35. Players Odd and Even play Low Person Wins, the rules of which are as follows:
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What is the Nash equilibrium of this game, based on the successive elimination of dominated strategies? Is there another Nash equilibrium? What are the similarities and differences between this game and the Prisoners’ Dilemma?
35.
The associated matrix is
| Even Player | ||||
| 2 | 4 | 6 | ||
| 1 | (2, 1) | (2, 1) | (2, 1) | |
| Odd Player | 3 | (2, 4) | (6, 3) | (6, 3) |
| 5 | (2, 4) | (4, 8) | (10, 5) | |
Even always does better playing 4 instead of 6. So, 6 is weakly dominated by 4. Once 6 is eliminated, Odd always prefers 3 to 1 and 3 to 5. So 1 and 5 are eliminated. After these eliminations, Even prefers playing 2 to 4. This yields the outcome of Odd playing 3 and Even player 2 as an equilibrium with payoffs of (2, 4). Another equilibrium has Odd playing 1 and Even playing 2. This game is similar to the Prisoners’ Dilemma because both players can do better by playing nonequilibrium strategies (e.g., when Odd plays 5 and Even plays 6). It is different from the Prisoners’ Dilemma because, although an outcome with the lowest payoff is an equilibrium, there is another equilibrium with a better payoff for Even.