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Game theory - A Tutorial |
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Dynamic Games: Example 1 |
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Two players play this game: P1 and P2 (see the left part of the figure). At the first step, P1 chooses L (left), M (middle), or R (right). At the second step, P2 gets to see whether the “state of the game” is L or not. Thus, P2 cannot distinguish whether P1 played M or R. Player P2 then plays L or R. The figure shows the rewards R1 of P1 and R2 of P2. The “information sets” show what P2 sees. |
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If P1 does not play L, then we can view P1 and P2 playing a static game with the reward matrix shown in the right-part of the figure. The only Nash equilibrium of this game is (R, R). It follows that if P1 does not play L, the two players play (R, R) and get the rewards (1, 0). If P1 plays L, then P2 should play L (to get the reward 1 instead of –1). We then end up with the two possibilities: (L, L) with rewards (0, 1) and (R, R) with rewards (1, 0). There is only one SPE: (R, R). Indeed, P1 knows that if it plays R it gets the reward 1 whereas if it plays L he gets the reward 0. This in an SPE because we considered all possible subgames after the first move. |
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The resemblance between the chess end game and the dynamic game on the right is purely fictitious. |