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COS 444 COS 444 Internet Auctions:Internet Auctions:Theory and PracticeTheory and Practice

Spring 2008

Ken Steiglitz ken@cs.princeton.edu

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181

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Quick FP equilibrium with reserve:

which gives us immediately:

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Example…

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181

Revenue at equilibrium:Revenue at equilibrium:

= = “marginal revenue”“marginal revenue” = = “virtual “virtual valuation”valuation”

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181Optimal choice of reserveOptimal choice of reserve

let let vv00 = value to seller = value to seller

Total revenue = Total revenue =

Differentiate wrt Differentiate wrt vv* * and set to zero and set to zero

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ReservesReserves

The seller chooses reserve The seller chooses reserve bb00 to to achieve a given achieve a given vv** . .

In first-price and second-price auctions In first-price and second-price auctions (but not in all the auctions in the Riley-(but not in all the auctions in the Riley-Samuelson class) Samuelson class) vv** = b = b00 . .

Proof: there’s no incentive to bid when Proof: there’s no incentive to bid when our value is below our value is below bb00 , and an , and an incentive to bid when our value is incentive to bid when our value is above above bb0 .0 .

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ReservesReserves Setting reserve in the second- and Setting reserve in the second- and

first-price increases revenue through first-price increases revenue through entirely different mechanisms: entirely different mechanisms:

o In first-price auctions bids are In first-price auctions bids are increased.increased.

o In second-price auctions it’s an In second-price auctions it’s an equilibrium to bid truthfully, but equilibrium to bid truthfully, but winners are forced to pay more.winners are forced to pay more.

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All-pay with reserveAll-pay with reserve

Set E[ pay ] from Riley & Samuelson 81 Set E[ pay ] from Riley & Samuelson 81 = = b ( v ) b ( v ) ! !

• For For n=2 n=2 and uniformand uniform v’s v’s this gives this gives b( v ) = v b( v ) = v 22/2 + v/2 + v**

22/2/2 • Setting E[ surplus at Setting E[ surplus at vv* * ] = 0 gives] = 0 gives b( vb( v* * )) = = vv**

22

• Also, Also, b( vb( v* * )) = b= b00 (we win only with no (we win only with no competition, so bid as low as possible)competition, so bid as low as possible)

Therefore, Therefore, bb0 0 = v = v**22 (not (not vv** as before)as before)

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Loser weeps auction, Loser weeps auction, n=2n=2

Winner gets item for free, loser pays his bid!Winner gets item for free, loser pays his bid!

Gives us reserve in terms of vGives us reserve in terms of v** (evaluate at (evaluate at vv** ):):

bb0 0 = v = v**22 / (1-v / (1-v**)) … using … using b( vb( v* * )) = b= b00

E[pay] of R&S 81 then leads directly to E[pay] of R&S 81 then leads directly to equilibriumequilibrium

))(1)(()(E[pay]E[value] vFvbvvF

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Santa Claus auction, Santa Claus auction, n=2n=2

Winner pays her bidWinner pays her bid Idea: give people their expected surplus Idea: give people their expected surplus

and try to arrange things so bidding and try to arrange things so bidding truthfully is an equilibrium.truthfully is an equilibrium.

Give peopleGive people

Prove: truthful bidding is a SBNE …Prove: truthful bidding is a SBNE …

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Santa Claus auction, Santa Claus auction, con’tcon’t

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Suppose 2 bids truthfully. Then

∂∕∂b = 0 shows b=v

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Matching auction: not in Matching auction: not in AArsrs

• Bidder 1 may tender an offer on a Bidder 1 may tender an offer on a house,house,

bb11 ≥ ≥ bb0 0 = reserve= reserve

• Bidder 2 currently leases house and Bidder 2 currently leases house and has the option of matching has the option of matching bb11 and and buying at that price. If bidder 1 buying at that price. If bidder 1 doesn’t bid, bidder 2 can buy at doesn’t bid, bidder 2 can buy at bb00 if if he wantshe wants

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Matching auction, con’tMatching auction, con’t

• To compare with optimal auctions, To compare with optimal auctions, choose choose vv** = ½= ½

• Bidder 2’s best strategy: Match Bidder 2’s best strategy: Match bb11 iffiff

vv2 2 ≥ ≥ bb1 1 ; else bid ½ iff ; else bid ½ iff vv2 2 ≥ ½≥ ½

• Bidder should choose Bidder should choose bb11 ≥ ½ so as to ≥ ½ so as to maximize expected surplus.maximize expected surplus.

This turns out to be This turns out to be bb11 = ½ … = ½ …

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Matching auction, con’tMatching auction, con’t

• Choose vChoose v** = ½ for comparison = ½ for comparison

Bidder 1 tries to maxBidder 1 tries to max

((vv11-b-b1 1 )·{prob. 2 chooses not to )·{prob. 2 chooses not to match} match}

= (= (vv11-b-b1 1 )·)·bb1 1

bb1 1 == 0 0 ifif v v1 1 << ½ ½

= = ½ ½ ifif v v1 1 ≥≥ ½ ½

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Matching auction, con’tMatching auction, con’t

Notice:Notice:

When ½ < vWhen ½ < v22 < v < v11 , bibber 2 gets the , bibber 2 gets the item, but values it less than bidder 1 item, but values it less than bidder 1 inefficient!inefficient!

E[revenue to seller] turns out to be E[revenue to seller] turns out to be 9/24 (optimal in A9/24 (optimal in Arsrs is 10/24; optimal is 10/24; optimal with no reserve is 8/24)with no reserve is 8/24)

Why is this auction not in AWhy is this auction not in Arsrs ? ?

Risk-averse Risk-averse biddersbidders

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Revenue ranking with risk Revenue ranking with risk aversionaversion

Result:Result: Suppose bidders’ utility is Suppose bidders’ utility is concave. Then with the concave. Then with the assumptions of Aassumptions of Ars , rs ,

RRFP FP ≥ R≥ RSPSP

Proof: Let Proof: Let γγ be the equilibrium be the equilibrium bidding function in the risk-bidding function in the risk-averse case, and averse case, and ββ in the risk- in the risk-neutral case.neutral case.

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Revenue ranking, con’tRevenue ranking, con’t

In first-price auction,In first-price auction,

E[surplus] = E[surplus] = W W ((z z )·)·u u ((x − x − γγ ((z z ) )) ) wherewhere we bid as if value =we bid as if value = z , W(z) z , W(z) is prob. of winning,is prob. of winning, … etc.… etc.

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Constant relative risk aversion Constant relative risk aversion (CRRA)(CRRA)

Defined by utilityDefined by utility u(t) = t u(t) = t ρρ , , ρρ << 11First-price equilibrium can be found by usual First-price equilibrium can be found by usual

methodsmethods ( ( u/u’ = t/u/u’ = t/ρρ helps): helps):

Very similar to risk-neutral form. Very similar to risk-neutral form. As if there were As if there were (n-1)/(n-1)/ρρ instead of (n-1) rivals! instead of (n-1) rivals!

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