Control Over the Agenda
• When cycling occurs, an additional rule can be imposed that losers in
sequential pair-wise contests are eliminated
— A voter who determines the order in which voting takes place — or sets
the agenda — then also determines the outcome of voting
— With the rankings of table 6.1b, the agenda may be set so that the
first pair-wise vote is on the choice between X and Y
— Then, {XPY } and Y is eliminated
— The next round of voting sets Z against X, with outcome {ZPX}
— Z is the winner, but the outcome was dependent on the order of voting
• If the first vote had been between Y and Z, then Y would have eliminated
Z
— In the next vote, X would win over Y
— Alternatively, if the first vote had been between Z and X, Y would
win
— We conclude:
Unless preferences are single-peaked, majority voting cannot be relied
on to result in a stable determinate collective decision — unless losers
are eliminated, in which case the outcome is determined by control over
the agenda
The Condorcet Winner and Cost-Benefit Analysis
• In table 6.1a, where preferences are single-peaked, the Condorcet winner
was project Y
— Table 6.2a shows an example of possible costs and benefits for the
three voters from project Y (we now disregard the connection between
table 6.1a and figure 6.2)
Table 6.2a. Benefits and Costs for Condorcet Winner Proposal Y
Voter 1
Voter 2
Voter 3
Total
Costs to the voter
of the proposal
100
100
100
Total cost = 300
Voter benefit from
Voter net benefit
the proposal
from the proposal
30
−70
130
30
110
10
Total benefit = 270 Total net benefit = − 30
• The total cost of the proposal is 300, financed by equal cost shares of 100
for each voter
— Each voter’s benefit is private information, known only to the voter
— Benefits for the voters from proposal Y are 30, 130 and 110
— A cost-benefit calculation of W = B −C reveals a negative value −30
for proposal Y
— Nonetheless, proposal Y was chosen as the Condorcet winner under
majority voting
• Table 6.2b shows possible voter benefits and costs for proposal X, which
lost to proposal Y by majority voting in table 6.1a
Table 6.2b. Benefits and Costs for Condorcet Loser Proposal X
Voter 1
Voter 2
Voter 3
Total
Costs to the voter
of the proposal
100
100
100
Total cost = 300
Voter benefit from
Voter net benefit
the proposal
from the proposal
500
400
80
−20
70
−30
Total benefit = 650 Total net benefit = 350
• The total cost of X is again 300, split equally among voters
— Proposal X provides a positive net benefit of 400 to voter 1
— It provides net losses to the two other voters, of 20 and 30 respectively
— Proposal X has a positive net benefit of 350 (650 − 300) and is
therefore efficient in satisfying the cost-benefit rule W = B − C > 0
• We define a Condorcet loser as a proposal that is defeated in pair-wise
majority voting by all other proposals
— Proposal X is a Condorcet loser in table 6.1a
• We therefore conclude:
An inefficient proposal can be a Condorcet winner
An efficient proposal can be a Condorcet loser
Voting Externalities
• There are externalities associated with voting
— Proposal X with positive net benefit to society (350) was defeated by
proposal Y , with negative net benefit (-30)
— In choosing Y over X, voters 2 and 3 imposed negative externalities on
voter 1 (who would have had a net benefit of 400 from X but instead
lost 70 from Y )
— Thus, majority voting resulted in the choice of an inefficient proposal
but externalities were also created because of the distributional consequences of voting
• Externalities arise through voting because voters base their voting decisions
on their personal rankings of the proposals, only
— The values of gains and losses — for themselves or for others — are not
taken into consideration
— In voting in favor of Y against X, voters 2 and 3 did not take into
account (or internalize) voter 1’s loss of 70 from X and the gain of
400 from Y
— However, it is voter’s values of gains and losses that determined whether
or not a proposal satisfies the efficiency condition W = B − C > 0
A proposal that is inefficient in not satisfying the cost-benefit rule can
be the Condorcet winner because voting decisions are insensitive to
how much losers lose and how much gainers gain
Pareto Improvement and Compensatory Payments
• Compensatory payments prevent choice of inefficient proposals
— If we compute the differential payments in going from proposal Y to
X, we see that voter 1 can readily compensate voters 2 and 3 for their
losses
— Voter 1 gains 470 in going from Y to X (400 − (−70))
— Voter 2 loses 50 (−20 − 30) and voter 3 loses 40 (−30 − 10) in going
from Y to X
— Thus, voter 1 can compensate the other two voters for agreeing to vote
for X rather than for Y
— When compensation (and over-compensation) is possible, Pareto improvement can be ensured by switching to public finance for proposal
X rather than the Condorcet winner Y
Markets in Votes
• In making compensatory payments, voter 1 would effectively be buying the
votes of voters 2 and 3
— However, markets in votes are illegal, for two reasons
1. One reason is the principle that personal income or wealth should
not determine how many votes a person controls
The principle of democratic equality of “one person, one vote” is
inconsistent with a market for votes which would allow people to
buy votes and vote more than once
2. The second reason is based on the value of aggregated votes
While the vote of one person from a large population has virtually
no effect on the outcome of an election, if increasingly many votes
are bought, the probability increases that they can be used decisively
There is value added through aggregation and the person who buys
sufficient votes in a market can undermine democracy
• There are thus justifiable reasons for markets in votes to be illegal
— Nonetheless,
Absence of markets for votes prevents the payments that would prevent
inefficient proposals being chosen by majority voting
1.3
Logrolling
• Although markets in votes are illegal, vote-trading among elected political
representatives can legally take place
— Vote-trading takes the form of an agreement that “If you vote for my
preferred proposal, I will vote for yours.”
— Thus, coalitions form to support designated proposals
— The formation of these voting coalitions is known as “logrolling” because, although one person can cut down a tree, it takes the cooperation of two to roll the log
Does logrolling result in efficient public spending?
Logrolling with Efficient Proposals
• In table 6.3, three political representatives (whom we call “politicians”)
face a vote on three proposals, D, E and F
— Taxes will be collected and each proposal will be publicly financed
through the government’s budget if there is majority support for it
— The proposals are therefore not alternatives as we previously considered:
each proposal can be financed if it receives majority support
Table 6.3. Logrolling with Socially Justified Proposals
Net benefit to
Net benefit to
Net benefit to
Total net benefit
Politician 1
Politician 2
Politician 3
Proposal D
Proposal E
Proposal F
110
−20
−30
−20
100
−30
−30
−30
100
60
50
40
• The final column in table 6.3 shows the total net benefit from a proposal
— Since all proposals provide positive net benefit, spending on each is
justified on cost-benefit grounds
• The proposals differ in distributional consequences:
— Each proposal provides positive net benefit to only one politician’s
constituencies
— In a vote on whether to finance each proposal separately, politician
1 supports only proposal 1, politician 2 only supports proposal 2 and
politician 3 only votes in favor of 3
— Though public spending on each proposal is justified on cost-benefit
grounds, when put to a vote, each proposal is rejected by majority
voting
• Two politicians can form a coalition to make a majority in favor of the
proposals that each want funded through public spending
— The three possible coalitions are (1, 2), (2, 3) and (1, 3) respectively
— These would provide majority support for pairs of proposals (D, E),
(E, F ) and (D, F ), respectively
— We denote a coalition between politician’s i and j as V (i, j) and the
net benefit to politician i’s constituency from the pair of coalitionsupported proposals as Bi (i = 1, 2, 3)
• If politician’s 1 and 2 combine to form a coalition to support proposals D
and E, we have:
V (1, 2) ⇒ projects (D, E) ⇒ B1 = 90, B2 = 80, B3 = −60,
X
Bi = 110
— If politicians 1 and 3 form a coalition,
V (1, 3) ⇒ projects (D, F ) ⇒ B1 = 80, B2 = −50, B3 = 70,
X
Bi = 100
X
Bi = 90
— If politicians 2 and 3 form a coalition,
V (2, 3) ⇒ projects (E, F ) ⇒ B1 = −50, B2 = 70, B3 = 70,
• Coalition V (1, 2) provides the greatest individual net benefit to politicians
1 and 2
— Politician 1 gains 90 from a coalition with politician 2 and 80 from a
coalition with politician 3
— Therefore, politician 2 is politician 1’s preferred coalition partner
— Likewise, the gain for politician 2 to associate with politician 3 is only
70, whereas he gains 80 by associating with politician 1, whom he thus
prefers as a coalition partner
• Politicians 1 and 2 thus combine their votes in favor of proposals D and
E and against proposal F
— The constituency of politician 1 gains 90 and that of politician 2 gains
80
— A loss of 60 is imposed on the constituents of politician 3
• Proposal F , which was not financed, provides positive net benefit and
should have been financed according to cost-benefit analysis
— Logrolling has resulted in the acceptance of two out of three efficient
proposals
• The two most valuable proposals are accepted for public finance in this
example
— The outcome is unfortunate for the constituents of politician 3, on
whom negative externalities have been imposed
— Constituents of politician 3 will have to pay taxes to finance two proposals from which they lose
• Therefore, in the example of table 6.3, where all projects are efficient in
satisfying the cost-benefit rule W = B − C > 0,
In the absence of logrolling, no proposal may be publicly financed; logrolling
results in public finance for some efficient projects and can leave other
efficient projects without public finance; and logrolling has arbitrary distributional effects through the externalities of collective decisions
Logrolling with Inefficient Proposals
• Table 6.4 shows three public spending proposals that have negative net
benefit and so are inefficient
— Each politician’s constituency again benefits from only one proposal
Table 6.4. Logrolling with Socially Unjustified Proposals
Proposal D
Proposal E
Proposal F
Net benefit to
Politician 1
Net benefit to
Politician 2
Net benefit to
Politician 3
Total net benefit
50
−40
−40
−40
50
−30
−30
−30
20
−20
−20
−50
• The three possible coalitions are:
V (1, 2) ⇒ projects (D, E) ⇒ B1 = 10, B2 = 10, B3 = −60,
X
V (1, 3) ⇒ projects (D, F ) ⇒ B1 = 10, B2 = −70, B3 = −10,
V (2, 3) ⇒ projects (E, F ) ⇒ B1 = −80, B2 = 20, B3 = −10,
Bi = −40
X
X
Bi = −70
Bi = −70
— Because only coalition V (1, 2) provides positive net benefit to its members, it is the only one that will be formed
— There will be majority support for proposals D and E, with proposal
F not financed
• This case illustrates that
Logrolling can result in majority support for proposals for which public
finance is not justified by cost-benefit analysis
Fortuitous Efficiency
• Logrolling could be fortuitously efficient:
— It is simple to formulate a case where a logrolling coalition of two voters
supports two efficient projects and a third, not financed, is inefficient
— The general conclusion is that:
Logrolling does not ensure efficient public spending — and indeed only
fortuitously (by luck) could public spending determined through logrolling
be efficient
Logrolling with Money Payments
• When logrolling takes place, votes are exchanged for votes and are not
traded for money
— We expect monetary payments among politicians — and among their
constituencies — to be illegal
— Nonetheless, there might be ways in which monetary payments could
be made
— What happens when monetary payments (also known as side payments)
can take place to entice politicians — or political parties — to leave an
existing coalition and join a new one?
• To consider this question, we return to the efficient public spending proposals in table 6.3
— We saw that coalition V (1, 2) forms when no monetary transfers are
possible
— When they are, politician 3 who does not take part in V (1, 2), can
entice either politician 1 or 2 to join him in a new coalition
Table 6.3a. The Instability of Logrolling for the Projects in Table 6.3
when Money Transfers Take Place
Coalition V (1, 3) , proposals
Coalition V (1, 2)
Coalition V (1, 3)
(D, F ) , with payment
Proposals (D, E)
Proposals (D, F )
from politician 3 to 1
Politician 1
Politician 2
Politician 3
Total net benefit
90
80
−60
110
80
−50
70
100
91
−50
59
100
• Table 6.3a shows the outcome of enticement of politician 1
— Politician 3 makes a monetary transfer payment of 11 to politician 1
— The transfer provides politician 1 with a benefit of 91 from the new
coalition-with-side-payments V (1, 3)
— Politician 1 is therefore better off than she was in the original coalition
V (1, 2) , where her benefit was 90
— After the payment to politician 1 to entice her to join V (1, 3), politician 3 has a benefit of 59, which is preferable to the loss of 60 from
V (1, 2)
• Politician 3 could have chosen to make the offer to politician 2 instead
— It would have cost him the same 11 units to raise the payoff of politician
2 from the 80 he was getting in V (1, 2) to 81, since politician 2’s payoff
in V (2, 3) is 70
• Just as the original logrolling coalition V (1, 2) is unstable, the new V (1, 3)
is unstable as well
— The now excluded politician 2, who loses 50 because of V (1, 3), can
approach either politician 1 or 3 with an offer of payment in exchange
for leaving V (1, 3)
— The least costly proposal is toward politician 1 since he is the politician
who experiences the greatest increase in personal benefit in moving
away from V (1, 3)
— In fact, politician 1’s personal benefit (from the proposals alone) would
increase to 90 if she switches to V (1, 2), whereas that of politician 3
would only remain at 70 should he join V (2, 3)
— Further, politician 2 also experiences a higher increase in payoff in
moving to V (1, 2) compared to V (2, 3)
— We conclude that
If money transfers can take place, any coalition formed through logrolling
is unstable
All-inclusive Logrolling Coalitions
• Monetary payments facilitate all-inclusive logrolling coalitions that provide
public finance for all efficient proposals
— In table 6.3, politician 3 could pay the other two in exchange for support
for proposal F
— All efficient proposals would then be financed
• When money transfers are not possible, the only gains possible through
logrolling and coalition formation are the gains directly obtained from the
proposals
— The benefits in table 6.3 to politicians 1 and 2 from forming a coalition without monetary payments were thus based on the benefits to
politician 1 from project D and to the benefits from E to politician 2
• Money transfers allow incentives to be introduced to finance efficient
project F as well
— Politician 3 gains 100 from project F whereas politicians 1 and 2 together lose 60, leaving a gain of 40 to be shared
— Money transfers compensate politicians 1 and 2 for their losses (or
compensate their constituencies)
• Quite generally, because of benefits to be shared, consensus can be found
in favor of public finance of any efficient proposal justified on the grounds
that W = B − C > 0
— This is similar to the Coase theorem which predicts that only efficient
outcomes will prevail in the presence of externalities if there are no
transactions costs and people can bargain over outcomes
• In table 6.4, no project satisfies the rule W > 0 but logrolling between
politicians 1 and 2 allows two inefficient projects to be financed at the
expense of the constituents of politician 3
— The latter lose 60 from the formation of V (1, 2) whereas the total
gain to the constituents of politicians 1 and 2 is only 40
— The constituency of politician 3 can gain by paying politicians 1 and 2
in order not to form V (1, 2)
— The possibility of making the payments results in benefits for politician
3’s constituency
— Nonetheless, politician 3’s constituency is paying to avoid losses from
majority-supported socially unwarranted proposals for public finance
Why is There Stability in Coalitions?
• The instability predicted by the inclusion of monetary payments into logrolling
is rarely observed
— This is largely associated with the fact that politicians are playing a
repeated game
— As such, there are incentives of reputation to maintain coalitions that
have been formed
— For example, defecting today from a majority coalition might have
future costs in terms of loss of reputation
— Politicians with low discount rates (high discount factors) are thus likely
not to defect
• Stability might also indicate lack of opportunity to gain from switching
coalitions
— There are high transactions costs of payments among different constituencies
— In addition to transactions costs and free-rider problems, monetary
payments are generally illegal and would be viewed as “bribes”
The Political Benefit of Logrolling
• The political benefit of logrolling without money transfers is precisely that
no money changes hands:
— There can be no accusation of bribery
— The resulting pure vote trading can be implicit through reciprocal support when the preferred projects of different politicians are put to a
vote
1.4
Checks and Balances
• The term tyranny of the majority describes the ability of the majority under
majority voting rule to dictate outcomes to the minority
— The tyranny of the majority can be countered through checks and
balances
Checks and balances take the form of legal and constitutional rules
that protect a minority of taxpayers from outcomes dictated by votes
of the majority
— Checks and balances are provided by an independent judiciary, duplicate
legislative bodies (such as a house of representatives and a senate), and
divisions of authority between executive and legislative branches of the
government
— Table 6.5 returns us to the (public good) prisoner’s dilemma and shows
how checks and balances ensure efficiency and social justice
Table 6.5. Voting and the Public Good Prisoner’s Dilemma
Group 2 pays Group 2 does not pay
Group 1 pays
3, 3
1, 4
Group 1 does not pay
4, 1
2, 2
• The outcome in table 6.5 is determined by majority voting by two groups
of voters
— At (2, 2), the group with a majority or both groups vote that no one
pays taxes to finance the public good
— As a consequence, the public good is not provided
— At (3, 3), the group with a majority or both groups vote that everyone
pays taxes to provide the public good
— There is further Pareto improvement relative to (2, 2)
• In the asymmetric outcomes (1, 4) and (4, 1), the majority votes that only
members of the minority group will pay taxes to finance the public good
— Majority voting in the latter asymmetric cases exhibits the tyranny of
the majority
• A rule requiring nondiscrimination though public finance rules out both
asymmetric outcomes and restricts the choice to a vote on the symmetric
cases (2, 2) or (3, 3)
— When confronted with a choice between (2, 2) and (3, 3), both groups
vote in favor of (3, 3)
— At (3, 3), there is consensus favoring everyone paying; at (2, 2), there
is consensus where nobody pays
— The outcome at (3, 3) is the efficient consensus
• Therefore,
A rule requiring a symmetric outcome of majority voting under the payoffs
of the prisoner’s dilemma is equivalent to a consensus voting rule
Checks and Balances in Fiscal Federal Systems
• Under fiscal federalism, taxation and public spending take place in hierarchical or federal systems of government
— The federal or highest level of government caters to the entire population
— State and regional governments cater to smaller groups; local governments cater to the smallest groups
• People may move to areas where the combination of taxes and provision
of public goods accords more with their preferences
— This locational choice responding to combinations of public goods and
taxes was first described by economist Charles Tiebout and is referred
to as “voting with the feet”
• Opportunities to choose public goods through the Tiebout locational choice
mechanism result in greater homogeneity of populations in jurisdictions at
lower levels of government
— At higher levels of government, there is correspondingly greater variety
of voter preferences and incomes
— Therefore, at higher levels of government there is also greater likelihood
of disagreement between majority and minority groups of voters and
thus greater need for checks and balances