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slides_commonvalueauction_modelII

# slides_commonvalueauction_modelII - Model II Common values...

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Model II Common values can also be modelled as a special case of interdependent values . In the interdependent values model v 1 = α s 1 + γ s 2 v 2 = α s 2 + γ s 1 where s 1 and s 2 are private signals of bidders 1 and 2, α 0 is the weight a bidder puts on her own signal and γ 0 is the weight she puts on her opponent’s signal. Rod Garratt ECON 177: Auction Theory With Experiments

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We consider the case where α = γ = 1 . In this case, v 1 = v 2 , which is a case of common values. The oil lease example fits this model when each signal s i is determined independently and s i = 0 or 3 with equal probability. In what follows we will consider a more general treatment of the private signals. We will assume throughout that the signals s i are drawn independently form the uniform distribution on [0 , 100] . Rod Garratt ECON 177: Auction Theory With Experiments
Claim: The first-price auction has a symmetric Nash equilibrium in which each bidder bids s i . Proof. Suppose bidder 2 bids s 2 and consider an arbitrary bid b 1 for bidder 1. We need to write down bidder 1’s expected payoff as a function of her bid b 1 and show that this is maximized at b 1 = s 1 . We derive bidder 1’s expected payoff as a function of her bid in three steps. Rod Garratt ECON 177: Auction Theory With Experiments

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Step 1. Compute the probability that bidder 1 wins with bid b 1 . Bidder 1 wins only if her bid is higher than bidder 2’s bid, i.e., b 1 > s 2 . Since we assume that signals are uniform on [0 , 100] , this happens with probability b 1 100 . Thus, bidder 1 wins the auction with bid b 1 with probability b 1 100 . Rod Garratt ECON 177: Auction Theory With Experiments
Step 2. Compute bidder 1’s expected value of the item if she wins at bid b 1 . Remember that each bidder’s common value for the good it equal to the sum of the private signals.

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