Consider a first-price, sealed-bid auction, and suppose there are only three feasible bids: A bidder can bid 1, 2, or 3. The payoff to a losing bidder is zero. The payoff to a winning bidder equals his valuation minus the price paid (which, by the rules of the auction, is his bid). What is private information to a bidder is how much the item is worth to him; hence, a bidder’s type is his valuation. Assume that there are only two valuations, which we’ll denote L and H, where H > 3 > L > 2. Assume also that each bidder has probability .75 of having a high valuation, H. The Bayesian game is then structured as follows: First, Nature chooses the two bidders’ valuations. Second, each bidder learns his valuation, but does not learn the valuation of the other bidder. Third, the two bidders simultaneously submit bids. A strategy for a bidder is a pair of actions: what to bid when he has a high valuation and what to bid when he has a low valuation. a. Derive the conditions on H and L whereby it is a symmetric BNE for a bidder to bid 3 when he has a high valuation and 2 when he has a low valuation. b. Derive the conditions on H and L whereby it is a symmetric BNE for a bidder to bid 2 when he has a high valuation and 1 when he has a low valuation. c. Derive the conditions on H and L whereby it is a symmetric BNE for a bidder to bid 3 when he has a high valuation and 1 when he has a low valuation. d. Derive the conditions on H and L whereby it is a symmetric BNE for a bidder to bid 1 when he has either a high valuation or a low valuation.
Consider a first-price, sealed-bid auction, and suppose there are only three feasible bids: A bidder can bid 1, 2, or 3. The payoff to a losing bidder is zero. The payoff to a winning bidder equals his valuation minus the price paid (which, by the rules of the auction, is his bid). What is private information to a bidder is how much the item is worth to him; hence, a bidder’s type is his valuation. Assume that there are only two valuations, which we’ll denote L and H, where H > 3 > L > 2. Assume also that each bidder has probability .75 of having a high valuation, H. The Bayesian game is then structured as follows: First, Nature chooses the two bidders’ valuations. Second, each bidder learns his valuation, but does not learn the valuation of the other bidder. Third, the two bidders simultaneously submit bids. A strategy for a bidder is a pair of actions: what to bid when he has a high valuation and what to bid when he has a low valuation.
a. Derive the conditions on H and L whereby it is a symmetric BNE for a
bidder to bid 3 when he has a high valuation and 2 when he has a low
valuation.
b. Derive the conditions on H and L whereby it is a symmetric BNE for a
bidder to bid 2 when he has a high valuation and 1 when he has a low
valuation.
c. Derive the conditions on H and L whereby it is a symmetric BNE for a
bidder to bid 3 when he has a high valuation and 1 when he has a low
valuation.
d. Derive the conditions on H and L whereby it is a symmetric BNE for a
bidder to bid 1 when he has either a high valuation or a low valuation.
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