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Internet Economics כלכלת האינטרנט. Class 4 – Optimal Auctions. Golden balls. Let’s warm up with some real-game theory: Reality games and game theory… Scene 1 Scene 2. Last week (1/4). How to sell a single item to n bidders? Seller doesn’t know how much bidders are willing to pay

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Internet economics

Internet Economicsכלכלת האינטרנט

Class 4 – Optimal Auctions

Golden balls
Golden balls

Let’s warm up with some real-game theory:

Reality games and game theory…

  • Scene 1

  • Scene 2

Last week 1 4
Last week (1/4)

  • How to sell a single item to n bidders?

  • Seller doesn’t know how much bidders are willing to pay

    • vi is the value of bidder i for the item.

  • Getting this information via an auction.

  • Game with incomplete information.

Last week 2 4
Last week (2/4)

  • The English Auction:

  • Price starts at 0

  • Price increases until only one

  • bidder is left.

  • Vickrey (2nd price) auction:

  • Bidders send bids.

  • Highest bid wins, pays 2nd highest bid.

  • Private value model: each person has a privately known value for the item.

  • We saw: the two auctions are equivalent in the private value model.

  • Auctions are efficient:dominant strategy for each player: truthfulness.

Last week 3 4
Last week (3/4)

  • The Dutch Auction:

  • Price starts at max-price.

  • Price drops until a bidder agrees to buy.

  • 1st-price auction:

  • Bidders send bids.

  • Highest bid wins, pays his bid.

  • Dutch auctions and 1st price auctions are strategically equivalent. (asynchronous vs simple & fast)

  • No dominant strategies. (tradeoff: chance of winning, payment upon winning.)

  • Analysis in a Bayesian model:

    • Values are randomly drawn from a probability distribution.

  • Strategy: a function. “What is my bid given my value?”

Last week 4 4
Last week (4/4)

  • We considered the simplest Bayesian model:

    • 2 bidders.

    • Values drawn uniformly from [0,1].


      In a 1st-price auction, it is a (Bayesian) Nash

      equilibrium when all bidders bid

  • An auction is efficient, if in (Bayesian) Nash equilibrium the bidder with the highest value always wins.

    • 1st price is efficient!

Remark efficiency
Remark: Efficiency

  • We saw that both 2nd –price and 1st –price auctions are efficient.

  • What is efficiency (social welfare)?The total utility of the participants in the game (including the seller).

    For each bidder: vi – pi

    For the seller: (assuming it has 0 value for the item)

  • Summing:

Optimal auctions
Optimal auctions

  • Usually the term optimal auctions stands for revenue maximization.

  • What is maximal revenue?

    • We can always charge the winner his value.

  • Maximal revenue: optimal expected revenue in equilibrium.

    • Assuming a probability distribution on the values.

    • Over all the possible mechanisms.

    • Under individual-rationality constraints (later).

Example spectrum auctions
Example: Spectrum auctions

  • One of the main triggers to auction theory.

  • FCC in the US sells spectrum, mainly for cellular networks.

  • Improved auctions since the 90’s increased efficiency + revenue considerably.

  • Complicated (“combinatorial”) auction, in many countries.

    • (more details further in the course)

New zealand spectrum auctions
New Zealand Spectrum Auctions

  • A Vickrey (2nd price) auction was run in New Zealand to sale a bunch of auctions. (In 1990)

  • Winning bid: $100000

    Second highest: $6 (!!!!)

    Essentially zero revenue.

  • NZ Returned to 1st price method the year after.

    • After that, went to a more complicated auction (in few weeks).

  • Was it avoidable?

1 st or 2 nd price
1st or 2nd price?

  • Assume 2 bidders, uniform distribution on [0,1].

  • Facts: (1) E[ max(v1,v2) ] = 2/3 (2) E[ min(v1,v2) ] = 1/3 (in general, k’th highest value of n is (n+1-k)/n+1)

  • Revenue in 2nd price:

  • Bidders bid truthfully.

  • Revenue is 2nd highest bid.

  • Expected revenue = 1/3

  • Revenue in 1st price:

  • bidders bid vi/2.

  • Revenue is the highest bid.

  • Expected revenue = E[ max(v1/2,v2/2) ]

  • = ½ E[ max(v1,v2)]

  • = ½ × 2/3 = 1/3

Revenue equivalence theorem
Revenue equivalence theorem

  • No coincidence!

    • Somewhat unintuitively, revenue depends only on the way the winner is chosen, not on payments.

  • Auction for a single good.

  • Values are independently drawn from distribution F (increasing)

    Theorem (“revenue equivalence”):

    All auctions where:

    • the good is allocated to the bidder with the highest value

    • Bidders can guarantee a utility of 0 by bidding 0.

      yield the same revenue!

      (more general: two auction with the same allocation rule yield the same revenue)

Remark individual rationality
Remark: Individual rationality

  • The following mechanism gains lots of revenue:

    • Charge all players $10000000

  • Bidder will clearly not participate.

  • We thus have individual-rationality (or participation) constraints on mechanisms:bidders gain positive utility in equilibrium .

    • This is the reason for condition 2 in the theorem.

All pay auction 1 3
All-pay auction (1/3)

  • Rules:

    • Sealed bid

    • Highest bid wins

    • Everyone pay their bid

  • Equilibrium with the uniform distribution: b(v)=

  • Does it achieve more or less revenue?

    • Note: Bidders shade their bids as the competition increases.

All pay auction 2 3
All-pay auction (2/3)

  • expected payment per each player: herbid.

  • Each bidder bids

  • Expected payment for each bidder:

  • Revenue: from n bidders

  • Revenue equivalence!

All pay auction 3 3
All-pay auction (3/3)

  • Examples:

    • crowdsourcing over the internet:

      • First person to complete a task for me gets a reward.

      • A group of people invest time in the task. (=payment)

      • Only the winner gets the reward.

    • Advertising auction:

      • Collect suggestion for campaigns, choose a winner.

      • All advertiser incur cost of preparing the campaign.

      • Only one wins.

    • Lobbying

    • War of attrition

      • Animals invest (b1,b2) in fighting.

What did we see so far
What did we see so far

  • 2nd-price, 1st-price, all pay: all obtain the same seller revenue.

  • Revenue equivalence theorem:Auctions with the same allocation decisions earn the same expected seller revenue in equilibrium.

    • Constraint: individual rationality (participation constraint)

  • Many assumptions:

    • statistical independence,

    • risk neutrality,

    • no externalities,

    • private values,

Next can we get better revenue
Next: Can we get better revenue?

  • Can we achieve better revenue than the 2nd-price/1st price?

  • If so, we must sacrifice efficiency.

    • All efficient auction have the same revenue….

  • How?

    • Think about the New-Zealand case.

Vickrey with reserve price
Vickrey with Reserve Price

  • Seller publishes a minimum (“reserve”) price R.

  • Each bidder writes his bid in a sealed envelope.

  • The seller:

    • Collects bids

    • Open envelopes.

  • Winner: Bidder with the highest bid, if bid is above R. Otherwise, no one winsPayment: winner pays max{2nd highest bid, R}

Yes. For bidders, exactly like an extra bidder bidding R.

Still Truthful?

Can we get better revenue
Can we get better revenue?


  • Let’s have another look at 2nd price auctions:

2 wins


1 wins


1 wins and pays x

(his lowest winning bid)






Can we get better revenue1
Can we get better revenue?


  • I will show that some reserve price improve revenue.

Revenue increased

2 wins


1 wins


Revenue increased





Revenue loss here (efficiency loss too)


Can we get better revenue2
Can we get better revenue?


We will be here with probability R(1-R)


  • Gain is at least 2R(1-R) R/2 = R2-R3

  • Loss is at most R2 R = R3

2 wins

We will be here with probability R2

1 wins

Average loss is R/2

Loss is always at mostR





  • When R2-2R3>0, reserve price of R is beneficial.(for example, R=1/4)

Reservation price
Reservation price

  • Can increase revenue!

  • 2 bidders, uniform distribution: optimal reserve price = ½

    • Revenue: 5/12=0.412 > 1/3

  • n bidders, uniform distribution: optimal reserve price = ½

    Theorem: (Myerson ‘81)Vickrey auction with a reserve price maximizes revenue.

    • For a general family of distributions (uniform, exponential, normal, and many others).

    • Reserve price is independent of n.

(Nobel prize, 2007)

Reservation price1
Reservation price

Let’s see another example:How do you sell one item to one bidder?

  • Assume his value is drawn uniformly from [0,1].

  • Optimal way: reserve price.

    • Take-it-or-leave-it-offer.

  • Let’s find the optimal reserve price:E[revenue] = ( 1-F(R) ) × R = (1-R) ×R

     R=1/2

  • Surprising? No. We said that the optimal reserve price does not depend on n.

  • Probability that the buyer will accept the price

    The payment for the seller

    Back to new zealand
    Back to New Zealand

    • Recall: Vickrey auction.Highest bid: $100000. Revenue: $6.

    • Two things to learn:

      • Seller can never get the whole pie.

        • “information rent” for the buyers.

      • Reserve price can help.

        • But what if R=$50000 and highest bid was $45000?

    • Of the unattractive properties of Vickrey Auctions:

      • Low revenue despite high bids.

      • 1st-price may earn same revenue, but no explanation needed…

    Summary efficiency vs revenue
    Summary: Efficiency vs. revenue

    Positive or negative correlation ?

    • Always: Revenue ≤ efficiency

      • Due to Individual rationality.

      • More efficiency makes the pie larger!

    • However, for optimal revenue one needs to sacrifice some efficiency.

    • Consider two competing sellers: one optimizing revenue the other optimizing efficiency.

      • Who will have a higher market share?

      • In the longer terms, two objectives are combined.

    Next week
    Next week

    • Designing dominant-strategy mechanisms for more general environments.

      • the magic of the VCG mechanism.