Question

Consider a game in which player 1 selects effort x and player 2 selects effort y, where x and y must be between 1 and 12 (inclusive). Payoffs are given by:

u₁(x,y) = cxy – x²,

u₂(x,y) = 2xy – y².

where c > 0 is a parameter.

1. What is the best response function for each player?

2. What is the B₁ and B₂ for each player if c = 1?

3. What is the rationalizable set if c = 4? What if c = 2?

Answer 1

(1) Each player will choose exertion to amplify utility:

Player 1: max cxy - x2

FOC wrt to x and likening to 0

cy - 2x = 0

=> x = cy/2 (BR1)

Player 1: max 2xy - y2

FOC wrt to y and likening to 0

2x - 2y = 0

=> y = x (BR2)

(2) if C=1 , we get

B1 => x = y/2

B2 => y = x

(3) when c=4

B1 => x = 4y/2 = 2y

B2 => x = y

Consequently we get x =y =0

at the point when c = 2

B1 => x = 2y/2 => x = y

B2 => y =x

Along these lines, the rationalizable set is all qualities with the end goal that x = y (45 degree line)

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answered by: anonymous