2. (15 points] Consider the following game Player 2 Yes No Player 1 No y. 0 1. 1

Question

2. (15 points] Consider the following game Player 2 Yes No Player 1 No y. 0 1. 1 a. Assume y= 1, For what values of x does this garne have n unique Nash equilibrium? What is that b. Assume y = 1 . 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. Consider the four payoff numbers in the payoff matrix namely, 0, 1, x and y. How should these numbers be ranked for this game to have the same structure as the Asurance Game? (Denote the (in)equality relation between 0. 1. x. y.) 3. [20 points) Marta has two sons: Arturo and Bernardo. She discovers a broken lamp in her living room and knows that one of her sons must have broken it at play, She cares more about finding out the truth than she does about punishing the child who broke the lamp, so Marta announces that her sns are to play the following game. Each child will write down his name on a piece of paper and write down either Yes, I broke the lamp, r No. I didn't break the lamp. Consider two alternative versions of this game with different payoff structures. Version 1 If at least one child claims to have broken the lamp, Marta will give an allowance of $2 to each child who claims to have broken the lamp, and $5 to the child who claims not to have broken the lamp. If both chikdren claim not to have broken the lamp, none of them receives any allowances (each receives $0) a. Write down the payoff matrix for the game. (Use the allowance amounts as payoffs.) b. Find the Nash equilibria of the game. Version 2: If only one child claims to have broken the lamp, Marta will give an allowance of $2 to the child who claims to have broken the lump, and 85 to the child who claims not to have broken the lamp If both chikdren claim to have broken the lamp, although she cannot learn the truth, Marta still wants to reward the children for their solidarity, and gives each child an allowance of $4· If both children claim not to have broken the lamp, each child receives an allowance of Sx. c. Write down the payoff matrix for the game. (Use the allowance amounts as payoff numbers, and take x as a constant.) d. Determine the range of values of "for which the game has the same structure as the Prisoner's Dilemma Hint: Consider the Yes, I broke the lamp as the coopeative strategy.)

Explanation / Answer

Yes

Assume y=1 then for x=1 we would have 2 nash equilibria at (Yes,Yes) & (No,No) hence for x>1 will give us pure nash equilibria of (Yes,Yes).

For x=1 when y=1 we will have 2 nash equilibria hence finding mixed nash equilibria as follows

Let player 1 plays with probability p & player 2 with probability q

Hence player 1 indifferent of player 2 choice of strategy to have expected payoff is

q(x)+(1-q)0=q(1)+(1-q)1

xq=1

q=1/x

Similarly

For player B

p(x)+(1-p)0= p(1)+(1-p)1

px=1

p=1/x

For x=1 we will have mixed strategy and p=q=1 then both players should play yes.

x>y>=1>0

For both the players

Answer 3

To replicate above game to prisoner s dilemma x should be less than 2

Yes No

Yes

(1,1) (0,1) No (1,0) (1,1)

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