Mechanism and Market Design · In 2012, the Economics Nobel Prize was awarded to Alvin Roth and...

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Mechanism and Market Design

George J. Mailath

University of PennsylvaniaAustralian National University

August 2, 2018

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Introduction

In 2007, the Economics Nobel Prize was awarded toLeonid Hurwicz, Eric Maskin, and Roger Myerson for“having laid the foundations of mechanism design theory.”

In 2012, the Economics Nobel Prize was awarded to AlvinRoth and Lloyd Shapley for “the theory of stable allocationsand the practice of market design.”

What are mechanism and market design? Do they haveany practical applications?

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Outline

An example of a simple mechanism

Markets and absent markets

How to sell a paintingMechanism design

revenue equivalencerevelation principlestrategic voting (Gibbard-Satterthwaite)Inefficiencies due to private information(Myerson-Satterthwaite)

Market Designauctionsschool choice

Conclusion

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Introduction

How to share a pie between two children?

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Introduction

How to share a pie between two children?

One child cuts the pie and the other chooses.

Result is fair: an even split.

Why does it work?

The chooser will choose the bigger slice, leaving thesmaller slice to the cutter.The cutter has an incentive to maximize the smaller slice,leading to an even split.

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Introduction

Suppose now pie has one cherry on top that both childrenwant (but parent does not know how much they want it).

Still have one child cut the pie and the other choose.

Result is still fair: split is uneven with the cherry slice justsmall enough that the children are indifferent.

Why does it work?

The chooser will choose the more attractive slice, leavingthe less attractive slice to the cutter.The cutter has an incentive to maximize the value of theless attractive slice, leading to an even split.

This is an example of an incentive-compatible mechanism.

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Absent Markets

Examples where there are no competitve markets:

Large-scale government purchases (such as defencesystems, bridges, roads, airports, and the provision ofother public goods).Sale of public assets:

Spectrum. Which telecom companies should be allocatedwhich spectrum (identified by frequency and location)?Public timber auctions.Natural resources, such as oil drilling rights.Airport slots.

Medical interns.

School choice.

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Selling a painting

Charlotte wants to sell a painting. There are two buyers,Alice and Bob.

Alice values the painting at $100, while Bob values thepainting at $90.

If Charlotte knows the values, then Charlotte offers thepainting to Alice for $100 (or maybe $99).

But what if Charlotte does not know the values?

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Mechanism DesignResource Allocation

An allocation is a description of who gets what (how bigand which piece of the pie do the children receive, whowins the auction and at what price, who builds the bridgeand under what terms)

The environment: the relevant characteristics, such as thepreferences, the bidder valuations, the cost of building thebridge (including firm-specific issues).

As the environment changes, we expect the allocation tochange. This description is called a social choice function:

Environment(Alice and Bob’s

valuations)

social choice function Allocation(winner,

payments)

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Mechanism DesignInverse game theory

Game theory can be thought of as taking a strategicsituation (game) as given, and investigating the resultingstrategic behavior (called an equilibrium).

Mechanism design takes a particular social choice function(such as the fair division of the pie, revenue maximization,least cost provision) and answers

What game (if any) has an equilibrium yielding the desiredoutcome?

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Selling the paintingSecond-Price Sealed-Bid Auction I

Each buyer (bidder) simultaneously submits a bid, thehighest bidder wins, and pays the second highest bid.In this auction, the best thing for the each buyer to do is tobid their valuation:

Suppose Bob submits a bid of $80.If Alice values the painting more than $80, then submitingher value guarantees she wins the auction, but only pays$80. Bidding more does not change anything, and biddingless only changes the outcome if she bids less than $80(and loses), but Alice wants to win the auction in thissituation.If Alice values the painting at less than $80, then Alice doesnot want to win the painting if she must pay $80.

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Selling the paintingSecond-Price Sealed-Bid Auction II

Note that this did not require Alice to know anything aboutBob’s bidding behavior (technically, bidding one’s valuationis a dominant strategy).

Interpretation of this auction (mechanism): outcome isdetermined on the basis of buyer reports of theirvaluations.

Important property of the auction is that the winner paysthe cost s/he imposes on the loser (because s/he pays theloser’s reported value) =⇒ efficiency.

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Which Auction?

There are many different auctions the seller could choose(all with or without a reserve price):

sealed bid first price,sealed bid second price (Vickrey auction),open outcry ascending price (English auction),open outcry descending price (Dutch or clock auction), orall pay auction.

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Which Auction?

There are many different auctions the seller could choose(all with or without a reserve price):

sealed bid first price,sealed bid second price (Vickrey auction),open outcry ascending price (English auction),open outcry descending price (Dutch or clock auction), orall pay auction.

Which is optimal?

Revenue equivalence =⇒ providing the reserve price isset appropriately, all auctions raise the same revenue!

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Revelation Principle 1


valuations)


payments)

What happens in a game?

Choice ofaction(bid)

Game Allocation(winner,

payments)

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valuations)


payments)

What happens in an equilibrium of a game?


valuations)

Optimalaction(bid)


payments)

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valuations)


payments)

What happens in an equilibrium of a game?


valuations)

Optimalaction(bid)


payments)ww

Specific game, actions = reports


valuations)

Optimalreport

(valuation)


payments)

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Revelation Principle

Revelation PrincipleThe equilibrium outcome of any game (mechanism) is thetruthful equilibrium of a game in which agents only report theircharacteristics.

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Strategic Voting

Charles Dodgson (aka Lewis Carroll, 1876):

This principle of voting makes an electionmore of a game of skill than a real test of thewishes of the electors.

Is this because electoral systems are poorly designed?

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Strategic Voting

Charles Dodgson (aka Lewis Carroll, 1876):

This principle of voting makes an electionmore of a game of skill than a real test of thewishes of the electors.

Is this because electoral systems are poorly designed?

NO: The Gibbard-Satterthwaite Theorem tells us that inmany situations, it is impossible to design any protocol sothat members will have an incentive to always honestlyreveal their characteristics (no matter how others behave).

Strategic behavior is an unavoidable feature of life, it is notthe result of people being “bad” or selfish.

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Inefficiency of private information

The privacy of information is a critical friction that can oftenpreclude efficient outcomes.

Sally owns a painting and is bargaining with Bob, who isinterested in buying it.

Efficiency requires that Sally sell the painting if and only ifshe values the painting more than Bob.

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Efficient Trade

0Sally’s value

0

Bob’s value

1

1

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If Sally knows Bob’s valuation, then obtaining efficienttrade is easy: Sally offers the painting to Bob at a price at(or just below) Bob’s valuation.

If neither Sally nor Bob know the others valuation, then it isimpossible to design a bargaining protocol to get efficienttrade.

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Equilibrium Trade

0Sally’s value

0

Bob’s value

1

1

.25

.75

equilibrium trade

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Market DesignAuctions for online ads

The following auctions have been used to sell ad spaceonline:

Generalized second price auctionVickrey-Clarke-Groves (each bidder is required to pay thecost their presence imposes on the other bidders, usingtheir stated bids as the value they place on the slots). VCGrequires bidders to submit bids on clickthrough rates (thesource of value)

Both auctions work similarly (bidding value is dominant)and raise the same revenue in simple scenarios.

But the real world is complicated. What counts as a matchin key word (broad or exact), and how to measure value?Turns out VCG is more flexible, and now used by Google.

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Matching 1

Every year, new medical doctors in the US look forpositions as medical residents.Doctors have preferences over hospitals and hospitalshave preferences over doctors.

Every year, (again in the US) new students must beassigned to schools with limited capacity. Students (andtheir parents) have preferences over schools and schoolshave priority lists.

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Matching 1

Every year, new medical doctors in the US look forpositions as medical residents.Doctors have preferences over hospitals and hospitalshave preferences over doctors.

Every year, (again in the US) new students must beassigned to schools with limited capacity. Students (andtheir parents) have preferences over schools and schoolshave priority lists.

An allocation is a description of who is matched with whom(which doctor is assigned to which hospital, which studentis assigned to which school). The matching is stable ifthere no unmatched pair who would prefer to match witheach other (rather than receive their assigned match).

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School ChoiceBoston School Mechanism 1

Intradistrict and interdistrict school choice have becomeprevalent in the US over the last twenty to thirty years.A common student assignment mechanism is the so-calledBoston Mechanism:

Each student submits a preference ordering over schools.Each school is assigned students who ranked that school atthe top (respecting the schools priority ordering, randomlyordered within priority).Remaining students are then assigned using their secondschool choice, and so on.

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Intradistrict and interdistrict school choice have becomeprevalent in the US over the last twenty to thirty years.A common student assignment mechanism is the so-calledBoston Mechanism:

Each student submits a preference ordering over schools.Each school is assigned students who ranked that school atthe top (respecting the schools priority ordering, randomlyordered within priority).Remaining students are then assigned using their secondschool choice, and so on.

In 2003, Abdulkadiroğlu and Sönmez published a paper inthe AER arguing that the Boston mechanism ismanipulable: students have an incentive not to truthfullyreport their preferences. The result may also not be stable.

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In 2005, Boston Public Schools switched to thestudent-proposing deferred acceptance algorithm:

Each student proposes to her first choice.Each school tentatively accepts students according to theirpriority list.Each rejected student proposes to her next choice.Each school tentatively accepts students from the newproposers, possibly withdrawing acceptances from earliertentatively accepted students, according to their priority list.And so on.

Students have no incentive to lie in the DA algorithm, andthe result is stable!

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In 2005, Boston Public Schools switched to thestudent-proposing deferred acceptance algorithm:

Each student proposes to her first choice.Each school tentatively accepts students according to theirpriority list.Each rejected student proposes to her next choice.Each school tentatively accepts students from the newproposers, possibly withdrawing acceptances from earliertentatively accepted students, according to their priority list.And so on.

Students have no incentive to lie in the DA algorithm, andthe result is stable!

This is not the end: the appropriate student assigmentmedchanism is still debated, both within economics andcommunities. One issue is that the Boston mechanism canhave better welfare properties in some circumstances.

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Conclusion

Markets allocate resources efficiently whenthere are many buyers and sellers,there are no externalities in consumption (eg, smoking) orproduction (eg, pollution), andthere is no private information.

Markets discover prices (value) when they work.

But markets don’t always work–this is where mechanismand market design is important.

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Conclusion 2

Mechanism design addresses the general question of howto efficiently allocate scarce resources.

The critical issue is to encourage individuals to revealprivate information to the mechanism.In simple situations, second price auctions (and itsgeneralization, Vickrey-Clarke-Groves mechanisms) workwell.Nonetheless, the privacy of information necessarily leads toinefficiency (bargaining).

Market design is mechanism design applied to specificeconomic problems, such as school choice, the allocationof medical interns to hospitals, and the sale of ads onwebpages.

Mechanism and Market Design · In 2012, the Economics Nobel Prize was awarded to Alvin Roth and...

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