Uncertainty and political participation

UNCERTAINTY AND POLITICAL PARTICIPATION

William H. Panning

The minimax regret model of decision making under uncertainty, which was proposed by Ferejohn and Fiorina (1974) as a model of the voting decision, is here generalized to encompass forms of political participation (such as contributing money or time to a campaign) in which the range of alternatives is continuous. A principal implication of the model is that aggregate campaign contributions may be substantially increased by limiting the amount that any one individual can contribute. The conditions under which the behavior of a minimax regret decision maker can be unambiguously distinguished from that of an expected utility maximizer are also specified.

According to the expected utility model of the voting decision suggested by Downs (1957), a citizen will vote only if the benefits to him (B) of having his preferred candidate elected ra ther than his opponent , dis- counted by the probabi l i ty (P) tha t his vote wilt be the decisive one, exceeds the cost (C) to him of v o t i n g - - t h a t is, if PB -- C > 0. But as Downs himself argued, the probabi l i ty that an individual 's vote will be decisive is so small that for plausible values of B and C one would expect PB -- C to be negative for most citizens. Even the modera te rates of voter turnout in recent Amer ican elections are therefore inconsistent wi th the implications of the expected utility model. More recently, Ferejohn and Fiorina (1974) offered a new model of the voting decision, based upon the minimax regret criterion, that appears to overcome this difficulty. Since the minimax regret model treats the decision to vote as a decision made under conditions of uncer ta inty ra ther than of risk, probabili t ies play no role in it, and it predicts tha t an individual will vote if B/4 > C. The empirical fruitfulness of the model is supported by Ferejohn and Fiorina's subsequent reanalysis

William H. Panning, Department of Political Science, University of Iowa, Iowa Cit)~

Political Behavior Vol. 4, No. I, 1982 © 1982 Agathon Press, Inc. 0190-9320/821030069-13501.50

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of presidential election survey data, which indicated that, while variations in B had a significant impact on turnout, variations in P, defined opera- tionally as the expected closeness of the outcome, did not (Ferejohn and Fiorina, 1975).

The apparent success of the minimax regret model in explaining the decision to vote suggests the possible fruitfulness of extending the model to other forms of political behavior. One step in this direction has already been taken by Aldrich (1976), who applied the model to data concerning forms of campaign participation other than voting and found strong empirical support for the minimax regret model but relatively little for the expected utility model. However, a principal limitation to Aldrich's analysis--one imposed by the nature of his data--is that he formulated the decision problem as a choice between "participating" and "not participating." In reality, however, decisions concerning contributions of money, time, or labor to a political campaign involve a choice among a continuous range of alternatives that vary in their cost to the individual. Although an individual can decide only whether to vote, he must decide how much he wishes to contribute to a campaign in these other ways.

In this paper, I extend Aldrich's (1976) analysis by undertaking the theo- retical task of applying the minimax regret model to choices among a continuous range of alternatives. The generalized model presented here has several striking implications pertinent to campaign participation. In particular, it implies that aggregate contributions to political campaigns can be substantially increased by limiting the amount any one individual can contribute.

ASSUMPTIONS

Although the model developed here might well be formulated as a choice among alternative contributions to the provision of any pure but "lumpy" public good [see Frohlich and Oppenheimer (1970) for an analo- gous formulation in the context of an expected utility model], I shall for convenience represent the problem as one involving contribution to a political campaign. I shall retain the following assumptions adopted by Fere- john and Fiorina (1974):

1. The utility gain to an individual from having his preferred candidate elected is B.

2. The utility gain to the individual from having the other candidate elected is zero.

3. If a tie occurs, a fair coin is flipped to determine the winner, so that the individual's expected gain is B/2.

UNCERTAINTY AND POUTICAL PARTICIPATION 71

. If the individual contributes some amount C to the election campaign, his payoff is reduced by the utility U(C) to him of his contribution. I shall assume that U(C) increases monotonically with C and that U(O) = O.

Although it can safely be assumed that campaigning affects the number of votes received by a candidate, election outcomes are not determined solely by the relative campaign expenditures of the two candidates. In many elections one of the candidates possesses an initial nonmonetary elec- toral advantage, e.g., by virtue of his greater visibility or support among constituents. His opponent must therefore spend more than he does in order to make the contest a close one. This fact is reflected in the following assumption of the model:

. Let C be the individual's contribution to his preferred candidate, Co the total contributions of others to that candidate, D the total contributions to the opposing candidate, and A the amount of contributions to the individual's preferred candidate that is necessary to offset the initial (nonmonetary) advantage of the opposing candidate. (Note that A can be positive or negative.) Then, a. if C + Co > D + A, then the individual's preferred candidate will

be elected, b. if C + Co < D + A, then the opposing candidate will be elected, c. if C + Co = D + A, then the election will result in a tie, so that by

assumption 3 the individual's expected gain is B/2.

Assumption 5 implies that contributions of money, time, and labor to a candidate bring about some increase in the number of votes he receives but that such contributions are not the only variable that affects the election outcome.

Finally, I shall assume that there are only two candidates competing for election, and I shall omit from further consideration all alternatives that are strictly dominated, such as (a) contributing any nonzero amount to the less preferred candidate, and (b) contributing any amount greater than or equal to C*, the value of C such Chat U(C*) = B. The problem, then, is to determine how much the individual will contribute to his preferred candidate in a two-candidate election.

UNRESTRICTED CONTRIBUTIONS

Let S ---- D -4- A -- Co, the amount by which contributions by others to the individual's preferred candidate fall short of matching the campaign

72 PANNING

chest and initial advantage of the opposing candidate. In order to apply the minimax regret model, it is necessary to determine the individual's maximum payoff for any given value of S. If S < 0, so that no further contributions are needed to obtain a favorable outcome, the individual will receive the maximum payoff of B by contributing nothing at all. If, at the other extreme S _> C*, the individual will again do best by contributing nothing, although his payoff in this case is zero. Now consider the ease in which 0 _< S < C*. Here the individual will obtain the greatest payoff by contributing an amount equal to S + e, where e is some arbitrarily small amount, and his payoff will be B - U(S + e). Unfortunately, the set of payoffs represented by this quantity has no maximum for given values of B and S, for it is an open set bounded from above by the value B -- U(S). No matter how small a value one chooses for e, one can find a still smaller value that exceeds zero. (It is necessary for e to be greater than zero, so that a tied election--and its accompanying lower payoff--is avoided.) Thus, unless one wishes simply to reject the minimax regret model as inappro- priate, it is necessary to use the supremum payoff (and regret) rather than the maximum payoff (and regret) in applying the model to the problem considered here. In the case at hand the supremum payoff is B -- U(S).

It must be emphasized that the results to be presented are not dependent in any crucial way on the use of the supremum payoff and supremum regret. In the Appendix to this paper I show that each of the principal results derived here can also be demonstrated for choices from a finite set of discrete alternatives. The use of the supremum simply permits the same results to be derived when alternatives are continuous rather than discrete.

In Figure 1, the horizontal axis represents the possible values of S, which correspond to the alternative states of the world that might occur, and the vertieal axis represents the payoff to the individual. The dotted line represents the supremum payoff to the individual--the best he can hope to do m for any particular state of the world (value of S). Now suppose that the individual contributes nothing. Then for S < 0 his payoff is B. For S = 0 his payoff is represented by the point at B/2, and for S > 0 his payoff is zero. If, on the other hand, the individual contributes some nonzero amount C, then his payoffs are represented by" the horizontal line that begins on the left at B -- U(C), continues until it intercepts the line of supremum payoffs, and then continues below the horizontal axis at - U(C) for S > C. (The midpoint of the dashed vertical line represents the payoff f o r S = C.)

The regret for a particular contribution is the difference between the supremum payoff and the actual payoff at each value of S. The regrets for each of the contributions in Figure 1 are shown in Figure 2, in which the vertical axis represents the regret rather than the payoff. From Figure 2 it

UNCERTAINTY AND POLITICAL PARTICIPATION

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can easily be seen that for each of the two contributions shown in Figure 1--and by extension for all contributions less than C*--the supremum regret is B. This is simply the difference between the supremum payoff, B - U(S) , and the infimum payoff for the same value of S, - U(S) . If an individual chooses the contribution which minimizes his supremum regret, then he will be indifferent among all contributions less than C*, for all such contributions have the same supremum regret. (All contributions of C* or more are, it will be remembered, dominated by a contribution of zero.)

Although this result may seem striking, its substantive interpretation is straightforward: No matter how large the individual's contribution, it may turn out to be just insufficient to transform loss into vietory. The possibility always exists that increasing one's contribution by some vanishingly small amount would bring about the election of one's preferred candidate and that failure to contribute this additional amount would result in his losing the election. No matter how many horseshoe nails are produced, a battle may be lost for the lack of just one more. The quantity produeed is therefore, the model implies, a matter of complete indifference, so long as one does not spend more than the objective is worth.

This first implication of the model, that individuals with unrestricted alternatives will be indifferent among all contributions less than C*, can be brought to bear on empirical data only if the model is augmented with some assumption about individual choice among equally valued alternatives. One possible assumption is that, in such eases of indifference, variables not yet included in the model will determine the individual's choice of a particular contribution. But were such variables included in the model, then considerations of parsimony would compel one to ask why the minimax regret framework is necessary at all, for it would be these other variables that explained observed behavior. A more compelling alternative is the assumption that all contributions less than C* are equally likely to be chosen when an individual is indiferent among them. This would imply that an individual's expected contribution is C*/2 and would thus render the model consistent with the data presented by Aldrich (1076), whieh shows that frequency of participation varies with B, and hence with C*.

RESTRICTIONS ON INDIVIDUAL CONTRIBUTIONS

So far we have assumed that the individual may contribute any amount he wishes to a political campaign. Now let us consider what the model predicts when the maximum contribution an individual can make is some- how restricted (e.g., by passing legislation that levies a heavy fine upon individuals who contribute more than the specified maximum) to some

UNCERTAINTY AND POLITICAL PARTICIPATION 75

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amount K which we shall assume is greater than zero but less than C*. The supremum payoffs for this problem are shown in Figure 3, where we see that for S > K a payoff of zero is the best the individual can hope to achieve. For a contribution of zero the line of actual payoffs is identical to that shown in Figure 1. If we let the contribution of C in Figure 1 be the maximum possible contribution K, then the line of actual payoffs for this strategy, too, is identical in Figure 1 and Figure 3. But when we turn to Figure 4, we see at once that there is a marked difference in the regrets for the two alternatives. For a contribution of zero and, by extension, for any contribution less than K the supremum regret is the same as before, B. However, for a contribution of exactly K the supremum regret is not B but U(K), which is by assumption less than B [since K < C* and U(C*) = B]. If the individual has contributed the maximum permissible amount, then the supremum regret for such a contribution will occur either when no contribution was necessary (S < 0) or when it was insufficient to bring about the desired outcome (S > K). The possibility no longer exists that by contributing just a bit more the individual might have transformed a loss into a victory for his preferred candidate.

76

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Fig. 4. Regrets when contributions are restricted to K or less.

PANNING

Thus, when contributions are limited to some amount K that is less than C* for an individual, the model predicts that he will contribute K, the maximum permissible amount.

INCREASING AGGREGATE CONTRIBUTIONS

The two results just presented, taken together with an additional assumption, have an important further implication: Placing a suitably chosen limit on individual campaign contributions will increase aggregate contributions. Recall that for each individual citizen there is some contribution C* whieh has utility to him equal to that of having his preferred candidate elected [i.e., U(C*) = B]. Let the distribution of C* among citizens be represented by the line HN in Figure 5, where the number of citizens is represented by the horizontal axis and the value of C* is represented by the vertical axis. Here I shall assume that the number of citizens is a monotonically decreasing function of C*. (For convenience, I represent the function as a linear one in Fig. 5, but this is inessential to the proof.) That is, the eleetion is assumed to be of low or moderate importance to most citizens and of great importance to a few, an assumption that is empirically plausible.

For the reasons given earlier it will here be assumed that all contributions less than C* are equally likely to be chosen when an individual is indifferent among them, so that an individual's expected contribution is C*/2. Hence it follows that aggregate expected contributions to a candidate, represented by the area JON, where ] = HI2, will equal half the

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78 PANNING

aggregate value to his supporters of having him elected (represented by area HON).

Now suppose that the maximum contribution an individual can make is restricted to some amount K, where K is assumed to exceed ]~ the largest amount that any individual is expected to contribute. In this case the model implies, as shown earlier, that individuals for whom C* > K will contribute K. Thus, the new distribution of individual contributions wilt be represented by the line KLMN in Figure 5, from which it is clear that aggregate contributions will be increased by an amount equal to the area JKLM. This result holds so long as there is at least one individual for whom C* > K. (The assumption that K exceeds ] is sufficient but not necessary for this result to hold.)

If we impose the further assumption that the distribution of C* is linear, as represented by line HN in Figure 5, then two further results can be derived. The first is that total contributions are maximized when the maximum permissible contribution K is two-thirds of the maximum value of C*. Second, the imposition of such a limit will increase total (expected) contributions by one-third. Proofs for these two results are straightforward and will not be presented here.

MINIMAX REGRET VERSUS EXPECTED UTILITY MAXIMIZATION

There is one further implication of the model that in principle should always permit one to distinguish between minimax regret decision makers and expected utility maximizers. Let us assume that individual contributions are restricted to some maximum amount K such that for both individuals B/4 < (7(14,) < B. The model predicts that the minimax regret decision maker will contribute K to his preferred eandidate. Let us assume that for the expected utility maximizer some nonzero contribution C is optimal. Suppose that now, having ascertained the preferred behavior of each individual, we restrict each of them to only two alternatives: contributing the amount each has already chosen (K for the one, C for the other), or nothing at all. Under these new circumstances the expected utility maximizer will still contribute C, but, as Ferejohn and Fiorina (1974) demonstrate for the case of voting, the minimax regret decision maker will now choose to eontribute nothing, since K > B/4.

This implication of the model illustrates an important property of the minimax regret decision rule that distinguishes it from expected utility maximization. What is important here is that in both circumstances the same two alternatives--contributing K and contributing nothing--were available to the individual. Yet the minimax regret model implies that, in the one case, in which still other options are available, he will choose the

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80 PANNING

first alternative, and in the other ease he will choose the second. That is, the minimax regret model, unlike the expected utility model, implies that the individual's choice between the two alternatives depends upon what other alternatives are available to him. From this it is clear that the minimax regret decision rule violates a condition that Sen (1969) calls "condition alpha" and that in other contexts has been called the "independence of irrelevant alternatives" (Chernoff, 1954; Luce and Raiffa, 1957, chap. 13). This implication of the decision rule can thus be used to distinguish empirically between behavior that conforms to the minimax regret criterion and behavior that is consistent with expected utility maximization.

CONCLUSIONS

My purpose in this paper has been to present some additional implications for political behavior of the minimax regret model of decision making under uncertainty To my knowledge, the model has never before been generalized to encompass choice from among a continuous range of alternatives. Although I have presented the model as one pertaining to contributions to a political campaign, it may well be applicable to other phe- nomena as well, for the problem discussed here can be regarded in the most general sense as one involving choice among alternative contributions to the provision of any pure but "lumpy" public good, of which contributions to a political campaign are but a special case.

The results derived from the model potentially have an important bear- ing on public policy, for they imply that public participation in campaign activities might be increased by establishing a suitably chosen limit on individual contributions. Moreover, quite apart from the possibility of al- tering existing laws concerning campaign finance, the model implies that candidates may find it a desirable strategy to limit the size of the contribution they will accept from individual supporters, as George McGovern did during his campaign for the presidency. Whether or not these policy and strategic implications are warranted depends, of course, upon the propor- tion in the population of individuals whose behavior conforms to the minimax regret decision rule. But the results presented here also make this pro- portion amenable to empirical study rather than speculation.

ACKNOWLEDGMENTS

I am particularly indebted to Don Moon and Russell Hardin for their extensive and valu- able comments on an earlier version of this paper.

APPENDIX

In this appendix I present a simple numerical example which demonstrates that

UNCERTAINTY AND POLITICAL PARTICIPATION 81

the results presented in this paper also hold for choices from among a finite set of discrete alternatives.

Let us here assume for the sake of simplicity- (and without loss of generality) that B, C, and S represent dollar values, and that B = 5. In the first ease to be considered the individual must choose among contributions of 0, 1, 2, 3, and 4 dollars (contributions of 5 dollars or more are dominated by a contribution of 0). The payoff matrix for this decision problem is presented in Table la, and the eorre- sponding regret matrix in Table lb. The maximum regret is the same for each of these alternatives, so that the individual will be indifferent among them. In Case II the individual is restricted to a eontribution of 3 dollars or less, and we see from the regret matrix in Table lc that a contribution of 3 dollars will minimize his maximum regret. Finally, in Case III he must choose between contributing nothing and contributing 3 dollars. The regret matrix, in Table ld, shows that he will minimize his maximum regret by contributing nothing. In each of these cases the alternatives of contributing 3 dollars and contributing nothing were available to the individual, but his choice between these two alternatives depended upon what other alternatives were available to him as well.

REFERENCES

Aldrich, John H. (1976). "Some Problems in Testing Two Rational Models of Par- ticipation." American ]ournal of Political Science 20: 713-733.

Chernoff, Herman (1954). "Rational Selection of Decision Funetions." Econo- metrica 22: 422-443.

Downs, Anthony (1957). An Economic Theory of Democracy. New York: Harper & Row.

Ferejohn, John A., and Morris E Fiorina (1974). "The Paradox of Not Voting: A Decision Theoretic Analysis." American Political Science Review 68: 525-536.

_ _ (1975). "Closeness Counts Only in Horseshoes and Dancing." American Political Science Review 69: 920-925.

Frohlich, Norman, and Joe A. Oppenheimer (1970). "I Get by with a Little Help from my Friends." World Politics 23: 104-120.

Luee, R. Duncan, and Howard Raiffa (1957). Games and Decisions. New York: John Wiley & Sons.

Sen, Amartya (1969). "Quasi-transitivity, Rational Choice and Collective Deci- sions." Review oJ Economic Studies 36: 381-393.

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Uncertainty and political participation