## ECON0027 Game Theory Home assignment 2

### 1.Twoplayers are to play the game Gwith payoffs as given below: 博弈论代写

 C D C 2, 2 −1, 3 D 3, −1 1, 1

Before they play this game, they each make a statement to the other, in the following sequence.

First, player 1 must choose between two statements, A and B: She can say either

(A)“I will play C”,or

(B)“It’s a niceday”. 博弈论代写

Player 2 must then choose between the same statements, A and B. Players then choose actions in the game simultaneously.

The payoffs of the players are those that accrue in the game G, minus a penalty of

> 0, which a player incurs if he or she makes a false statement. (It so happens, it is a nice day).

Assume that ∆ is commonly known to both players. Solve for the subgame perfect equilibria of this game for:

(a)  ∆ = 0.5;

(b) ∆ = 1.5;

(c) ∆ = 2.5. Explain the difference between these three cases.

### 2.Considerthe prisoner’s dilemma with payoffs as given below:  博弈论代写

 C D C 1, 1 −A, 1 + g D 1 + g,-A 0, 0

g > 0, A > 0

Suppose that the game is repeated twice, with the following twist. If a player chooses an action in period 2 which differs from her chosen action in period 1, then she incurs a cost of ε. Players maximize the sum of payoffs over the two periods, with discount factor δ = 1博弈论代写

(a)Suppose that g < 1 and 0 < g < ε < A.Suppose each player plays the strategy which chooses C at t = 1, and chooses C at t = 2 if (C, C) has been played at t = 1, and D Show that this is a subgame perfect equilibrium.

(b)Letε > 0 be  Show that there is always a subgame perfect equilibrium where (D, D) is played in both periods.

### 3.Two players are playing a game “21”. At the beginning of the game player 1 starts bysaying number “1”.

Then players take turns and increase the number by either 1, 2 or 3. A player that is forced to say a number that is larger or equal to 21 loses and his opponent wins.

(a)Formalize this game. Define the most appropriate solution concept for it.

(b)Solvefor all equilibria of this game.

(c)Considera modification of this game in which a player that is forced to say a number that is larger or equal to 2001 loses. Which player wins the game in equilibrium? Explain youranswer.

### 4.Thethree musketeers, Aramis, Athos and Porthos (players 1,2 and 3 respectively), decide that they are no longer “all for one, one for all” and agree to a truel (a truel is a duel among three opponents).  博弈论代写

According to the rules of the truel, each opponent has a musket loaded with one  The musketeers act sequentially, in alphabetic order starting with Aramis. When his turn comes, a musketeer chooses his target and shoots. In normal circumstances, a musketeer i ∈ {1, 2, 3} hits a target with probability pi [0, 1] (assume that a musketeer cannot miss a target on purpose). Once a musketeer is wounded in the truel, he cannot shoot and cannot be chosen as a target by another musketeer. A wounded musketeer has to pay his own medical expenses that are equal to C > 0 (the three musketeers are so tough that they cannot be killed). Assume that the musketeers are risk-neutral and that they minimise their expenses.  博弈论代写

(a)Formalize this situation as a game.

(b)Assumingthat a musketeer who is indifferent between two targets picks either with probability 0.5, find a subgame perfect Nash equilibrium.

(c)Forthe equilibrium that you found, provide ex ante probabilities of becoming wounded for all three duelists. Explain intuitively what would change if you drop the assumption that a musketeer cannot miss a target on purpose.