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Backward Induction for a Sequential Closest-Number Game

Article Quant Q&A · Author: Xu Shan
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Summary

The document poses a sequential game in which players choose distinct numbers from a bounded interval before a uniformly distributed target is revealed. Later players observe earlier choices, and the winner is the player whose choice lies closest to the target. The question asks how to reason about optimal choices, including a case where the first player selects an endpoint.

The sole response recommends solving the smaller two-player version first. Determine the second player’s best reply to any first choice, then use that response to evaluate which initial choice gives the first player the best chance. This is a backward-induction outline rather than a full solution: it does not calculate the optimal choices, winning probabilities, or the effect of adding the third player. The discussion is a probability and game-theory exercise, with no direct trading method or empirical evidence, so its relevance to quantitative trading is limited to strategic reasoning under sequential information.

Key ideas

  • Analyze the two-player version before tackling the game with an additional player.
  • Find the later player’s optimal response conditional on the earlier choice.
  • Use the later player’s best response to assess the earlier player’s optimal choice.
  • The response provides a method outline but does not solve the game or quantify winning chances.

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# Gaming strategy for "closest number" game


# Gaming strategy for "closest number" game












Suppose there are 3 people A, B, C and a referee. A, B, C individually takes one number from [0,1] with the order A->B->C. B could see the choice of A, C could see the choice of A and B. After that, the referee randomly take a number $Y$ from U(0,1). People who chooses the number which is most closest to $Y$ wins. But people who did a later choice cannot take the same number as the previous one chooses.

So:

- what's the strategy of A, B, and C to be the final winner, if it exists? if not, please state the reason?

- if A takes 0, what is the strategy of B? and is there a strategy to guarantee B is the winner? and the strategy of C?

For question 2), I think as long as B takes a positive number and makes it close to 0, B would be closer to the final number than A. But not sure how he could defeat C...is there anybody who can help me? Thanks!

## Answer by Bob Jansen (score 3, accepted)

https://quant.stackexchange.com/a/75255

First try to work out the case with 2 players, for that case figure out what Player B does given player A's choice. Given this you can determine the optimal choice for A, that is the choice that maximizes their chance of winning given they know B will choose optimally.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.