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In game theory, coordination games are a class of games in which all pure strategy Nash equilibria exist when players choose the same or corresponding strategies. The classic example for a coordination game is the 2-player, 2-strategy game, with a payoff matrix as shown on the right (Fig. 1). In a coordination game the following inequalities in payoffs hold for player 1 (rows): A > B, D > C, and for player 2 (columns): a > b, d > c. In this game the strategy profiles and are pure Nash equilibria, marked in yellow. This setup can be extended for more than two strategies, where strategies are usually sorted so that the Nash equilibria are in the diagonal from top left to bottom right, as well as game with more than two players. The opposite of a coordination game is a discoordination game.
Examples A typical case for a coordination game is choosing the side of the road upon which to drive. In a simplified example, assume that two drivers meet on a narrow dirt road. Both have to swerve in order to avoid a head-on collision. If both choose the same side they manage to pass each other but if they choose different sides they will collide. In the payoff matrix in Fig. 2, "pass" is represented by a payoff of 10, and "collide" by a payoff of 0. In this case there are two pure Nash equilibria: either both swerve to the left, or both swerve to the right. In this example, it doesn't matter which side both players pick, as long as they both pick the same. Both solutions are Pareto efficient. This is not true for all coordination games, as the pure coordination game in Fig. 3 shows. Pure (or common interest) coordination is the game where the player both prefer the same Nash equilibrium outcome, here both players partying over both watching TV. The outcome Pareto dominates the outcome, just as both Pareto dominate the other two outcomes, and . This is different in another type of coordination game commonly called battle of the sexes (or conflicting interest coordination), as seen in Fig. 4. In this game both players prefer engaging in the same activity over going alone, both their preferences differ over which activity they should engage in. Player 1 prefers that both party while player 2 prefers that they both watch TV. Mixed Nash equilibrium Coordination games also have mixed strategy Nash equilibria. In the generic coordination game above, a mixed Nash equilibrium is given by probabilities p = (d-c)/(a-b-c+d) to play Up and 1-p to play Down for player 1, and q = (D-C)/(A-B-C+D) to play Left and 1-q to play Right for player 2. Since d > c and d-c < a-b-c+d, p is always between zero and one, so existence is assured (similarly for q). The reaction correspondences for 2×2 coordination games are shown in Fig. 5. The pure Nash equilibria are the points in the bottom left and top right corners of the strategy space, while the mixed Nash equilibrium lies in the middle, at the intersection of the dashed lines. Unlike the pure Nash equilibria, the mixed equilibrium is not an evolutionarily stable strategy (ESS). The mixed Nash equilibrium is also Pareto dominated by the two pure Nash equilibria (since the players will fail to coordinate with non-zero probability), a quandary that led Robert Aumann to propose the refinement of a correlated equilibrium. Coordination and equilibrium selection Games like the driving example above have illustrated the need for solution to coordination problems. Often we are confronted with circumstances where we must solve coordination problems without the ability to communicate with our partner. Many authors have suggested that particular equilibria are focal for one reason or another. For instance, some equilibria may give higher payoffs, be naturally more salient, may be more fair, or may be safer. Sometimes these refinements conflict which lead to some of the other interesting coordination games (e.g. Stag hunt and Battle of the sexes). Other coordination games | ||||||||
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