# 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).

 Left Right Up A, a C, c Down B, b D, d Fig. 1: 2-player coordination game

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 > c, d > b. 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 (strategies are usually sorted so that the Nash equilibria are in the diagonal from top left to bottom right), as well as for a game with more than two or more players.

## Examples

A typical case for a coordination game is choosing the sides 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 execute the same swerving maneuver they will manage to pass each other, but if they choose differing maneuvers they will collide. In the payoff matrix in Fig. 2, successful passing is represented by a payoff of 10, and a collision 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}.

 Left Right Left 10, 10 0, 0 Right 0, 0 10, 10 Fig. 2: Choosing sides
 Party Home Party 10, 10 0, 0 Home 0, 0 5, 5 Fig. 3: Pure coordination game
 Party Home Party 10, 5 0, 0 Home 0, 0 5, 10 Fig. 4: Battle of the sexes
 Stag Hare Stag 10, 10 0, 8 Hare 8, 0 7, 7 Fig. 5: Stag hunt

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 strategy 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-b)/(a+d-b-c) to play Up and 1-p to play Down for player 1, and q = (D-C)/(A+D-B-C) to play Left and 1-q to play Right for player 2. Since d > b and d-b < a+d-b-c, 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. 6.

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

## 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 makes certain coordination games especially complicated and interesting (e.g. the Stag hunt, in which {Stag,Stag} has higher payoffs, but {Hare,Hare} is safer).

## Other games with externalities

Coordination games are closely linked to the economic concept of externalities, and in particular positive network externalities, the benefit reaped from being in the same network as other agents. Conversely, game theorists have modeled behavior under negative externalities where choosing the same action creates a cost rather than a benefit. The generic term for this class of game is anti-coordination game. The best-known example of a 2-player anti-coordination game is the game of Chicken (also known as Hawk-Dove game). Using the payoff matrix in Figure 1, a game is an anti-coordination game if B > A and C > D for row-player 1 (with lowercase analogues for column-player 2). {Down, Left} and {Up, Right} are the two pure Nash equilibria. Chicken also requires that A > C, so a change from {Up, Left} to {Up, Right} improves player 2's payoff but reduces player 1's payoff, introducing conflict. This counters the standard coordination game setup, where all unilateral changes in a strategy lead to either mutual gain or mutual loss.

The concept of anti-coordination games has been extended to multi-player situation. A crowding game is defined as a game where each player's payoff is non-increasing over the number of other players choosing the same strategy (i.e., a game with negative network externalities). For instance, a driver could take U.S. Route 101 or Interstate 280 from San Francisco to San Jose. While 101 is shorter, 280 is considered more scenic, so drivers might have different preferences between the two independent of the traffic flow. But each additional car on either route will slightly increase the drive time on that route, so additional traffic creates negative network externalities, and even scenery-minded drivers might opt to take 101 if 280 becomes too crowded. A congestion game is a crowding game in networks. The minority game is a game where the only objective for all players is to be part of smaller of two groups. A well-known example of the minority game is the El Farol Bar problem proposed by W. Brian Arthur.

A hybrid form of coordination and anti-coordination is the discoordination game, where one player's incentive is to coordinate while the other player tries to avoid this. Discoordination games have no pure Nash equilibria. In Figure 1, choosing payoffs so that A > B, C < D, while a < b, c > d, creates a discoordination game. In each of the four possible states either player 1 or player 2 are better off by switching their strategy, so the only Nash equilirium is mixed. The canonical example of a discoordination game is the matching pennies game.