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Published in: Social Choice and Welfare 1/2024

19-09-2023 | Original Paper

The “invisible hand” of vote markets

Authors: Dimitrios Xefteris, Nicholas Ziros

Published in: Social Choice and Welfare | Issue 1/2024

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Abstract

This paper studies electoral competition between two non-ideological parties when voters are free to trade votes for money. We find that allowing for vote trading has significant policy consequences, even if trade does not actually take place in equilibrium. In particular, the parties’ equilibrium platforms are found to converge (hence, there is no reason for vote trading) to the ideal policy of the mid-range voter, instead of converging to the peak of the median voter (as they do when vote trading is forbidden). That is, a market for votes may not change the outcome only by redistributing the political power among voters when the parties’ policy proposals are fixed (e.g., Casella et al. in J Polit Econ 120:593–658, 2012, etc.), but also by acting as an invisible hand—modifying parties’ incentives when platform choice is endogenous.

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Appendix
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Footnotes
1
See, for instance, Philipson and Snyder (1996); Casella et al. (2012); Casella and Turban (2014); Casella et al. (2014); Xefteris and Ziros (2017, 2018); Tsakas et al. (2021); Casella and Macé (2021); Casella and Sanchez (2022), among others.
 
2
The main criticisms stem from the fact that vote trading affects the payoffs of all voters. This externality complicates the existence of equilibrium and, moreover, leads to ambiguous results concerning social welfare. For an account of the criticisms, one is referred to the papers cited above, which study vote trading with exogenously fixed policy platforms.
 
3
Similar approaches are found in many other works. See, for instance, Lijphart (1984); Ortuño-Ortín (1997); Grossman and Helpman (1999); Llavador (2006); Merrill and Adams (2007); De Sinopoli and Iannantuoni (2007); Saporiti (2014); Matakos et al. (2016), among others.
 
4
In light of the equilibrium features, our work is associated with Hirata and Kamada (2020), which studies a two-party election where a party’s winning probability depends on the contributions it raises.
 
5
Considering that a voter’s utility declines with the distance between her ideal policy and the implemented one is essential for our main result about platform convergence. This is also the case in standard electoral competition models without vote trading; hence, such a modeling allows for a direct comparison with the results when vote trading is not allowed, which is the main inquiry of this paper.
 
6
Most assumptions employed (e.g., two parties, quasilinear preferences) are standard in the vote-trading literature (e.g., Casella et al. 2012, 2014; Xefteris and Ziros 2017) and provide a convenient way for solving the problem of nonexistence of equilibrium in a market for votes. This literature also abstracts from several empirically relevant factors in real-world settings (e.g., elite influence, lobbies), which future research should consider.
 
7
While abstention is not allowed in our model, our main results are robust to considering voluntary participation.
 
8
That is, if at least one other individual sells her vote, the utility function is well-defined, differentiable, and strictly concave in [0, 1].
 
9
Choosing not to trade is always a best response of an individual when all other individuals choose not to trade.
 
10
See the Appendix.
 
11
We notice that the equilibrium bids of the two buyers are increasing in the size of the electorate, but they are not affected by the ideal policies of all other voters. Moreover, one can easily show that \({\bar{b}}^{\alpha }>\bar{ b}^{\beta }\) (\({\bar{b}}^{\alpha }<{\bar{b}}^{\beta }\)) whenever \(\frac{\alpha +\beta }{2}>\frac{1}{2}\) (\(\frac{\alpha +\beta }{2}<\frac{1}{2}\)).
 
12
See the Appendix.
 
13
See the Appendix.
 
14
In this paper the policy outcome is deterministic (i.e., a weighted average of the parties’ platforms, with weights being equal to their vote shares). When voters’ utility functions are strictly concave (as they are here), then the probabilistic setup is not equivalent to the deterministic one: in the former case, the relationship between a voter’s utility and the vote share of her preferred party is linear, while in the latter, the relationship is strictly concave, making the problem distinctly more complicated. For this reason, we cannot simply refer to earlier arguments to establish Proposition 1.
Arguably, the current deterministic setup, where policy is a compromise of the two platforms and not the product of a “random dictatorship”, is a better assumption in many ways and it aligns with how several papers in the literature treat policy formation in the presence of divergent platforms (e.g., Ortuño-Ortín 1997; Merrill and Adams 2007; Matakos et al. 2016). Moreover, by employing this assumption, the current paper, beyond its main contribution (i.e., to show how allowing for vote trading affects policy outcomes when platforms are endogenous), makes a secondary point: it establishes that earlier results provided in probabilistic settings, are also valid in more standard settings of deterministic policy formation.
 
15
Future research could consider additional factors (e.g., concave utilities in money or uncertainty regarding the ideal policies of other individuals) that might provide further justification for a larger number of voters.
 
16
This is very reminiscent of actual behaviors in legislatures. Indeed, empirical research provides evidence that the relevant party ideology in legislatures is more extreme than the ideology of the median party legislator (see, for instance, Grofman et al. 2002).
 
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Metadata
Title
The “invisible hand” of vote markets
Authors
Dimitrios Xefteris
Nicholas Ziros
Publication date
19-09-2023
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 1/2024
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-023-01485-z

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