Mixed-Strategy Equilibrium
Fifty-fifty is right in matching pennies because the game is symmetric. Almost nothing else is, so the mix has to be computed.
The indifference condition
Suppose you are mixing between two actions in equilibrium. Then both actions must pay you exactly the same, because if one paid more you would abandon the other and stop mixing. A player only randomises when they have nothing to choose between.
That gives the condition that pins everything down: the opponent's mix is whatever makes you indifferent, and yours is whatever makes them indifferent.
The crossed structure is the part people get backwards. Your own mixing probability does nothing for your payoff. It exists solely to remove the opponent's profitable pure response. Solve for the mix by writing down the other player's indifference.
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