# Gaming strategy for "closest number" game [closed]

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:

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

2. 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!

• I think you need to show some work yourself. I gave a hint below. Commented Apr 18, 2023 at 11:05
• I am voting to close this question as it is not among the topics to be discussed in QF. Commented Apr 18, 2023 at 12:25
• A cannot go into the interval because she would be squeezed between B and C, so A takes 0 (or 1), then B takes 2/3 (or 1- 2/3). This way, B guarantees her at least 1/3 chance of winning. C goes either immediately right or left of B giving her 1/3 chance. Equal chances for all and A has avoided a move were she would get 0 chance! Commented Apr 18, 2023 at 12:26
• Thanks for the reply @MatsLind, but why B takes 2/3 or 1-2/3? Commented Apr 18, 2023 at 12:51
• A takes 0: if B takes a point right of 2/3 it gets less chance than 1/3 and C takes a point left thereof and gets more than 1/3 chance. If B takes point left of 2/3 C takes a point immediately right thereof and gets more than 1/3 chance and B gets less than 1/3 Commented Apr 18, 2023 at 13:35

• In the case that A chooses $a = 0.5$, B should choose $0.5 \pm \varepsilon$ to get a chance of winning arbitrarily close to 50%. A is an epsilon more likely to win with that strategy. Commented Apr 18, 2023 at 13:07