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Problem 982

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Problem 982

The Third Dice

Alice and Bob play the following game with two six-sided dice (numbered $1$ to $6$):

  1. Alice rolls both dice; she can see the rolled values but Bob cannot
  2. Alice chooses one of the dice and reveals it to Bob
  3. Bob chooses one of the dice: either the one he can see, or the one he cannot
  4. Alice pays Bob the value shown on Bob’s chosen dice

Each player devises a (possibly non-deterministic) strategy. An example strategy for each player could be:

  • Alice chooses to reveal the dice with value closest to $3.5$, or if both are equidistant she chooses randomly with equal probability
  • Bob chooses the revealed dice if its value is at least $4$; otherwise he chooses the hidden dice

In fact, these two strategies together form a Nash equilibrium. That is, given that Bob is using his strategy, Alice’s strategy minimises the expected payment; and given that Alice is using her strategy, Bob’s strategy maximises the expected payment.

With these strategies the expected payment from Alice to Bob is $\frac{145}{36}\approx 4.027778$.

To make the game more interesting, they introduce a third (six-sided) dice:

  1. Alice rolls three dice; she can see the rolled values but Bob cannot
  2. Alice chooses two of the dice and reveals both to Bob
  3. Bob chooses one of the three dice: either one of the two visible dice, or the one hidden dice
  4. Alice pays Bob the value shown on Bob’s chosen dice

Supposing they settle on a pair of strategies that form a Nash equilibrium, find the expected payment from Alice to Bob, and give your answer rounded to six digits after the decimal point.

第三枚骰子

爱丽丝和鲍勃正在用两枚六面骰子（刻有点数$1$到$6$）玩如下的游戏：

  1. 爱丽丝掷两枚骰子；她能看到所有掷出的点数，但鲍勃看不到
  2. 爱丽丝选择其中一枚骰子并展示给鲍勃
  3. 鲍勃选择一枚骰子：可以选他看到的那枚，也可以选他没看到的那枚
  4. 爱丽丝向鲍勃支付所选骰子所掷出的点数

每位玩家各自制定一个（可以是非确定性的）策略；如下是两位玩家各自可能选择的一种策略示例：

  • 爱丽丝选择展示点数最接近$3.5$的骰子，如果两枚骰子的点数与$3.5$的距离相同，则以相等概率随机选择
  • 如果展示的骰子点数至少是$4$，鲍勃选择展示的骰子；否则选择隐藏的骰子

事实上，这两种策略组合起来构成了一个纳什均衡：也就是说，若鲍勃采用上述策略，则爱丽丝的上述策略能最小化支付点数的期望；而若爱丽丝使用上述策略，则鲍勃的上述策略能最大化支付点数的期望。

双方均采用上述策略时，爱丽丝向鲍勃支付点数的期望为$\frac{145}{36}\approx 4.027778$。

为了使游戏更有趣，他们引入了第三枚（六面）骰子：

  1. 爱丽丝掷三枚骰子；她能看到所有掷出的点数，但鲍勃看不到
  2. 爱丽丝选择其中两枚骰子展示给鲍勃
  3. 鲍勃选择三枚骰子中的一枚：可以选两枚可见骰子中的任一枚，也可以选隐藏的那枚
  4. 爱丽丝向鲍勃支付所选骰子所掷出的点数

假设他们确定了一对构成纳什均衡的策略，求爱丽丝向鲍勃支付点数的期望，并四舍五入保留六位小数作为你的答案。