Costly Voting and Why the Order of Voting Matters

There is a natural reason why behavior should depend on the order of the voting procedure: when voters can observe previous voters' behavior, they can be influenced by previous choices that may signal private information. In a recent contribution, however, Deckel and Piccione 2000 have questioned this reasoning. They show that, under general conditions, any symmetric equilibrium of the simultaneous voting game in which players use their information is in fact a sequential equilibrium in any sequential voting game and that there always exists equilibrium behavior in the sequential game that is completely independent of the order of voting.

Their argument is based on the observation that a rational voter would realize that he is influential only when pivotal. To see the intuition, assume that voters ignore the sequential order of the voting protocol and behave as if they were in an equilibrium of a simultaneous voting game. In this case, the expected benefit of voting for alternative A for a voter i who votes at stage t after a history h_t and an observed signal s_i=a can be represented as

where

is the probability of being pivotal; and

is the expected utility obtained if A wins conditional on being pivotal and on a signal s_i.¹

is non-negative, his choice will be determined by the sign of

, which implies that he finds it optimal to make a choice that is independent of the history. An informational cascade will not occur.

Dekel and Piccione's (2000) result does not imply that the set of symmetric informative equilibria of the simultaneous and sequential voting games are identical, only that the first is a subset of the second. This result leaves open the possibility that the sequential voting game has additional equilibria that are not in the equilibrium set of the simultaneous game.²

An attempt to solve this indeterminacy is provided in Battaglini 2005, by introducing voting costs. When there is a cost of voting c and the agent can abstain, the decision depends on the sign of

In this case the decision is determined by the magnitude of

, which depends on h_t. We should therefore expect to see rates of abstention that depend on the history and that increase as the probability of being pivotal decreases. This strategic abstention phenomenon also suggests that the set of equilibria and the informational properties of the two elections will also differ: the set of equilibria are disjoint and simultaneous voting should be superior when the size of the election is large enough.⁴

In an election this cost straightforwardly represents the cost of the physical time and effort of voting and can also be interpreted as the cost of mobilizing a group of voters to participate. In a legislative situation the cost can be interpreted as the opportunity cost of engaging in other legislative activities–the cost of leaving a meeting of a committee, constituents, or executive officials to cast a ballot in a roll call vote—or even the risk of taking an unpopular stand on an issue. Legislators are often aware of the progress of voting on contested matters while engaging in other activities and can and do choose whether to return to the chamber. The cost could also be interpreted as a cost of position taking if we assume that these costs are independent of the position taken or the outcome of the voting; that is, legislators may see it as desirable to not take any positions on issues. A number of researchers have found evidence that members of Congress, both House and Senate, avoid voting either because of the demands of campaigning or a desire to not to take a policy position (see Thomas 1991; Rothenberg and Sanders 1999, 2000; and Jones 2003).⁵

With costly voting, the net utility function of a voter who votes is therefore u(x, θ)−c: in state θ if option x is chosen. We assume that a voter who decides alone would always prefer to pay the cost and determine the outcome of the election: so

, where

is the expected utility of voting for A (B) conditional on a a (b) signal. It is therefore convenient to re-parametrize the cost as

, where γ∈(0, 1).

The Voting Games

We consider two game forms, which we call the simultaneous voting game and the sequential voting game. In both games the outcome is chosen by majority rule and we assume that when A receives the same votes as B, or when all voters abstain, then one of the two alternatives is chosen with probability 1/2. In what follows we assume n=3. In the simultaneous voting game all voters vote simultaneously. In this case, a (pure) voting strategy for voter i is a map

: that is, given the signal, the voter may vote for A, B or abstain. A mixed strategy assigns a probability of abstaining σ_i(ϕ; s_i), and, conditional on voting a probability of voting for each alternative, σ_i(x; s_i), x=A, B. In the second game form, voters vote sequentially. In this case, a strategy is a function

, where H_i is the set of histories that voter i can observe. In this case too we will denote σ_i(ϕ; s_i, h_i) the probability that voter i abstains after observing a signal s_i and a history h_i; and σ_i(x; s_i, h_i) the respective probability of voting for x, conditional on not abstaining.

An equilibrium of the sequential game (resp. simultaneous game) is symmetric if σ_i(θ; s, h_t)=σ_j(θ; s, h_t) for all i,j and all h_t∈H_t, and

, s∈{a, b} (resp. if σ_i(θ; s)=σ_j(θ; s) for all i,j and for

, s∈{a, b}). In this symmetric environment there is no a priori difference between state A and state B: it is therefore natural to assume that the names associated with these two states are irrelevant for the strategic considerations of the agents. Let us define N_a(h_t) (N_b(h_t)) the number of a (b) votes in a history h_t; and let

. After any of these histories the states continue to be symmetric. We define an equilibrium of the sequential game (resp. simultaneous game) to be neutral if two requirements are satisfied: (i) σ_i(ϕ; a, h_t)=σ_i(ϕ; b, h_t) for any

(resp. σ_i(ϕ; a)=σ_i(ϕ; b)); and (ii)

for any

, h_t+1={h_t, ϕ}, and θ=A, B.⁶

Equilibrium Characterization

The characterization of equilibria in the simultaneous game is simplified by two observations. First, because we focus on equilibria that are neutral and symmetric and π=0.5, voters never vote against their signal; they either vote sincerely or abstain. Therefore, to characterize the equilibrium we only need to determine the abstention probabilities,

. Second, neutrality implies that σ_i(ϕ; a)=σ_i(ϕ; b)=σ_i(ϕ), and symmetry implies σ_i(ϕ)=σ_j(ϕ)=σ(ϕ) for all i,j. Therefore we can focus on one variable only: σ(ϕ), and we drop the dependence on ϕ, simply writing it as σ.

The equilibrium value of σ is determined by the cost of voting c and the equilibrium expected benefit of voting, which is balanced against the expected utility of not voting, so the usual cost-benefit calculus applies by conditioning on pivotal events. Consider voter i with a signal s_i=a. His vote is pivotal only in three events. First, when no other voter participates, event P₀. This event occurs with probability σ²; and, in this event, the expected benefit of voting for A is equal to pv and the expected benefit of not voting is simply

. Hence the expected gain from voting in event P₀ equals

, where p is the posterior probability of state A after one signal a. Second, a voter is pivotal when exactly one other player votes, and this player votes B, event P₁. In this case, however, the posterior is

because in P₁ there are exactly two opposite signals that offset on the other, so the expected gain from voting is 0. The third possibility is when the two other voters vote, and they vote for opposite alternatives, event P₂. In this case, voter i knows that there are two a signals and one b signal. The posterior is

, and the expected benefit of voting is, again,

. From the point of view of i, this event occurs with probability:

. The expected utility of voting for A for agent i is therefore

Comparing with the cost of voting we have a pure strategy equilibrium in which all agents vote when EU(0)=(2p−1)p(1−p)v>c; and we have a mixed equilibrium at any value of σ∈ (0, 1), such that EU

. Using these conditions we can characterize the set of symmetric equilibria in the simultaneous game.

Proposition 1. In the simultaneous voting game, when n=3:

• i. If
  there is a unique pure strategy equilibrium σ*=0.
• ii. If
  there are three equilibria: one pure strategy equilibrium σ*=0, and two mixed strategy equilibria.
• iii. If
  , there is a unique mixed strategy equilibrium σ*∈(0, 1)

In the sequential game the action of an agent affects the outcome in two ways. First, we have a direct effect: given the vote of the others, a vote in favor of an option increases its plurality. But the vote of early voters has an indirect influence on later voters as well: the vote signals the voter's information to the remaining voters. This allows information to be leaked in a way that is not possible with simultaneous voting, and this leakage may lead to efficiency gains because later voters will rationally (and efficiently) abstain after some sequences of decisions by earlier voters. We focus on sincere equilibria in which no voter votes against his own signals. Although there can exist equilibria where early voters vote against their signals, they are intuitively implausible, inefficient, and not observed in our experiments.⁸

Proposition 2. In the sequential voting game, when n=3 there exists a unique sincere, neutral equilibrium path, for all voting costs, and this equilibrium is in pure strategies. The equilibrium path is as follows:

• i. If c∈[0, p(1−p)2p−1)v], the first voter votes (sincerely), the second voter votes only if the first voter has voted and he has a different signal than the first voter; and the last voter only if the first and second voters vote for opposite alternatives or if no voter votes before. All voters vote informatively when they vote;
• ii. If
  , the first and second voters abstain and the third voter votes (sincerely).

Theoretical Implications for Efficiency and Equity

Propositions 1 and 2 present a clear characterization of the equilibria. When

and c>p(1−p)(2p−1)v, we have a unique equilibrium in both the simultaneous and in the sequential models, and these equilibria are different. In particular:

• When
  there is a unique equilibrium of the simultaneous game in which the voters vote informatively and never abstain. In the sequential game there is a unique equilibrium in pure strategies as described in point i of Proposition 2.
• When c>p(1−p)(2p−1)v, there is a unique equilibrium of the simultaneous game in which the voters abstain with probability EU⁻¹(c)∈(0, 1) and vote informatively with the complementary probability. In the sequential game there is a unique equilibrium in pure strategies in which only the last voter votes in equilibrium, as described in point ii of Proposition 2.

In the rest of the paper we focus on parameters only in these two regions to avoid multiplicity of equilibria. We refer to the first case as the low-cost case and to the second case as the high-cost case. Given this, we should expect very different behavior between simultaneous and sequential elections, and given the voting mechanism between low and high costs. In particular:

• In simultaneous elections, we should expect the probability of abstention to be decreasing in the cost of voting: the probability should be zero in the low-cost region and positive in the high-cost region.
• In sequential elections with low costs, the first voter should always vote and late voters should vote only if they find it optimal to correct the choice of earlier voters and if they are pivotal. In sequential elections with high costs the opposite should occur—they should be characterized by free riding from early voters who should abstain counting on the participation of late votes.

These differences have an impact on the theoretical efficiency and equity properties of the voting mechanisms as well as noted above. With respect to equity, in a symmetric equilibrium under simultaneous voting all voters obtain the same expected utility, in the sequential mechanism expected utility depends on the stage in which the agent votes. When the cost is low and the later voters free ride, early voters receive a lower utility level than later voters; in a high-cost regime, on the contrary, early voters free ride on the participation of later voters and obtain higher expected utility. The predictions with respect to efficiency will be discussed in greater details where we develop the appropriate benchmark case for efficiency: here we note that when the cost is low we should expect lower abstention than with high voting costs: therefore, we should expect a more efficient collective choice when the cost of voting is low.

Individual Choices: Does Sequence Matter?

Simultaneous Voting Choices

Our theoretical analysis of simultaneous voting suggests that we should see zero abstention in the low-cost treatment and positive abstention in the high-cost treatment. Table 2 summarizes the voting choices of participants in the simultaneous voting games. Of the 900 individual voting decisions in the simultaneous voting games, only 17 (< 2%) were votes against a subject's signal, and of these 11 were cast by two subjects in the low-cost treatment. Abstention was significantly higher in the high cost treatment than in the low cost games (67.86% compared to 38.96%). As is clear from Table 2, we find little support for the exact quantitative Nash equilibrium predictions in simultaneous voting: low-cost voters abstain significantly more than predicted, and high-cost voters abstain significantly less than predicted.¹²

We also estimated a multinomial probit to determine if there were learning effects where we find significant evidence that voters increase the probability of abstaining in the high-cost treatment but little evidence of learning in the low-cost case. We performed similar estimations for voters in the sequential treatments and found limited evidence of learning.

However, the Nash solution assumes voters behave perfectly rationally with no error. Given the complexity of the game they are playing, such a strong assumption seems implausible.

An alternative approach, following McKelvey and Palfrey 1995, 1998, is to consider a statistical version of Nash equilibrium where, for each actor, all possible actions have a positive probability with the probabilities ordered by the expected payoffs of the actions. The specification of these probabilities uses a quantal response function, which is a statistical version of a best response function. Of course, these “quantal” responses will also be influenced by the probability distribution of choices by the other players in the game and so on. A QRE is the fixed point of this iterative process, just as Nash equilibrium is a fixed point of the best response iteration. To simplify computations, we consider QRE of only the simplified version of the simultaneous voting game in which players choose either to vote sincerely or to abstain. The conclusions are unchanged if we use a QRE model where subjects vote against their signal with some probability. To provide parametric estimates in our analysis, we use the logit specification of QRE, where the quantal response functions are logit curves and λ is the response parameter. When λ=0, the response curves are flat and all strategies are used with equal probability, or zero rationality. As λ approaches ∞, the logit response curves converge to the best response curves, or perfect rationality. Thus, the Nash equilibrium predictions correspond to a boundary case of the QRE model.

Our estimates of the QRE for the simultaneous voting games are given in Table 2. We estimated three values of λ, one where λ is constrained to be equal across cost treatments (corresponding predicted abstention rates are given in columns 5 and 9) and two unconstrained values of λ by cost treatment, λ_H and λ_L for high- and low-cost treatments, respectively. For all rounds, using a likelihood ratio test, the difference between λ_H and λ_L is not significant at the 5% level (the χ² equals 2.9). This finding suggests that a unique parameter can explain behavior of the subjects in the simultaneous game, even though the Nash equilibria are extremely different in the high- and low-cost treatments. We find little change in the values of λ_H and λ_L over time, except for some apparent convergence toward each other (and to the constrained value). For the last 10 rounds we find the difference not significant at any conventional level (the χ² equals 1.2).

Figure 1 presents the relationship between the probability of abstaining and the equilibrium values of λ for both the low- and high-cost treatments along with the estimated values for our treatments. The two curves show the equilibrium abstention rates for each cost treatment associated with given values of λ. For λ=0, the QRE predicted abstention rates for both low- and high-cost treatments is equal 0.5. As λ increases, the equilibritum abstention rate in the low-cost treatment approaches zero, whereas the equilibrium abstention rate in the high-cost treatment approaches the Nash equilibrium prediction of 0.89. The vertical lines denote the values of λ for both the constrained and unconstrained estimations and the small circles the observed abstention rates in the treatments.

Simultaneous Voters' QRE Abstention Probabilities

Sequential Voting Choices

In the sequential voting games theory implies two types of strategic abstention: in the low-cost treatment, the equilibrium predicts that later voters will strategically abstain when they are not pivotal, voting sincerely otherwise and in the high-cost treatment, the equilibrium predicts that early voters will strategically abstain, leaving the choice for later voters. If early voters do vote, later voters will choose sincerely if pivotal, otherwise they will strategically abstain. Table 3 summarizes the aggregate abstention rates at all information sets as well as our predictions for both equilibrium and off the path behavior. We pool observations for voters with a and b signals. In the history column, “S” indicates that a previous voter voted for the alternative consistent with the current voter's signal (i.e., the same as the current voter's signal), and “D” indicates that a previous voter voted for the alternative inconsistent with the current voter's signal (i.e., different from the current voter's signal). “A” represents abstention by a previous voter. For the histories facing the third voter, the first character refers to the voting choice of the first voter with respect to the third voter's signal, and the second character refers to the voting choice of the second voter with respect to the third voter's signal. Out of 1,860 voting decisions, we observed only 27 (< 1.5%) cases where voters voted against their signal, and these were scattered randomly across the information sets. We discuss the results of the table in the reverse order of voting.

QRE Estimation

As with the simultaneous voting game, we estimate the QRE for the simplified sequential voting game (where voters either vote their signals or abstain); the results from that estimation is also presented in Table 3. We did not estimate the QRE model where voters could vote against their signal, due to computational limitations. As in the QRE estimation of the simultaneous game, the assumption is that voters use a logit response function and we solve for the QRE fixed point of the sequential game. As earlier, λ is our measure of voter responsiveness, where higher values of λ corresponding to behavior that is more consistent with perfect best responses. We report the estimate where λ is constrained to be the same for both low- and high-cost sessions as in the simultaneous voting game analysis, and also report the separate estimates. Figures 2a, 2b, and 2c display the logit equilibrium correspondences for the sequential game for both low and high cost treatments with unconstrained values of λ.¹³

The constrained λ lies between the two and is not shown on the figures.

Figure 2a displays the correspondences for the first voter; Figure 2b, for the second voter; and Figure 2c, for the fhird voter. Note that in Figures 2b and 2c, the equilibrium correspondences depend on the voter's information set with the histories defined as in Table 3 described earlier.

First Voter's QRE Abstention Probabilities

Second Voter's QRE Abstention Probabilities

Third Voter's QRE Abstention Probabilities

As in the simultaneous voting analysis, we find a lack of significant difference between λ_H and λ_L, and an apparent convergence over time. For all rounds the likelihood ratio test the χ² statistic equals 4.88 which is barely significant, but for the last 10 rounds the χ² statistic is less than 0.01. As with the simultaneous game, this fact suggests that a single value of the QRE parameter, λ, can explain behavior in quite different strategic environments. That is, just one parameter explains behavior at different nodes of the game in which subjects are in different stages of voting and information sets.

We also estimated a constrained value of λ for all the data (simultaneous and sequential, low and high cost). Because we had more observations of sequential voting for a greater number of information sets, the resulting λ was almost identical to the constrained λ for the sequential voting games, that is 0.155. A likelihood ratio test shows that the estimates for the sequential and simultaneous games are significantly different. This is not surprising, because the simultaneous and sequential game forms are completely different. The sequential game form has many different information sets and is a signaling game, which requires players to make subtle inferences from earlier player's choices. This has been observed elsewhere, even when comparing two simultaneous-move games.¹⁴

See, for example, Camerer, Palfrey, and Rogers 2006. How to endogenize λ and to characterize how it varies across games are interesting and challenging open theoretical questions.

Besides providing a much better quantitative fit to the data than the Nash equilibrium, the QRE model also makes a number of successful qualitative predictions about treatment effects, where Nash equilibrium predicts no effect at all. For the second and third voters, for any value of λ, the QRE abstention probabilities are higher in the high-cost treatment than the low-cost treatment.

Furthermore, Nash equilibrium predicts no effect at any history for the second and third voters, whereas QRE predicts such effects, independent of the value of λ. This is borne out in the data too, for the most part. For all three histories, the second voter abstains more often in the high-cost treatment (t statistic = 3.05). In fact, for the high-cost treatment, after a contradictory vote by the first voter, the second voter chooses to abstain more often than voting, which is consistent with QRE, but grossly inconsistent with the Nash prediction of always voting. The reason is that, given the actual behavior by the third voter, the second voter is actually better off abstaining than voting in that history (contrary to Nash equilibrium). For the third voter, the positive cost effect on abstention conditional on history is generally not significant, but goes in the direction predicted by QRE in 5 out of 9 histories. These directional predictions are independent of λ.

As the previous analysis shows, we can conclude the following about strategic abstention:

1. We find weak evidence of strategic abstention by early voters. First voters do abstain more under the high-cost treatment, passing the choice on to later voters, but abstain less than simultaneous voters facing the same cost. First voters respond significantly to expected utility gains from voting.
2. We find strong evidence of strategic abstention by later (third) voters when they are not pivotal and second voters “passing” on voting when first voters' choices agree with their signals.

Informational Efficiency: How Accurate are Decisions?

As noted in the introduction, we distinguish two different kinds of efficiency: informational and economic. First we consider the informational efficiency of the simultaneous and sequential voting games. Informational efficiency is simply defined as decision accuracy, without consideration for the deadweight loss of voting costs. What fraction of the time does the committee make the right decision?

To answer this question and allow comparison with a benchmark, we construct two indices of accuracy. The “optimal” voting mechanism from the standpoint of informational efficiency is a full information mechanism, where all voters always vote their signal. For the parameters of our experiment, the best the committee can do on average is to vote correctly with ex ante probability

. Conditional on the actual signal draws, the best possible decision accuracies are (.96, .75) depending on whether three or two of the committee member's signals agreed with each other, respectively. Using this as a benchmark, we compute an empirical measure of decision accuracy (DA) for each treatment and each combination of signals for both the predicted Nash equilibrium strategies and the actual strategies used in the experiment. DA is the fraction of actual decisions that match the decision that would have been made in the full information mechanism, given the committee members' actual signal draws.¹⁵

Table 4 presents comparisons of informational efficiency across treatments according to whether voting is sequential or simultaneous, by computing the difference in scores (ΔDA). In the Nash equilibrium, decision accuracy should depend on both costs and the voting mechanism. When we hold the voting mechanism constant, we expect that an increase in cost should decrease informational efficiency except when three signals agree and voting is sequential. Not surprisingly, we find significant support for this prediction in our comparisons, which are omitted from the table. When voting costs are low, both sequential and simultaneous voting should have almost the same informational efficiency (a slight difference is predicted when we combine across all signal realizations because of differences in signal realizations in the treatments), but when voting costs are high, sequential voting should provide more informational efficiency. We test the five possible comparisons for all signal configurations as well as cases broken down by the distribution of signals for 15 total comparisons (the comparisons reported and not reported). Because the probability of false significance is higher when making such multiple comparisons, we used the nonparametric procedure described by Benjamini and Hochberg (1995). Define q* as the desired minimum false discovery rate or FDR. If we rank the comparisons by their corresponding p-values, where 1 denotes the smallest and 15 the greatest and the rank is denoted by i, Benjamini and Hochberg show that rejection of only null hypotheses such that the p -value is less than

(which we label the

value in Table 4) controls the FDR at q* when the test statistics are independent. Benjamini and Hochberg (2001) further show that rejection of only null hypotheses such that the p-value is less than \smash{

} controls the FDR at q* when the tests have dependencies. We report results using both procedures in Table 4 (q*=0.05).

We find mixed results in our comparisons of sequential and simultaneous voting on informational efficiency. As expected, we find that in the low-cost case, there is no significant difference between sequential and simultaneous informational efficiency except when all three signals agree and sequential voting is slightly more efficient, although the result is only significant if we assume that the multiple tests are independent and the magnitude of the difference is very small (0.05). The effect is due to greater than equilibrium abstention by low-cost voters in simultaneous voting. However, although we expect that in the high-cost case there will be a significant difference between sequential and simultaneous informational efficiency, we again find only a significant difference when all three signals agree and only if we assume that the multiple tests are independent. This reflects the fact that high-cost voters vote more frequently than predicted in simultaneous voting. Our results suggest that informational efficiency is somewhat affected by the predicted variables but is also affected by behavioral factors that lead voters to diverge from Nash equilibrium predictions and, as we found earlier, is better explained by the quantal response model.

Economic Efficiency

We use as a benchmark in evaluating economic efficiency the total expected payoffs received by the groups. To compare the differences in economic efficiency between the sequential and simultaneous voting mechanisms with their predicted differences we calculated the predicted net expected group payoffs given the realized signals and expected Nash equilibrium behavior. Note that these are calculated before the realization of the state A or B so that any randomness in the state, conditional on signal draws, that might benefit a particular treatment, does not affect our comparisons. Furthermore, we calculated the payoffs received using the frequency of signal realizations and before the realization of the state in the same way. Finally, as with informational efficiency, we calculated the efficiency measure for the two different signal configurations (three agree and two agree). Similarly, we calculated the actual net expected group payoffs in the same fashion. Table 4 also presents the predicted differences in net expected benefits and our statistical comparisons (again we only report comparisons of voting mechanisms as the comparisons between low- and high-cost treatments are, although highly significant and in the predicted directions, not surprising). As with informational differences, we controlled for a false rejection rate of 0.05 under both the assumption that the multiple test statistics are independent and that they are dependent.

In general, the Nash predicted differences in economic efficiency are supported by the comparisons and we find stronger differences in economic efficiency between the voting mechanisms than for informational efficiency. The Nash equilibrium predictions on the effects of voting mechanism on economic efficiency are different from those with respect to informational efficiency. That is, when three signals agree, sequential voting is predicted to be more economically efficient than the simultaneous mechanism, greatly so when voting costs are high. We find significant support for these predictions. But when only two signals agree, simultaneous voting is predicted to be more economically efficient when voting costs are low and very little difference in economic efficiency by voting mechanism is predicted when voting costs are high in this situation. In our empirical analysis, we find no significant difference by voting mechanism when only two signals agree regardless of the cost of voting. Again, these divergences from the Nash equilibrium prediction support the quantal response model of voter behavior as voters vote more than predicted when voting costs are high and less than predicted in the low-cost case.

Summarizing, the two main findings about efficiency are:

1. Sequential voting is slightly more informationally and economically efficient than simultaneous voting for both high and low voting costs, but the difference is only significant when all three signals agree and, in the case of informational efficiency, if we assume that the multiple tests are independent.
2. The most informationally efficient outcomes are observed in the low-cost sequential voting game, and the least informationally efficient in the high-cost simultaneous game. The difference in efficiency between the two is estimated to be 13 percentage points across all cases, 17 percentage points when all three signals agree, and 12 percentage points when only two signals agree.