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(Single-item) auctions. v(. ) = $5. Vincent Conitzer conitzer@cs.duke.edu. v(. ) = $3. A few different 1-item auction mechanisms. English auction: Each bid must be higher than previous bid Last bidder wins, pays last bid Japanese auction:

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Single item auctions l.jpg

(Single-item) auctions

v(

) = $5

Vincent Conitzer

conitzer@cs.duke.edu

v(

) = $3


A few different 1 item auction mechanisms l.jpg
A few different 1-item auction mechanisms

  • English auction:

    • Each bid must be higher than previous bid

    • Last bidder wins, pays last bid

  • Japanese auction:

    • Price rises, bidders drop out when price is too high

    • Last bidder wins at price of last dropout

  • Dutch auction:

    • Price drops until someone takes the item at that price

  • Sealed-bid auctions (direct revelation mechanisms):

    • Each bidder submits a bid in an envelope

    • Auctioneer opens the envelopes, highest bid wins

      • First-price sealed-bid auction: winner pays own bid

      • Second-pricesealed bid (or Vickrey) auction: winner pays second-highest bid


The vickrey auction is strategy proof l.jpg
The Vickrey auction is strategy-proof!

  • What should a bidder with value v bid?

Option 1: Win the item at price b, get utility v - b

Would like to win if and only if v - b > 0 – but bidding truthfully accomplishes this!

b = highest bid among other bidders

Option 2: Lose the item, get utility 0

0


Collusion in the vickrey auction l.jpg
Collusion in the Vickrey auction

  • Example: two colluding bidders

v1 = first colluder’s true valuation

v2 = second colluder’s true valuation

price colluder 1 would pay when colluders bid truthfully

gains to be distributed among colluders

b = highest bid among other bidders

price colluder 1 would pay if colluder 2 does not bid

0


First price sealed bid auction bne l.jpg
First-price sealed-bid auction BNE

  • Supposeevery bidder (independently) draws a valuation from [0, 1]

  • What is a Bayes-Nash equilibrium for this?

  • Say a bidder with value vi bids vi(n-1)/n

  • Claim: this is an equilibrium!

  • Proof: suppose all others use this strategy

  • For a bid b < (n-1)/n, the probability of winning is (bn/(n-1))n-1, so the expected value is (vi-b)(bn/(n-1))n-1

  • Derivative w.r.t. b is - (bn/(n-1))n-1 + (vi-b)(n-1)bn-2(n/(n-1))n-1 which should equal zero

  • Implies –b + (vi-b)(n-1) = 0, which solves to b = vi(n-1)/n


Analyzing the expected revenue of the first price and second price vickrey auctions l.jpg
Analyzing the expected revenue of the first-price and second-price (Vickrey) auctions

  • First-price auction: probability of there not being a bid higher than b is (bn/(n-1))n (for b < (n-1)/n)

    • This is the cumulative density function of the highest bid

  • Probability density function is the derivative, that is, it is nbn-1(n/(n-1))n

  • Expected value of highest bid is

    n(n/(n-1))n∫(n-1)/nbndb = (n-1)/(n+1)

  • Second-price auction: probability of there not being two bids higher than b is bn + nbn-1(1-b)

    • This is the cumulative density function of the second-highest bid

  • Probability density function is the derivative, that is, it is nbn-1 + n(n-1)bn-2(1-b) - nbn-1 = n(n-1)(bn-2 - bn-1)

  • Expected value is (n-1) – n(n-1)/(n+1) = (n-1)/(n+1)


Revenue equivalence theorem l.jpg
Revenue equivalence second-price (Vickrey) auctions theorem

  • Suppose valuations for the single item are drawn i.i.d. from a continuous distribution over [L, H] (with no “gaps”), and agents are risk-neutral

  • Then, any two auction mechanisms that

    • in equilibrium always allocate the item to the bidder with the highest valuation, and

    • give an agent with valuation L an expected utility of 0,

      will lead to the same expected revenue for the auctioneer


As an aside what if bidders are not risk neutral l.jpg
(As an aside) what if bidders are not risk-neutral? second-price (Vickrey) auctions

  • Behavior in second-price/English/Japanese does not change, but behavior in first-price/Dutch does

  • Risk averse: first price/Dutch will get higher expected revenue than second price/Japanese/English

  • Risk seeking: second price/Japanese/English will get higher expected revenue than first price/Dutch


As an aside interdependent valuations l.jpg
(As an aside) second-price (Vickrey) auctionsinterdependent valuations

  • E.g. bidding on drilling rights for an oil field

  • Each bidder i has its own geologists who do tests, based on which the bidder assesses an expected value vi of the field

  • If you win, it is probably because the other bidders’ geologists’ tests turned out worse, and the oil field is not actually worth as much as you thought

    • The so-called winner’s curse

  • Hence, bidding vi is no longer a dominant strategy in the second-price auction

  • In English and Japanese auctions, you can update your valuation based on other agents’ bids, so no longer equivalent to second-price

  • In these settings, English (or Japanese) > second-price > first-price/Dutch in terms of revenue


Redistribution cavallo 06 l.jpg
Redistribution second-price (Vickrey) auctions[Cavallo 06]

  • Suppose we are in a setting where we do not want to have high revenue

  • We want to allocate the item efficiently, but we do not actually like to take money from the agents

  • Can we redistribute some of the (Vickrey mechanism’s) revenue back to the agents without affecting the incentives?

  • To maintain strategy-proofness, agent’s redistribution payment should not depend on agent’s own bid

  • Also need to make sure that we do not redistribute more than there is

  • So: redistribute, to each agent, 1/n of the second-highest other bid


Expected revenue maximizing optimal auctions myerson 81 l.jpg
Expected-revenue maximizing (“ second-price (Vickrey) auctionsoptimal”) auctions [Myerson 81]

  • Vickrey auction does not maximize expected revenue

    • E.g. with only one bidder, better off making a take-it-or-leave-it offer (or equivalently setting a reserve price)

  • Suppose agent i draws valuation from probability density function fi (cumulative density Fi)

  • Bidder’s virtual valuationψ(vi)= vi - (1 - Fi(vi))/fi(vi)

    • Under certain conditions, this is increasing; assume this

  • The bidder with the highest virtual valuation (according to his reported valuation) wins (unless all virtual valuations are below 0, in which case nobody wins)

  • Winner pays value of lowest bid that would have made him win

  • E.g. if all bidders draw uniformly from [0, 1], Myerson auction = second-price auction with reserve price ½


Other settings l.jpg
Other settings second-price (Vickrey) auctions

  • Reverse auction: auctioneer wants to buy an item, bidders who own (a unit of) the item submit their valuations for the item, (typically) lowest bid wins

    • Application: task allocation (auctioneer wants to “buy” an agent’s services)

    • In many ways similar to normal (“forward”) auctions

  • Exchanges/double auctions: buyers and sellers both submit values (and potentially quantities)


Myerson satterthwaite impossibility 1983 l.jpg
Myerson-Satterthwaite impossibility second-price (Vickrey) auctions[1983]

  • Simple setting:

) = x

) = y

v(

v(

  • We would like a mechanism that:

    • is efficient (trade if and only if y > x),

    • is budget-balanced (seller receives what buyer pays),

    • is BNE incentive compatible, and

    • is ex-interim individually rational

  • This is impossible!