Homework Problem Set 1

1. Consider the following game: a>0, b>0, c>0, d>0, e>0, f>0, g>0, h>0.

Player 2

Left Right

Top a, b c, d

Player 1

Bottom e, f g, h

(a) If (Top, Left) is a dominant strategy equilibrium, then what inequalities must hold among a,…,h?

(b) If (Top, Left) is a Nash equilibrium, then which of the above inequalities must be satisfied?

2. Consider a game show ‘Fastest Fingers First’ played between two contestants where all

that they have to do is press the buzzer as soon as the music stops but within two

milliseconds. The one presses the buzzer first gets one point but the one who is late gets a

penalty. If both press at the same time, both get nothing. If both press after two

milliseconds, they get nothing.

a) What is the Nash equilibrium of this game?

b) What is the minimax strategy for this game?

3. In the following constant-sum game, find the relationship between the values of a, b, c d,

where a, b, c, d and k are non-negative constants:

Player 2

α β

Pla

yer

1

α k, 0 a, k-b

β c, kc d, k+d

4.Consider an extension of the game “Rock-Paper-Scissors-Lizard-Spock”: Scissors cut paper, paper covers rock, rock crushes lizard, lizard poisons Spock, Spock smashes scissors, scissors decapitate lizard, lizard eats paper, paper disproves Spock, Spock vaporizes rock, rock crushes scissors.

(for a vivid illustration of the rule, watch http://www.youtube.com/watch?v=Kov2G0GouBw)

Can you describe the game in normal form? Suppose winning gives a payoff of 1, losing generates payoff of -1 and tying gives 0, how would the equilibrium look like?