Questions -

Q1. Show that the two-player game illustrated in the table has a unique equilibrium. Hint: show that it has a unique Pure Strategy Nash Equilibrium; then prove that there can be no equilibrium in strictly mixed strategies.

Q2. Consider the following auction conducted under somewhat unusual rules:  A dollar is auctioned to two bidders who take turns to bid. Bids must be multiples of 5 cents. The highest bidder gets the dollar and pays his final bid, but the loser must also pay his (the second highest) final bid. If the bidding must also be in 5 cent increments (5, 10, 15 etc) is there a Nash Equilibrium of this game? If the bidders can name any (higher) bid so long as it is a multiple of 5 cents does that change your answer? Provide all Nash equilibria for both mechanisms. Are there any Nash equilibria to this game that are not subgame perfect?

Q3. Two Sellers, 1 and 2, face the following situation. Seller i has a loyal customer base of l_i, 0 < l₁ < l₂ such that l₁ + l₂ < 1. The remaining 1- l₁ - l₂ customers buy from whichever firm offers the lower price, dividing their purchases evenly if both firms offer the same price. All customers buy precisely one unit from their preferred provider provided the price offered is less than or equal to R. Sellers choose prices simultaneously to maximize their expected profits, and face zero marginal cost.

a. Derive the profits of the two sellers as a function of their prices p₁ and p₂.

b. What is the highest price pH and lowest price pL that each seller would ever charge?

c. Derive the best response of firm i : β_i(p_j)

d. Show that there is no pure strategy Nash equilibrium in prices.

e. Consider two candidate prices x₁ and x₂ for seller 1. What conditions must be satisfied for the seller to be indifferent between them?

f. Using your result in e derive the Nash Equilibrium pricing strategies.