An introduction to Mechanism design: Single Item...

Mechanism designAuctions:Context and Definitions

Single item auctionsAnalyzing auctions

Revenue in auctions

An introduction to Mechanism design:Single Item Auctions

Maria Serna

Fall 2017

AGT-MIRI Single item auctions

Revenue in auctions

1 Mechanism design

2 Auctions:Context and Definitions

3 Single item auctions

4 Analyzing auctions

5 Revenue in auctions

Revenue in auctions

Mechanism design

Players have hidden information and try to manipulate theworld

Mediator wants to avoid some type of social misbehavior

An example: resource sharing

How to split a pizza among two kids so that they do not envyeach other?

mediator has to design an algorithm (mechanism) to split thepizza and to allocate the two pieces.Ask one kid to split the pizza and let the other choose theportion

This mechanism provides an envy-free splitting

Revenue in auctions

Mechanism design

Revenue in auctions

Mechanism design

mediator has to design an algorithm (mechanism) to split thepizza and to allocate the two pieces.

Ask one kid to split the pizza and let the other choose theportion

Revenue in auctions

Mechanism design

Revenue in auctions

Mechanism design

Revenue in auctions

Mechanism design

One way to design mechanism combines prizing withallocations.

We will focus on the study of some auctions:How to sell items to potential buyers with private valuations.

Objective: truth-telling

Revenue in auctions

Mechanism design

Revenue in auctions

Mechanism design

Revenue in auctions

Mechanism design

Revenue in auctions

PricesAuctions

1 Mechanism design

Revenue in auctions

PricesAuctions

Prices

What is the right price for objects?

The government wants to sell a big building.

Buyer 1: willing to pay 200 Billion.Buyer 2: willing to pay 100 Billion.Buyer 3: does not need the building, but wants to buy it if hecan resale it with a profit.

But, all those values are private, known only by the buyer.

how to get the building worth?

Revenue in auctions

PricesAuctions

Prices

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PricesAuctions

Prices

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PricesAuctions

Prices

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PricesAuctions

Prices

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PricesAuctions

Prices

Even when the true preferences of the buyers are knownit is not clear how to price the objects and how to allocatethem.

Buyers might lie and manipulate to get better prices and/orbetter allocation.

How can the true preferences be revealed?

At which cost?

Revenue in auctions

PricesAuctions

Prices

At which cost?

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PricesAuctions

Prices

At which cost?

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PricesAuctions

Prices

At which cost?

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PricesAuctions

Auction theory

Auction theory is a sub-field of Mechanism Design.

Aim: Design and analyze the rules and properties of anauction.

Goal: Design an auction so that in equilibrium we get theresults we want.

As in Game theory we rely on rationality.

Revenue in auctions

PricesAuctions

Auction theory

Revenue in auctions

PricesAuctions

Auction theory

Design an auction so that in equilibrium we get theresults we want.

Revenue in auctions

PricesAuctions

Auction theory

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PricesAuctions

Auction theory

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PricesAuctions

What is an Auction?

An auction is a mechanism to allocate resources among agroup of bidders.

An auction model includes three major parts:

The set of possible resource allocations.The number (or portion) of goods of each type including legalor other restrictions on how the goods may be allocated.Rules for bidding and clearing.A procedure to determine who wins what (allocation) and howmuch pays (payment) on the basis of the received information.

Revenue in auctions

PricesAuctions

What is an Auction?

Revenue in auctions

PricesAuctions

What is an Auction?

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PricesAuctions

What is an Auction?

The set of possible resource allocations.

The number (or portion) of goods of each type including legalor other restrictions on how the goods may be allocated.Rules for bidding and clearing.A procedure to determine who wins what (allocation) and howmuch pays (payment) on the basis of the received information.

Revenue in auctions

PricesAuctions

What is an Auction?

The set of possible resource allocations.The number (or portion) of goods of each type including legalor other restrictions on how the goods may be allocated.

Rules for bidding and clearing.A procedure to determine who wins what (allocation) and howmuch pays (payment) on the basis of the received information.

Revenue in auctions

PricesAuctions

What is an Auction?

The set of possible resource allocations.The number (or portion) of goods of each type including legalor other restrictions on how the goods may be allocated.Rules for bidding and clearing.

A procedure to determine who wins what (allocation) and howmuch pays (payment) on the basis of the received information.

Revenue in auctions

PricesAuctions

What is an Auction?

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PricesAuctions

Strategic component?

Bidders decide the information that is revealed in theinteraction.

To whom?

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PricesAuctions

To whom?

Revenue in auctions

PricesAuctions

To whom?

Revenue in auctions

Open auctionsSealed-Bid auctions

1 Mechanism design

Revenue in auctions

Single item auctions

For today’s lecture assume that we have a single item or good tosell.

Revenue in auctions

Open auction

The auctioneer and the bidders interact physically.

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English Auction

Revenue in auctions

English Auction

The auctioneer starts the bidding at some reservation price.

The bidders then shout out ascending prices.

Once bidders stop shouting, the highest bidder gets the goodat the declared price.

Revenue in auctions

English Auction

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English Auction

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English Auction

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Japanese Auction

Revenue in auctions

Japanese Auction

The auctioneer calls out ascending prices.

All bidders start out standing, when the price reaches a levelthat a bidder is not willing to pay, that bidder sits down.

Once a bidder sits down, they can’t get back up

The only action that a bidder may take is to drop out of theauction

The last person standing gets the good at the last priceshouted.

analytically more tractable than English because jump biddingcan’t occur.

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Japanese Auction

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Japanese Auction

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Japanese Auction

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Japanese Auction

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Japanese Auction

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Japanese Auction

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Dutch Auction

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Dutch Auction

The auctioneer starts a clock at some high value; it descends

At some point, a bidder shouts mine! and gets the good atthe price shown on the clock.

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Dutch Auction

Revenue in auctions

Dutch Auction

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Sealed-Bid auctions

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Sealed bid auctions

The auctioneer and the bidders do not interact physically.

The bidders submit their bid privately to the auctioneer.

The bidder on the basis of the bid sets allocation and price.

Revenue in auctions

First price (FP) Auction

The bidders write down a price and send it to the auctioneer.

The auctioneer awards the good to the bidder with thehighest bid.

The winner pays the amount of his bid.

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Second price (SP) Auction

The winner pays the amount bid by the second-highest bidder.

Second price auctions are also known as Vickrey auctions.defined by William Vickrey in 1961. Vickrey won the Nobelprize in Economics in 1996.

Revenue in auctions

All-Pay Auction

Everyone pays the amount of their bid regardless of whetheror not they get the good.

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All-Pay Auction

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All-Pay Auction

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All-Pay Auction

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GoalsSP AuctionsFP AuctionsEnglish and Japanese auctions

1 Mechanism design

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Auctioneer goals

A seller (“auctioneer”) may have several goals.

Revenue: maximize profit.

Efficiency: maximize social welfare:Give the item to the buyer that wants it the most. (regardlessof payments.)

Fairness:An auction that greedily maximizes the total utility of all users(i.e., the social welfare) in each round could lead to a subsetof secondary users starving for products.

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Auctioneer goals

Efficiency: maximize social welfare:

Give the item to the buyer that wants it the most. (regardlessof payments.)

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Auctioneer goals

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Auctioneer goals

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Bidder goal

Auctions are used precisely because the seller is unsure aboutthe values that bidders attach to the object being sold.

A valuation is the maximum amount each bidder is willing topay.

If each bidder knows the value of the object to himself at thetime of bidding, the situation is called one of private values.

If the object has a value and the players have some believe onsuch value, the situation is called one of common values.

A bidder wants to get the object as cheap as possible.

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Bidder goal

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Bidder goal

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Bidder goal

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Strategic equivalence

Definition

Two games with the same set of players and the same strategyspace are said to be strategically equivalent if each player’sexpected profits under one of the games are identical to hisexpected profits in the other game.

Revenue in auctions

SP-Auctions: strategic equivalences

SP-auctions are equivalent to Japanese auctions.

Given that bidders bid truthfully, the outcomes in the twoauctions are the same.

Actually, in Japanese auctions bidders observe additionalinformation: valuations of other players.

This might create a herd phenomena.

But do bidders bid truthfully in SP-auctions?

Revenue in auctions

SP-Auction: Modeling

n bidders

Each bidder has value vi for the item willingness to pay.Known only to him – private value.

If Bidder i wins and pays pi , his utility is vi–pi .Her utility is 0 when she loses.

Bidders prefer losing than paying more than their value.

Revenue in auctions

SP-Auction: Modeling

n bidders

Each bidder has value vi for the item willingness to pay.Known only to him – private value.

If Bidder i wins and pays pi , his utility is vi–pi .Her utility is 0 when she loses.

Bidders prefer losing than paying more than their value.

Revenue in auctions

SP-Auction: Strategies

A strategy for each bidder:

how to bid given your value?

Examples for strategies:

bi (vi ) = vi truthful!bi (vi ) = vi/2bi (vi ) = vi/n if you have information on the number of bidders.If vi < 50, bi (vi ) = vi ; otherwise, bi (vi ) = vi + 17.

The auction is a strategic game, where these strategies arethe pure strategies (infinitely many).

Revenue in auctions

SP-Auctions: Equilibrium behaviour

Theorem

In SP-price auctions truth-telling is a dominant strategy.

In Japanese auctions with private values too.

Let’s prove now that truthfulness is a dominant strategy.

The proof is by case analysis.

We have to show that Bidder 1 will never benefit from biddinga bid that is not v1.

Revenue in auctions

Theorem

In SP-price auctions truth-telling is a dominant strategy.

In Japanese auctions with private values too.

Let’s prove now that truthfulness is a dominant strategy.

The proof is by case analysis.

We have to show that Bidder 1 will never benefit from biddinga bid that is not v1.

Revenue in auctions

Case 1: Bidder 1 wins when bidding v1.

v1 is the highest bid and b2 is the 2nd highest.

Bidder 1 utility is v1 − b2 > 0.

Bidding above b2 will not change anything.

Bidding less than b2 will turn him into a loser. From positiveutility to zero!

No gain from lying!.

Revenue in auctions

Case 2: Bidder 1 loses when bidding v1.

Let b2 be the 2nd highest bid now.

Bidder 1 utility is 0.

Any bid below b2 will gain him zero utility.

Any bid above b2 will gain him either 0 (still not wining)or an utility smaller that v1 − b2 < 0 (losing is better).

Again, no gain from lying!.

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SP-Auctions: Efficiency

Since SP-auction is truthful, we can conclude it is efficient.

That is, in equilibrium,

the auctioneer allocates the item tothe bidder with the highest value.

With the actual highest value, not just the highest bid.Without assuming anything on the values.

However the seller does not get maximum revenue.

Revenue in auctions

That is, in equilibrium, the auctioneer allocates the item tothe bidder with the highest value.

Revenue in auctions

That is, in equilibrium, the auctioneer allocates the item tothe bidder with the highest value.

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FP-Auctions Strategic equivalences

FP-price auctions are strategically equivalent to Dutch auctions.

Strategies:

FP: Given that no one has a higher bid,what is the maximum I am willing to pay?

Dutch: Given that no body has raised their hand,when should I raise mine?No new information is revealed during the auction!

Revenue in auctions

Strategies:

Dutch: Given that no body has raised their hand,when should I raise mine?

No new information is revealed during the auction!

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Strategies:

Dutch: Given that no body has raised their hand,when should I raise mine?No new information is revealed during the auction!

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FP-Auctions: properties

FP and Dutch auctions are

Equivalent.

Efficient?Yes, in equilibrium the good will be allocated to the playerwith a higher valuation.

Truthful?v1 = 100 and other’s highest bid b2 = 30.Player 1 by bidding 31 gets the good and a positive benefit.No truthfulness in the strategic setting.We continue the analysis on Bayesian games.

Revenue in auctions

Equivalent.

Truthful?

v1 = 100 and other’s highest bid b2 = 30.Player 1 by bidding 31 gets the good and a positive benefit.No truthfulness in the strategic setting.We continue the analysis on Bayesian games.

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Equivalent.

Truthful?v1 = 100 and other’s highest bid b2 = 30.

Player 1 by bidding 31 gets the good and a positive benefit.No truthfulness in the strategic setting.We continue the analysis on Bayesian games.

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Equivalent.

Truthful?v1 = 100 and other’s highest bid b2 = 30.Player 1 by bidding 31

gets the good and a positive benefit.No truthfulness in the strategic setting.We continue the analysis on Bayesian games.

Revenue in auctions

Equivalent.

Truthful?v1 = 100 and other’s highest bid b2 = 30.Player 1 by bidding 31 gets the good and a positive benefit.

No truthfulness in the strategic setting.We continue the analysis on Bayesian games.

Revenue in auctions

Equivalent.

Truthful?v1 = 100 and other’s highest bid b2 = 30.Player 1 by bidding 31 gets the good and a positive benefit.No truthfulness in the strategic setting.

We continue the analysis on Bayesian games.

Revenue in auctions

Equivalent.

Truthful?v1 = 100 and other’s highest bid b2 = 30.Player 1 by bidding 31 gets the good and a positive benefit.No truthfulness in the strategic setting.We continue the analysis on Bayesian games.

Revenue in auctions

FP auctions: Bayesian analysis

How do people behave?

They have beliefs on the valuations of the other players!

As usual beliefs are modeled with probability distributions.

Bidders do not know their opponent’s values, i.e., there isincomplete information.Each bidder’s strategy must maximize her expected payoffaccounting for the uncertainty about opponent values.

Revenue in auctions

Bidders do not know their opponent’s values, i.e., there isincomplete information.

Each bidder’s strategy must maximize her expected payoffaccounting for the uncertainty about opponent values.

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Auctions with uniform distributions

A simple Bayesian auction model:

2 buyersValues are between 0 and 1.Values are distributed uniformly on [0, 1]

What is the equilibrium in this game of incompleteinformation?

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Simple FP: Equilibrium

2 bidders uniform distributionBidding b(v) = v/2 is an equilibrium

Assume that Bidder 2’s strategy is b2(v) = v2/2.

Let us show that b1(v) = v1/2 is a best response to Bidder 2.(clearly, no need to bid above v1).

Bidder 1’s utility is:

Prob[b1 > b2] (v1 − b1) =

=Prob[b1 > v2/2] (v1 − b1)

=2b1 (v1 − b1)

maximizing for b1 we have: [2b1 (v1 − b1)]′ = 2v1 − 4b1 = 0which gives b1 = v1/2

Revenue in auctions

Prob[b1 > b2] (v1 − b1) =

=Prob[b1 > v2/2] (v1 − b1)

=2b1 (v1 − b1)

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Prob[b1 > b2] (v1 − b1) =

=Prob[b1 > v2/2] (v1 − b1)

=2b1 (v1 − b1)

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Prob[b1 > b2] (v1 − b1) =

=Prob[b1 > v2/2] (v1 − b1)

=2b1 (v1 − b1)

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Prob[b1 > b2] (v1 − b1) =

=Prob[b1 > v2/2] (v1 − b1)

=2b1 (v1 − b1)

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Prob[b1 > b2] (v1 − b1) =

=Prob[b1 > v2/2] (v1 − b1)

=2b1 (v1 − b1)

maximizing for b1 we have:

[2b1 (v1 − b1)]′ = 2v1 − 4b1 = 0which gives b1 = v1/2

Revenue in auctions

Prob[b1 > b2] (v1 − b1) =

=Prob[b1 > v2/2] (v1 − b1)

=2b1 (v1 − b1)

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FP: uniform values

We consider the simple Bayesian model

n biddersValues drawn uniformly form [0, 1]

Theorem

In a FP auction with n bidders under the uniform values model, thestrategy bi = n−1

n vi , for 1 ≤ i ≤ n, is a Bayesian Nash equilibrium.

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FP: uniform values

Theorem

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FP: uniform values

Theorem

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FP uniform values: Efficiency

An auction is efficient if, in a Bayesian Nash equilibrium, thebidder with the highest value always wins.

Thus, in the uniform value model FP is efficient.

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English and Japanese auctions

A much more complicated strategy space than sealed bidauctions

extensive form gamebidders are able to condition their bids on information revealedby othersin the case of English auctions, the ability to place jump bids

Independent private values model (IPV): the n bidders havevalues v1, . . . , vn identically and independently distributedwith cdf F (·).

Theorem

Under the IPV model, it is a dominant strategy for bidders to bidup to (and not beyond) their valuations in both Japanese andEnglish auctions.

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Theorem

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Theorem

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Theorem

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Theorem

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Maximal revenueSP auctionsFP auctionsRevenue equivalenceAll-pay auctions

1 Mechanism design

Revenue in auctions

Optimal auctions

Usually the term optimal auctions stands for revenuemaximization.

What is maximal revenue? optimal expected revenue inequilibrium.

Assuming a probability distribution on the values.Over all the possible mechanisms.Under individual-rationality constraints.

Revenue in auctions

Optimal auctions

Revenue in auctions

Optimal auctions

What is maximal revenue?

optimal expected revenue inequilibrium.

Revenue in auctions

Optimal auctions

Revenue in auctions

Optimal auctions

Revenue in auctions

Simple auctions with uniform distributions

2 biddersValues drawn uniformly form [0, 1] x , y

What is the expected revenue gained by SP and FP auctions?

Revenue in auctions

Simple auctions with uniform distributions

2 biddersValues drawn uniformly form [0, 1] x , y

What is the expected revenue gained by SP and FP auctions?

Revenue in auctions

Revenue in SP Simple auctions with uniform distributions

In a SP auction, the payment is the minimum of the twovalues.

E [revenue] = E [min{x , y}]

Claim: When x , y ≡ U[0, 1] we have E [min{x , y}] = 1/3

Revenue in auctions

Revenue in SP Simple auctions with uniform distributions

In a SP auction, the payment is the minimum of the twovalues.

E [revenue] = E [min{x , y}]

Claim: When x , y ≡ U[0, 1] we have E [min{x , y}] = 1/3

Revenue in auctions

Claim’s proof

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

2+ (1− x)x = x − x2

The expectation over all possible x is:

E [min{x , y}] =

0(x − x2

2− x3

Revenue in auctions

Claim’s proof

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

2+ (1− x)x = x − x2

The expectation over all possible x is:

E [min{x , y}] =

0(x − x2

2− x3

Revenue in auctions

Order statistics

Let 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 n-th-order statistic.

Example: the uniform distribution, 2 samples.

The expected 1st-order statistic: 2/3Expected efficiency.The expected 2nd-order statistic: 1/3Expected revenue.

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Order statistics

Revenue in auctions

Order statistics

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Order statistics

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Order statistics

The expected 1st-order statistic: 2/3Expected efficiency.

The expected 2nd-order statistic: 1/3Expected revenue.

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Order statistics

Revenue in auctions

Order statistics

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 auctions

Revenue in FP simple auctions with uniform distributions

In a FP simple auction, vi/2 is a bayesian Nash

Revenue is the highest bid.

Thus, expected revenue is

E [revenue] =E [max{v1/2, v2/2}] =1

2E [max{v1, v2}]

Same revenue as in SP simple auctions!

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2E [max{v1, v2}]

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2E [max{v1, v2}]

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Revenue: FP vs. SP auctions with uniform distributions

Revenue in SP:Bidders bid truthfully.Revenue is 2nd highest bid:

E [revenue] =n − 1

Revenue in FP:Bidders bid.Revenue is highest bid:

E [revenue] =E

{n − 1

nv1, . . . ,

n − 1

}]=n − 1

nE [max{v1, . . . , vn}] =

n − 1

n + 1=

n − 1

Revenue in auctions

Revenue in SP:

Bidders bid truthfully.Revenue is 2nd highest bid:

E [revenue] =E

{n − 1

nv1, . . . ,

n − 1

}]=n − 1

nE [max{v1, . . . , vn}] =

n − 1

n + 1=

n − 1

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E [revenue] =E

{n − 1

nv1, . . . ,

n − 1

}]=n − 1

nE [max{v1, . . . , vn}] =

n − 1

n + 1=

n − 1

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Revenue in FP:

Bidders bid.Revenue is highest bid:

E [revenue] =E

{n − 1

nv1, . . . ,

n − 1

}]=n − 1

nE [max{v1, . . . , vn}] =

n − 1

n + 1=

n − 1

Revenue in auctions

E [revenue] =E

{n − 1

nv1, . . . ,

n − 1

}]=n − 1

nE [max{v1, . . . , vn}] =

n − 1

n + 1=

n − 1

Revenue in auctions

E [revenue] =E

{n − 1

nv1, . . . ,

n − 1

}]=n − 1

nE [max{v1, . . . , vn}] =

n − 1

n + 1=

n − 1

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Revenue Equivalence Theorem

Assumptions:

vi ‘s are drawn independently from some F on [a, b].F is continuous and strictly increasing.Bidders are risk neutral: utility is a linear function of hiswealth.

Theorem (The Revenue Equivalence Theorem)

Consider two auction such that:

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

(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.

Revenue depends most on allocation than on valuations!

Revenue in auctions

Assumptions:vi ‘s are drawn independently from some F on [a, b].F is continuous and strictly increasing.Bidders are risk neutral: utility is a linear function of hiswealth.

Revenue in auctions

Revenue depends most on allocation than on valuations!AGT-MIRI Single item auctions

Revenue in auctions

Revenue in All-pay auction

Sealed bidHighest bid winsEveryone pay their bid

Equilibrium with the uniform distribution is

b(v) =n − 1

Revenue?

Revenue in auctions

b(v) =n − 1

Revenue?

Revenue in auctions

b(v) =n − 1

Revenue?

Revenue in auctions

b(v) =n − 1

Revenue?

Revenue in auctions

b(v) =n − 1

Revenue?

Revenue in auctions

Expected payment per each player: her bid

Each bidder bids b(v) = n−1n vn.

Expected payment for each bidder:∫ 1

n − 1

nvndv =

n − 1

Revenue for n bidders

n + 1.

Again revenue equivalence!

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n − 1

nvndv =

n − 1

n + 1.

Revenue in auctions

n − 1

nvndv =

n − 1

n + 1.

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n − 1

nvndv =

n − 1

n + 1.

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n − 1

nvndv =

n − 1

n + 1.

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n − 1

nvndv =

n − 1

n + 1.

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All-pay auctions: 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.

War of attrition

Animals invest (b1,b2) in fighting.Maynard Smith, J. (1974) Theory of games and the evolutionof animal conflicts.

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War of attrition

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War of attrition

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War of attrition

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War of attrition

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War of attrition

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Revenue

At equilibrium neither SP nor FP lead the maximum possiblebenefit to the seller.

Can we get a better understanding of revenue?

Can we have a truthful auction giving maximum revenue?

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Revenue

Revenue in auctions

Revenue

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