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COS 444 Internet Auctions: Theory and Practice

COS 444 Internet Auctions: Theory and Practice. Spring 2008 Ken Steiglitz ken@cs.princeton.edu. Theory: Riley & Samuelson 81. Quick FP equilibrium with reserve:. which gives us immediately:. Example …. Theory: Riley & Samuelson 81. Revenue at equilibrium:

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COS 444 Internet Auctions: Theory and Practice

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

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

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

  4. Theory: Riley & Samuelson 81 Optimal choice of reserve let v0 = value to seller Total revenue = Differentiate wrt v* and set to zero 

  5. Reserves • The seller chooses reserve b0 to achieve a given v* . • In first-price and second-price auctions (but not in all the auctions in the Riley-Samuelson class) v* = b0 . Proof: there’s no incentive to bid when our value is below b0 , and an incentive to bid when our value is above b0 .

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

  7. All-pay with reserve Set E[ pay ] from Riley & Samuelson 81 = b ( v ) ! • For n=2 and uniform v’s this gives b( v ) = v 2/2 + v*2/2 • Setting E[ surplus at v* ] = 0 gives b( v* )= v*2 • Also, b( v* )= b0 (we win only with no competition, so bid as low as possible) Therefore, b0 = v*2(not v*as before)

  8. Loser weeps auction, n=2 Winner gets item for free, loser pays his bid! Gives us reserve in terms of v* (evaluate at v*): b0 = v*2 / (1-v*) … using b( v* )= b0 E[pay] of R&S 81 then leads directly to equilibrium

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

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

  11. Matching auction: not in Ars • Bidder 1 may tender an offer on a house, b1 ≥ b0 = reserve • Bidder 2 currently leases house and has the option of matching b1 and buying at that price. If bidder 1 doesn’t bid, bidder 2 can buy at b0 if he wants

  12. Matching auction, con’t • To compare with optimal auctions, choose v*= ½ • Bidder 2’s best strategy: Match b1iff v2 ≥ b1 ; else bid ½ iff v2 ≥ ½ • Bidder should choose b1 ≥ ½ so as to maximize expected surplus. This turns out to be b1 = ½ …

  13. Matching auction, con’t • Choose v* = ½ for comparison Bidder 1 tries to max (v1-b1 )·{prob. 2 chooses not to match} = (v1-b1 )·b1  b1 = 0 if v1 < ½ = ½ if v1 ≥½

  14. Matching auction, con’t Notice: When ½ < v2 < v1 , 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 Ars is 10/24; optimal with no reserve is 8/24) Why is this auction not in Ars ?

  15. Risk-averse bidders

  16. Revenue ranking with risk aversion Result: Suppose bidders’ utility is concave. Then with the assumptions of Ars , RFP ≥ RSP Proof: Let γbe the equilibrium bidding function in the risk-averse case, and β in the risk-neutral case.

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

  18. Constant relative risk aversion (CRRA) Defined by utility u(t) = t ρ, ρ<1 First-price equilibrium can be found by usual methods ( 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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