Game theory -  A Tutorial

Jean Walrand

Dynamic Games: Example 2

Player P1 plays, then P2 (who does not see the first move of P1), then P1 (who sees the move of P2 and remembers his first move). 

The right-part of the figure shows the rewards that the players get for each move of P2. For instance, if P1 starts with L, then if P2 chooses L, we know that P1 will choose R and that the rewards will be (2, 0). We can then simplify the game by replacing the bottom part of the tree by the actual choices of P1.  This results in a static game shown in the right-part of the figure. This static game has only one NE: (L, R). It follows that the only SPE is L, R, L with rewards (1, 2).  More precisely, the strategies describe all the possible eventualities:

P1: L
P2: R
P1: R if P2 is L, L otherwise.