yale - information economics

7/30/2019 Yale - Information Economics

1/86

Dirk Bergemann

Department of Economics

Yale University

Information Economics

March 2009

Springer-Verlag

Berlin Heidelberg NewYorkLondon Paris TokyoHongKong BarcelonaBudapest


2/86


3/86

Contents

Part I. Introduction

1. Game Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91.1 Game theory and parlor games - a brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Game theory in microeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Information Economics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11

3. Akerlofs Lemon Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 133.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4. Wolinskys Price Signals Quality : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 174.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Part II. Static Games of Incomplete Information

5. Harsanyis insight : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21

6. Denition of a Bayesian Game : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23

7. Purication : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25

8. Global Games : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27

Part III. Dynamic Games of Incomplete Information

9. Job Market Signalling : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 319.1 Pure Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2 Perfect Bayesian Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.3 Equilibrium Domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.4 Informed Principal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9.4.1 Maskin and Tiroles informed principal problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10. Spence-Mirrlees Single Crossing Condition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3910.0.2 Separating Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

10.1 Supermodular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.2 Supermodular and Single Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.3 Signalling versus Disclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


4/86

4 Contents

10.4 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Part IV. Moral Hazard

10.5 Introduction and Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.6 Binary Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.6.1 First Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.6.2 Second Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.7 General Model with nite outcomes and actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10.7.1 Optimal Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.7.2 Monotone Likelihood Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.7.3 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10.8 Information and Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.8.1 Informativeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.8.2 Additional Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10.9 Linear contracts with normally distributed performance and exponential utility . . . . . . . . . . . 5210.9.1 Certainty Equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310.9.2 Rewriting Incentive and Participation Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10.10Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Part V. Mechanism Design

11. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 57

12. Adverse selection: Mechanism Design with One Agent : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5912.1 Monopolistic Price Discrimination with Binary Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

12.1.1 First Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.1.2 Second Best: Asymmetric information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12.2 Continuous type model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.2.1 Information Rent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.2.2 Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.2.3 Incentive Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.3 Optimal Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6512.3.1 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.3.2 Pointwise Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13. Mechanism Design Problem with Many Agents : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.2 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.3 Mechanism as a Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.4 Second Price Sealed Bid Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

14. Dominant Strategy Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 75

15. Bayesian Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 77

15.1 First Price Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7715.2 Optimal Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

15.2.1 Revenue Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7915.2.2 Optimal Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

15.3 Additional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


5/86

Contents 5

15.3.1 Procurement Bidding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215.3.2 Bilateral Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8215.3.3 Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

16. Eciency : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8316.1 First Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8316.2 Second Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17. Social Choice : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8517.1 Social Welfare Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.2 Social Choice Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


6/86


7/86

Part I

Introduction


8/86


9/86

1. Game Theory

Game theory is the study of multi-person decision problems. The focus of game theory is interdependence,situations in which an entire group of people is aected by the choices made by every individual within thatgroup. As such they appear frequently in economics. Models and situations of trading processes (auction,bargaining) involve game theory, labor and nancial markets. There are multi-agent decision problemswithin an organization, many person may compete for a promotion, several divisions compete for investmentcapital. In international economics countries choose taris and trade policies, in macroeconomics, the FRBattempts to control prices.

Why game theory and economics? In competitive environments, large populations interact. How-ever, the competitive assumption allows us to analyze that interaction without detailed analysis of strategicinteraction. This gives us a very powerful theory and also lies behind the remarkable property that ecient

allocations can be decentralized through markets.In many economic settings, the competitive assumption does not makes sense and strategic issues must

addressed directly. Rather than come up with a menu of dierent theories to deal with non-competitiveeconomic environments, it is useful to come up with an encompassing theory of strategic interaction (gametheory) and then see how various non-competitive economic environments t into that theory. Thus this sec-tion of the course will provide a self-contained introduction to game theory that simultaneously introducessome key ideas from the theory of imperfect competition.

1. What will each individual guess about the other choices?2. What action will each person take?3. What is the outcome of these actions?

In addition we may ask

1. Does it make a dierence if the group interacts more than once?2. What if each individual is uncertain about the characteristics of the other players?

Three basic distinctions may be made at the outset

1. non-cooperative vs. cooperative games2. strategic (or normal form) games and extensive (form) games3. games with perfect or imperfect information

In all game theoretic models, the basic entity is a player. In noncooperative games the individual playerand her actions are the primitives of the model, whereas in cooperative games coalition of players and their

joint actions are the primitives.

1.1 Game theory and parlor games - a brief history

1. 20s and 30s: precursors

a) E. Zermelo (1913) chess, the game has a solution, solution concept: backwards induction


10/86

10 1. Game Theory

b) E. Borel (1913) mixed strategies, conjecture of non-existence

2. 40s and 50s: core conceptual development

a) J. v. Neumann (1928) existence in of zero-sum gamesb) J. v. Neumann / O. Morgenstern (1944) Theory of Games and Economic Behavior: Axiomatic

expected utility theory, Zero-sum games, cooperative game theoryc) J. Nash (1950) Nonzero sum games

3. 60s: two crucial ingredients for future development: credibility (subgame perfection) and incompleteinformation

a) R. Selten (1965,75) dynamic games, subgame perfect equilibriumb) J. Harsanyi (1967/68) games of incomplete information

4. 70s and 80s: rst phase of applications of game theory (and information economics) in applied eldsof economics

5. 90s and on: real integration of game theory insights in empirical work and institutional design

For more on the history of game theory, see Aumanns entry on Game Theory in the New PalgraveDictionary of Economics.

1.2 Game theory in microeconomics

1. decision theory (single agent)2. game theory (few agents)3. general equilibrium theory (many agents)


11/86

2. Information Economics

We shall start with two very simple, but classic models and results to demonstrate (i) how asymmetry ofinformation changes classical eciency results of markets and (ii) how asymmetry of information changesclassical arguments about the role of prices and the equilibrium process.

Economics of information examines the role of information in economic relationship. It is therefore aninvestigation into the role ofimperfectand incomplete information. In the presence of imperfect information,learning, Bayesian or not, becomes important, and in consequence dynamic models are prevalent. In thisclass we will focus mostly on incomplete or asymmetric information.

The nature of information economics as a eld is perhaps best understood when contrasted with thestandard general equilibrium theory. Information consists of a set of tools rather than a single methodology.Furthermore, the choice of tools is very issue driven. Frequently we will make use of the following tools:

small number of participants institutions may be represented by constraints noncooperative (Bayesian) game theory simple assumptions on bargaining: Principal-Agent paradigm

We refer to the Principal-Agent paradigm as a setting where one agent, called the Principal, can make allthe contract oers, and hence has (almost) all bargaining power and a second agent, called the Agent, canonly choose whether to accept or reject the oer by the principal. With this general structure the interactivedecision problem can often by simplied to a constrained optimization problem, where the Principal hasan objective function, and the Agent simply represents constraints on the Principals objective function.Two recurrent constraints will be the participation constraint and the incentive constraint.

We refer to the Principal-Agent paradigm as a setting where one agent, called the Principal, can make all

the contract oers, and hence has (almost) all bargaining power and a second agent, called the Agent, canonly choose whether to accept or reject the oer by the principal. With this general structure the interactivedecision problem can often by simplied to a constrained optimization problem, where the Principal hasan objective function, and the Agent simply represents constraints on the Principals objective function.Two recurrent constraints will be the participation constraint and the incentive constraint.

The following scheme organizes most of the models prevalent in information economics according totwo criteria: (i) whether the informed or an uniformed agent has the initiative (makes the rst move, oersthe initial contract, arrangement; and (ii) whether the uniformed agent is uncertain about the action orthe type (information) of the informed agent.

hidden action hidden informationinformed agent signalling

uninformed agent moral hazard adverse selection


12/86


13/86

3. Akerlofs Lemon Model

3.1 Basic Model

This section is based on ?. Suppose an object with value v s U[0; 1] is oered by the seller. The valuationsare

us = sv

and

ub = bv

with b > s, and hence trading is always pareto-optimal. But trade has to be voluntary. We then ask isthere a price at which trade occurs. Suppose then at a price p trade would occur. What properties wouldthe price have to induce trade. The seller sells if

sv p

and thus by selling the object he signals that

v p

s(3.1)

The buyer buys the object if

bE [v] p (3.2)

and as he knows that (3.1) has to hold, he can form a conditional expectation, that

bE [v jp ] p , bp

2s p (3.3)

Thus for the sale to occur,

b 2s. (3.4)

Thus unless, the tastes dier substantially, the market breaks down completely:

market mechanism in which a lower prices increases sales fails to work as lowering the price decreasesthe average quality, lower price is bad news.

market may not disappear but display lower volume of transaction than socially optimal.


14/86

14 3. Akerlofs Lemon Model

3.2 Extensions

In class, we made some informed guesses how the volume of trade may depend on the distribution ofprivate information among the sellers. In particular, we ventured the claim that as the amount of privateinformation held by the sellers decreases, the possibilities for trade should increase. We made a secondobservation, namely that in the example we studied, for a given constellation of preferences by buyer andseller, represented by b and s, trade would either occur with positive probability for all prices or it would

not occur at all. We then argued that this result may be due to the specic density in the example, butmay not hold in general. We now address both issues with a uniform density with varying support andconstant mean:

v~U

1

2 ";

1

2+ "

with " 2

0; 12

. We can formalize the amount of private information by the variance of f() and how it

aects the eciency of the trade.We may redo the analysis above where the seller signals by selling the object at a xed price p that

v p

s.

The buyer buys the object if

bE [v jp ] p,

The expected value is now given by

E [v jp ] =12 " +

ps

2

and hence for sale to occur

b

12 " +

ps

2 p (3.5)

and inequality prevails if, provided that " 2 [0; 1

2

) and 2s > b,

p 1

2bs

1 2"

2s b

Consider next the eciency issue. All types of sellers are willing to sell if

p = s

1

2+ "

(3.6)

in which case the expected conditional value for the buyer is

1

2b s

1

2+ "

or equivalently

b s (1 + 2") (3.7)

Thus as the amount of private information, measured by ", decreases, the ineciencies in trade decrease.Notice that we derived a condition in terms of the preferences such that all sellers wish to sell. However


15/86

3.2 Extensions 15

even if condition (3.7) is not met, we can in general support some trade. This is indicated already in theinequality (3.5):

b

12 " +

ps

2 p (3.8)

as the lhs increases slower in p, then the rhs of the inequality above. Thus there might be lower prices,

which will not induce all sellers to show up at the market, yet allow some trade. We show how this can arisenext. For a positive mass of seller willing to sell the price may vary between p 2 (s 12 " ; s 12 + "]. Ifwe then insert the price as a function of x 2 ("; "], we nd

b

12 " +

s( 12+x)s

2 s

1

2+ x

or

b1 " + x

2 s

1

2+ x

For a given b; s and ", we may then solve the (in-)quality for x, to nd the (maximal) price p where thevolume of trade is maximized, or:

x =b (1 ") s

2s b:

Thus x is increasing in b and decreasing in ";conrming now in terms of the volume of trade our intuitionabout private information and eciency. In fact, we can verify that for a given " 2 [0; 1

2), the volume of

trade is positive, but as " ! 12 becomes arbitrarily small and converges as expected to zero for b < 2s.


16/86


17/86

4. Wolinskys Price Signals Quality

This section is based on ?. Suppose a continuum of identical consumers. They have preferences

u = v p

and the monopolist can provide low and high quality: v = f0; 1g at cost 0 < c0 < c1. The monopolistselects price and quality simultaneously. Assume that > c1 so that it is socially ecient to produce thehigh quality good. Assume that the consumers do not observe quality before purchasing. It is clear that anequilibrium in which the monopolist sells and provides high quality cannot exist.

Suppose now that some consumers are informed about the quality of the product, say a fraction .Observe rst that if the informed consumers are buying then the uniformed consumer are buying as well.

When is the seller better o to sell to both segments of the market:

p c1 (1 ) (p c0)

or

p c1 (1 ) c0. (4.1)

We can then make two observations:

high quality is supplied only if price is suciently high, high price can signal high quality. a higher fraction, , of informed consumers favors eciency as it prevents the monopolist from cutting

quality the informational externality favors government intervention as individuals only take private benet and

cost into account.

4.1 Conclusion

We considered hidden information or hidden action models and suggested how asymmetry in informa-tion may reduce or end trade completely. In contrast to the canonical model of goods, which we may callsearch goods, where we can assert the quality by inspection, we considered experience goods (?, ?),where the quality can only be ascertained after the purchase. The situation is only further acerbated withcredence goods (?).

In both models, there was room for a third party, government or other institution, to induce paretoimprovement. In either case, and improvement in the symmetry of information lead to an improvement

in the eciency of the resulting allocation. This suggest that we may look for optimal or equilibriumarrangements to reduce the asymmetry in information, either through:

costly signalling optimal contracting to avoid moral hazard, or optimal information extraction through a menu of contract (i.e. mechanism design).


18/86

18 4. Wolinskys Price Signals Quality

4.2 Reading

The lecture is based on Chapter 1 in ?, Chapter 2 in ?, and Chapter 5 in ?.


19/86

Part II

Static Games of Incomplete Information


20/86


21/86

5. Harsanyis insight

Harsanyis insights is illustrated by the following example.

Example: Suppose payos of a two player two action game are either:

H TH 1,1 0,0T 0,1 1,0

or

H T

H 1,0 0,1T 0,0 1,1

i.e. either player II has dominant strategy to play H or a dominant strategy to play T. Suppose thatII knows his own payos but player I thinks there is probability that payos are given by the rstmatrix, probability 1 that they are given by the second matrix. Say that player II is of type1 if payos are given by the rst matrix, type 2 if payos are given by the second matrix. Clearlyequilibrium must have: II plays H if type 1, T if type 2; I plays H if > 12 , T if 0

and thus type and strategic variable are complements (supermodular).

9.1 Pure Strategy

A pure strategy for the worker is a function

be : f1; 2g ! R+A pure strategy for the rm is a function

w : R+ ! R+

where w (e) is the wage oered to a worker of educational level e.

9.2 Perfect Bayesian Equilibrium

The novelty in signalling games is that the uninformed party gets a chance to update its prior belief onthe basis of a signal sent by the informed party. The updated prior belief is the posterior belief, dependingon e and denoted by p (a je ). The posterior belief is a mapping

^p : R+ ! [0; 1]


32/86

32 9. Job Market Signalling

In sequential or extensive form games we required that the strategy are sequential rational, or time consis-tent. We now impose a similar consistency requirement on the posterior beliefs by imposing the Bayes rulewhenever possible. Since p (a je ) is posterior belief, and hence a probability function, it is required that:

8e; 9p (a je ) ; s.th. p (a je ) 0; andX

a2f1;2g

p (a je ) = 1: (9.1)

Moreover, when the rm can apply Bayes law, it does so, or

if 9a s.th. at e (a) = e; then p (a je ) =p (a)P

fa0je(a)=egp (a0)

: (9.2)

We refer to educational choice which are selected by some worker-types in equilibrium, or if 9a s.th. ate (a) = e, as on-the-equilibrium path and educational choices s.th. @a with e (a) = e as o-the-equilibrium-path.

As before it will be sometimes easier to refer to

p (e) , p (a = 2 je )

and hence

p (1 je ) = 1 p (e) :

Denition 9.2.1 (PBE). A pure strategy Perfect Bayesian Equilibrium is a set of strategiesfe (a) ; w (e)gand posterior beliefs p (e) such that:

1. 8e; 9p (a je ) ; s.th. p (a je ) 0; andP

2f1;2gp (a je ) = 1;

2. 8i; wi (e) =P

ap (a je ) a;

3. 8a; e (a) 2 arg max

w (e) ea

4. if 9a s.th. at e (a) = e; then:

p (a je ) =p (a)P

fa0je(a)=egp (a0)

:

Denition 9.2.2 (Separating PBE). A pure strategy PBE is a separating equilibrium if

a 6= a0 ) e (a) 6= e (a0) .

Denition 9.2.3 (Pooling PBE). A pure strategy PBE is a pooling equilibrium if

8a; a0 ) e (a) = e (a0) .

[Lecture 3]

Theorem 9.2.1.

1. A pooling equilibrium exists for all education levels e = e (1) = e (2) 2 [0; p] :2. A separating equilibrium exists for all e (1) = 0 and e (2) 2 [1; 2].

Proof. (1) We rst construct a pooling equilibrium. For a pooling equilibrium to exist it must satisfy thefollowing incentive compatibility constraints

8e; 1 +p (e) e 1 +p (e) e (9.3)

and


33/86

9.2 Perfect Bayesian Equilibrium 33

8e; 1 +p (e) e

2 1 +p (e)

e

2(9.4)

Consider rst downward deviations, i.e. e < e, then (9.3) requires that

p (e) p (e) e e: (9.5)

Then consider upward deviations, i.e. e > e, then (9.4) requires that

p (e) p (e) 1

2(e e) : (9.6)

We can then ask for what levels can both inequalities be satised. Clearly both inequalities are easiest tosatisfy if

e < e ) p (e) = 0

e > e ) p (e) = 0;

which leaves us with

e < e ) p e e (9.7)

e > e ) p 1

2

(e e) :

We may then rewrite the inequalities in (9.7) as

e < e ) e p + e (9.8)

e > e ) e p + 2e

As the inequality has to hold for all e, the asserting that 0 p e holds, follows immediately.(2) Consider then a separating equilibrium. It must satisfy the following incentive compatibility con-

straints

8e; 1 +p (e1) e1 1 +p (e) e

and

8e; 1 +p (e2) e2

2 1 +p (e) e

2: (9.9)

As along the equilibrium path, the rms must apply Bayes law, we can rewrite the equations as

8e; 1 e1 1 +p (e) e

and

8e; 2 e22

1 +p (e) e

2:

Consider rst the low productivity type. It must be that e1 = 0. This leaves us with

p (e) e: (9.10)

But if e = e2 is supposed to be part of a separating equilibrium, then p (e2) = 1. Thus it follows further

from (9.10) that e2 1, for otherwise we would not satisfy 1 = p (e2) e

2. Finally, we want to determine

an upper bound for e2. AS we cannot require the high ability worker to produce too much eort otherwisehe would mimic the lower ability, we can rewrite (9.9) to obtain:


34/86


1 p (e) 1

2(e2 e) ;

which is easiest to satisfy if

p (e) = 0;

and hence

8e; 2 + e e2

which implies that:

e2 2.

which completes the proof.

The theorem above only denes the range of educational choices which can be supported as an equi-librium but is not a complete equilibrium description as we have not specied the beliefs in detail. The

job market signalling model suggests a series of further questions and issues. There were multiple equilib-ria, reducing the predictive ability of the model and we may look at dierent approaches to reduced themultiplicity:

rened equilibrium notion dierent model: informed principal

The signal was modelled as a costly action. We may then ask for conditions:

when do costly signals matter for Pareto improvements (or simply separation): Spence-Mirrlees singlecrossing conditions

when do costless signals matter: cheap-talk games.

In the model, education could also be interpreted as an act of disclosure of information through theverication of the private information by a third party. It may therefore be of some interest to analyze thepossibilities of voluntary information disclosure.

9.3 Equilibrium Domination

The construction of some of the equilibria relied on rather arbitrary assumptions about the beliefs of therms for educational choice levels o the equilibrium path. Next we rene our argument to obtain somelogical restrictions on the beliefs. The restrictions will not be bases on Bayes law, but on the plausibilityof deviations as strategic choices by the agents. The notions to be dened follow ?.

Denition 9.3.1 (Equilibrium Domination). Given a P BE, the message e is equilibrium-dominatedfor typea, if for all possible assessments p (a je ) (or simply p(e)):

w (e (a)) e (a)

a> w (e)

e

a

or equivalently

1 +p (e (a)) e (a)

a> 1 +p (e)

e

a


35/86

9.3 Equilibrium Domination 35

Denition 9.3.2 (Intuitive Criterion). If the information set following e is o the equilibrium pathand e is equilibrium dominated for type a, then, if possible,

p (a je ) = 0: (9.11)

Remark 9.3.1. The qualier, if possible, in the denition above refers to the fact that p (a je ) has to be awell-dened probability, and thus if the eort e is equilibrium dominated for all a, then the requirementimposed by (9.11) is vacuous.

Denition 9.3.3. A PBE where all beliefs o the equilibrium path satisfy the intuitive criterion is said tosatisfy the intuitive criterion.

Theorem 9.3.1 (Uniqueness). The unique Perfect Bayesian equilibrium outcome which satises theintuitive criterion is given by

fe1 = 0; e2 = 1; w

(0) = 1; w (1) = 2g : (9.12)

The beliefs are required to satisfy

p (e) =

8 2 e;

but any message in the interval is not equilibrium dominated for the high productivity worker, as

1 +p e

2

< 2

e

2

and thus for e = e + (1 p), we have

p < 1 (1 p)

2,

1

2p 1, any e 2 (1; e2) is equilibrium dominated for the low

ability worker as

1 > 2 e;

but is not equilibrium dominated for the high ability worker, as

2

e2

2 < 2

e

2 .

It follows that p (e) = 1 for all e 2 (1; e2). But then e2 > 1, cannot be supported as an equilibrium as the

high ability worker has a protable deviation by lowering his educational level to some e 2 (1; e2), and stillreceive his full productivity in terms of wages. It remains e2 = 1 as the unique PBE satisfying the intuitivecriterion.


36/86


Criticism. Suppose p ! 1:

Remark 9.3.3. Add the Kreps critique of equilibrium domination.

[Lecture 4]

9.4 Informed PrincipalAs plausible as the Cho-Kreps intuitive criterion may be, it does seem to predict implausible equilibriumoutcomes in some situations. For example, suppose that, p, the prior probability that a worker of typea = 2 is present, is arbitrarily large, (p ! 1). In that case, it seems a rather high cost to pay to incur aneducation cost of

c(e = 1) =1

2

just to be able to raise the wage by a commensurately small amount w = 2 (1 + p) ! 0 as p ! 1.Indeed, in that case the pooling equilibrium where no education costs are incurred seems a more plausibleoutcome. This particular case should serve as a useful warning not to rely too blindly on selection criteria(such as Cho-Kreps intuitive criterion) to single out particular PBEs.

9.4.1 Maskin and Tiroles informed principal problem

Interestingly, much of the problem of multiplicity of PBEs disappears when the timing of the game ischanged to letting agent and principal sign a contract before the choice of signal. This is one importantlesson to be drawn from ?.

To see this, consider the model of education as specied above and invert the stages of contractingand education choice. That is, now the worker signs a contract with his/her employer before undertakingeducation. This contract then species a wage schedule contingent on the level of education chosen by theworker after signing the contract.

Let fw(e)g denote the contingent wage schedule specied by the contract.Consider the problem for the high productivity worker. Suppose he would like to make an oer by

which he can separate himself from a low ability worker

maxfe;w(e)g

nw (e)

e

2

osubject to

w (e1) e1 w (e2) e2 (IC1)

and

w (e2) e22

w (e1) e12

(IC2)

and

a1 w (e1) 0 (IR1)

a2 w (e2) 0 (IR2)

Thus to make incentive compatibility as easy as possible he suggests e1 = 0 and w (e1 = 0) = 1. Asw (e2) = 2, it follows that after setting e2 = 1, he indeed maximizes his payo.


37/86

9.4 Informed Principal 37

Suppose instead he would like to oer a pooling contract. Then he would suggest

maxe;w

nw

e

2

o1 +p w 0 (IR1)

which would yield w = 1 +p for all e.

This suggest that there are two dierent cases to consider:

1. 1 +p 2 12 : a high productivity worker is better o in the least cost separating equilibrium thanin the ecient pooling equilibrium.

2. 1 +p > 2 12 : a high productivity worker is better o in the ecient pooling equilibrium.

It is easy to verify that in the case where p 12 , the high productivity worker cannot do better thanoering the separating contract, nor can the low productivity worker. More precisely, the high productivityworker strictly prefers this contract over any contract resulting in pooling or any contract with more costlyseparation. As for the low productivity worker, he has everything to lose by oering another contract whichwould identify himself.

In the alternative case where p > 12 , the unique equilibrium contract is the one where:

w(e) = 1 +p for all e 0.

Again, if the rm accepts this contract, both types of workers choose an education level of zero. Thus, onaverage the rm breaks even by accepting this contract, provided that it is as (or more) likely to originatefrom a high productivity worker than a low productivity worker. Now, a high productivity worker strictlyprefers to oer this contract over any other separating contract. Similarly, a low productivity worker haseverything to lose from oering another contract and thus identifying himself. Thus, in this case again thisis the unique contract oer made by the workers.


38/86


39/86

10. Spence-Mirrlees Single Crossing Condition

10.0.2 Separating Condition

Consider now a general model with a continuum of types:

A R+

and an arbitrary number of signals

E R+

with a general quasilinear utility function

u (t;e;a) = t + v (e; a)

where we recall that the utility function used in Spence model was given by:

u (t;e;a) = t e

a

We now want ask when is it possible in general to sustain a separating equilibrium for all n agents, suchthat

a 6= a0 ) e 6= e0

Suppose we can support a separating equilibrium for all types, then we must be able to satisfy for a0 > a

and without loss of generality e0 > e:

t + v (e; a) t0 + v (e0; a) , t t0 v (e0; a) v (e; a) (10.1)

and

t0 + v (e0; a0) t + v (e; a0) , t t0 v (e0; a0) v (e; a0) ; (10.2)

where t , t (e) and t0 , t (e0), and by combining the two inequalities, we nd

v (e0; a) v (e; a) v (e0; a0) v (e; a0) ; (10.3)

or similarly, again recall that e < e0

v (e; a0) v (e; a) v (e0; a0) v (e0; a) : (10.4)

We then would like to know what are sucient conditions on v (; ) such that for every a there exists esuch that (10.3) can hold.


40/86

40 10. Spence-Mirrlees Single Crossing Condition

10.1 Supermodular

Denition 10.1.1. A function v : E A ! R has increasing dierences in (e; a) if for any a0 > a,v(e; a0) v(e; a) is nondecreasing in e.

Note that ifv has increasing dierences in (e; a), it has increasing dierences in (a; e). Alternatively, wesay the function v is supermodular in (e; a).

If v is suciently smooth, then v is supermodular in (e; a) if and only if @2

v=@e@a 0.As the transfers t; t0 could either be determined exogenously, as in the Spence signalling model throughmarket clearing conditions, or endogenously, as in optimally chosen by the mechanism designer, we want toask when we can make a separation incentive compatible. We shall not consider here additional constraintssuch as a participation constraints. We shall merely assume that t (a), i.e. the transfer that the agent of typea gets, t (a), conditional on the sorting allocation fa; e (a) ; t (a)g to be incentive compatible, is continuouslydierentiable and strictly increasing.

Theorem 10.1.1. A necessary and sucient condition for sorting ((10.1) and (10.2)) to be incentivecompatible for all t (a) is that

1. v (e; a) is strictly supermodular2. @v

@e< 0 everywhere

Proof. We shall also suppose that v (; ) is twice continuously dierentiable. The utility function of theagent is given by

t (a) + v (e (a) ; a) ,

where a is the true type of agent a and e (a) is the signal the informed agent sends to make the uninformedagent believe he is of type a.(Suciency) For e (a) to be incentive compatible, a must locally solve for the rst order conditions of theagent at a = a, namely the true type:

t0 (a) +@v (e (a) ; a)

@e

de

da= 0, at a = a (10.5)

But Fix the payments t (a) as a function of the type a to be t (a) and suppose without loss of generality

that t (a) is continuously dierentiable. The equality (10.5) denes a dierential equation for the separatingeort level

de

da=

t0 (a)@v(e(a);a)

@e

(10.6)

for which a unique solution exists given a initial condition, say t (0) = 0. The essential element is that therst order conditions is only an optimal condition for agent with type a and nobody else, but this followsfrom strict supermodularity. As

t0 (a)

and

de

da

are independent of the true type, a, it follows that if

@2v (e; a)

@e@a> 0,


41/86

10.3 Signalling versus Disclosure 41

for all e and a, then (10.5) can only identify truthtelling for agent a.(Necessity) For any particular transfer policy t (a), we may not need to impose the supermodularity con-dition everywhere, and it might often by sucient to only impose it locally, where it is however necessaryto guarantee local truthtelling, i.e.

@v (e (a) ; a)

@e@a> 0

at e = e (a). However, as we are required to consider all possible transfers t (a) with arbitrary positive slopes,we can guarantee that for every a and every e there is some transfer problem t (a) such that e (a) = e by(10.6), and hence under the global condition on t (a), supermodularity becomes also a necessary conditionfor sorting.

10.2 Supermodular and Single Crossing

In the discussion above we restricted our attention to quasilinear utility function. We can generalize all thenotions to more general utility functions. We follow the notation in ?. Let

U(x; y; t) : R3 ! R

where x is in general the signal (or allocation), y an additional instrument, such as money transfer and tis the type of the agent.

Denition 10.2.1 (Spence-Mirrlees). The function U is said to satisfy the (strict) Spence-Mirrleescondition if

1. the ratio

UxjUyj

is (increasing) nondecreasing in t, and Uy 6= 0, and2. the ratio has the same sign for every (x; y; t).

Denition 10.2.2 (Single-Crossing). The function U is said to satisfy the single crossing property in(x; y; t) if for all (x0; y0) (x; y)

1. wheneverU(x0; y0; t) U(x; y; t), then U(x0; y0; t0) U(x; y; t0) for all t0 > t;2. wheneverU(x0; y0; t) > U (x; y; t), then U(x0; y0; t0) > U (x; y; t0) for all t0 > t;

? show that the notions of single crossing and the Spence Mirrlees conditions are equivalent.Thus the supermodularity condition is also often referred to as the single-crossing or Spence-Mirrlees

condition, where Mirrlees used the dierential form for the rst time in ?. The notions of supermodu-larity and single-crossing are exactly formulated in ? and for some corrections ? Some applications tosupermodular games are considered by ? and ?. The mathematical theory is, inter alia, due to ? and ?.

10.3 Signalling versus Disclosure

Disclosure as private but certiable information. Signalling referred to situations where private informationis neither observable nor veriable. That is to say, we have considered private information about such thingsas individual preferences, tastes, ideas, intentions, quality of projects, eort costs, etc., which cannot reallybe measured objectively by a third party. But there are other forms of private information, such as an


42/86

42 10. Spence-Mirrlees Single Crossing Condition

individuals health, the servicing and accident history of a car, potential and actual liabilities of a rm,earned income, etc., that can be certied or authenticated once disclosed. For these types of informationthe main problem is to get the party who has the information to disclose it. This is a simpler problemthan the one we have considered so far, since the informed party cannot report false information. It canonly choose not to report some piece of information it has available. The main results and ideas of thisliterature is based on ? and ?.

10.4 Reading

The material of this lecture is covered by Section 4 of ? and for more detail on the Spence-Mirrleesconditions, see ?.


43/86

Part IV

Moral Hazard


44/86


45/86

10.6 Binary Example 45

Today we are discussing the optimal contractual arrangement in a moral hazard setting

10.5 Introduction and Basics

In contractual arrangements in which the principal oers the contract, we distinguish between

hidden information - adverse selection

and

hidden action - moral hazard

The main trade-o in adverse selection is between ecient allocation and informational rent. In moralhazard settings it is between risk sharing and wok incentives. Today we are going to discuss the basicmoral hazard setting. As the principal tries to infer from the output of the agent about the eort choice,the principal engages in statistical inference problems. Today we are going to develop basic insights intothe optimal contract and use this is an occasion to introduce three important informational notions:

1. monotone likelihood ratio2. garbling in the sense of Blackwell

3. sucient statistic

The basic model goes as follows. An agent takes action that aect utility of principal and agent. Principalobserves outcome x and possible some signal s but not action a of agent. Agent action in absenceof contract is no eort. Inuence agents action by oering transfer contingent on outcome.

Example 10.5.1. Employee employer eect (non observable), Property insurance (re, theft)

F B: agents action is observable (risk sharing)SB : agents action is unobservable (risk sharing-incentive)

10.6 Binary Example

The agent can take action a 2 fal; ahg at cost c 2 fcl; chg. The outcomes x 2 fxl; xhg occur randomly,where the probabilities are governed by the action as follows.

pl = Pr (xh jal ) < ph = Pr(xh jah )

The principal can oer a wage, contingent on the outcome w 2 fwl; whg to the agent and the utility of theagent is

u (wi) ci

and of the principal it is

v (xi wi)

where we assume that u and v are strictly increasing and weakly concave.


46/86

46

10.6.1 First Best

We consider initially the optimal allocation of risk between the agents in the presence of the risk and withobservable actions. If the actions are observable, then the principal can induce the agent to choose thepreferred action a by

w (xi; a) = 1

if a 6= a for all xi. As the outcome is random and the agents have risk-averse preference, the optimalallocation will involve some risk sharing. The optimal solution is characterized by

maxfwl;whg

fpiv (xh wh) + (1 pi) v (xl wl)g

subject to

piu (wh) + (1 pi) u (wl) ci U; ()

which is the individual rational constraint. Here we dene

w (xi; a) , wi

This is a constrained optimization problem, and the rst order conditions from the Lagrangian

L (wl; wh; ) = piv (xh wh) + (1 pi) v (xl wl) + (piu (wh) + (1 pi) u (wl) ci)

are given by

V0 (xi wi)

U0 (wi)= ;

which is Borchs rule.

10.6.2 Second Best

Consider now the case in which the action is unobservable and therefore

w (xi; a) = w (xi)

for all a. Suppose the principal wants to induce high eort, then the incentive constraint is:

phu (wh) + (1 ph) u (wl) ch plu (wh) + (1 pl) u (wl) cl

or

(ph pl) (u (wh) u (wl)) ch cl (10.7)

as pl ! ph wh wl must increase, incentives become more high-powered. The principal also has to respecta participation constraint (or individual rationality constraint)

phu (wh) + (1 ph) u (wl) ch U (10.8)

We rst show that both constraints will be binding if the principal maximizes

maxfwh;wlg

ph (xh wh) + (1 ph) (xl wl)


47/86

10.7 General Model with nite outcomes and actions 47

subject to (10.7) and (10.8). For the participation constraint, principal could lower both payments and bebetter o. For the incentive constraint, subtract

(1 ph) "

u0 (wh)

from wh and add

ph"u0 (wl)

to wl. Then the incentive constraint would still hold for " suciently small. Consider then participationconstraint. We would substract utility

(1 ph) "

from u (wh) and add

ph"

to u (wl), so that the expected utility remains constant. But the wage bill would be reduced for the principalby

"ph (1 ph)

1

u0 (wh)

1

u0 (wl)

> 0;

since wh > wl by concavity. The solution is :

u (wl) = U chpl phcl

ph pl

and

u (wh) = U chpl phcl

ph pl+

ch clph pl

10.7 General Model with nite outcomes and actions

Suppose

xi 2 fx1;:::;xIg

and

aj 2 fa1;:::aJg

and the probability

pij = Pr (xi jaj )

the utility is u (w) a for agent and x w for the principal.


48/86

48

10.7.1 Optimal Contract

The principals problem is then

maxfwig

Ii=1

;j

(IXi=1

(xi wi)pij

)

given the wage bill the agent selects aj if and only if

IXi=1

u (wi)pij aj IXi=1

u (wi)pik ak (k)

and

IXi=1

u (wi)pij aj U ()

Fix aj , then the Lagrangian is

L (wij; ; ) = (I

Xi=1

(xi wi)pij)+Xk6=j

(I

Xi=1

u (wi) (pij pik) (aj ak))+

(IXi=1

u (wi)pij aj

)Dierentiating with respect to wij yields

1

u0 (wi)= +

Xk6=j

k

1

pikpij

; 8i (10.9)

With a risk-avers principal the condition (10.9) would simply by modied to:

v0 (xi wi)

u0 (wi) = +Xk6=j

k 1 pikpij ; 8i (10.10)In the absence of an incentive problem, or k = 0, (10.10) states the Borch rule of optimal risk sharing:

v0 (xi wi)

u0 (wi)= ; 8i.

which states that the ratio of marginal utilities is equalized across all states i.We now ask for sucient conditions so that higher outcomes are rewarded with higher wages. As wi

increases as the right side increases, we might ask for conditions when the rhs is increasing in i. To dothis we may distinguish between the downward binding constraints (k; k < j) and the upward bindingconstraints (k; k > j). Suppose rst then that there were only downward binding constraints, i.e. either

j = J, or k = 0 for k > j . Then a sucient for monotonicity in i would clearly be that

pikpij

is decreasing in i for all k < j .


49/86

10.7 General Model with nite outcomes and actions 49

10.7.2 Monotone Likelihood Ratio

The ratio

pikpij

are called likelihood ratio. We shall assume a monotone likelihood ratio condition, or 8i < i0; 8k < j :

pikpij

>pi0kpi0j

or reversing the ratio

pijpik

pij

The monotonicity in the optimal contract is then established if we can show that the downward bindingconstraints receive more weight than the upward binding constraints. In fact, we next give conditions whichwill guarantee that k = 0 for all k > j .

10.7.3 Convexity

The cumulative distribution function of the outcome is convex in a: for aj < ak < al and 2 [0; 1] suchthat for

ak = aj + (1 ) al

we have


50/86

50

Pik Pij + (1 ) Pil

The convexity condition then states that the return from action are stochastically decreasing. Next weshow in two steps that monotone likelihood and convexity are sucient to establish monotonicity. Theargument is in two steps and let aj be the optimal action.

First, we know that for some j < k, j > 0. Suppose not, then the optimal choice of ak would be sameif we were to consider A or fak;:::;aJg. But relative to fak;:::;aJg we know that ak is the least cost action

which can be implemented with a constant wage schedule. But extending the argument to A, we know thata constant wage schedule can only support the least cost action a1.Second, consider fa1;:::;akg. Then ak is the most costliest action and by MLRP wi is increasing in i.

We show that ak remains the optimal choice for the agent when we extend his action to A. The proof isagain by contradiction. Suppose there is l > k s.th.

IXi=1

u (wi)pik ak 0 and hence

I

Xi=1u (wi)pik ak =

I

Xi=1u (wi)pij aj :

Then there exists 2 [0; 1] such that

ak = aj + (1 ) al

and we can apply convexity, by rst rewriting pik and using the cumulative distribution function

IXi=1

u (wi)pik ak =IXi=1

(u (wi) u (wi+1)) Pik + u (wI) ak

By rewriting the expected value also for aj < ak < al, we obtain

I

Xi=1 (u (wi) u (wi+1)) Pik + u (wI) ak

IXi=1

(u (wi) u (wi+1)) Pij + u (wI) aj

!

+ (1 )

IXi=1

(u (wi) u (wi+1)) Pil + u (wI) al

!The later is of course equivalent to

IXi=1

u (wi)pij aj

!+ (1 )

IXi=1

u (wi)pil al

!

The inequality follows from the convexity and the fact that we could assume wi to be monotone, so thatu (wi) u (wi+1) 0.

A nal comment on the monotonicity of the transfer function : Perhaps a good reason why the functionwi must be monotonic is that the agent may be able to articially reduce performance. In the aboveexample, he would then articially lower it from xj to xi whenever the outcome is xj .


51/86

10.8 Information and Contract 51

10.8 Information and Contract

10.8.1 Informativeness

We now ask how does the informativeness of a signal structure aect the payo of the agent. Consider astochastic matrix Q which modies the probabilities pij such that

^pkj =IXi=1

qkipij; (10.11)

such that qki 0 and

KXk=1

qki = 1

for all i. The matrix is called stochastic as all its entries are nonnegative and every column adds up toone. Condition (10.11) is a generalization of the following idea. Consider the information structure givenby pij. Each time the signal xi is observed, it is garbled by a stochastic mechanism that is independent ofthe action aj , which may be interpreted here as a state. It is transformed into a vector of signals

bxk. The

term qki

can be interpreted as a conditional probability of bxk given xi. Clearly ^pkj are also probabilities.Suppose the outcomes undergo a similar transformation so thatKXk=1

xk^pkj =IXi=1

xipij;

for all j. Thus the expected surplus stays the same for every action. It can be achieved (in matrix language)by assuming that x = Q1x.

In statistical terms, inferences drawn on aj after observing bx with probabilities bp will be less precisethan the information drawn from observing x with probability p.

Consider now in the (^p; x) model a wage schedule bwi which implements aj . Then we nd a new wageschedule for the (p; x) problem based on

bw such that aj is also implementable in (p; x) problem, yet we

will see that it involves less risk imposed on the agent and hence is better for the principal. Let

u (wi) =KXk=1

qkiu (bwk) (10.12)We can then write

IXi=1

piju (wi) =IXi=1

pij

KXk=1

qkiu (bwk)!

=KXk=1

^pkju (bwk)But the implementation in (p; x) is less costly for the principal than the one in (^p; x) which proves the

result. But (10.12) immediately indicates that it is less costly for the principal as the agent has a concaveutility function. This shows in particular that the optimal contract is only a second best solution as therst best contains no additional noise.


52/86

52

10.8.2 Additional Signals

Suppose besides the outcome, the principal can observe some other signal, say y 2 Y = fy1;::::;yl;:::;yLg,which could inform him about the performance of the agent. The form of the optimal contract can thenbe veried to be

1

u0 wli = +Xk6=j k1

plik

pl

ij! .Thus the contract should integrate the additional signal y if there exists some xi and yl such that for ajand ak

plikplij

6=pl0

ik

pl0

ij

;

but the inequality simply says that

x

is not a sucient statistic for (x; y) as it would be if we could write the conditional probability as follows

f(xi; yl jaj ) = h (xi; yl) g (xi jaj ) :

Thus, Hart-Holmstrm [1986] write:

The additional signal s will necessarily enter an optimal contract if and only if it aects theposterior assessment of what the agent did; or perhaps more accurately if and only if s inuencesthe likelihood ratio.

10.9 Linear contracts with normally distributed performance and exponentialutility

This constitutes another natural simple case. Performance is assumed to satisfy x = a + ", where " isnormally distributed with zero mean and variance 2. The principal is assumed to be risk neutral, whilethe agent has a utility function :

U(w; a) = er(wc(a))

where r is the (constant) degree of absolute risk aversion (r = U00=U0), and c(a) = 12ca2:

We restrict attention to linear contracts:

w = x + :

A principal trying to maximize his expected payo will solve :

maxa;;

E(x w)

subject to :


53/86

10.9 Linear contracts with normally distributed performance and exponential utility 53

E(er(wc(a))) U(w)

and

a 2 arg maxa

E(er(wc(a)))

where U(w) is the default utility level of the agent, and w is thus its certain monetary equivalent.

10.9.1 Certainty Equivalent

The certainty equivalent w of random variable x is dened as follows

u (w) = E [u (x)]

The certainty equivalent of a normally distributed random variable x under CARA preferences, hence wwhich solves

erw = E erxhas a particularly simple form, namely

w = E [x] 1

2r2 (10.13)

The dierence between the mean of random variable and its certain equivalent is referred to as the riskpremium:

1

2r2 = E [x] w

10.9.2 Rewriting Incentive and Participation Constraints

Maximizing expected utility with respect to a is equivalent to maximizing the certainty equivalent wealthbw(a) with respect to a, where bw(a) is dened byer bw(a) = E(er(wc(a)))

Hence, the optimization problem of the agent is equivalent to:

a 2 arg maxfbw(a)g =2 arg max

a +

1

2ca2

r

222

which yields

a =

c

Inserting a into the participation constraint

c+

1

2c

c

2

r

222 = w


54/86

54

yields an expression for ,

= w +r

222

1

2

2

c

This gives us the agents eort for any performance incentive . The principal then solves :

max

c

( w + r2

22 + 12

2c

)The rst order conditions are

1

c (r2 +

c) = 0;

which yields :

=1

1 + rc2

Eort and the variable compensation component thus go down when c (cost of eort); r (degree of risk aver-sion), and 2 (randomness of performance) go up, which is intuitive. The constant part of the compensationwill eventually decrease as well as r; c or 2 become large, as

= w +

1

2

r2 1c

(1 + rc2)2

!:

10.10 Readings

The following are classic papers in the literature on moral hazard: ?, ?, ?, ?, ?, ?,?,?.

[Lecture 7]


55/86

Part V

Mechanism Design


56/86


57/86

11. Introduction

Mechanism design. In this lecture and for the remainder of this term we will look at a special class ofgames of incomplete information, namely games ofmechanism design. Examples of these games include: (i)monopolistic price discrimination, (ii) optimal taxation, (iii) the design of auctions, and (iv) the provisionof public goods.

The adverse selection model when extended to many agent will ultimately form the theoretical center-piece of the lectures. In the later case we refer to it as the

many informed agents:mechanism design

(social choice)

mechanism design problem which encompasses auctions, bilateral trade, public good provision, taxationand many other general asymmetric information problems. The general mechanism design problem can berepresented in a commuting diagram:

private information social choice(type) of agent i 2 I (allocation, outcome)

fi 2 igi2I ! f : ! X ! x 2 X&

si : i ! Mi g : M ! X

& %fmi 2 Migi2I

messages:from agentto principal

The upper part of the diagram represents the social choice problem, where the principal has to decide onthe allocation contingent on the types of the agent. The lower part represents the implementation problem,where the principals attempts to elicit the information by announcing an outcome function which mapsthe message (information) he receives from the privately informed agents into allocation. From the pointof view of the privately informed agents, this then represents a Bayesian game in which they can inuencethe outcome through their message.

Information extraction, truthtelling, and incentives.All these examples have in common there isa principal (social planner, monopolist, etc.) who would like to condition her action on some informationthat is privately known by the other players, called agents. The principal could simply ask the agentsfor their information, but they will not report truthfully unless the principal gives them an incentive to doso, either by monetary payments or with some other instruments that she controls. Since providing theseincentives is costly, the principal faces a trade-o that often results in an ecient allocation.

Principals objective. The distinguishing characteristic of the mechanism design approach is thatthe principal is assumed to choose the mechanism that maximizes her expected utility, as opposed to usinga particular mechanism for historical or institutional reasons. (Caveat: Since the objective function of the


58/86

58 11. Introduction

principal could just be the social welfare of the agents, the range of problem which can be studies is ratherlarge.)

Single agent. Many applications of mechanism design consider games with a single agent (or equivalentwith a continuum of innitesimal small agents). For example, in a second degree price discrimination bya monopolist, the monopolist has incomplete information about the consumers willingness to pay for thegood. By oering a price quantity schedules he attempts to extract the surplus from the buyer.

Many agents. Mechanism design can also be applied to games with several agents. The auctions we

studied are such an example. In a public good problem, the government has to decide whether to supplya public good or not. But as it has incomplete information about how much the good is valued by theconsumers, it has to design a scheme which determines the provision of the public good as well as transferto be paid as a function of the announced willingness to pay for the public good.

Three steps. Mechanism design is typically studied as three-step game of incomplete information,where the agents type - e.g. their willingness to pay - are private information. In step 1, the principaldesigns a mechanism, contract, or incentive scheme. A mechanism is a game where the agents sendsome costless message to the principal, and as a function of that message the principal selects an allocation.In step 2, the agents simultaneously decide whether to accept or reject the mechanism. An agent who rejectsthe mechanism gets some exogenously specied reservation utility. In step 3, the agents who acceptedthe mechanism play the game specied by the mechanism.

Maximization subject to constraints. The study of mechanism design problems is therefore formallya maximization problem for the principal subject to two classes of constraints. The rst class is called

the participation or individual rationality constraint, which insures the participation of the agent.The second class are the constraints related to thruthtelling, what we will call incentive compatibilityconstraints.

Eciency and distortions. An important focus in mechanism design will be how these two setof constraints in their interaction can prevent ecient outcomes to arise: (i) which allocation y can beimplemented, i.e. is incentive compatible?, (ii) what is the optimal choice among incentive compatiblemechanisms?, where optimal could be ecient, revenue maximizing.


59/86

12. Adverse selection: Mechanism Design with One Agent

Often also called self-selection, or screening. In insurance economics, if a insurance company oers atari tailored to the average population, the tari will only be accepted by those with higher than averagerisk.

12.1 Monopolistic Price Discrimination with Binary Types

A simple model of wine merchant and wine buyer, who could either have a coarse or a sophisticated taste,which is unobservable to the merchant. What qualities should the merchant oer and at what price?

The model is given by the utility function of the buyer, which is

v (i; qi; ti) = u (i; qi) ti = iqi ti; i 2 fl; hg (12.1)

where i represent the marginal willingness to pay for quality qi and ti is the transfer (price) buyer i hasto pay for the quality qi. The taste parameters i satises

0 < l < h < 1. (12.2)

The cost of producing quality q 0 is given by

c (q) 0; c0 (q) > 0; c00 (q) > 0. (12.3)

The ex-ante (prior) probability that the buyer has a high willingness to pay is given by

p = Pr (i = h)

We also observe that the dierence in utility for the high and low valuation buyer for any given quality q

u (h; q) u (l; q)

is increasing in q. (This is know as the Spence-Mirrlees sorting condition.). If the taste parameter i werea continuous variable, the sorting condition could be written in terms of the second cross derivative:

@2u (; q)

@@q> 0,

which states that taste and quality q are complements. The prot for the seller from a bundle (q; t) is

given by

(t; q) = t c (q)


60/86

60 12. Adverse selection: Mechanism Design with One Agent

12.1.1 First Best

Consider rst the nature of the socially optimal solution. As dierent types have dierent preferences, theyshould consume dierent qualities. The social surplus for each type can be maximized separately by solving

maxqi

fiqi c (qi)g

and the rst order conditions yield:

qi = qi () c

0 (qi ) = i ) ql < q

h.

The ecient solution is the equilibrium outcome if either the monopolist can perfectly discriminatebetween the types (rst degree price discrimination) or if there is perfect competition. The two outcomesdier only in terms of the distribution of the social surplus. With a perfectly discriminating monopolist,the monopolist sets

ti = iqi (12.4)

and then solves for each type separately:

maxfti;qig (ti; qi) () maxfti;qig f

iqi c (qi)g ;

using (12.4). Likewise with perfect competition, the sellers will break even, get zero prot and set prices at

ti = c (qi )

in which case the buyer will get all the surplus.

12.1.2 Second Best: Asymmetric information

Consider next the situation under asymmetric information. It is veried immediately that perfect discrim-ination is now impossible as

hql tl = (h l) ql > 0 = hqh th (12.5)

but sorting is possible. The problem for the monopolist is now

maxftl;ql;th;qhg

(1 ) tl c (ql) + (th (c (qh))) (12.6)

subject to the individual rationality constraint for every type

iqi ti = 0 (IRi) (12.7)

and the incentive compatibility constraint

iqi ti = iqj tj (ICi) (12.8)

The question is then how to separate. We will show that the binding constraint are IRl and ICh, whereasthe remaining constraints are not binding. We then solve for tl and th, which in turn allows us to solve forqh; and leaves us with an unconstrained problem for ql.Thus we want to show

(i) IRl binding, (ii) ICh binding, (iii) qh ql (iv) qh = qh (12.9)


61/86

12.1 Monopolistic Price Discrimination with Binary Types 61

Consider (i). We argue by contradiction. As

hqh th =ICh

hql tl =h>l

lql tl (12.10)

suppose that lql tl > 0, then we could increase tl; th by a equal amount, satisfy all the constraints andincrease the prots of the seller. Contradiction.

Consider (ii) Suppose not, then as

hqh th > hql tl =h>l

lql tl(IRl)

= 0 (12.11)

and thus th could be increased, again increasing the prot of the seller.(iii) Adding up the incentive constraints gives us (ICl) + (ICl)

h (qh ql) = l (qh ql) (12.12)

and since:

h > l ) qh ql = 0: (12.13)

Next we show that ICl can be neglected as

th tl = h (qh ql) = l (qh ql) : (12.14)

This allows to say that the equilibrium transfers are going to be

tl = lql (12.15)

and

th tL = h (qh qL) ) th = h (qh qL) + lql.

Using the transfer, it is immediate that

qh = qh

and we can solve for the last remaining variable, ql.

maxql

f(1 p) (lql (c (ql)) +p (h (qh qL) + lql c (q

h)))g

but as qh is just as constant, the optimal solution is independent of constant terms and we can simplifythe expression to:

maxql

f(1 p) (lql c (ql)) p (h l) qlg

Dividing by (1 p) we get

maxql

lql c (ql)

p

1 p(h l) ql

for which the rst order conditions are

l c0 (ql)

p

1 p(h l) ql = 0

This immediately implies that the solution ql:


62/86


() c0 (ql) < l () ql < ql

and the quality supply to the low valuation buyer is ineciently low (with the possibility of completeexclusion).

Consider next the information rent for the high valuation buyer, it is

I(ql) = (h l) ql

and therefore the rent is increasing in ql which is the motivation for the seller to depress the quality supplyto the low end of the market.

The material is explicated in ?, Chapter 2.

[Lecture 8]

12.2 Continuous type model

In this lecture, we consider the adverse selection model with a continuous type model and a single agent.This will set the state for the multiple agent model.

12.2.1 Information RentConsider the situation in the binary type model. The allocation problem for the agent and principal canwell be represented in a simple diagram.

The information rent is then

I(h) = (h l) ql

and it also represents the dierence between the equilibrium utility of low and high type agent, or

I(h) = U(h) U(l)

If we extend the model and think of more than two types, then we can think of the information rent asresulting from tow adjacent types, say k and k1. The information rent is then given by

I(k) = (k k1) qk1

As the information rent is also the dierence between the agents net utility, we have

I(k) = U(k) U(k1) ;

well as we see not quite, but more precisely:

I(k) = U(k jk ) U(k1 jk ) ;

where

U(k jl )

denotes in general the utility of an agent of (true) type l, when he announces and pretends to be of typek. As we then extend the analysis to a continuous type model, we could ask how the net utility of theagent of type k evolves as a function of k. In other words as k1 ! k:

U(k jk ) U(k1 jk )

k k1=

(k k1)

(k k1)qk1


63/86

12.2 Continuous type model 63

and assuming continuity:

qk1 ! qk;

we get

U0 () = q()

which is identical using the specic preferences of our model to

U0 () =@u

@(q() ; ) :

Thus we have a description of the equilibrium utility as a function of q() alone rather than (q() ; t ()).The pursuit of the adverse selection problem along this line is often referred to as Mirrlees trick.

12.2.2 Utilities

The preferences of the agent are described by

U(x;;t) = u (x; ) t

and the principals

V (x;;t) = v (x; ) + t

and u; v 2 C2. The type space 2 R+.The social surplus is given by

S(x; ) = u (x; ) + v (x; ) :

The uncertainty about the type of the agent is given by:

f() ; F() .

We shall assume the (strict) Spence-Mirrlees conditions

@u

@ > 0;@2u

@@x > 0:

12.2.3 Incentive Compatibility

Denition 12.2.1. An allocation is a mapping

! y () = (x () ; t ()) .

Denition 12.2.2. An allocation is implementable if y = (x; t) satises truthtelling:

u (x () ; ) t () u

xb ; tb ; 8;b 2 :

Denition 12.2.3. The truthtelling net utility is denoted by

U() , U( j ) = u (x () ; ) t () ; (12.16)

and the net utility for agent , misreporting by telling is denoted by

U

j

= u

xb ; tb .


64/86


We are interested in (i) describing which contracts can be implemented and (ii) in describing optimalcontracts.

Theorem 12.2.1. The direct mechanism y () = (x () ; t ()) is incentive compatible if and only if:

1. the truthtelling utility is described by:

U() U(0) =

Z0

u (x (s) ; s)) ds; (12.17)

2. x (s) is nondecreasing.

Remark 12.2.1. The condition (12.17) can be restated in terms of rst order conditions:

dU

d=

@u

@y

@y

@+

@u

@= 0: (12.18)

Proof. Necessity. Suppose truthtelling holds, or

U() U

j

; 8;

b;

which is:U() = U

b j M= Ub+ uxb ; uxb ;b ;and thus

U() Ub uxb ; uxb ;b : (12.19)

A symmetric condition gives us

U

U() u

x () ;

u (x () ; ) : (12.20)

Combining (12.19) and (12.20), we get:

u (x () ; ) ux () ; U() Ub uxb ; uxb ;b :Suppose without loss of generality that > , then monotonicity of x () is immediate from

@2u

@@x.

Dividing by ; and taking the limit as

!bat all points of continuity of x () yields

dU()

d ()

= u (x () ; ) ;

and thus we have the representation of U. As x () is nondecreasing, it can only have a countable number ofdiscontinuities, which have Lebesgue measure zero, and hence the integral representation is valid, indepen-dent of continuity properties ofx (), which is in contrast with the representation of incentive compatibilityby the rst order condition (12.18).


65/86

12.3 Optimal Contracts 65

Suciency. Suppose not, then there exists and b s.th.Ub j = U() : (12.21)

Suppose without loss of generality that . The inequality (12.21) implies that the inequality in (12.19)is reversed:

uxb ; uxb ;b > U () Ubintegrating and using (12.17) we get

Z

us

xb ; s ds > Z

us (x (s) ; s) ds

and rearranging

Z^ h

us

x

b

; s

us (x (s) ; s)

ids > 0 (12.22)

but since

@2u

@@x> 0

and the monotonicity condition implied that this is not possible, and hence we have the desired contradic-tion.

[Lecture 9]

12.3 Optimal Contracts

We can now describe the problem for the principal.

maxy()

E [v (x () ; ) + t ()]

subject to

u (x () ; ) t () u

x

;

t

; 8;

and

u (x () ; ) t () u.

The central idea is that we restate the incentive conditions for the agents by the truthtelling utility asderived in Theorem 12.2.1.


66/86


12.3.1 Optimality Conditions

In this manner, we can omit the transfer payments from the control problem and concentrate on the optimalchoice of x as follows:

maxx()

E [S(x () ; ) U()] (12.23)

subject to

dU()

d= u (x () ; ) (12.24)

and

x () nondecreasing (12.25)

and

U() = u; (12.26)

which is the individual rationality constraint. Next we simplify the problem by using integration by parts.Recall, that integration by parts, used in the following formZdU(1 F) = U(1 F) +

ZU f

and henceZU f = U(1 F) +

ZdU(1 F)

As

E [U()] = Z1

0

U() f() d = [U() (1 F())]10 + Z1

0

dU()

d

1 F()

f()f() d (12.27)

we can rewrite (12.23) and using (12.24), we get

maxE

S(x () ; )

1 F()

f()u (x () ; ) U(0)

(12.28)

subject to (12.25) and (12.26).and thus we removed the transfers out of the program.

12.3.2 Pointwise Optimization

Consider the optimization problem for the principal and omitting the monotonicity condition, we get

(x; ) = S(x; ) 1 F()

f()u (x; ) : (12.29)

Assumption 12.3.1 (x; ) is quasiconcave and has a unique interior maximum in x 2 R+ for all 2 :

Theorem 12.3.2. Suppose that Assumption 12.3.1 holds, then the relaxed problem is solved at y = y ()if:


67/86

12.3 Optimal Contracts 67

1. x (x () ; ) = 0:2. the transfer t () satisfy

t () = u (x () ; )

0@U(0) + Z0

u (x (s) ; s) ds)

1AThen y = (x; t) solve the relaxed program.

The idea of the proof is simple. By integration by parts we removed the transfers. Then we can choosex to maximize the objective function and later choose t to solve the dierential equation. Given that@u=@ > 0; the IR contract can be related as

U(0) = u.

It remains to show that x is nondecreasing.

Assumption 12.3.3 @2 (x; ) =@x@ 0.

Theorem 12.3.4. Suppose that assumptions 12.3.1 and12.3.3 are satised. Then the solution to the relaxedprogram satises the original solution in the unrelaxed program.

Proof. Dierentiating

x (x () ; ) = 0 for all

and hence we obtain

=)dx ()

d=

xxx

and as

xx 0

has to be satised locally, we know that x () has to be increasing, which concludes the proof.

Next we interpret the rst order conditions and obtain rst results:

Sx (x () ; ) =1 F()

f()ux (x() ; ) = 0

implies ecient provision for = 1 and underprovision of the contracted activity for all but the highesttype = 1: To see that x () < x (), we argue by contradiction. Suppose not, or x () x (). Observerst that for all x and < 1

@ (x; )

@x 0. (12.31)

Thus at x (), by denition

@S(x () ; )

@x= 0


68/86


and using (12.30)

@ (x () ; )

@x>>:x1 = 1; t1 = m2

x2 = 0; t2 = 0; ifm1 m2

x1 = 0; t1 = 0x2 = 1; t2 = m1

; ifm1 < m2

A variety of other examples can be given:

Example 13.4.1 (Income tax). x is the agents income and t is the amount of tax paid by the agent; isthe agents ability to earn money.

Example 13.4.2 (Public Good). x is the amount of public good supplied, and ti is the consumer is monetarycontribution to nance it; i indexes consumer surplus from the public good.

The example of the second price sealed bid auction illustrates that as a general matter we need notonly to consider to directly implementing the social choice function by asking agents to reveal their type,but also indirect implementation through the design of institutions by which agents interact. The formalrepresentation of such institutions is known as a mechanism.

[Lecture 12]


74/86


75/86

14. Dominant Strategy Equilibrium

Example 14.0.3 (Voting Game). Condorcet paradox

Next we consider implementation in dominant strategies. We show that in the absence of prior restric-tions on the characteristics, implementation in dominant equilibria is essentially impossible.

General Environments. Next, we prove the revelation principle for the dominant equilibrium concept.Equivalent results can be proven for Nash and Bayesian Nash equilibria.

Denition 14.0.1. The strategy prole m = (m1 (1) ;:::;mI (I)) is a dominant strategy equilibrium of

mechanism = (M; g ()) if for all i and all i 2 i :

ui (g (mi (i) ; mi) ; i) ui (g (m

0i; mi) ; i)

for all m0i 6= mi, 8mi 2 Mi.

Theorem 14.0.1 (Revelation Principle). Let = fM; g ()g be a mechanism that implements thesocial choice function f() for the dominant equilibrium concept, then there exists a direct mechanism0 = f; f()g that implements f() by revelation.

Proof. (Due to ?.). Let m () = (:::;mi (i) ;:::) be an Ituple of dominant messages for (M; g ()). Deneg to be the composition of g and m.

g , g m

or

g () , g (m ()) . (14.1)

By denition

g () = f() .

In fact

= (; g ())

is a direct mechanism. Next we want to show that the mechanism implements f() as an equilibriumin dominant strategies. The proof is by contradiction. Suppose not, that is the transmission of his true

characteristics is not a dominant message for agent i. Then there exists

i; i

such that

ui

g

i; i

; i

> ui (g

() ; i) :

But by the denition (14.1) above, the inequality can be rewritten as

ui

g

m

i; i

; i

> ui (g (m

()) ; i) :

But this contradicts our initial hypothesis that = (M; g ()) implements f() in a dominant strategyequilibrium, as m is obviously not an equilibrium strategy for agent i, which completes the proof.


76/86

76 14. Dominant Strategy Equilibrium

The notion of implementation can then be rened in Denition ?? in a suitable way. This is imple-mentation in a very robust way, in terms of strategies and in informational requirements as the designerdoesnt need to know P() for the successful implementation. But can we always implement in dominantstrategies:

Denition 14.0.2. The social choice function f() is dictatorial if there is an agent i such that for all 2 ,

f() 2 fx 2 X : ui (x; i) ui (y; i) for all y 2 Xg .

Next, we can state the celebrated result from ? and ?.

Theorem 14.0.2 (Gibbard-Satterthwaite). Suppose that X contains at least three elements, and thatRi = P for all i, and that f() = X. Then the social choice function is truthfully implementable if andonly if it is dictatorial.

Quasi-Linear Environments. The quasilinear environment is described by

vi (x;t;i) = ui (x; i) ti.

Denote the ecient allocation by x (). Then the generalization of the Vickrey auctions states: ?, ?, ?.

Theorem 14.0.3 (Vickrey-Clark-Groves). The social choice function f() = (x; t1 () ;:::;tI ()) istruthfully implementable in dominant strategies if for all i:

ti (i) =

24Xj6=i

uj (x () ; j)

3524Xj6=i

uj

xi (i) ; j35 : (14.2)

Proof. If truth is not a dominant strategy for some agent i, then there exist i;bi; and isuch thatui

xbi; i ; i+ ti bi; i > vi (x (i; i) ; i) + ti (i; i) (14.3)

Substituting from (14.2) for ti bi; i and ti (i; i), this implies thatIXj=1

uj

xbi; i ; j > IX

j=1

uj (x () ; j) ; (14.4)

which contradicts x () being an optimal policy: Thus, f() must be truthfully implementable in dominantstrategies.

This is often referred to as the pivotal mechanism. It is ex-post ecient but it may not satisfy budgetbalance:X

i2I

ti (i) = 0.

We can now weaken the implementation requirement in terms of the equilibrium notion.


77/86

15. Bayesian Equilibrium

In many cases, implementation in dominant equilibria is too demanding. We therefore concentrate next onthe concept of Bayesian implementation. Note however that with a single agent, these two concepts areequivalent.

Denition 15.0.3. The strategy prole s = (s1 (v1) ;:::;sI (vI)) is a Bayesian Nash equilibrium of mech-

anism = (S1; : : ;SI; g ()) if for all i and all v 2 :

Evi

ui

g

si (vi) ; si (vi)

; vi

jvi

Evi

ui

g

si; si (vi)

; vi

jvi

for all si 6= si (v), 8si (v).

15.1 First Price Auctions

We replace the type notation v by the vector of private valuations

v = (v1;:::;vI)

and denote the distribution over types by symmetric

f(vi) ; F(vi) .

For agent i to win with value v, he should have a higher value than all the other agents, which happenswith probability

G (v) , (F(v))I1

and the associated density is

g (v) = (I 1) f(v) (F(v))I2 :

This is often referred to as the rst order statistic of the sample of size I 1.

Theorem 15.1.1. The unique symmetric Bayes-Nash equilibrium in the rst price auction is given by

b (v) =

Zv0

y

g (y)

G (v)

dy

Remark 15.1.1. The expression in the integral is thus the expected value of

yale - information economics

Documents