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请教各位高手一道博弈论的问题：

Jerry can hide in the bedroom, the den or the kitchen. Tom can search in one and only one of these locations. If he searches where Jerry is hiding, he catches Jerry. Otherwise Jerry escapes. If Jerry is caught, Tom's utility is 1 and Jerry's is 0. If Jerry gets arMay, Tom's utility is 0 and Jerry's is 1.

(a) Draw the game tree for the case when Tom can see where Jerry hides before searching. Explain why Tom has 27 strategies to choose from while Jerry has only 3. What are the subgame perfect equilibria?

(b) Draw the game tree for the case when Jerry can see where Tom searches before hiding. How many possible strategies does each player have? Write down 4 different pure-strategy subgame perfect equilibria. How many pure-strategy subgame perfect equilibria are there?

(c) Draw a game tree for the case where the players make their decisions in ignorance of the other's choice. Write down the strategic form. Does this game have a mixed-strategy equilibrium?

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关键词：博弈论 equilibrium Equilibria Strategies otherwise 博弈论

game tree omitted

indicating the Jerry's choice is b(hide in bedroom), d(hide in den), k(hide in kitchen)

(a)

Jerry has three choices: b, d, k.

Tom has 27 different strategies, because he has three decision nodes which have three choices respectively. The number of strategies of Tom's is 3*3*3=27.

There are three subgame perfect eq. in pure strategy:

Sj={b} St={b in b, d in d, k in k}

Sj={d} St={b in b, d in d, k in k}

Sj={k} St={b in b, d in d, k in k}

(b)

Tom has three strategies, while Jerry has 27 strategies.

St={b} Sj={d in b, b in d, b in k}

St={b} Sj={k in b, b in d, b in k}

St={d} Sj={d in b, b in d, b in k}

St={k} Sj={d in b, b in d, b in k}

There are totally 24 pure-strategy subgame perfect eq.

(c)

This is a game of imperfect information. Tom and Jerry both have three strategies, which are b, d and k.

It is apparent that there is no NE in pure strategies. And neither of them will randomize two of their strategies, because the other one will certainly deviate to a pure strategy immediately. There exists mixed-strategy eq, when both randomize all of their strategies with the same probabilities. It is indifferent for both of them between to choose a pure a strategies and to choose a mixed strategy.

谢谢大侠，太nice了。

麻烦大侠能再解释一下

There are three subgame perfect eq. in pure strategy:

Sj={b} St={b in b, d in d, k in k}

Sj={d} St={b in b, d in d, k in k}

Sj={k} St={b in b, d in d, k in k}

的含义吗？谢谢

对于一个博弈树，我们要看清每个参与人的信息集（information state）是什么，每个信息集上的行动（move）是什么，每个信息集有几个节点（node）。

如果所有参与人的所有信息集有且只有一个节点，则博弈是完全信息博弈（perfect information）。

“纯策略”是“信息集”到“行动”的映射，某参与人的一条纯策略描述了该参与人在其各信息集上将采取的行动。策略与行动是不同的概念。

在（a）中，显然博弈是完全信息的，即每个参与人的每个信息集有且只有一个节点。不过从“Tom can see where Jerry hides before searching”看，Tom应该只有三个信息集（每个信息集只有一个节点，且有三个行动）吧？这样Tom的纯策略应该是9个吧？

对于一个博弈树，我们要看清每个参与人的信息集（information state）是什么，每个信息集上的行动（move）是什么，每个信息集有几个节点（node）。

如果所有参与人的所有信息集有且只有一个节点，则博弈是完全信息博弈（perfect information）。

“纯策略”是“信息集”到“行动”的映射，某参与人的一条纯策略描述了该参与人在其各信息集上将采取的行动。策略与行动是不同的概念。

在（a）中，显然博弈是完全信息的，即每个参与人的每个信息集有且只有一个节点。不过从“Tom can see where Jerry hides before searching”看，Tom应该只有三个信息集（每个信息集只有一个节点，且有三个行动）吧？这样Tom的纯策略应该是9个吧？

（b）中“Jerry can see where Tom searches before hiding”中的“see”如何理解？是“清楚”还是“看见”，两人谁最先行动？

对于（c），个人的理解是，Jerry先行动，她有且仅有一个单节点（即博弈开始的节点）信息集，该信息集上她有三个行动，故她有且仅有三条纯策略。Tom后行动，他有且仅有一个三节点信息集（故该博弈是imperfect information game），该信息集上他有三个行动，故他也有且仅有三条纯策略。

该博弈没有纯策略均衡，而有混合策略均衡。

版主能给划个图吗，不用Intensive  form，简单形式的就好。谢谢

从一个出发点引出三条路，每条路又可以再分出三条路。