vote trading and pareto efficiency

Vote Trading and Pareto EfficiencyAuthor(s): Thomas SchwartzSource: Public Choice, Vol. 24 (Winter, 1975), pp. 101-110Published by: SpringerStable URL: http://www.jstor.org/stable/30022848 .

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9 Articles

VOTE TRADING AND

PARETO EFFICIENCY

Thomas Schwartz,

Tullock (1959) and Riker and Brams (1973) have shown that vote trading can lead to Pareto inefficient legislative outcomes - outcomes to which other possible outcomes are unanimously preferred. They offer examples of legislative decision

making1, highly simplified but purportedly realistic in essential respects, in which vote trading makes every legislator worse off than he would have been in the absence of vote trading.2

I will show that vote trading also can have the opposite effect, making every legislator better off than he would have been in the absence of vote trading. Besides

offering an example to establish this possibility, I will prove a theorem that

generalizes the example: In every legislature meeting certain apparently realistic conditions, the outcome that would prevail in the absence of vote trading has got to be Pareto inefficient; there must exist at least one potential vote trade that would make every legislator better off than he would be in the absence of vote trading.

I will argue that the anomalies exhibited by my example and those of Tullock and Riker and Brams are not really attributable either to vote trading or to its absence, but to another legislative practice - roughly, that of legislating the transfer of assets from parts of a population to other parts.3

*Associate Professor of Philosophy and Urban and Public Affairs, and Chairman, Philosophy Program, Carnegie-Mellon University.

1In Tullock's admittedly simplified example, the 'legislators' are the citizens themselves. But that is supposed to be inessential.

2Tullock's argument is criticized by Downs (1961) and defended against Downs' criticisms by Tullock (1961). Riker and Brams's argument is criticized by Tullock (19 74) and Bernholz (1974) and defended against these criticisms by Riker and Brams (1974).

3This supports the basic point of Tullock (1959), which is that majority rule can easily lead to Pareto inefficient spedning in the public sector, against Downs (1961), who claims that



102 PUBLIC CHOICE

I. An Example

Here is an example of legislative decision making, using simple majority rule, in which vote trading makes every legislator better off than he would have been were there no vote trading:

Three legislators, Messrs. 1, 2 and 3, must vote on three bills, a, b and c. There are eight possible outcomes:

abc - a, b and c all pass.

abc - a and b pass, and c is defeated.

abc - a and c pass, and b is defeated;

abc - a passes, and b and c are defeated.

abc - a is defeated, and b and c pass.

abc - a and c are defeated, and b passes.

abc - a and b are defeated, and c passes.

abc - a, b and c are all defeated.

Messrs. 1, 2 and 3 assign dollar values to a, b and c, as follows:

Mr. 1 Mr. 2 Mr.3

a 4,000,000 3,000,000 - 9,00,000 b - 9,000,000 4,000,000 3,000,000 c 3,000,000 - 9,000,000 4,000,000

In effect, each bill prescribes a transfer of priceable assets from one legislator (or his

constituents) to the others, and perhaps a transformation of these assets from money into other forms. In each case, the cost to the non-beneficiary, measured in dollars, is greater than the total benefit to the beneficiaries, due to transfer costs.

Each legislator evaluates the eight possible outcomes according to their total dollar values for him, preferring one outcome to another whenever the bills passed under the first outcome have a higher total dollar value for him than those passed vote trading on certain issues (and therewith seriatum voting on those issues) is essential to Tullock's example but not prevalent in the real world. Tullock's argument is directed at the use of majority rule to make certain types of public finance decision, not at vote trading per se.

My conclusion weakens that of Riker and Brams (1973), who claim that their example exhibits an infirmity in vote trading - rather than in majority voting per se, in the public provision of certain types of goods, or whatever.

Whether Pareto inefficient legislative outcomes necessarity exhibit an infirmity in anything - whether they are necessarily objectionable, or predictably rejectable under reasonable constitutional arrangements - is called in question by Schwartz (1975).



VOTE TRADING 103

under the second. This means Messrs. 1, 2 and 3 rank the eight outcomes in order of preference as follows:

Mr. 1 Mr. 2 Mr. 3

abc abc abc

abc abc abc

abc abc abc

abc abc abc

abc abc abc

abc abc abc

abc abc abc

abc abc abc

In the absence of vote trading, Messrs. 1 and 2 would vote for a, Messrs. 2 and 3 for b, and Messrs. 3 and 1 for c, insuring passage of all three bills. So abc is the outcome that would prevail were there no vote trading. But abc is Pareto inefficient. Every legislator prefers abc to abc.

As it happens, Messrs. 1, 2 and 3 trade votes, each agreeing to vote against the two bills he favors in return for a like agreement by the other two legislators. As a result, each legislator votes against all three bills, and abc - which each prefers to abc - is the outcome that actually prevails. Vote trading has made every legislator better off than he would have been had no vote trading taken place.

II. The Legislative Setting. The Consequence to be Derived

Imagine a legislature in which a great number of proposed bills have sufficient

support to pass in the absence of vote trading. Of these, it seems realistic to assume there are, say, k bills,

al,...., ak, fulfilling the three conditions listed in the next

section, whence I will deduce the following consequence: Whether or not there is any vote trading involving bills other than

al,...,ak, there must exist at least one potential vote trade involving al, . . ,ak that would make every legislator better off than he would be in the absence of any vote trading involving al,... , ak.

Before listing the three conditions, I will introduce some useful notation and restate this consequence in a more formal way.

The final outcome of legislative activity may be identified with the set of bills that actually pass. So a possible outcome is any set of proposed bills (including the empty set, which represents defeat of every bill).

Let there be n legislators, Messrs. 1, 2, ..., n, with strict preference relations

P1 '''. Pn among possible outcomes.



104 PUBLIC CHOICE

Let 7r be the set of bills, other than al,.. . ,ak, that actually pass.

Then the actual outcome is one of the 2k sets consisting each of r Ua for some a C ( al, . . ., ak ) . I will call them the feasible outcomes. Of these, the actual outcome is I U ( a ,... , ak } if al,..., ak all pass, 7ifal,...., ak are

all defeated, r U (al,a3 } if a1 and a3 pass and a2, a4,....,

ak are defeated, and so on.

Since al, ... , ak have sufficient support to pass in the absence of vote trading, I7r U I a,..., ak is the outcome that would prevail if there were no vote trading involving al, . . . , ak. What the consequence states, then, is that another of the 2 feasible outcomes, obtainable by vote training involving

a1, ..., ak, is unanimously preferred to r U { a1,

... ,akj There is no need to assume that the legislature uses simple majority rule. We

need only assume that the voting rule used is not so perverse as to permit passage of a bill that every legislator votes against. Given this assumption, if all legislators who favored passage of any of the k bills traded votes with one another, each agreeing to vote against the bill(s) whose passage he favored, then all k bills would be defeated, making 7r the actual outcome. So 7r is obtainable by a vote trade involving al, ..., ak

Hence, to show that some outcome obtainable by vote trading involving al

...., ak is unanimously preferred to iT U { a, ..., ak } , it sufficies to show

that 7 is unanimously preferred to r U { al, ... , ak , i.e.,

" Pi TT U al,...,ak , i = 1, 2, ..., n.

This is exactly what I will deduce from the three conditions below.

III. The Three Conditions

The first condition is that each legislator can assign a dollar -value to

al, ..., ak and 7r in such a way that he evaluates the 2k feasible outcomes

according to their total dollar values for him. Where vi (x) is the dollar value Mr. i assigns to x, for i=

1,2,....,n and x e { al,..., ak,Tr ), this condition can be

stated formally as follows:

CONDITION I. If c ( al,.".,ak)'

B al,...,ak, 1 < i < 1 and

S v.(x) > v. (x), then

Xe&TTI PU U 1

x1srr) UB

'1 U ci aP.7 U B. 1



VOTE TRADING 105

If xe Ial... , ak , let B ( ) be the set of legislators who would benefit from x (who assign positive v-values to x) and C (~) the set of those who find x costly (who assign negative v-values to x), Le.,

DEFINITION. For xe(al,...,ak), define:

B(x) = (ill < i < n and vi(x) > 0) , 3.

C(x) = (ill < i < n and v.(x) < 0)

The second condition is that every legislator would benefit from as many of the k bills as every other legislator, while finding the remaining bills costly. In other words, there is a number k*, between 1 and k (inclusive), such that every legislator assigns a positive v-value to k* of the k bills and a negative v-value to k-k* of them:

CONDITION II. Let A = al,

....,

an . Then there is a k* = 1,2, .. . ,k such that, for all i = 1,2, . . . , n and for some k*-fold subset A* of A,

ier B(x) n C(x)

x eA* xeA-A*

To formulate the final condition, we need another definition: DEFINITION: For i = 1,2, ... , n, j = 1,2, ... , k, define:

max max v (a ), if ieB(a.); p=l meB(a) m p

p

M.i(aj ) =

k max max v (a ), otherwise. p=l meC(ap)

m p p

Suppose 1 <i <n and 1 <j <k. Then if v.(a.)is positive, M(a.) is the.maximum 11J 1 j net benefit to anyone of any of the k bills (the maximum v-value). And if vi(a.) is

negative, Mi(a.) is the inverse minimum net cost to anyone of any of the k bills 'the

maximum negative v-value). The idea behind the third condition is that each of the k bills is designed

primarily to transfer assets from non-beneficiaries to beneficiaries, and perhaps to transform these assets from money into other forms - river and harbor



106 PUBLIC CHOICE

improvements, subsidized air fares for rural areas, educational television programs, statues of famous men, etc. The net benefits to beneficiaries would therefore be financed by the non-beneficiaries, who would receive no benefits (or virtually none). Because the cost of each bill includes, not just the value of the assets it would transfer and transform, but also the substantial costs of carrying out the transfer and transformation (the transfer costs, for short), the total net cost to non-beneficiaries far exceeds the total net benefit to beneficiaries (in dollars, not

'utility'). To be precise, the total net cost to non-beneficiaries not only is greater than

the total net benefit to beneficiaries, but would remain greater even if net benefits to beneficiaries were raised to the maximum received under any of the k bills (i.e., if positive v-values were raised to the corresponding M-values) and net costs to non-beneficiaries were reduced to the minimum incurred under any of the k bills (i.e., if negative v-values were raised to the corresponding M-values):

CONDITION III.

B(aMi

If there is little disparity among net benefits to beneficiaries of the k bills

(i.e., among positive v-values) or among net costs to non-beneficiaries (i.e., among negative v-values), then there is little difference between M-values and v-values, and so Condition III is not very restrictive: it says little more than that the total net cost of each bill to non-beneficiaries exceeds the total net benefit to beneficiaries.

Only if there is a significant disparity among net benefits to beneficiaries or among net costs to non-beneficiaries does Condition III constrain total net costs to

non-beneficiaries to be far in excess of total net benefits to beneficiaries. Even if a, ak simply prescribe direct transfers of money from

non-beneficiaries to beneficiaries, there are bound to be significant transfer costs,

causing total net costs to non-beneficiaries to exceed total net benefits to

beneficiaries by a significant amount. And, of course, transfer costs are likely to be

much greater if al, ak prescribe relatively indirect transfers of money - as in

the case of tax subsidies - or transformations of money into non-monetary (though

priceable) assets - as in the case of pork barrel legislation. There is another reason, suggested by Tullock (1959), why total net costs to

non-beneficiaries of each of the k bills are likely to be far in excess of total net

benefits to beneficiaries. Let x be any one of the k bills. In sponsoring and

supporting x in the absence of vote trading, the beneficiaries of x would

presumably be seeking far more costly benefits for themselves than they would seek if they had to pay the cost themselves. Therefore, the benefits provided by x are far

less valuable to the beneficiaries themselves than the dollar-cost of x. Since the

latter is borne entirely (or nearly so) by the non-beneficiaries, who receive little or

no benefit, the net benefit to beneficiaries, measured in dollars, is far less than the

net cost to non-beneficiaries.



VOTE TRADING 107

Conditions I - III are strong; they are not likely to be satisfied by k

randomly selected bills. My assumption, however, is not that these conditions are satisfied by every k bills, by k randomly selected bills or by k specific bills, but by some k bills or other - for some k or other.

IV. The Consequence Deduced

THEOREM. Assume Conditions I - III. Then

Tr P r U aiT,...,ak0, i=1,2,...,n.

PROOF. Let b be the maximum positive v-value and c the maximum negative v-value, i.e.,

k b = max max v (a.),

j=1 ieB (a.) k

c = max max vi.(a.) j=1 ieC(a.)

Then by definition of M, if 1 -<i -<n and 1 -<j <k, Mi(aj)is b or c according as

vi(aj) is positive or not. But by Condition II, there exists a k* = 1,2,,..., k such

that, for all i = 1,2,, ..., n, k* of vi(al), ... , vi(ak) are positive and the remaining k - k* negative, whence k* of Mi(al)' ... , Mi(ak) are b and the rest c, so that

k E Mi(aj) = bk* + c(k-k*), i=1,2,...,n.

j=1

Adding ieC( Mi(aj) to both sides of Condition III, we obtain:

E M.(a.) + T Mi(a.) < 0, j=,2,...,k. ieB(a.) ieC(a.)

But B (a.) and C (a.) are disjoint, j - 1,2,..., k, by definition of B and C; and ieB (aj) ) C(aj), i= 2, ..., n, j = 1,2 ..., k, by Condition II. Therefore,

D M (a.) + 0

M.i(a.) = M.(a.), j=1,2,...,k, ieB(a.) ieC(aj) i=l

whence

n

SM(a) < , j=1,2,...k i=1



108 PUBLIC CHOICE

by (2), so that

k n SM. (a .) < 0,

j= 1 i= 1

i.e., n k

D Mi(a.) 0,

i=1 j=1 3

and thus k SM. (a.) < 0 for some i.

j=1 1

Hence, k

bk* + c(k-k*) = M.(a.) < 0, i=1,2,...,n, j=1

by (1). But by definition of M,

v.(a.) < Mi.(a.),

i=1,2,...,n, j=1,2,...,k.

So k E v.(a.) < 0, i=,2,...,n,

j=l whence

k E v.(a.) + v.i() < v.i((), i=1,2,...,n,

j=1 Le.,

v.i () > r v.(x), i=1,2,...,n.

xe,(,al,..,a k) 1

Consequently, by Condition I, IrP ir U ai, ... .

ak i, i=1,2, ..., n, q.e.d.

V. To Trade or Not to Trade: That is not the Question

With a, ...., ak interpreted as bills, each enjoying sufficient support to pass in the absence of vote trading, the Theorem just proved generalizes the example given in Section 1:-Assuming Conditions I - III, some vote trade would make every legislator better off then he would be in the absence of vote trading.

But there is another way to interpret al, . . . , ak, one that reverses this

reading of the Theorem. Suppose we interpret each ai, not as a single bill, but as a

legislative package, whose component bills enjoy sufficient support to pass only as a result of a vote trade involving all the bills in the package. Then the Theorem



VOTE TRADING 109

generalizes the Tullock and Riker-Brams examples: Assuming Conditions I - III, some set of vote trades would make every legislature worse off than he would be in the absence of vote trading.

What all this shows is that the Pareto inefficient legislative outcomes discussed here are not really attributable either to vote trading or to its absence. So

long as there exist positions al, ..., ak fulfilling Conditions I - III, it makes little difference whether these positions are legislative packages that can be passed only by vote trading, or single bills that can be defeated only by vote trading. It matters little whether you buy a barrel-full of pork, or enough pieces of pork to fill a barrel.

One way to insure that there do not exist positions al,...., ak fulfilling Conditions I - III is to prohibit constitutionally the transfer of assets from parts of the population to other parts. Another is to impose a constitutional restriction on transfers of assets to insure that not everyone gets to be a beneficiary (contrary to Condition II), e.g., by allowing transfers only from rich people to poor - or only from poor to rich, for that matter.

Everyone wants a free lunch. It is not possible for everyone to have one. The

attempt by everyone to get one through legislation only insures that everyone over-spends on lunch.

RE FE RENCE S

Bernholz, Peter. Communication. American Political Science Review, 68 (1974). Downs, Anthony. "In Defense of Majority Voting." Journal of Political Economy,

69 (1961). Reprinted in Schwartz (1973). Riker, William H., and Steven J. Brams. "The Paradox of Vote Trading." American

Political Science Review, 66 (1973). . Communication. American Political Science Review, 68 (1974).

Schwartz, Thomas. Freedom and Authority: An Introduction to Social and Political Philosophy. Encino, California: Dickenson Publishing Company, Inc., 1973.

0 "Collective Choice, Separation of Issues and Vote Trading." Carnegie-Mellon University: Photo-copied, 1975. Presented at Conference on Foundations of Political Economy, University of Texas, Austin, February 1975, and at meetings of Public Choice Society, Chicago, April 1975.

Tullock, Gordon. "Problems of Majority Voting." Journal of Political Economy, 67

(1959). Reprinted in Schwartz (1973). " "Reply to a Traditionalist." Journal of Political Economy, 69

(1961). _ Communication. American Political Science Review, 68 (1974).



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