Coordination game

A coordination game is a type of simultaneous game found in game theory. It describes the situation where a player will earn a higher payoff when they select the same course of action as another player. The game is not one of pure conflict, which results in multiple pure strategy Nash equilibria in which players choose matching strategies. Figure 1 shows a 2-player example.

Both (Up, Left) and (Down, Right) are Nash equilibria. If the players expect (Up, Left) to be played, then player 1 thinks their payoff would fall from 2 to 1 if they deviated to Down, and player 2 thinks their payoff would fall from 4 to 3 if they chose Right. If the players expect (Down, Right), player 1 thinks their payoff would fall from 2 to 1 if they deviated to Up, and player 2 thinks their payoff would fall from 4 to 3 if they chose Left. A player's optimal move depends on what they expect the other player to do, and they both do better if they coordinate than if they played an off-equilibrium combination of actions. This setup can be extended to more than two strategies or two players.

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 8, 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 game is called a pure coordination game. This is not true for all coordination games, as the assurance game in Fig. 3 shows.

An assurance game describes the situation where neither player can offer a sufficient amount if they contribute alone, thus player 1 should defect from playing if player 2 defects. However, if Player 2 opts to contribute then player 1 should contribute also. An assurance game is commonly referred to as a “stag hunt” (Fig.5), which represents the following scenario. Two hunters can choose to either hunt a stag together (which provides the most economically efficient outcome) or they can individually hunt a Rabbit. Hunting Stags is challenging and requires cooperation. If the two hunters do not cooperate the chances of success is minimal. Thus, the scenario where both hunters choose to coordinate will provide the most beneficial output for society. A common problem associated with the stag hunt is the amount of trust required to achieve this output. Fig. 5 shows a situation in which both players (hunters) can benefit if they cooperate (hunting a stag). As you can see, 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.

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. Assume that a couple argues over what to do on the weekend. Both know that they will increase their utility by spending the weekend together, however the man prefers to watch a football game and the woman prefers to go shopping.

Since the couple want to spend time together, they will derive no utility by doing an activity separately. If they go shopping, or to football game one person will derive some utility by being with the other person, but won’t derive utility from the activity itself. Unlike the other forms of coordination games described previously, knowing your opponent’s strategy won’t help you decide on your course of action. Due to this there is a possibility that an equilibrium will not be reached.

In social sciences, a voluntary standard (when characterized also as de facto standard) is a typical solution to a coordination problem. The choice of a voluntary standard tends to be stable in situations in which all parties can realize mutual gains, but only by making mutually consistent decisions.
In contrast, an obligation standard (enforced by law as "de jure standard") is a solution to the prisoner's problem.

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).