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先手价值

First Right

专题
Probability / 概率
难度
L2

题目详情

你和朋友玩游戏：你先抛一枚公平硬币。

  • 若你抛出正面（H），朋友付给你 30 美元，游戏结束。
  • 若你抛出反面（T），硬币交给朋友由他来抛。
  • 若朋友抛出正面，你付给他 30 美元，游戏结束。
  • 若朋友抛出反面，硬币再交回给你。

如此往复，直到第一次出现正面为止。

问：你愿意为“先手权”最多支付多少钱（美元）？

英文原题

You and a friend are playing a game. You start with a coin, if you flip heads your friend gives you $30, but if you flip tails your friends gets the coin. Then he flips, if he gets heads you give him $30 and if he gets tails you get the coin, and the process repeats. What is the maximum amount (in dollars) you would pay for the right to go first?

解析

设 VV 为轮到你抛时，从你视角的期望收益。

你抛：以 1/21/2 概率立刻赢 30；以 1/21/2 概率进入朋友回合。

设 WW 为轮到朋友抛时（从你视角）的期望收益，则

V=12⋅30+12W,W=12⋅(−30)+12V.V=\frac12\cdot 30+\frac12 W, \quad W=\frac12\cdot(-30)+\frac12 V.

解得 W=−15+12VW=-15+\frac12V，代入得

V=15+12(−15+12V)=7.5+14V⇒34V=7.5⇒V=10.V=15+\frac12\left(-15+\frac12V\right)=7.5+\frac14V \Rightarrow\frac34V=7.5\Rightarrow V=10.

因此你最多应支付 10\boxed{10} 美元来获得先手权。

英文解析

Set VV as the expected benefit from your perspective when it's your turn to toss.

You throw: immediately wins 30 with a chance of 1/21/2; enters a friend's turn with a chance of 1/21/2.

If WW is the expected return (from your perspective) when the friend's turn is tossed, then

V=12⋅30+12W,W=12⋅(−30)+12V.V=\frac12\cdot 30+\frac12 W, \quad W=\frac12\cdot(-30)+\frac12 V.

Understand W=−15+12VW=-15+\frac12V, substitute

V=15+12(−15+12V)=7.5+14V⇒34V=7.5⇒V=10.V=15+\frac12\left(-15+\frac12V\right)=7.5+\frac14V \Rightarrow\frac34V=7.5\Rightarrow V=10.

So you should pay up to$10\boxed{10} USD to get the first right.