Game Theory

Other Games

There are many types of games. Some have dominant strategies, and others do not. Some games may not have a Nash equilibrium.
The Prisoner's Dilemma, in which both players have a dominant strategy that leads to a worse outcome than if they had chosen the opposite strategy, is one type of game. For a different example, consider Quik Mart and Shop & Save, two convenience stores located down the street from one another in a small city. Both are considering adding an ice cream shop to their stores to draw in more customers. Adding the section may increase sales, but it will also lead to higher costs of running the store. The payoffs that each store would receive can be shown in a payoff matrix displaying their daily profits if they build or do not build the additions to their stores.
In this game, only Quik Mart has a dominant strategy-the best response for a player regardless of what the other player chooses, shown by the green check marks. Shop & Save does not have dominant strategy because its best option depends on what Quik Mart decides to do.
The best response analysis (in which a payoff maximizes a player's well-being, given the other player's choice) can be used to solve this game and look for a Nash equilibrium, the outcome in which no participant can gain by a change of strategy if the strategies of others remain unchanged.

If Quik Mart believes that Shop & Save is not going to build an addition, it is better off building an addition, because the payoff for building ($9,000) is greater than the payoff for not building ($7,000). If Quik Mart believes that Shop & Save is going to build the addition, it is still better off building an addition ($10,000 > $9,000). Here, building the addition is a dominant strategy for Quik Mart, because it is the best strategy regardless of what Shop & Save chooses to do.

If Shop & Save believes that Quik Mart will decide not to build the addition, Shop & Save would be better off building its own addition ($7,000 > $5,000). However, if Shop & Save believes that Quik Mart will build the addition, its best option is to not build, because its payoff for building the addition ($6,000) is less than the payoff for not building ($7,000). Shop & Save does not have a dominant strategy because its best option depends on what Quik Mart chooses.

There is a Nash equilibrium in this case. We know that Quik Mart will follow its dominant strategy and build the ice cream shop addition. Given that decision, Shop & Save will choose its best option and not build an addition to its store. This is a Nash equilibrium because each player is choosing its best option, given the choice made by the other. However, this game is different from the Prisoner's Dilemma because only one player has a dominant strategy.

Some games will not have a Nash equilibrium, while others may have more than one. Consider the following example. Rachel and Monica are roommates. Rachel loves spending time with Monica, but Monica would rather be alone. Each is deciding what she is going to do this evening, and the payoff matrix represents the level of happiness the roommates receive from going to dinner or a movie, conditional on what the other chooses.
In this game, what each player does depends on what the other chooses. There is no Nash equilibrium, in which players have the best outcome even if the other player changes their mind. Neither player has a dominant strategy, in which they have a best choice regardless of the other player's choice. There is no outcome that leaves both players happy.
If Rachel believes that Monica is going to dinner, she should also choose to go to dinner because her payoff is larger (8 > 6). If Rachel believes that Monica is going to a movie, she should also choose to go to a movie (3 > 2). Rachel does not have a dominant strategy because her best option changes when Monica changes her decision.

If Monica believes that Rachel will go to dinner, her best option is to go to a movie (5 > –1). If she thinks that Rachel will opt for a movie, Monica should choose to go out to dinner (4 > –8). Monica has no dominant strategy either. Her best choice depends on the option that Rachel chooses.

In this game, there is no Nash equilibrium. No matter what Monica and Rachel choose, one of them will always want to alter her choice. For example, if both choose dinner, Monica will wish she had opted to go to a movie. However, if Monica opted for a movie when Rachel opted for dinner, Rachel will wish she had chosen to go to dinner. There is no outcome in which both players are happy, so there is no Nash equilibrium.