Risk dominance

Risk dominance
Payoff dominance
Risk dominance; Payoff dominance
A Non-cooperative games
Example	Stag hunt

Risk dominance and payoff dominance are two related refinements of the

basin of attraction

(i.e. is less risky). This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.

The

credible commitments

.

	Hunt	Gather
Hunt	5, 5	0, 4
Gather	4, 0	2, 2
Fig. 1: Stag hunt example

	H	G
H	A, a	C, b
G	B, c	D, d
Fig. 2: Generic coordination game

Formal definition

The game given in Figure 2 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.

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 product of the deviation losses is highest for (G, G) (Harsanyi and Selten, 1988, Lemma 5.4.4). In other words, if the following inequality holds: (C – D)(c – d)≥(B – A)(b – a). If the inequality is strict then (G, G) strictly risk dominates (H, H).²(That is, players have more incentive to deviate).

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.

Another way to calculate the risk dominant equilibrium is to calculate the risk factor for all equilibria and to find the equilibrium with the smallest risk factor. To calculate the risk factor in our 2x2 game, consider the expected payoff to a player if they play H: $E[\pi _{H}]=pA+(1-p)C$ (where p is the probability that the other player will play H), and compare it to the expected payoff if they play G: $E[\pi _{G}]=pB+(1-p)D$ . The value of p which makes these two expected values equal is the risk factor for the equilibrium (H, H), with $1-p$ the risk factor for playing (G, G). You can also calculate the risk factor for playing (G, G) by doing the same calculation, but setting p as the probability the other player will play G. An interpretation for p is it is the smallest probability that the opponent must play that strategy such that the person's own payoff from copying the opponent's strategy is greater than if the other strategy was played.

Equilibrium selection

A number of evolutionary approaches have established that when played in a large population, players might fail to play the payoff dominant equilibrium strategy and instead end up in the payoff dominated, risk dominant equilibrium. Two separate evolutionary models both support the idea that the risk dominant equilibrium is more likely to occur. The first model, based on

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 strategies. 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.³

Notes

^1 A single Nash equilibrium is trivially payoff and risk dominant if it is the only NE in the game.
^2 Similar distinctions between strict and weak exist for most definitions here, but are not denoted explicitly unless necessary.
^3 Harsanyi and Selten (1988) propose that the payoff dominant equilibrium is the rational choice in the stag hunt game, however Harsanyi (1995) retracted this conclusion to take risk dominance as the relevant selection criterion.

References

Samuel Bowles: Microeconomics: Behavior, Institutions, and Evolution, Princeton University Press, pp. 45–46 (2004)
ISBN 0-691-09163-3

Drew Fudenberg and David K. Levine: The Theory of Learning in Games, MIT Press, p. 27 (1999)
ISBN 0-262-06194-5

John C. Harsanyi: "A New Theory of Equilibrium Selection for Games with Complete Information", Games and Economic Behavior 8, pp. 91–122 (1995)
John C. Harsanyi and Reinhard Selten: A General Theory of Equilibrium Selection in Games, MIT Press (1988)
ISBN 0-262-08173-3

Rafael Rob: "Learning, Mutation, and Long-run Equilibria in Games", Econometrica 61, pp. 29–56 (1993) Abstract

Roger B. Myerson: Game Theory, Analysis of Conflict, Harvard University Press, pp. 118–119 (1991)
ISBN 0-674-34115-5

ISBN 0-262-19382-5

H. Peyton Young: "The Evolution of Conventions", Econometrica, 61, pp. 57–84 (1993) Abstract
H. Peyton Young: Individual Strategy and Social Structure, Princeton University Press (1998)
ISBN 0-691-08687-7

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v t e Topics in game theory
Definitions	Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game
Equilibrium concepts	Bayes correlated equilibrium Bayesian Nash equilibrium Berge equilibrium Core Correlated equilibrium Epsilon-equilibrium Evolutionarily stable strategy Gibbs equilibrium Mertens-stable equilibrium Markov perfect equilibrium Nash equilibrium Pareto efficiency Perfect Bayesian equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Risk dominance Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfection Trembling hand
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Key figures	Albert W. Tucker Amos Tversky Antoine Augustin Cournot Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin John Conway Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
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