## Introduction

As the name 2×2 games suggests, these games describe various types of interactions between two individuals with two strategical options to choose from. In order to illustrate the characteristics of such games, we consider the most prominent representative, the prisoner's dilemma (PD) (Axelrod, 1984). The PD provides a simple yet powerful mathematical framework to study the emergence ofcooperative and altruistic behavior among unrelated selfish individuals.

In the PD, two players have to simultaneously decide whether to cooperate (C) or defect (D). Their joint decisions then determine the payoffs for each player. Mutual cooperation pays a reward R while mutual defection results in a punishment P. If one player opts for D and the other for C, then the former obtains the temptation to defect T and the latter is left with the sucker's payoff S. From the rank ordering of the four payoff values T>R>P>S follows that a player is better off by defecting, regardless of the opponents decision. Consequentially, rational players always end up with the punishment P instead of the higher reward for cooperation R - hence the dilemma.

### 2×2 Games

In the general formulation, a 2×2 game is determined by the payoff matrix (for the column player):

 Row player Column player C D C R S D T P

where the rank ordering of the four payoff values R, S, T, P determines the characteristics of the game. Without loss of generality we may assume R>P (if this does not hold, we simply interchange C and D) and normalize the payoff values such that R=1, P=0 holds. This leads to 12 different rank orderings corresponding to very different strategic situations \cite[see e.g.][]{rapoport:76,binmore:92,colman:95}. Each game corresponds to a region in the S, T-plane:

Two of these games attracted the vast majority of both theoretical as well as experimental investigations. First, this is the prisoner's dilemma to study the evolution of cooperative behavior, as mentioned above. Second, the hawk-dove game models intra-species competitions.

### Hawks and Doves

 Updated on Tuesday, January 5, 2002 by Christoph Hauert.