Question

Consider a chess tournament in which you play one game with each of 3 opponents, but you get to choose the order in which you play your opponents, knowing the probability of a win against each. You win the tournament if you win two games in a row, and you want to maximize the probability of winning. Assume that it is optimal to play the weakest opponent second, and that the order of playing the other two opponents doesn't matter. Suppose in the tournament, x < y < z (x is weaker than y is weaker than z), whereby your chance of winning against x is higher than that of winning against y which is in turn higher than that of winning against z, no matter when you play them.

Now consider scenarios A and B:

Scenario A 1st game: probabilities of winning against x, y, and z are 1, 0.4, and 0.3 respectively
2nd game: probabilities of winning against x, y, and z are 0.8, 0.6, and 0.3 respectively (independent of what happened in the first game or who your opponent was in the first game).
3rd (last) game: probabilities of winning against x, y, and z are 0.8, 0.6, and 0.3 respectively (independently of what happened or who your opponents were in the other games.

Q1. What is the order in which you would play your opponents to maximize your chances of winning the prize?
Q2. For the order of opponents you chose, what is the probability you will win the prize?

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Scenario B Unknown to you, your chess skills really depend on whether you play black or white. In this tournament, after you choose the order of your opponents, the referee tosses a coin.

If heads, you will play white in all games,

if tails you play black in all games.

If you play white, your chances of winning against a, b, and c in any game are 0.8, 0.5, and 0.3 respectively independent of all other games.

If you play black, your chances against a, b, and c in any game are 0.6, 0.4, and 0.3 respectively independent of all other gaes.

Q1. What order would you pick your opponents? (This choice must be made before the coin toss)

Q2. What is the probability you will win the tournament (before you know anything about the coin toss)?

Detailed explanations very much appreciated! :)

Answer 1