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week 9 1 COS 444 Internet Auctions: Theory and Practice Spring 2008 Ken Steiglitz ken@cs.princeton.edu

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week 92 Theory: Riley & Samuelson 81 Quick FP equilibrium with reserve: which gives us immediately: Example …

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week 93 Theory: Riley & Samuelson 81 Revenue at equilibrium: = “marginal revenue” = “virtual valuation”

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week 94 Theory: Riley & Samuelson 81 Optimal choice of reserve let v 0 = value to seller let v 0 = value to seller Total revenue = Total revenue = Differentiate wrt v * and set to zero

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week 95 Reserves The seller chooses reserve b 0 to achieve a given v *. The seller chooses reserve b 0 to achieve a given v *. In first-price and second-price auctions (but not in all the auctions in the Riley- Samuelson class) v * = b 0. In first-price and second-price auctions (but not in all the auctions in the Riley- Samuelson class) v * = b 0. Proof: there’s no incentive to bid when our value is below b 0, and an incentive to bid when our value is above b 0. Proof: there’s no incentive to bid when our value is below b 0, and an incentive to bid when our value is above b 0.

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week 96 Reserves Setting reserve in the second- and first- price increases revenue through entirely different mechanisms: Setting reserve in the second- and first- price increases revenue through entirely different mechanisms: o In first-price auctions bids are increased. o In second-price auctions it’s an equilibrium to bid truthfully, but winners are forced to pay more.

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week 98 All-pay with reserve Set E[ pay ] from Riley & Samuelson 81 = b ( v ) ! For n=2 and uniform v’s this gives For n=2 and uniform v’s this gives b( v ) = v 2 /2 + v * 2 /2 b( v ) = v 2 /2 + v * 2 /2 Setting E[ surplus at v * ] = 0 gives Setting E[ surplus at v * ] = 0 gives b( v * ) = v * 2 b( v * ) = v * 2 Also, b( v * ) = b 0 (we win only with no competition, so bid as low as possible) Also, b( v * ) = b 0 (we win only with no competition, so bid as low as possible) Therefore, b 0 = v * 2 (not v * as before)

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week 910 Loser weeps auction, n=2 Winner gets item for free, loser pays his bid! Gives us reserve in terms of v * (evaluate at v * ): b 0 = v * 2 / (1-v * ) … using b( v * ) = b 0 b 0 = v * 2 / (1-v * ) … using b( v * ) = b 0 E[pay] of R&S 81 then leads directly to equilibrium

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week 912 Santa Claus auction, n=2 Winner pays her bid Winner pays her bid Idea: give people their expected surplus and try to arrange things so bidding truthfully is an equilibrium. Idea: give people their expected surplus and try to arrange things so bidding truthfully is an equilibrium. Give people Give people Prove: truthful bidding is a SBNE … Prove: truthful bidding is a SBNE …

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week 913 Santa Claus auction, con’t Suppose 2 bids truthfully. Then ∂∕∂b = 0 shows b=v

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week 914 Matching auction: not in A rs Bidder 1 may tender an offer on a house, Bidder 1 may tender an offer on a house, b 1 ≥ b 0 = reserve b 1 ≥ b 0 = reserve Bidder 2 currently leases house and has the option of matching b 1 and buying at that price. If bidder 1 doesn’t bid, bidder 2 can buy at b 0 if he wants Bidder 2 currently leases house and has the option of matching b 1 and buying at that price. If bidder 1 doesn’t bid, bidder 2 can buy at b 0 if he wants

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week 915 Matching auction, con’t To compare with optimal auctions, choose v * = ½ To compare with optimal auctions, choose v * = ½ Bidder 2’s best strategy: Match b 1 iff Bidder 2’s best strategy: Match b 1 iff v 2 ≥ b 1 ; else bid ½ iff v 2 ≥ ½ v 2 ≥ b 1 ; else bid ½ iff v 2 ≥ ½ Bidder should choose b 1 ≥ ½ so as to maximize expected surplus. Bidder should choose b 1 ≥ ½ so as to maximize expected surplus. This turns out to be b 1 = ½ … This turns out to be b 1 = ½ …

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week 916 Matching auction, con’t Choose v * = ½ for comparison Choose v * = ½ for comparison Bidder 1 tries to max Bidder 1 tries to max (v 1 -b 1 )·{prob. 2 chooses not to match} (v 1 -b 1 )·{prob. 2 chooses not to match} = (v 1 -b 1 )·b 1 = (v 1 -b 1 )·b 1 b 1 = 0 if v 1 < ½ b 1 = 0 if v 1 < ½ = ½ if v 1 ≥ ½ = ½ if v 1 ≥ ½

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week 917 Matching auction, con’t Notice: Notice: When ½ < v 2 < v 1, bibber 2 gets the item, but values it less than bidder 1 inefficient! When ½ < v 2 < v 1, bibber 2 gets the item, but values it less than bidder 1 inefficient! E[revenue to seller] turns out to be 9/24 (optimal in A rs is 10/24; optimal with no reserve is 8/24) E[revenue to seller] turns out to be 9/24 (optimal in A rs is 10/24; optimal with no reserve is 8/24) Why is this auction not in A rs ? Why is this auction not in A rs ?

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Risk-averse bidders

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week 919 Revenue ranking with risk aversion Result: Suppose bidders’ utility is concave. Then with the assumptions of A rs, Result: Suppose bidders’ utility is concave. Then with the assumptions of A rs, R FP ≥ R SP R FP ≥ R SP Proof: Let γ be the equilibrium bidding function in the risk-averse case, and β in the risk-neutral case. Proof: Let γ be the equilibrium bidding function in the risk-averse case, and β in the risk-neutral case.

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week 920 Revenue ranking, con’t In first-price auction, E[surplus] = W (z )·u (x − γ (z ) ) E[surplus] = W (z )·u (x − γ (z ) ) where we bid as if value = z, W(z) is prob. of winning, … etc.

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week 921 Constant relative risk aversion (CRRA) Defined by utility u(t) = t ρ, ρ < 1 u(t) = t ρ, ρ < 1 First-price equilibrium can be found by usual methods ( u/u’ = t/ρ helps): ( u/u’ = t/ρ helps): Very similar to risk-neutral form. As if there were (n-1)/ρ instead of (n-1) rivals!

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