### Game Theory & Nash Equilibrium

Game theory deals with mathematical models of situations involving conflict, cooperation and competition. Such situations are central in the social and behavioural sciences. Game Theory is a framework for making rational decisions in many fields: economics, political science, psychology, computer science and biology. It is also used in industry, for decisions on manufacturing, distribution, consumption, pricing, salaries, etc.

Theory of Games and Economic Behavior.
Centre: John von Neumann. Right: Oskar Morgenstern.

During the Cold War, Game Theory was the basis for many decisions concerning nuclear strategy that affected the well-being of the entire human race.

Origins

The term “game” suggests a limitation to recreational activities so, in the social and political sciences, alternative terms like Decision Theory or Rational Choice Theory are used. Though important in social and political contexts, Game Theory was originally developed with economic applications in mind, and it has had a major influence in this field. Indeed, more than ten Nobel prizes in economics have been awarded to game theorists.

Game theory gained prominence following the publication in 1944 of the book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. In 1928 John von Neumann had proved his famous “minimax theorem” which showed that every two-person zero-sum game has an equilibrium point (described below). The proof of this result involved Brouwer’s fixed point theorem.

An example of a zero-sum game is poker, where the total sum of money is fixed, so one player’s win is another’s loss. However, most real-life situations are more complex, and are better modelled by non-zero-sum games. However, an n-player non-zero-sum game can be extended to an (n+1)-player zero-sum game. For example, in a national lottery, the agency running the lottery is the additional player (and has a winning strategy!).

Equilibrium

An equilibrium point corresponds to a collection of choices or strategies, one for each player, such that no single player can improve his position by changing strategy. Thus, at equilibrium, each player’s choice is the best response to the strategies of the other players. At equilibrium, no player has a reason for regret: given the strategies of the other players, he cannot do better by changing.

A game involves a number of players, a collection of choices or strategies, and a set of pay-offs for each choice. Simple two-player games may be presented in tabular form. Suppose two ticket agents, Acme and Bril, are selling tickets, either at full price €100 or reduced price €75. The total number of tickets is limited, so the more that Acme sells, the fewer available to Bril and vice versa. The table below shows the profits of the two agencies for possible combinations of choice.

The numbers in each box are the profits of Acme and Bril (they are chosen more-or-less arbitrarily).

•  Suppose Acme sells at full price. Then Bril should do likewise to maximise profits.
• If Acme sells at reduced price, Bril should also do likewise.
• Now suppose Bril sells at full price. Then Acme must do the same.
• Finally, if Bril offers a reduction, Acme should still ask for full price.

But now Bril will see that the decision to offer a discount was a bad one: given Acme‘s choice, Bril can increase profits by asking for full rate. The solution in the bottom right cell is unstable: a player knowing the strategy of his opponent will wish to change.

The solution at top left is a stable equilibrium: neither player has a reason to change his strategy, given the strategy of the other.

John Nash

John Nash’s doctoral studies were on non-cooperative games. His dissertation, published in 1950 was just 28-pages long. The thesis contained the definition and properties of what came to be called the ‘Nash equilibrium’. It’s a central concept in non-cooperative games. In 1994, Nash was awarded the Nobel prize in economics for this work.

Nash greatly extended the results of von Neumann and Morgenstern. They had considered cooperative games, in which players can communicate with each other and make binding coalitions. Nash considered non-cooperative games, in which coalitions are prohibited. His work was concerned with non-zero-sum games with more than three players.

Nash showed that every game with a finite number of players has an equilibrium point. However, Nash’s solution can sometimes appear irrational. This is illustrated by the paradoxical result known as The Prisoners’ Dilemma. But that is another story.

Sources:

Nash, J. F. Jnr., 1950: Equilibrium points in n-person games. Proc. Nat. Acad. Sci., USA, 36, 48–49.

Nash, John Forbes (1951). “Non-cooperative Games” (PDF). Annals of Mathematics 54 (2): 286–95.

Von Neumann, John; Morgenstern, Oskar (1944): Theory of games and economic behavior, Princeton University Press.