Indian grandmaster Vishwanathan Anand describes his preparations for an upcoming world championship match against Russia’s Vladimir Kramnik:
I have been studying Kramnik since the end of April, up to ten hours a day, here at home in my cellar, where I have my office. I have a database and construct game plans. I try to neutralise positions in which Kramnik is strong. He is doing the same thing with my game, which I must of course take into consideration. Let me put it this way: I must remember that he is thinking about what I am thinking about him.
Anand is focusing his preparations on positions where Kramnik is known to be strong based on previous performance. But knowing this, Kramnik could choose to play positions that he hasn’t much used in the past or ones where his previous performance wasn’t good; Anand might be unprepared for these openings and Kramnik could win by taking him by surprise. Of course, Anand could take account of that possibility and therefore spend more preparation time on positions where Kramnik is weak then he otherwise would have. Knowing this, Kramnik might….
By now you get the idea. There doesn’t seem to be any clear dominant strategy here. Whether Anand should focus his preparations on positions where Kramnik is strong depends on what Kramnik is going to do, and vice versa.
The optimal strategy, most likely, is what game theorists call a mixed strategy. Anand should construct an equation under which he will focus on openings where Kramnik is strong with X% probability and not do so with probability 1-X%. The value of X would be determined by the relative payoffs of the various possible scenarios that could arise depending on what Kramnik does with his preparations. Perhaps that’s what Anand really is doing, and this is what he means when he says that “I must remember that he is thinking about what I am thinking about him.” Obviously, calculating the right value of X might not be an easy task.
The whole situation reminds me of this famous dialogue from The Princess Bride:
Man in Black: All right. Where is the poison? The battle of wits has begun. It ends when you decide and we both drink, and find out who is right… and who is dead.
Vizzini: But it’s so simple. All I have to do is divine from what I know of you: are you the sort of man who would put the poison into his own goblet or his enemy’s? Now, a clever man would put the poison into his own goblet, because he would know that only a great fool would reach for what he was given. I am not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool, you would have counted on it, so I can clearly not choose the wine in front of me.
Man in Black: You’ve made your decision then?
Vizzini: Not remotely. Because iocane comes from Australia, as everyone knows, and Australia is entirely peopled with criminals, and criminals are used to having people not trust them, as you are not trusted by me, so I can clearly not choose the wine in front of you.
Man in Black: Truly, you have a dizzying intellect.
Vizzini: Wait till I get going! Now, where was I?
Man in Black: Australia.
Vizzini: Yes, Australia. And you must have suspected I would have known the powder’s origin, so I can clearly not choose the wine in front of me.
Man in Black: You’re just stalling now.
Vizzini: You’d like to think that, wouldn’t you? You’ve beaten my giant, which means you’re exceptionally strong, so you could’ve put the poison in your own goblet, trusting on your strength to save you, so I can clearly not choose the wine in front of you. But, you’ve also bested my Spaniard, which means you must have studied, and in studying you must have learned that man is mortal, so you would have put the poison as far from yourself as possible, so I can clearly not choose the wine in front of me.
Man in Black: You’re trying to trick me into giving away something. It won’t work.
Vizzini: IT HAS WORKED! YOU’VE GIVEN EVERYTHING AWAY! I KNOW WHERE THE POISON IS!
Man in Black: Then make your choice.