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The Science of Strategic Thinking: Strategize for Zero-Sum Situations, with Kevin Zollman, Game Theorist and Co-author, The Game Theorist’s Guide to Parenting
Mixed Strategy
When one is confronted with a situation that’s truly zero-sum, where one party’s going to win and the other party’s going to lose, the situation is very complicated and sometimes difficult to analyze. Game theory spent much of its early days analyzing zero-sum games and trying to figure out what’s the best strategy. It’s a little complicated because it depends critically on how sophisticated you think the other party is. If they’re very, very, very smart, the chances that you’re going to outthink them are not very high. In such a situation, oftentimes the best strategy is very counterintuitive because it involves flipping a coin or rolling a dice or doing something random. Professional poker players know this, and they oftentimes advocate in poker strategy books that one should occasionally do something completely counterintuitive in order to keep your opponents off guard. And in fact, game theory has shown that this is good, solid, mathematically well-founded advice. That oftentimes what you want to do is engage in a kind of random strategy. Game theorists call this a mixed strategy. The nice thing about these random strategies is that they ensure that your opponent can never outthink you. So even if you think your opponent’s a little smarter than you, or a little bit more sophisticated than you, or has a little more information than you do, the fact that you’re being random to a certain extent means that they can’t outthink you.
Minimax Strategy
Now, how do you figure out how to be random? I’m not saying flip a coin all the time, or whatever. What game theorists have figured out is that in zero-sum games, the best strategy to pursue when you’re against a sophisticated opponent is to adopt the strategy which minimizes your maximum loss. This is sometimes called the minimax strategy. So the idea is you think, what’s the worse-case scenario for me? What could my opponent do that would make me worse off? And then you figure out, what’s the best strategy against that? So you’re minimizing your maximum loss. Game theorists prove that if you use this way of thinking, minimizing your maximum loss, you ensure that no matter how sophisticated your opponent is you’ve guarded against the worst-case scenario, and not only that, but in zero-sum games you’ve done the best you could possibly do. That’s not true in games that aren’t zero-sum, so one has to be very careful about employing this strategy because if you’re mistaken, and you’re not in a zero-sum interaction, you can end up ruining it for everybody. But if you’re truly in a zero-sum interaction, this is one of the strategies you can use.
Suppose you’re dealing with an opponent who’s not sophisticated – you’re smarter than they are. There it depends very much on how smart are they, can you outthink them, and what’s the individual interaction that you’re engaged in. Game theorists have actually proven, although it’s not very helpful, that there’s no one-size fits all strategy in a situation where you’re dealing with an opponent who is not very sophisticated.