `C' with time resource 340. + +10. He/She sent request `C' to 19896 with time resource 20. + +Round 2: + +1. He/She played `C,' and player 19902 played `C' with time resource 720. + +2. He/She sent request `C' to 19890 with time resource 720. + +3. He/She played `C,' and player 19863 played `C' with time resource 360. + +4. He/She sent request `C' to 19868 with time resource 360. + +5. He/She played `C,' and player 19868 played `C' with time resource 360. + +Round 3: + +1. He/She played `C,' and player 19902 played `C' with time resource 720. + +2. He/She denied the request from player 19863. + +3. He/She sent request `C' to 19863 with time resource 720. + +4. He/She played `C,' and player 19863 played `C' with time resource 720. + +Round 4: + +1. He/She played `C,' and player 19902 played `C' with time resource 720. + +2. He/She sent request `D' to 19890 with time resource 720. + +3. He/She denied the request from player 19863. + +4. He/She sent request `C' to 19863 with time resource 720. + +5. He/She played `C,' and player 19863 played `C' with time resource 720. + +6. He/She had to deny playing with player 19873 due to a lack of time. + +Round 5: + +1. He/She played `C,' and player 19902 played `C' with time resource 720. + +2. He/She sent request `D' to 19890 with time resource 720. + +3. He/She sent request `D' to 19873 with time resource 720. + +4. He/She denied the request from player 19863. + +5. He/She sent request `D' to 19863 with time resource 720. + +6. He/She played `D,' and player 19873 played `C' with time resource 720. + +7. He/She denied the request from player 19863. + +Round 6: + +1. He/She sent request `C' to 19863 with time resource 720. + +2. He/She played `C,' and player 19863 played `C' with time resource 720. + +3. He/She played `C,' and player 19902 played `C' with time resource 720. + +4. He/She denied the request from player 19890. + +Round 7: + +1. He/She sent request `D' to 19890 with time resource 360. + +2. He/She played `C,' and player 19902 played `C' with time resource 720. + +3. He/She sent request `C' to 19863 with time resource 360. + +4. He/She played `C,' and player 19863 played `C' with time resource 360. + +5. He/She sent request `D' to 19896 with time resource 360. + +6. He/She sent request `D' to 19844 with time resource 360. + +7. He/She sent request `D' to 19868 with time resource 360. + +Round 8: + +1. He/She played `C,' and player 19902 played `C' with time resource 720. + +2. He/She played `C,' and player 19863 played `C' with time resource 720. + +3. He/She denied the request from player 19890. + +Round 9: + +1. He/She sent request `D' to 19890 with time resource 360. + +2. He/She played `C,' and player 19902 played `C' with time resource 720. + +3. He/She denied the request from player 19863. + +4. He/She sent request `C' to 19863 with time resource 360. + +5. He/She played `D,' and player 19890 played `C' with time resource 360. + +6. He/She denied the request from player 19896. + +7. He/She played `C,' and player 19863 played `C' with time resource 360. + +Round 10: + +1. He/She sent request `D' to 19896 with time resource 720. + +2. He/She played `C,' and player 19902 played `C' with time resource 720. + +3. He/She played `D,' and player 19896 played `D' with time resource 720. + +4. He/She denied the request from player 19863. + +Round 11: + +1. He/She played `C,' and player 19902 played `C' with time resource 720. + +2. He/She sent request `C' to 19863 with time resource 720. + +3. He/She played `C,' and player 19863 played `C' with time resource 720. + +Round 12: + +1. He/She played `C,' and player 19863 played `C' with time resource 720. + +2. He/She played `C,' and player 19902 played `C' with time resource 720. + +Round 13: + +1. He/She sent request `D' to 19844 with time resource 720. + +2. He/She played `C,' and player 19863 played `C' with time resource 720. + +3. He/She denied the request from player 19902. + +4. He/She sent request `C' to 19902 with time resource 720. + +5. He/She played `C,' and player 19902 played `C' with time resource 720. + +Round 14: + +1. He/She sent request `D' to 19863 with time resource 1080. + +2. He/She sent request `C' to 19902 with time resource 360. + +3. He/She played `D,' and player 19863 played `C' with time resource 1080. + +4. He/She played `C,' and player 19902 played `C' with time resource 360. + +\textbf{Strategy 3} + +Round 1: + +1. He/She sent request `C' to 19840 with time resource 300. + +2. He/She played `C,' and player 19866 played `C' with time resource 360. + +3. He/She sent request `C' to 19862 with time resource 360. + +4. He/She played `C,' and player 19904 played `C' with time resource 360. + +5. He/She played `C,' and player 19897 played `C' with time resource 360. + +6. He/She played `C,' and player 19862 played `C' with time resource 360. + +7. He/She denied the request from player 19840. + +Round 2: + +1. He/She sent request `C' to 19840 with time resource 700. + +2. He/She played `C,' and player 19897 played `C' with time resource 360. + +3. He/She played `C,' and player 19904 played `C' with time resource 360. + +4. He/She had to deny playing with player 19866 due to a lack of time. + +5. He/She played `C,' and player 19840 played `C' with time resource 720. + +6. He/She denied the request from player 19862. + +7. He/She denied the request from player 19862. + +Round 3: + +1. He/She sent request `C' to 19897 with time resource 360. + +2. He/She sent request `C' to 19904 with time resource 360. + +3. He/She played `C,' and player 19840 played `C' with time resource 720. + +4. He/She played `C,' and player 19904 played `C' with time resource 360. + +5. He/She played `C,' and player 19897 played `C' with time resource 360. + +6. He/She denied the request from player 19866. + +7. He/She denied the request from player 19884. + +8. He/She denied the request from player 19862. + +Round 4: + +1. He/She sent request `C' to 19904 with time resource 360. + +2. He/She played `C,' and player 19904 played `C' with time resource 360. + +3. He/She sent request `C' to 19897 with time resource 360. + +4. He/She played `C,' and player 19840 played `C' with time resource 720. + +5. He/She played `C,' and player 19897 played `C' with time resource 126. + +6. He/She sent request `C' to 19884 with time resource 234. + +7. He/She denied the request from player 19862. + +8. He/She sent request `C' to 19862 with time resource 234. + +9. He/She played `C,' and player 19862 played `C' with time resource 234. + +Round 5: + +1. He/She sent request `C' to 19840 with time resource 1440. + +2. He/She had to deny playing with player 19862 due to a lack of time. + +3. He/She sent request `C' to 19862 with time resource 700. + +4. He/She sent request `C' to 19868 with time resource 740. + +5. He/She played `C,' and player 19862 played `C' with time resource 700. + +6. He/She had to deny playing with player 19877 due to a lack of time. + +7. He/She sent request `C' to 19884 with time resource 740. + +8. He/She played `C,' and player 19840 played `C' with time resource 720. + +Round 6: + +1. He/She sent request `C' to 19890 with time resource 20. + +2. He/She played `C,' and player 19866 played `C' with time resource 720. + +3. He/She sent request `C' to 19840 with time resource 700. + +4. He/She denied the request from player 19897. + +5. He/She sent request `C' to 19897 with time resource 20. + +6. He/She played `C,' and player 19897 played `C' with time resource 20. + +7. He/She played `C,' and player 19877 played `C' with time resource 40. + +8. He/She had to deny playing with player 19862 due to a lack of time. + +9. He/She sent request `C' to 19862 with time resource 660. + +10. He/She played `C,' and player 19862 played `C' with time resource 660. + +Round 7: + +1. He/She sent request `C' to 19904 with time resource 360. + +2. He/She played `C,' and player 19904 played `C' with time resource 360. + +3. He/She played `C,' and player 19897 played `C' with time resource 720. + +4. He/She denied the request from player 19866. + +5. He/She sent request `C' to 19866 with time resource 360. + +6. He/She played `C,' and player 19866 played `C' with time resource 360. + +7. He/She denied the request from player 19840. + +8. He/She denied the request from player 19862. + +Round 8: + +1. He/She sent request `C' to 19840 with time resource 720. + +2. He/She played `C,' and player 19904 played `C' with time resource 360. + +3. He/She played `C,' and player 19840 played `C' with time resource 720. + +4. He/She denied the request from player 19866. + +5. He/She sent request `C' to 19866 with time resource 360. + +6. He/She played `C,' and player 19866 played `C' with time resource 360. + +7. He/She denied the request from player 19862. + +8. He/She denied the request from player 19877. + +9. He/She denied the request from player 19897. + +10. He/She denied the request from player 19877. + +11. He/She denied the request from player 19897. + +Round 9: + +1. He/She played `C,' and player 19897 played `C' with time resource 1000. + +2. He/She sent request `C' to 19840 with time resource 440. + +3. He/She played `C,' and player 19862 played `C' with time resource 360. + +4. He/She denied the request from player 19877. + +5. He/She denied the request from player 19890. + +6. He/She sent request `C' to 19890 with time resource 80. + +7. He/She played `C,' and player 19890 played `D' with time resource 80. + +Round 10: + +1. He/She played `C,' and player 19866 played `C' with time resource 360. + +2. He/She played `C,' and player 19897 played `C' with time resource 1000. + +3. He/She denied the request from player 19862. + +4. He/She sent request `C' to 19862 with time resource 80. + +5. He/She played `C,' and player 19862 played `C' with time resource 80. + +6. He/She denied the request from player 19877. + +Round 11: + +1. He/She played `C,' and player 19862 played `C' with time resource 360. + +2. He/She played `C,' and player 19897 played `C' with time resource 1000. + +3. He/She denied the request from player 19884. + +4. He/She sent request `C' to 19884 with time resource 80. + +5. He/She denied the request from player 19840. + +6. He/She sent request `C' to 19840 with time resource 80. + +7. He/She sent request `C' to 19904 with time resource 80. + +8. He/She sent request `C' to 19890 with time resource 80. + +9. He/She denied the request from player 19866. + +Round 12: + +1. He/She played `C,' and player 19897 played `C' with time resource 1440. + +2. He/She denied the request from player 19884. + +3. He/She denied the request from player 19862. + +4. He/She denied the request from player 19877. + +5. He/She denied the request from player 19866. + +6. He/She denied the request from player 19840. + +Round 13: + +1. He/She played `D,' and player 19897 played `C' with time resource 1440. + +2. He/She denied the request from player 19866. + +3. He/She denied the request from player 19862. + +4. He/She denied the request from player 19840. + +5. He/She denied the request from player 19877. + +6. He/She denied the request from player 19866. + +Round 14: + +1. He/She sent request `D' to 19840 with time resource 700. + +2. He/She sent request `D' to 19866 with time resource 500. + +3. He/She had to deny playing with player 19904 due to a lack of time. + +4. He/She sent request `D' to 19904 with time resource 240. + +5. He/She played `D,' and player 19866 played `D' with time resource 500. + +6. He/She played `D,' and player 19904 played `C' with time resource 240. + +7. 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