Auction Theory

Class 2 – Revenue equivalence

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This class: revenue• Revenue in auctions

– Connection to order statistics

• The revelation principle

• The revenue equivalence theorem– Example: all-pay auctions.

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English vs. Vickrey

• 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.

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.

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Dutch vs. 1st-price

• 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?”

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.

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Bayes-Nash eq. in 1st-price auctions• We considered the simplest Bayesian model:

– n 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 (Bayes) Nash equilibrium the bidder with the highest value always wins.– 1st price is efficient!

ivn

n 1

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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).

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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)

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New Zealand Spectrum Auctions• A Vickrey (2nd price) auction was run in New Zealand to

sale a bunch of auctions. (In 1990)

• Winning bid: $100000Second 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?

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Auctions with uniform distributionsA simple Bayesian auction model: • 2 buyers• Values are between 0 and 1.• Values are distributed uniformly on [0,1]

What is the expected revenue gained by 2nd-price and 1st price auctions?

Revenue in 2nd-price auctions• In 2nd-price auction, the payment is the minimum of

the two values.– E[ revenue] = E[ min{x,y} ]

• Claim: when x,y ~ U[0,1] we have E[ min{x,y} ]=1/3

Revenue in 2nd-price auctions• Proof:

– assume that v1=x. Then, the expected revenue is:

• We can now compute the expected revenue (expectation over all possible x):

xxx

x 12 2

2xx

1

0

32

62

xx

1

0

21

0

2

)2

()()2

(y}]E[min{x, dxx

xdxxfx

x

0 1x

3

1

Order statisticsLet v1,…,vn be n random variables.

– The highest realization is called the 1st-order statistic.– The second highest is the called 2nd-order statistic.– ….– The smallest is the nth-order statistic.

Example: the uniform distribution, 2 samples.– The expected 1st-order statistic: 2/3

• In auctions: expected efficiency

– The expected 2nd-order statistic: 1/3• In auctions: expected revenue

Expected order statistics

0 1

0 1

0 1

1/2

1/3 2/3

2/41/4 3/4

One sample

Two samples

Three samples

In general, for the uniform distribution with n samples: • k’th order statistic of n variables is (n+1-k)/n+1)• 1st-order statistic: n/n+1

Revenue in 1st-price auctions• We still assume 2 bidders, uniform distribution

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

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Same revenue as in 2nd-price auctions.

1st vs. 2nd priceRevenue in 2nd price:• Bidders bid truthfully.• Revenue is 2nd highest bid:

Revenue in 1st price:• bidders bid• Expected revenue is

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1

1E[revenue]

n

n

nvvn

1-n,...,

n

1-nmaxE 1

nvv ,...,maxEn

1-n1

1n

n

n

1-n

1n

1-n

What happened? Coincidence?

This class• Revenue in auctions

– Connection to order statistics

• The revelation principle

• The revenue equivalence theorem– Example: all-pay auctions.

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ImplementationOur general goal: given an objective (for example,

maximize efficiency or revenue), construct an auction that achieves this goal in an equilibrium.– "Implementation”– Equilibrium concept: Bayes-Nash

For example: when our goal is maximal efficiency– 2nd-price auctions maximize efficiency in a Bayes-Nash

equilibrium• Even stronger solution: truthfulness (in dominant strategies).

– 1st price auctions also achieve this goal.• Not truthful, no dominant strategies.

– Many other auctions are efficient (e.g., all-pay auctions).

TerminologyDirect-revelation mechanism: player are asked to report

their true value.– Non direct revelation: English and Dutch auction, most

iterative auctions, concise menu of actions.

– Concepts relates to the message space in the auction.

Truthful mechanisms: direct-revelation mechanisms where revealing the truth is (a Bayes Nash) equilibrium.– Other solution concepts may apply.

– Alternative term: Incentive Compatibility.

• What’s so special about revealing the truth? – Maybe better results can be obtained when people report half

their value, or any other strategy?

The revelation principle• Problem: the space of possible mechanisms is often too

large.

• A helpful insight: we can actually focus our attention to truthful (direct revelation) mechanisms.– This will simplify the analysis considerably.

• “The revelation principle”– “every outcome can be achieved by truthful mechanism”

• One of the simplest, yet trickiest, concepts in auction theory.

Theorem (“The Revelation Principle”):

Consider an auction where the profile of strategies s1,…,sn is a Bayes-Nash equilibrium.

Then, there exists a truthful mechanism with exactly the same allocation and payments (“payoff equivalent”).

The revelation principle

Recall: truthful = direct revelation + truthful Bayes-Nash equilibrium.

• Basic idea: we can simulate any mechanism via a truthful mechanism which is payoff equivalent.

The revelation principle• Proof (trivial): The original mechanism:

v1 v1s1(v1)

Auction protocol

Allocation (winners)

payments

v2 v2 s2(v2)

v3 v3 s3(v3)

v4v4 s4(v4)

s1(v1)

s2(v2)

s3(v3)

s4(v4)

Bidders Auction mechanism

The revelation principle• Proof (trivial): A direct-revelation mechanism:

v1 v1s1(v1)

Auction protocol

Allocation (winners)

payments

v2 v2 s2(v2)

v3 v3 s3(v3)

v4v4 s4(v4)

v1

v2

v3

v4

Bidders reports their true types, The auction simulates their equilibrium strategies.

Equilibrium is straightforward: if a bidder had a profitable deviation here, he would have one in the original mechanism.

The revelation principle• Example:

– In 1st-price auctions with the uniform distribution: bidders would bid truthfully and the mechanism will “change” their bids to be

– In English auctions (non direct revelation):people will bid truthfully, and the mechanism will raise hands according to their strategy in the auction.

• Bottom line: Due to the revelation principle, from now on we will concentrate on truthful mechanisms.

ivn

n 1

This class• Revenue in auctions

– Connection to order statistics

• The revelation principle

• The revenue equivalence theorem– Example: all-pay auctions.

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Revenue equivalence• We saw examples where the revenue in 2nd-price and

1st-price auctions is the same.

• Can we have a general theorem?

• Yes.

Informally:What matters is the allocation. Auctions with the same allocation have the same revenue.

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Revenue Equivalence TheoremAssumptions:

– vi‘s are drawn independently from some F on [a,b]

– F is continuous and strictly increasing

– Bidders are risk neutral

Theorem (The Revenue Equivalence Theorem):

Consider two auction such that:

1. (same allocation) When player i bids v his probability to win is the same in the two auctions (for all i and v) in equilibrium.

2. (normalization) If a player bids a (the lowest possible value) he will pay the same amount in both auctions.

Then, in equilibrium, the two auctions earn the same revenue.

Proof• Idea:

we will start from the incentive-compatibility (truthfulness) constraints.

We will show that the allocation function of the auction actually determines the payment for each player.– If the same allocation function is achieved in equilibrium, then the

expected payment of each player must be the same.

• Note: Due to the revelation principle, we will look at truthful auctions.

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Proof• Consider some auction protocol A, and a bidder i.• Notations: in the auction A,

– Qi(v) = the probability that bidder i wins when he bids v.

– pi(v) = the expected payment of bidder i when he bids v.

– ui(v) = the expected surplus (utility) of player i when he bids v and his true value is v.

ui(v) = Qi(v) v - pi(v)

• In a truthful equilibrium: i gains higher surplus when bidding his true value v than some value v’.– Qi(v) v - pi(v) ≥ Qi(v’) v - pi(v’)

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=ui(v’)+ ( v – v’) Qi(v’)=ui(v)

We get: truthfulness ui(v) ≥ ui(v’)+ ( v – v’) Qi(v’)

Proof• We get: truthfulness

ui(v) ≥ ui(v’)+ ( v – v’) Qi(v’) or

• Similarly, since a bidder with true value v’ will not prefer bidding v and thus

ui(v’) ≥ ui(v)+ ( v’ – v) Qi(v) or

Let dv = v-v’

Taking dv 0 we get:

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)'(– v’ v

)(v’u- (v)u ii vQi

)(– v’ v

)(v’u- (v)u ii vQi

)'(dv

)(v’u- dv)(v'u)'( ii dvvQvQ ii

)'(dv'

)(v'du i vQi

Proof• We saw:

• We know:

• And conclude:

• Of course:

• Interpretation: expected revenue, in equilibrium, depends only on the allocation.– same allocation same revenue (as long as Q() and ui(a) are the same).

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)'(dv'

)(v'du i vQi dvvQv

a

i'

ii )((a)u)(v'u

)'()'(')(v'u i vpvQv ii )(v'u)'(')'( i vQvvp ii

dvvQvQvvpv

a

iii '

)()'(')'(

integrating

n

iii vprevenueE

1

)(

Assume ui(a)=0

dvvQv

a

i'

i )()(v'u

Picture

31

)(vQi

v'v

)'(vQi

dvvQvQvvpv

a

iii '

)()'(')'(

a

Example: 2 players, uniform dist.

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dvvQvQvvpv

a

iii '

)()'(')'(

)(vQi

v

Q1(v)= v

pi(1/2)=

1/2

pi(v)=

6

1

62

1

0

31

0

2

v

dvv

The expected revenue from bidder 1:

For 2 bidders: E[revenue]=1/6+1/6=1/3

1/2*1/2*1/2

v*v*1/2=v2/2

1/2

Revenue equivalence theorem• No coincidence!

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

– Since 2nd-price auctions and 1st-price auctions have the same (efficient) allocation, they will earn the same revenue!

• One of the most striking results in mechanism design

• Applies in other, more general setting.

• Lesson: when designing auctions, focus on the allocation, not on tweaking the prices.

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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.

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This class• Revenue in auctions

– Connection to order statistics

• The revelation principle

• The revenue equivalence theorem– Example: all-pay auctions.

35

Example: All-pay auction (1/3)• Rules:

– Sealed bid– Highest bid wins– Everyone pay their bid

• Claim: Equilibrium with the uniform distribution:

b(v)=

• Does it achieve more or less revenue?– Note: Bidders shade their bids as the competition

increases.36

nvn

n 1

All-pay auction (2/3)• expected payment per each player: her bid.

• Each bidder bids • Expected payment for each bidder:

• Revenue: from n bidders

• Revenue equivalence!37

1

111

01

1111 1

0

1

0 n

n

nn

nv

n

ndvnv

n

ndvnv

n

n

1

1E[revenue]

n

n

nvn

nvb

1)(

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.

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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,– … 39

Next topic• Optimal revenue:

which auctions achieve the highest revenue?

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