Evolutionary games in the multiverse

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Gokhale CS, Traulsen A.

Evolutionary games in the multiverse.

Proceedings of the National Academy of Sciences.

Mar 23;107(12):5500–4, 2010

…human life is a many-person game and not just a dis-joined collection of two-person games – William D. Hamilton

Stander (1992) has described the hunting behaviour of lionesses on the open semi-arid plains of Namibia. Individual hunting tactics of lionesses were analysed from 486 independent group hunts. The lionesses hunt in packs and employ the flush and ambush technique. Some lie in ambush while others flush out the prey from the flanks and drive them towards the ones waiting in ambush. This technique needs an interaction of more than two players to succeed.

Flush and Ambush. Taken from [Stander (1992)](#stander:1992) this sketch describes the flush and ambush technique used by the lionesses in Etosha National Park. Top panel reflects the position of the lionesses from the point of view of the prey. The bottom panel shows the attack positions of the lionesses. A-B form the left flank while F-G are the right flank. C-D-E take the centre position. This hunting set-up is not possible with just two lionesses..

Similarly, we interact with innumerable people at the same time, directly or indirectly. Some interactions may be pair-wise, but others are not. In real life, we may typically be engaged in many person games instead of a disjoined collection of two person games (Hamilton, 1975). Evolutionary game theory which we have discussed until now has been about two player games. It becomes mathematically more demanding when we try to include more players. Hamilton (1975) illustrates the theoretical challenges of multiplayer games as, “The theory of many-person games may seem to stand to that of two-person games in the relation of sea-sickness to a headache.” A special class of multiplayer games has been experimentally and theoretically studied by economists and sociologists to study social behaviour of individuals. Such a typical “public goods game” consists of participants who have an option of contributing to a common pot. The sum is shared equally amongst all participants. Numerous variants of this basic set-up have been explored (Wedekind and Milinski, 2000; Milinski et al., 2001; Anderson and Franks, 2001; Hauert et al., 2002; Semmann et al., 2003; Milinski et al., 2006; Rockenbach and Milinski, 2006; Hauert et al., 2007;Milinski et al., 2008; Santos et al., 2008; Pacheco et al., 2009; Souza et al., 2009; van Veelen, 2009; Traulsen et al., 2010; Connor, 2010). We develop some general conditions for multiplayer games with multiple strategies with simplicity in mind (Kurokawa and Ihara, 2009; Gokhale and Traulsen, 2010). To refrain from repeating the word “multi” for player and strategies we use the short form “games in the multiverse” for these kind of games. Let us begin with the well known scenario of two player games with two strategies and add one more player to this setting. The changes which happen are reviewed below,

  • The payoff matrix for a 2 x 2 games is a square matrix whereas for a 2 x 2 x 2 player game it is an extended table of permutations.
  • The dynamics for a 2 x 2 proceeds on a simplex which is one dimensional, a single line. Even for 2 x 2 x 2 games there are only two strategies and thus the simplex is a single line.
  • There are five possible outcomes for a 2 x 2 game, as shown in Fig. 4.2. As the number of player increases the possible internal equilibrium points also increase. For 2 x 2 x 2 games all the scenarios from 2 x 2 games are possible and in addition there is a possibility of having two equilibria in the interior, one stable one unstable.

Adding a player to the usual setting increases the complexity. The level of complexity increases as more and more players are added. In this project we examine this complexity and extract simple relations from it. There were two main analytical advances in evolutionary game theory in the study of finite populations for two player games with two strategies.

Comparing two player games with three player games for two strategies. Writing the payoffs for three player games cannot be done in a square payoff matrix as two player games. Instead it is a table of permutations of a player playing with 2 other players. For two player games there are five possible outcomes. As the payoffs are linear in x there can be at most a single internal equilibrium. For three player games the payoffs are not linear in $x$ but of degree 2 leading to at most two possible solutions in the interior of the simplex.
  • When is the fixation probability of a strategy is greater than neutral (One Third rule)?
  • When is the fixation probability of a strategy is greater than the fixation probability of the other strategy (Risk Dominance)?

We develop general conditions for multiplayer games and two strategies without compromising on simplicity. Using the infinite population size assumption we also calculate the maximum number of internal equilibria of a given game with multiple player and multiple strategies.

Link to Paper

P. E. Stander. Cooperative hunting in lions: the role of the individual. Behavioral Ecology and Sociobiology, 29:445–454, 1992.

W. D. Hamilton. Innate social aptitudes of man: an approach from evolutionary genetics. In R. Fox, editor, Biosocial Anthropology, pages 133–155. Wiley, New York, 1975.

C. Wedekind and M. Milinski. Cooperation through image scoring in humans. Science, 288: 850–852, 2000.

M. Milinski, D. Semmann, T. C. Bakker, and H. J. Krambeck. Cooperation through indirect reciprocity: image scoring or standing strategy? Proc. R. Soc. Lond. B, 268:2495–2501, Dec 2001. ISSN 0962-8452.

C. Anderson and N. R. Franks. Teams in animal societies. Behavioral Ecology, 12(5):534–540, 2001.

C. Hauert, S. De Monte, J. Hofbauer, and K. Sigmund. Volunteering as red queen mechanism for cooperation in public goods games. Science, 296:1129–1132, 2002.

D. Semmann, H. J. Krambeck, and M. Milinski. Volunteering leads to rock-paper-scissors dynamics in a public goods game. Nature, 425(6956):390–393, 2003.

M. Milinski, D. Semmann, H. J. Krambeck, and J. Marotzke. Stabilizing the earth’s climate is not a losing game: Supporting evidence from public goods experiments. Proc. Natl. Acad. Sci. USA, 103:3994–3998, 2006.

B. Rockenbach and M. Milinski. The efficient interaction of indirect reciprocity and costly punishment. Nature, 444:718–723, 2006.

C. Hauert, A. Traulsen, H. Brandt, M. A. Nowak, and K. Sigmund. Via freedom to coercion: the emergence of costly punishment. Science, 316:1905–1907, 2007.

M. Milinski, R. D. Sommerfeld, H.-J. Krambeck, F. A. Reed, and J. Marotzke. The collective- risk social dilemma and the prevention of simulated dangerous climate change. Proc. Natl. Acad. Sci. USA, 105(7):2291–2294, 2008.

F. C. Santos, M. D. Santos, and J. M. Pacheco. Social diversity promotes the emergence of cooperation in public goods games. Nature, 454:213–216, 2008.

J. M. Pacheco, F. C. Santos, M. O. Souza, and B. Skyrms. Evolutionary dynamics of collective action in N-person stag hunt dilemmas. Proc. R. Soc. B, 276:315–321, 2009.

M. O. Souza, J. M. Pacheco, and F. C. Santos. Evolution of cooperation under n-person snowdrift games. J. Theor. Biol., 260:581–588, 2009.

M. van Veelen. Group selection, kin selection, altruism and cooperation: When inclusive fitness is right and when it can be wrong. J. Theor. Biol., 259:589–600, 2009.

A. Traulsen, D. Semmann, R. D. Sommerfeld, H-J. Krambeck, and M. Milinski. Human strategy updating in evolutionary games. Proc. Natl. Acad. Sci. U.S.A., 107(7):2962–2966, 2010.

R. C. Connor. Cooperation beyond the dyad: on simple models and a complex society. Phil. Trans. Roy. Soc. London B, 365:2687–2697, 2010.

S. Kurokawa and Y. Ihara. Emergence of cooperation in public goods games. Proc. R. Soc. B, 276:1379–1384, 2009.

C. S. Gokhale and A. Traulsen. Evolutionary games in the multiverse. Proc. Natl. Acad. Sci. U.S.A., 107(12):5500–5504, 2010.

Evolutionary games in the multiverse was last modified on Chaitanya S. Gokhale