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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. When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered risk dominant if it has the largest 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 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 individual 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. Like the Prisoner's dilemma, it provides a reason why collective action might fail in the absence of credible commitments. == 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. 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: . If the inequality is strict then (G, G) strictly risk dominates (H, H).(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: , or simply . 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: (where ''p'' is the probability that the other player will play H), and compare it to the expected payoff if they play G: . The value of ''p'' which makes these two expected values equal is the risk factor for the equilibrium (H, H), with 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Risk dominance」の詳細全文を読む スポンサード リンク
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