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Erschienen in: Public Choice 1-2/2017

10.08.2017

Crafting consensus

verfasst von: Jan Zápal

Erschienen in: Public Choice | Ausgabe 1-2/2017

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Abstract

The paper analyzes the problem of a committee chair using favors at her disposal to maximize the likelihood that her proposal gains committee support. The favors increase the probability of a given member approving the chair’s proposal via a smooth voting function. The decision-making protocol is any quota voting rule. The paper characterizes the optimal allocation of any given level of favors and the optimal expenditure-minimizing level of favors. The optimal allocation divides favors uniformly among a coalition of the committee members. At a low level of favors, the coalition comprises all committee members. At a high level, it is the minimum winning coalition. The optimal expenditure level guarantees the chair certain support of the minimum winning coalition if favors are abundant and uncertain support of all committee members if favors are scarce; elitist or egalitarian committees are compatible with a strategic chair. The results are robust to changing the chair’s objectives and to alternative voting functions, and reconcile theoretical predictions with empirical observations about legislative bargaining experiments, lobby vote buying and executive lawmaking.

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Fußnoten
1
Others, for example Dougherty and Edward (2012), use the term k-majority rule.
 
2
Our voting functions are formally related to the contest success functions. See Amegashie (2002) on the use of contest success functions in a bargaining model and Corchon and Dahm (2010) on the connection between contests and bargaining. Moreover, the voting functions are related to probabilistic voting (Lindbeck and Weibull 1987) in electoral competition models. We study distribution (of favors), while electoral models typically study redistribution (of income) from one group of voters to another and predict no redistribution with homogeneous voters.
 
3
Supermajorities arise in legislative bargaining in non-stationary equilibria (Norman 2002; Dahm et al. 2014; Dahm and Glazer 2015), when proposing is costly (Glazer and McMillan 1990, 1992) or while bargaining over public goods (Volden and Wiseman 2007).
 
4
Supermajorities arise in government formation when proposers are uncertain about which coalitions will form (Weingast 1979; Shepsle and Weingast 1981), under preference for ideologically connected coalitions (Axelrod 1970), under uncertainty about the size of the voting body created by abstention (Koehler 1975), because supermajorities make legislative logrolls self-enforcing (Carrubba and Volden 2000), because supermajorities allow for extraction of rents from excess coalition members (Baron and Diermeier 2001) and in the presence of the agenda setter’s aspirations for political stability and greatness (Doron and Sherman 1995).
 
5
Most of the vote-buying literature restricts attention either to vote (yes versus no) contingent payments or to outcome (acceptance versus rejection) contingent payments. Dal Bo (2007) introduces ‘pivotal bribes’—payments contingent on a vote being outcome-pivotal—and shows that these allow a single vote-buyer to purchase all votes almost for free. Felgenhauer and Gruner (2008) and Morgan and Vardy (2011) use pivotal bribes, at least in part of their analyses. The pivotal bribes are related but distinct from the observation that if voters can commit to vote for a certain action in return for a transfer, the resulting Bertrand competition among the voters benefits the vote-buyer (Ferejohn 1986).
 
6
Tyutin and Zaporozhets (2017), assuming identical payments to the members of the paid coalition, generalize this insight to multiple players.
 
7
We assume that the chair is not a voting member of the committee. This is not important. If the chair were to vote, we could relabel n and q, if necessary, and proceed without further alterations.
 
8
The full specification is required for the discussion of alternative voting functions in Sect. 5.
 
9
Section 5 discusses voting functions generated by different distributions of \((o_{i})_{i\in N}\).
 
10
The upfront payments and campaign promises terminology comes from the vote-buying literature, specifically from Dekel et al. (2008).
 
11
Section 5 discusses the addition of an outside option the chair receives if her proposal fails.
 
12
This observation greatly simplifies the subsequent analysis, as it implies that the uncertainty remaining in the number of votes the chair receives has a Binomial distribution with \(n-r^{*}-s^{*}\) trials, not a Poisson Binomial distribution. The Poisson Binomial random variable represents the number of successes in n independent Bernoulli trials with success probabilities \(p_{1},\ldots ,p_{n}\). The Binomial distribution is the special case when all of the probabilities are equal.
 
13
Universal coalitions, in which all members receive favors, can be seen as equally extreme and, hence, implausible prediction as the minimum winning coalitions. However, the model predicts universal coalitions among members who are willing to sell their votes, leaving members who are not willing to sell outside the model. Moreover, not every favor-receiver votes yes.
 
14
We conjecture that \(r^{*}\) with increasing \(b\in [q-1,q)\) visits any integer in \(\{0,\ldots ,n-q\}\) and is non-decreasing. We have (numerically) confirmed the conjecture for \(n\le 13\) and any \(q\le n\). Formal proof, however, eludes us.
 
15
An alternative intuition uses the fact that the mean number of votes given any \({\mathbf {p}}=(p_{i})_{i\in N}\) is \(\sum _{i\in N}p_{i}\le b\). When \(b\ge q\), the minimum winning coalition clearly is optimal. When \(b<q-1\), the chair’s proposal on average loses and her optimal favor allocation maximizes the variance in the number of votes, \(\sum _{i\in N}p_{i}(1-p_{i})\), which is done by setting \({\mathbf {p}}=(\frac{b}{n})_{i\in N}\). When \(b\in [q-1,q)\), the chair increases the variance only partially.
 
16
A Gaussian or ordinary hypergeometric function \(\,_{2}F_{1}\) is a special function that includes many other (special) functions as special cases. See the preliminary section to the proofs in Appendix 1 or Olver et al. (2010) for details.
 
17
Propositions 2 and 3 reflect a large increase in B. The former proposition shows that B and \(b^{*}\) are positively related in general.
 
18
Because \(\frac{n}{2}+\frac{1}{2}\,_{2}F_{1}(1,1+n,1+\frac{n}{2},\frac{1}{2})\) equals \(\frac{n}{2}+\frac{1}{2}+\frac{\sqrt{\pi }}{2}z(n)\), where, using \(\Gamma \), the Euler gamma function, \(z(n)=\frac{\Gamma \left( 1+\frac{n}{2}\right) }{\Gamma \left( \frac{1+n}{2}\right) }\) and we have \(\lim _{n\rightarrow \infty }z(n)-\frac{\sqrt{n}}{\sqrt{2}}=0\).
 
19
The objective \(B{\mathbb {R}}_{q|n}[b]-b\) assumes that, for all \(i\in N\), the chair pays \(x_{i}\) to i and i votes yes with probability \(p_{i}(x_{i})=x_{i}\). This requires a behavioral model of the voting functions, for example, i voting yes out of gratitude to the chair for having received \(x_{i}\). Generating the voting functions through vote-contingent payments and vote motivation implies that the chair pays only for the votes she actually receives.
 
20
Assuming that the chair pays only for the votes she actually receives considerably complicates the analysis. In the \({{\mathbb {CE}}}\) problem, the chair now maximizes, by choosing \({\mathbf {p}}\in {\mathbb {R}}^{n}\) subject to the ex-post budget constraint \({\mathbf {p}}*{\mathbf {1}}\le B\) and \(p_{i}\in [0,1]\) for \(\forall i\in N\), either \((B-{\mathbf {p}}*{\mathbf {p}}){\mathbb {P}}_{q|n}[{\mathbf {p}}]\) or \(B{\mathbb {P}}_{q|n}[{\mathbf {p}}]-{\mathbf {p}}*{\mathbf {p}}\), depending on whether she pays for the votes only when her proposal passes or not. The complication arises because \({\mathbb {P}}_{q|n}[{\mathbf {p}}]\) and \(-{\mathbf {p}}*{\mathbf {p}}\) must be maximized jointly, so that the nested nature of the \({{\mathbb {FA}}}\) and \({{\mathbb {CE}}}\) problems is lost. The only observation that does not require extended additional analysis is that, for \(B<q-1\), any solution to the \({{\mathbb {CE}}}\) problem must involve uniform allocation of favors among committee members. This is because, for any \(b'<B\), \(p_{i}=\frac{b'}{n}\) for \(\forall i\in N\) maximizes both \({\mathbb {P}}_{q|n}[{\mathbf {p}}]\) and \(-{\mathbf {p}}*{\mathbf {p}}\), the former by Proposition 1 and the latter by a simple argument.
 
21
Because B in \(B{\mathbb {R}}_{q|n}[b]-b\) captures the importance of the proposal for the chair, the second effect does not arise when the proposal is important for the chair.
 
22
Setting \(n=q=1\) and \(B\in (0,1)\cup (1,2)\), \(b^{*}=\frac{B}{2}\) in the benchmark \({{\mathbb {CE}}}\) while \(b^{*}=0\) if \(B<1\) and \(b^{*}=1\) if \(B>1\) with upfront payments. The change in \(b^{*}\) under upfront payments illustrates the first effect when \(B>1\) and the second effect when \(B<1\).
 
23
Upfront payments lower the amount of favors distributed for a small overall budget and increase the amount of favors distributed for a large overall budget. The first effect is similar to the effect of upfront payments in Dekel et al. (2008) and is driven by the all-pay nature of the upfront payments. The second effect is specific to our model and arises because the upfront payments can be considered as an investment, and the chair ensures that her investment returns a profit (results in acceptance) by increasing the investment itself.
 
24
As an example, consider binary voting over alternatives yielding utility \(x_{i}\) and y in a quantal response model such as the Quantal Response Equilibrium of McKelvey and Palfrey (1995). Then the probability that i votes for \(x_{i}\) is \(p_{i}(x_{i})=\frac{\exp {\lambda x_{i}}}{\exp {\lambda x_{i}}+\exp {\lambda y}}\), where \(\lambda \ge 0\) measures the precision in i’s best response. Thus, \(p_{i}\) is increasing as a function of \(x_{i}\), convex for \(x_{i}\le y\) and concave otherwise.
 
25
For the Quantal Response Equilibrium mentioned in footnote 24, we have constructed an example with \(n=3\) and \(q=2\) to illustrate how concavity creates incentives for \(x_{i}\approx x_{j}\) and convexity for \(x_{i}\gg x_{j}\). As y increases from 0 to 1, the voting functions change in a smooth manner from concave to convex, assuming \(b=1\). Correspondingly, the optimal favor allocation focuses first on all 3 players and then switches to focusing on 2, minimum winning, players.
 
26
The jump in the \({{\mathbb {CE}}}\) solution is driven by points where \({\mathbb {R}}_{q|n}\) is not differentiable. These points correspond to the values of b at which the \({{\mathbb {FA}}}\) solution switches between different coalition sizes. This shows that analysis of the \({{\mathbb {CE}}}\) problem cannot be conducted using calculus since \({\mathbb {R}}_{q|n}\) is endogenous.
 
27
The empirical tests of Grossman and Helpman (1994) by Goldberg and Maggi (1999) and of Groseclose and Snyder (1996) by Wiseman (2004) are notable exceptions.
 
28
Campos and Giovannoni (2007), Richter et al. (2009), Dorsch (2013) and Kang (2016) study lobbying empirically (see Binderkrantz 2014 and Gregor 2017, for surveys).
 
29
The theorem is stated for \(t\in (0,1-p)\), but its discussion clarifies that for \(t=1-p\) the bound is \(p^{n}\) and \(p^{n}\le \exp {[-2n(1-p)^2]}\) \(\forall n\in {\mathbb {N}}_{\ge 1}\) and \(\forall p\in (0,1)\) is standard to verify.
 
30
Hoeffding (1956, Theorem 2) shows that if \({\mathbf {p}}^{*}\) has two interior coordinates \(p_{i}^{*}\ne p_{j}^{*}\), then there exists another solution to \({{\mathbb {FA}}}\), \({\mathbf {p}}'\), identical to \({\mathbf {p}}^{*}\) except that \(p_{i}'=p_{j}'\). Lemma 4 is stronger, it shows that \({\mathbf {p}}^{*}\) with interior \(p_{i}^{*}\ne p_{j}^{*}\) does not exist.
 
31
Theorem 1 in Darroch (1964) relies on all coordinates of \({\mathbf {p}}\) to lie strictly between zero and unity. To see we can apply the result, note that \({\mathbb {P}}^{*}_{s|n-3}[{\mathbf {p}}^{\{ijk\},*}]={\mathbb {P}}^{*}_{s-u|n-3-u-z}[{\mathbf {p}}^{\{ijkuz\},*}]\) for \(s\in \{q-3,q-2,q-1\}\). We have, \(\forall s\in \{q-3,q-2,q-1\}\), \(s-u\ge q-3-u\ge 0\) by \(u\le q-3\), \(n-3-u-z\ge 2\) by \(z+u\le n-5\) and \(s-u\le n-3-u-z\Leftrightarrow z\le n-3-s\le n-q-2\) by \(z\le n-q-2\).
 
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Metadaten
Titel
Crafting consensus
verfasst von
Jan Zápal
Publikationsdatum
10.08.2017
Verlag
Springer US
Erschienen in
Public Choice / Ausgabe 1-2/2017
Print ISSN: 0048-5829
Elektronische ISSN: 1573-7101
DOI
https://doi.org/10.1007/s11127-017-0470-8

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