Single item auctions
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(Single-item) auctions - PowerPoint PPT Presentation

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(Single-item) auctions. v(. ) = $5. Vincent Conitzer 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


) = $5

Vincent Conitzer


) = $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


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


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

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(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

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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)

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Myerson-Satterthwaite impossibility second-price (Vickrey) auctions[1983]

  • Simple setting:

) = x

) = y



  • 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!