Elsevier

Journal of Theoretical Biology

Volume 356, 7 September 2014, Pages 98-112
Journal of Theoretical Biology

Optional games on cycles and complete graphs

https://doi.org/10.1016/j.jtbi.2014.04.025Get rights and content

Highlights

  • Optional games on cycles and complete graphs are analyzed.

  • We explore various limits for weak selection and large population analytically.

  • Some analytic results for strong selection limits are obtained.

  • Numerical analysis is performed on optional PD games on simple graphs.

Abstract

We study stochastic evolution of optional games on simple graphs. There are two strategies, A and B, whose interaction is described by a general payoff matrix. In addition, there are one or several possibilities to opt out from the game by adopting loner strategies. Optional games lead to relaxed social dilemmas. Here we explore the interaction between spatial structure and optional games. We find that increasing the number of loner strategies (or equivalently increasing mutational bias toward loner strategies) facilitates evolution of cooperation both in well-mixed and in structured populations. We derive various limits for weak selection and large population size. For some cases we derive analytic results for strong selection. We also analyze strategy selection numerically for finite selection intensity and discuss combined effects of optionality and spatial structure.

Introduction

In the typical setting of evolutionary game theory, the individual has to adopt one of several strategies (Hofbauer and Sigmund, 1988, Hofbauer and Sigmund, 1998, Weibull, 1997, Friedman, 1998, Cressman, 2003, Nowak, 2004, Vincent and Brown, 2005, Gokhale and Traulsen, 2011). For example in a standard cooperative dilemma (Hauert et al., 2006, Nowak, 2012, Rand and Nowak, 2013, Débarre et al., 2014), the individual can choose between cooperation and defection. Natural selection tends to oppose cooperation unless a mechanism for evolution of cooperation is at work (Nowak, 2006a). In optional games there is also the possibility not to play the game (Kitcher, 1993, Batali and Kitcher, 1995, Hauert et al., 2002, Hauert, 2002, Szabó and Hauert, 2002a, De Silva et al., 2009, Rand and Nowak, 2011). The individual player has to choose whether to participate in the game (by cooperating or defecting) or to opt out. Opting out leads to fixed “loner׳s payoff”. This loner׳s payoff is forfeited if one decides to play the game. Thus there is a cost for playing the game. Optional games tend to lead to relaxed social dilemmas (Michor and Nowak, 2002, Hauert et al., 2006). They have also been used to study the effect of costly punishment (by peers and institutions) on evolution of cooperation (Boyd and Richerson, 1992, Nakamaru and Iwasa, 2005, Hauert et al., 2007, Sigmund, 2007, Traulsen et al., 2009, Hilbe and Sigmund, 2010). There is also a relationship between optional games and empty places in spatial settings (Nowak et al., 1994).

Here we study the effect of optional games on cycles and on complete graphs (van Veelen and Nowak, 2012). Cycles and complete graphs are on opposite ends of the spectrum of spatial structure. Most graphs will lead to an evolutionary dynamics between these two extremes. Evolutionary graph theory (Lieberman et al., 2005, Santos and Pacheco, 2005, Ohtsuki et al., 2006, Szabó and Fáth, 2007, Fu et al., 2007a, Fu et al., 2007b, Santos et al., 2008, Perc and Szolnoki, 2010, Perc, 2011, Allen et al., 2013, Maciejewski, 2014; Allen and Nowak, 2014) is an approach to study the effect of population structure on evolutionary dynamics (Nowak and May, 1992, Nakamaru et al., 1997, Tarnita et al., 2009a, Tarnita et al., 2009b, Tarnita et al., 2011, Nowak et al., 2010). Using stochastic evolutionary dynamics for games in finite populations (Foster and Young, 1990, Challet and Zhang, 1997, Taylor et al., 2004, Nowak et al., 2004, Imhof and Nowak, 2006, Traulsen et al., 2006), we notice that the number of different loner strategies has an important effect on selection between strategies that occur in the game. Increasing the number of ways to opt out (or, increasing mutational bias toward Garcia and Traulsen, 2012 loner strategies) in general favors evolution of cooperation.

Our paper is organized as follows. In Section 2 we give an overview of the basic model and list our key results. In Section 3 we calculate the abundance in the low mutation limit. It is used to investigate the conditions for strategy selection in the weak selection limit in Section 4 and in the strong selection limit in Section 5. We calculate these conditions for optional games with simplified prisoner׳s dilemma games in Section 6. We then analyze strategy selection numerically for finite mutation rate as well as finite selection intensity in low mutation in Section 7. In our concluding remarks in Section 8, we summarize and discuss the implications of our findings.

Section snippets

Model and main results

We consider stochastic evolutionary dynamics of populations on graphs. In particular, we investigate the condition for one strategy to be favored over the others in the limit of low mutation and for two different reproduction processes, birth–death (BD) updating and death–birth (DB) updating on cycles. We compare the results with those for the Moran Process (MP) on the complete graph. The fitness of an individual is determined by the payoff from the non-repeated matrix games with its nearest

Derivation of general expressions for fixation probability and abundance

We now begin our derivation of the results presented above. We begin by obtaining general expressions for fixation probability and abundance that are valid for any population size and selection intensity. These expressions are obtained first for a general 3×3 matrix game, and then for the optional prisoners׳ dilemma game.

When there are mutations, the population will not evolve to an absorbing state of one kind. Yet, in many cases, it is expected for them to evolve to a steady state in which the

Analysis of the wN limit

We now consider the results of Section 3 under the wN limit. This limit is obtained by taking the w0 limit for fixed N, and then taking the N limit of the result. We calculate the abundance in terms of fixation probabilities in the wN limit and analyze the condition for the cooperators are more abundant than defectors.

Analysis of the Nw limit

Here, we consider the results of Section 3 under the Nw limit. We first calculate fixation probability in the large N limit using Eq. (19). The Nw limit is obtained by taking the w0 limit of the result. Once we obtain fixation probability in this limit, we calculate the abundance and find that the condition for strategy Si is more abundant than strategy Sj for three strategy games.

Optional game with simplified prisoner׳s dilemma

To further clarify how spatial structure and optionality of the game affect the success of cooperation, we study an optional version of a simplified prisoner׳s dilemma, in which cooperators pay a cost c to generate a benefit b for the other player. This simplified prisoner׳s dilemma is also known as the donation game or the prisoner׳s dilemma with equal gains from switching. Here, we consider the n=1 optional game with a simplified prisoner׳s dilemma, whose payoff matrix is given byCDLCDL(bccg

Numerical analysis

We have analyzed the conditions for strategy selection analytically in the two extreme limits of selection intensity, w0 and w in the zero mutation rate. Here, we first obtain conditions for xC>xD in the simplified game (72) numerically for finite values of w (with low mutation rate), using calculated abundance from fixation probabilities. Then, we perform a series of Monte Carlo simulations with small but finite mutation rates. The condition for strategy selection is obtained numerically

Conclusion

We have analyzed strategy selection in optional games on cycles and on complete graphs and found a non-trivial interaction between volunteering and spatial selection.

For 2×2 games on cycles using exponential fitness, we have presented a closed form expression for the fixation probability for any intensity of selection and any population size. Using this fixation probability, we have found the conditions for strategy selection analytically in the limits of weak intensity of selection and large

Acknowledgments

Support from the program for Foundational Questions in Evolutionary Biology (FQEB), the National Philanthropic Trust, the John Templeton Foundation and the National Research Foundation of Korea grant (NRF-2010-0022474) is gratefully acknowledged.

References (59)

  • M. Nakamaru et al.

    The evolution of cooperation in a lattice-structured population

    J. Theor. Biol.

    (1997)
  • M.A. Nowak

    Evolving cooperation

    J. Theor. Biol.

    (2012)
  • M. Perc et al.

    Coevolutionary games—a mini review

    Biosystems

    (2010)
  • D.G. Rand et al.

    Human cooperation

    Trends Cogn. Sci.

    (2013)
  • K. Sigmund

    Punish or perish? Retaliation and collaboration among humans

    Trends Ecol. Evol.

    (2007)
  • G. Szabó et al.

    Evolutionary games on graphs

    Phys. Rep.

    (2007)
  • C.E. Tarnita et al.

    Strategy selection in structured populations

    J. Theor. Biol.

    (2009)
  • C. Taylor et al.

    Evolutionary game dynamics in finite populations

    Bull. Math. Biol.

    (2004)
  • M. van Veelen et al.

    Multi-player games on the cycle

    J. Theor. Biol.

    (2012)
  • B. Allen et al.

    Games on Graphs

    EMS Surv. Math. Sci.

    (2014)
  • B. Allen et al.

    Spatial dilemmas of diffusible public goods

    eLife

    (2013)
  • R. Cressman

    Evolutionary Dynamics and Extensive Form Games

    (2003)
  • F. Débarre et al.

    Social evolution in structured populations

    Nat. Commun.

    (2014)
  • H. De Silva et al.

    Freedom, enforcement, and the social dilemma of strong altruism

    J. Evol. Econ.

    (2009)
  • D. Friedman

    On economic applications of evolutionary game theory

    J. Evol. Econ.

    (1998)
  • J. Garcia et al.

    The structure of mutations and the evolution of cooperation

    Plos One

    (2012)
  • C. Hauert

    Volunteering as red queen mechanism for cooperation in public goods games

    Science

    (2002)
  • C. Hauert et al.

    Via freedom to coercionthe emergence of costly punishment

    Science

    (2007)
  • C. Hilbe et al.

    Incentives and opportunismfrom the carrot to the stick

    P. R. Soc. B

    (2010)
  • Cited by (0)

    View full text