(CH.7.U9.) Consider the following game:

COLUMN

yes         no

yes

no

(a) For what values of x does this game have a unique Nash equilibrium? What is this equilibrium?

(b) For what values of x does this game have a mixed-strategy Nash equilibrium? With what probability, expressed in terms of x, does each player play YES in this mixed- strategy equilibrium?

(c) For the values of x found in the previous part, is the game an example of an assurance game, a game of chicken, or a game similar to tennis? Explain.

2.  (CH.10.U1.)  In a scene from the movie Manhattan Murder Mystery, Woody Allen and Diane Keaton are at a hockey game in Madison Square Garden.  She is obviously not enjoying herself, but he tells her: “Remember our deal.  You stay here with me for the entire hockey game, and next week I will come to the opera with you and stay until the end.”Later, we see them coming out of the Met into the deserted Lincoln Center Plaza while inside the music is still playing. Keaton is visibly upset: “What about our deal? I stayed to the end of the hockey game, and so you were supposed to stay till the end of the opera.”Allen answers: “You know I can’t listen to too much Wagner. At the end of the first act, I already felt the urge to invade Poland.”

Comment on the strategic choices made here by using your knowledge of the theory of strategic moves and credibility.

3.  (CH.11.U3.) A town council consists of three members who vote every year on their own salary increases. Two Yes votes are needed to pass the increase. Each member would like a higher salary but would like to vote against it herself because that looks good to the voters. Specifically, the payoffs of each are as follows:

Raise passes, own vote is No: 10

Raise fails, own vote is No: 5

Raise passes, own vote is Yes: 4

Raise fails, own vote is Yes: 0

Voting is simultaneous. Write down the (three-dimensional) payoff table, and show that in the Nash equilibrium the raise fails unanimously.  Examine how a repeated relation- ship among the members can secure them salary increases every year if (1) every member serves a 3-year term, (2) every year in rotation one of them is up for reelection, and (3) the townspeople have short memories, remembering only the votes on the salary-increase motion of the current year and not those of past years.

 (CH.12.U5.) A group of 12 countries is considering whether to form a monetary union.

They differ in their assessments of the costs and benefits of this move, but each stands to gain more from joining, and lose more from staying out, when more of the other countries choose to join. The countries are ranked in order of their liking for joining, 1 having the highest preference for joining and 12 the least. Each country has two actions, IN and OUT.

Let

B(i,n) = 2.2 + n − i

be the payoff to country with ranking i when it chooses IN and n others have chosen IN.

Let

S(i,n) = i − n

be the payoff to country with ranking i when it chooses OUT and n others have chosen IN.

(a) Show that for country 1, IN is the dominant strategy.

(b) Having eliminated OUT for country 1, show that IN becomes the dominant strategy

for country 2.

(c) Continuing in this way, show that all countries will choose IN.

(d) Contrast the payoffs in this outcome with those where all choose OUT. How many countries are made worse off by the formation of the union?

 (Binmore, 2007) The rhyming triplets, Boris, Horace, and Maurice, are the membership

committee of the very exclusive Dead Poets Society. The final item on their agenda one morning is a proposal that Alice should be admitted as a new member.  Now mention is made of another possible candidate called Bob, so an amendment to the final item is proposed.  The amendment says that Alice’s name should be replaced with Bob’s.  The rules for voting in committees call for amendments to be voted on in the reverse order to which they are proposed. The committee therefore begins by voting on whether Bob should replace Alice. If Alice wins, they then vote on whether Alice or Nobody should be made a new member. If Bob wins, they then vote on whether Bob or Nobody should be made a new member. The table below shows how the three committee members rank the three possible outcomes. The figure on the next page represents the order in which voting takes place.

(a) Who will win the vote if everybody just votes according to their rankings?              (b) Why should Horace switch to voting for the candidate he likes least at the first vote?

(c) What happens if everybody votes strategically? (Hint: Focus on equilibria in (weakly) dominant strategies.)