Chess tournament

Probability Level 5

You've just made it into the finals of a chess tournament. Your opponent is stronger than you and you are trying to think of a way to beat him. There will be 2 matches, and if it results in a tie, another match will be played to decide the winner. 1 point is awarded for winning, 0.5 for drawing, and 0 for losing. You will beat your opponent if you score more than him in these 2 matches, or if both players score the same, if you win the play-off, you will win the match. You can play a daring game, in which you win 45% and lose 55% of the time, or you can play a defensive game, in which you draw 90% and lose 10% of the time. If you play optimally, the probability that you beat your opponent is $a/b,$ where a and b are coprime integers. Find $a+b.$

The answer is 12293.

1 solution

ZhiJie Goh
Sep 17, 2014

Your optimal strategy is to play daringly if you are behind or tied, and defensively if you are ahead. Thus, you have a .55 probability of losing and a .45 probability of winning the first game. The probability that you lose both matches and lose the match is 0.55×0.55=0.3025. The probability that you lose the first game and win the second to proceed to the play-off is 0.55×0.45=0.2475.

However, you have a .45 chance of winning the first game. If that happens, you play defensively, thus the chance which you win the first match and tie the second to win the tournament is 0.45×0.9=0.405. The probability of you losing the second match but winning the first to proceed to play-off is 0.45×0.1=0.045.

The probability in which you enter the play-off is 0.045+0.2475=0.2925. As you are tied, you play daringly, thus the probability in which you lose the play-off and the match is 0.55×0.2925=0.160875. And the probability that you win the play-off and the match is 0.45×0.2925=0.131625.

Thus the overall probability of you winning the match is 0.405+0.131625=0.536625, which is 4293/8000. And 4293+8000 is 12293.

It's surprising that the answer is greater than $1/2$ here; the key seems to be that you have control over the type of game that is played, so that e.g. your opponent cannot play defense if he wins game 1, but you can if you do.

Patrick Corn - 6 years, 8 months ago

Can you define "beat your opponent" in the question?

Can you explain why that is the optimal strategy? It makes sense verbally, but you need to construct the probability trees to explain.

Calvin Lin Staff - 6 years, 8 months ago

The strategy here is to choose your playing style carefully. It doesn't make sense to play defense if you are behind or tied, as you have a 0% chance of winning, and a 10% chance of losing, so you will never gain an advantage over your opponent, and will eventually lose. Thus you should play daring on game 1 and the play-off, if any, and also game 2 if you lost game 1.

If you play daringly all the way, it will not be the optimal strategy as your opponent is still stronger than you, so you will only have a 45% of winning. Thus, the "optimal" strategy is to play daringly if you are behind or tied, and ahead if you are ahead.

ZhiJie Goh - 6 years, 8 months ago

Chess tournament

The answer is 12293.

1 solution

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