Not for Sale

• #### Index 249-264

rules for strategic behavior that covers all eventualities. And in fact, there is—for two-person zero-sum games. You can find the best strategy using the minimax theorem that von Neumann published in 1928. His proof of that theorem was notoriously complicated. But its essence can be boiled down into something fairly easy to remember: When playing poker, sometimes you need to bluff.

### MASTERING MINIMAX

The secret behind the minimax approach in two-person zero-sum games is the need to remember that whatever one player wins, the other loses (the definition of zero sum). So your strategy should seek to maximize your winnings, which would have the effect of minimizing your opponent’s winnings. And of course your opponent wants to do the same.

Depending on the game, you may be able to play as well as possible and still not win anything, of course. The rules and stakes may be such that whoever plays first will always win, for instance, and if you go second, you’re screwed. Still, it is likely that some strategies will lose more than others, so you would attempt to minimize your opponent’s gains (and your losses). The question is, what strategy do you choose to do so? And should you stick with that strategy every time you play?

It turns out that in some games, you may indeed find one pure strategy that will maximize your winnings (or minimize your losses) no matter what the other player does. Obviously, then, you would play that strategy, and if the game is repeated, you would play the same strategy every time. But sometimes, depending on the rules of the game, your wisest choice will depend on what your opponent does, and you might not know what that choice will be. That’s where game theory gets interesting.

Let’s look at an easy example first. Say that Bob owes Alice \$10. Bob proposes a game whereby if he wins, his debt will be reduced. (In the real world, Alice will tell Bob to take a hike and fork over the \$10.) But for purposes of illustrating game theory, she might agree.

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