chapter  3
Solving a game
Pages 26

Being acquainted with the notions of game structure and information, we are now in a position to make a crucial step, which consists in identifying the solution concepts for noncooperative games. These are usually defined as equilibrium concepts, or simply equilibria. What do we mean here (i.e., in the social sciences) by equilibrium? Our concept of equilibrium indeed reflects the corresponding concept that dominates physics. Trivially, any given system (no matter whether it is physical, biological, economic or social) is in a state of equilibrium if, in the absence of perturbations, it remains in its current state. With this in mind, of course, the investigation of equilibria in economic and social systems requires the analysis of the behaviour of agents driven by their respective incentives. What do people, parties or countries want to attain when they behave in a specific manner? The answer to this question, as we know from the previous chapter, is given by the payoff function assigned to each of them. The remaining issue, then, is to figure out the criterion informing their behaviour. This criterion is individual rationality. In game theory (or better, in the social sciences across the board), an outcome is individually rational if it gives each agent at least as much as his/her security level, the latter being the ‘maximin’ payoff – the maximum of all minimum payoffs – that this player can guarantee to him/herself irrespective of what the other agents do. But this is usually not enough. That is to say, what game theory assumes is that each player will rationally choose a strategy in order to pursue the maximization of his/her payoff, being aware of the relevant solution concept for the game being played, and the structure of the game itself. This leads to another crucial assumption, that of common knowledge across agents. This establishes that each player must be fully aware of the rules of the game, the nature of information in the game, and the game structure (including thus also the objectives of all other players); but this is not enough, as each player must know that every other player knows the same things, as well as that each player knows that everybody else knows, and so on recursively. Should the common knowledge assumption fail to apply consistently, then the game would not be correctly specified and the resulting analysis would yield completely unreliable predictions.1