Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered payoff dominant if it is Pareto superior to all other Nash equilibria in the game.¹ When faced with a choice among equilibria only all players would agree on the payoff dominant equilibrium since it offers at least as much payoff as each player's best alternative. Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction, meaning the more uncertainty players have about the actions of the other player(s), the more likely they will choose the risk dominant strategy. In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... Game theory is most often described as a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ... John Charles Harsanyi (Hungarian: Harsányi János) (May 29, 1920 – August 9, 2000) was a Hungarian-Australian-American business and economics professor who contributed to the study of game theory in mathematics by developing the quite revolutionary analysis of games of incomplete information, so-called Bayesian games. ... Reinhard Selten (born October 5, 1930) is a German economist. ... In game theory, a non-cooperative game is a one in which players can cooperate, but any cooperation must be self-enforcing. ... In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ... In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... Game theory is most often described as a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... John Charles Harsanyi (Hungarian: Harsányi János) (May 29, 1920 – August 9, 2000) was a Hungarian-Australian-American business and economics professor who contributed to the study of game theory in mathematics by developing the quite revolutionary analysis of games of incomplete information, so-called Bayesian games. ... Reinhard Selten (born October 5, 1930) is a German economist. ... Pareto efficiency, or Pareto optimality, is an important notion in neoclassical economics with broad applications in game theory, engineering and the social sciences. ... In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...

The payoff matrix in Figure 1 provides a simple two-player, two-strategy example of a game with two pure Nash equilibria. The strategy pair (Hunt, Hunt) is payoff dominant since payoffs are higher for both players compared to the other pure NE, (Gather, Gather). On the other hand, (Gather, Gather) risk dominates (Hunt, Hunt) since if uncertainty exists about the other player's action, gathering will provide a higher expected payoff. The game in Figure 1 is a well-known game-theoretic dilemma called stag hunt. The rationale behind it is that communal action (hunting) yields a higher return if all players combine their skills, but if it is unknown whether the other player helps in hunting, gathering might turn out to be the better indvidual strategy for food provision, since it does not depend on coordinating with the other player. In addition, gathering alone is preferred to gathering in competition with others. This game is similar to the Prisoner's dilemma in that it provides a rationale why collective action might fail in the absence of credible commitments. It has been suggested that this article or section be merged with normal form game. ... In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ... Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma is a type of non-zero-sum game in which two players can cooperate with... The economic theory of collective action is concerned with the provision of public goods (and other collective consumption) through the collaboration of two or more individuals, and the impact of externalities on group behavior. ...

In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ...

Formal definition

The game given in Figure 2 is a coordination game if the following payoff inequalities hold for player 1 (rows): A > B, D > C, and for player 2 (columns): a > b, d > c. The strategy pairs (H, H) and (G, G) are then the only pure Nash equilibria. In addition there is a mixed Nash equilibrium where player 1 plays H with probability p = (d-c)/(a-b-c+d) and G with probability 1–p; player 2 plays H with probability q = (D-C)/(A-B-C+D) and G with probability 1–q. In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ...

Strategy pair (H, H) payoff dominates (G, G) if A ≥ D, a ≥ d, and at least one of the two is a strict inequality: A > D or a > d.

Strategy pair (G, G) risk dominates (H, H) if the following inequality holds: (B – A)(b – a) ≥ (C – D)(c – d). If the inequality is strict then (G, G) strictly risk dominates (H, H).²

If the game is symmetric, so if A = a, B = b, etc., the inequality allows for a simple interpretation: We assume the players are unsure about which strategy the opponent will pick and assign probabilities for each strategy. If each player assigns probabilities ½ to H and G each, then (G, G) risk dominates (H, H) if the expected payoff from playing G exceeds the expected payoff from playing H: ½ B + ½ D ≥ ½ A + ½ C, or simply B + D ≥ A + C.

Equilibrium selection

Even though rational choice theory would suggest that players converge on the payoff dominant equilibrium³ a number of evolutionary approaches have established that when played in a large population, players might fail to do so and instead end up in the payoff dominated, risk dominant equilibrium. Two separate evolutionary models come to different predictions how likely this is. The first model, based on replicator dynamics, predicts that a population is more likely to adopt the risk dominant equilibrium than the payoff dominant equilibrium. The second model, based on best response strategy revision and mutation, predicts that the risk dominant state is the only stochastically stable equilibrium. Both models assume that multiple two-player games are played in a population of N players. The players are matched randomly with opponents, with each player having equal likelihoods of drawing any of the N−1 other players. The players start with a pure strategy, G or H, and play this strategy against their opponent. In replicator dynamics, the population game is repeated in sequential generations where subpopulations change based on the success of their chosen stratregies. In best response, players update their strategies to improve expected payoffs in the subsequent generations. The recognition of Kandori, Mailath & Rob (1993) and Young (1993) was that if the rule to update one's strategy allows for mutation⁴, and the probability of mutation vanishes, i.e. asymptotically reaches zero over time, the likelihood that the risk dominant equilibrium is reached goes to one, even if it is payoff dominated. Rational choice theory assumes human behaviour as guided by instrumental reason. ... The replicator equation is a differential equation that defines the dynamics of evolutionary games. ... In game theory, the best response, is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players strategies as given. ... In biology, mutations are changes to the base pair sequence of genetic material (either DNA or RNA). ...

Notes

• Note 1: A single Nash equilibrium is trivially payoff and risk dominant if it is the only NE in the game.
• Note 2: Similar distinctions between strict and weak exist for most definitions here, but are not denoted explicitly unless necessary.
• Note 3: Harsanyi and Selten (1988) propose that the payoff dominant equilibrium is the rational choice in the stag hunt game.

References

• Samuel Bowles: Microeconomics: Behavior, Institutions, and Evolution Princeton University Press, N.J., (2004) pp. 45–46
• Drew Fudenberg and David Levine: The Theory of Learning in Games MIT Press, (1999) p. 27
• John C. Harsany and Reinhard Selten: A General Theory of Equilibrium Selection in Games, MIT Press (1988)
• Michihiro Kandori, George J. Mailath & Rafael Rob: "Learning, mutation, and long-run equilibria in games", Econometrica 61, pp. 29–56 (1993)
• Roger B. Myerson: Game Theory, Analysis of conflict, Cambridge, Harvard University Press (1991) pp. 118–119
• Larry Samuelson: Evolutionary Games and Equilibrium Selection, MIT Press (1997) ISBN 026219382-5
• H. Peyton Young: "The evolution of conventions", Econometrica, 61, pp. 57–84 (1993)
• H. Peyton Young: Individual Strategy and Social Structure Princeton University Press (1998) ISBN 0691086877