Focal point (game theory)
Focal point (game theory)
In game-theory, a focal point (or Schelling point) is a solution that people tend to choose by default in the absence of communication in order to avoid coordination failure. The concept was introduced by the American economist thomas-schelling in his book The Strategy of Conflict (1960). Schelling states that "[p]eople can often concert their intentions or expectations with others if each knows that the other is trying to do the same" in a cooperative situation (p. 57), so their action would converge on a focal point which has some kind of prominence compared with the environment. However, the conspicuousness of the focal point depends on time, place and people themselves. It may not be a definite solution.
Existence The existence of the focal point is first demonstrated by Schelling with a series of questions. Here is one example: to determine the time and place to meet a stranger in New York City, but without being able to communicate in person beforehand. In this coordination-game, any place and time in the city could be an equilibrium solution. Schelling asked a group of students this question and found that the most common answer was "noon at (the information booth at) Grand Central Terminal". There is nothing that makes Grand Central Terminal a location with a higher payoff because people could just as easily meet at another public location, such as a bar or a library, but its tradition as a meeting place raises its salience and therefore makes it a natural "focal point".
The existence of focal points can help explain the use of social norms, including traditional gender roles, in order to ensure coordination, and why changing said norms can be difficult. Because of the limit of players' expectation level and players' priors, it is possible to reach an equilibrium in games without communication.
The cognitive hierarchy theory The cognitive hierarchy (CH) theory is a derivation of level-n theory. A level-n player from the CH model would assume that their strategy is the most sophisticated and that the levels 0, 1, 2, ..., n − 1 on which their opponents play follow a normalized Poisson distribution. This model works well in multi-player games where the players need to estimate a number in a given range, such as the guess-2/3-of-the-average game. A player would be able to determine the value which they should play based on the assumed distribution of lower-level players described by the Poisson distribution. With the identity changed, the player follows the prescription of an imaginary group leader to maximize the group interest.
Examples ### Schelling's questions Here is a subset of the questions raised by Schelling to prove the existence of a focal point. In this case, the decision to swerve right can serve as a focal point which leads to the winning right–right outcome. It seems a natural focal point in places using right-hand traffic.
This idea of anti-coordination game is also apparent in the game of chicken, which involves two cars racing toward each other on a collision course and in which the driver who first decides to swerve is seen as a coward, while no driver swerving results in a fatal collision for both.
“Guess 2/3 of the average” game The guess-2/3-of-the-average game shows the level-n theory in practice. In this game, players are tasked with guessing an integer from 0 to 100 inclusive which they believe is closest to 2/3 of the average of all players’ guesses. A Nash equilibrium can be found by thinking through each level: * Level 0: The average can be in [0, 100]. * Level 1: The average can be in [0, 67], which is 2/3 of the maximum average of level 0. * Level 2: The average can be in [0, 45], which is 2/3 of the maximum average of level 1. * Level N: Assuming all other players reason similarly, 2/3 of the maximum average will never be higher than <math alt="two thirds to the power of N">100 \cdot (2/3)^N.</math> As N grows, 2/3 of the average will trend towards zero. At this point, the only Nash equilibrium is for all players to guess 0.
Adding repetition to the game introduces a focal point at the Nash equilibrium solution of 0. This was shown by Camerer as, “[when] the game is played multiple times with the same group, the average moves close to 0”. Introducing the iterative aspect to the game forces all players onto higher levels of thinking, which allows them all to play guesses trending towards 0.