1 . Individual Problems 15-1

Mr. Ward and Mrs. Ward typically vote oppositely in elections, so their votes "cancel each other out." They each gain 4 units of utility from a vote for their positions (and lose 4 units of utility from a vote against their positions). However, the bother of actually voting costs each 2 units of utility. The following matrix summarizes the strategies for both Mr. Ward and Mrs. Ward.

Using the given information, fill in the payoffs for each cell in the matrix. For example, in the top left cell, fill in the payoffs for Mr. Ward and Mrs. Ward if they both vote. (Hint: Be sure to enter a minus sign if the payoff is negative.)

	Mrs. Ward
Vote	Don't Vote
Mr. Ward	Vote	Mr. Ward: ,Mrs. Ward	Mr. Ward: ,Mrs. Ward
Don't Vote	Mr. Ward: ,Mrs. Ward	Mr. Ward: ,Mrs. Ward

2 . Individual Problems 15-2

Mr. and Mrs. Ward typically vote oppositely in elections and so their votes "cancel each other out." They each gain 24 units of utility from a vote for their positions (and lose 24 units of utility from a vote against their positions). However, the bother of actually voting costs each 12 units of utility. The following matrix summarizes the strategies for both Mr. Ward and Mrs. Ward.

	Mrs. Ward
Vote	Don't Vote
Mr. Ward	Vote	Mr. Ward: -12,Mrs. Ward: -12	Mr. Ward: 12,Mrs. Ward: -24
Don't Vote	Mr. Ward: -24,Mrs. Ward: 12	Mr. Ward: 0,Mrs. Ward: 0

The Nash equilibrium for this game is for Mr. Ward to and for Mrs. Ward to . Under this outcome, Mr. Ward receives a payoff of

units of utility and Mrs. Ward receives a payoff of

units of utility.

Suppose Mr. and Mrs. Ward agreed not to vote in tomorrow's election.

True or False: This agreement would decrease utility for each spouse, compared to the Nash equilibrium from the previous part of the question.

True

False

This agreement not to vote a Nash equilibrium.

3 . Individual Problems 15-3

Microsoft and a smaller rival often have to select from one of two competing technologies, A and B. The rival always prefers to select the same technology as Microsoft (because compatibility is important), while Microsoft always wants to select a different technology from its rival. If the two companies select different technologies, Microsoft's payoff is 4 units of utility, while the small rival suffers alossof utility of 2. If the two companies select the same technology, Microsoft suffers alossof utility of 2 while the rival gains 2 units of utility.

Using the given information, fill in the payoffs for each cell in the matrix, assuming that each company chooses its technology simultaneously.

	Microsoft
Technology A	Technology B
Rival	Technology A	Rival: ,Microsoft	Rival: ,Microsoft
Technology B	Rival: ,Microsoft	Rival: ,Microsoft

True or False: There is an equilibrium for this game in pure strategies.

True

False 4 . Individual Problems 15-4

After graduation, you enter salary negotiations for your first job. Suppose the potential employer (employer A) has two choices: to offer you a high salary or to offer you a low salary. You may then accept or reject whatever offer is made. The payoffs, as well as the decision tree, are depicted in the following figure.

Assume this is a sequential game.

If employer A offers a low salary, you, as the employee, are best served by the offer. In this case, you would earn a payoff of , and employer A would earn a payoff of . Alternatively, if employer A offers a high salary, you are best served by the offer. In this case, you would earn a payoff of , and employer A would earn a payoff of . With this information, employer A will choose to make a offer, since it will yield a higher payoff for him, based on what you (the employee) will subsequently choose.

Suppose you have a competing job offer from employer B. Accepting this job offer gives a payoff of 86.5. During your negotiations with employee A, you have the option of taking this offer from employer B, and employer A is aware of this offer (as well as the payoff to you). Given this competing offer, the negotiation with employer A is depicted in the following figure:

True or False: With this competing job offer, your threat to reject employer A's offer, if it is low, is now credible.

True

False 5 . Individual Problems 15-5

Every year, management and labor renegotiate a new employment contract by sending their proposals to an arbitrator, who chooses the best proposal (effectively giving one side or the other $5 million). Each side can choose to hire, or not hire, an expensive labor lawyer (at a cost of $200,000) who is effective at preparing the proposal in the best light. If neither hires a lawyer or if both hire lawyers, each side can expect to win about half the time. If only one side hires a lawyer, it can expect to win nine tenths, or 0.9, of the time.

Use the given information to fill in theexpected payoff, in dollars, for each cell in the matrix. (Hint: To find the expected payoff, multiply the probability of winning by the dollar amount of the payoff. Be sure to account for lawyer costs, which are incurred with certainty if a lawyer is hired.)

	Management (M)
No Lawyer	Lawyer
Labor (L)	No Lawyer	L: ,M:	L: ,M:
Lawyer	L: ,M:	L: ,M:

The Nash equilibrium for this game is for Management to a lawyer, and for Labor to a lawyer.

6 . Individual Problems 15-6

Consider a sequential-move game in which an entrant is considering entering an industry in competition with an incumbent firm. If the entrant does not enter ("Out"), the incumbent firm earns a payoff of 10, while the entrant earns a payoff of 0. If the entrant enters ("In"), then the incumbent can either accommodate or fight. If the incumbent accommodates, both earn a payoff of 5. If the incumbent fights, then the entrant can either leave the industry ("Withdraw") or remain in it ("Stay"). If the entrant stays, both earn a payoff of -5. If the entrant withdraws, the entrant earns a payoff of -1, and the incumbent earns a payoff of 8. The extensive form of the game is depicted in the following figure, where the payoffs are of the form (Entrant Payoff, Incumbent Payoff).

True or False: The equilibrium for this game is {In, Fight, Stay}.

True

False