Econ 20740: Analysis of Collective Decision-Making
Richard Van Weelden∗
May 22, 2017
1
Aggregation of Preferences
In this course we will be interested in environments in which decisions need to be made
by a group. We begin with the topic of social choice theory. When looking at decisions
made by groups, there are two principal components: the aggregation of preferences and the
aggregation of information. Social choice theory largely deals with the first of these, and
we’ll consider informational issues later in the course. In the first part of the class we ask the
questions: If we knew everyone’s preferences, how would we determine which outcome(s) are
preferable to others. For this analysis we will follow Chapters 1 and 7 of Taylor and Pacelli
(2008).
What are some of the issues inherent in aggregating preferences? Recall that in previous
economics classes (Econ 200, Econ 201) we represented individual’s preferences by a utility
function. One alternative is preferred to another if it gives the individual a higher utility.
What happens when there are multiple individuals and they disagree on which alternative
is preferred? If one alternative pareto dominates another the comparison is easy: if all
individuals would prefer A to B then clearly A should be preferred to B by the group.
However we have seen in Econ 200/201 that there are generally many pareto dominant
allocations, so we won’t always be able to rely on a pareto criterion when comparing different
allocations. Allocations that are pareto dominated should be rejected, but how do we choose
among different pareto optimal choices?
Part of the complexity is that it is difficult to make inter-personal comparisons. Recall
that a utility function is meant to represent preferences. This allows us to analyze the
consumer’s problem by maximizing an objective function using calculus. Recall that in
order to represent an individual’s preferences by a utility function we need the individual’s
preferences to be complete and transitive:
1. Complete: For any two alternatives A and B either A B or B A.
2. Transitive: If A B and B C then A C.
∗
Assistant Professor of Economics, University of Chicago. Email: [email protected].
1
Complete means simply that the individual is always capable of comparing the two alternatives and determining which alternative she prefers. Transitive means that, if A is
preferred to B, and B is preferred to C, then A must be preferred to C. This is necessary since, if C were preferred to A then the individual would not be able to choose when
A, B and C are all available. If an individual has transitive preferences then we can assign a higher utility level to more preferred alternatives and compare alternatives by their
associated utility levels.
In this class we will consider the case where there are n > 1 individuals/voters, and each
individual i = 1, . . . , n has preferences i over the some set of alternatives X. We will be
interested in finding a social choice function f which maps the individual’s preferences and
the the available alternatives into the outcome that is chosen or preferred. That is, a social
choice function is a mapping,
f : X × (1 , . . . , n ) → X
If there is not a unique alternative preferred then this mapping would not be a function but
rather a correspondence with a set of preferred alternatives. For this reason we will often
talk about a social choice rule.
Notice that one way to choose an alternative is to take a utilitarian approach. This
approach is often associated with the philosopher Jeremy Bentham. If individual preferences
i are represented by a utility function ui we can define a social welfare function
U (x) = Σni=1 ui (x)
and select x ∈ X to maximize U (x). However, while we can (and sometimes do) use a
utilitarian approach it has some drawbacks. As we have seen in Econ 200/201, if preferences
can be represented by a utility function u the same preferences can also be represented
by any monotonic transformation of u. So which utility function to use? Generally with
heterogenous preferences we need different utility functions to represent the preferences of
different individuals, and how to choose which of the utility functions to use? Depending
on which utility function we choose would give a different utilitarian social welfare function
which would lead to a different choice. We will explore these issues in problem set 1.
2
Collective Choice with Two Alternatives
We begin, by supposing there are only two alternatives, so X = {A, B}. As mentioned above
one approach is to construct a utilitarian social welfare function, and select the alternative
that maximizes the sum of all voters utilities. Another approach is to consider which alternative is preferred by a majority of the individuals. We could then define the social choice
function that always selects the alternative that a majority of the voters prefer. This is
perhaps the most common way of making decisions at least in democratic societies: vote
by majority rule. Assume there are n individuals/voters and assume that n is odd. Also
assume that no individual is indifferent between A and B. By assuming an odd number of
individuals and no assuming there is no indifference we ensure that there will be no ties.
Definition 1. A social choice function satisfies majority rule if alternative A is preferred to
B if and only if A i B for a majority of individuals.
2
While majority rule is not perfect—it does not take into account the intensity of preferences in any way—it has several advantages. Moreover, if the individuals have binary
preferences, so either they prefer A or they prefer B, it is not clear what intensity of preference means. To say that person 1 has a stronger preference for one alternative than person
2 requires some other dimension that the individuals care about so that we can consider
the differing willingness to sacrifice in the other dimension of the sake of this dimension.
For example, person 1 might have a greater willingness to pay money for their preferred
alternative or to spend time showing up to vote or attending a meeting to get their preferred
alternative, etc. We will discuss such environments later in the class, but for now consider
only a setting in which there are only two alternatives and no other relevant dimensions with
which to compare intensity of preferences.
The first advantage of majority rule is that it always allows us to make a choice: if there
are an odd number of individuals and no individual is indifferent then one alternative is
preferred to the other.1 Second, it treats all voters equally: if the preferences of two individuals are reversed the same alternative would be preferred. Third, it treats the alternatives
equally, and is not biased in favor of one alternative or the other. Finally, if one alternative
becomes more popular (more people prefer it than before), it can only become more likely
to be chosen. While there are many other decision making rules we could consider (appoint
a dictator, weighted voting, super-majority requirements, etc), May’s theorem shows that
majority rule is the unique decision making mechanism to satisfy all of these criteria.
Theorem 1. (May 1952) Suppose there are n (odd) individuals and two alternatives, X =
{A, B}. Then majority rule is the unique social choice function to satisfy the following
properties:
• Anonymity: For any i, j and preferences i , j , 0i , 0j such that A i B, A ≺j B, A ≺0i
B, A 0j B, then f ({A, B}, (1 , . . . , n )) = f ({A, B}, (1 , . . . , 0i , . . . , 0j , . . . , n )).
• Neutrality: For each x ∈ {A, B}, if f ({A, B}, (1 , . . . , n )) = x and, B 0i A if and
only if A i B for all i = 1, . . . , n, then f ({A, B}, (01 , . . . , 0n )) 6= x.
• Monotonicity: Consider two preference profiles i , 0i for some individual i such that
A ≺i B, and A 0i B. Then if f ({A, B}, (1 , . . . , n )) = A then
f ({A, B}, (1 , . . . , i−1 , 0i , i+1 , . . . , n )) = A. Similarly, if
f ({A, B}, (1 , . . . , i−1 , 0i , i+1 , . . . , n )) = B then f ({A, B}, (1 , . . . , n )) = B.
Anonymity says that it only matters the number of people who prefer A to B, not who
those individuals are, in order to determine whether A is preferred to B. This is usually
considered desirable since it treats everyone equally and gives them an equal weight. However
it is violated in many rules used in practice, from systems that give only some members voting
rights to the electoral college used to elect U.S. Presidents. Neutrality reflects that neither
alternative is advantaged over the other, so if all preferences are reversed this would reverse
which alternative is chosen. Neutrality is natural if we don’t have an ex-ante reason to
think one alternative is preferred to another; it is violated, for example, with super-majority
1
And if we had an even number of voters split evenly over which alternative they preferred it would be
natural to be indifferent over those alternatives. Similarly if some voters are indifferent.
3
requirements or when unanimity is required (e.g. jury deliberations). Finally, monotonicity
says that increasing the number of individuals who prefer one alternative can never cause
that alternative not to be chosen.
So we see that one way to make decisions between two alternatives is by majority rule.
And if we want to avoid making interpersonal utility comparisons May’s theorem tells us that,
in environments in which neutrality, anonymity, and monotonicity are desirable properties,
majority rule is the “correct” way to make decisions. As we will see when there are more
than two alternatives things get more complicated.
3
The Condorcet Paradox
Now we consider collective choice with more than two alternatives. For simplicity, we assume
none of the voters are indifferent over any of the alternatives. In general, having three
alternatives is sufficient to illustrate the difficulties. When there are three alternatives there
will not always exist one alternative that is the first choice of a majority of voters. One way
to compare alternatives is to compare each two alternatives in X by majority rule, and say
that one alternative x ∈ X is preferred to another, x0 , if a majority of voters prefer x to x0 .2
This was suggested by the Marquis de Condorcet, and any alternative that is not defeated
by any other alternative is referred to as a Condorcet winner.
Definition 2. An alternative x ∈ X is a Condorcet winner if, for all x0 ∈ X, x i x0 for at
least n/2 individuals.
If a Condorcet winner exists it should (arguably) be preferred to any alternative that
is not a Condorcet winner: a non-Condorcet winner means that more than half of the
individuals could be made better off. However, a Condorcet winner may not exist.
Let X = {A, B, C}, and assume that n = 3. So we have three voters and three alternatives. Consider the following preferences:
• Voter 1: A 1 B 1 C
• Voter 2: B 2 C 2 A
• Voter 3: C 3 A 3 B
Notice that, voter 1 and voter 2 both prefer B to C. So, since a majority prefers B
to C we should prefer B to C. And since a majority (1 and 3) prefer A to B we should
prefer A to B. So, for preferences to be transitive we must have that A is preferred to C.
However, voters 2 and 3 both prefer C to A, so a majority prefers C to A. This means that
aggregating preferences by Condorcet’s method violates transitivity and a Condorcet winner
may not exist.
This is the famous voting paradox or Condorcet paradox, discovered by the Marquis de
Condorcet in the late eighteenth century. It shows that, even if each voter has transitive
preferences, aggregating preferences by majority rule can produce non-transitive preferences.
2
Another reason we might apply this method is that there could be three alternatives but only two of
them would be available at each time.
4
And we know from Econ 200/201 that having non-transitive preferences means that we
can’t use Condorcet’s method to represent the preferences with something akin to a utility
function.
4
Aggregating Preferences with 3 (or more) Alternatives
The Condorcet Paradox hints at the difficulties associated with constructing a social choice
function with more than two alternatives. We can use the Condorcet method and compare
each tuple. However, as the above example demonstrates a Condorcet winner need not exist.
This means that we wont always be able to make a choice. However, it will exist in many
circumstances.
Example 1. Let X = {A, B, C}, and assume that n = 5. Consider the following preferences:
• Voter 1: A 1 B 1 C
• Voter 2: B 2 C 2 A
• Voter 3: C 3 A 3 B
• Voter 4: A 4 C 4 B
• Voter 5: B 5 A 5 C
Notice that since A is preferred to B by voters 1, 3, and 4, and preferred to C by voters 1,
4, and 5, that A is the Condorcet winner here.
In the above example a unique Condorcet winner exists. This is not always the case:
there could exist multiple Condorcet winners. If we add a sixth voter with the following
preferences there are two Condorcet winners.
• Voter 6: B 6 A 6 C
Notice that since A is preferred to B by three voters (1, 3, 4) and to C by four voters (1, 4,
5, 6), A is not defeated by either B or C and so remains a Condorcet winner. B is preferred
to A by three voters (2, 5, 6), and to C by four voters (1, 2, 5, 6) and B is also a Condorcet
winner.
Several other methods have been suggested. Perhaps the most common (e.g. in most
elections in the United States and Westminster Democracies) is to choose by plurality rule.
Under plurality rule, the alternative that is the first choice of the most voters is the one
selected (if tied we could declare multiple winners then decide by some other criteria such a
flipping a coin). Note however that plurality rule does not guarantee that a Condorcet winner (if it exists) will be selected. This was arguably the case in 2000 when George W. Bush
defeated Al Gore but, given that a majority of the Ralph Nader voters probably preferred
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Gore to Bush, it is possible that Al Gore was the Condorcet winner.3 In the literature this
often described as saying that plurality rule violates “Independence of Irrelevant Alternatives”. Since Ralph Nader was clearly not going to be elected, individuals’ ranking of Nader
is in some sense irrelevant relative to their ranking of Bush and Gore. Arguably, whether
Bush or Gore would be selected should depend only each voter’s comparison of Bush and
Gore. However, plurality rule does not guarantee this, so one alternative could be selected
even if a majority prefers a different alternative.
Example 2. Suppose 49% of individuals have preferences B G N , 48% of individuals
have preferences G B N , and 3% of individuals have preferences N G B. Then B
is chosen by plurality rule, but if N was removed from the choice set G would defeat B with
51% of the individuals preferring G. Note that here G is the Condorcet winner.
While Bush and Gore had pretty even support and Nader received few votes similar issues
can arise with three “serious” candidates. Former professional wrestler Jesse Ventura was
elected Governor of Minnesota in 1999 as a third party candidate with 37% of the popular
vote, and exit polls indicated that he would have lost a head to head election with either
of the major party candidates (Norm Coleman and Hubert Humphrey III). Similarly, in the
2016 Republican primary, it is possible (though far from certain) that Donald Trump would
have been defeated in many of the early states if he were running in a two candidate election:
during the early states many different candidates divided the anti-Trump vote. Any choice
rule that satisfies independence of irrelevant alternatives will select a Condorcet winner—if
a Condorcet winner exists—but plurality rule does not satisfy independence of irrelevant
alternatives.
Another well-know approach to aggregating preferences is the Borda rule. In the Borda
rule, if there are k alternatives, each individual ranks the alternatives in order of desirability.
Each alternative is then given k points if it is the first choice of an individual, k − 1 points
if it is second, down to 1 point for the individual’s least preferred alternative. The points
are then added up across the individuals, and the alternatives are compared based on the
sum of their points. While its use in political settings is rare, some variation of this is used
to rank NCAA college sports teams, determine baseball MVPs, and in the Eurovision song
contest. Obviously the Borda rule does not satisfy independence of irrelevant alternatives
and will not guarantee that a Condorcet winner, if it exists, is selected. The Borda rule
embraces that it doesn’t satisfy independence of irrelevant alternatives since it regards the
ranking as informative. That is, knowing whether a voter considers a candidate the second
best alternative or the worst alternative provides information about the voter’s estimation
of this alternative. But the concern is that changing the choice set can have an unnatural
effects on the outcome.
Consider the case with 5 voters from Example 1 above. In that example, under the
Borda rule, A receives 11 points, B receives 10 points, and C receives 9 points. So A would
be selected under Borda rule. Now suppose we add a new alternative B 0 in which we get
outcome B but everyone pays a penny. This is a pareto dominated alternative since all
3
US presidential elections, because of the electoral college, is not exactly a plurality rule election, and
Gore received more votes than Bush but still lost the election. However, each state is determined by plurality
rule and Gore may well have been the Condorcet winner in Florida.
6
individuals would prefer B to B 0 . Since a penny is a small amount, we would expect the
preferences to be
• Voter 1: A 1 B 1 B 0 1 C
• Voter 2: B 2 B 0 2 C 2 A
• Voter 3: C 3 A 3 B 3 B 0
• Voter 4: A 4 C 4 B 4 B 0
• Voter 5: B 5 B 0 5 A 5 C
Adding this alternative doesn’t reveal any information about preferences (since a penny is
small) and the alternative B 0 would never be chosen. Still adding B 0 changes the outcome
under Borda rule: now B receives 15 points, A receives 14 and C would receive 11. So, after
changing the choice set to allow for alternative B 0 , B would be chosen instead. Notice that
A remains the Condorcet winner.
Another approach, more common than the Borda rule in practice, is the Hare system.
This is often referred to as single-transferrable vote, and a variation of the Hare system is
used to run elections in countries such as Ireland and Australia and for Party leadership
elections in some countries (e.g. the New Democratic Party in Canada). Under the Hare
system, the alternative(s) that is/are the top choice of the fewest number of individuals
is removed.4 This is continued until all remaining alternatives are the first choice of the
same number of individuals. The Hare system, and the closely related approach of run-off
elections, is a popular proposal among those who favor electoral reform. However the Hare
system is not a panacea. Not only does it often fail to select a Condorcet winner (if it
exists), but it can also fail to satisfy monotonicity: if individuals become more favorable
to one alternative this can cause it to no longer be selected. This is demonstrated in the
following example.
Example 3. Suppose there are 17 individuals and the preferences are as follows:
• Voters 1-7: A B C
• Voters 8-12: B C A
• Voters 13-16: C B A
• Voter 17: C B A
As there are seven with A their most preferred, and five each with B and C most preferred,
the Hare procedure removes both B and C in the first round meaning that A is selected [Note:
though A is selected, B is the Condorcet winner]. Now suppose we change the preferences of
Voter 17 so that she likes alternative A more than C and B.
4
A related approach of having a run-off between the top two vote getters is used in other systems such
as electing the French president and in Louisiana Senate races. It has also recently been introduced in other
states, such as California, as an alternative to party primaries.
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• Voter 17: A C B
Now there are five for whom B is most preferred and four for whom C is. So C is removed at
the first stage. Now we have a head-to-head comparison on A and B: as eight voters prefer
A and nine voters who prefer B, B is chosen over A. So making A more desirable to voter
17 causes it to no longer be chosen.
Another approach, that is sometimes used, known as sequential pairwise voting, could
violate an even more fundamental property, as it may not produce a pareto optimal outcome.
Under sequential pairwise voting, there is a fixed order with which the alternatives are
compared to each other by majority rule, with the winner compared to the next alternative
in sequence. For example, if X = {A, B, C, D}, we could compare A and B, then the winner
with C, and finally compare the winner of that comparison with D. As the following example
shows, this approach may not lead to an outcome that is pareto optimal.
Example 4. Suppose preferences are as follows:
• Voter 1: A 1 B 1 D 1 C
• Voter 2: B 2 D 2 C 2 A
• Voter 3: C 3 A 3 B 3 D
Notice that, with these preferences, B pareto dominates D. If we use the sequential pairwise
procedure then A would defeat B in the first round (voters 1 and 3 prefer A to B), then C
would defeat A (voters 2 and 3 prefer C to A) subsequently, and finally D would defeat C
(voters 1 and 2 prefer D to C). So alternative D would be chosen even though it is pareto
dominated by alternative B.
Finally, one alternative that has been used in many cases historically (e.g. absolute
monarchy) is dictatorship. This approach works in terms of ensuring complete transitive
preferences, and the choice is independent of irrelevant alternatives. Choosing the most
preferred alternative of one agent is a simple problem that we have experience solving.
However it has the obvious drawback that only one individual’s preferences matter. It
is then an very strong rejection of the anonymity property we discussed regarding May’s
theorem.
It should be noted that when considering each of these choice rules we are assuming
that individuals’ true preferences are known. If the preferences are not known, individuals
must reveal their preferences through voting or in some other manner. Different social
choice rules are more prone to manipulation by individuals mis-reporting their preferences—
independence of irrelevant alternatives and monotonicity will be important for ensuring
that individuals have an incentive to report truthfully. We will discuss the incentives for
individuals to truthfully reveal their preferences later in the course.
5
Desirable Properties for Social Choice
As the above analysis shows, while there are many different ways to aggregate preferences,
there are criticisms of each of the ways we have suggested so far. It is then worth asking
8
whether a social choice rule that satisfies all the properties we have suggested as desirable
exists or not. If so then such a rule would be a candidate for the “correct” way to aggregate
preferences. If not then there is no “correct” way to aggregate preferences, and whichever
approach we take will involve compromises. While we have so far been looking for a choice
rule that maps the preferences of the individuals into an alternative that is selected, we now
look for a way to rank all the alternatives. We now define a social choice ordering (so as
not to conflict with the terminology used above). A social choice ordering is a function from
the set of alternatives, and the preferences of the individual voters, into a social preference
relation,
f (X, (1 , . . . , n )) =
The social preference relation ranks each of the alternatives in X in order of their desirability (with the possibility of ties). This allows us to order all the alternatives, which will
be useful for precisely defining the conditions we want the social choice ordering to satisfy,
and will simplify the proof of our main result. Note, of course, that to determine which
alternative would be selected is the same as identifying the most preferred alternative.
Before proceeding let us state a list of properties that we often want a social choice
ordering to satisfy precisely. The first thing we want to do is ensure that it is always possible
to make a choice: while we may identify multiple alternatives as equally good, and so we can
be indifferent between certain alternatives, we can’t reject all alternatives or find a pair that
we can’t compare. That is, for any preference profile, we want the resulting social preference
relation to be complete and transitive. Second, at a minimum we want to make sure that the
alternative chosen is pareto optimal—we know that if something is pareto dominated that
there exists a different alternative that is preferred by all voters. We insist on a very weak
notion of pareto optimality: that it is not possible to make all individuals strictly better
off.5 We also want to make sure that the social choice rule is monotonic: if one alternative
is made more desirable to at least one voter then this shouldn’t cause it to be ranked lower
from a social perspective. Perhaps more controversially, we want the social choice rule to
satisfy independence of irrelevant alternatives. Finally, we have a preference for treating
different individuals equally and so are averse to dictatorial choice rules. Formally we define
these conditions as follows.
Definition 3. (Unrestricted Domain) A social choice order satisfies unrestricted domain
if, for all preference rankings (1 , . . . , n ), f (X, (1 , . . . , n )) =, where is a complete,
transitive preference relation over X.
Definition 4. (Pareto Optimality) A social choice ordering satisfies pareto optimality if
x y for all x, y ∈ X for which x i y for all i = 1, . . . , n.
Definition 5. (Monotonicity) A social choice ordering satisfies monotonicity if, for all
(1 , . . . , n ) such that x i y for some i, if the preferences of i changes so that now y 0i x,
then if y x where = f (X, (1 , . . . , n )) this implies that y 0 x where 0 = f (X, (1
, . . . , i−1 , 0i , i+1 , . . . , n )).
5
This is a very weak notion of pareto optimality. We don’t even require that if one alternative makes
everyone weakly better off, and one person strictly better off, that that alternative is preferred. Since we are
leading to an impossibility result it makes sense to ask for the weakest conditions possible.
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Definition 6. (Independence of Irrelevant Alternatives) A social choice ordering satisfies
independence of irrelevant alternatives if, for all preference profiles (1 , . . . , n ) and
(01 , . . . , 0n ), if x, y ∈ X such that x i y if and only if x 0i y and x ≺i y if and only
if x ≺0i y, then x y if and only if x 0 y and x ≺ y if and only if x ≺0 y, where
= f (X, (1 , . . . , n )) and 0 = f (X, (01 , . . . , 0n )).
Definition 7. (Non-Dictatorial) A social choice ordering is dictatorial if there exists an j
such for all x, y ∈ X, x j y implies that x y. A social choice function is non-dictatorial
if it is not dictatorial.
The most complicated, and controversial condition is Independence of Irrelevant Alternatives. The simplest way to think about what it means is to imagine a group choosing
between two alternatives {A, B}. Suppose A is preferred to B. Independence of Irrelevant
Alternatives requires that if another option C was available then the presence of C should
not make B preferred to A. This is considered desirable since why should the ranking of A
and B depend on an “irrelevant” factor such as whether or not C is available or what what
features C has? Also, if the ranking of A and B could be affected by whether or not C is
available, it could create the incentive to provide fewer alternatives to voters.6
The search for a choice ordering that is guaranteed to satisfy the above conditions leads
to one of the seminal results in economics, the Arrow Impossibility Theorem. This theorem
resulted in Kenneth Arrow being awarded the Nobel prize in 1972. The Arrow Impossibility
Theorem states that there does not exist any social choice ordering that satisfies these five
conditions. It is not simply that no such procedure for aggregating preferences has been
found, but rather it will never be possible to find such a choice rule. This does not mean,
of course, that we cannot talk about aggregating preferences. It does mean, however, that
there is no “correct” or perfect way to aggregate preferences and the choice of mechanism
will involve trade-offs. This is a common theme in economics: it typically isn’t possible to
have everything we want so there will be tradeoffs. While we will omit the proof of many of
the results in the class, given the importance of the Impossibility theorem in economics we
will go over its proof.
6
Arrow’s Impossibility Theorem
Let N = {1, . . . , n} be the set of voters, and X be the set of alternatives. We now state and
prove the Arrow Impossibility Theorem.
Theorem 2. (Arrow 1950) Suppose that X at least three elements. Then there does not exist
a social choice ordering satisfying unrestricted domain, pareto optimality, monotonicity, and
independence of irrelevant alternatives that is non-dictatorial.
Before proceeding with the proof we introduce the concept of a decisive set of voters for
each pair of alternatives. A set Vyx ⊆ N is a decisive set for a pair of alternatives (x, y)
6
For example, in the 2016 election it was believed that Michael Bloomberg decided not to run, at least
in part, because plurality rule elections violate Independence of Irrelevant Alternatives and he thought that
running would make it more likely Trump would defeat Clinton. Bloomberg preferred Clinton and ended up
endorsing her instead.
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if x i y for all i ∈ Vyx implies that x y regardless of the preferences of those not in
Vyx . What constitutes a decisive set depends on the social choice ordering (e.g. if applying
Condorcet’s method a decisive set is any set containing a majority of the voters, under
dictatorship a set is decisive if and only if the dictator is included). It is immediate that a
decisive set exists for each (x, y) for any social choice ordering that satisfies pareto optimality
since N = {1, . . . , n} is a decisive set for any pair: if x is preferred to y by all individuals
it must be socially preferred. We then say that a set V is a minimally decisive set if it is
a decisive set for some pair of alternatives, but no subset of V is a decisive set for any pair
0
of alternatives. That is, there does not exist any (x0 , y 0 ) such that Vyx0 is a decisive set for
0
(x0 , y 0 ) and Vyx0 ⊂ V . A minimally decisive set must exist since the empty set is not a decisive
set for any pair of alternatives—again, by the Pareto criterion.
Proving Arrow’s theorem will then consist of showing that for any choice rule that satisfies unrestricted domain, pareto optimality, monotonicity, and independence of irrelevant
alternatives, there exists some individual j ∈ N such that {j} is a decisive set for all pairs of
alternatives in X. This will then imply that j is a dictator and so no choice ordering that satisfies unrestricted domain, pareto optimality, monotonicity, and independence of irrelevant
alternatives can also be non-dictatorial.
We now proceed with the proof of Arrow’s Impossibility Theorem. The version of the
proof we present comes from Geanakoplos (2005).
Proof. Let f be a social choice ordering that satisfies unrestricted domain, monotonicity,
independence of irrelevant alternatives and pareto optimality. Let V be a minimally decisive
set, which, due to pareto optimality must exist. Then there exist two alternatives x, y ∈ X
such that V is a decisive set for the alternative pair (x, y). Let z be any element of X that
does not equal either x or y. Since X has at least three elements, such a z must exist.
Let j be any individual in V . Since we have assumed unrestricted domain we must be
able to account for all possible preferences. Consider the following preference profile:
• Voter j: x j y j z
• Any voter i 6= j in V : z i x i y
• Any voter i ∈
/ V : y i z i x
By independence of irrelevant alternatives and the definition of V we know that x y.
We must consider the comparison of y and z. Note that, by independence of irrelevant
alternatives and monotonicity, if z y then V \{j} is a decisive set for (z, y). However, since
V is minimally decisive, V \{j} cannot be a decisive set for any alternative pair. Hence,
y z and, by transitivity, x z.
Since, in the specified preferences, voter j is the only individual who prefers x to z
and all other individuals strictly prefer z, by monotonicity and independence of irrelevant
alternatives {j} is a decisive set for (x, z). Moreover, since {j} ⊆ V is a decisive set, and V
is minimally decisive, V = {j}, and so {j} is a decisive set for (x, y) as well.
Since we allowed z to be any element of X not equal to x or y, we can then conclude that
{j} is a decisive set for (x, z) for all z ∈ X. This is true for a fixed x. It remains to show
that, for all w 6= x, {j} is a decisive set for (w, x) and a decisive set for (w, z), where z 6= x..
11
We first consider the case in which z 6= x. Consider the following preference profile:
• Voter j: w j x j z
• Voters i 6= j: z i w i x
Since {j} is decisive for (x, z), x z and, by pareto optimality, w x. So by transitivity
w z. Since in this preference profile w z even though only voter j prefers w to z and
all other voters prefer z to w, by independence of irrelevant alternatives and monotonicity
we can conclude that {j} is a decisive set for (w, z). Since (w, z) are arbitrary we can then
conclude that {j} is a decisive set for all (w, z) where z 6= x.
Finally we conclude by showing that, {j} is a decisive set for (w, x) for any w. To do so
take any z not equal to w or x and consider the preference profile
• Voter j: w j z j x
• Voters i 6= j: z i x i w
Since {j} is a decisive set for (w, z), w is preferred to z. By pareto, z is preferred to x, so by
transitivity w x. As j is the only individual who prefers w to x, independence of irrelevant
alternatives and monotonicity then imply that {j} is a decisive set for (w, x).
We have now established that for any social choice ordering that satisfies unrestricted
domain, monotonicity, independence of irrelevant alternatives, and pareto optimality, there
must exist one individual whose preferences are decisive for the ranking of every pair of alternatives. This individual is a dictator, and so the choice ordering must be dictatorial. This
completes the proof that there cannot exist a social choice ordering that satisfies unrestricted
domain, monotonicity, independence of irrelevant alternatives, and pareto optimality that is
non-dictatorial.
This shows that, regardless of the choice rule used, there always exists some preference
profile for which at least one of the above conditions will be violated. Some choice rules
are guaranteed to satisfy some of the properties, but no choice rule is guaranteed to satisfy
all of them. Of course, the above properties are only violated for some preference profiles.
The decision of which choice rule to use then depends largely on: (1) which violations are
most troubling in the environment considered, (2): how likely are the individuals to have
preferences for which the above conditions are violated. In the next section we consider what
properties we might expect the preferences of the individuals to satisfy.
7
Single Peaked Preferences
One of the demands of the Arrow’s Impossibility Theorem is unrestricted domain: that no
matter what the preferences of the voters are it is possible to aggregate preferences. This is
important because we do not know, ex-ante, when designing a system what the preferences
of the individuals who will use the system to make decisions might be. However we might
expect certain preference profiles to be more common than others. This is the case when
preferences can be ordered in some way—for example, in terms of their ideology.
12
Think back to the 2000 election, with Bush, Gore, and Nader. We can clearly rank these
three candidates in terms of ideology: Bush was the most conservative and Nader was the
most liberal. Now consider a voter whose first choice is Nader: her second choice could
either be Bush or Gore. Which is more likely? Since a Nader voter is probably on the
left of the political spectrum, and Gore is ideologically to the left of Bush, it is likely that
they would prefer Gore to Bush. This is the reason why we think Gore may have been a
Condorcet winner in the 2000 election: since we think Gore would have been preferred to
Bush by a large majority of Nader voters. This suggests that certain preferences would be
more common than others: in particular N B G and B N G should be very rare.
This suggests that a natural assumption about individual preferences is that they are
single peaked. Suppose we can order the alternatives in X and let x1 be the smallest alternative up to xk as the largest alternative: x1 < x2 < . . . < xk . In this case we are associating
each alternative with a real number (e.g., ideological position on a left-right scale, what is
the tax rate, how much public good to provide, etc.).7 Since the magnitudes are arbitrary
assume that X ⊆ [0, 1]. We can then define single-peaked preferences as follows:
Definition 8. Suppose individual i’s most preferred alternative is xk̂ ∈ X = {x1 , . . . , xk },
when the alternatives have been ordered so that x1 < x2 < . . . < xk . Individual i’s preferences
are single-peaked if, for all k 0 > k̂, xk0 i xk0 +1 and, for all k 0 < k̂, xk0 i xk0 −1 .
As an example of this definition, consider the Bush-Gore-Nader example in which X =
{B, G, N }. In this case, since N is on the left and B on the right, then x1 = N , x2 = G,
and x3 = B. To be single peaked any individual i for whom Bush (x3 ) is the most preferred
alternative must prefer Gore to Nader (i.e. if x3 is the first choice then x2 i x1 ), and any
individual with Nader (x1 ) as their first choice must prefer Gore to Bush (i.e. if x1 is the
first choice then x2 i x3 ). The assumption of single peaked preferences would rule the
preferences N B G and B N G.
A common example of preferences that are single-peaked are when each individual i has
a bliss point ti and her utility from any alternative xk is
u(ti , xk ) = −(ti − xk )2 .
More preferred alternatives are then clearly associated with higher utility levels. The interpretation of these preferences is that each person has a ideal policy in ideological space—an
ideal set of policy positions a candidate could have, her ideal policies that she would implement if she were a dictator—and her utility is maximized the closer the policy is to her
ideal point. We will see that focusing attention on single-peaked preferences can help in
aggregating preferences.
Suppose that no two voters share the same bliss point and that there are 2n+1 voters (an
odd number of voters). We can re-index the voters {1, . . . , 2n + 1} so that t1 < . . . < t2n+1 .
In that case voter n + 1 is the median voter. Then we can compare each pair of alternatives
in X by Condorcet’s method. Now take any two alternatives x, y ∈ X, and without loss of
generality assume that x < y. Notice that the median voter prefers x to y if and only if
−(tn+1 − x)2 > −(tn+1 − y)2
7
For example, political scientists place individuals on scale from 1 to 7 in terms of how conservative or
liberal they are.
13
or equivalently
tn+1 − x < y − tn+1
which is equivalent to
x+y
.
2
In this case ti reflects the individual’s most preferred policy (where the locate on the political
spectrum, their ideal tax rate, the amount of public good provision they think is optimal,
etc.) Notice that, since for all i < n + 1, ti < tn+1 , this implies that all voters with ideal
point lower than the median would also prefer x to y. So if the median voter prefers x
to y then x is preferred to y under Condorcet’s method. Similarly, if tn+1 > x+y
then
2
the median voter, and all voters with ideal point higher than the median, would prefer y
to x. So if the median voter prefers y to x then y is preferred to x under Condorcet’s
method. Hence the alternative that is preferred by the median voter is the alternative that
is socially preferred under Condorcet’s method.8 This means that there is no possibility of
non-transitive preferences when preferences are single peaked and the voters’ preferences are
transitive. We can then use the Condorcet method to aggregate preferences when all voters
have single-peaked preferences.
tn+1 <
Theorem 3. (Black 1958) Suppose that there are an odd number of voters and that the
preferences of the all voters are single-peaked. Then the social preference ordering induced
by Condorcet’s method is a transitive preference ordering over X that satisfies independence of
irrelevant alternatives, pareto optimality, monotonicity, and non-dictatorship. In particular,
the most preferred alternative of the median voter is the Condorcet winner in X and so is
the most preferred alternative according to the induced social ordering.
This shows that if we can relax the requirement of unrestricted domain, and instead focus
attention on preference profiles that are single peaked, it is possible to aggregate preferences
with Condorcet’s method. Single-peaked preferences are not necessary for a Condorcet
winner to exist, but are sufficient, and when preferences are not single-peaked it is not
guaranteed that a Condorcet winner will exist. While we know that no method will work
for all preferences, Dasgupta and Maskin (2008) show that, among all approaches that could
be used, Condorcet’s method satisfies the other axioms for the largest set of preferences.
However, unlike other methods, it doesn’t always tell us how to make a choice.
The assumption of single-peaked preferences can more easily be applied in some settings
than others, however. If the alternatives were to build a bridge, not build a bridge, or to
build half a bridge, individuals would probably not have single-peaked preferences. Similarly
when voting on host cities for the Olympics it may not be possible to order the alternatives
and Condorcet cycles are possible. However, in electoral competition, given that ideological
considerations are of central importance, we can usually order the alternatives (candidates)
and the voters in terms of ideology and the voters are likely to have preferences that are (or
at least are close to) single-peaked. However there are important caveats that we will return
to later. That the median voter is decisive under Condorcet’s method will be very important
when discussing electoral competition, which we turn to in the next section. A detailed
treatment of electoral competition is available in Roemer (2009) on the Chalk website.
8
This does not mean that the median voter is a dictator. If other individuals’ preferences change the
median voter’s preferred alternative would no longer be socially preferred.
14
8
Electoral Competition
So far we have not discussed strategic considerations: we have assumed that individuals’
preferences are known and that the alternatives are fixed and exogenous. We will consider
the incentives for individuals to reveal their preferences later in the course, and we turn
to determining the set of alternatives now. Often the alternatives that individuals vote
over are strategically determined: for example, in an election the candidates for office run
campaigns in which they offer different platforms to the voters, and the voters then decide
which candidate to vote for depending (at least in part) on what the candidates have promised
to do. Given that the set of alternatives is endogenous, we want to get a sense of which
alternatives will be offered to voters in elections. Moreover, Black’s theorem shows that there
is a natural policy alternative to select when preferences are single-peaked, so we want to
get a sense of whether competitive elections are likely to select this alternative. This leads
us to the model of electoral competition due to Downs (1957). In the Downsian model two
candidates compete for office, and we solve for the Nash equilibrium platforms proposed by
each candidate. Recall from Econ 201 that a Nash equilibrium is when both candidates are
optimizing given the strategy of the other candidate.
Suppose there are two candidates for office, 1 and 2. The two candidates i ∈ {1, 2}
compete by choosing any policy xi ∈ X = [0, 1] to implement if elected. This policy is
observed by the voters and the candidates commit to the policy before the election. Both
candidates 1 and 2 are purely office motivated and receive a payoff of 1 if elected and a
payoff of 0 if not elected. There are 2n + 1 voters with single-peaked preferences over the
policy implemented with ideal points t1 < . . . < t2n+1 . The candidates compete by majority
rule. Assume that if a voter is indifferent between the two candidates she flips a balanced
coin and votes for each candidate with probability 1/2. This game has a unique equilibrium:
both candidates propose the bliss point of the median voter.
Theorem 4. (Downs 1957) In the unique Nash equilibrium x1 = x2 = tn+1 .
This is frequently called the Median Voter Theorem, that electoral competition creates
an incentive for both parties to locate at the ideal point of the median voter. You may notice
the similarity between the Downsian model and Hotelling’s linear city which you may have
seen in Econ 201 or an IO class. In order to attract the greatest number of customers, the
firms located at the median location of the customers. The same logic leads candidates to
locate at the median of the electorate.
Notice how important the assumption that preferences are single-peaked is for the median
voter theorem. While the voters will only choose between two alternatives, the candidates
can choose any two alternatives in X to offer them. When a Condorcet winner exists, both
candidates must propose a Condorcet winner: otherwise the other candidate could propose
a Condorcet winner and defeat them for sure. However, if a Condorcet winner does not
exist, then there cannot exist a pure strategy equilibrium. If a Condorcet winner does not
exist then whatever policy 1 offers 2 could find a policy to defeat it, but then 1 would
want to change the policy it implemented to something that defeats 2 and so on. If we had
considered electoral competition with X = {A, B, C}, three voters, and the preferences from
the Condorcet paradox in section 3, then there cannot not exist a pure strategy equilibrium;
15
only a mixed strategy equilibrium could exist.9 In this class we will focus on pure strategy
equilibria.
This also leads to an important caveat about single-peakedness and the median voter
theorem. Suppose that instead of having an ideal point in one dimensional space the voters
each have an ideal point ti = (t1i , t2i ) in two dimensional space, and the alternatives in X also
consist of two dimensions. In this case a median voter does not exist, ordering the set of
alternatives is more difficult, and a Condorcet winner will not always exist.10 To guarantee
that a Condorcet winner exists requires strong symmetry assumptions about the distribution
of voter ideal points. So, while the Downsian logic is compelling, we also need to be aware of
the caveats about the theoretical prediction. In the next section we evaluate the prediction
empirically.
9
The Empirical Relevance of the Median Voter Theorem
The Median Voter Theorem is the starting point for modeling political competition. However
its prediction that both candidates converge to median voter is often criticized, especially
in recent years, as counter-factual. We now consider some of the empirical work testing the
predictions of the median voter theorem and look at the evidence that the median voter
theorem may not hold. In the next section we will consider some theories of why candidates’
platforms may diverge.
Arguably the Downsian equilibrium was a close approximation of reality when it was written in 1957. A 1950 report of the American Political Science Association titled “Towards A
More Responsible Two-Party System” called on political parties to offer clear, differentiated
alternatives to the voters. Democrat John F. Kennedy criticized the Republican Eisenhower
administration for being weak on defense in during the election of 1960 and proposed substantial tax cuts soon after taking office. Moreover, in 1964 presidential candidate Barry
Goldwater promised that unlike the candidates in previous elections he would offer the voters “a choice, not an echo” before being defeated in one of the most lopsided elections in
American history.
In recent elections, however, few would argue that the parties are interchangeable. The
concern now is that the parties have become extremely polarized. For instance, most people
perceived a large difference between Obama and Romney in 2012 and Clinton and Trump
in 2016. Similarly, the parties were considered very polarized in the second half of the twentieth century. We want to discuss the evidence regarding whether or not parties platforms
converge. A series of papers by Nolan McCarty, Howard Rosenthal, and Keith Poole address
this by estimating the ideology of legislators—known as NOMINATE scores—from different
parties. This literature is summarized in their book, McCarty et al. (2008) and the relevant
chapter is available under Library Reserves on the Chalk website.
9
In the mixed strategy equilibrium each candidate randomizes with equal probability over each policy in
X.
10
If there are two dimensions but only one of them involves disagreements (e.g. one dimension is ideology and second dimension is quality of “valence” of the candidate) then a Condorcet winner still exists.
Alternatively if candidates have fixed characteristics in the second dimension, existence is again guaranteed.
16
McCarty, Poole and Rosenthal estimate the policy positions of members of the legislature.
The approach is to assume that the legislator (in the House or Senate) is voting as if they were
maximizing a utility function with an ideal point xi —this could be their personal ideal policy
or it could be the policy they have promised to the voters—then estimate this objective point
based on the voting record of the legislator. Namely, those who vote similarly are considered
to have ideologically close positions. We are interested in how much variation there is across
legislators and how much the voting records vary within and across parties. How much of
ideology appears to be driven by constituency preferences? Do Senators from the same state
vote the same way?
Consider three politicians who are currently serving in the U.S. Senate: Bernie Sanders
(I-VT), Ted Cruz (R-TX), and Susan Collins (R-ME). Cruz is considered to be very conservative, Sanders very liberal, and Collins as a Republican in a Democratic state is viewed
as a moderate. For example, she was one of two Republican senators to vote against the
confirmation of Betsey DeVos and was one of the few to meet with Obama’s Supreme Court
Nominee, Merrick Garland. How would we see this relationship in the data? Suppose we
were to see votes that looked something like this:
Bill 1: Cruz and Collins YEA, Sanders NAY
Bill 2: Cruz NAY, Collins and SandersYEA
Looking that this data we would see that Collins voted with Cruz once and Sanders once
on bills that Cruz and Collins disagreed on. We could interpret this as a sign that Collins
was ideologically between Cruz and Sanders—provided of course that this pattern was borne
out in the larger sample of bills (recent congresses have had 500 to 1200 votes a year). We
could also conclude that yes on bill 1 fits with the Cruz side and yes on bill 2 fits on the
Sanders side. Nothing in the data would tell us which side was conservative and which side
was liberal, though we can make that judgment based on who was estimated to be on which
side of the spectrum. Let’s represent the conservative side with higher numbers since the
real line is ordered from left to right.
Of course we would need to look at the whole range of votes, in which case not all votes
would fit so neatly. We would likely see some bills with Cruz and Sanders voting against
Collins. However we would still estimate that Collins is between Cruz and Sanders if such
an outcome is less common than Collins voting with either Sanders or Cruz. In this case we
assume that the legislators make “errors” —that is, there are elements outside the model
such as local conditions, interest groups, or a personal connection to some issue lead them to
cast votes different from the model predictions. The approach is then to estimate legislative
ideal points based on minimizing the “errors” so that the the model fits the data as well as
possible. This approach is known as maximum likelihood estimation.
Assume there are K legislators, and M bills that the legislators will vote on. Assume to
begin with that we know which side represents the conservative side of each bill and which
represents the liberal side. We must estimate the ideology of each of the K legislators and
the M bills, so that is K + M parameters to estimate. But we have KM data points since
each legislator votes on each bill (excepting votes that are missed). As we have many more
data points than parameters to estimate we have enough data to estimate to estimate the
parameters we are interested in. We assume that on each bill m legislator k voters for the
conservative side if and only if
xk + εk,m > ym
17
where εkm are independent and identically distributed draws from the distribution (normally
distributed for example). We can then calculate the probability that the legislator k would
cast the vote they did, if their ideology is xk and the cutoff on bill m is ym . This probability
of the observed vote is then
P r(εk,m > ym − xk ) vote on conservative side,
p(xk , ym ) =
P r(εk,m < ym − xk ) otherwise.
But then, of course, we observed KM different votes, and the probability of all KM votes
happening, given that each ε is independent, is just the product of the different probabilities.
We then define the Likelihood Function
L(x1 , . . . , xK , y1 , . . . , yM ) =
K Y
M
Y
p(xk , ym ).
k=1 m=1
That is, the likelihood function is the probability that, if the actual ideologies of the legislators were (x1 , . . . xK ) and the actual vote cutoffs were (y1 , . . . , yM ) what the probability of
observing the profile of votes is. We then find the legislator ideologies and vote cutoffs for
which the actual vote profile we observed would be as likely as possible, by maximizing
L(x1 , . . . , xK , y1 , . . . , yM )
over (x1 , . . . xK ) and (y1 , . . . , yM ). This provides an estimate for the ideal point of each
legislator—conditional on correctly identifying which bills are conservative and which are
liberal. Of course, we may not know ex-ante which side is conservative and which side is
liberal, and the vote profile will only match the data well if we have correctly identified
which side is which. This means we have an additional M things to estimate: whether a
yes vote is conservative or liberal on each bill. In principal, we could then compare over the
2M different combinations to see which combination of bills being conservative matches the
data the best by giving us the highest value of the Likelihood function. This gives us an
estimate of which vote on each bill represents the “conservative” position, as well as how
far to the left or right each vote is. In practice, since 2M is an extremely large number, this
a very computationally hard problem, so computational techniques are necessary to avoid
calculating the likelihood function for every possible permutation. For example, on some
bills it is clear what the conservative position is from who is voting on which side—so we
may not need to check every permutation. Using this approach we get what we are really
interested in: an estimate of the ideal point each legislator is maximizing. We can then plot
how the ideal points vary within, and across, the parties based on how they vote.
We can also compare polarization over time by comparing legislators who are in different
Congresses. Also, while it becomes more complicated, we can add additional dimensions and
see how much the explanatory power of the model is improved. Determining whether the
additional dimensions adds enough to make sense including is known as Factor analysis.
This estimation leads to the following conclusion:
1. The voting patterns of legislators elected from different parties is very different.
2. This polarization has increased since the 1960s: there used to be substantial overlap
between the legislators from each party, but such Representatives and Senators have
largely disappeared.
18
3. We can extend the methodology to allow for multiple dimensions, but, excepting the
civil rights era, more dimensions do not add additional explanatory power. The voting
records of legislators is largely captured by a one dimensional Liberal-Conservative
axis.
This approach establishes (subject to all the caveats about the data and all the assumptions underlying the estimation) that legislators from different parties have very different
voting records. This does not in and of itself prove that the policies taken by the winning
candidate diverge from those of the median member of the district (at least in House districts
where only one representative is elected). It is possible that the winning candidate in each
district locates at the median in their district and that districts that elect Republicans are
simply more conservative than those who elect Democrats. While it is certainly true that
Republicans are elected in more conservative districts than Democrats, some evidence that
this is not the whole story is provided by Lee et al. (2004).11 This paper is posted on Chalk.
They take a regression discontinuity approach, and look at elections in which the share
of the two-party vote is almost exactly 50-50. The idea is that when an election is that close,
whether a Republican or Democrat is elected in these districts is more or less random. This
can, at least partially, be tested by looking at how districts that barely elected a Republican
compare to those who barely elected a Democrat. They then look to see whether there
is discontinuity—a discrete jump in the voting record of the of the legislator—at exactly
50% of the two-party vote. Figure VI (on page 840) provides evidence of such a jump at
50%. In fact, it indicates that the vote share has little effect on the voting record of the
legislator, except in determining which party is elected. This indicates that a Republican
and a Democrats would vote very differently, even if they were elected to represent the same
constituency. This, provides evidence against the median voter theorem.
10
10.1
Models of Divergent Platforms
Policy Motivation and Uncertainty
As there is strong empirical evidence refuting the Downsian prediction that platforms converge, it is important to ask what is missing from Downs’s model. Many different proposals
have been put forward to explain why candidates may diverge in equilibrium. One natural
element of the Downs model to question is the assumption that parties are purely office motivated. What happens if instead we assume that parties/candidates are policy motivated?
This leads to the Calvert-Wittman model.
Consider the Downs model, but instead of assuming that parties care only about holding
office assume that parties care about policy. Assume that party 1 has utility function −x2 ,
and party 2 has utility function −(1−x)2 , where x is the policy of the candidate who is elected
by the voters. In this case x = 0 is the bliss point of Party 1 (the Democrats) and x = 1 is
the bliss point of party 2 (the Republicans). Here the parties don’t care about who holds
office at all—they only care about the policy outcome—though, of course, we could all a mix
11
Another piece of evidence that candidates do not converge to the median in the district comes from
Senate. In the Senate two candidates represent the same state at each time, but typically have different
voting records (particularly when they come from different parties).
19
of both. We maintain the assumption from the Downsian model that parties can perfectly
commit to the policies they propose. Unlike in the case with office motivated candidates,
this assumption is not innocuous: the parties may not propose their most preferred policy
but we assume they will follow through on their promises. What is the equilibrium of this
game?
First note that xL = xR = tn+1 is still an equilibrium. If either party deviates then it
has no effect on the implemented policy (since the deviating party is defeated), so neither
party has an incentive to deviate. It is, in fact, the only equilibrium. To see this, note that,
although party 1 would prefer to implement a policy to the left of the median, and party
2 would prefer a policy to the right of the median, in order for the party’s platform to be
implemented it must win the election. If we were to have an equilibrium in which one party
always wins with a non-median policy, the other party would “undercut” them by locating
closer to the median voter’s ideal point but in the direction of that party’s ideal policy. If
the two candidates were to tie by locating an equal distance from the median, then one party
could move slightly closer to the median voter and win for sure. This is similar to Bertrand
competition from Econ 201. While the parties would like to pull the policy towards their
ideal point (and firms would like to set price above cost) since the either party can take the
election by locating slightly closer to the median voter’s ideal point (either firm could win
the entire market by undercutting its rivals price slightly), competition forces both parties
to locate at the median. This means that more than just policy preferences are necessary to
explain divergent platforms.
The simplest model of policy divergence comes from a combination of policy preferences
and uncertainty by the parties about the ideal point of the voter. Suppose that there are
2n + 1 voters and two policy motivated parties with ideal points 0 and 1. Suppose now
that the parties don’t know the ideal point of the median voter, but believe that the ideal
point of the median voter is tM = 1/2 + ε, where ε is Normally distributed with mean 0
and variance σ 2 . Now note that we cannot have an equilibrium with xL = xR = 1/2: either
party could deviate and choose a policy closer to their ideal point. This has no effect on
the policy if they aren’t elected, but moves the policy closer to their ideal point if they are.
Since the location of the median voter’s ideal point is unknown, the election is random, and
the deviating party wins with positive probability. In equilibrium the parties must locate
away from the median.
Theorem 5. There exists a unique pure strategy symmetric equilibrium. In this equilibrium
xL = 1/2 − d, xR = 1/2 + d where d ∈ (0, 1/2). The degree of divergence d is increasing in
σ with
lim d = 0,
σ→0
lim d = 1/2.
σ→∞
This indicates that a combination of policy motivation and uncertainty can generate the
prediction of divergent platforms. If the parties were office motivated then both parties
would locate at the expected median even if there is uncertainty. This uncertainty could
be about the median voter’s ideal point or anything else (i.e. whether the voters will like
the candidate of the party personally, uncertainty about which voters will turn out to vote,
etc.) However, the pressure to move to the center is still apparent, and parties only move a
20
non-trivial distance from the expected median if the uncertainty is not too small. For more
details on this model see the first two chapters of Roemer (2009).
10.2
Electoral Accountability
The Downs and Calvert-Wittman models both assume that candidates can make binding
commitments to specific policies and will always follow through on them. This assumption
seems unrealistic: there is no formal mechanism underlying this commitment and candidates
may make promises that are impossible to fulfill. The question of which promises can be
committed to is considered in Alesina (1988), available on Chalk.
Suppose there are two candidates, L and R, who are concerned only with policy. The
ideal point of the L is 0 and the ideal point of R is 1, and they have quadratic loss as before.
Let’s assume there is no uncertainty about the median voter’s ideal point: it’s tn+1 = 1/2.
We assume that these candidates (parties) are long-lived and care about the policy in
each period. Future outcomes are usually discounted because of people are impatient or
because it is uncertain whether the candidates will be running again. Let β ∈ (0, 1) reflect
the weight placed on future periods.12 Suppose the game is infinitely repeated so the is no
known last period; we can interpret this as the players not being certain when the game will
end. If the policy chosen by the elected candidate at each period s = 0, 1, 2, ... is x(s) the
payoff to the left candidate is
∞
X
β s x(s)2 ,
−
s=0
and to the right candidate it’s
−
∞
X
β s (1 − x(s))2 .
s=0
Note that in this formulation, the payoff in period 0 is not discounted, one period in the
future is weighted β, two periods in the future are weighted at β 2 , and so on.
We look for a symmetric equilibrium in which, if elected, candidate L chooses xL = 1/2−d
and candidate R chooses xR = 1/2 + d in every period if they are elected. That is, we look
for an equilibrium in candidate behavior is stationary: that is, the policy they choose if
elected is the same in every period. If d = 0 then we have that the candidates converge to
the median, and d > 0 then they don’t. One equilibrium is for each candidate to choose their
ideal policy in every period (d = 1/2). However, voters can do better if the median voter
conditions her vote on the incumbent’s behavior in office: Suppose the median voter votes
to re-elect the incumbent if, and only if, the policy they chose was no further than d from
the median voter’s ideal point. We now ask what the most moderate (smallest d) outcomes
that would not give the candidates an incentive not to deviate. To do this we have to make
sure the candidate wouldn’t be willing to deviate to their ideal policy and lose election.
12
For example, it could be that in every period, with probability 1 − β the candidate is prevented from
running due to health reasons; assume that if the candidate is prevented from running the game ends and a
new game that is unaffected by the play in this game starts.
21
As everything is symmetric, we only need to look how the R candidate would behave if
elected. For these strategies to be an equilibrium, so we must have
∞
X
s
2
−β (1 − (1/2 + d)) ≥ 0 +
∞
X
s=0
−β s (1 − (1/2 − d))2 .
s=1
The left hand side reflects the payoff from choosing xR = 1/2 + d every period and being
re-elected forever. The right hand since reflects the payoff from choosing xR = 1 today then
losing office and having the rival win office forever with xL = 1/2−d. To have an equilibrium
the left hand side must be at least is large as the right.
The above condition gives an infinite sum which is, in general, difficult to calculate.
However since the payoff is the same in each period, it becomes easy. Notice that
!
∞
∞
∞
∞
X
X
X
X
βs = 1 +
βs = 1 + β
β s−1 = 1 + β
βs ,
s=0
s=1
which implies that
s=1
∞
X
βs =
s=0
and
∞
X
s=1
s
β =β
∞
X
s=0
s=0
1
,
1−β
βs =
β
.
1−β
Using this we can calculate the infinite sums, and the condition to be an equilibrium reduces
to
−β(1 − (1/2 − d))2
− (1 − (1/2 + d))2
≥
.
1−β
1−β
To characterize the minimum level of divergence we must have this hold with equality, so we
look for d that solves
2
2
β
1
1
1
−d =
+d .
1−β 2
1−β 2
This implies that the most moderate policy that can be supported solves
1/2 − d p
= β,
1/2 + d
and so
√
1− β
1
√ ∈ 0,
d=
.
2
2(1 + β)
This leads to the following result.
Theorem 6. Let d =
rium:
√
1− √β
2(1+ β)
∈ 0, 21 . The following strategies constitute a Nash Equilib-
• Candidate L chooses 1/2 − d if elected, and candidate R chooses 1/2 + d.
22
• All voters to the left of the median vote for L, and all to the right of the median vote
for R. The median voter re-elects the incumbent if and only if the policy they chose is
in [1/2 − d, 1/2 + d].
Moreover, this is the lowest level of polarization that can be supported in any stationary
equilibrium.
The above result says that, although elections can incentivize compromise, politicians
cannot commit to the median policy unless they become extremely patient. Extreme patience
would only emerge if elections are extremely frequent: if we think of elections as happening
every four years it is likely the future is significantly discounted. This is one reason why the
median voter theorem breaks down: it simply isn’t possible for candidates to make binding
commitments in campaigns and so we have to look at the policies they will have an incentive
to choose if elected. In this way the identity not just the promises of candidates matter.
However, even though promises may not be binding it is possible to create incentives for
compromise by the candidates.
While we do not have time to to go into it here, we can make the model richer in many
different ways. If we include other forms of “shirking” by elected officials (lack of effort or
forms of rent-seeking) the same basic logic applies—we consider such a model in Problem Set
2. In addition we can incorporate other features such as uncertainty about the median voter’s
preferences or imperfect observability of the elected official’s behavior. These possibilities,
aside from ensuring the incumbent will not be re-elected forever, will generally increase the
amount of shirking by the official. The reason for this is that “good” behavior may not be
observed and so will not necessarily be rewarded with re-election and “bad” behavior will
not necessarily result in losing office. This is general problem of “moral hazard” and it is
typically the case that individuals will shirk on their responsibilities more when they are less
easily monitored. Some empirical evidence of such shirking by politicians, focused more on
politician effort, is available in the Snyder and Stromberg (2010) paper on Chalk.
10.3
Citizen Candidates
One question, particularly if we think polarization is driven (at least partly) by the preferences of the candidates, is where those preferences come from. The citizen candidate model
views candidates for office as citizens with policy preferences of their own who decided to
run for office. They assume the opposite of what the papers in the Downs/Calvert/Wittman
framework assume. Namely that candidates always implement their ideal policy if they are
elected.13 The candidates are strategic, however, in their decision of whether or not to run
for office. There are two papers that introduced this literature—one of them, Osborne and
Slivinski (1996) is available on Chalk. This gives a model of the ideology of those who run
for office.
There are a continuum of citizens with ideal points t ∈ [0, 1], and they all have singlepeaked preferences over the implemented policy x. Assume that the ideal points of the
citizens are uniformly distributed from [0, 1]. All citizens care about the policy implemented,
13
It is possible to consider a repeated citizen candidate model but we will not do that in this class.
23
and can also become candidates by paying a cost c > 0. If the citizen becomes a candidate
and wins election they receive a utility benefit of b > 0. Hence the utility of each citizen is

elected,
 b−c
−(x − t)2 − c ran and lost and candidate with ideal point x won,

−(x − t)2
didn’t run for office, and candidate with ideal point x elected.
The election takes place by plurality rule: the candidate with the largest share of the vote
is elected. We assume that all citizens who do not become candidates vote sincerely in that
they vote for the candidate that would provide them with the highest utility. Voters who are
indifferent randomize with equal probability across the candidates they are indifferent over.
First note that, if b > 2c, there cannot exist an equilibrium with only one candidate: in
that case, another candidate with the same preferences could also run for office and win with
probability 1/2 and earn strictly higher payoff.
Is there an equilibrium with two candidates? If so, are both candidates located at the
median? To have an equilibrium requires that none of the candidates for office who ran for
office would receive a higher (expected) payoff from not running, and that no citizen could
receive a higher expected utility from running for office. If there are two candidates, both
with the median policy, then each candidate receives half the vote and wins with probability
1/2.
Now consider a candidate of type x that is slightly higher than 1/2. Note that, all citizens
of type t > x would prefer this candidate to the median candidate, so this candidate would
receive at least 1 − x of the vote. As the other voters are indifferent between the median
candidates their votes split evenly: hence each candidate receives less than x/2 votes. So if
x is close enough to 1/2, they would win the election and have an incentive to run for office.
This shows that we cannot have an equilibrium with two candidates and converging ideal
points. So any two candidate equilibrium must involve non-converging platforms. Since a
candidate who would lose office for sure could avoid the cost of running for office could save
the cost of running for office by not declaring as a candidate, for both candidates to run for
office, the candidates must tie.
So any two-candidate equilibrium must involve two candidates, x1 = 1/2 − d and x2 =
1/2 + d running for office. As the median voter is indifferent between them each candidate is
elected with probability 1/2. To determine when this constitutes an equilibrium we need to
check whether (1) neither candidate who is running for office could do better by not running
(2) no citizen who did not declare themselves a candidate could increase their utility by
running for office. The first part is immediate: since each candidate wins with probability
1/2 and b > 2c, and the expected policy is better for each candidate than from not running,
there is no incentive for either candidate not to run. Next note that there is no incentive for
any citizen of type t < x1 or t > x2 to run for office: they could never win office and would
only siphon off votes from the candidate on their side, leading to a worse policy outcome.
So we need to check whether a candidate of type t ∈ 12 − d, 12 + d would have an incentive
to run. The most successful such candidate would be one with t = 1/2.
So for which values of d would a median
citizen have no incentive to run for office. Note
1+d
that all voters with type t ∈ 1−d
,
would vote for candidate 1/2, all voters of type
2
2
1+d
t > 2 would vote for candidate x2 and all voters of type t < 1−d
would vote for candidate
2
x1 . Hence the median candidate would receive d share of the votes, and candidates x1 and
24
x2 would each receive 1/2 − d. So if d < 1/3, a median citizen would not have an incentive
to become a candidate.
Theorem 7. If b > 2c then there exist multiple equilibria with two candidates. For any
d ∈ (0, 1/3), there exists an equilibrium in which two candidates run for office x1 = 1/2 − d
and x2 = 1/2 + d and both candidates win with probability 1/2.
There are clearly many elements of electoral competition missing from the citizen candidate model—in particular, candidates always pursue their most preferred policy if elected—
but there are several interesting results. In particular, notice that the candidates can be as
polarized as 1/6 and 5/6. This is a long way from the median voter theorem. A median
candidate, even though they are the Condorcet winner, would not run for office, because the
non-centrist candidates crowd out the votes for the centrist candidates. As we saw at the
beginning of the course, plurality rule elections may not select the Condorcet winner when
there are three candidates. Arguably, the Republican and Democratic party do this in U.S.
elections, making it difficult for centrist parties to form. Note however that, as long as the
costs aren’t too high, there would also exist an equilibrium with three candidates for office,
x = 1/6, x = 1/2, x = 5/6. Each candidate would receive 1/3 of the votes and each win
office with 1/3 of the probability. So it is difficult to make tight predictions from a citizen
candidate framework.
11
Strategic Voters
Central to the strategic consideration of whether to run for office or not is how voters decide
who to vote for if there are more than two candidates. In Osborne and Slivinski (1996) it
is assumed that voters vote for their first choice. This may not aways be true—think of the
voters who worried that a vote for Nader was a “wasted vote” in the 2000 election or during
the 2016 Republican primary the “never Trump” faction rallying around Ted Cruz even if
they preferred another candidate such as John Kasich.14 The set of equilibria can change
a lot if voters are strategic about which candidate to support, and it will often complicate
the analysis further. Strategic considerations about which candidate to vote for, or which
alternative to support, takes on even greater significance in a small committee with a few
voters. We first illustrate the effect of strategic behavior by voters in a plurality rule election,
and then return to our considerations from the first part of the class about the incentive for
voters to report their preferences truthfully under different decision making mechanisms.
Consider the citizen candidate model, but allowing for strategic voters. This is the
approach taken in Besley and Coate (1997). We now assume that voters seek to affect the
election, which requires, of course, that one vote can influence the outcome. We assume now
that there are 2n + 1 citizen types, that the median type’s ideal point is at tn+1 = 1/2 and
the furthest left and right citizens are t1 = 0 and t2n+1 = 1. Suppose also that there are
several citizens with each of the ideal points (so there are many voters with the same ideal
point as the median voter) and that there are an equal number of citizens of each type. If
voters care only about the outcome and not who they vote for intrinsically, then who each
14
For example, South Carolina Senator Lindsey Graham supported Ted Cruz saying he was his sixteenth
choice out of the seventeen candidates, but he was the most likely to candidate to defeat his last choice.
25
vote only matters if pivotal. We can then construct several equilibria based on the voter’s
expectations on when the election is likely to be pivotal.
First note that, unlike in the case in which voters are sincere, we can have an equilibrium
in which there are two candidates for office, x1 , x2 both who have the median ideal point
(assume the costs are such that a third candidate wouldn’t run). That is x1 = x2 = tn+1 .
Such an equilibrium can be supported with the following voting strategies. If there are two
candidates and the voter is indifferent between the candidates, flip a coin. If there are two
median candidates and a third candidate enters to the left or right of the median, abandon
candidate 2 and each voter votes for candidate 1 or 3 based on which candidate they prefer.
Note that then there would be no incentive for a non-median citizen to run—they would
surely lose. In this equilibrium, the voters on the left respond to the threat of a candidate
on the right by rallying around one of the two candidates at the center, and voters on the
right respond to a threat from the left in the same way. (Note: this coordination is very
difficult for a large number of voters to achieve, but campaigns, endorsements, and polls can
facilitate this coordination by telling the voters which candidate to support).
While the median outcome is an equilibrium with strategic voters, but not with sincere
ones, when the voters behaved sincerely this places an upper bound on how much polarization
is possible. This is not true when voters are strategic. We can have an equilibrium in which
two candidates x1 = t1 = 0 and x2 = t2n+1 = 1 run for office. Voters to the right of the
median vote for candidate 2 regardless of who else runs for office, and those on the left vote
for candidate 1 regardless of who runs for office, and median voters flip a coin between parties
1 and 2. In this case, both candidates win with probability 1/2 and each voter’s vote could
affect the outcome. Since the voters care only about who wins the election, and since nobody
else is voting for any other candidate their vote can never cause a different candidate (for
example a citizen with median policy preferences who ran for office) to be elected, if another
candidate were to enter the race, no voter would have an incentive to deviate and vote for
them. Basically, this is saying that, if voters (similarly activists, volunteers, fundraisers)
are skeptical that a new candidate is likely to win they will not support it. Arguably this
coordination contributes to stability of the two-party structure in the United States. For a
third party to be able to win office it needs to not only be desirable to the voters, but also
be able to convince them that it can win.
The coordination on which candidate to vote for is part of the large class of coordination
games. Other examples include figuring out where to meet for lunch (e.g. we would like to
go to the same place so as to meet each other) and which side of the road to drive on (it
works best of we all drive on the same side of the road, but whether that’s the left or right
may not matter as much). We look for an equilibrium of a game because they are stable—we
if we get to this point, there is no incentive for anyone to change what they are doing—and
in coordination games there are often many equilibria. When there are many equilibria it is
difficult to know which equilibrium is more likely to be played. We often think that some
equilibria are “focal” and so more likely to be played (Thomas Schelling articulated this
concept and won a Nobel Prize in 2005 for his work). For example, we might think that it
is focal to meet in a location we have met at in the past, drive on the side of the road the
street signs say, or vote for the parties that were competitive in the last election. One form
of behavior that is often thought to be focal is to tell the truth, so if it is an equilibrium
for everyone to sincerely report their preferences we expect this equilibrium to be played.
26
If reporting your preferences sincerely isn’t an equilibrium, then it becomes less clear which
equilibrium to expect to be played.
This has shown that plurality rule can cause voters to, for strategic reasons, vote for
alternatives other than their first choice. This raises many issues: (1) We can’t necessarily
interpret vote totals in multi-candidate elections as reflecting the true preferences (2) The
set of equilibria are different (often larger) when voters behave strategically so it is difficult
to make tight predictions about what will happen. If sincere behavior constitutes an equilibrium, even one that isn’t unique, this is likely to a “focal” equilibrium. (3) It is possible
that the most preferred alternative may not be selected because of voters’ expectation that
it won’t win. In fact, it is possible for the election to come down to any two candidates.
This means that, due to the strategic coordination issues, it is difficult to predict the outcome when the voters are behaving strategically. We expect voters to abandon third place
candidates, but can’t be sure ex-ante which candidate(s) will be abandoned. So if we design
an electoral system with considerable room for strategic manipulation we can’t make tight
predictions about what the outcome will be with more than two candidates. When there
are only two candidates there is no reason for individuals (at least if they know which alternative they prefer) to behave strategically. We now consider how the incentive to behave
strategically varies with the electoral rules, and study whether there exists a mechanism that
is immune to strategic manipulation.
12
The Gibbard-Satterthwaite Theorem
It has been well known for a long time that individual voters may have an incentive to
strategically misrepresent their preferences. When Borda proposed the Borda method it
was quickly pointed out to him that individuals would have an incentive to mis-report their
preferences. He famously dismissed the concern, saying that his method was meant for
“honorable men”. To an economist Borda’s response is unsatisfying. First, it is not clear
that it is dishonorable to mis-report ones preferences in an effort to get an outcome the
individual thinks is more desirable. Second, even if we accept that doing so is dishonorable,
if it is also advantageous to misreport ones preferences this would put the “honorable” people
at a disadvantage. So such a procedure would require that everyone who might use this
procedure would be “honorable”. We then want to discuss whether it is possible to design
a system for aggregating preferences in which no voter finds it beneficial to mis-report their
preferences. This leads to the Gibbard-Satterthwaite theorem.
When considering the Arrow’s Impossibility Theorem we assumed that individual preferences are known. But it is rare that we would know the exact preferences of the individuals.
Some of the conditions of Arrow’s theorem—namely independence of irrelevant alternatives
and monotonicity—play a central role in ensuring that individuals do not benefit from misreporting their preferences. Independence of irrelevant alternatives is probably the most
controversial requirement of Arrow’s Impossibility Theorem and some would argue that it
is not always desirable—this is the rationale for the Borda method, for example, which embraces its violation of IIA when calculating its ranking. However, independence of irrelevant
alternatives and monotonicity both play a key role in ensuring that individuals cannot profit
from misreporting their preferences. It is the violation of independence of irrelevant alter27
natives that creates the scope for strategic behavior in plurality rule elections, for example.
Similarly, the following example shows the scope for strategic manipulation in Borda.
Example 5. Suppose there are three voters with preferences
• Voter 1: A 1 B 1 C 1 D
• Voter 2: A 2 B 2 C 2 D
• Voter 3: B 3 A 3 C 3 D
Under the Borda procedure A receives 11 points and B receives 10 points, so A is chosen.
Note, however, that if Voter 3 misreports her preferences as
• Voter 3: B 03 C 03 D 03 A
then A receives 9 points and B still receives 10 points and so B is chosen.
Monotonicity is also important for individuals to have an incentive to report their preferences truthfully. It is easy to see why: if monotonicity is violated it is possible that A would
be selected if person i prefers B to A, but B would be selected if i prefers A to B. Clearly
in that situation person i would have an incentive to mis-report her preferences.15
Before proceeding we define precisely what it means for a choice rule to be “manipulable”
by individuals. Basically manipulable means that is possible to get an alternative you like
more by misreporting your true preferences. If a choice rule is not manipulable we call it
non-manipulable or strategy proof: in that their is no benefit to any individual from behaving
strategically rather than truthfully reporting her own preferences. Unlike in Arrow, we look
for a social choice rule that selects one alternative from the choice set, and we assume that
we can uniquely select one outcome for each reported preference profiles. Individuals could
still be indifferent, but the choice rule must specify which alternative is selected in the event
of a tie in a way that is non-random. The key is that this rules out an approach such
as randomly appointing a dictator, which would clearly be strategy proof, but which are
arguably no more desirable than appointing a dictator. Note that in Arrow’s theorem we
also assumed that the ranking was not random.
Definition 9. A social choice rule, f , is manipulable if there exists a voter i and a profile
of preferences, (1 , . . . , n ), such that for some preference profile 0i ,
f (X, (1 , . . . , i−1 , 0i , i+1 , . . . , n )) i f (X, (1 , . . . , n )).
As social choice rule non-manipulable if it is not manipulable.
To be non-manipulable, it must be that no voter can ever benefit from misreporting their
preferences. So it is always a best response to report truthfully, regardless of the preferences
of the other individuals and whether or not they are reporting truthfully. The individuals
then have a very simple decision: they don’t have to know the distribution of others’ preferences or worry about their strategies, reporting truthfully is always optimal. Clearly if a
15
For empirical evidence of how this can arise in practice see Spenkuch (2015), who looks at strategic
voting in an election to the German Bundestag where monotonicity was violated.
28
social choice rule is non-manipulable then, if the decision is made by this choice rule, it will
always be a Nash equilibrium for all individuals to truthfully report their preferences.
We have seen that if independence of irrelevant alternatives or monotonicity are violated
this can sometimes cause individuals to have an incentive to misreport their preferences. As
such, if instead of looking for a social choice rule that satisfies independence of irrelevant
alternatives and monotonicity we look for a choice rule that is non-manipulable, then, analogously to Arrow, we will find that no such rule exists. This leads to the Gibbard-Sattherwaite
theorem, proven independently by Allan Gibbard and Mark Satterthwaite.
Theorem 8. (Gibbard and Satterthwaite 1973) Suppose X has at least three alternatives and
there are a finite number of voters. Then there does not exist a non-manipulatable choice
rule that yields a deterministic winner for any preferences and satisfies pareto optimality that
is non-dictatorial.
The Gibbard-Satterthwaite Theorem demonstrates that any non-dictatorial, pareto optimal, voting system is potentially manipulable. The requirement that the choice rule be
deterministic rules out, for example, the random dictator choice rule, where one individual
is selected at random with her most preferred alternative selection. The random dictator
method is non-manipulable and pareto optimal but does not give a deterministic winner.
While the incentives to misreport preferences are clear in plurality rule elections, in other
settings the incentives are more subtle. In fact, proponents of electoral reform sometimes
argue in the press that a desirable property of other systems, such proportional representation
and single transferable vote (Hare), is that individuals would not have an incentive to vote
strategically. This is incorrect: if all preference profiles are possible then single transferrable
vote, proportional representation, or any other system devised, would be manipulable.16
As with Arrow’s Impossibility Theorem, the Gibbard-Satterthwaite Theorem depends
heavily on the possibility of any preference combination.17 As we have discussed, the logic of
Gibbard and Satterthwaite theorem is similar to Arrow’s result, and the proof closely mimics
the proof of Arrow’s Impossibility Theorem, replacing the use of independence of irrelevant
alternatives and monotonicity with non-manipulatability at the appropriate point in the
proof. As such we will not go through the proof in this class. The Gibbard-Satterthwaite
theorem is arguably as important, or even more important than, the Impossibility Theorem.
It was around the time that economists started taking information, and the fact that individuals may have private information: about their preferences, what they have done, their
risk or ability level, or about an object they may be selling. It was a core result establishing
the limit on what can be accomplished given the strategic incentives of the individuals in
the setting we are analyzing.
In the application we have considered, it is a voting mechanism that determines which
of a set of alternatives to select. Presumably we care about which alternative is selected
16
The incentives to mis-report in proportional representation is more complicated, since the resulting
policy results from bargaining between the different parties in the legislature depending on the number of
seats each party received. However, no matter what the mapping between seat share and policy outcomes is,
Gibbard-Satterthwaite demonstrates that there would be room for manipulation at least for some preferences.
For example, in the Austen-Smith and Banks (1988) model of government formation the influence of a party
can be monotonic in the share of votes it receives, so it is possible to gain from mis-reporting preferences.
17
When preferences are single-peaked, Condorcet’s method is non-manipulable and always yields a deterministic winner.
29
because it affects the allocation in the economy. In the remainder of the class we consider
how to design systems for allocating objects taking into account that individuals may have an
incentive to misreport their preferences. We will consider this in a simpler environment than
designing voting rules. As Arrow’s Impossibility Theorem demonstrates, it is not clear what
the objective should be even if we knew everyone’s preferences. In many other environments,
however, there is a natural objective. We will consider the limits on what is possible in those
environments.
13
Designing Institutions: What Can We Achieve?
In Econ 201, and in the models of political competition we have considered, we looked at
a specific game and solved for the equilibrium. A different question is: suppose we could
choose the game we could have the individuals play. How would we design the system to
ensure a desirable outcome is obtained? This is similar to the question asked in the Arrow
and Gibbard-Satterthewaite Theorems when we considered whether it is possible to design
a choice procedure that satisfies what we consider to be desirable properties. There we are
asking, if we could choose the electoral system to be used, could we design one that is immune
to the issues we have discussed. We saw that this was not possible, and such theorems serve
to establish the limits of what we can accomplish. This is important for evaluating any
procedure we might use.
If the preferences of all individuals were known, then conditional on determining which
outcome is desirable (itself a difficult problem as we have seen), designing the game to induce
this behavior is trivial—we could specify the game where the only action available to each
player is the action that we want them to take. However, if preferences are unknown, then
we won’t know what the most desirable action is and the problem becomes more difficult.
Moreover, when there are information asymmetries, decentralized markets may not ensure
a socially efficient outcome.
Of course, as we’ve seen, even if we were to know the preferences of all individuals it is not
always clear which alternative should be selected. However, there may be some alternatives
that are clearly suboptimal (e.g. if the alternative is pareto dominated) or some sensible
objective to focus on. For example, when determining how much of a public good to provide
we want to ensure that the outcome is pareto optimal—which as we saw in Econ 201 may not
be the case when left to voluntary contributions of individuals. Another application would
be when selling one (or multiple) object(s). One goal could be to generate as much revenue
for the seller as possible. In this case we are considering second-degree price discrimination
or the design of auctions and we could be trying to generate revenue for the government or a
company. Alternatively we might try to allocate objects to the individuals who value them
most highly, ensure an efficient match between individuals and institutions, or mediate a
dispute to prevent a destructive outcome such as war or the costs of a trial. We begin by
looking at a setting in which there is one object that must be allocated to either the buyer
or the seller. We will then consider the problem of public good provision.
We now turn to the problem of facilitating trade. In this case we would think of the
mechanism designer as a broker, mediator, or arbitrator whose objective is to make efficient
trades happen (assume that this mediator is disinterested third party who only wants to
30
ensure that efficient trades happens and inefficient ones don’t). The deal in question could
be a salary negotiation, one firm could be considering acquiring another (privately held)
firm, or a buyer could be considering a major purchase such as a home. In each case the
seller of the worker’s time, the firm, or the house, has some valuation from walking away and
the buyer has some valuation of owning it, and this valuation is private information. The
designer’s task is to design the rules by which the bargaining should take place for a given
objective.
Suppose the buyer has valuation vb for the object, and the seller has valuation vs for
the object. As we may not know the valuation of the buyer or the seller, assume that
vs ∈ [v s , v̄s ] and vb ∈ [v b , v̄b ], where v s < v̄s and v b < v̄b . We assume that the distribution
for both the seller and the buyer has full support on their interval. That is, P r(vb < v) is
strictly increasing in v on [v b , v̄b ] and P r(vs < v) is strictly increasing in v on [v s , v̄s ]. In
essence, full support says that the density from which vs and vb are drawn from is always
strictly positive.
The buyer and seller each knows their own valuation, but both only know the distribution
of the other’s valuation. Similarly, whoever is designing the mechanism only knows the
distribution of valuations for each individual. Assume also for simplicity, that vb and vs are
independent: this means that there is no information revealed to the buyer from the seller’s
valuation. There are then two things to determine: (1) does the buyer get the object or does
it stay with the seller? (2) how much should the seller receive from the buyer? Let x ∈ [0, 1]
denote the probability the object is transferred to the buyer. That is, if x = 1 sale happens
with certainty, x = 0 means it never does, and x ∈ (0, 1) means the object is sold with some
positive probability. Suppose that the buyer transfers amount t to the seller. The surplus of
the buyer and seller from participating in the transaction are
ub (vb , x, t) = vb x − t,
us (vs , x, t) = t − vs x.
Note that we are normalizing the payoff to not trading to be equal to 0, and assuming that
both the buyer and seller are risk neutral. In this simple setting, a natural goal is to ensure
that trade happens if and only if vb > vs (i.e. x = 1 if vb > vs and x = 0 if vb < vs . If vb > vs
then it cannot be pareto efficient for trade not to happen: if the buyer and seller agree to
trade at any price t ∈ (vs , vb ) both are made better off. Similarly, if vs > vb it cannot be
pareto efficient for trade to happen: if the buyer were to sell the object back to the seller for
any price in (vb , vs ) this would make both better off. From the perspective of the mechanism
designer the price may not be important part of the objective as it simply reflects a transfer
from one player to another and so different prices cannot be pareto ranked.
Since don’t know ex-ante, when designing the mechanism, the realized valuations of the
buyer and seller respectively we seek to design an allocation rule such that the individuals
trade if and only if vb > vs . We say that the mechanism is ex-post efficient if it ensures
trade whenever trade is socially efficient given the realized vs and vb . Ex-post means that
the mechanism we designed before the valuations of the individuals are determined will be
efficient after the valuations have been realized. Note that if we cannot compel the buyer
or the seller to trade then each must prefer to do so rather than walk away. So, for any
valuation the seller/buyer could have they cannot receive negative utility. We call this
31
individual rationality, and our goal is to find an individually rational trading mechanism
under which the final allocation is ex-post efficient.
One mechanism to allocate the object is to allow the seller to set a price, then the buyer
decides whether to accept the price or not. We can quickly see that this may not generate an
equilibrium that is ex-post efficient. In order to calculate the equilibrium we need to assume
the specific distribution of buyer and seller valuations. Let us assume that the buyer and
seller valuations are both uniform on [0, 1]. If we allow the seller to set the price then the
seller would set a price t(vs ): t is the transfer or price she receives from the buyer in the
event of a sale. Note that the seller knows her own valuation and so can condition the price
she sets on her valuation. We assume that the buyer and seller valuations are independent so
the seller’s valuation does not affect the buyer’s valuation: this is different from the market
for lemons problem in which the seller’s valuation reveals information to the buyer—while it
is possible to address that case, it is more complicated and so we focus on the simplest case
in which valuations are independent.
What price would the seller set? Note that if she sets price t then a buyer will purchase
the object if and only vb ≥ t. Assuming that vb is uniformly distributed on [0, 1] this happens
with probability 1−t. Hence, if the seller sets price t she sells the object at price t, generating
surplus t − vs , with probability 1 − t, and with probability t she keeps the object generating
0 surplus. Hence the expected utility of the seller from setting price t is
us = (1 − t)(t − vs ).
The seller sets the price t to maximize her utility. This is a straightforward maximization
problem and taking derivative and solving for the maximum shows that she would set price
vs + 1
.
2
But note that then not all efficient trades are realized: the object is sold to the buyer if and
only if vb ≥ vs2+1 , so if vb ∈ (vs , vs2+1 ) trade would be efficient but would not take place. This
is an example of market-power: if the seller prices at cost (i.e. her valuation), all efficient
trades take place, but the seller would receive no surplus. If the seller prices above cost, this
prevents some, but not all, efficient trades from occurring, but allows the seller to extract
positive surplus from the trades that do happen.
We have seen an example in which the game induced by the seller setting the price is not
ex-post efficient. But this is just one possible mechanism, and an example with one specific
distribution of valuations. We can imagine many other ways to facilitate trade. Is there
any way that will ensure an ex-post efficient outcome? Myerson and Satterthwaite (1983)
shows that it is not possible to construct a mechanism that is both individuality rational and
ex-post efficient. This impossibility theorem does not depend on the assumption that buyer
and seller valuations are uniformly distributed, but extends to any distribution with strictly
positive density on the respective intervals [v s , v̄s ] and [v b , v̄b ] in which there is overlap in
the possible valuations of the buyer and seller. It is important that the two intervals have a
non-empty intersection: if v̄s ≤ v b then having a third party set price any t ∈ [v̄s , v b ] would
lead to efficient trade.
t(vs ) =
Theorem 9. (Myerson and Satterthwaite 1983) Suppose vs is drawn from a distribution with
full support on [v s , v̄s ] and vb from a distribution with full support on [v b , v̄b ]. Suppose that vs
32
and vb are independent and (v s , v̄s ) ∩ (v b , v̄b ) 6= ∅. Then there does not exist an individually
rational bilateral trading mechanism that is ex-post efficient.
The Myerson-Satterthwaite theorem shows that, when individuals have private information, it is not always possible to ensure a pareto optimal allocation no matter how we design
the system. However, this does not mean any mechanism is equally good, and we can compare different mechanisms by how close they come to ex-post efficiency. For example, we
have seen what happens when the trading mechanism is to allow the seller to make a take it
or leave it offer to the buyer. This is not ex-post efficient, but no mechanism will be ex-post
efficient. Is simply allowing the seller to make a take it or leave it offer to the buyer the
best we can do, or is it possible to increase efficiency with a different mechanism, and, if so,
what mechanism? We turn to answering that question in the next section. While we will
not prove the Myerson-Satterthwaite theorem, we will prove a simpler discretized version.
In this environment we will demonstrate that it is not possible to ensure the ex-post efficient
allocation and will solve for the limit on what we can achieve.
The Myerson-Satterthwaite theorem is applied much more generally than to the sale of
an object. It basically says that when there is private information, it is not possible to
guarantee that an efficient outcome will be chosen. This is often applied to study socially
inefficient events such as war, strikes, or delay in reaching deals on the debt ceiling or the
sequester. It says that in environments where a small number of individuals are interacting,
and preferences are unknown, it will not necessarily be possible to ensure the outcome is
pareto efficient. This shows a limit of the Coase Theorem, since even if we could design
the optimal way for the trade to take place the outcome may be inefficient. When there
are a small number of players (so each player has “market power”) information can prevent
efficiency. We consider this difference in problem set 3.
14
The Revelation Principle and Optimal Mechanisms
We now turn to considering what can be accomplished. We consider what game we could
have the buyer and seller play in order to maximize the total surplus from trade. This is
a complicated problem: we could specify any game we could imagine and need to consider
what the equilibrium would be in that game. Our life is made a lot easier due to an important
result introduced by Roger Myerson known as the revelation principle (Myerson, 1979, 1981).
This was the basis of Myerson’s nobel prize in 2008. The revelation principle, in essence,
says that when designing the mechanism (i.e. choosing the game to have the individuals
play) it is sufficient to restrict attention to games in which the only action the individuals
take is to announce their own valuation. This does not mean we should always use a direct
mechanism, but it is useful for determining the limits on what we can and cannot achieve,
and for evaluating the efficiency of different mechanisms.
In mechanism design we are specifying the rules of the game the individuals will play.
Assume there are n players, and that each player i has a type vi ∈ Vi . This type could
reflect the valuation placed on an object or a public good, or something different such as
the individuals ideological ideal point or preferred school. Each individual i knows their
own type, and everyone else only knows the distribution individual i’s type is drawn from.
We specify the game for each individual to play, G, and each individual has a strategy si
33
which determines the action they would take as a function of their type (e.g. In the price
setting game, the strategy for the seller is what price to set for any valuation, and the buyer’s
strategy is the decision, for any valuation, of whether to accept the seller’s offer). As each
individual (may) not know the types of other players, and so may not know which action
they take (e.g. the seller doesn’t know the buyer’s valuation and so doesn’t know if her offer
will be accepted), each player seeks to maximize their expected utility. A strategy profile
s∗ = (s∗1 , . . . , s∗n ) constitutes a (Bayesian) Nash Equilibrium if, for all individuals i, all types
vi , and all strategies s0i ,
Ev−i [ui (s∗1 (v1 ), . . . , s∗n (vn ))] ≥ Ev−i [ui (s∗1 (v1 ), . . . , s∗i−1 (vi−1 ), s0i (vi ), s∗i+1 (vi+1 ), s∗n (vn ))].
That is, a strategy profile constitutes a Nash equilibrium if each player is maximizing their
expected utility given their type and the probability distribution over the other players’
types.
For any game G we can then calculate the set of equilibria, and resulting outcome of
the game. The revelation principle is the key insight that, if there exists a game G with an
equilibrium s∗ , then we can define a new game G0 as follows. The allowed strategy of each
player is to report their type vi ∈ Vi . We call G0 a direct mechanism, since all we allow the
individuals to do is report their type. Under the game G0 the outcome if each individual
reports vi is the allocation (s∗1 (v1 ), . . . , s∗n (vn )) from the game G. Note that if s∗ is a Nash
equilibrium, then no individual i could do better under a different strategy: including the
strategy of playing s∗i (vi0 ) for some vi0 6= vi when the individual’s valuation is vi . Hence if
s∗ is an equilibrium of G then it is an equilibrium for each individual to truthfully report
their type in the game G0 . We refer to a mechanism for which it is an equilibrium for each
individual to report their type truthfully as incentive compatible.
The above argument shows that we can restrict attention to a mechanism or game, G, in
which each player simply reports their true type, vi , and truthful revelation is incentive compatible. We apply the revelation principle to study the optimal bilateral trading mechanism.
In order to be able to calculate the optimal mechanism, we consider a simpler environment in
which the seller and buyer’s valuations are both binary. Assume that the buyer’s valuation
could be either 1 or 5 each with equal probability, and the seller’s valuation could be 0 or
4, each with equal probability. The buyer and seller’s valuation are independent. To solve
this we seek to maximize our objective—the total gains from trade—given the individual rationality and incentive compatibility constraints. This is an, admittedly more complicated,
version of the constrained maximization problems we saw in Econ 200 and 201.
Such a mechanism can then be represented by the probability of trade x(vs , vb ) for any
reported valuation, and the transfer from the buyer to the seller from this reported valuation t(vs , vb ). As there are two types for each the seller and the buyer, the mechanism
must specify and probability of trade and a transfer for each of the four type combinations.
Hence the mechanism consists of solving for eight numbers: x(0, 1), x(0, 5), x(4, 1), x(4, 5)
and t(0, 1), t(0, 5), t(4, 1), t(4, 5).
Since the buyer doesn’t know the seller’s valuation, if the buyer announces that she
b)
and the exhas valuation vb then the expected trade probability is xb (vb ) = x(0,vb )+x(4,v
2
t(0,vb )+t(4,vb )
pected payment is tb (vb ) =
. Similar calculations apply for the seller, xs (vs ) =
2
x(vs ,1)+x(vs ,5)
t(vs ,1)+t(vs ,5)
and ts (vs ) =
. We need to ensure the incentive compatibility and
2
2
34
individual rationality constraints for the buyer and the seller of each type in any mechanism.
Taking the buyer first we have
5xb (5) − tb (5) ≥ 5xb (1) − tb (1),
xb (1) − tb (1) ≥ xb (5) − tb (5),
and
5xb (5) − tb (5) ≥ 0,
xb (1) − tb (1) ≥ 0.
The first two inequalities are the IC constraints for each type, and the last two inequalities
are the IR constraints. Similarly, for the seller we get
ts (0) ≥ ts (4),
ts (4) − 4xs (4) ≥ ts (0) − 4xs (0),
and
ts (0) ≥ 0,
ts (4) − 4xs (4) ≥ 0.
Note that, to be ex-post efficient the mechanism must involve
x(4, 1) = 0,
and
x(0, 1) = x(0, 5) = x(4, 5) = 1.
We first show that there is no incentive compatible (IC) and individually rational (IR)
mechanism that is ex-post efficient.
Theorem 10. (Discrete Myerson-Satterthwaite) Suppose the buyer’s valuation is vb ∈ {1, 5},
both with equal probability, the seller’s valuation is vs ∈ {0, 4} both with equal probability, and
the buyer and seller valuations are independent. Then there does not exist an individually
rational bilateral trading mechanism that is ex-post efficient.
Proof. By the revelation principle, it is sufficient to show that there is no incentive compatible
and individually rational direct mechanism that is ex-post efficient. To have an ex-post
efficient allocation, the allocation is pinned down for any realization of valuations. So we
must show that there do not exist a set of transfers t(0, 1), t(0, 5), t(4, 1), t(4, 5) consistent
with IC and IR and the ex-post efficient allocation. To show this we will demonstrate that
it is not possible to satisfy the IC constraints when vb = 5 and vs = 0 and the IR constraints
when vb = 1 and vs = 4 simultaneously. To show this we proceed by contradiction, and
will show that it is not possible to simultaneously satisfy the IC constraints for the high
valuation buyer and the low valuation seller while also satisfying the IR constraints for the
low-valuation buyer and the high valuation seller.
Suppose there were, and so both the buyer and seller are reporting truthfully. As the
buyer and seller have independent valuations, regardless of their own valuation they believe
35
each valuation is equally likely for the other player. First consider the buyer, and focus on
the IC constraint of the high valuation buyer. If a seller reports vb = 5 she gets the object
with probability
1
xb (5) = (x(0, 5) + x(4, 5)) = 1
2
and pays
1
tb (5) = (t(0, 5) + t(4, 5))
2
in expectation. If she reports vb = 1 she gets the object with probability
1
xb (1) = (x(0, 1) + x(4, 1)) = 1/2
2
and pays
1
tb (1) = (t(0, 1) + t(4, 1)).
2
So to have the buyer report truthfully when her valuation is vb = 5 we must have
xb (5)5 − tb (5) ≥ xb (1)5 − tb (1)
or equivalently
5
≥ tb (5) − tb (1).
(1)
2
We next consider the IR constraint for the buyer with vb = 1. The mechanism we must
have
xb (1) − tb (1) ≥ 0
or equivalently
tb (1) ≤ 1/2.
Now consider the seller. If she reports vs = 4 the object is sold with probability
1
xs (4) = (x(4, 1) + x(4, 5)) = 1/2
2
and she receives
1
ts (4) = (t(4, 1) + t(4, 5))
2
in expectation. If she reports vs = 0 the object is sold with probability
1
xs (0) = (x(0, 1) + x(0, 5)) = 1
2
and she receives
1
ts (0) = (t(0, 1) + t(0, 5)).
2
Now consider the IC constraint when vs = 0. To be satisfied we must have
ts (0) − xs (0)0 ≥ ts (4) − xs (4)0,
36
(2)
or equivalently
ts (0) − ts (4) ≥ 0.
(3)
For the IR constraint when vs = 4, seller is willing to participate if and only if
ts (4) − xs (4)4 = ts (4) − 2 ≥ 0.
(4)
We now show that it is not possible to simultaneously satisfy (1)–(4). By (2)
t(0, 1) + t(4, 1)
1
≤
2
2
so
t(0, 1) + t(4, 1) ≤ 1.
And from (1) and (2),
5
t(0, 5) + t(4, 5) 1
≥ tb (5) − tb (1) ≥
−
2
2
2
or equivalently
t(0, 5) + t(4, 5) ≤ 6.
Adding up these two constraints, to get the IC satisfied for the high valuation buyer and the
IR for the low valuation buyer we must have that
t(0, 1) + t(0, 5) + t(4, 1) + t(4, 5) ≤ 7.
Now note that by (4),
t(4, 1) + t(4, 5)
≥2
2
so
t(4, 5) + t(4, 1) ≥ 4.
Further by (3) and (4),
ts (0) =
t(0, 1) + t(0, 5)
≥ ts (4) ≥ 2
2
so
t(0, 1) + t(0, 5) ≥ 4.
Adding up these two conditions, to get the IR satisfied for the high valuation seller and the
IC for the low valuation seller we must have
t(0, 1) + t(0, 5) + t(4, 1) + t(4, 5) ≥ 8.
This, obviously contradicts that t(0, 1) + t(0, 5) + t(4, 1) + t(4, 5) ≤ 7 and so the constraints
for the buyer and seller cannot be satisfied simultaneously. Hence there cannot exist an
ex-post efficient mechanism.
37
The proof of the discrete Myerson-Satterthwaite Theorem illustrates the fundamental
friction which makes it impossible to ensure ex-post efficiency. The IR constraints put an
upper bound on how much the lowest-valuation buyer can be expected to pay, and the IC
constraints but a bound on how fast the expected payment can be increasing in the buyer’s
valuation. In particular, since a buyer with a higher valuation could always mimic one with
a lower valuation, such a buyer must expect to pay less than their valuation when from
reporting truthfully. In the discrete setting,
xb (5)5 − tb (5) ≥ xb (1)5 − tb (1) > xb (1) − tb (1) ≥ 0,
and so
tb (5) < 5xb (5).
Reversing these arguments, the seller must expect to receive more than her valuation from
reporting truthfully (unless she has the highest possible valuation). This means that when
the seller’s valuation is slightly lower than the buyer’s valuation, so trade will be efficient, it
will not be possible to ensure it happens.
We now turn to the optimal mechanism. We define the objective of the designer to be
to maximize the total gains from trade:
X
E[x(vs , vb )(vb − vs )] =
p(vs , vb )x(vs , vb )(vb − vs )
vs ,vb
x(0, 1)(1 − 0) + x(0, 5)(5 − 0) + x(4, 1)(1 − 4) + x(4, 5)(5 − 4)
4
5x(0, 5) + x(0, 1) + x(4, 5) − 3x(4, 1)
.
=
4
=
Now that we have our objective we now look at the constraints that must be satisfied. We
need to look at the IC and IR constraints for the seller and buyer of each type. Combining
the IC and IR constraints we have a total of eight constraints. In addition we need to
make sure that x(vs , vb ) ∈ [0, 1] for each vs and vb . We then must maximize a our objective
5x(0, 5) + x(0, 1) + x(4, 5) − 3x(4, 1) subject to (1) the IC and IR constraints (2) x(vs , vb ) ∈
[0, 1] for each vs and vb .
This is now a standard problem like we have seen in Econ 200 and Econ 201: We can
set up the Lagrangian to maximize the objective, with a Lagrange multiplier for each of
the (sixteen) constraints. While not all constraints bind, we can solve for the Kuhn-Tucker
conditions to determine which constraints bind which don’t. Alternatively we can reason
out which constraints will bind, use this to reduce the number of constraints. Even with
these simplifications, however, since there are so many constraints the computations become
rather involved. So, in the interest of time, we will not explicitly work through solving the
optimal mechanism, but rather turn to a simpler problem in the next section.
While we have set up the problem with discrete valuations, we could do the same thing
when the preferences of the buyer and the seller are both uniformly distributed as in the
previous section. While it is possible for us to solve for the surplus maximizing mechanism, it
is more complicated than the constrained maximization problems we have seen. The reason
for this this is that we now must maximize over the functions x(vs , vb ) and t(vs , vb ) which
38
each have a continuum as their domain. For this reason we will not set up or solve this
problem in this class. The optimal mechanism given the incentive constraints is solved for
in Myerson and Satterthwaite (1983) and the solution is quite intuitive.
The optimal mechanism involves allocation rule
1 if vb > vs + 1/4,
x(vs , vb ) =
0 otherwise.
This allocation rule is implemented with transfers
vs +vb +0.5
if vb > vs + 1/4,
3
t(vs , vb ) =
0
otherwise.
There are two important take-away points from this solution. First, we can see that
it is not possible to design a mechanism that is guaranteed to generate an ex-post efficient
allocation: when vs < vb < vs +1/4 it would be efficient to have the object sold, but under the
optimal mechanism it isn’t. This is shows the Myerson-Satterthwaite theorem for the case
with uniform distribution. That we cannot achieve the first best shows us that informational
constraints are real constraints applying to the problem that must be taken seriously and
that limit what we can achieve. We often refer to the an optimal mechanism, given the
incentive constraints, as “second best”. Second, for any mechanism that is proposed, since
we can calculate the optimal mechanism, we can determine whether that mechanism achieves
the “second best” or not. We can see that the trading mechanism of having the seller make
a take it or leave it offer to the buyer does not achieve the second best. When the seller
makes it a take it or leave it offer, the result is that
1 if vb > vs2+1 ,
x(vs , vb ) =
0 otherwise.
As this differs from the optimal, we can see that having the seller make a proposal to buyer
is not the “second best” mechanism.
While the revelation principle says that there is a direct mechanism that implements the
second-best allocation, it is possible to implement this allocation with a mechanism that
seems more likely in practice. Suppose we use the following mechanism: the buyer and seller
b
each simultaneously set a price, ps and pb , and the mechanism is to trade at price t = ps +p
2
if ps ≤ pb and to not trade otherwise. As we will see on problem set 4, that mechanism is
not incentive compatible, and the seller will set a price ps > vs and the buyer will set a price
pb < vb . However, in the equilibrium of this game, the second-best allocation characterized
above is attained: we will solve for the equilibrium of this game in problem set 4. So we
can verify that the seller making a take it or leave it offer to the buyer is not optimal, but
allowing the buyer and seller to each set price and trading at the average of the two prices
is.
What are the economics behind why it is better to give both the buyer and the seller
an opportunity to set a price rather than simply allow the seller to set the price? Recall
that, when the seller sets the price, she has market power and so will set a price higher than
her valuation in an effort to extract surplus. As we have seen in Econ 201, the deadweight
loss associated with market power is quadratic in the amount of market power the seller
39
possesses. By giving both the buyer and the seller proposal power this is split between the
two individuals, meaning that the marginal effect of each agent exerting market power is
decreased.
15
Regulating a Monopolist
In the last section we considered the design of an institution to facilitate trade between a
buyer and a seller. We now consider the possibility of regulating a market. As we have seen
in Econ 201, a market in the absence of government intervention may or may not be efficient:
Efficiency may break down due to externalities (which we consider in the next section by
looking at public good provision) or when there is market power, which we consider in
this section. If the government has all relevant information, and the authority to regulate
the market, it can simply force everyone to take actions that ensure the outcome is pareto
efficient. If there is some information the government lacks (as is generally the case) then its
problem becomes more complicated. Here we consider a simplified version of the problem,
introduced in Baron and Myerson (1982), of regulating a monopolist with unknown costs.
Suppose the monopolist could either have high or low cost, each with probability 1/2.
To make the problem as simple as possible, assume the firm has linear costs
c(q) = cq
where c ∈ {1, 2} is the firm’s cost of producing each unit and q is the quantity produced.
We say that a firm is low cost if c = 1 and high cost if c = 2. The firm knows its own cost, c,
but the government only knows the distribution of costs. Both the firm and the government
know the distribution of buyer preferences. The demand for the product the monopolist sells
is
q(p) = a − p
where a > 3.18
From Econ 201 we know that the monopolist, if not regulated, will set p > c, generating
deadweight loss and the possibility that government regulation could improve efficiency. We
assume that the government can choose the price to regulate the firm at, p, and also the
transfer (positive or negative), t, to make to the firm. If it sets t > 0 then it is subsidizing
the firm, whereas if t < 0 it is taxing (charging a fee) for the firm to be able to operate. The
price in the market then determines the quantity based on consumer demand. We assume
the objective of the government is to maximize the consumer surplus less the amount of the
transfer paid to the monopolist,
Z q(p)
(a − p)2
− t.
CS(p) − t =
(a − q − p)dq − t =
2
0
Hence the government doesn’t care about the firm’s profit, but the firm has the option to
shut down if it were to earn negative profits.19
18
Assuming that a > 3 will ensure that it will be optimal for the government to allow both the high and
low cost firm to operate; when a < 3 it will be optimal to force the high cost firm to shut down. Why a > 3
is the necessary condition will become clear later.
19
We could allow the government to care about firm profits/producer surplus as well. However we want to
incorporate that the government would prefer not to make transfers to the firm if would operate anyways.
40
The firm’s profits are then
π(c, p, t) = t + (p − c)q(p) = t + (p − c)(a − p)
and the IR constraint is that π(c, p, t) ≥ 0. If the government knew the firm’s cost c then it
would choose −t = (p − c)(a − p) to capture all surplus, and then choose p to maximize
(a − p)2
+ (p − c)(a − p).
2
Taking FOC,
−(a − p) + (a − p) − (p − c) = 0
so it would be optimal to set p = c (hence t = 0). This is familiar from Econ 201: to maximize
the total surplus we would want to regulate the firm to price at cost, and here we can make
sure all of that surplus goes to consumers. We now consider the optimal mechanism when
the government does not know the firm’s cost.
By the revelation principle it is sufficient to restrict attention to direct mechanisms which
are incentive compatible and individually rational. In the direct mechanism the government
simply asks the firm to report its cost c ∈ {1, 2} then chooses the price pc and transfer tc .
Since each cost is equally likely the government’s objective is then to maximize
(a − p2 )2
1 (a − p1 )2
− t1 +
− t2 .
(5)
2
2
2
The first thing the government must do is ensure the firm will continue to operate. This
gives the IR constraints for the low and high cost firms respectively:
π(1, p1 , t1 ) = t1 + (p1 − 1)(a − p1 ) ≥ 0,
(6)
π(2, p2 , t2 ) = t2 + (p2 − 2)(a − p2 ) ≥ 0,
(7)
The government also needs to make sure that both types of firm truthfully report their cost.
That is we must have π(1, p1 , t1 ) ≥ π(1, p2 , t2 ) and π(2, p2 , t2 ) ≥ π(2, p1 , t1 ). Substituting
into these equations we get the IC constraints:
t1 + (p1 − 1)(a − p1 ) ≥ t2 + (p2 − 1)(a − p2 ),
(8)
t2 + (p2 − 2)(a − p2 ) ≥ t1 + (p1 − 2)(a − p1 ).
(9)
We are then left with four constraints. Note, however, that some of them are redundant.
In particular, we do not have to worry about the low cost firm dropping out of the market:
the payoff to a low cost firm of selling at the high cost firm’s price is always higher than
to the high cost firm, but the high cost firm’s profits are non-negative or its IR constraint
would be violated. So combining (8) and (7) we have
t1 + (p1 − 1)(a − p1 ) ≥ t2 + (p2 − 1)(a − p2 ) ≥ t2 + (p2 − 2)(a − p2 ) ≥ 0
so the IR constraint for the low cost firm is automatically satisfied if the other constraints
are satisfied. Next note that, since the IR constraint for the low cost firm is redundant, the
41
IC constraint must hold with equality. Why? If not, the government could decrease t1 and
still have the firm operate and select the correct contract. But if the IC constraint for the
low cost firm is satisfied with equality and p2 ≥ p1 , the IC for the high cost firm holds for
free. When (8) holds with equality,
t1 + (p1 − 2)(a − p1 ) = t1 + (p1 − 1)(a − p1 ) − (a − p1 )
= t2 + (p2 − 1)(a − p2 ) − (a − p1 )
≤ t2 + (p2 − 1)(a − p2 ) − (a − p2 )
= t2 + (p2 − 2)(a − p2 )
so the IC constraint for the high cost firm is automatically satisfied as well. This argument
means that only two of the constraints, the IC for the low cost firm and the IR for the high
cost firm, are relevant and we only have two constraints to worry about.20
So we are left maximizing (5) subject to (7) and (8). Letting λ1 and λ2 be the Lagrange
multipliers associated with (8) and (7) the Lagrangian for this problem is,
1 (a − p1 )2
(a − p2 )2
L(t1 , t2 , p1 , p2 , λ1 , λ2 ) =
− t1 +
− t2 +
2
2
2
λ1 (t1 + (p1 − 1)(a − p1 ) − t2 − (p2 − 1)(a − p2 )) + λ2 (t2 + (p2 − 2)(a − p2 )).
Taking First Order Conditions,
1
∂L(t1 , t2 , p1 , p2 , λ1 , λ2 )
= λ1 − = 0,
∂t1
2
∂L(t1 , t2 , p1 , p2 , λ1 , λ2 )
1
= −λ1 + λ2 − = 0,
∂t2
2
a − p1
∂L(t1 , t2 , p1 , p2 , λ1 , λ2 )
=−
+ λ1 (a + 1 − 2p1 ) = 0,
∂p1
2
∂L(t1 , t2 , p1 , p2 , λ1 , λ2 )
a − p2
=−
− λ1 (a + 1 − 2p2 ) + λ2 (a + 2 − 2p2 ) = 0,
∂p2
2
∂L(t1 , t2 , p1 , p2 , λ1 , λ2 )
= (t1 + (p1 − 1)(a − p1 ) − t2 − (p2 − 1)(a − p2 ) = 0,
∂λ1
∂L(t1 , t2 , p1 , p2 , λ1 , λ2 )
= t2 + (p2 − 2)(a − p2 ) = 0.
∂λ2
From the first two FOCs we get that
λ1 = 1/2,
λ2 = 1.
20
The argument for why we need only consider the IR for the high cost firm and the IC for the low cost
firm is the same as the argument for why, in the Discrete Myerson-Satterthwaite proof, we looked at the
IR constraints for only the low valuation buyer and the high valuation seller, and the IC constraints for the
high valuation buyer and the low valuation seller.
42
Substituting this into the third and fourth FOC we see
p1 = 1,
and
p2 = 3.
Finally from the last two FOCs we have
π(2, t2 , p2 ) = t2 + (3 − 2)(a − 3) = 0,
so t2 = 3 − a, and
π(1, t1 , p1 ) = t1 = 3 − a + (3 − 1)(a − 3) = a − 3 > 0.
Comparing the optimal mechanism when the government is unsure of the firm’s cost to
the case where costs are known we see the following:
1. It is not possible for the government to implement the full information solution when
the monopolist privately knows its own cost.
2. The optimal regulation involves the government setting the efficient price for the low
cost firm (p1 = 1) but giving the low-cost monopolist a subsidy, t1 = a − 3 > 0, that
allows them to earn positive profits. This is because, otherwise, the low cost firm would
mimic the high cost firm and earn positive profits that way.
3. In the optimal regulation, the high cost firm is made exactly indifferent between operating and not operating, so receives none of the surplus. However the price set for the
high cost firm is not efficient. The government forces the firm to price above marginal
cost, then takes the surplus in the form of a fee collected for the firm to be able to
operate. The reason for this that, by increasing the price/decreasing the quantity it
reduces the surplus the low cost firm would receive when the high cost firm earns 0
profits. So the government creates an inefficiency when the firm is high cost so as to
be able to extract more surplus when the firm is low cost.
For those who studied second degree price discrimination in Econ 201 the logic is similar.
In that case the firm extracts all surplus from the low valuation consumer, but has to give
the consumer with a higher valuation some surplus in order for them to select the contract
meant for them. The seller also degrades the quality offered to the low valuation consumer
to reduce the amount of surplus it must share with the high valuation consumer. Here the
government extracts all surplus from the high cost (i.e. low profitability) firm, and forces it
to produce less than optimal, to reduce the amount of surplus it has to share with the low
cost (i.e. high profitability) firm.
43
16
The Provision of Public Goods
We now consider another classic problem: how to ensure an efficient amount of public good
provision. The provision of public goods is difficult since, as we saw in Econ 201, there is a
positive externality associated with contributing to a public good, and so each individual will
contribute less than the socially optimal amount. Hence we do not expect that voluntary
contributions of individuals to lead to a pareto efficient level of public good provision. If we
know how much each individual values the public good then calculating the efficient level
of public good provision is straightforward. Then we can specify the game in which each
individual must contribute the appropriate amount in order to pay for the efficient level of
the public good. However, it is difficult to get people to reveal their valuation of the public
good truthfully. Since all individuals value the public good, they must pay the part of the
cost or else they would have an incentive to announce a very high valuation. However if
the amount the individual must contribute increases too much based on their announced
valuation, they will have an incentive to under-report their true valuation and free-ride on
the contributions of others. This was the case in the example we studied on problem set 3.
A class of mechanisms that allows us to ensure that the social efficient level of public good
is provided are the Vickrey-Clarke-Groves mechanisms. William Vickrey who developed the
idea—that was later generalized independently by Clarke and Groves—in the context of
auctions was awarded a Nobel prize for this work in 1996. The idea behind the VickreyClarke-Groves (VCG) mechanism is simple. In order to make it incentive compatible for
individuals to truthfully report their preferences their contribution to the public good must
reflect their own value of increasing the public good, but not force the individual to pay for
the externality that increasing the public good has on others. If the game can be set up
in this way we can have individuals reveal their valuation and provide the efficient level of
public good.
Suppose there are n individuals i = 1, . . . , n with utility functions
ui (G, t) = vi log(G) + w − ti ,
where G is the amount of public good, and ti is the amount of the transfer individual i
must make the government to pay for the provision of the public good. Assume that each
individual’s valuation of the public good vi is their own private information. Note that, as
we have discussed, the pareto optimal allocation are those which maximize the sum of the
individuals’ utilities. This depends heavily on the fact that it is possible to make transfers
between the individuals (adjust the ti ’s). If this were not the case, allocations which do
not maximize the sum of utilities could still be pareto optimal. If we set up the problem of
maximizing
n
X
vi log(G) + w − ti
i=1
subject to the constraint
G≤
n
X
i=1
44
ti
we see immediately that any solution must have
G=
n
X
vi .
i=1
P
If our goal is to choose the socially efficient level of public good provision G = ni=1P
vi then
we need to look for an incentive compatible direct mechanism under which G = ni=1 vi .
The difficulty lies in constructing the transfers to make this mechanism incentive compatible.
Under a VCG mechanism each individual
Pn reports their valuation vi of the public good, the
level of public good provision is G = i=1 vi and each individual makes transfer
"
!
!#
n
X
X
X
X
ti (v1 , . . . , vn ) = t̄ + vi −
vj log
vk −
vj log
vk
.
j6=i
k=1
j6=i
k6=i
The amount t̄ is a constant determined by the mechanism designer that must be paid regardless of the reported valuation and so does not affect the individual’s incentives. Note
that vi reflects the increase in the optimal amount
of the public
good because individual i
P
P
Pn
P
values it, and j6=i vj log ( k=1 vk ) − j6=i vj log
k6=i vk is the amount that individuals
other than i would be willing to pay to increase the amount of public good by vi . The idea of
the VCG mechanism is as follows. Because the higher individual i’s valuation is the higher
the level of public good, and because increasing the amount of the public good provides a
benefit to other individuals, the VCG mechanism requires each individual to pay for their
own value of increasing the public good, but not the effect of this increase on the utility of
other individuals. In general, in a world with externalities, in order to make a mechanism
incentive compatible each individual must only pay for their own marginal valuation and not
the externality component. Note that here, regardless of the reports of the other individuals,
person i has an incentive to truthfully report her own preferences, vi . If individual i with
valuation vi reports valuation vi0 her payoff is
!
X
vi log
vj + vi0 + w − ti (v1 , . . . , vi−1 , vi0 , vi+1 , . . . , vn ) =
j6=i
!
vi log
X
j6=i
vj + vi0
!
+ w − t̄ − vi0 +
X
vj log
j6=i
X
vk + vi0
!
−
k6=i
X
j6=i
vj log
X
vk
.
k6=i
Taking derivative with respect to vi0 , and setting equal to 0 we get first order condition
P
vi
j6=i vj
P
−1+ P
= 0.
0
0
k6=i vk + vi
k6=i vk + vi
So we can see that the vi0 that maximizes individual i’s payoff is
vi0 = vi .
This means that it is a Nash equilibrium for all individuals to truthfully report their true
valuation, and so the VCG mechanism is incentive compatible. In fact, since individual i
45
has an incentive to report her valuation truthfully regardless of other’s valuation, the VCG
mechanism is non-manipulable or strategy-proof in the sense we described in section 12.
Note that a mechanism being strategy proof is stronger than it being a Nash equilibrium for
each individual to truthfully report their valuation: to be a Nash equilibrium no individual
can gain from misreporting their preferences in expectation, but to be strategy proof no
individual can benefit from misreporting their valuation no matter what the valuation of
the other individuals happens to be. That is, a mechanism is incentive compatible if it is a
Nash equilibrium for each person to report truthfully, it is strategy-proof if each person has
a dominant strategy to report truthfully. We often like mechanisms that are strategy-proof
since, especially with a large number of individuals in the economy, it may not be clear what
each individual believes about the others’ valuations.
One important feature to note of the VCG mechanism is that the total amount of public
good provided,
n
X
G=
vi ,
i=1
may not equal to the total amount collected
!#
!
"
n
n
n
n
X
X
X
X
X
X
X
.
vj log
vk
vj log
vk −
ti = nt̄ +
vi +
i=1
i=1
i=1
j6=i
k6=i
j6=i
k=1
This is because the amount each individual pays is different from their valuation and
"
!
!#
n
n
X
X
X
X
X
vj log
vk −
vj log
vk
i=1
j6=i
k6=i
j6=i
k=1
depends on the valuation of the individuals. Given that t̄ must be chosen before the vi are
reported we don’t know its value when designing the mechanism. And if t̄ depended on
the reported valuation this would change the incentives of the players and cause them to
misreport their valuations. In fact, there is no way to construct a mechanism that guarantees
both an efficient level of public good provision and that the amount collected will equal to
the amount spent on the public good. We call this budget balancedness.
Theorem 11. There does not exists an incentive compatible, budget-balanced mechanism
that guarantees the efficient level of public good provision.
So there are positive and negative conclusions from the study of public good provision
problem. While we can ensure the socially efficient level of public good, any mechanism that
ensures the socially optimal level of public good provision runs the risk of running budget
deficit or a budget surplus. We can construct mechanisms that ensure that the allocation
will be budget balanced, but those mechanisms will not ensure the socially efficient level
of public good. So again we see that there are tradeoffs. We can construct t̄ so that the
budget is balanced in expectation, but if we insist that budget balancedness must occur for
any profile (v1 , . . . , vn ) then we won’t be able to implement the efficient level of the public
good.
46
17
The Design of Markets and the Gale Shapley Algorithm
We conclude the course by looking at the design of markets in which two distinct sets of
individuals or institutions must be matched with individuals/institutions from the other set,
and both sides have preferences over who they match with. In the most environments the
seller of an object doesn’t care who they sell to but only the price they receive. However
there are many settings where who an individual matches with is very important. Some
examples include: matching students with schools (either college or more commonly due to
the centralized structure, public school systems), matching medical residents with hospitals,
matching kidney donors with those in need of a transplant, and marriage markets. Moreover,
in many of those settings, monetary transfers are either restricted or forbidden.
We begin by defining the Gale-Shapley algorithm for matching, a commonly used approach that is guaranteed to produce a “stable” match introduced in Gale and Shapley
(1962). We will then conclude the class by considering some applications of market design.
The Gale-Shapley algorithm is covered in Chapter 1 of the Gura and Maschler (2009) textbook, available on the library reserves section of Chalk. Lloyd Shapley shared the Nobel
Prize in economics in 2012 with Al Roth: Shapley for his theoretical work on developing this
algorithm, and Roth for applying it and adapting it to the design of different markets. A
survey article on market design written by Roth that highlights many practical issues and
applications of market design is available on Chalk. We will discuss these applications after
describing the algorithm.
Assume there are n individuals/institutions on each side of the market, and that each
individual/institution has strict preferences over the potential matches on the other side. For
concreteness, and to be consistent with the Gale-Shapley terminology, suppose there are n
men, a1 , . . . , an , and n women, b1 , . . . , bn , and that each side of the market matches with one
individual from the other side of the market (all heterosexual monogamous matches with no
monetary transfers). Assume that each individual can rank all potential matches as well as
not being matched. For simplicity, assume that each individual has strict preferences over
each potential match. Let ai and bj be the preferences of the ith man and jth woman
respectively. This preference order is over n + 1 alternatives: the n potential matches of the
other gender and the option to remain unmatched.
In a matching problem each individual/institution is matched with an individual/institution
on the other side of the market.21 Since an individual can’t unilaterally decide to match with
someone else we look for “stable” matches, in which no pair of individuals could agree to
match with each and both do better. If the matching outcome is not stable then two individuals could benefit by leaving their matches and matching with each other. Stability
is important since it ensures that no pair of individuals could decide, after the algorithm
has assigned matches for each pair, to leave the market and match with each other. This is
important when it is not possible to compel people to participate in the clearinghouse. We
define a stable match as follows.
21
A related problem, often referred to as the roommate assignment problem, is similar but instead of
assuming there are two different pools of individuals to match with the other pool, everyone is in the same
pool and we assign an efficient match.
47
Definition 10. A match is stable if there do not exist a man ai and a woman bj who are
not matched together but who prefer each other to the individual (if any) they are matched
with.
The Gale-Shapley Algorithm, sometimes called the Deferred Acceptance Algorithm, is a
simple procedure that works as follows. In the first round, each man “proposes” to the woman
who is his first choice. Any woman who received only one proposal becomes “engaged” to that
man unless she prefers to be unmatched, and any woman who receives multiple proposals
becomes engaged to her first choice among those who proposed to her, and “rejects” the
others. In each future round, every man who was rejected in the previous round proposes
to his most preferred woman who has not rejected him so far. Each woman chooses her
most preferred alternative among those who proposed to her in this round (if any) and the
individual she was engaged to in the previous period (if any) and rejects all but her most
preferred alternative. When we reach a stage at which no man is rejected, because there
is one man proposing to each woman (or deciding not to propose to any of the remaining
who have not rejected), the algorithm terminates. At this point the engagements become
“marriages” and the individuals are matched.
Obviously real marriage/dating markets do not work exactly like this, but this is literally
how students are assigned seats in a school choice program: they (or rather their parents)
submit a choice form detailing their preferences, and the school board puts this list in the
computer with the each school’s ranking of which students have priority for that seat, and the
Gale-Shapley, or a related, algorithm, is applied to assign students to different schools. That
problem is a little different than the one we considered here—in that case several students
are matched to each school and the school may not have strict preferences. However the
basic algorithm can be adapted to such a setting, and that problem is known as the college
assignment problem.
Note that, by construction, the Gale-Shapley algorithm cannot terminate without reaching a stable match: if any man prefers a different woman to the one he matches with he
would have proposed to her, and if that woman prefers him to the man she matched with
she would have accepted his proposal instead (once a woman is engaged this engagement
can only be broken if she gets a proposal she prefers). Moreover, since there are n men, and
each man can only propose to each woman at most once, it is clear that the Gale-Shapley
algorithm must terminate in no more than n2 rounds and so ultimately settles into a stable
match in finite time.
A more difficult issue is whether individuals have an incentive to truthfully reveal their
preferences and whether the outcome is efficient. Roth (1982) showed that each man has
an incentive to truthfully report his preferences, but that a woman may benefit from misreporting her true preferences. The reason for this is that the Gale-Shapley algorithm produces the optimal stable allocation from the perspective of the men, but not necessarily from
the perspective of the women. Intuitively, it gives the side that makes the proposals (men)
the ability to propose to anyone, but the women can only choose among those who have
proposed to them. We can see this from the following simple example.
Example 6. Suppose there are two men and two women. Man 1 prefers woman 1, man 2
prefers woman 2. The women have opposite preferences (woman 1 prefers man 2 to man
1 and woman 2 prefers man 1 to man 2), and everyone prefers matching with their second
48
choice to not matching. Under the Gale-Shapley algorithm, man 1 proposes to woman 1 and
man 2 proposes to woman 2. Since each man has only proposed to one woman, and the
women prefer to match, neither proposal is rejected. This means that we have a stable match
and the algorithm terminates. This is the optimal stable match for the men, but not for the
women: man 1 matching with woman 2 and man 2 with woman 1 is also stable, and both
women prefer it.
This also demonstrates that it is possible for the women to benefit from misreporting their
preferences. If woman 1 reported that she preferred not matching to matching with man 1,
then man 1 would be rejected and unmatched after round 1 and propose to woman 2. Woman
2 would then reject man 2 for man 1, and man 2 would propose to woman 1, who accepts and
the algorithm terminates. Note that by misreporting her preferences woman 1 gets a match
she prefers to the case where she reports truthfully.
While women may have an incentive to misreport their preferences, and there may exist
a different stable match that would make all women better off, there is no scope for men
to benefit from misreporting. Moreover, there is no other stable match that is preferred
by anyone on the men’s side. We say that a stable match is the optimal stable match
for the proposers if no proposer receives a higher utility in any other stable match. The
Gale-Shapley algorithm terminates at the proposer optimal stable match, since each man
proposes in decreasing order of preference, and once any stable match is reached the algorithm
terminates.
Theorem 12. The Gale-Shapley algorithm terminates in finite time and, if both sides have
reported their preferences truthfully, always produces a stable match. Moreover it is strategyproof for the proposers and generates the optimal stable match for the proposers.
The Gale-Shapley algorithm is commonly used in practice because it has many properties
that are considered desirable: namely that it always produces a stable match and is strategy
proof for the proposers. It has the drawback that it is not immune to manipulation on the
woman side, and that the match is only optimal on the proposer side. How the market
designer trades off different objectives, and how concerned we’d be about this, depends on
the specifics of the market considered. We consider some of these applications in the next
section.
18
Applications of Market Design
We now consider some applications of market design, beginning with school choice. See
Abdulkadiroğlu and Sönmez (2003), available on Chalk, for a discussion of some of these
issues related to school choice. In school choice there will be many students matched to each
school. While schools do not have utility functions, they have priority rankings over students.
These priority rankings typically take the form of giving higher priorities to students who
have a sibling currently attending the school, who lives in the walk zone, and to lower
income students (e.g. those who qualify for free or reduced lunch). As there are usually
many students in each priority ranking group, the students are also assigned a random
lottery number to generate a strict priority ranking over students.
49
How concerned we are about the possibility of misreporting on the “woman” side depends
on the application. If we are considering a setting in which students propose to schools,
which have publicly stated priority protocols, and the matching algorithm administered by
the school district which sets the priority, then we might not be too concerned about the
priority rankings being mis-reported.
It is however very important that the mechanism used be strategy proof on the “man”
or student side. First, when considering such a large market (such as the market for public
schools in New York city) it is very difficult to anticipate the beliefs individuals will have
about which schools others will apply to, and so determine the equilibrium of the game.
Second, if we want to know the preferences of individuals—either because we want to see
what characteristics of schools are valued by parents, or because we want to evaluate how well
the program is doing in assigning students to schools they want to attend—it is important
for individuals to have an incentive to report truthfully.22 Finally, when school districts have
used mechanisms that are not strategy proof (such as the so-called Boston mechanism that
gave higher priority to those who ranked a certain school as their first choice) websites and
parent groups have sprung up collecting data about admissions rates and trying to figure
out the optimal way to rank schools.23 Many school districts insist on a strategy-proof
mechanism since they are afraid that the children of less involved or strategic parents will
be placed at a disadvantage in terms of getting their students into their preferred school.
In school choice we generally evaluate the effectiveness of the match based on how well it
does for students, not on whether schools admit students with high priority rankings. So we
are unlikely to be troubled that the student optimal rather than school optimal mechanism
is used. We would also like the mechanism to ensure pareto optimality for the students, but
unfortunately Gale-Shapley does not guarantee that. In particular, pareto optimality may
conflict with stability, and the Gale-Shapley algorithm selects the stable allocation in this
case.
Example 7. Suppose there are three Students and three Schools. The preferences of Students
1, 2, and 3 are
b2 a1 b1 a1 b3
b1 a2 b2 a2 b3
b1 a3 b2 a3 b3
The Priority Rankings at schools 1, 2, and 3 are
a1 b1 a3 b1 a2
a2 b2 a1 b2 a3
a2 b3 a1 b3 a3
22
See Abdulkadiroğlu et al. (2015) for an empirical study of the welfare gains from improving the design
of markets in NYC. They find very large welfare gains from moving from a Gale Shapley style mechanism.
23
Under the Boston mechanism, the first slots in a school are allocated to those with a high priority who
ranked the school first. By advantaging those who rank an alternative first students have an incentive to
conserve their first place ranking by not using it on a school they know they are unlikely to get accepted by.
50
Under the Gale-Shapley algorithm, Student 1 proposes to School 2, and Student 2 and 3 each
propose to School 1, who accepts Student 3. Student 2 is then unmatched and so applies to
his second choice, School 2, who then rejects Student 1 and accepts Student 2. Student 1 then
proposes to School 1 who then rejects Student 3 for School 1. Finally Student 3 matches with
School 3, and the algorithm terminates. This leaves matches (a1 , b1 ), (a2 , b2 ) and (a3 , b3 ).
Notice however that, from the students’ perspective (a1 , b2 ), (a2 , b1 ) and (a3 , b3 ) is a pareto
improvement. However it is not stable: Student 3 prefers School 1 to the school it’s matched
with and has a higher priority for School 1 than student 2 does.
We are, of course, concerned about pareto optimality—there are, other algorithms that
are guaranteed to be pareto optimal—see, in particular, the top trading cycles mechanism
described in Abdulkadiroğlu and Sönmez (2003). But, as usual with these problems, there
are other tradeoffs. In particular, for a student assignment mechanism where only the
proposers have preferences, there is no strategy-proof algorithm that guarantees a stable,
pareto efficient match.24 While we have extensively discussed the value of pareto optimality,
stability is also considered important: it ensures no student is rejected by a school that they
preferred to the one they were assigned to, when they have a higher priority for acceptance
to the rejecting school than at least one of the accepted students. This is sometimes referred
to as “justified envy” and school boards may be constrained to satisfy it.
Another market in which matching is important, and monetary payments are forbidden, is
the market for kidneys. The demand for kidneys far exceeds the supply of kidney donations,
and an individual who needs a kidney will probably die if they don’t receive one.25 Donations
typically either come from cadavers or from individuals willing to done to a specific individual
they are close to. The standard economic response when demand exceeds supply is that the
price in the market should adjust to the market clearing price. While some economists (most
notably Gary Becker) have advocated for it to be legal to buy and sell kidneys there is federal
law that forbids the sale of kidneys or other organs,26 and we must deal with this constraint.
The kidney exchange market is an important area for market design because lives are at
stake, monetary transfers are prohibited, and matching process is complicated because the
donor and the recipient must be an appropriate match for each other or the recipient will
reject the transplant.
In some cases an individual would be willing to donate a kidney to an individual but may
not be a match to donate to that individual. For example, person A would like to donate to
B, and person C would like to donate to D, but A is only compatible with D and C is only
compatible with B. While monetary payments are illegal, pairwise donations can be done
legally. In a pairwise donation, A and C agree that they would each donate a kidney to the
person they are compatible with. Obviously, in practice, finding a suitable pair with which
to make such an exchange is difficult. But matching algorithms can and have been used to
help match sets of non-compatible donors and facilitate pairwise donation. It can also be
24
When both sides of the market have strict preferences the Gale-Shapley algorithm ensures the there is
no pareto improvement possible on both sides.
25
Approximately 7000 people on the waitlist for a kidney either day or are removed from the waitlist for
being to sick for surgery. It is likely a significant fraction of those people could have been saved if kidneys
were available earlier.
26
The market for kidneys is an example of what is called a “repugnant market” because people are
uncomfortable or disgusted by the idea of internal organs being sold for a price.
51
to identify cycles involving more than two simultaneous donations (e.g. A donates to D, C
donates to F, and E donates to B).
The New England Program for Kidney Exchange, created in 2004 in order to facilitate
matching kidney donations. Other similar programs to facilitate kidney exchange have since
been created. There are a lot of additional logistical complications that come up in this
setting of course. For example, since donors have the right to back out of the donation at
any time, the surgeries all have to take place simultaneously. A method for finding matches
is based on the top trading cycles algorithm. Each potential recipient has a ranking of the
possible kidneys they could receive and the have a preference ordering over them (closer
genetics to the donor make the match more likely to succeed). Simultaneously each kidney
is “pledged” to an individual. Essentially each potential recipient points at their first choice
kidney, and each kidney points at the person it is pledged to. If we find a cycle, we remove
those individuals and execute the exchange. Of course many issues come up, including
merging the live donations with the waitlist from cadavers. See the lecture notes by Tayfun
Sömnez posted on Chalk for additional details.
Until recently the most common and famous application of market design was to the
matching of medical residents to hospitals. Residents have preferences for which hospital to
work at (hospitals differ in terms of quality but also in areas of expertise and in terms of the
location, so residents have heterogenous preferences), and since residents have specialized
in different areas in their schooling, hospitals have heterogenous preferences over residents.
The difficulty in the market for medical residents was that each hospital would attempt to
hire many new graduates at once, over a short period of time. Since they didn’t know who
would accept, and if offers were left open for too long the hospital would lose their next
choice, hospitals tended to make exploding offers forcing the residents to make a decision
before they knew which other hospitals would make them an offer. As it was understood
by the interested parties—namely medical schools and hospitals—that the market was not
working well, medical schools and hospitals agreed to create a clearinghouse 1952. This
clearinghouse was set up to match all residents and hospitals at once through a variation of
the Gale-Shapley algorithm with residents proposing to hospitals. Today this is called the
National Resident Matching Program (NRMP). Over time this program has been redesigned
to accommodate other needs (a large increase in the number of couples who graduate at the
same time and want to be sent to the same location) and to improve the way the mechanism
works in this specific market. One difficulty relative to the school choice example however is
that hospitals have preferences over residents, and we know that the Gale-Shapley algorithm
is only strategy proof on one side, so hospitals could potentially benefit from mis-reporting
their preferences. However more recent research has shown that the probability of benefitting
from misreporting on the “woman” side of the market becomes limited as the market becomes
large, so this may not be that great a concern. See Roth (2008) for a discussion.
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