SEQUENTIAL GAMES with PERFECT INFORMATION Example
SEQUENTIAL GAMES WITH PERFECT INFORMATION Example 4.9 (page 105) Consider the sequential game given in Figure 4.9. We want to apply backward induction to the tree. 0 Vertex B is owned by player two, P2. The payoffs for P2 are 1 and 3, with 3 > 1, so the player picks R . Thus, the payoffs at B become (0, 3). 00 Next, vertex C is also owned by P2 with payoffs 1 and 0. Since 1 > 0, P2 picks L , and the payoffs are (4, 1). Player one, P1, owns A; the choice of L gives a payoff of 0 and R gives a payoff of 4; 4 > 0, so P1 chooses R. The final payoffs are (4, 1). 0 00 We claim that this strategy profile, { R } for P1 and { R ,L } is a Nash equilibrium. Notice that the 0 00 strategy profile gives a choice at each vertex. For the strategy { R ,L } fixed for P2, P1 has a maximal payoff by choosing { R }, ( 0 00 0 00 π1(R, { R ,L }) = 4 π1(R, { R ,L }) = 4 ≥ 0 00 π1(L, { R ,L }) = 0. 0 00 In the same way, for the strategy { R } fixed for P1, P2 has a maximal payoff by choosing { R ,L }, ( 00 0 00 π2(R, {∗,L }) = 1 π2(R, { R ,L }) = 1 ≥ 00 π2(R, {∗,R }) = 0, where ∗ means choose either L0 or R0. Since no change of choice by a player can increase that players own payoff, the strategy profile is called a Nash equilibrium. Notice that the above strategy profile is also a Nash equilibrium on each branch of the game tree, mainly starting at either B or starting at C.
[Show full text]