Articles

THE BORDA GAME

Roy Gardner

Recently a number of authors have constructed axiomatic defenses of Borda's rule. 1 Such defenses assume that voters mark their ballots honestly, in accordance with their preferences. Elsewhere, it has been shown how Borda's rule can reward individual voters who misrepresent their preferences. 2 This result is in the same spirit as, but not a consequence of, the Gibbard-Satterthwaite 3 Theorem, since Borda's rule allows ties. That the assumption of honest voting may be unrealistic was known to Borda himself; indeed, it was Condorcet who brought it to his attention. 4

This paper investigates the strategic differences between Borda and Condorcet elections in a setting where there are many voters, and parties are allowed to form. In such a case misrepresentation of preferences by individuals is unlikely to be significant unless it is part of a party's overall strategy. In the language of game theory, then, elections are being modeled as cooperative games in normal form without side-payments. When we pass to the characteristic function form of these games, we find that the set of winning coalitions of a Borda game is a proper subset of that of a Condorcet game, and shrinks monotonely with an increase in the number of alternatives. Strategic Borda voting with many alternatives ultimately requires a two-thirds majority coalition. There exists a tension between honest and dishonest Borda voting, in that the outcome of honest Borda voting is not necessarily a voting equilibrium.

*Department of Economics, Iowa State University. This reseazch was supported in part by the National Science Foundation, I would like to thank Arnold Faden, Alan Gibbard, Mark Satterthwaite, Henry Wan and especially Kenneth Shepsle for their helpful comments.

1For example, see B. Fine and K. Fine, P. Gardenfors and H. P. Young.

2For results along these lines, see P. G'ardenfors and the author.

3For precise statement and proofs, see A. Gibbard and M. Satterthwaite.

4For the history behind Borda's rule, see D. Black and [. M. Blin.

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This tension grows with the number of alternatives. The evidence presented suggests that, however appealing when voters vote honestly, Borda elections will generally lead to quite different results when voters are aware of their strategic capabilities.

II. FORMAL PRELIMINARIES

We consider a finite set of voters T, where T = (1,2 . . . . . n } and n may be very large. 5 Voting takes place over a Fixed, finite set o f alternatives M, denoted by { % 2 . . . . . m }, with m at least two. Each voter t in T has an irreflexive, com- plete, and.transitive preference ordering o f M, denoted P(t). A voter is never in- different between two alternatives; 6 i f voter t prefers alternative i to alternative j, we write this as iP(t)j. Let ti be the set o f all such preference orderings on M. A distribution of opinion p tells how many voters have any given preference ordering P in II; formally, p(V) = # {t: eft) = V }.

An election maps a distribution o f opinion into a subset o f the set o f alter- natives, the choice set. Formally, if f is an election; then f: (II, p) -+ 2 M, the power set o f M. We denote the choice set o f an election, given a distribution o f opinion p, b y f(p). The two elections o f immediate concern are those o f Condorcet (fc) and

Borda (fB)" Let Li.(p ) = # (t: ie(t)j } when the distribution o f opinion is p. An alter-

native i is aJCondorcet choice for the distribution of opinion p if and only if Lij!p ) Lji(P ) for every j different from i in M. In a word, a Condorcet choice has a strict

majority against every other alternative. It is easy to show that fc(P) is either a

singleton or the empty set. The Borda score of alternative i based on the distribution of opinion p is

given by "/

~. Lij (p) - Lji (p)

i < j < m (I)

j ~i

In Borda's original formulation, the highest alternative in an individual's ordering gets m votes, the next highest gets m - 1 votes, etc.; then votes for each alternative are summed. Let b(i) be the vote total for alternative i under Borda's counting scheme. Given m alternatives and n voters, formula (1) equals 2b(i) - (m + 1)n. This linear transformation does not affect the results of the election, and it has the

5It is easy to generalize all the results to the case where N is a non-atomic measure space of voters. In that conceptualization, individual misrepresentation has in finitesimat importance.

6Relaxing this by allowing for individual indifference between alternatives does not materially affect the results, although there are a number of ways of defining Borda election in this event.

7This formula is due to H. P. Young.

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virtue of showing how a Borda election can be conducted solely on the basis of pairwise comparisons. In the past, this informational symmetry between Borda and Condorcet elections has not always been stressed. An alternative i is a Borda choice if and only if no other alternative gets a higher Borda score. It is easy to show that fB(P) can be any subset o f M except the empty set.

Elections are defined in terms of the distribution of opinion p as it actually exists. I f each voter votes honestly, p is indeed the input to the electoral process; however, i f some individual or group of individuals happens not to vote honestly, the input to the electoral process will not in general equal p. This observation is basic to all that follows.

IlL STRATEGIC VOTING AND CRITICAL SIZE

If voters are free to do with their votes as they wish, then their freedom surely includes that o f making "mistakes," either intended or unintended, when they vote. The intentional case is more interesting: the motivation here is that, by voting dishonestly, a voter may swing the outcome of the election to one he prefers. This aspect o f elections can be captured by modeling them as non-cooperative games in normal form. It is then natural to ask whether honest voting constitutes a Nash equilibrium point. The Gibbard-Satterthwaite theorem shows how dim are the prospects along these lines for honest voting: elections whose outcomes are neither dictatorial nor imposed can be swung by individuals voting dishonestly. Now let blocs or coalitions of voters be free to form. This at once raises the possibility of dishonest voting, well organized and on a large scale. Non-cooperative game theory no longer applies; the questions become ones for cooperative game theory. The game remains one in normal form without side-payments. The strategies open to an individual voter are just the preference orderings in II; the strategies open to a coalition, the Cartesian product o f the voting strategies open to its members.

It is usually more convenient to work with a cooperative game in characteristic function form. We shall make the transition to characteristic function form using Aumann's alpha-characteristic function, We suppose that the only reason for a bloc o f voters to form is to carry the election. The ability of a bloc of voters to carry the election depends, among other things, on how large it is and how it yotes. I f the bloc of voters is large enough and votes cleverly enough that it can guarantee the election of any alternative it sees fit, we shall call it a winning coalition. -s A coalition which is not winning is losing; a losing coalition cannot guarantee the election of a n y alternative it sees fit against all possible voting strategies by the rest o f the voters. 9 In this section we ask how big a coalition must be to be winning in Condorcet or Borda elections.

8Tlais term is due to L. S. ShapIey. Technically speaking, the Borda and Condorcet games in alpha-characteristic function form are simple games.

9A coalition which is losing in terms of the alpha-characteristic function may turn out to be important in terms of some other characteristic function. Passing from normal to characteristic function form inevitably involves some loss of information.

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Let S, a subset o f N, be a coalition consisting o£ s voters. We shall measure the size of S by the proport ion of all voters that S contains, namely-~. For an election f, the critical size, c(f), is the size a coalition must exceed in order to be winning. The following propositions specify the critical sizes of Condorcet and Borda Elections.

1 Proposition 1. c(fc) - 2 "

Proof. Suppose s > ~' Let every t in S adopt the strategy 1P(t) j, for all j in

M different from 1. Then S guarantees the Condorcet choice of 1. Since the choice of I is arbitrary, S can guarantee the election o f any alternative it sees fit, and so S is winning.

An immediate consequence of Proposition 1 is that the set of winning coalitions o f a Condorcet election are the {S: s 1 f i ' > 2 )" Note that the critical size

for a Condorcet election dods not depend on the number of alternatives, m.

Proposition 2. c(fB) = ~ for n large enough.

Proof. Suppose coalition S wants to guarantee ~he Borda election of, say, alternative 1. S has no control over how the counter-coalition N-S votes. Thus, S can only be sure o f electing alternative 1 if l ' s Borda score is greater than j ' s for all other j and ali possible votes of the counter coalition. Thus, coalition S does best to spread its votes between alternative 1 and all the other alternatives as far as possible. There are two cases to consider, depending on whether s is even or odd.

Case 1. S has.an even number o f voters. Let one-half o f the voters in S adopt the strategy.

1 P 2 P . . . P m

and the other half adopt the strategy

1 P m P . . . P 2 .

(2)

(3)

From (1), S gives alternative 1 the Borda score ( m - 1)s and each other alternative the Borda score -s.

Now the best the counter coalition can do against this strategy is to have all its members rank alternative 1 last and some other alternative j f'rrst, giving j the Borda number (m - 1) (n - s) and 1 the Borda number - (m - 1) (n - s).

Alternative 1 bests alternative j if and only if

( m - 1)s- ( m - 1) (n- s) > - s + (m- 1) (n- s) (4)

which implies

s > 2 m Lme_kk. (5) ~- 3m - 2

Since the choice of alternative 1 is arbitrary, S can guarantee the election of any

alternative it sees fit when its size exceeds 2 (m - 1) (3m- 2)"

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Case 2. S has an odd number o f voters. Suppose further that S cannot avail itself o f a strategy of balancing votes among alternatives 2 through m as an even- numbered coalition can. 10

s + 1 members o f S adopt the strategy (2) and let the remaining members Let

o f S adopt the strategy (3). The counter coalition retains its strategy in case 1. By a parallel argument to case 1, the odd-numbered coalition wins so long as

1 s_>(2 +n~ (m- 1) - n (6)

n 3m- 2 (m-2) The discrepancy between (5) and (6) equals n 3m 2 ' which tends to zero

for n large enough, t 1

An immediate consequence of Proposition 2 is that, given m and large n, the

set o f winning coalitions o f a Borda election are the ( S: s ~ 2_(m 5 ~) ) . A 1"1 ~ m

consequence o f Propositions 1 and 2 together is that, for m ~ 3 , C(fb) ~C( fc ) ; the set o f winning coalitions o f a Borda game is a proper subset o f those of the corresponding Condorcet game. It also foUows from Proposition 2 tha t c(fB; m) c(fB; m - 1); the set o f winning coalitions shrinks monotonely with ~n increase in the number o f alternatives. The limit o f this process is the critical size two-thirds. Strategic voting in a Borda election with many voters and many alternatives eventually requires a two-thirds majority to ensure success.

IV. VOTING EQUILIBRIUM AND STRATEGIC TENSION

A coalition which exceeds the critical size has the power to carry an election, but not necessarily the will. A coalition has the will to upset the election of alternative i if there exists another alternative j which all its members prefer to L It is not enough that a coalition can keep alternative i from winning; it must also be strong enough to guarantee the election o f alternative j in its place. Indeed, a coalition with a simple majority is large enough to prevent the Borda election o f a given alternative; but a larger majority is required to ensure the election of some other alternative in its place. An alternative which no coalition has both the Mll and the power to replace we shall call a voting equilibrium. 12 The next two propositions compare the outcome of honest voting with voting equilibrium for Condorcet and Borda elections.

10Under some circumstances, for instance if m = 4 and s is divisible by 3, the odd- membered coalition can spread its votes evenly over the remaining alternatives.

l l ln practice, n may have to be fairly large to negate the difference. For instance, when m = 3 and n = 5, it takes 4 voters, not 3, to form a winning coalition. But even for small n, the error introduced by ignoring the difference between formula (5) and formula (6) is never larger than one voter.

12This solution concept is the same as R. Aumann's "alpah-core." An analogous concept for games in normal form is R. Farquharson's "collective equilibrium."

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Proposition 3. If alternative i belongs to fc(P)' then i is a voting equilibrium.13

Proof. Suppose alternative i is not a voting equilibrium. Then there exists a coalition S with the power (~-> ~) and the will (for all t in S, jP(t)i for some j) to

replace the election of i with j. This contradicts the hypothesis that Lij(P ) ) h i ( p ),

since Lij(P ) + Lji(P)~ 1. This result highlights one of the attractive features of Condorcet election. By

contrast, in Borda elections the necessary connection between honest voting and voting equilibrium is lost. The following proposition constructs the most glaring example of this.

Proposition 4. Given m, an alternative can have almost an ~ majority

against every other alternative and not belong to fB(p). However, such an alter- native is a Borda voting equilibrium; indeed, uniquely so.

Proof. No generality is lost by considering the distribution of opinion such that, for ~-n <~s ~ n, s voters have the preferences 1 P 2 P . . .P m and n - s voters

have the preferences 2 P . . . P 1. (Only the rankings of alternatives 1 and 2 matter here). Thus, alternative 1 has an S-majority against every other alternative.

n

The Borda number of alternative 1 is m(2s - n) + n - 2s, while the Borda number of alternative 2 is nm- 2s - n. Alternative 2 beats alternative 1 under honest voting when

(m- 1) > s (7) I13 n "

On the other hand, for m ~ 2, and s large enough

(m- 1)~ L ~ 2 ( m - 1) , (8) m n (3m- 2)

so that alternative i is a unique Borda voting equilibrium. We say that strategic tension exists whenever the outcome of an election

under honest voting is not a voting equilibrium. Proposition 4 shows that a Borda election gives rise to strategic tension. We propose to measure strategic tension by the difference between the largest majority that loses under honest voting and the critical size for winning. Then the strategic tension of Conclorcet election is zero, whereas the strategic tension of Borda voting is

i ~ - 2 ) (m- 1) , (9) m(3m- 2)

which one obtains by subtracting the right hand side o f (8) from the left. The strategic tension grows with the number of alternatives, eventually reaching one- third. Since the critical size approaches two-thirds, this means a coalition may exceed the critical size for winning by 50% and still see its favorite outcome lose, if

13This result is not new. It has been noticed by 1k. B. Wilson, among others.

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it votes honestly. Such a coalition would indeed be a sleeping giant, unaware of its strategic invincibility.

There are two other strategic aspects o f Borda voting worth noting in this connection. First, it may turn out that no alternative has a critical majority for or against it. This happens for distributions o f opinion similar to the uniform distribution of opinion on II. In such an event, every alternative is a voting equ~ibrium. Again, it may turn out that every alternative has a critical majority against it. This happens for distributions of opinion similar to cyclic preferences. In such an event, no alternative is a voting equilibrium. As with strategic tension, both of these eventualities lead one to expect results rather different from those of the hypothesis of honest voting.

V. CONCLUSION

One issue these results for Borda's rule raise is whether similar results hold for voting rules which differ from Borda's only in their rank weights. It turns out that qualitatively, the results are quite similar. The critical sizes for such elections range from one-half to one, and depend in general on the number of alternatives. All o f these elections display strategic tension to some degree.

The case of three alternatives wilt serve to illustrate the general result. Let an election give three votes to ftrst place, v votes to second place, and one vote to third place in a ranking. One can show by an argument like that for proposition 2, that the critical number for the election parameterized by v, fv' is given by

1

C(fv)- 2- (v- 1)/4

The case v = 1 is tantamount to plurality election, which has critical size one-half. Plurality election thus shares the same winning coalitions as Condorcet election. The case v = 2 is o f course Borda's rule. The critical size reaches a maximum of two-thirds in the case v = 3. For larger numbers of alternatives, one gets an even greater range of critical sizes.

In an assembly with coalitions of strategic voters, whether one opts for qualified majority rule or for positional voting with the same winning coalitions, seems more a matter o f style than o f substance.

REFERENCES

Auman, R. J. "A Survey of Cooperative Games Without Side Payments." Essays in Mathematical Economics. Edited by M. Shubik. Princeton: University Press, 1967.

Blin, J. M. "How Relevant are 'Irrelevant' Alternatives." Theory and Decision 7 (1976), 95-105.

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Black, D. The Theory of Committees and Elections. Cambridge: University Press, 1958.

Farquharson, R. Theory of Voting. New Haven: Yale University Press, 1969. Fine, B. and Fine, K. "Social Choice and Individual Ranking." Review of Economic

Studies 41 (1974), 303-322, 459-475. G~denfors, P. "Manipulation o f Social Choice Functions." Journal of Economic

Theory 13 (1976), 217-228. "Positionalist Voting Functions." Theory and Decision 4

(1973), 1-24. Gardner, R. "Studies in the Theory of Social Institutions." Unpublished Ph.D.

thesis, CorneU University, 1975. Gibbard, A. "Manipulation of Voting Schemes: A General Result." Econometrica

41 (1973), 587-601. Satterthwaite, M. "Strategy*proofness and Arrow's Conditions: Existence and

Correspondence Theorems for Voting Procedures and Social Welfare Funcdons."Journat of Economic Theory 10 (1975), 187-217.

Shapley, L. S. "Simple Games: An Outline o f the Descriptive Theory." Behavioral Science 7 (1962), 59-66.

Wilson, R. B. "A Game-Theoretic Analysis o f Social Cholce." Social Choice. Edited by B. Lieberman. New York: Gordon and Breach, 1971.

Young, H. P. "An Axiomatization o f the Borda Rule." Journal of Economic Theory 9 (1974), 43-52.

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