Question

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 sons 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,” or “No, I didn’t break the lamp.” Consider two alternative versions of this game with different payoff structures, and answer the corresponding questions.

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 lamp, and $5 to the child who claims not to have broken the lamp. If both children 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 $x.

d. Write down the payoff matrix for the game. (Use the allowance amounts as payoffs.)

e. Determine the range of values of “x” for which the game has the same structure as the Prisoner’s

Dilemma. Consider the “Yes, I broke the lamp” as the cooperative strategy.

Answer 1

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

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

	Bernardo(Yes)	Bernardo(No)
Arturo(yes)	(4,4)	(2,5)
Arturo(No)	(5,2)	(x,x)

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