Game Theory Basics

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

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

Assuming a set of players I with |I| = I and a finite space S = x_iS_i of pure strategy profiles s ∈ S. Derive the mathematical expression for the utility of player i ∈ I if a mixed strategy profile σ =(σ₁, σ₂,…, σ_I ) is played!

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:

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

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

In a mixed strategy Nash Equilibrium: Why must a player be (a priori)  indifferent between all pure strategies to which he assigns positive probability  (not regarding the other player‘s choice)? (1 sentence)

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!

In the Cournot competition, each firm i ’s profit is given by u_i(q₁, q₂) = q_i p(q) -­ c_i (q_i ). Derive the defining equation for the reaction function r₂(.)

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

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