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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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Presentation Transcript
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].

Then:

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?

1

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

2 wins

v2

1 wins

x

1 wins and pays x

(his lowest winning bid)

0

x

0

v1

1

can we get better revenue1
Can we get better revenue?

1

  • I will show that some reserve price improve revenue.

Revenue increased

2 wins

v2

1 wins

R

Revenue increased

0

0

v1

1

Revenue loss here (efficiency loss too)

R

can we get better revenue2
Can we get better revenue?

1

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

v2

  • 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

0

v1

0

1

  • 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.
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