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Game Theory

Game Theory

Game theory is a mathematical theory concerned with the optimum choice of strategy in situations involving a conflict of interest pioneered by John von Neumann (Rowland, 1944). Evolutionary game (Wang et al, 2015), including coevolutionary game, (Perc et al, 2010) will no longer model into a rational game, but achieve the game equilibrium by trial and error method. It has wide real-world applications in many fields, including economics, biology, computer science, aerospace, and so on. The Nash equilibrium is a fundamental concept in game theory, which means that each player has made his or hers stable choice. In (1951) Nash proved that if mixed strategies were allowed, then every game with a finite number of players and strategies had at least one Nash equilibrium. It has been shown that every congestion game and every finite potential game possess a pure Nash equilibrium (PNE) in (Nash, 1951; Harks et al, 2012; Monderer et al, 1996).

According to the difference of the decision order made among the players, a game can be divided into static case and dynamic case (Gibbons, 1992). A static game means that all players take actions at the same time or if they do not take actions together, they know nothing about what others have chosen. A dynamic game means that players making decisions in order and players can get part or complete information of the game’s history. Besides, a game can also be distinguished as cooperative or non-cooperative (Basar et al, 1998), complete information or incomplete information (Watson, 2013). One of the most classic game models in game theory is Prisoner’s Dilemma (Kreps et al, 1982), which is a non-cooperative, complete information and static game.

A static game of complete information is a fundamental kind of games, and the method of finding a solution to such game is called List Method (Watson, 2013). Such method is easy to understand and convenient, but powerless when facing big data (Wang et al, 2018).

Reference

  • Basar, T., Olsder. G.J. (1998). “Dynamic Noncooperative Game Theory”. SIAM.
  • Gibbons, R. (1992). “A Primer in Game Theory”. Harvester Wheatsheaf.
  • Harks, T., Klimm, M. (2012). “On the existence of pure Nash equilibria inweighted congestion games”. Math. Oper. Res., 37, pp. 419-436.
  • Kreps, D.M., Milgrom, P., Roberts, J., Wilson, R. (1982). “Rational cooperation in the finitely repeated prisoners’ dilemma”. J. Econ. Theory, 27, pp. 245-252.
  • Monderer, D., Shapley, L.S. (1996). “Potential games”. Games Econ. Behav., 14, pp. 124-143.
  • Nash, J. (1951). “Non-cooperative games”. Ann. Math., 54, pp. 286-295.
  • Perc, M., Szolnoki, A. (2010). “Coevolutionary gamesa mini review”. BioSystems, 99, pp. 109-125.
  • E. (1944). “The Theory of Games and Economic Behavior”. Princeton University Press.
  • Wang, L., Liu, Y., Wu, ZH., E.Alsaadi, F. (2018). “Strategy optimization for static games based on STP method”. Applied Mathematics and Computation, Volume 316, Pages 390-399.
  • Wang, Z., Wang L., Szolnoki, A., Perc, M. (2015). “Evolutionary games on multilayer networks: a colloquium”. Eur. Phys. J. B, 88, p. 124
  • Watson, J. (2013). “Strategy: An Introduction to Game Theory”. WW Norton.

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