Focal point (game theory)

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. == Theories ==

Although the concept of a focal point has been widely accepted in game theory, it is still unclear how a focal point forms. The researchers have proposed theories from two aspects. The level-n theory Stahl and Wilson argue that a focal point is formed because players would try to predict how other players act. They model the level of "rational expectation" players have by their ability to • form priors (models) about the behavior of other players; • choose the best responses given these priors. A level-0 player will choose actions regardless of the actions of other players. A level-1 player believes that all other players are level-0 types. A level-n player estimates that all other players are level-0, 1, 2, ..., n − 1 types. Based on experimental data, most of the players only use one model to predict the behavior of all the other players. Although the hierarchy of types could be indefinite, the benefits of higher levels would decrease substantially while incurring a much greater cost. 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 100 \cdot (2/3)^N. 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. == See also ==