One game or best of three against a stronger player?

Question

You are the weaker player: you win any individual game against your opponent with probability 0.4, independently each time. You are offered a choice: play one game, or play a best-of-three match (first to two wins). 1. Which should you choose to maximise your chance of winning? 2. What would the stronger player choose, and why does the answer flip? 3. What about best of five, or best of a hundred?

Accepted Answer

Choose the single game. As the weaker player, you want as few games as possible. Best of three. You win the match if you win two games out of two, or two out of three with the loss in game one or two. The probability is p squared + 2 p squared (1 minus p) = p squared (3 minus 2p). With p = 0.4: 0.16 × 2.2 = 0.352, lower than the single-game 0.4. Why. A longer match averages out luck. The more games, the closer the outcome tracks who is actually better, and that is bad news for the underdog. In the limit of a very long match the stronger player wins almost surely: best of a hundred with p = 0.…

One game or best of three against a stronger player?

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