Does rational choice theory really imply that voting in mass elections is "irrational"? Though commonly made, this assertion is considerably overdrawn. The basic framework for considering "rational" voting choices by a focal voter i in a two-candidate (or party) contest is set out in Figure 1 of Aldrich's "Rational Choice and Turnout" (AJPS, 1993), which is also presented in abbreviated form below.

        Voter i has three choices: "Vote for Preferred Candidate P" (VP), "Vote for Non-preferred Candidate N" (VN), and "Abstain" (A). The effect of each choice differs according to the contingency that voter i may face, i.e., whether and how other citizens vote and, in particular, the outcome of the election in the absence of i's vote. There are five distinct contingencies: (1) P wins by more than one vote (in the remainder of the electorate); (2) P wins by exactly one vote; (3) P is tied with N; (4) P loses by exactly one vote; and (5) P loses by more than two votes.

        The utility benefit to voter i of P's winning (as opposed to N's winning) is B. (Downs calls B the voter's "party differential. Aldrich normalizes B at 1.) B is by definition non-negative for every voter. The cost to i of voting (as opposed to abstaining) is C.   C is assumed to be positive for every voter. As a practical matter (and very plausibly), B is assumed to exceed substantially C for most voters, specifically including the focal voter i whose choice we are analyzing. Conventionally, it is assumed that a tie election is resolved by flipping a fair coin, so the utility benefit to i of a tie outcome (as opposed a clear defeat of P) is B/2.

        The utility payoffs to each of i's choices in each contingency are therefore as follows:

                                 (1)              (2)               (3)             (4)               (5)
                VP           B - C         B - C            B - C          B/2 - C         0 - C
                VN           B - C           B/2 - C       0 - C            0 - C          0 - C
                A                B               B                B/2                0                 0

        We observe that VN gives i a lower payoff than VP (whatever the values of B and C, consistent with the stipulations given above) in contingencies (2), (3) and (4), and gives the same payoff in the remaining contingencies (1) and (5), so VP dominates VN. (Abstention also dominates VN.) Thus (unsurprisingly) in a two-candidate contest, a rational voter either votes for his preferred candidate or abstains, and we can restrict subsequent analytic attention to just two choices: V (= VP) and A.

        Given the stipulations above, V never dominates A, and A dominates V if and only if C > B/2. (Hence the "practical" assumption that B substantially exceeds C.)

        Suppose B/2 > C. Should a "rational" voter chose V or A. We can consider three standard decision contexts: individual decision making under uncertainty, individual decision making under risk, and collective decision making with strategic interaction.

        Given individual decision making under uncertainty, voter i has no sense at all as to which contingency may occur. One decision making criterion that may be employed in this information-poor context is maximin -- that is, i determines the worst possible (minimum utility) outcome that may result from each choice and makes the choice that gives the best (maximum utility) among these worst outcomes. If i votes, the worst that can happen is that P loses anyway and i bears the cost of voting (payoff = 0 - C). If i abstains, the worst that can happen is that P loses but i at least avoids the cost of voting (payoff = 0). So maximin prescribes abstention and, if we identify "rationality" with maximin, the common assertion is true. But, while maximin may be a reasonable criterion for a player interacting with a rational opponent with totally opposed interests (i.e., in a two-person zero-sum game), it has little appeal or justification in other circumstances, including the present one, if only because its prescription is unreasonably insensitive to parameter values. (Suppose for $1 you can buy a lottery ticket that gives you a 50% chance of winning $100: maximin says don't buy the ticket -- and it says the same even if the lottery ticket gives you a 99% chance of winning $1 million.)

        Another decision making criterion that a totally uncertain voter can use (as suggested by Ferejohn and Fiorina, APSR, 1974) is minimax regret. If a decision maker does not have a choice that dominates all others, he may experience some "regret" whatever choice he does make, where his regret is the utility loss he experiences from his actual choice as opposed to that which he would experience from his optimal choice in the contingency that actually occurred. In the case of voting, the "regret matrix" is as follows:

                                                               (1)           (2)           (3)          (4)       (5)
                                            V                 C             C             0             0         C
                                            A                  0              0         B/2 - C    B/2 - C    0

        For example, in contingency (1), abstention is the citizen's optimal choice and, if he actually votes, he experiences a utility loss of C relative to what abstention would have given him. The minimax regret criterion prescribes the choice that minimize the maximum regret. Thus in the voting case, minimax regret prescribes voting if B/2 - C > C (and abstention if the reverse is true). The condition for voting simplifies to B/4 > C, which -- as Ferejohn and Fiorina observe -- seems to be an easy condition for most citizens to meet. So minimax regret prescribes voting for most citizens and, if we identify "rationality" with minimax regret, the common assertion is false. But, while minimax regret may be a reasonable criterion in a few circumstances, it has little appeal or justification in other circumstances, including the present one. Like maximin, its prescription is unreasonably insensitive to parameter values. (Suppose for $10 you can buy a lottery ticket that gives you a 1% chance of winning $100: minimax regret says buy the ticket -- as it says for any lottery ticket that offers a prize worth more than twice the cost of the ticket, regardless of the odds of winning the prize. Of course, all lottery tickets have expected values less than their cost, yet many consumers buy them, but this behavior is more often characterized as "irrational" than "rational.")

        Decision making under risk implies that the decision knows the probability that each contingency occurs or at least has some subjective sense of these probabilities. (Since the contingencies in the voting case not generated by a random mechanism, the probabilities here must be subjective in some sense. In this circumstance, the commonly accepted criterion for choice is expected utility maximization. That is, the decision maker assesses the expected utility of each choice by multiplying the resulting utility in each contingency by the probability that the contingency occurs and adding up the results; he then makes the choice that gives the greatest expected utility.

        Applying this context to the voting case produces the famous "calculus of voting" expression. Let p(1) be the (subjective) probability that contingency (1) occurs, and so forth. Then define p = p(3) + p(4) -- that is, p is the probability that i's vote "makes a difference," either tipping an election that would otherwise by tied in favor of P or producing a tie in an election that N would otherwise win by. Once the number of other voters is fixed as either odd or even, either p(3) or p(4) is zero while the other is non-zero. Suppose this number is even, so p(1) + p(3) + p(5) = 1 and p = p(3). (The opposite assumption leads to the same result.) Thus we have the following expected utilitie

An expected utility maximizing citizen votes if EU(V) > EU(A) or

Simplifying (and noting that p(1) × C + p × C + p(5) × C = C), we get

which is the famous "calculus of voting" expression, except that B/2 is commonly written simply as B. (This is because p is commonly described as the probability that the citizen's vote will convert a loss for his preferred candidate into a victory, which ignores the fact the a single vote cannot do this but can only create or break a tie.)

        Then it is commonly observed that p (while not zero) must be vanishingly small in a mass election, with the result that the product p × B/2 is also extremely small even if B is quite large, so p × B/2 < C even if C is very small and EU(V) < EU(A), from which it seems to follow that an expected utility maximizing citizen abstains from voting in mass elections.

        While this simple algebra is correct, the substantive prescription for voting behavior is fundamentally incoherent because this whole setup treats what in truth is a collective decision-making problem as an individual decision-making problem. This incoherence actually was acknowledged by Downs (1957, p. 267) as the problem of "conjectural variation" (and is also noted by Aldrich, p. 257), but it is perhaps best conveyed by that famous expositor of incoherent statements, Yogi Berra, when he observed that "Nobody goes to Coney Island any more, because it's too crowded." In parallel fashion, the fundamental (il)logic of the standard assertion that rational choice theory implies zero-turnout is that "nobody goes to polling places any more, because they're too crowded."

        The lesson, surely, is that we must treat the voting turnout question in a framework that is appropriate to the nature of the problem -- that is, as a collective decision-making problem that entails strategic interaction and is analyzed in a game-theoretical framework. When we do this, one thing is immediately clear: a zero-turnout election cannot be a Nash equilibrium if B/2 > C for any potential voter. Thus, if we are willing to assert (for example, in the manner of Austen-Smith and Banks, APSR, 1996) that rational behavior must result in a Nash equilibrium, it follows that zero-turnout elections cannot result from rational behavior provided there are any differences between the candidates that lead any citizen to have a substantial preference for one candidate over the other. (Ledyard, Public Choice, 1984, shows that a zero-turnout election can result from rational behavior, but only because candidate positions are not fixed exogenously and the candidates fully converge in general equilibrium.)

        While it is easy to see that zero-turnout elections cannot generally result from rational behavior in the Nash equilibrium sense, it is much harder to determine what does result from rational behavior. It quickly becomes apparent that analyzing the voting problem in game-theoretical terms is extremely complicated (though actual voters do not find the practical problem complicated). Some years ago, Palfrey and Rosenthal (Public Choice, 1983 and APSR, 1985) made some limited progress along these lines and identified equilibria (generally in mixed strategies) than entail positive turnout, but much more work needs to be done. But is should be apparent that the simple assertion that rational choice theory implies that voting in mass elections is "irrational" is considerably overdrawn.