.\'Il Sprint 5:51 PM 80% last handout game theory.pdf Problem 1 There are three

Question

.'Il Sprint 5:51 PM 80% last handout game theory.pdf Problem 1 There are three players. Windom has 2 cups of coffee, Bob has 3 cups, and Dale has ble. All players want to get as many cups as possible. 2 cups. Assume that cups of coffee are not divisi- a) Consider the following game. First, Windom proposes to redistribute the cups. For example, he may propose that he gets 5 cups and Bob and Dale each get only I cup. Next, Bob and Dale vote Yes or No. If at least one of them votes Yes, the cups are redistributed as Windom proposed. Oth- erwise, nothing changes. Find the subgame-perfect equilibrium. Assume that if a player is indifferent between the two outcomes, he turns down the b) Consider a different version of this game. First, Windom proposes to redistribute the cups and Bob and Dale vote Yes or No. If both vote No, the game ends and nothing changes. If at least one of them votes Yes the cups are redistributed as Windom proposed, and this becomes the new status quo-Then Windom gets to make another proposal. If it is accepted. then the cups are redistributed again and the game ends, Otherwise, the new status quo remains in place. Find at least one Subgame-Perfect equilibrium. Assume again that if a player is indifferent between the two outcomes, he turns down the offer. Problem 2 Audrey. Donna, Shelly, and Laura are four students who must form pairs to do a school project. The preferences are as follows: t from the mathching prob- lems we have seen before. Specifically, the players are not divided into two groups. Dividing them into two groups and running an algorithm does not guarantee a stable matching. a) Is it possible to match the girls into pairs such that no two girls who pairs would like to leave their current partners and form a are in different pair? Give an example or show that it is impossible. b) Now that the Audrey Audrey h the girls into pairs such that no two girls who are partners and form a pair? pairs would like to leave their current in different Give an example or show that it is impossible Problem 3 There are three voters and a 100 golden coins. Each voter

Explanation / Answer

Windom = W

Bob = B

Dale = D

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Part a)

For a subgame-perfect equilibrium state, W has to make a proposal that will be accepted by atleast 1 of the players. He is looking for his own gain, as is any other player. Hecne, the redistribution he suggests needs to satisfythe condition that two people profit (including W himself).

To maximise his own gains, W should choose the player who is most easy to satisfy, i.e. currently has the lowest cups, amongst all players except W. => He chooses D.

Process: In W's proposal, he will give D one cup extra and keep all remaining cups (From B) for himself, leaving us at the subgame-perfect equilibrium state:-

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Part b)

Second process, W has the same objective. But, not the player with lowest number of cups (amongst players except W) is B (0 cups). So, for B to accept W's next proposal, W needs to offer him just 1 cup, leaving us at the subgame-perfect equilibrium state:-

W B D 4 0 3

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