Suppose there are two bidders in a 1st price (sealed-bid) auction for a single object. Each...

 

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Suppose there are two bidders in a 1st price (sealed-bid) auction for a single object. Each bidder i privately observes his type ti that is independently drawn from a uniform distribution over [0, 1]. The value of the object to each bidder is t1 + t2, so this is a common value auction. (a) Analogous to how we verified a Bayesian Nash Equilibrium (BNE) in a private value first price auction in class, verify that this problem has a BNE in which each bidder bids using the strategy si(ti) = ti. Specifically:Taking the opponent's strategy as given, write out the expected payoff, call it ?i (b,ti) for player i from bidding b given his type ti. Use calculus to show that, as function of b, ?i (b, ti) is maximized at b = ti. (b) In the above setting, conditional on only his own private information, ti, a player's expected valuation for the object 1/2 + ti. Now consider a private values setting in which each player values the object at 1/2 +ti. The types are drawn just as before. Verify that there is a BNE in which each player i bids according to si(ti) = 1/2+ti /2. (c) Compare the bidding strategies in parts (a) and (b) and explain the economic intuition. Suppose there are two bidders in a 1st price (sealed-bid) auction for a single object. Each bidder i privately observes his type ti that is independently drawn from a uniform distribution over [0, 1]. The value of the object to each bidder is t1 + t2, so this is a common value auction. (a) Analogous to how we verified a Bayesian Nash Equilibrium (BNE) in a private value first price auction in class, verify that this problem has a BNE in which each bidder bids using the strategy si(ti) = ti. Specifically:Taking the opponent's strategy as given, write out the expected payoff, call it ?i (b,ti) for player i from bidding b given his type ti. Use calculus to show that, as function of b, ?i (b, ti) is maximized at b = ti. (b) In the above setting, conditional on only his own private information, ti, a player's expected valuation for the object 1/2 + ti. Now consider a private values setting in which each player values the object at 1/2 +ti. The types are drawn just as before. Verify that there is a BNE in which each player i bids according to si(ti) = 1/2+ti /2. (c) Compare the bidding strategies in parts (a) and (b) and explain the economic intuition.

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