Research Master Thesis:
Reference Points and Voter Turnout:
Experimental Evidence
Oliver Herrmann, s2356503
Faculty of Economics and Business
University of Groningen
The Netherlands
October 30, 2018
Abstract
I examine the role of reference points in voting behavior as suggested by Prospect Theory, and develop an experimental protocol to change individ-uals’ reference points to either ’voting’ or ’abstaining’ through a framing manipulation that changes the salience of each reference point. I further conduct a small-scale pilot study to test the effect of the framing ma-nipulation in an experimental election. I find limited evidence that the framing manipulation changes individuals’ propensity to vote, and thus provide the first empirical verification of the role of Prospect Theory in voting behavior.
1
Introduction
Models of instrumental voting are generally regarded as insufficient to explain observed turnout in experiments and real world elections. As a consequence, behavioral theories, such as expressive behavior (Hillman, 2010) and altruism (Fowler, 2006) have been advanced. Given that these theories feature outcome-independent utility, they predict significant turnout no matter how low the probability of being pivotal in an election, thereby resolving the ’Downsian paradox’ (Downs, 1957).
terms of gains and losses relative to some reference point. In addition, individ-uals are assumed to exhibit subjective probability weighting. Herrmann, Jong-A-Pin, and Schoonbeek (2018) implement the assumptions of Prospect Theory into the standard game-theoretic model of instrumental voting of Levine and Palfrey (2007), which assumes that agents are rational expected utility maxi-mizers. The work by Herrmann, Jong-A-Pin, and Schoonbeek (2018) thus forms the theoretical basis of the predictions tested here. This companion paper can be found in Appendix C.
The authors find that the Prospect Theory model is able to explain signif-icant turnout even in very large electorates. since due to probability weight-ing, individuals overestimate the probability of being decisive so that turnout decreases more slowly as the electorate size grows. Furthermore, the model predicts that an individual’s reference point (i.e. voting or abstaining) has a considerable impact on their turnout decision. Specifically, if having ’voting’ as the reference point, individuals are much more likely to vote (up to twice as likely) compared to the alternative reference point of ’abstaining’. ”This is due to Prospect Theory’s assumption of loss aversion, so that under the reference point of voting, the election payoff (a relatively large potential loss) is weighed more heavily than voting costs (a relatively small potential gain). As a result, to change one’s strategy from voting to abstaining is less attractive compared to the alternative change of strategy when the reference point is to abstain.” (Herrmann, Jong-A-Pin, and Schoonbeek, 2018).
The prediction that reference points matter for behavior is unique to the Prospect Theory model since standard models based on expected utility max-imization do not allow for differing behavior based on how the problem is framed. Furthermore, the setup in election experiments used in Levine and Palfrey (2007) and Herrmann, Jong-A-Pin, and Schoonbeek (2018) excludes the possibility that expressive or altruistic motives influence behavior. Thus, the experimental finding that reference points influence individuals’ propensity to vote would provide strong evidence for the Prospect Theory model of voting.
reference points are based on their expectations regarding outcomes. This idea has been applied, for example, in the context of cab drivers who tend to stop working as soon as they reach some target level of income that has been found to be consistent with ex ante expected daily income (Koszegi and Rabin, 2006). There exist different but related concepts of how reference points are deter-mined. The status quo bias (also referred to as endowment bias) observed in Kahneman, Knetsch, and Thaler (1990) can be explained by Prospect Theory, with the reference point being the status quo that was assigned to experimental subjects. Similarly, some researchers have argued that choice salience is impor-tant in determining people’s reference points. Bhatia and Golman (2015) argue that the more salient an option, the more likely that it is chosen as the reference point, and further that expectations (of, say, daily cab driver income) tend to be more salient as well.
The question is which of these theories of reference point formation is the most appropriate in context of voting behavior. In a Prospect Theory model of instrumental voting, there are only 2 options for the reference point, voting or abstaining. In that sense, it is less difficult to determine the reference point than in the case of (continuous) variables like income.
Let’s consider the different theories of reference point determination in turn. First, it is hard to imagine that decision makers would be able to form an expectation of whether or not it is generally better to vote or abstain in the experimental setting of Levine and Palfrey (2007). In the absence of a clear expectation, it appears unlikely that this mechanism has a strong influence on reference point formation.
On the other hand, the status quo could play a large role: First, behavior of parents, friends and participants themselves in real world elections could con-stitute a default choice that acts as the reference point in the first round of experimental elections. Second, the choice made in previous rounds of exper-iments (which is partially influenced by a randomly drawn voting cost) could create a status quo bias as well.
Lastly, choice salience is likely to determine reference point formation in the context of the voting experiment as well. When framed neutrally (i.e. without making explicit references to choices of voting and abstaining), none of the two options should be more salient. However, in real world elections, advertisements, (social) media , and social interactions are likely to increase the salience of the voting choice, in addition to influencing other factors like expressive utility.
find evidence for the role of reference points in turnout decisions. The first approach is to check if the voting decision in previous voting rounds has an impact in the propensity to vote, after controlling for other factors such as voting costs. Consider an individual who in the first round of an experimental election draws a very low voting costs and thus decides to vote, independent of their initial reference point. Then in the subsequent round, it is more likely for this person to have the reference point of voting, given the exogenously assigned voting cost in the previous round. I refer to this as the status quo effect.
Hypothesis 1: In a regression with the decision to vote as the dependent variable, the coefficient estimate of the lag of the dependent variable is positive.
The second, and arguably more robust method is to experimentally manip-ulate the salience of the voting and abstaining options in an effort to change subjects’ reference points, and to do so in a way that does not influence other factors relevant to people’s voting decision. In other words, the way the deci-sion is ’framed’ is differed among participants, thereby manipulating the salience of either reference point. If such a framing manipulation causes differences in turnout, this would provide strong evidence that reference points matter in the decision making of voters. It is worth noting that framing experiments have a long history in the fields of Psychology and Consumer Research. Kuhberger (1998) conducts a meta-analysis of 136 experimental framing studies and finds that when framing is used to manipulate reference points, studies tend to reli-ably find treatment effects. I refer to this as the framing effect.
Hypothesis 2: A framing manipulation that increases the salience of the ’vot-ing’ (’abstain’vot-ing’) reference point has a positive (negative) impact on individu-als’ propensity to vote.
changing the salience of the ’voting’ and ’abstaining’ reference points via a framing manipulation changes individuals’ propensity to vote.
The paper proceeds as follows. In the next section, the research design is presented and discussed. Section 3 describes the data, and Section 4 details the regression analysis. Section 5 concludes.
2
Research Design
The experimental design closely follows that of Levine and Palfrey (2007), with a number of key differences in the experimental instructions.
The general set up is as follows: There is an odd number of participants in the experiment. They are then randomly divided into either group A or group B, with NA = NB + 1 describing the group sizes. Next, an election is held
to determine which of the two group’s members get the higher election payoff of 105. The losing group only gets a payoff of 5. There is a voting cost that is randomly drawn for each individual. These voting costs are drawn from a uniform distribution in the range of 0 to 55. Each individual is informed of their group assignment and voting cost and decides whether to vote or abstain, and subsequently which group to vote for.
The main difference in the experimental setup compared to Levine and Pal-frey (2007) is the way in which the individual’s decision to vote or abstain is framed. In previous work the choice has been presented in a neutral manner, without mention of words such as ’election’, ’vote’ or ’abstain’. In this research on the other hand, I explicitly describe the experiment as an election and fur-ther denote the choice as one to vote or abstain. Based on standard models of instrumental voting, where people are assumed to be expected utility maxi-mizers, this difference in wording should not make a difference. But even based on behavioral models such as expressive voting, the wording should be incon-sequential since there is no ’citizen duty’ to vote and no expressive utility in terms of ideology that can be obtained.
Prospect Theory, however, does predict that the way the decision is framed will impact individuals’ propensity to vote. Thus, in this research, I use a framing manipulation to elicit either the voting or abstaining reference point.
would like to abstain. As described in the introduction, this manipulation changes the salience of the two options and in this way changes the likelihood that participants will adopt either reference point. I consider this to be a clean and unambiguous way of eliciting different reference points. Alternative manip-ulation approaches were rejected due to their ability to introduce confounding motivations. For example, one could ask participants to reflect on their past voting behavior, or show them pictures of people in voting booths versus neutral images. But for both these alternative approaches, considerations such as civic duty and social pressure may confound the results. The simple manipulation of differently stating the problem does not raise these concerns. To further elicit associations with real life voting, the experiment was conducted in paper form rather than on computers.
There were a total of three different sessions, with the two last sessions including revisions in experimental protocol and instructions. In what follows here, I will describe the structure of these experimental sessions.
The structure of the main experimental session was as follows. First, partic-ipants were seated apart in a large room and group instructions (see Appendix A) were distributed and read out loud by the experimenter. As part of the instruction, participants were told not to communicate with each other from that point. Next, participants were allowed to ask questions about the exper-iment. This was followed by participants filling out six short questions that tested their understanding of the setup, and the answers being discussed by the experimenter.
Subsequently, the ’choice sheets’ were distributed, which detail the indi-vidual participant’s voting cost and group membership. These numbers were randomly generated prior to the experiment. There were two types of ’choice sheets’ (see Appendix B): one version asked the question of whether a partic-ipant wants to vote, whereas the other version asked if they want to abstain. This treatment stayed constant over all subsequent rounds of elections. That is, participants faced the same reference point manipulation in each round played. In the main experiment, a total of 9 rounds were played. Importantly, due to time constraints, election results were not announced at the end of each round. As a consequence, the impact of learning on outcomes over several rounds is limited.
participants to act optimally and truthfully, they were informed that their per-formance would form a small part of their assignment grade for that course. In the first experiment on the other hand, the participants were advised in advance that they would not be remunerated, but asked to act as if they were to receive the rewards described in the instructions. In the final experimental session, however, participants did receive the rewards of the experiment in monetary form and were informed of this fact in advance and during the instructions.
After all rounds had been played a survey was distributed that includes basic demographic information, such as age, gender, nationality and field of study. Finally, participants were debriefed.
3
Data
There were a total of three experimental sessions conducted. The number of participants varied per session, as did their study background. Table 1 summa-rizes these differences. The main experiment (session 2) with 21 participants was performed in a class at the University College Groningen, whereas the other sessions were held with PhD and Master level students from the Faculty of Eco-nomics and Business at the University of Groningen.
There is a number of differences between sessions 1 and subsequent ones in terms of the experimental setup. First, election results were announced only in session 1, but due to time constraints, not in subsequent sessions. Second, group assignment in the first session was fixed per participant over all rounds played, whereas in the other sessions group assignment was randomly assigned each round to avoid the possibility of strategic behavior such as retaliation. Third, there were minor differences in the way the instructions described the election setting. These instructions were made more clear and easier to understand for session 2 and later. Lastly, there was no survey conducted in the first experimental round, which limits the sample size for which demographic data is available.
Experimental session 1 2 3 Number participants 7 21 5 Study 6 PhD, 1 MSc BSc MSc Election result announced Yes No No
Rounds played 5 9 7
Group assignment changes No Yes Yes Table 1: Overview of experimental sessions
4
Regression analysis
I conduct a regression analysis to test the hypotheses outlined in the introduc-tion. The first thing to note is that treatment is randomly assigned between individuals. However, due to the relatively small number of participants, ran-domization may not be completely successful. To check for this, I calculate summary statistics for each treatment group, given in Table 2. As can be seen, despite randomization, there are sizable differences in term of voting cost and group assignment between treatment groups. While these two confounders can be controlled for in a regression framework, there may in principle be omitted variables that differ despite randomization and thus introduce bias. However, these individual level differences such as differing risk preferences would likely be time-invariant and thus be controlled for in a panel regression with random effects.
Treatment Voting cost group A Age Frequency Voting 27.72 .4872 20.1 117 Abstaining 23.75 .5634 20.5 126
Table 2: Summary statistics per treatment group
I perform a number of different regressions to test the hypotheses developed earlier. First, I run a panel logistic regression with cluster robust standard errors to account for possible autocorrelation in individual’s choices. See equation (1). A random effects specification was used which assumes that any omitted variables are uncorrelated with the included regressors. Arguably, this is a realistic assumption since potential omitted variables such as a person’s risk preferences are uncorrelated with the randomly assigned regressors voting cost and group assignment.
voted in a given round, and 0 otherwise. The main regressor is the treatment variable Vote mani, which takes on value 1 if an individual received the voting reference point treatment and 0 otherwise. In a linear regression, the coeffi-cient estimate hence describes how much higher turnout is for those individuals who received the voting reference point treatment versus the abstaining refer-ence point treatment. The control variables used are the voting cost, group membership, and the size of the experiment the individual took part in.
choiceit= α + β1V ote manii+ β2costit+ β3grpAit+ β4ex sizei+ ci+ (1)
Model (2) is identical to equation (1), other than including a lag of the dependent variable in order to test hypothesis 1. Given the loss of sample size when including lags, these equations were estimated separately.
Next, model (3) includes the control variables age and male, where the latter is a dummy taking on value one if the subject is a male, and 0 otherwise. As explained earlier, there was no survey conducted in the first experimental session, so that the sample size when controlling for these demographic variables is smaller. To not further restrict sample size, and due to the non-significance of the lagged dependent variable, it is excluded from this specification.
Model (4) is identical to model (1), but is instead estimated using a linear probability model to facilitate the interpretation of coefficient estimates.
Lastly, equation (5) is estimated using logistic regression on the pooled data. I run this regression only as a robustness check, since as pointed out earlier, omitted variable bias may be present so that a panel random effects approach is a better choice to estimate equation (1).
Table 3 summarizes the results. The first thing to note is that the results are similar across the different specifications. The estimates for voting cost and the group dummy are significant at the 1 percent level in all specifications. For ease of interpretation, consider the random effects linear probability model (4). Given that the dependent variable is a dummy for whether or not someone voted, it can be interpreted as the probability to cast a vote, or put differently, as the propensity to vote.
(1) (2) (3) (4) (5) VARIABLES choice choice choice choice choice Vote mani 0.908 0.824 1.030 0.102 0.684 (0.610) (0.674) (0.640) (0.0700) (0.489) cost -0.137*** -0.139*** -0.142*** -0.0189*** -0.106*** (0.0286) (0.0329) (0.0319) (0.00200) (0.0178) groupA 1.765*** 1.594*** 1.928*** 0.210*** 1.448*** (0.431) (0.459) (0.452) (0.0615) (0.365) exp size 0.0556 0.0603 0.150 0.00605 0.0506 (0.0486) (0.0532) (0.114) (0.00594) (0.0387) male 0.509 (0.683) Age 0.147 (0.251) L.choice -0.389 (0.3658) Constant 1.611 1.963 -3.396 0.779*** 1.086 (1.138) (1.310) (6.660) (0.149) (0.900) Observations 243 212 208 243 243 Number of i 31 31 24 31
Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1
Table 3: Regression results
of course, in line with theoretical predictions: the higher the voting cost, the lower the expected net benefit of voting.
Being a member of group A rather than group B increases the propensity to vote by 21 percentage points, ceteris paribus. That is, turnout in the larger group is higher, which is not in line with the theoretical predictions of the standard and Prospect Theory model, which both say that the voting propensity in the larger group should be relatively smaller. This is also a result found in the experiment conducted by Levine and Palfrey (2007). This discrepancy between the results of my experiment and that of Levine and Palfrey (2007) can be explained by the fact that election results for the experiments conducted by me were not announced in most of the sample. Furthermore, less rounds per individual were played compared to Levine and Palfrey (2007). Hence, the opportunity for learning is limited.
free-riders once they observe after several rounds that their group keeps winning. Indeed, in my experiment, the larger group A won 17/21 rounds, whereas the theoretical models would predict it to be closer to even odds for both groups. Thus, I conclude that learning effects are very important in reaching the the-oretical equilibrium. The limited learning opportunity should have no impact on the effect of the reference point manipulation. However, if individuals are allowed to learn, they may make less mistakes (such as to abstain when the voting cost is extremely low) and play more consistently. This could result in reduced unexplained variance, and thereby make it easier to reach statistical significance in the coefficient estimates of the variables of interest.
We now turn to the hypotheses developed earlier. First, recall hypothesis 1, which states that the coefficient estimate of the lag of the dependent variable is positive. That is, if a person voted in the last round, then due to a status quo effect, their reference point is more likely to be ’voting’, resulting in a higher propensity to vote. However, as can be seen in column (2) of Table 3, there is no statistically significant effect (p-value 0.288) of the lag of choice. We therefore do not find evidence for hypothesis 1.
5
Conclusion
Prospect Theory predicts that an individual’s propensity to vote depends on their reference point. That is, the way people weigh costs and benefits of voting and abstaining depends on the way they frame the problem in their mind. In this paper, I develop an experimental protocol to test this prediction. I use the concept of reference point salience to argue how reference points could be manipulated in the context of turnout decisions in elections. Namely, by manip-ulating how the choice between voting and abstaining is framed. In addition, I argue for a way in which status quo bias could affect reference points in repeated experimental elections and how this can be detected in a regression framework. Furthermore, I undertook a pilot study with limited sample size to refine the experimental design as well as to obtain preliminary results. Due to the small scale of the pilot experiment, the coefficients are estimated with relative wide standard errors. Even though the estimated coefficients have the expected sign, they are insignificant at conventional significance levels. In a (future) large scale experiment, the hypothesized effect can be addressed more thoroughly, which will give a more conclusive answer to the research question.
6
References
Barberis, N.C. (2013). Thirty Years of Prospect Theory in Economics: A Re-view and Assessment. Journal of Economic Perspectives, vol. 27, pp. 173-196. Bhatia, S. and Golman, R. (2013). Attention and reference dependence. Work-ing paper.
Downs, A. (1957). An Economic Theory of Democracy. New York: Harper & Brothers.
Fowler, J.H. (2006). Altruism and Turnout. The Journal of Politics, vol. 68, pp. 674-683.
Herrmann, O.M., Jong-A-Pin, R. and Schoonbeek, L. 2018. Prospect Theory peferences and voter participation. Mimeo.
Hillman, A.L. (2010). European Journal of Political Economy, vol. 26, pp. 403-418.
Kahneman, D., Knetsch J.L. and Thaler, R.H. (1990). Experimental Tests of the Endowment Effect and the Coase Theorem. Journal of Political Economy, vol. 98, pp. 1325-1348.
Kahneman, D. and Tversky, A. (1976). Prospect Theory: An Analysis of Deci-sion under Risk. Econometrica, vol. 47, pp. 263-292.
Kszegi, B. and Rabin, M. (2006). A Model of Reference-Dependent Preferences. The Quarterly Journal of Economics, vol. 121, pp. 11331165.
Kuhberger, A. (1998). The Influence of Framing on Risky Decisions: A Meta-analysis. Organizational Behavior and Human Decision Processes, vol. 75, pp. 23-55.
Appendix A
Instructions
Instructions
Thank you for participating in this experiment. Please read these instructions carefully. If you need clarification or explanation, ask the experimenter(s). Do not communicate with each other from now until the experiment ends. This is economic research; we are interested in the decisions you make under the conditions we set.
General
This is an election experiment, in which your payoff depends on both your individual decisions and the decisions of other participants. Please follow the instructions carefully when making your decisions. Your attendance earns you 4.50 (the show-up fee). Your final earnings are somewhere between 4.5 and 15, depending on the choices made in the experiment. Your earnings are unknown to the other participants. The money will be transferred to your bank account within 1 week.
Explanation of the election
Before each round of elections, participants are assigned to either group A or group B. Group A has NA members and Group B has NB members, where
NA = NB+ 1. Next, an election will be held which decides which of the two
groups gets the higher election payoff. The group with the most votes has won the election and each member gets the higher payoff of 105 ECU (1 ECU = 0.01EUR). The other groups members receive a lower payoff of 5 ECU. Ties are broken with a coin flip.
Each individual has to decide if they want to vote for their group or if they want to abstain (i.e. not vote). The ballot is secret, so an individuals decision is unknown to the other participants.
(uniform distribution). Note that you have unlimited credit, so you can always afford to pay the voting cost.
In total there will be 10 rounds of elections. Each time, you draw a different random voting cost and each time you are randomly assigned to either group A or B. That is, your group membership and voting cost are variable and need to be considered before making your choice in a given round.
Hence, please read carefully each round the information and choice presented to you on the ballot. In addition, please denote your seat number on each ballot. You have 1 minute to make your choice per round. Once you completed the ballot, please turn it face down until collected.
Due to time constraints, the election results for each round will not be an-nounced. Instead, the election outcome for each round will be determined after the experiment.
In summary
- There are 2 groups, A and B. An election is held to determine who gets the higher payoff.
- You have to decide to vote or abstain, given your voting costs and the group you belong to for that round
- The outcome of the election depends both on your decision and the decision of other participants.
- You will get an election payoff that depends on whether your group won or lost the election. If you decided to abstain, then you dont pay any costs. If you decided to vote, then the voting cost you draw that round will be subtracted from the election payoff.
- You receive the sum of payoffs of all 10 rounds at the end of the experiment, in addition to the show up fee.
- You have unlimited credit, so you can always afford to pay the voting cost.
Test questions
- How many members are in group A, how many in group B? - Do you have to pay the voting cost if you abstain?
- What are the payoffs for the winner and loser of the election? - What is the range of possible voting costs?
Appendix B
Appendix C
Prospect-theory preferences and voter
participation
Oliver Herrmann, Richard Jong-A-Pin, Lambert Schoonbeek
Department of Economics, Econometrics and Finance
Faculty of Economics and Business
University of Groningen
The Netherlands
October 30, 2018
Abstract
We study prospect-theoretic risk preferences to address the Downsian paradox of voting. Using a game-theoretic rational choice model, we find that prospect-theoretic risk preferences predict higher voter turnout dur-ing elections than standard risk preferences. We find that voter turnout is driven by agents’ reference point with respect to the vote or abstain decision. We also study the role of probability weighting in the deci-sion making process and find that under this assumption voter turnout decreases less quickly (relative to other models) as the electorate grows.
1
Introduction
The seminal work of Downs (1957) states that rational voters will only partici-pate in elections when the expected benefits of voting exceed the costs. As such, its prediction is that in large elections, when the probability of being decisive is close to 0, we should observe large scale abstention. However, in practice, we do observe significant turnout in large scale elections. The discrepancy between theory and practice has become (in)famous as the ”paradox that ate rational choice theory” (Fiorina, 1990).
2010). Alternatively, and much related to the Expressive voting argument, Fowler (2006) argues that altruism explains turnout during elections, whereas Funk (2010) finds evidence in favor of social incentives. Ferejohn and Fiorina (1974) take on a different argument and argue that voters consider the cost of regret in case of being decisive in elections.
Apart from these behavioral explanations, a number of studies have focused on the role of strategic behavior in the voting decision. Palfrey and Rosenthal (1983), for example, argue that the probability of being decisive is not exogenous and model the voting decision accordingly. Palfrey and Rosenthal (1985) extend that model and incorporate the role of asymmetric information and heteroge-neous voting costs. Levine and Palfrey (2007) also assume bounded rationality when they model the strategic decision to vote leading to model outcomes with higher turnout.
In this paper we address the Downsian paradox by studying voters’ risk pref-erences. Standard models of rational choice, both game-theoretic and decision-theoretic, assume that voters are risk neutral expected utility (EU) maximizers in that they only care about the expected value of net payoffs. While this ’as if’-type of assumption has the benefit of simplifying analyses, experimental re-search in psychology and economics has shown that, most often, this is a poor description of actual human behavior. An alternative, but more realistic, way of formalizing risk preferences is suggested by Kahneman and Tversky (1976) and Kahneman and Tversky (1992). Their Prospect Theory states that individuals evaluate options based on gains and losses relative to a reference point and, ad-ditionally, that agents evaluate probabilities based on non-standard weighting functions.
voters.
To preview our main findings, we find that our model provides new insight in the voting decision individuals make. In particular, we find that the reference point of an individual has a particularly large impact on the propensity to vote. That is, individuals who have voting as their reference point have a much higher likelihood to vote than those that have abstention as their reference point. This is due to Prospect Theory’s assumption of loss aversion, so that under the reference point of voting, the election payoff (a relatively large potential loss) is weighed more heavily than voting costs (a relatively small potential gain). As a result, to change one’s strategy from voting to abstaining is less attractive compared to the alternative change of strategy when the reference point is to abstain.
Furthermore, we find that Prospect Theory’s assumption of probability weight-ing (as opposed to the assumption of voters usweight-ing objective probabilities) leads turnout to decrease at a slower pace, consequently leading to higher turnout pre-dictions in (very) large electorates. This is because with probability weighting, individuals are assumed to overweight small probabilities and to underweight large probabilities. As a consequence, the chance of being pivotal in large elec-tions is overestimated and turnout is increased.
2
Related literature
In this section we review the existing literature on explaining voter turnout. The existing approaches can be summarized as follows. First, outcome-independent utility could be allowed for in the voter’s utility function. That is, rather than only obtain utility from changing the election outcome, voters may receive a consumption-like utility from the act of voting itself. This could include a ’warm glow’ from doing one’s civic duty of voting, or some form of expressive utility that confirms the voter’s identity as, for example, a conservative or demo-crat. Wiese and Jong-A-Pin (2017) find empirical support for expressive voting behavior in a laboratory experiment and observe that the expressive behavior increases with the size of the electorate. This makes sense intuitively since with small electorate sizes, the probability of being decisive is non-negligible, so that the instrumental aspect of voting can be expected to matter more. Thus, the ex-pressive voting approach is able to explain high turnout in large elections while maintaining the comparative statics predictions of instrumental voting models in small elections.
Alternatively, one can allow for altruistic behavior, such that people take the welfare consequences of other individuals into account. In that case, while the probability of being decisive decreases with electorate size, so does the potential payoff (Fowler, 2006), since the altruistic agent obtains incremental utility with every additional voter joining the electorate.
Ferejohn and Fiorina’s (1974) ’minimax regret’ model simply assumes that probabilities do not matter at all in the voters’ decision process. Instead, voters are assumed to act so as to avoid the regret felt had they voted and been decisive. This model predicts that participation would be the same independent of electorate size, and could thus explain why significant turnout is observed in the real world. However, the approach has been heavily criticized on theoretical grounds, and numerous observational and experimental studies indicate that probabilities do matter for turnout.
behavior in a game-theoretic rather than theoretic model. A decision-theoretic model of voting considers the decision process of an arbitrary individ-ual among the electorate under the ceteris paribus assumption. That is, the behavior of all other people in the electorate (and thus the probability of being decisive) is taken as given. However, as Palfrey and Rosenthal (1983) point out, ”the voting probabilities and the turnout decisions are simultaneously de-termined”. In other words, the probability of being decisive is not exogenous, but also determined by one’s own strategy and others’ beliefs about it. Pal-frey and Rosenthal (1983) set up a game-theoretic model of voting, where every individual is assumed to have the same voting cost. They show that for large electorates, there exist equilibria with significant participation.
As a refinement of the above model, Palfrey and Rosenthal (1985) make the -arguably more realistic- assumptions of heterogeneous voting costs and imperfect information about others’ voting costs. This adaptation of the model gets rid of the multiple equilibria problem that the previous model suffered. However, the unique equilibrium in turnout tends to zero as the electorate grows large. Given this prediction of negligible turnout in large elections together with the model’s underlying assumption of rational voters, this leaves us right back with the paradox of voting.
Levine and Palfrey (2007) consider the same model with heterogeneous vot-ing costs, but rely on a different equilibrium concept to derive a prediction about voter behavior, namely that of Quantal Response Equilibrium (QRE). Under-lying this solution concept is that voters are only boundedly rational. More precisely, individuals are assumed to make unsystematic random errors when choosing the optimal strategy, where the likelihood of making a specific mistake is decreasing with the cost of making that mistake. Allowing for these sort of mistakes on the part of voters leads to higher predicted turnout, and the QRE model still predicts significant turnout for arbitrarily large electorates.
better on such a manageable scale. In their paper, Levine and Palfrey (2007) also conduct an experiment in which they assign up to a total of 51 individuals to either of two groups and allow them to vote or abstain on the distribution of payoffs. While the comparative statics predictions of their model are broadly confirmed, compared to the model’s predictions, there is significant under-voting for small electorate sizes and over-voting for larger ones. On first sight one may blame this failure on some of the other factors affecting voters, namely expres-sive and altruistic behavior. However, the experimental setup in Levine and Palfrey (2007) is such that we can rule out these factors, so there must be some-thing else. Furthermore, as we shall see the model not only has shortcomings in explaining aggregate outcomes, but it also fails in explaining the behavior of individuals. In a sense, the failure of rational choice models of voting to ac-curately predict the magnitude of turnout in small-scale experimental elections could be considered a paradox in its own right.
The adapted model is able to explain the apparent ’over-voting’ and individual-level deviations noted in Levine and Palfrey (2007). However, the model is not able to predict the precise patterns of voting for specific group sizes, underesti-mating the gap in participation between groups of differing sizes. We argue that this shortcoming may be remedied by a more involved model that endogenizes voters’ reference points. Furthermore, even though the model can predict much larger turnout in large elections than the standard model, turnout still converges to zero as the electorate grows arbitrarily large. It should therefore be consid-ered as complementary to other approaches mentioned in the introduction, such as expressive behavior.
3
The standard model
We consider individuals who belong to group A or group B. Group A and group B have, respectively, NA and NB members, with NB > NA ≥ 1. Note that
All members of group A prefer candidate A, the members of group B prefer candidate B. Whichever candidate has more votes has won the election, with ties being broken by the flip of a fair coin. We say that a group wins the election if the candidate preferred by this group wins the election. If a group wins, each member of the group receives a reward of (money) value H, while the members of the losing group each receive a reward L, with H > L ≥ 0. An individual i who votes incurs a voting cost ci. The value of ciis drawn by Nature, using the
density function f (ci), which is positive everywhere on its support [0, ¯c], with
¯
c > 0. The corresponding distribution function is denoted by F (·). We assume that H − L > ¯c. Each individual is privately informed about the size of her own voting cost before deciding whether to abstain or vote, but only knows that the voting costs of the other individuals are drawn with density function f (·). The voting costs of the different individuals are drawn independently. This setup is common knowledge.
Let us first consider the standard model of Levine and Palfrey (2007). We denote the number of voters excluding individual i that vote for the candidate of group A or group B by n−iA and n−iB , respectively. Tables 1 and 2 describe the expected payoffs of different outcomes for members of the two groups. The payoffs depend both on the action chosen by the individual herself, as well as on the actions of all other individuals. The payoff matrix is unique for each individual, since ci is drawn separately for every individual.
Table 1: Expected payoff matrix for individual i of group A. Vote Abstain n−iA > n−iB + 1 H − ci H n−iA = n−iB + 1 H − ci H n−iA = n−iB H − ci H+L2 n−iA = n−iB − 1 H+L 2 − ci L n−iA < n−iB − 1 L − ci L
Table 2: Expected payoff matrix for individual i of group B. Vote Abstain n−iA > n−iB + 1 L − ci L n−iA = n−iB + 1 H+L2 L n−iA = n−iB H − ci H+L2 n−iA = n−iB − 1 H − ci H n−iA < n−iB − 1 H − ci H
(Bayesian-Nash) equilibrium in which all members of a group employ the same strategy.1 We denote equilibrium values with superindex ∗. In a quasi-symmetric equilibrium, each individual i of group A uses a cut-point strategy such that she votes if and only if ci < c∗A, where the equilibrium threshold c∗A is shared
by everyone in her group. If this individual votes, we say that τA(ci) = 1, else
τA(ci) = 0. Similarly, and using obvious notation, every individual i of group
B votes (so that τB(ci) = 1) if and only if ci < c∗B. Thus, the equilibrium is
described by the pair of thresholds (c∗A, c∗B).2
Given that everybody in a group has the same cut-point strategy, the equi-librium aggregate voting probabilities of the two groups, (p∗A, p∗B), are
p∗A = Z ¯c 0 τA(c)f (c)dc = Z c∗A 0 f (c)dc = F (c∗A), (1) p∗B = Z ¯c 0 τB(c)f (c)dc = Z c∗B 0 f (c)dc = F (c∗B). (2) Given these voting probabilities for each group, we can calculate the following
1OLIVER: are there any other equilibria? 2We assume that c∗
A< ¯c and c ∗
probabilities that each voter uses to calculate the benefit of casting her vote: PA,break∗ = P rob(voter in group A breaks a tie) (3)
= NA−1 X k=0 NA− 1 k NB k (p∗A)k(1 − p∗A)NA−1−k(p∗ B) k(1 − p∗ B) NB−k,
PA,create∗ = P rob(voter in group A creates a tie) (4) = NA−1 X k=0 NA− 1 k NB k + 1 (p∗A)k(1 − p∗A)NA−1−k(p∗ B) k+1(1 − p∗ B) NB−1−k,
PB,break∗ = P rob(voter in group B breaks a tie) (5) = NA X k=0 NA k NB− 1 k (p∗A)k(1 − p∗A)NA−k(p∗B)k(1 − p∗B)NB−1−k,
PB,create∗ = P rob(voter in group B creates a tie) (6)
= NA−1 X k=0 NA k + 1 NB− 1 k (p∗A)k+1(1 − p∗A)NA−1−k(p∗ B) k(1 − p∗ B) NB−1−k.
In equilibrium, individuals in groups A and B are indifferent between abstaining and voting if and only if the expected benefit of voting equals the cost of voting, i.e. we have PA,break∗ × H −H + L 2 + PA,create∗ × H + L 2 − L = c∗A, (7) PB,break∗ × H −H + L 2 + PB,create∗ × H + L 2 − L = c∗B. (8) The equilibrium pair of thresholds (c∗A, c∗B) simultaneously solves (1) − (8), and in turn yields the equilibrium aggregate voting probabilities for each group.
For later use, we also define
PA∗ = PA,break∗ + PA,create∗ , (9)
PB∗ = PB,break∗ + PB,create∗ . (10)
4
Prospect-theory model
reference point. In our setup the reference point can be to either abstain or to vote. A priori it is not clear which reference point is used. We therefore look at both cases in our prospect-theory model (PT-model).
Tables 3 and 4 detail, for individuals of group A, the state-dependent gains and losses, respectively, when deviating from one’s reference point. Since H − L > ¯c, payoffs can be unambiguously determined as either gains or losses. Each individual in the group receives an independent draw from the distribution of voting costs. Hence, the gains and loss matrices are unique for each individual i in their respective group.
Consider individual i of group A, who has the reference point of abstaining (see the second column in Tables 3 and 4). She evaluates the alternative action of voting in terms of the state-dependent gains and losses that result from that action. If group A would lead in the election by more than one vote before taking into account individual i (see the third row in Tables 3 and 4), then her action of voting would not result in any gain received by her group winning the election. Yet, she would incur the individual voting cost ci, so that a net loss
would result. The same calculation is performed for the other four states of the world. Note that (H − L − ci,12; 0,12) in Table 3 denotes that the gain equals
H − L − ciwith probability 12, and 0 with probability 12. The entry (0,12; ci,12)
in the third column of this table, where voting is the reference point, can be interpreted analogously. The entries in Table 4 can be interpreted in a similar way, but then in terms of losses. For brevity, we omit here the gains and loss matrices of individuals of group B.
Let us return to the case with abstaining as the reference point. Take indi-vidual i of group A and suppose that she votes. Tables 3 and 4 imply that she will gain H − L − ci with probability PA∗/2 and incur a loss ci with probability
1 − PA∗/2, with PA∗ given by (9). Following Tversky and Kahneman (1992)3,
individual i attaches the following value to voting in the PT-model: VA(ci) = π+ P∗ A 2 × u(H − L − ci) + π− 1 −P ∗ A 2 × u(−ci). (11)
Table 3: Gains matrix for individual i of group A. Reference point: Abstain Vote
Strategy: Vote Abstain
n−iA > n−iB + 1 0 ci
n−iA = n−iB + 1 0 ci
n−iA = n−iB (H − L − ci,12; 0,12) (0,12; ci,12)
n−iA = n−iB − 1 (H − L − ci,12; 0,12) (0,12; ci,12)
n−iA < n−iB − 1 0 ci
Table 4: Loss matrix for individual i of group A. Reference point: Abstain Vote
Strategy: Vote Abstain
n−iA > n−iB + 1 ci 0
n−iA = n−iB + 1 ci 0
n−iA = n−iB (0,12; ci,12) (H − L − ci,12; 0,12)
n−iA = n−iB − 1 (0,12; ci,12) (H − L − ci,12; 0,12)
n−iA < n−iB − 1 ci 0
Here the utility function for outcome x is given by u(x) =
xα x ≥ 0,
−λ(−x)β x < 0, (12)
with 0 < α < 1, 0 < β < 1 and λ > 1. We have a gain if x > 0 and a loss if x < 0. The probability weighting functions for the objective probability p are given by π+(p) = p γ [pγ+ (1 − p)γ]γ1 , (13) π−(p) = p δ [pδ+ (1 − p)δ]1δ, (14) with 0 < γ < 1 and 0 < δ < 1.
(here to abstain or to vote), so that the same action may be assigned a different utility value for distinct reference points; (ii) diminishing sensitivity, i.e. the utility function is concave in gains and convex in losses, which implies that voters are risk-averse in gains and risk-loving in losses; (iii) loss aversion, i.e. the increase in utility from gaining a given payoff is smaller than the decrease in utility from a loss of the same size. The individual evaluates the utility for each outcome and calculates the corresponding expected value given her subjective probabilities. The subjective probabilities are given by the weighting functions π+(·) and π−(·) (associated with gains and losses, respectively), which
depend on the objective probabilities of each state. The weighting functions imply that small probabilities are overestimated, while large probabilities are underestimated. We allow that the weighting functions are different for gains and losses. In the special case with γ = δ = 1, the individual calculates the standard expected value based on objective probabilities.
Next, consider the case where voting is the reference point, and suppose that individual i of group A abstains from voting. Using Tables 3 and 4, we see that her value of voting is then given by
WA(ci) = π+ 1 −P ∗ A 2 × u(ci) + π− P∗ A 2 × u(−(H − L − ci)). (15)
We can define the values VB(ci) and WB(ci) for individual i of group B in
the same manner, by replacing PA∗ with PB∗, defined by (10), in (11) and (15), respectively. Note that we assume that all individuals of both groups have the same utility function and probability weighting functions.
Turning to the quasi-symmetric equilibrium of the PT-model, we use Ta-bles 3 and 4 in order to determine the thresholds of the voting costs in the cut-point strategies used by the members of each group. First, assume that all voters use abstaining as their reference point. Then in order to be indifferent between switching or not switching to voting, the value of voting must equal zero. The equilibrium pair of thresholds of the voting costs, (c∗A, c∗B), simultane-ously solves equations (1) - (6), (9), (10), VA(c∗A) = 0 and VB(c∗B) = 0.
4 Second,
assume that voting is the reference point. The equilibrium pair (c∗A, c∗B) then si-multaneously solves equations (1) - (6), (9), (10), WA(c∗A) = 0 and WB(c∗B) = 0.
In the special case with α = β = λ = 1 and γ = δ = 1, the equilibrium of the PT-model coincides with the equilibrium of the standard model of Section 3.
5
Comparison of turnouts
We want to compare numerically the turnouts predicted by the standard model and PT-model, given by their equilibrium aggregate voting probabilities. We also compare these predicted turnouts with the observed turnouts in the exper-iment of Levine and Palfrey (2007).5 Following Levine and Palfrey (2007), we
look at cases where NA = NB− 1 or NA = NB/2, H = 105 and L = 5, and
the voting cost ci is uniformly distributed on [0, 55].6 We set the preference
pa-rameter values in the PT-model equal to the median estimates of experimental studies.7 Following Abdellaoui (2000), we take α = 0.89, β = 0.92, γ = 0.6 and δ = 0.7. These values are very close to those of Tversky and Kahneman (1992). Using Kahneman and Tversky (1979), we set λ = 1.69.
Table 5 gives the turnouts observed in the experiment of Levine and Palfrey (2007, Table 2) and the corresponding turnouts predicted by the standard model of Section 3. The first seven rows with results represent the group sizes used by Levine and Palfrey, while the last row shows results for larger group sizes. We see that, except for two cases, the predicted turnouts are smaller than the observed turnouts. Hence, the standard model almost always underestimates voters’ turnout. Further, except for NA= 1 and NB= 2, the predicted turnouts
are smaller for group A than for group B, which is always true for the observed turnouts. As a simple measure of the fit between the predicted and observed turnouts, we calculate for each group across the first seven rows with results of Table 5 the average of the absolute differences between the predicted turnouts
superindex *. The context makes clear to which model the equilibrium applies.
5Calculations were performed using the program ’Mathematica’. The code is available upon request.
6The appendix reports results for larger group sizes and larger values of H as a robustness check, which yields similar ??? results.
and corresponding observed turnouts. We call these the average prediction errors associated with the standard model. They amount to 0.071 for group A, and 0.061 for group B.
Tables 6 and 7 detail predicted turnouts for the PT-model of Section 4. We also examine results for a special case of this model, referred to as PT1-model, in which both probability weighting functions are equal to the identity function (i.e. δ = γ = 1), and voters thus use the objective probabilities to calculate expected values.
Table 5: Turnouts in the experiment of Levine and Palfrey (2007), and predicted turnouts in the standard model, for groups A and B and different group sizes.
experiment standard model NA NB group A group B p∗A p ∗ B 1 2 0.539 0.573 0.537 0.640 3 6 0.436 0.398 0.413 0.374 4 5 0.479 0.451 0.460 0.452 9 18 0.377 0.282 0.270 0.228 13 14 0.385 0.356 0.302 0.297 17 34 0.333 0.266 0.206 0.171 25 26 0.390 0.362 0.238 0.235 100 101 - - 0.145 0.145
Table 6: Predicted turnouts in the PT-model and PT1-model, with abstaining as the reference point, for groups A and B and different group sizes.
Table 7: Predicted turnouts in the PT-model and PT1-model, with voting as the reference point, for groups A and B and different group sizes.
PT-model PT1-model NA NB p∗A p∗B p∗A p∗B 1 2 0.465 0.916 0.496 0.802 3 6 0.556 0.608 0.488 0.478 4 5 0.651 0.815 0.592 0.650 9 18 0.424 0.399 0.334 0.287 13 14 0.670 0.696 0.425 0.421 17 34 0.338 0.304 0.254 0.211 25 26 0.568 0.571 0.327 0.325 100 101 0.374 0.373 0.191 0.191
the number of voters. Hence, while initially reference dependence, diminishing sensitivity and loss aversion offset the impact of probability weighting, the latter has a higher impact when the group size gets larger. For large group sizes (100+), the PT-model shows slightly higher turnouts than the standard model. Notice that the PT-model has the desirable quality that turnout in group A will be larger than in group B.
In Table 7 the reference point is voting. In this situation, the PT1-model shows that reference dependence, diminishing sensitivity and loss aversion lead, except for one case, to higher predicted turnout than in the standard model. This is due to the fact that, if members of group A abstain in this situation, then opposite to the previous case, c∗A is considered as a potential gain and H − L − c∗A as a potential loss. The same holds for group B. Moving to the PT-model, we see that the effect of probability weighting is to further increase turnout. The intuition behind this result is that the probability of receiving a gain from abstaining is now underweighted, while the probability of a loss is overweighted. The attractive feature of the PT-model with the reference point voting is that it leads to higher turnout than the standard model. In fact, the predicted turnouts are higher now (except for one case) than the turnouts observed in the experiment of Levine and Palfrey (2007). Also note that for the PT-model in most cases the turnout of group A is smaller than that of group B.
In the next section we investigate the case with a mixture of reference points in the population. This is a natural extension, given that neither reference point is an objectively better of more rational choice.
6
Allowing for a mixture of reference points
We consider again individuals with prospect-theory preferences. Further, in ad-dition to being randomly assigned a voting cost, each individual is independently assigned (by Nature) either the reference point 1 (abstaining) or the reference point 2 (voting) with probability q and 1 − q, respectively. The threshold for the voting cost in an individual’s cut-point strategy then varies with both group and reference point type. Let c∗A,1 and c∗A,2 denote the equilibrium thresholds for a member of group A who has the reference point 1 or 2, respectively. In the same way, we define c∗B,1and c∗B,2for members of group B. Hence, individ-ual i with the reference point 1 in group A votes if and only if ci < c∗A,1, and
so on. The two types per group have a distinct threshold cost level, but the equilibrium aggregate voting probability per group is simply their probability weighted average, i.e.
p∗A = q × F (c∗A,1) + (1 − q) × F (c∗A,2), (16) p∗B = q × F (c∗B,1) + (1 − q) × F (c∗B,2). (17) The equilibrium set of thresholds (c∗A,1, c∗A,2, c∗B,1, c∗B,2) simultaneously solves (3) - (6), (9), (10), (16), (17), VA(c∗A,1) = 0, VA(c∗A,2) = 0, VB(c∗B,1) = 0 and
WB(c∗B,2) = 0.
Table 8: Predicted turnouts in the PTM-model, with a mixture of reference points, for groups A and B and different group sizes.
PTM-model NA NB p∗A p∗B 1 2 0.566 0.598 3 6 0.437 0.416 4 5 0.490 0.488 9 18 0.311 0.281 13 14 0.371 0.368 17 34 0.249 0.220 25 26 0.314 0.312 100 101 0.219 0.218
the experimental observations.8 It turns out that the best result is obtained
with q = 0.7. We refer to the model with this value of q, and with the other parameters having the same values as in Section 5, as the PTM-model.9
Table 8 reports the predicted turnouts of the PTM-model. We see that, except for one case, the turnout in the PTM-model is higher than in the stan-dard model. The PTM-model is further able to explain the turnouts observed in the experiment of Levine and Palfrey (2007), although it tends to perform somewhat worse the larger the electorate gets. This is due to the fact that the proportion of individuals with abstaining as the reference point is larger than that of individuals with voting as the reference point. When the relative pro-portions are reversed, the model does worse in predicting turnout for smaller elections, and better for explaining larger elections. However, a drawback of a higher proportion of individuals with voting as the reference point is that the comparative statics observed in the experiment no longer hold, i.e. group A does not have a higher turnout than group B. With a proportion of q = 0.7, the comparative statics are still correct, except when NA= 1 and NB = 2. Similar
8We have calculated the average prediction errors of group A and B for q = 0.1, 0.2, . . . , 0.9. Next, ****. OLIVER: can you describe which procedure you have followed to find the best choice q = 0.7?
to Section 5, we have also calculated the average prediction errors associated with the PTM-model. For group A it amounts to 0.040, and for group B it equals 0.027. Hence, based on this measure, we can conclude that the model with mixed reference points has much better predictive performance than the standard model in terms of explaining observed turnout in the experiment of Levine and Palfrey (2007).
7
Conclusion
We have argued that while able to explain significant turnout in arbitrarily large elections, game-theoretic models of instrumental voting fail to explain both aggregate and individual turnout in small elections. Existing alternatives to classical rational choice models, most prominently expressive and altruistic behavioral models are unable to explain these deviations since these explana-tions are excluded by the experimental design of voting games in the laboratory. The novel approach of this paper is to include assumptions from Prospect The-ory in the standard models, which reflect the empirically verified boundedly rational behavior of people. We find that the adapted model is able to explain a wide range of aggregate turnout, depending on the reference point of voters. In particular, depending on the assumptions about reference points, the model predicts much larger turnout than the standard model, and probability weight-ing leads turnout to decrease more slowly as the electorate grows large. Yet, predicted turnout still converges to zero as the electorate grows to infinity, so the model could only be considered an approach in explaining the ’paradox of voting’ that is complementary to other explanations such as expressive behavior. Importantly, the model could also explain the individual level differences in voting behavior observed in experimental studies, something that the standard model has a hard time explaining.
theory is incorrect, or it means that the model needs to be more complex in terms of endogenizing reference points. The next step should therefore be to confirm empirically that individuals’ risk preferences as well as reference points do influence their voting behavior, which would increase the confidence in the general approach of the model, thereby justifying theoretical extensions.
There are a number of avenues available to verify the predictive power of the model in small elections, as well as the importance of risk preferences and reference points generally for voting behavior.
The adapted model of voter participation that takes into account prospect-theoretic risk preferences makes a number of testable predictions. First, one could judge it by the ability to correctly predict the turnout observed in an experimental setting such as in Levine and Palfrey (2007). As stated earlier, the theory fails in this respect since depending on the assumed reference point, predictions are either too low or too high. However, this may be due to the simplifying assumption that all players have either of the reference points, so relaxing this assumption in the model may improve its performance in that aspect.
Second, in an experimental setting, subjects could be randomly assigned to different samples, with each sample containing people with a certain range of risk preferences. The experiment can then be performed separately for each of the two samples. Subsequently, the model’s predictions about the relative turnout in the two samples of subjects can be compared to the data. For example, if one sample contains people below the 50th percentile w.r.t. the level of probability weighting, while the other contains the people above the 50th percentile, then to observe differences in turnout levels would serve to confirm the importance of probability weighting for voting behavior.
be-tween them) on an individual’s propensity to vote.
There is a large literature on experimental methods of eliciting risk prefer-ences of experimental subjects, see for example Abdellaoui (2007). Since we can only observe but not set individual’s risk preference parameters (whereas we can set voting costs), this would not prove a causal relationship. However, it is hard to think of any omitted variable that may explain the effect instead.
Similarly, one could test for the impact of reference points on turnout. While this variable is unobservable (and not consciously known by subjects), two meth-ods could be employed to solve that issue. First, one could use variables that can be hypothesized to correlate with the individual’s choice of reference point, e.g. whether someone has voted in the last national election, or how they answer the question of whether there is a moral obligation to vote in general elections. Second, one could use experimental manipulations, in particular priming tech-niques. For example, in the experimental instructions one could expressively tell subjects that they have to make the decision whether they want to abstain or not (vote or not). Subsequently, the impact of the proxy or manipulation dummy on turnout can be measured.
Appendix: larger rewards and group sizes
H larger makes voting more attractive, leading to higher turnouts in many cases. For example, in the standard model with NA= 25 and NB= 26, turnout
increases from 0.238 to 0.646 for group A, and from 0.235 to 0.808 for group B. Turnouts also increase in many cases for the PT-model, regardless of the reference point. Second, in the standard model with H = 1005, turnouts in group A are almost always smaller than in group B. This is opposite to the results for this model with H = 105, see Table 5. A similar reversal can be seen for the PT-model with abstaining as the reference point, but not for the PT-model with voting as the reference point. Given the significant sensitivity of the standard model’s predictions to the size of H, it may be valuable to do further empirical testing with different sizes of this reward. Third, turnouts in the standard model with larger electorate sizes of up to twenty thousand quickly become much smaller. Depending on the reference point assumed, the PT-model’s turnout predictions are between two to five times larger than the standard model at an electorate size of twenty thousand. Thus, in principle, this model is able to explain significant turnouts in large elections. While due to computational limits no further calculations were made, the small rate at which turnout decreases in the PT-model with voting as the reference point suggests that even in elections with millions of voters, this model would predict significant turnout.10
Table A.1: Predicted turnouts with H = 1005 in the standard model and PT-model with the reference points abstaining or voting, denoted with (a) and (v), respectively, for groups A and B and different group sizes.
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