# Game theory

### From Lesswrongwiki

**Game theory** attempts to mathematically model interactions between individuals.[1] Individuals are seen as *rational agents* with a set of alternative actions, a set of preferable *outcomes* and a function that chooses the action which better fits its preferences. They interact in a way that an agent must anticipate other agent’s responses to his actions in order to choose the action which maximizes his preferences; this interaction is called a *game*. If the agent only has to consider his own actions, then decision theory best suits to model his behavior and the situation is no longer a game. Each agent in a game has to choose from a set of previously established algorithms called *strategies*, each strategy is a complete list of responses to every possible situation he might encounter during the game.[2]

The most famous example of a game is the Prisoner's Dilemma: “Suppose that the police have arrested two people whom they know have committed an armed robbery together. Unfortunately, they lack enough admissible evidence to get a jury to convict. They do, however, have enough evidence to send each prisoner away for two years for theft of the getaway car. The chief inspector now makes the following offer to each prisoner: If you will confess to the robbery, implicating your partner, and she does not also confess, then you'll go free and she'll get ten years. If you both confess, you'll each get 5 years. If neither of you confess, then you'll each get two years for the auto theft.”[3]
Let the utility function of both the prisoners, which ascribes a numerical *payoff* to each outcome, be:

- Go free: 4
- 2 years in prison: 3
- 5 years in prison: 2
- 10 years in prison: 0

If both prisoners confess, each gets a payoff of 2. If neither of them confesses, each gets a payoff of 3. If only one of them confesses but not the other, one gets a payoff of 4 and the other of 0. Since they are ideal rational agents, prisoner I evaluates his actions by its consequences given each possible action of prisoner II, and so does prisoner II. For prisoner I, if prisoner II confesses then prisoner I gets a payoff of 2 by confessing and a payoff of 0 by refusing. If prisoner II refuses, then prisoner I gets a payoff of 4 by confessing and a payoff of 3 by refusing. Therefore, prisoner I is better off confessing regardless of what prisoner II does. The same reasoning goes to prisoner II, he is always better off confessing. One could argue that if both prisoners refuse they would each get a payoff of 4 and be better off. However, one must analyze each possible consequence given each possible action of the other prisoner. Remember that if prisoner I refuses and prisoner II confesses, prisoner I gets a payoff of 0. Hence, of all possible strategies, in average, always confessing is the one with better outcomes and greater payoffs. This strategy *dominates* over the other possible strategies, and is a solution to the game. In this state, the game is said to be at *equilibrium* or *equilibria*.

Game theory is an extremely powerful and robust tool in analyzing much more complex situations, such as: mergers and acquisitions, political economy, voting systems, war bargaining and biological evolution. Eight game-theorists have won the Nobel Prize in Economic Sciences.

## Blog posts

- Fair Division of Black-Hole Negentropy: an Introduction to Cooperative Game Theory by Wei Dai
- A gentle video introduction to game theory by Konkvistador