Game Theory with Applications to Finance and Marketing

31. Because the extensive form specifies two more things than the normalform, there does not exist a one-to-one correspondence between thetwo. In fact, a normal form can correspond to more than one extensiveform. Consider the following extensive game. Player 1 first choosesamong pure strategies A, B, and C. If A is chosen, then player 1 gets 2and player 2 gets 1. Otherwise, upon seeing player 1’s choice of action,player 2 can react by choosing either L or R. The payoffs resulting fromthese strategies are summarized in the following bimatrix.player 1/player 2 L RB 4,2 1,03C 0,1 2Draw the extensive form for this game.32. So, how do we construct the strategic form from its extensive counterpart?The complication is that we need to define pure and mixedstrategies for the normal form game, while all we know is what eachplayer can do at each of his information sets in the extensive game.First, we define a pure strategy for player i as a complete descriptionof which action player i will take (with probability one) at each of hisinformation sets. Second, if two pure strategies of player i always (“always”means when put together with any profile σ −i ∈ Σ −i ) generatethe same payoff for player i, then they are said to be equivalent, and areduced normal form game is obtained from the original extensive gameif for each player, equivalent pure strategies are identified (remove allof them but one). 8 A mixed strategy for player i is then defined as aprobability distribution over all feasible pure strategies in the reducednormal form game. To fix terminology, mixed strategies in the originalextensive game are referred to as behavior strategies, and each behaviorstrategy of player i specifies a set of probability distributions for playeri, where each probability distribution corresponds to one information8 For example, consider an extensive game where player 1 first chooses between A and B,and if A is chosen then the game ends; or else, following B, players 1 and 2 simultaneouslychoose between a and b. Player 1 has four pure strategies, and two of them, (A,a) and(A,b), are equivalent strategies; for another example, see the game Battle of Sex below.18