llA NormalForm Representation of Games
In the normal-form representation of a game, each player simultaneously chooses a strategy, and the combination of strategies chosen by the players determines a payoff for each player. We illustrate the normal-form representation with a classic example — The Prisoners' Dilemma. Two suspects are arrested and charged with a crime. The police lack sufficient evidence to convict the suspects, unless at least one confesses. The police hold the suspects in
separate cells and explain the consequences that will follow from the actions they could take. If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. If both confess then both will be sentenced to jail for six months. Finally, if one confesses but the other does not, then the confessor will be released immediately but the other will be sentenced to nine months in jail—six for the crime and a further three for obstructing justice.
The prisoners' problem can be represented in the accompanying bi-matrix. (Like a matrix, a bi-matrix can have an arbitrary number or rows and columns; "bi" refers to the fact that, in a two-player game, there are two numbers in each cell—the payoffs to the two players.)
Prisoner 1
Fink
The Prisoners' Dilemma
In this game, each player has two strategies available: confess (or fink) and not confess (or be mum). The payoffs to the two players when a particular pair of strategies is chosen are given in the appropriate cell of the bi-matrix. By convention, the payoff to the so-called row player (here, Prisoner 1) is the first payoff given, followed by the payoff to the column player (here, Prisoner 2). Thus, if Prisoner 1 chooses Mum and Prisoner 2 chooses Fink, for example, then Prisoner 1 receives the payoff -9 (representing nine months in jail) and Prisoner 2 receives the payoff 0 (representing immediate release).
We now turn to the general case. The normal-form representation of a game specifies: (1) the players in the game, (2) the strategies available to each player, and (3) the payoff received by each player for each combination of strategies that could be chosen by the players. We will often discuss an «-player game in which the players are numbered from 1 to n and an arbitrary player is called player i. Let Sj denote the set of strategies available to player i (called i's strategy space), and let s, denote an arbitrary member of this set. (We will occasionally write s, <5 S, to indicate that the
Prisoner 2 Mum Fink
Prisoner 1
Fink
|
-1,-1 |
-9, 0 |
|
0,-9 |
-6,-6 |
strategy s,- is a member of the set of strategies S,.) Let (sj,... ,s„) denote a combination of strategies, one for each player, and let Uj denote player i's payoff function: w,(s i,...,s„) is the payoff to player i if the players choose the strategies (s^,... ,s„). Collecting all of this information together, we have:
Definition The normal-form representation of an n-player game specifies the players' strategy spaces Si..... S„ and their payoff functions hj, ..., u„. We denote this game by G = {Si,..., S„;uj,..., u„}.
Although we stated that in a normal-form game the players choose their strategies simultaneously, this does not imply that the parties necessarily act simultaneously: it suffices that each choose his or her action without knowledge of the others' choices, as would be the case here if the prisoners reached decisions at arbitrary times while in their separate cells. Furthermore, although in this chapter we use normal-form games to represent only static games in which the players all move without knowing the other players' choices, we will see in Chapter 2 that normal-form representations can be given for sequential-move games, but also that an alternative—the extensive-form representation of the game—is often a more convenient framework for analyzing dynamic issues.
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