10 Basic Game Theory

Question 1

Prove via iterated strict dominance, that for the Prisoner’s Dilemma (D,D) is a (pure, strict)
Nash-Equilibrium!

Answer 1

• No matter what player 2 does: D gives player 1 a strictly higher payoff than C
• Player 2 knows that player 1 will not play C -> D is better than C
• -> (D, D) is the best option for both players

Question 2

What is the rational behavior in an infinitely repeated Prisoner’s Dilemma? Give a short
reasoning!

Answer 2

• Cooperation is rational because of two reasons
• If one defects the other player might punish by defecting in the next run
• Testing cooperation (and possibly getting the sucker’s payoff) is not tragic,
  because on the long run a few sucker’s payoffs are statistically not important

Question 3

Explain why a pure strategy may be dominated by a mixed strategy even if it is not strictly
dominated by any pure strategy, using the following example game:

Answer 3

• For player 1: M not dominated by U and M not dominated by D
• If player 1 plays σ1 = (½, 0, ½) the expected utility u is u(σ1) = ½ regardless of
  how player 2 plays
• Compared to the pure strategy of playing M, where the payoff is always 0 shows
  that M gets dominated by σ1

Question 4

Prove that a mixed strategy may be dominated even if it assigns positive probabilities to
pure strategies that are not even weakly dominated, using the following example:

Answer 4

• U and M are not dominated by D for player 1
• Playing σ1 = (½, ½, 0) gives expected utility u(σ1) = -½, no matter what player 2
  plays
• D dominates σ1

Question 5

What is a Vickrey Auction? Name the rational strategy in a Vickrey auction (no
explanation required)!

Answer 5

• An auction “game” where the good’s valuation v_i is assumed to be common
  knowledge
• Bids s_i
• Winning condition is to bid higher than everyone else
• The price r_i to be paid when you win is the highest bid that was not the winning
  bid
• Winners utility: u_i = v_i – r_i
• Other players utility = 0
• E.g. ebay
• The rational strategy is to bid the true valuation

Question 6

Assume that in a 2-player game the mixed strategy profile ((a,b,0),(c,d,0)) is a mixed
strategy NE. Does the Indifference Condition in a mixed strategy NE imply that a = b = ½ ?
Give a short reasoning!

Answer 6

• No, this depends on the payoff of each strategy. (E.g. could be a = 2/3 and b =
  1/3

Question 7

Explain, why ((1/2, 1/2); (1/2, 1/2)) is a mixed strategy NE in the Matching Pennies game!

Answer 7

• If player 2 plays (1/2, ½) then player 1’s expected payoff is ½1 + ½(-1) = 0 when
  playing H and ½(-1) + ½1 = 0 when playing T
• -> player 1 is also indifferent

Question 8

Derive the mixed strategy NE in the battle of the sexes, using the Indifference Condition!

Answer 8