Weingast B.R., Wittman D. The Oxford Handbook of Political Economy

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chapter 21

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SOCIAL CHOICE

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1 Introduction

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Social choice is part and parcel of the formal and axiomatic revolution that took

over economic analysis and, to a lesser degree, political and other social sciences in

the middle of the twentieth century. In the shadow of game theory and of the theory

of general equilibrium, social choice theory focuses on the normative foundations of

political and economic institutions.

Understanding the depth of Arrow’s celebrated impossibility theorem was its sole

agenda until the beginning of the 1970s, a line of research that has all but died out in

the last three decades. Saari (this volume) oﬀers a refreshing viewpoint on a whole

range of impossibility results. The current research retains the social engineering

spirit of Arrow’s aggregation of preferences, while reaching out to a much broader

set of resource allocation problems. A good example of this enlarged vision is “fair

division” (Brams, this volume): developed by mathematicians such as Steinhaus in

the 1940sand1950s, then by economists like Foley, Kolm, and Varian in the 1960sand

1970s, it is now a mainstream subject within the ﬁeld of social choice.

In economics the social engineering approach is called “mechanism design.” It

inspires a large body of research in microeconomic theory, addressing mostly positive

(descriptive, predictive) issues, e.g. the inﬂuence of auction design on the seller’s rev-

enue. The social choice component of mechanism design specializes in the normative

aspects of mechanism design, and has evolved into a broad theory of distributive

justice.

This brief overview concentrates on the interface between traditional notions of

endstate justice such as welfarism (Blackorby and Bossert, this volume) and the more

modern theme of strategic implementation, bringing the full methodological beneﬁt

∗

I am very grateful to the editors for critical comments on an earlier draft.

374 social choice

of game theory to the discussion of procedural justice. I propose a simple explanation

for the overwhelming success of this cross-fertilization.

Section 2 outlines a general model of mechanism design. Section 3 deﬁnes the

classical notions of procedural and endstate justice in the context of that model,

and Section 4 argues that a single-minded concern for only one mode of justice

is untenable. Section 5 submits that the property of strategy-proofness captures the

attractive mechanisms for which the two interpretations of justice coincide. Section 6

surveys the extensive research of the last three decades identifying strategy-proof

mechanisms in a variety of microeconomic problems.

2 The Microeconomic Approach

to Distributive Justice

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When political or economic resources are distributed, the two criteria by which a

selﬁsh Homo economicus evaluates the process are his rights and his welfare. When

voting, I wish to maximize the extent to which my vote could inﬂuence the ﬁnal

outcome, and I also wish that the candidate ﬁnally elected be one whom I like better

than other names on the ballot. Most of the normative discussion of allocation mech-

anisms thus revolves around two main questions. With what rights are individual

agents endowed, what inﬂuence can they have on the ﬁnal allocation, by what free,

unconstrained actions? To what degree does the actual distribution of resources, the

outcome, or endstate, fulﬁll the individual values and aspirations of the agents, what

level of welfare does it allow them to reach?

The conceptual dichotomy between rights and welfare has deep roots in

political philosophy: a good example is Berlin’s richly nuanced distinction (Berlin

1969) between negative freedom (the extent of my rights), and positive freedom

(the freedom of choosing my own goals and aspirations, and my ability to ful-

ﬁll them). In a microeconomic formulation the former is captured by a descrip-

tion of the legal moves/messages each participant is allowed to make/send (e.g.

the set of admissible ballots in an election). The latter is the choice of my own

measure of welfare (my preferences) and my ability to reach a high level of

welfare.

Ignoring several subtle aspects of the interaction between individual rights and

welfare (on which more, three paragraphs below), many canonical models of resource

allocation are described as a 4-tuple (N, X,

R, G)where:

the “society” is the set N = {1, 2,...n} of concerned agents, with generic ele-

ment i, i ∈ N.

the set X describes all feasible states of the world, or endstates; in particular X

embodies the set of resources and technological constraints.

hervé moulin 375

the set R contains all possible “welfare functions.” Such a function takes into

account all relevant components of an individual welfare, such as preferences,

values, aspirations, but also tastes and needs, endowments of talents and skills,

disabilities, etc. One possible representation of individual welfare is as a utility

function, associating to each possible endstate (each element of X)anumerical

measure of welfare. Here we adopt the more general “ordinal” description of

welfare as a preference ordering of X:eachelementR of

R is a binary relation

comparing pairs of endstates (deciding which of any two endstates gives the

higher welfare level), but refraining from associating a cardinal measure to each

endstate. Thus

R is the set of all possible welfare orderings R.Theadvantages

of the ordinal approach over the cardinal one are well known (see for instance

Mas-Colell, Whinstone, and Green 1990).

the game form G describing individual rights, and their interaction. It consists

of a set M

of actions among which agent i freely chooses one element m

,anda

mapping f from M

×...× M

into X, specifying which endstate z results for

each proﬁle (m

,...,m

) of individual actions. The notation m

reminds us that

the action often consists of sending a message. When this message is a report of

agent i’s preference ordering R

,wespeakofadirect game form.

The model (N, X, R, G) encompasses many familiar examples. Voting is the sim-

plest one. The set of voters is N,andX is an arbitrary ﬁnite set of candidates. Prefer-

ences over X are only restricted by the standard rationality postulates of transitivity

and completeness (see Mas-Colell, Whinston, and Green 1990). In plurality voting,

each voter writes the name of a single candidate on his ballot, thus the set of actions—

in this case messages—available to voter i is the set of candidates, that is M

= X.The

candidate with the largest number of votes is elected, which deﬁnes unambiguously

(except for the tie-breaking rule) the mapping of G from proﬁles of ballots into X.

More complicated voting rules such as the Borda Count require a much larger set

of actions M

; here each voter reports his entire preference ordering of X, therefore

= R. Multistage voting rules (such as voting by successive eliminations) give rise

to even bigger and more complex action spaces, because the participants are involved

in a multiperiod strategic game.

The Divide and Choose mechanism is another simple illustration of our general

model. The set X describes all possible divisions of a given “cake” between two agents,

N = {1, 2}. The preference ordering R

compares all shares of the cake from the point

of view of agent i. Besides the postulates of transitivity and completeness, further

restrictions may apply, for instance it is natural to assume that a larger piece of cake

is always desirable. The agent with the role of Divider, say agent 1, must choose a

division of the cake in two pieces, therefore M

= X. The Chooser, on the other hand,

can only pick one of the two pieces cut by the Divider, therefore M

speciﬁes which

piece is chosen from every possible division left by the Divider.

It should be clear from the above examples that the model (N, X,

R, G)encom-

passes a huge variety of applications from market mechanisms to political processes

376 social choice

and everything in between. However, the model rests on a handful of heroic assump-

tions worth repeating. The three basic components, welfare (

R), rights (M

), and the

set of endstates (X), are three clearly deﬁned and separate entities whose interaction

is limited to two aspects. First, the endstate z only depends on the proﬁle of actions

m via the mapping f , namely the game form. Second, the ﬁnal welfare level only

depends upon the endstate z, and not directly upon the individual actions.

Conspicuously absent from the model are two important elements of the interac-

tion of rights and welfare. First, the fact that the nature and extent of my rights aﬀect

my welfare: which game I play, and what role I have in it matters to me. Formally the

whole set M

inﬂuences R, not just my particular action m

. Second, I bear some

responsibility in shaping my own welfare because I am free to choose my values

and goals.

The former point inspires the literature on freedom of choice, where the central

question is to extend an ordinary welfare relation comparing single outcomes, to

another welfare relation comparing choice sets (subsets of outcomes) (see Bossert,

Pattanaik, and Xu 1994; Dutta and Sen 1996; Laslier et al. 1998; Pattanaik and Xu 1998,

2000a, 2000b;Sugden1998). The latter point explains the paradox of adaptive pref-

erences: agents who adapt their expectations to their lot enjoy a higher welfare than

those who do not, and this responsibility should be accounted for when we deﬁne fair

endstates. One solution is to introduce additional variables in the microeconomic

formulation, so as to diﬀerentiate between those characteristics for which an agent

is responsible, and those for which she is not: Roemer and Fleurbaey initiated this

fruitful line of research (Bossert 1995;Fleurbaey1995a, 1995b;Roemer1993, 1996,

2000;Sen1991).

Another simplifying assumption of the general model is the identiﬁcation of the

game form G with an allocation of rights. Since Gardenfors (1981), this is an approach

endorsed by many authors (e.g. Gaertner, Pattanaik, and Suzumura 1992). The more

traditional approach insists that a right confers on an individual the ability to ﬁx some

features of the ﬁnal outcome; that the two approaches are not identical is discussed in

particular by Deb, Pattanaik, and Razzolini (1997).

3 Endstate Justice versus

Procedural Justice

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These shortcomings notwithstanding, the logical separation of welfare and rights

has considerable advantages. By separating ends—the distribution of welfare—from

means—the set of actions and the game form G—we can contrast and combine two

familiar one-sided interpretations of justice: on the one hand endstate justice (e.g.

classical utilitarianism), on the other hand procedural justice, rooted in the liberal

tradition.

hervé moulin 377

Endstate justice: fair outcome. Endstate justice is a judgment on the proﬁle

,...,R

) of individual characteristics and the state of the world z, deciding

whether this particular endstate is equitable at this particular proﬁle. The equity

criteria bear on the individual welfare levels, as well as on the endstate.

The most popular criterion is an index of collective utility, aggregating cardinal

measures of individual welfare (Blackorby and Bossert, this volume). An alternative

route eschews cardinal measures of welfare and relies only on the proﬁle of ordinal

preference orderings, together with some physical characteristics of the endstate. A

very inﬂuential instance of this ordinal interpretation of endstate justice is the test of

“No Envy.” We are dividing a given bundle of resources (a “cake”) among participants

endowed with identical property rights over the resources, and diﬀerent preferences

over the various shares of resources. Everyone cares only about the share she receives.

We say that a particular division of the cake is “envy free” if no participant prefers

the share of someone else to her own share. For instance, the outcome of the Divide

and Choose procedure is envy free if the Divider prudently cuts two pieces of equal

value (to him). For very general division problems, a systematic way to obtain an

envy free and Pareto optimal allocation is to endow each participant with an equal

share of the resources and compute a competitive equilibrium from these egalitarian

property rights. See Moulin and Thomson 1997 for a review of the No Envy test and of

competing ordinal interpretations of endstate justice in microeconomic fair division

problems.

In some important problems of fair division, individual characteristics capture

the pattern of responsibilities in the production of a joint surplus or cost. A typical

example involves a scholar delivering lectures in several diﬀerent cities: the question

is to divide fairly her total travel cost between her successive hosts. We know the cost

of her entire trip, as well as the (presumably lower) cost of all trips in which she

would have visited only a subset of the original cities. The celebrated formula known

as the “Shapley value” proposes to compute the cost share of each city as follows:

order the cities arbitrarily and compute the incremental cost share of a given city (the

cost of adding this city to the set of cities preceding it in the given order); then take

the average of these incremental shares over all orderings, each with equal probability.

The normative properties of the Shapley value, and of several extensions and variants,

are the subject of much axiomatic research (see Moulin 2002; Moulin and Sprumont

2005;Sprumont1998;Thomson2003).

The normative statement of endstate justice is oblivious to any particular pro-

cedure we may use to implement a particular endstate. The information about

individual characteristics relevant to the computation of a just endstate is pri-

vate to each participant, and must be somehow retrieved from these agents. As

discussed in the following sections, this elicitation process is not straightforward,

because rational selﬁsh agents cannot be expected to report their private informa-

tion truthfully, if they can improve their welfare by not doing so. This is the fa-

miliar “incentive compatibility” issue: because crucial information is dispersed, the

process of gathering it to determine a just endstate is a strategic game G,exactly

as in the previous section. This game must be carefully designed to allow proper

378 social choice

discovery of the necessary information. Yet in the context of endstate justice, neither

the design of G, nor the behavior of its players has any normative content. The

judgement of justice is entirely ex post, oblivious to the means by which the endstate

is reached.

Procedural justice: fair play. Procedural justice is a judgement on the game form G

itself, independent of the way the game is subsequently played by the agents, and of

the particular endstate that results. The normative discussion is limited to the rules of

the game (the procedure); no outcome (endstate) can be called unfair if the rules have

been scrupulously observed while the game was played, just as a fair lottery cannot

produce an unfair outcome. This time the judgement of justice is entirely ex ante;it

bears only on the means regardless of the ends.

Consider Divide and Choose. If we deem this procedure fair to divide a certain

cake among two given agents, then it does not matter if the Divider plays the game

well or not. If she has some idea about the preferences of the Chooser, she may be

able to obtain a share worth more than one half (to her), provided her expectation of

the chooser’s behavior is correct; if she is wrong and ends up with a share worth less

than one half, she can only blame herself, not the procedure.

Naturally, a fair procedure may yield an unfair endstate, and a fair endstate may

result from an unfair procedure.

Examples of the former include provision of a public good by voluntary subscrip-

tion, where a single agent often ends up bearing the entire cost of the good (see

Ledyard, this volume). Indeed, if Ann likes the good enough that she is willing to

pay the full price of the public good, every other agent will gladly get a free ride and

this is a Nash equlibrium conﬁguration. Another instance is when fair competition

results in a blatantly unfair endstate. The canonical example is the celebrated “gloves

market” with 101 owners of a right glove and 100 owners of a left glove. Assume a

pair of gloves is worth $1 but a single glove is worthless. If the price of a right glove is

positive, there is excess supply of these gloves; a price of zero for the right gloves is the

only way to balance demand and supply. Then the owners of a left glove each get one

right glove so that the competitive equilibrium gives all the surplus to the left glove

owners.

Examples of the latter include the choice of the “just” candidate by a benevolent

dictator; or the envy-free division of the cake between two agents by the Divide and

Choose game form. In this asymmetric game form, the Divider cuts the pie in two

shares among which the Chooser picks the one she prefers. In this game both agents

can guarantee that their share is at least as valuable (to them) as the other share: the

Divider can cut to shares of equal value to him.

Endstate justice and procedural justice pursue similar goals. They look for

“solutions” to simple problems of resource allocations and justify these solutions

by a suitable set of axioms (normative requirements). Under procedural justice this

solution is a distribution of rights, a game form; under endstate justice it is a social

choice function, associating an outcome deemed just to every proﬁle of individual

characteristics.

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4 The Complementarity of Endstate

and Procedural Justice

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We develop the commonplace “means matter as well as ends” in the context of the

general model of Section 2.

Procedural justice is not concerned with the strategic properties of the procedure.

Whether or not the game has a zero, many, or single equilibrium outcome does not

matter. Even irrational actions by the agents are of no concern: my negative freedom

includes the privilege of being unpredictable (Berlin 1969). A serious objection to

this position comes from the familiar concept of eﬃciency, also known as Pareto

optimality. An outcome z (endstate) is Pareto optimal (eﬃcient) if there exists no

otherfeasibleoutcomewhichisweaklypreferredtoz by all agents, and strictly

preferred to z by at least one agent. This is the only rationality test of the endstate

that is common to all doctrines of the social order derived from methodological

individualism.

In our model, the requirement of eﬃciency means that upon playing the game G,

an eﬃcient outcome should result. This forces our attention to the strategic properties

of G. In particular,

suppose that the game G has no equilibrium for some proﬁle of individual

characteristics. This precludes rational behavior by the participants: anything can

happen, including ineﬃciencies. A typical example is majority voting when the

majority relation has a cycle. Where we stop in the cycle may be a matter of

pure luck, or of who is able to control the agenda of the relevant committee. The

cycles of the majority relation encompass large subsets of ineﬃcient outcomes so

there is the very real possibility of an ineﬃcient endstate. This point is emphati-

cally demonstrated by McKelvey’s theorem (McKelvey 1976) in the spatial voting

context.

symmetrically, suppose our game G has a large set of equilibrium outcomes,

then the selection of a particular endstate is a diﬃcult coordination problem.

If the coordination fails, ineﬃciency is to be expected. The standard example

is the division of a surplus by means of the Nash demand game. Each player

independently requests a share of a dollar. If the demands are compatible, they

are satisﬁed; otherwise everyone goes home empty handed. Any division of the

dollar between the players is a legitimate (and eﬃcient) equilibrium of the game.

However, the deﬁnition of the game does not help the participants to coordinate

their demands on a particular division of the dollar. If they choose their demands

independently, chances are they will not be compatible, resulting in an ineﬃcient

endstate.

To meet the compelling requirement of endstate eﬃciency, the game form must

have a unique equilibrium outcome (or at least a small set of such outcomes), more-

over this outcome must pass the test of Pareto optimality.

380 social choice

A good example where these demanding properties are met is the exchange and

distribution of private goods in a competitive environment: many agents with convex

preferences each with a “small” initial endowment relative to the market size. The

set of (eﬃcient) competitive allocations is then small and coincides with the set

of equilibrium outcomes under a variety of strategic scenarios, both cooperative

(core stability) and non-cooperative (market bidding games) (see e.g. Mas-Colell,

Whinston, and Green 1990).

On the other hand, many procedures commonly used to allocate certain scarce

resources display either a unique but ineﬃcient equilibrium outcome, or a large or

empty set of such equilibria.

Examples of the former include the exploitation of common property resources

under free access (Hardin 1968), and the provision of public goods by voluntary

subscription, the classic exposition of which is by Olson (1965)(seealsoLedyard,

this volume).

Examples of the latter include voting by a qualiﬁed majority: the quota is the

relative size of the majority required to pass a motion. With a small quota of 51 per

cent, we only require simple majority and cycles are likely to appear: in this case there

is no equilibrium when coalitions of agents can easily coordinate their actions. If the

quota is large, say 90 per cent, then the equilibrium set is likely to be large: there are

many outcomes z that cannot be defeated by any other outcome z



because this would

require that 90 per cent of the voters prefer z



to z. (See Nakamura 1979.)

We can speak of private ownership as the natural system of rights when the issue

is to produce and exchange private goods, because the resulting trading game has all

the desirable strategic features, namely a small set of eﬃcient equilibrium outcomes.

But for many important resource allocation problems, no natural system of rights, no

canonical game form stands out to our attention. Examples include:

the quota (size of the qualiﬁed majority) we should require to amend the consti-

tution;

how we should regulate access to common property exhaustible resources, so as

to avoid the tragedy of the commons;

how we should produce and trade under decreasing marginal costs, as is the case

for “natural monopolies;”

how we should share costs under widespread externalities.

The point is that any reasonable answer to these questions must take into account

the endstate consequences of the procedure, which procedural justice alone is delib-

erately ignoring.

We turn to the central objections to “pure” endstate justice and return to the imple-

mentation problem alluded to in Section 3. As some of the individual characteristics

included in R

are privately known only to the concerned agent (and perhaps to

some of her fellow citizens), the mechanism designer relies on a game form to elicit

these private characteristics, either directly by asking each agent to simply report this

piece of information, or indirectly by designing a game form of which the equilib-

rium outcome (or outcomes) is (are) just in the endstate sense. Such a game form,

hervé moulin 381

whether direct or indirect, “implements” the social choice function recommended by

endstate justice.

If endstate justice is the only motivation of our benevolent dictator, the choice of

the game form G is “value free” and we can take an opportunistic attitude toward

the resolution of the implementation problem. Any mechanism with the correct

equilibrium outcomes (as speciﬁed by the social choice function) is legitimate, even

if this involves one or more of the following features:

endowing agents with unequal rights (a non-anonymous game form), even

though the social choice function to be implemented is anonymous and treats

equal agents equally;

using coercive threats to extract private information from the agents and/or to

inﬂuence their strategic choices; metaphorically speaking, burning the agents’

feet to make them reveal their characteristics under duress;

using more subtle, yet no less objectionable, twists to force the agents to

reveal mutually their private information, as in the mechanisms introduced

by Maskin (1985) amounting to a sophisticated version of “unanimity at gun-

point:” the agents must unanimously agree in their report about the entire

proﬁle of characteristics, lest the dictator enforces an endstate very bad for

everyone;

changing the game form frequently, as dictated by circumstantial information,

for instance about the statistical distribution of individual characteristics.

These features run contrary to the simple and pervasive intuition that agents do

care about their rights and about the decision process. In the case of the implementa-

tion problem, this means that the benevolent dictator should not be allowed to extract

the private information held by individual agents in any way she pleases. She should

not bully them into revealing information without giving them some real inﬂuence

on the process itself. Not to place any limits on the procedures that can be used

aﬀords her too much power. Means matter as well as ends.

5 Endstate Justice cum

Procedural Justice

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Granted that means matter as well as ends, what is the next logical step? How can we

combine endstate and procedural justice in the general model of Section 2?

In this section we deﬁne formally the concepts of strategy-proofness and group

strategy-proofness. We submit that they provide a general answer to the above ques-

tion. We postpone until the next section the discussion of examples showing these

two concepts at work.

Assume from now on that R

represents a preference ordering over X, and varies

within the domain

R, known to the mechanism designer, who however is not aware