Coordination game

In game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies. Coordination games are a formalization of the idea of a coordination problem, which is widespread in the social sciences, including economics, meaning situations in which all parties can realize mutual gains, but only by making mutually consistent decisions. A common application is the choice of technological standards.

For a classic example of a coordination game, consider the 2-player, 2-strategy game, with the payoff matrix shown on the right (Fig. 1).

If this game is a coordination game, then 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 {Left, Up} and {Right, Down} are pure Nash equilibria, marked in gray. 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.

Examples

A typical case for a coordination game is choosing the side of the road upon which to drive, a social standard which can save lives if it is widely adhered to. 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 swerve to the same side they will 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 players both prefer the same Nash equilibrium outcome, here both players prefer partying over both staying at home to watch TV. The {Party, Party} outcome Pareto dominates the {Home, Home} outcome, just as both Pareto dominate the other two outcomes, {Party, Home} and {Home, Party}.

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, but their preferences differ over which activity they should engage in. Player 1 prefers that they both party while player 2 prefers that they both stay at home.

Finally, the stag hunt game in Fig. 5 shows a situation in which both players (hunters) can benefit if they cooperate (hunting a stag). However, cooperation might fail, because each hunter has an alternative which is safer because it does not require cooperation to succeed (hunting a hare). This example of the potential conflict between safety and social cooperation is originally due to Jean-Jacques Rousseau.

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)/(d-c+a-b) to play Up and 1-p to play Down for player 1, and q = (D-C)/(D-C+A-B) 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.

Fig.5 - Reaction correspondence for 2x2 coordination games. Nash equilibria shown with points, where the two player's correspondences agree, ie. cross