Suppose a single item will be sold to one of two bidders. Each bidder's value is drawn uniformly at random from the interval [0, 1]. Each bidder will simultaneously name a price and the item will be sold to the higher bidder for the price that bidder named. If this price is p and the winning bidder has a value of v then the utility of the winning bidder is v-p and the utility of the losing bidder is 0. This situation is more complex than the normal form games we studied in class for two reasons. First, the set of strategies is infinite. Second, rather than known utilities there is uncertainty over them, so we need a Bayesian model and in general the equilibrium strategy will depend on (i.e. be a function of) v. This assignment walks you through the derivation of an equilibrium in this setting. (a) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. If you bid p, what is your expected utility? (b) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. Show that your optimal bid is 0.5v. (c) Suppose you bid following your strategy from part (b). What is the distribution of your bids? (Recall that your value is uniform on [0, 1]). (d) Hee the analysis from the preceding nort bidding half their luo in o
Suppose a single item will be sold to one of two bidders. Each bidder's value is drawn uniformly at random from the interval [0, 1]. Each bidder will simultaneously name a price and the item will be sold to the higher bidder for the price that bidder named. If this price is p and the winning bidder has a value of v then the utility of the winning bidder is v-p and the utility of the losing bidder is 0. This situation is more complex than the normal form games we studied in class for two reasons. First, the set of strategies is infinite. Second, rather than known utilities there is uncertainty over them, so we need a Bayesian model and in general the equilibrium strategy will depend on (i.e. be a function of) v. This assignment walks you through the derivation of an equilibrium in this setting. (a) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. If you bid p, what is your expected utility? (b) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. Show that your optimal bid is 0.5v. (c) Suppose you bid following your strategy from part (b). What is the distribution of your bids? (Recall that your value is uniform on [0, 1]). (d) Hee the analysis from the preceding nort bidding half their luo in o
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.7P
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![1. Equilibria in an auction
Suppose a single item will be sold to one of two bidders. Each bidder's value is drawn uniformly at
random from the interval [0, 1]. Each bidder will simultaneously name a price and the item will be sold
to the higher bidder for the price that bidder named. If this price is p and the winning bidder has a
value of v then the utility of the winning bidder is v-p and the utility of the losing bidder is 0. This
situation is more complex than the normal form games we studied in class for two reasons. First, the
set of strategies is infinite. Second, rather than known utilities there is uncertainty over them, so we
need a Bayesian model and in general the equilibrium strategy will depend on (i.e. be a function of)
v. This assignment walks you through the derivation of an equilibrium in this setting.
(a) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. If you
bid p, what is your expected utility?
(b) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. Show
that your optimal bid is 0.5v.
(c) Suppose you bid following your strategy from part (b). What is the distribution of your bids?
(Recall that your value is uniform on [0, 1]).
(d) Use the analysis from the preceding parts to argue that each player bidding half their value is a
(Bayes) Nash equilibrium of this game.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F672bf286-8abe-4b07-9ca1-0d5b2612956c%2Fadcbb8f4-1394-4bff-bde8-26f13abef1ac%2Fh0m8gm7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Equilibria in an auction
Suppose a single item will be sold to one of two bidders. Each bidder's value is drawn uniformly at
random from the interval [0, 1]. Each bidder will simultaneously name a price and the item will be sold
to the higher bidder for the price that bidder named. If this price is p and the winning bidder has a
value of v then the utility of the winning bidder is v-p and the utility of the losing bidder is 0. This
situation is more complex than the normal form games we studied in class for two reasons. First, the
set of strategies is infinite. Second, rather than known utilities there is uncertainty over them, so we
need a Bayesian model and in general the equilibrium strategy will depend on (i.e. be a function of)
v. This assignment walks you through the derivation of an equilibrium in this setting.
(a) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. If you
bid p, what is your expected utility?
(b) Suppose your value is v and the bid of the other bidder is uniformly distributed on [0,0.5]. Show
that your optimal bid is 0.5v.
(c) Suppose you bid following your strategy from part (b). What is the distribution of your bids?
(Recall that your value is uniform on [0, 1]).
(d) Use the analysis from the preceding parts to argue that each player bidding half their value is a
(Bayes) Nash equilibrium of this game.
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