Mechanism and Market Design George J. Mailath University of - - PowerPoint PPT Presentation

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Mechanism and Market Design

George J. Mailath

University of Pennsylvania Australian National University

August 2, 2018

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Introduction

In 2007, the Economics Nobel Prize was awarded to Leonid Hurwicz, Eric Maskin, and Roger Myerson for “having laid the foundations of mechanism design theory.” In 2012, the Economics Nobel Prize was awarded to Alvin Roth and Lloyd Shapley for “the theory of stable allocations and the practice of market design.” What are mechanism and market design? Do they have any practical applications?

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Outline

An example of a simple mechanism Markets and absent markets How to sell a painting Mechanism design

revenue equivalence revelation principle strategic voting (Gibbard-Satterthwaite) Inefficiencies due to private information (Myerson-Satterthwaite)

Market Design

auctions school 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 the smaller 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 children want (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 just small enough that the children are indifferent. Why does it work?

The chooser will choose the more attractive slice, leaving the less attractive slice to the cutter. The cutter has an incentive to maximize the value of the less 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 defence systems, bridges, roads, airports, and the provision of

• ther public goods).

Sale of public assets:

• Spectrum. Which telecom companies should be allocated

which 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 the painting at $90. If Charlotte knows the values, then Charlotte offers the painting to Alice for $100 (or maybe $99). But what if Charlotte does not know the values?

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Mechanism Design

Resource Allocation

An allocation is a description of who gets what (how big and which piece of the pie do the children receive, who wins the auction and at what price, who builds the bridge and under what terms) The environment: the relevant characteristics, such as the preferences, the bidder valuations, the cost of building the bridge (including firm-specific issues). As the environment changes, we expect the allocation to

• change. 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 Design

Inverse game theory

Game theory can be thought of as taking a strategic situation (game) as given, and investigating the resulting strategic 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 desired

• utcome?

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Selling the painting

Second-Price Sealed-Bid Auction I

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

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

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Selling the painting

Second-Price Sealed-Bid Auction II

Note that this did not require Alice to know anything about Bob’s bidding behavior (technically, bidding one’s valuation is a dominant strategy). Interpretation of this auction (mechanism): outcome is determined on the basis of buyer reports of their valuations. Important property of the auction is that the winner pays the cost s/he imposes on the loser (because s/he pays the loser’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),

• pen outcry ascending price (English auction),
• pen outcry descending price (Dutch or clock auction), or

all 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),

• pen outcry ascending price (English auction),
• pen outcry descending price (Dutch or clock auction), or

all pay auction.

Which is optimal? Revenue equivalence = ⇒ providing the reserve price is set appropriately, all auctions raise the same revenue!

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Revelation Principle 1

Environment (Alice and Bob’s valuations) social choice function Allocation (winner, payments) What happens in a game? Choice of action (bid) Game Allocation (winner, payments)

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Revelation Principle 2

Environment (Alice and Bob’s valuations) social choice function Allocation (winner, payments) What happens in an equilibrium of a game? Environment (Alice and Bob’s valuations) Optimal action (bid) Game Allocation (winner, payments)

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Revelation Principle 3

Environment (Alice and Bob’s valuations) social choice function Allocation (winner, payments) What happens in an equilibrium of a game? Environment (Alice and Bob’s valuations) Optimal action (bid) Game Allocation (winner, payments)

• Specific game, actions = reports

Environment (Alice and Bob’s valuations) Optimal report (valuation) Game Allocation (winner, payments)

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Revelation Principle

Revelation Principle

The equilibrium outcome of any game (mechanism) is the truthful equilibrium of a game in which agents only report their characteristics.

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Strategic Voting

Charles Dodgson (aka Lewis Carroll, 1876): This principle of voting makes an election more of a game of skill than a real test of the wishes 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 election more of a game of skill than a real test of the wishes of the electors. Is this because electoral systems are poorly designed? NO: The Gibbard-Satterthwaite Theorem tells us that in many situations, it is impossible to design any protocol so that members will have an incentive to always honestly reveal their characteristics (no matter how others behave). Strategic behavior is an unavoidable feature of life, it is not the 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 often preclude efficient outcomes. Sally owns a painting and is bargaining with Bob, who is interested in buying it. Efficiency requires that Sally sell the painting if and only if she values the painting more than Bob.

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Efficient Trade

Sally’s value Bob’s value 1 1

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If Sally knows Bob’s valuation, then obtaining efficient trade 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 is impossible to design a bargaining protocol to get efficient trade.

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Equilibrium Trade

Sally’s value Bob’s value 1 1 .25 .75 equilibrium trade

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Market Design

Auctions for online ads

The following auctions have been used to sell ad space

• nline:

Generalized second price auction Vickrey-Clarke-Groves (each bidder is required to pay the cost their presence imposes on the other bidders, using their stated bids as the value they place on the slots). VCG requires bidders to submit bids on clickthrough rates (the source 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 match in 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 for positions as medical residents. Doctors have preferences over hospitals and hospitals have preferences over doctors. Every year, (again in the US) new students must be assigned to schools with limited capacity. Students (and their parents) have preferences over schools and schools have priority lists.

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Matching 1

Every year, new medical doctors in the US look for positions as medical residents. Doctors have preferences over hospitals and hospitals have preferences over doctors. Every year, (again in the US) new students must be assigned to schools with limited capacity. Students (and their parents) have preferences over schools and schools have priority lists. An allocation is a description of who is matched with whom (which doctor is assigned to which hospital, which student is assigned to which school). The matching is stable if there no unmatched pair who would prefer to match with each other (rather than receive their assigned match).

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School Choice

Boston School Mechanism 1

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

Each student submits a preference ordering over schools. Each school is assigned students who ranked that school at the top (respecting the schools priority ordering, randomly

• rdered within priority).

Remaining students are then assigned using their second school choice, and so on.

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School Choice

Boston School Mechanism 1

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

Each student submits a preference ordering over schools. Each school is assigned students who ranked that school at the top (respecting the schools priority ordering, randomly

• rdered within priority).

Remaining students are then assigned using their second school choice, and so on.

In 2003, Abdulkadiro˘ glu and S¨

• nmez published a paper in

the AER arguing that the Boston mechanism is manipulable: students have an incentive not to truthfully report their preferences. The result may also not be stable.

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School Choice

Boston School Mechanism 2

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

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

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

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School Choice

Boston School Mechanism 2

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

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

Students have no incentive to lie in the DA algorithm, and the result is stable! This is not the end: the appropriate student assigment medchanism is still debated, both within economics and

• communities. One issue is that the Boston mechanism can

have better welfare properties in some circumstances.

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Conclusion

Markets allocate resources efficiently when

there are many buyers and sellers, there are no externalities in consumption (eg, smoking) or production (eg, pollution), and there is no private information.

Markets discover prices (value) when they work. But markets don’t always work–this is where mechanism and market design is important.

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Conclusion 2

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

The critical issue is to encourage individuals to reveal private information to the mechanism. In simple situations, second price auctions (and its generalization, Vickrey-Clarke-Groves mechanisms) work well. Nonetheless, the privacy of information necessarily leads to inefficiency (bargaining).

Market design is mechanism design applied to specific economic problems, such as school choice, the allocation

• f medical interns to hospitals, and the sale of ads on

webpages.