Complete information pivotal-voter model with asymmetric group size and asymmetric benefits

https://doi.org/10.1016/j.ejpoleco.2020.101961Get rights and content

Highlights

  • We analyze a standard pivotal-voter model with costly voting and majority rule.

  • The benefit of the favorite alternative differs across the two groups.

  • The number of players differs across the two groups.

  • We pin down a unique equilibrium through an intuitive refinement.

  • We provide some comparative statics of the unique equilibrium.

Abstract

We analyze a standard pivotal-voter model under majority rule, with two rival groups of players, each preferring one of two public policies and simultaneously deciding whether to cast a costly vote, as in Palfrey and Rosenthal (1983). We allow the benefit of the favorite public policy to differ across groups and impose an intuitive refinement, namely that voting probabilities are continuous in the cost of voting to pin down a unique equilibrium. The unique cost-continuous equilibrium depends on a key threshold that compares the sizes of the two groups.

Introduction

Consider a stylized economy with two rival groups of players; group “m”, with m players, and group “n”, with n players. Each player may cast a costly vote in a majoritarian election between two public policies; policy M and policy N. Players of group m receive a higher payoff if policy M is implemented, while players of group n receive a higher payoff if policy N is implemented. In a standard pivotal-voter model, we study the equilibria of such a voting game, focusing on the case relevant for applications where both the group size and the payoffs could be different across groups.

A number of applications can be analyzed under this setup. For example, an economy made of some resource-poor and some resource-rich players is called to vote between no redistribution and full redistribution of resources, under the assumption that the poor players outnumber the rich. Each single rich player has more at stake in the election than a single poor player, as the amount of resources each individual would obtain under full redistribution is closer to the level of resources of the poor than to that of the rich. Another application is a university faculty consisting of several economists and a few lawyers, all called to vote over who to hire between two job market candidates: an economist and a lawyer.1 Both economists and lawyers are better-off if the newly hired candidate is of their same type. Furthermore, the benefit for a lawyer from hiring another lawyer is greater than the one for an economist from hiring another economist because of the asymmetric size of the two groups; that is, since lawyers are fewer, having another lawyer in the department sharply increases each lawyer's coauthoring possibilities, whereas the benefit for an economist from having a new economist in the faculty is lower because they are already plenty. In other words, the benefit is asymmetric across players of different groups. A third application is that of residents of two neighborhoods are called to vote over the location of a new school in one of the two neighborhoods. In neighborhood N there is already a school, in neighborhood M there is none: thus, despite the fact that each resident strictly prefers the school to be located in her neighborhood, residents in neighborhood N “care less” than residents in neighborhood M about the location of the school since there is already a school in neighborhood N.

In all these applications, we may have an asymmetry in both size of each group and benefits, it is therefore an asymmetric-asymmetric setup which, to the best of our knowledge, has not been studied before. To study it, we adopt a complete-information pivotal-voter model with costly voting. 2 This class of models is far from novel. The seminal contribution dates back to Palfrey and Rosenthal (1983), where two groups of players each preferring one of two alternatives simultaneously decide between abstaining or voting for their preferred alternative. The winner is decided by majority rule.3 Technical difficulties and multiplicity issues allowed Palfrey and Rosenthal to analyze only special cases.4 The analysis of this model has been pushed forward by two other works. First, Nöldeke and Peña (2016) focus on the two groups having symmetric number of members and symmetric benefit from the favored alternative winning the election. Second, Mavridis and Serena (2018) focus on the two groups having asymmetric number of members and symmetric benefit from the favored alternative winning the election.

After analysing the asymmetric-asymmetric case, we deploy a continuity refinement, “cost-continuity”, pinning down a unique equilibrium which we discuss. We illustrate our results focusing on the application of redistribution of resources as a running example throughout the paper, as this application has a clear and convenient parametrization of the levels of payoffs between the two groups. In particular, after describing the model in Section 2, in Section 3 we fully characterize the simple (m; n) = (3; 2) model in which there are multiple equilibria.5 We characterize analytically the equilibria where members of at least one group play a pure strategy in participation (Section 4.1, Section 4.2), and we characterize in part analytically and in part numerically the equilibria where members of both groups play a mixed strategy (Section 4.3). Our numerical analysis, which complements our analytical results, characterizes the redistribution trade-off. Despite the multiplicity of equilibria, only one equilibrium survives a novel and intuitive cost-continuity refinement; that is, the equilibrium probability of voting is continuous in the cost of voting. In fact, our numerical analysis shows that the cost-continuity refinement turns out to single out a unique equilibrium in the general model with an arbitrary number of poor and rich players (see Section 5). We study the properties of the unique cost-continuous equilibrium; if the size of the poor group is sufficiently small (namely, the number of poor players is lower than the square of the number of the rich players), then the poor players may vote and redistribution has a chance of winning, while if the size of the poor group is large (namely, the number of poor players is greater than the square of the number of the rich players), the unique cost-continuous equilibrium dictates that poor players abstain with certainty, and thus poor players are doomed to lose the election and redistribution is not implemented.

The above threshold suggests an answer to the redistribution trade-off. Consider an increase in the number of poor players. On the one hand, it makes the individual resources under redistribution closer to the resources of a poor and further apart from those of a rich. This makes the stakes of the rich in the election increase, and that of the poor decrease.6 On the other hand, an increase in the number of poor players makes the poor group bigger and thus stronger in the election; that is, if all players were to vote with the same exogenous probability full redistribution would be the most likely outcome of the election as there are more poor players than rich. All in all, the former effect of an increase in the number of poor players (making rich players vote “more”) is stronger than the latter effect (making rich players relatively less numerous) the greater is the number of poor players; in fact, when the number of poor players is greater than the square of the number of the rich players, the former dominates the latter and in the unique cost-continuous equilibrium poor players have no chance of winning the election. The opposite happens when the number of poor players is low, and hence poor players have a chance of democratically redistributing resources.

After their 1983 paper, Palfrey and Rosenthal (1985) analyzed a version of the model under private information on the cost of voting and from then onwards, much of the literature has developed under private information, as the model is often more tractable and allows more elegant and neat results (e.g., Borgers, 2004; Taylor and Yildirim, 2010). Nevertheless, the general idea that voters compare the benefits of voting with its costs is older and has been long the interest of economists, dating back at least to Downs' (1957) seminal work. We consider only benefits that accrue from changing the policy to the voters' preferred outcome, despite a number of other benefits playing an important role in real-life; for instance, Wiese and Jong-A-Pin (2017) empirically examine the benefits arising from “expressive” motives of voting. In our model, we follow the main strand of the literature on complete-information pivotal-voter model with costly voting, started by Palfrey and Rosenthal (1983), and further developed by Nöldeke and Peña (2016) and Mavridis and Serena (2018). And as we generalize this strand of the literature to the asymmetric-asymmetric setup described above, we abstract away from other interesting forces that typically play an important role in real life voting. An example is communication. In the laboratory, Palfrey and Pogorelskiy (2019) find that communication increases turnout when messages are public within one's own party, and decreases turnout with a low voting cost when subjects exchange public messages through computers. In the analysis of correlated equilibria, Pogorelskiy (2020) finds that communication helps sustain equilibria with high levels of turnout. Kalandrakis (2007, 2009), under heterogeneous voting costs, finds that almost all Nash equilibria are robust to small amounts of incomplete information. Social pressure is also a key aspect of voters' turnout which we abstract away from (see e.g. Gerber et al., 2008; Gerber et al., 2016; and DellaVigna et al., 2017). Finally, polls are known to have a significant effect on voters' turnout (e.g., Großer and Schram, 2010; Morton et al., 2015).

Section snippets

Model

There are two types i ∈{m, n} of players; m > 1 poor players and n > 1 rich players. Assume poor players are more numerous; m > n.7 Players are simultaneously called to cast a vote between two alternatives, M and N. When we specialize the model to redistribution of resources, M is full redistribution and N is no redistribution of resources. Poor players prefer alternative M, in

m,n=3,2 in the (Redistribution-Parametrized Example)

We start with a simple example where we assume that m = 3 and n = 2 in our (Redistribution-Parametrized Example).11

General m,n

The probability of being pivotal is crucial for the voting/abstention choice in (4). While we discussed it in the previous section for the special case of m,n=3,2, it is useful to provide the expression for a general pair m,n. The following table shows the calculations that a single poor player must make when she knows that she faces m − 1 poor players and n rich with (m¯,n¯) signifying how many of the rest actually vote in each instance.

Poor player: pivotal if voting?(m¯,n¯)ProbabilityΔπi
Yes(n

Application - voting over redistribution of resources

From Proposition 4 we know that the “Partially Mixed” pm(Bm)=0 and pn(Bn)=1Bn1n1 exist if and only if Bm>nBn(n1)Bnnn1. Recall from 5 and 6 that Bm = 2c(n + m)/n and Bn = 2c(n + m)/m. Finally, recall that the notation for our (Redistribution-Parametrized Example) is Bn = B and Bm=mnB. Plugging these expressions intoBmnBn(n1)Bnnn1,we obtain(mn2)B+n(n1)Bnn1>0B>n2mn2nn1.And in fact, the lowest value of B for which the “Partially Mixed” exists is 0 when n2 ≤ m, while when n2 > m it

Conclusions

We study a standard pivotal-voter model under majority rule, with two rival groups of players, each preferring one of two public policies and simultaneously deciding whether to cast a costly vote. We further the existing literature, in particular Palfrey and Rosenthal (1983), Nöldeke and Peña (2016) and Mavridis and Serena (2018), by allowing the benefit of the favorite public policy to differ across groups. We impose an intuitive refinement, namely that voting probabilities are continuous in

Declaration of competing interest

None.

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  • Cited by (0)

    We are grateful to Luis Corchón and Ignacio Ortuño-Ortín for their advice and support. We would like to thank the Editor, an anonymous referee, Kai Konrad, Diego Moreno and Georg Nöldeke for their useful comments as well as participants in seminars in Universidad Carlos III de Madrid, the Max Planck Institute for Tax Law and Public Finance, PET 2015 in Luxemburg and the MOMA Meeting 2016. All errors are our own.

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