Reference no: EM132653700

Question 1. Two candidates, A and B, vie for an office. A committee of 11 people conducts an election to decide whom to choose. Abstention is not possible; each committee member has to vote for either candidate A or candidate B. The committee members cast their vote simultaneously. The candidate with most votes wins. A committee member's payoff is 2 if the candidate she supports wins the election; and -2 otherwise.
- How many players are there in this game?
- How many strategies does each player have?
- Suppose a committee member supports candidate A. How many strategies of this committee member are strictly dominated?
- Suppose a committee member supports candidate A. How many strategies of this committee member are weakly dominated?

Question 2. John Jackson makes the following offer to his two sons, Dave and Thomas. He says that each of them is to submit a written request for an amount that they want (in whole dollars). They can request $0. Let $ D (respectively, $T ) denote the amount requested by Dave (respectively, Thomas). Each son is to decide on the amount to request without the knowledge of the amount requested by the other brother. If D + T ≤ 20, then they get their requested amounts. In addition, John Jackson doubles the remaining amount, 20 - D - T , and splits the doubled amount equally between the two brothers. If D + T > 20, then both get zero dollars.
- Is D=10,T=10 a Nash equilibrium?
- Is D=20,T=20 a Nash equilibrium?
- Is D=15,T=20 a Nash equilibrium?
Question 3. Two firms simultaneously decide whether or not to enter a market, and if yes, when to enter a market. The market lasts for 5 periods: starting in period 1 and ending in period 5. A firm that chooses to enter can enter in any of the five periods. Once a firm enters the market in any period it has to stay in the market through period 5. In any period tt that the the firm is not in the market, it earns a zero profit. In any period tt, if a firm is a monopolist in the market, it makes the profit 10t-24. In any period tt if a firm is a duopolist in the market it makes a profit of 7t-24. A firm's payoff is the total profit it earns in all the periods it is in the market.
- How many strategies does each firm have?
- Firm 1's best response to Firm 2's choice Do not enter is to enter in period:
- In a Nash equilibrium, Firm 1 enters in period _______ (if there is more than one answer, write anyone)

Question 4. John Jackson makes the following offer to his two sons, Dave and Thomas. He says that each of them is to submit a written request for an amount that they want (in whole dollars). They can request $0. Let $ D (respectively, $T ) denote the amount requested by Dave (respectively, Thomas). Each son is to decide on the amount to request without the knowledge of the amount requested by the other brother. If D + T ≤ 20, then they get their requested amounts. In addition, John Jackson tripless the remaining amount, 20 - D - T , and splits the tripled amount equally between the two brothers. If D + T > 20, then both get zero dollars.
- Is D=10,T=10 a Nash equilibrium?
- Is D=20,T=20 a Nash equilibrium?
- Is D=15,T=20 a Nash equilibrium?
Question 5. There are 100 residents in city Lonis who work in another city called Pardon. The residents can can take one of the two routes, A or B, to commute to Pardon. Both routes are free, however route A is wider, whereas route BB is narrower. A commuters net payoff from traveling on route A is y-2k if a total of kk drivers take route AA. Similarly, the payoff of a commuter from traveling on route B is y-3k, if a total of kk drivers take route B.
- In a Nash equilibrium, how many drivers take Route A?

Question 6. Consider a country with 10 citizens. Let vk=k denote the value that citizen kk attaches to protesting. That is, citizen 1 attaches a value of 1, citizen 2 attaches a value of 2 to protesting, and so on with citizen 10 attaching a value of 10 to protesting. There is also a cost to protesting, which equals 20/m, where mm is number of citizens protesting. If citizen kk protests, her payoff is
k-20/m
where m is the total number of people who protest.
Payoff to a citizen who does not protest is zero.
- How many Nash equilibria does the game have?
- Is it a Nash equilibrium if everyone protests?
- How many people protest in the equilibrium? If there are 2 more Nash equilibria, write the number of protesters from any one of those equilibria.