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弹珠游戏中选择 A 还是 B

A or B

专题
Strategy / 策略
难度
L4

题目详情

玩家 AA 和 BB 玩一个弹珠游戏。每个玩家各有一颗红弹珠和一颗蓝弹珠，并同时向对方展示一颗。如果双方都出红色，AA 赢 $3；如果双方都出蓝色，AA 赢 $1；如果颜色不同，BB 赢 $2。奖金来自外部来源，并不是由另一名玩家支付。

选择成为 AA 更好，还是成为 BB 更好，还是无所谓？

英文原题

Two players AA and BB play a marble game. Each player has both a red and a blue marble. They present one marble to each other. If both present red, AA wins $3. If both present blue, AA wins $1. If the colors do not match, BB wins $2. The winnings come from an external source, not from the other player.

Is it better to be AA, or BB, or does it matter?

解析

解析需会员查看。

英文解析

Assuming that the players have fifty- fifty probabilities of playing Red or Blue, <sup>8</sup> each player has the same expected payoff: 11. Player BB has a variance of payoffs given by

[(0−1)2×12]+[(2−1)2×12]=1,\left[(0 - 1)^2 \times \frac{1}{2}\right] + \left[(2 - 1)^2 \times \frac{1}{2}\right] = 1,

whereas player AA has a variance of payoffs given by

[(1−1)2×14]+[(3−1)2×14]+[(0−1)2×12]=1.5.\left[(1 - 1)^2 \times \frac{1}{4}\right] + \left[(3 - 1)^2 \times \frac{1}{4}\right] + \left[(0 - 1)^2 \times \frac{1}{2}\right] = 1.5.

Thus, if you are risk averse, player BB 's position is favored (it offers the same expected return, but less risk).