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382 social choice

of the actual preferences of real agents. Choosing a single outcome in X for every

proﬁle R of individual preferences determines a mapping ϕ from the set of actual

preference proﬁles

into X, and provides an exhaustive interpretation of endstate

justice. On the other hand this mapping can be interpreted as a direct game form,

namely one where an action by agent i is simply to report his preference ordering, so

that the only strategic move available to an agent is to misreport his preferences. In

most cases, we should expect the agents strategically to misreport. Think of plurality

voting when my favorite candidate is a sure loser, but my second choice is in a close

race to win: the rational use of my ballot is then to support my second choice, instead

of truthfully casting my vote for my ﬁrst choice. Now if the strategic equilibrium of

the direct game form does not, in general, consist of reporting one’s true preferences,

there is no common ground to compare the mapping ϕ qua social choice function,

and qua game form (no matter what criterion of procedural justice we apply to the

game form).

This is why we are particularly interested in those game forms where all agents

always want to tell the truth, namely report their true preference ordering, irrespective

of the reports (truthful or not) sent by other agents. Formally the mapping ϕ from

into X has the property that for all proﬁles R =(R

,...,R

) reporting one’s true

preferences is for every participant a “best response” strategy to the strategic choices

of the other participants.

The strategy “report truthfully any preferences” is then a dominant strategy, and

implements our social choice function in a simple and robust way, irrespective of the

degree of dispersion of the information on private characteristics. Knowing what the

other agents’ preferences are, and/or what they will report, is worthless to any single

participant (we provide examples in the next section).

The convenience of the direct game form and the robustness of the dominant

strategy equilibrium as a rationality scenario are two positive properties making

implementation very credible. But the appeal of a strategy-proof social choice func-

tion runs deeper. As the game form and social choice functions are the same for-

mal object, the same tests of equity apply to both. For instance a property like

anonymity—equal treatment of equal reports—of the mapping ϕ is at the same time

a statement of endstate justice and of procedural justice. In Rawls’s terminology, we

have a situation of “perfect procedural justice,” where the means are justiﬁed precisely

to the extent that the ends are. Endstate and procedural justice coincide.

Next we observe that an even more demanding property than strategy-proofness is

required to pass the most thorough tests of strategic determinism and predictability.

Group strategy-proofness states that it is never proﬁtable for any subgroup or coalition

of agents jointly to misreport their preferences.

Thus not only is the knowledge of other agents’ actions worthless to me, even the

opportunity to coordinate my strategy with that of other agents is useless. Under

group strategy-proofness the prediction that the participants will report their prefer-

ences truthfully is robust against both non-cooperative and cooperative moves. It is

oblivious not only to the dispersion of information mentioned earlier, but also to the

possibility that some players may enter binding agreements.

hervé moulin 383

The advantages of (group) strategy-proof social choice functions are so clear that

the search for such social choice functions in a variety of contexts has been active and

multifarious.

Note that group strategy-proof social choice functions always exist. Two obvious—

and equally repugnant—instances are the imposed social choice functions, always

choosing the same endstate irrespective of the proﬁle of messages, and the dictatorial

social choice functions, choosing for all proﬁles an endstate that maximizes the

preferences of a certain agent—the dictator—ﬁxed once and for all (and if the dictator

is indiﬀerent between several endstates, choosing one that suits best the preferences

of the second person in line, and so on). The imposed social choice functions ignore

eﬃciency entirely, and the dictatorial ones violate even the mildest requirements of

endstate/procedural justice.

The interesting and diﬃcult question is how far can a social choice function go

toward eﬃciency and fairness while preserving the (group) strategy-proofness prop-

erty? The question is mathematically diﬃcult and still widely open in many contexts.

Yet recent research provides a large number of partial answers from which much may

be learned.

6 Strategy-proof Allocation

Mechanisms

.............................................................................

Two excellent surveys on the material of this section are Barberà (2001) and Sprumont

(1995). For the literature review below, the following taxonomy of results on strategy-

proofness will be useful:

positive results: strategy-proofness is compatible with both equity and eﬃciency.

Examples include voting with single-peaked preferences, house barter, matching,

and fair division under single-peaked preferences;

negative results: strategy-proofness imposes a loss of eﬃciency and/or equity so

severe as to force us to drop one or the other requirements altogether. Examples

include voting between three or more alternatives where strategy-proofness is

only compatible with imposed or dictatorial social choice functions, trading of

multiple private goods, and production under decreasing marginal costs;

“neutral” results: strategy-proofness is not compatible with full eﬃciency; yet we

can ﬁnd equitable group strategy-proof social choice functions for which the

eﬃciency loss is not too severe. Examples include public decisions with monetary

compensations under quasi-linear utilities, house assignment with lotteries, and

cost- and surplus-sharing under increasing marginal cost.

The concept of strategy-proofness was invented in the early 1970s, and the ﬁrst two

decades of the literature are dominated by two seminal negative results.

In the voting problem with unrestricted domain of preferences, the Gibbard–

Satterthwaite theorem (Gibbard 1973) conveys the striking result that a strategy-proof

384 social choice

rule among three or more candidate/outcomes must be dictatorial. This result is

technically equivalent to Arrow’s theorem, hence establishes a direct link between

the problems of aggregating and of truthfully revealing individual preferences.

In the problem of exchanging private commodities, a conjecture by Hurwicz in

1972 (and only recently proven in full generality, see Schummer 1997;Serizawa2002)

states a similarly disheartening message: a strategy-proof exchange mechanism in

which agents trade voluntarily cannot be eﬃcient.

Yet even in the basic problems of voting and exchange, the situation is not as

negative as these benchmark results suggest.

When candidates can be arranged along a one-dimensional line and preferences

of the voters are single peaked, majority voting is (group) strategy-proof, eﬃcient,

and fair. This result goes back to Black’s 1948 observation that the median peak is the

non-manipulable Condorcet winner: it does not pay for a voter whose peak is to the

left of the median to report a peak even more to the left, because the median will be

unaﬀected; if this voter reports a peak to the right of his true peak, the only time this

move aﬀects the median is when it shifts the median to the right, hence further away

from the true peak. The entire family of group strategy-proof voting rules in the one-

dimensional model is well understood: all such rules are derived from the median

rule by adding a number of ﬁxed ballots (Moulin 1980). Extending this result to the

popular “spatial voting” model, where candidates live in a Euclidean vector space and

preferences are separable in each coordinate, yields a handful of positive or neutral

results (Barberà, Gu, and Stacchetti 1993;Barberà,Jackson,andNeme1997;Barberà,

Massó, and Neme 1997; Barberà, Massó, and Serizawa 1998; Peters, van den Stel, and

Storcken 1993a, 1993b).

The one-dimensional problem of fairly dividing a single commodity under single-

peaked preferences without free disposal yields similarly a rich family of (group)

strategy-proof social choice functions, interpretable as rationing methods. If the

general family is complex (Barberà, Jackson, and Neme 1997; Moulin 1999b), the uni-

form rationing method stands out as its unique member that is also fair (Sprumont

1991). This positive result proves helpful in the more general and as yet uncompleted

search for (group) strategy-proof methods in the classic Arrow–Debreu economy

with multiple commodities (Barberà and Jackson 1995).

By far the most important family of strategy-proof social choice functions uncov-

ered at the outset of the literature is the versatile class of Vickrey–Clarke–Groves

mechanisms (hereafter VCG). The prototype of these mechanisms is the familiar

second price auction duetoVickrey.Buyersbidfortheobjectandthehighestbidder

wins the object but pays only the second highest price. Because of this feature, the

simple strategy of bidding one’s true valuation of the object (reservation price) is

“dominant:” it is the best reply to any proﬁle of bids by other participants. Bidding

below my true valuation will not change the price I pay if I win, and will only make

it less likely that I win the object. Bidding above my true valuation is consequential

only if I win the object at a price higher than my true valuation. Contrast this with

the more common ﬁrst price auction, where the winner pays his own bid: surely he

hervé moulin 385

will submit a bid lower than his reservation price, since the latter bid eliminates the

possibility of making any proﬁt.

The classic exposition of the VCG mechanism is Green and Laﬀont (1979). These

versatile mechanisms can be used in any resource allocation problem where indi-

vidual preferences are quasi-linear with respect to a numeraire good (the same for

every agent). The corresponding social choice functions associate to every preference

proﬁle a feasible allocation and a proﬁle of monetary transfers. Strategy-proofness

holds if the transfers are chosen in such a way that the net charge to any individual

agent equals the marginal social worth of her own message, plus an arbitrary term

independent of her message. This yields a partial answer to the free rider problem in

the provision of a public good (see Ledyard, this volume), as well as a general compen-

sation method for public decision-making. The appeal of the VCG mechanisms (or

social choice functions) is limited, severely in some contexts, for two reasons: they are

ineﬃcient, and they are manipulable by coalitions, i.e. strategy-proof but not group

strategy-proof.

VCG mechanisms are particularly well suited to assign indivisible objects when

monetary transfers are feasible, as in an auction. Indeed they originate in Vickrey’s

1961 observation that the second price auction is strategy-proof on the buyer side.

They play a growing role in the design of auctions to assign multiple objects (Ausubel

and Milgrom 2002; Bikhchandani and Ostroy 2002; Bikhchandani et al. 2002; Pápai

2003). Other recent applications include cost-sharing (Moulin and Shenker 2001)and

the scheduling of jobs (Mitra, 2001, 2002;Suijs1996).

An early observation in Shapley and Scarf (1974) shows that the barter of indivisible

goods, without monetary transfers, leads to a unique core stable allocation when each

agent is only interested in consuming one object. This unique outcome obtains by the

simple “top trading cycle” algorithm, and deﬁnes a group strategy-proof mechanism

(Ma 1994). Substantial generalizations of this positive result (Ehlers, Klaus, and Pápai

2002; Pápai 2000, 2001; Svensson 1999) throw much light on the rich structure of the

set of group strategy-proof mechanisms for assigning indivisble goods in the absence

of initial property rights (distribution as opposed to exchange).

Bilateral matching is a natural generalization of the assignment problem (without

money), where each agent seeks a match on the other side of the market, and every-

one has preferences over potential mates. The celebrated Gale–Shapley algorithm

provides an elegant and eminently practical resolution of the matching problem,

also a group strategy-proof mechanism from the point of view of one side of the

market. Concrete applications of the algorithm keep emerging, from the assignment

of students to schools, of tenants to housing units, to the exchange of kidneys for

transplant (Roth and Sotomayor 1990;Sönmez1996).

Much new ground has been broken in the models of cooperative productions

where a group of users “share” a given technology in the sense that they all contribute

some input and receive a share of the output. Two well-known examples are the

provision of a public good and the exploitation of a “commons,” namely, a technology

producing a private good under increasing marginal costs. Unrestricted access to the

386 social choice

technology leads to severe ineﬃciencies due to the free rider problem in the former

model, and to the tragedy of the commons in the latter.

The most interesting characterization results combine (group) strategy-proofness

with normative requirements of fairness, in particular the fundamental horizontal

equity (often called anonymity), requiring that equal messages yield equal allocations.

An anonymous group strategy-proof game form always produces an “envy-free”

allocation (Moulin 1993). The nature of the results (positive, negative, or neutral)

depends much on the nature of the good produced by the shared technology. In

the provision of a pure public good problem we have negative or at best neutral

results: for instance, strategy-proofness is not compatible with the natural stand-

alone lower bound, computed as the welfare level of an agent who pays the full cost

of the public good (Saijo 1991;Serizawa1999). On the other hand, the production

of a private good yields strikingly positive results: the serial cost-sharing method

emerges as uniquely fair, (group) strategy-proof, and not too ineﬃcient (Moulin and

Shenker 1992;Shenker1995). This method can also be adapted to the provision of an

excludable public good—a non-rival private good (Deb and Razzolini 1999; Moulin

1994;Olszewski2004). A general characterization of (group) strategy-proof social

choice functions that does not use simplifying requirements of fairness may not be

too far from our reach (Leroux 2004; Moulin 1999a).

7 Concluding Comments

.............................................................................

Group strategy-proofness is a tractable requirement in a number of special-

ized allocation problems: majority voting with single-peaked preferences, uniform

rationing of a single commodity, assignment of indivisble goods and matching,

serial cost-sharing of a private or excludable public good with increasing marginal

cost. Yet the initial negative results continue to place tight limits on the ﬂexibility

that the strategy-proofness property allows to the mechanism designer. In many

cases we must settle for a less demanding equilibrium concept, hence a less plau-

sible prediction of implementation, in order to secure an eﬃcient and endstate-

just outcome. The question is now to strike a compromise between two conﬂicting

objectives:

we want a simple mechanism, with natural message spaces, easily understandable

rules, and a fair distribution of rights (as required by procedural justice).

we want a mechanism implementing the desired social choice function in a

manner as strategically robust as possible (as required by endstate justice).

The precise nature of this trade-oﬀ is still far from clear at this point of a research

eﬀort that started nearly three decades ago. For recent surveys of the implementation

problem, the reader may consult Moore (1992) and Jackson (2001).

hervé moulin 387

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chapter 22

..............................................................

A TOOL KIT FOR

VOTING THEORY

..............................................................

donald g. saari

1 Introduction

.............................................................................

For someone outside of social choice, this area may seem to be discouragingly

consumed with voting paradoxes and/or the impossible such as Arrow’s and Sen’s

theorems. Just thumbing through journals reveals negative results stating that what

we might want to do cannot be done, and all of this is described in a technical

language that even a mathematician would ﬁnd diﬃcult to parse. But in recent years,

positive conclusions have been found and negative assertions put to rest or placed

in perspective. In this chapter I describe some results that could be useful tools to

analyze actual elections. An outline follows:

1. During any close election we might wonder whether the outcome would have

changed had a diﬀerent election rule been used. But the diﬃculty and time

required to compute outcomes often limit us to checking only a couple of

methods. There now exist ways to quickly ﬁnd all outcomes for most of the

voting methods that have ever been considered for actual use. This includes

“positional methods” where weights are assigned to candidates according to

theirpositiononaballot,aswellasapprovalvoting.

2. To adopt a voting rule, we want to identify all of its properties. Using terms

associated with the axiomatic approach, we want to know “what we are getting.”

How can this be done? After a cautionary warning about the axiomatic method,

two approaches are described.

3. The voting literature is ﬁlled with commentary about strategic behavior and

other disturbing anomalies such as where a winning candidate loses because

donald g. saari 391

12 11

(a) Names

(b) “Chair election” profile

9 + 2s

6 + 12s

8 + 9s

Fig. 22.1 Proﬁle representation and tallies

she receives more support. The pragmatic question is to ﬁnd simple ways to

analyze speciﬁc rules to understand whether and when such problems can occur.

I indicate how to do this.

4. Do Arrow’s and Sen’s theorems, partially responsible for two Nobel prizes, really

mean what we have believed for several decades? I indicate why this is not the

case.

As my emphasis is to show how to use these results to analyze and understand daily

concerns, theoretical support and speciﬁc details about the approaches are left to the

references.

2 Representing Profiles

.............................................................................

A proﬁle lists the number of voters who have each ranking of the candidates. But when

computing election outcomes, moving up and down the list to see who is ranked

above whom is tedious. To avoid this problem, I developed a geometric approach

(Saari 1994, 2001a) to describe proﬁles and compute election outcomes. (See Nurmi’s

2003 use of my methods to analyze elections.) The approach uses an equilateral

triangle (Figure 22.1) where the vertices represent candidates {Ann, Barb, Connie}

(denoted by A, B,andC). The ﬁrst step is to assign a ranking to each point in the

triangle. Using a “closer is better” sense, let points closer to the A than the B vertex

represent “A preferred to B”denotedbyA  B. Because the vertical line connecting

the midpoint of the A–B edge with the C vertex identiﬁes all points equidistant from

the A and B vertices, it represents the tied A ∼ B ranking. Points to the left of this

vertical line represent A  B, while those to the right represent B  A.

Similarly, the three interior lines connecting vertices with the midpoint of the

opposite side represent the three binary indiﬀerences. They divided the triangle into

six smaller triangles or ranking regions: each region represents the strict transitive

three-candidate ranking determined by its proximity to each vertex. In Figure 22.1a,

for instance, points in region “3” are closest to the C vertex, next closest to the

A, and furthest from B:theydeﬁnetheC  A  B ranking. The numbers in