982. The Third Dice

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

982. 第三颗骰子

爱丽丝和鲍勃正在用两颗六面体骰子（六个面标有 $1$$1$ 至 $6$$6$ 的数字）进行下述博弈：

1. 爱丽丝掷出两颗骰子；她能看到掷出的点数，但鲍勃看不到。

2. 爱丽丝从中选择一颗骰子，并向鲍勃公开展示其点数。

3. 鲍勃从两颗骰子中选择一颗（已公开或未公开的骰子均可选）。

4. 爱丽丝向鲍勃支付的金额等于鲍勃所选骰子的点数。

两人需各自制订一个（可以是非确定性的）策略。例如，两人可以采取如下策略：

• 爱丽丝将点数最接近 $3.5$$3.5$ 的骰子公开给鲍勃，若两颗骰子的点数与 $3.5$$3.5$ 的差相同，则等概率选择一个公开。

• 若已公开的骰子的点数至少是 $4$$4$，则鲍勃选择已公开的那颗骰子；否则鲍勃选择未公开的那颗骰子。

事实上，这两个策略共同构成了一个 纳什均衡。也就是说，若已知鲍勃采用对应策略，爱丽丝的策略能够 最小化 支付金额的期望；若已知爱丽丝采用对应策略，鲍勃的策略能够 最大化 支付金额的期望。

若双方采取如上策略，则爱丽丝付给鲍勃的金额的期望为 $\frac{145}{36}\approx 4.027778$$\frac{145}{36}\approx 4.027778$。

为了使博弈更加有趣，两人向这个游戏中加入了第三颗（六面体）骰子，博弈规则改为：

1. 爱丽丝掷出 三颗 骰子；她能看到掷出的点数，但鲍勃看不到。

2. 爱丽丝从中选择 两颗 骰子，并向鲍勃公开展示其点数。

3. 鲍勃从三颗骰子中选择一颗（已公开或未公开的骰子均可选）。

4. 爱丽丝向鲍勃支付的金额等于鲍勃所选骰子的点数。

假设二人最终采取的策略构成纳什均衡，求爱丽丝付给鲍勃的金额的期望。将其四舍五入至小数点后第六位作为你的答案。

点 这个链接 回到源站。

点 这个链接 回到详细版题目目录。