Why Pure Strategies Fail
Every game so far has had at least one cell where neither player wanted to move. Not all games do.
A game with nowhere to settle
Matching pennies. You each show heads or tails. If they match, you win a unit; if they differ, they do.
| Your payoff | They show heads | They show tails |
|---|---|---|
| You show heads | +1 | −1 |
| You show tails | −1 | +1 |
Try to find a stable cell. If you both show heads, they want to switch to tails. Once they do, you want to switch to tails as well. Now they want heads. The best responses chase each other around the matrix forever and never land.
There is no equilibrium in fixed choices here. That is not a defect in the game or a gap in the analysis: it is a property of any situation where one side wants to be predicted and the other wants to avoid it.
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