This chapter introduces the central ideas in game theory. It begins by showing how rational players can logically weed out strategies which are strategically inferior (sections 2.3 and 2.4). Such elimination of strategies relies on what game theory refers to as dominance reasoning and it sometimes requires the assumption of common knowledge of rationality (CKR). It is important because it yields clear predictions of what instrumentally rational players will do in some games by means of a step-by-step logic. However, in many games dominance reasoning offers no clear (or useful) predictions of what might happen. In these circumstances, game theorists commonly turn to the Nash equilibrium solution concept, named after its creator John Nash (section 2.5). The basic idea behind this concept is that rational players should not want to change their strategies if they knew what each of them had chosen to do.
This solution concept helps to refine the predictions of game theory. However, there is a cost in terms of generality. The step to Nash seems to require rather more than the assumptions of rationality and CKR. In section 1.2.2 of the previous chapter we described the essence of the extra requirement: the assumption that players’ beliefs will be consistently aligned (CAB). In some games even this move does not generate predictions adequately because there are some games in which no specific set of strategies is recommended by the Nash equilibrium. In the jargon, there are games in which there is either no Nash equilibrium in pure strategies, or there are many. 1 Thus predictions made using the Nash equilibrium concept can be either non-existent or indeterminate.
As a result game theorists have attempted to refine the Nash equilibrium concept. We present two such refinements: the Bayesian Nash equilibrium concept for games of incomplete information (section 2.6) and the idea of trembling hand perfect equilibria (section 2.7). They embody two of the central ideas which have been at play in the project of refining the Nash equilibrium to overcome the problems it encounters in many games.