Managerial Economics Ch.10 Game Theory Problems: 1. Consider a game in which the

Question

Managerial Economics Ch.10 Game Theory Problems:

1. Consider a game in which there are two players, Annie and Ben. Annie can choose one of three strategies to play: A, B or C. Ben can choose one of three strategies to play: 1, 2 or 3. Their payoffs in every situation are as follows:

(a) Is there a Nash equilibrium (or more than one) for this game, if it played simultaneously? If so, what is it (or what are they, if there is more than one)?

(b) Suppose that this game could be played sequentially. Does Annie prefer for herself to play first, or for Ben to do so? Explain.

(c) Suppose that this game could be played sequentially. Does Ben prefer for himself to play first, or for Annie to do so? Explain.

2. Imagine a simultaneous-play, one-shot game between Alice and Bob. Alice can play one of three strategies, H, M or L. Bob can play one of three strategies, 1, 2, or 3. The payoff matrix of this game looks as follows:

(a) If this game is played simultaneously, how many Nash equilibria are there?

(b) If this game is played simultaneously, what will Bob’s payoff be?

(c) If this game is played sequentially and Bob plays first, what will Bob’s payoff be?

Explanation / Answer

b) If Annie plays first,

if she chooses A, Ben chooses 2. Payoff = (12,8)

if she chooses B, Ben chooses 1. Payoff = (7,10)

if she chooses C, Ben chooses 1. Payoff = (5,12)

Since Annie is the first player she will maxmises her payoff and choose A, with payoff (12,8)

If Ben plays first,

if he chooses 1, Annie chooses A. Payoff = (10,6)

if he chooses 2, Annie chooses A. Payoff = (12,8)

if he chooses 3, Annie chooses C. Payoff = (8,11)

Since Ben is the first player he maxmises his payoff by choosing 3, with payoff (8,11)

Hence Annie should choose to be the first player because that gives her a higher payoff of 12 rather than 8 when Ben is the first player.

c) ) If Annie plays first,

if she chooses A, Ben chooses 2. Payoff = (12,8)

if she chooses B, Ben chooses 1. Payoff = (7,10)

if she chooses C, Ben chooses 1. Payoff = (5,12)

Since Annie is the first player she will maxmises her payoff and choose A, with payoff (12,8)

If Ben plays first,

if he chooses 1, Annie chooses A. Payoff = (10,6)

if he chooses 2, Annie chooses A. Payoff = (12,8)

if he chooses 3, Annie chooses C. Payoff = (8,11)

Since Ben is the first player he maxmises his payoff by choosing 3, with payoff (8,11)

Hence Ben should choose to be the first player because that gives him a higher payoff of 11 rather than 8 when Annie is the first player.

b) 5

c) If Bob plays first,

if he chooses 1, Alice chooses H. Payoff = (7,8)

if he chooses 2, Alice chooses M. Payoff = (8,5)

if he chooses 3, Alice chooses M. Payoff = (10,3)

Since Bob is the first player he maxmises his payoff by choosing 1, with payoff 8.

b) If Annie plays first,

if she chooses A, Ben chooses 2. Payoff = (12,8)

if she chooses B, Ben chooses 1. Payoff = (7,10)

if she chooses C, Ben chooses 1. Payoff = (5,12)

Since Annie is the first player she will maxmises her payoff and choose A, with payoff (12,8)

If Ben plays first,

if he chooses 1, Annie chooses A. Payoff = (10,6)

if he chooses 2, Annie chooses A. Payoff = (12,8)

if he chooses 3, Annie chooses C. Payoff = (8,11)

Since Ben is the first player he maxmises his payoff by choosing 3, with payoff (8,11)

Hence Annie should choose to be the first player because that gives her a higher payoff of 12 rather than 8 when Ben is the first player.

c) ) If Annie plays first,

if she chooses A, Ben chooses 2. Payoff = (12,8)

if she chooses B, Ben chooses 1. Payoff = (7,10)

if she chooses C, Ben chooses 1. Payoff = (5,12)

Since Annie is the first player she will maxmises her payoff and choose A, with payoff (12,8)

If Ben plays first,

if he chooses 1, Annie chooses A. Payoff = (10,6)

if he chooses 2, Annie chooses A. Payoff = (12,8)

if he chooses 3, Annie chooses C. Payoff = (8,11)

Since Ben is the first player he maxmises his payoff by choosing 3, with payoff (8,11)

Hence Ben should choose to be the first player because that gives him a higher payoff of 11 rather than 8 when Annie is the first player.

b) 5

c) If Bob plays first,

if he chooses 1, Alice chooses H. Payoff = (7,8)

if he chooses 2, Alice chooses M. Payoff = (8,5)

if he chooses 3, Alice chooses M. Payoff = (10,3)

Since Bob is the first player he maxmises his payoff by choosing 1, with payoff 8.

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