The Meeting Game

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The Meeting Game

The meeting game is a popular example of a two-person game of strategic interaction, and it's a common introductory example in many game theory textbooks. The logic of the game is as follows:

  • The two players in the game are trying to meet up with each other but have lost their cell phones and can't remember where they had agreed to meet.
  • Each player decides independently whether he is going to go to the opera or the baseball game.
  • Because each of the two players has two possible options (strategies), there are four possible outcomes to the game.
  • If both players choose the same event, they meet up and each gets a positive outcome.  (The specific values of the outcomes doesn't matter and don't have to be the same either across events or across individuals.)
  • If one player chooses one event and the other chooses the other event, they fail to meet up and both get a payout of zero.  (Technically, the payout doesn't have to be zero, but it does have to be less than the payoffs if they managed to meet up at either event.)

In the game itself, rewards are represented by utility numbers. Positive numbers represent good outcomes, negative numbers represent bad outcomes, and one outcome is better than another if the number associated with it is greater. (Be careful, however, of how this works for negative numbers, since -5, for example, is greater than -20!)

In the table above, the first number in each box refers to the outcome for player 1 and the second number represents the outcome for player 2. These numbers represent just one of many sets of numbers that are consistent with the meeting game setup.

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Analyzing the Players' Options

Once a game is defined, the next step in analyzing the game is to assess the players' strategies and try to understand how the players are likely to behave. Economists make a few assumptions when they analyze games- first, they assume that both players are aware of the payoffs both for themselves and for the other player, and, second, they assume that both players are looking to rationally maximize their own payoff from the game.

One easy initial approach is to look for what are called dominant strategies- strategies that are best regardless of what strategy the other player chooses. In the example above, however, there are no dominant strategies for the players:

  • Opera is better for player 1 if player 2 chooses opera since 5 is better than 0.
  • Baseball is better for player 1 if player 2 chooses baseball since 10 is better than 0.
  • Opera is better for player 2 if player 1 chooses opera since 5 is better than 0.
  • Baseball is better for player 2 if player 1 chooses baseball since 10 is better than 0.

Given that what is best for one player depends on what the other player does, it's not surprising that the equilibrium outcome of the game can't be found by just looking at what strategy is dominant for both players. Therefore, it's important to be a bit more precise with our definition of an equilibrium outcome of a game.

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Nash Equilibrium

The concept of a Nash Equilibrium was codified by mathematician and game theorist John Nash. Simply put, a Nash Equilibrium is a set of best-response strategies. For a two-player game, a Nash equilibrium is an outcome where player 2's strategy is the best response to player 1's strategy and player 1's strategy is a best response to player 2's strategy.

Finding the Nash equilibrium via this principle can be illustrated on the table of outcomes. In this example, player 2's best responses to player one are circled in green. If player 1 chooses opera, player 2's best response is to choose opera, since 5 is better than 0. If player 1 chooses baseball, player 2's best response is to choose baseball, since 10 is better than 0. (Note that this reasoning is very similar to the reasoning used to identify dominant strategies.)

Player 1's best responses are circled in blue. If player 2 chooses opera, player 1's best response is to choose opera, since 5 is better than 0. If player 2 chooses baseball, player 1's best response is to choose baseball, since 10 is better than 0.

The Nash equilibrium is the outcome where there is both a green circle and a blue circle, since this represents a set of best response strategies for both players. In general, it is possible to have multiple Nash equilibria or none at all (at least in pure strategies as described here).  As such, we see above a case where the game has multiple Nash equilibria.

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Efficiency of the Nash Equilibrium

You may have noticed that not all of the Nash equilibria in this example seem entirely optimal  (specifically, in that it is not Pareto optimal), since it is possible for both players to get 10 rather than 5 but both players get 5 by meeting at the opera.  It's important to keep in mind that a Nash equilibrium can be thought of as an outcome where no player has an incentive to unilaterally (i.e. by himself) deviate from the strategy that led to that outcome. In the example above, once the players both choose opera, neither player can do better by changing his mind by himself, even though they could do better if they switched collectively.