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The value of formalism: re-examining external costs and decision costs with multiple groups

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Several authors have examined the optimal k-majority rule using a variety of criteria. We formalize and extend the original argument laid out by Buchanan and Tullock (The calculus of consent: logical foundations of constitutional democracy, 1962) using a decision theoretic analysis from the perspective of an individual voter. Unlike previous formalizations, voters in our study are members of one or more groups. This allows us to examine cases wherein different voters have starkly different interests. Furthermore, voters in our study can err in their judgments of proposals allowing us to model potential irrationalities in the choice of an optimal k-majority rule. We consider both up or down votes on a single proposal as well as votes over a series of proposals. We find that the optimal k-majority rule depends on a number of parameters, most notably the number of rounds needed to create a proposal that will pass. Group membership has almost no affect. Furthermore, if two groups are at odds, then the external cost function can actually rise over some range of \(k\); if voters err systematically in their judgment, more inclusive k-majority rules, such as unanimity rule, can fail to pass Pareto preferred proposals. Our results should help advance a classic work in Public Choice.

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  1. More precisely, they examine the expected number of BT-losers, which they define as someone who votes for a BT-preferred status quo or a BT-preferred proposal and society chooses the other alternative (Dougherty and Edward 2011, p. 60).

  2. There are notable exceptions, such as cases with a small dissident group.

  3. See Heckelman and Dougherty (2010) for a crude test of whether larger k-majority rules have negative effects on tax increases.

  4. Buchanan and Tullock (1962, p. 69) also include the “opportunity costs of bargains never made” in their definition, a notion which is open to various interpretations including the cost of negotiating concessions. Under at least one interpretation, including the one we adopt here, these costs can be measured by the time and effort needed to reach an agreement.

  5. Mueller bases his claim on Caplin and Nalebuff (1988), who show that voting rules with \(k \ge 0.64(N)\) will not produce vote cycles if preferences are Euclidean and the density of ideal points is concave in all directions.

  6. Note that our model can be made non-probabilistic by setting all standard deviations to zero.

  7. External costs are incurred only by voters who have negative utility when a proposal passes. When the proposal passes, external costs for a voter are \(|u_{i}|\) if \(u_{i} \le 0\), and \(0\) if \(u_{i} > 0\).

  8. You can find the Web Appendix and the code needed to replicate all of the findings in the paper at:

  9. In describing our minimums, we treat differences less than 0.001 as non-discernable.

  10. In our model, improvements in the proposals serve as proxy for negotiations. Furthermore, assuming proposals continually improve is consistent with Mueller’s understanding of Buchanan and Tullock’s work. Mueller (2003, p. 138) writes, “Thus, the political process implicit in a defense of the unanimity rule is one of discussion, compromise, and amendment, continuing until a formulation of the issue is reached benefiting all” (emphasis in original).

  11. For the first round, we assume that \(-1 \le \mu _{g,1} \le 1\), but we relax this restriction for later rounds. Furthermore, the program made available with our web appendix does not allow \(m_{g,r}\) to vary in \(r\), nor does it vary in \(r\) in the examples below.

  12. Buchanan and Tullock (1962) discuss the effect of preference homogeneity on the optimal k-majority rule. Our framework allows for several formalizations of preference homogeneity, such as the distribution of initial utility and the relative size of each group. In our model, the homogeneity of the group has little effect on the optimal k-majority rule independent of its effect on the round in which a proposal passes and the losses incurred in those rounds – themes we have examined in the text. For example, three groups sized 30, 35, and 30 with initial utility \(N(0,0.2)\), \(N(0,0.2)\), \(N(0,0.2)\), and \(\alpha =0.01\) might produce larger external costs and smaller decision costs for \(k > 45\) than the same groups with initial utility \(N(-0.9,0.2)\), \(N(0,0.2)\), \(N(0.9,0.2)\). The difference is mainly because the first set passes a proposal more quickly than the second set.

  13. For \(k > 50\), decision costs total are \(c \cdot R = 0.01 \cdot 10,000 = 100\).

  14. With \(r^*\) rounds of voting, the proposal has been improved only \((r^*-1)\) times, thus the factor \((r^*-1)\) in this sentence.


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The authors would like to thank Roger Congleton, Andrei Gomberg, Georg Vanberg, Arthur Zillante and the participants in West Virginia University’s Department of Economics seminar series for their helpful feedback and suggestions.

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Correspondence to Keith L. Dougherty.



In this section, we show analytical results for a specialized case. We assume that the population consists of three groups and that each group has very homogeneous preferences. Group X has \(x\) voters, each with \(u_{i,1}=\beta\); Group Y has \(y\) voters, each with \(u_{i,1}=\gamma\); Group Z has \(z\) voters, each with \(u_{i,1}=\delta\). Thus we are thereby removing all of the variance within groups and setting \(\sigma _{i,r}=0\) for all \(i,r\). We also assume that there is no voter error, i.e., \(e_{i,r}=0\). These assumptions might model a parliamentary system with three parties (with the members of each party having uniform preferences), or a three-person committee (with each member’s vote having a different weight).

We also assume that proposals improve by a constant \(\alpha\) for all groups, the per round decision costs are \(c\), and rounds continue until a proposal passes. We assume that \(\beta >\gamma >\delta\), and also assume

$$\begin{aligned} \beta >0>\delta . \end{aligned}$$

In what follows, we deduce the optimal voting rule \(k\) as a function of \(c\), from the point of view of the worst group’s total costs (WGTC). Recall that total costs are the sum of external costs and decision costs, i.e., \(WGTC=WGEC+DC\). We will use the notation \(WGTC_-\) to denote the \(WGTC\) for \(k\le x\), \(WGTC_0\) to denote the \(WGTC\) for \(k\in [x+1,x+y]\), and \(WGTC_+\) to denote the \(WGTC\) for \(k>x+y\).

Case 1: \(\gamma >0\).

In this case, the proposal will pass in one round for \(k\le x+y\). For such \(k\), \(WGEC=-\delta\), and decision costs are \(DC=c\), so

$$\begin{aligned} WGTC_-=WGTC_0=-\delta +c. \end{aligned}$$

For \(k>x+y\), passage will require \(r^*\ge 2\) rounds of voting, with \(r^*\) solving \((r^*-1)\alpha +\delta >0\) and \((r^*-2)\alpha +\delta <0\).Footnote 14 Thus

$$\begin{aligned} r^*=\left\lfloor \frac{-\delta }{\alpha }\right\rfloor +2. \end{aligned}$$

For such \(k\), we have \(WGEC=0\) and \(DC=r^*c\), so

$$\begin{aligned} WGTC_+=r^*c. \end{aligned}$$

We now compute \(c\) such that \(WGTC_-=WGTC_+\). This c is that it is the unique c such that WGTC is the same for all k. Solving for \(c\) using  (1), (3), and (2) we get

$$\begin{aligned} c=c_*:=\frac{-\beta }{r^*-1}=\frac{-\beta }{1+\lfloor \frac{-\delta }{\alpha }\rfloor }. \end{aligned}$$

Thus for \(c<c_*\), the optimal k-majority rule for the worst group will be any \(k\) in the interval \([x+y+1,x+y+z]\). For \(c>c_*\), the optimal k-majority rule for the worst group will be any \(k\) in the interval \([0, x+y]\).

Case 2: \(\gamma <0\).

Suppose first that \(k\le x\). Then the proposal will pass in the first round, and, as above, we have for such \(k\)

$$\begin{aligned} WGTC_-=-\delta +c. \end{aligned}$$

For \(x+y<k\), the measure will pass in \(r^*\) rounds, with \(r^*\) as in (2), and arguing as in Case 1,

$$\begin{aligned} WGTC_+=r^*c. \end{aligned}$$

For \(x<k\le x+y\), the measure will pass in \(r_*\ge 2\) rounds, where \((r_*-1)\alpha +\gamma >0\) and \((r_*-2)\alpha +\gamma <0\). Thus

$$r_*=\left\lfloor \frac{-\gamma }{\alpha }\right\rfloor +2.$$

For such \(k\), the \(WGEC\) will be \(-\delta -(r_*-1)\alpha\), and \(DC=r_*c\). Hence,

$$\begin{aligned} WGTC_0=-\delta -(r_*-1)\alpha +r_*c. \end{aligned}$$

It is intuitively clear that for \(c\) sufficiently large, the optimal \(k\) should be any \(k\le x\). To find the value of \(c\) where the optimal voting rule becomes \(x<k\le x+y\), we solve for \(c\) when \(WGTC_0=WGTC_-\). By (7), (4), and (6) we get \(-\gamma -(r_*-1)w+mc=-\gamma +c\), hence

$$\begin{aligned} c=c^{*}{:}\,{=}\,\alpha . \end{aligned}$$

Thus for \(c\ge c^*\), the optimal voting rule will be any \(k\le x\), while for \(c\) slightly under \(c^*\) the optimal voting rules will be \(k\in [x+1, x+y]\). To find a lower bound on the \(c\) associated with this interval in \(k\), we consider two sub-cases.

Case 2i: \(\lfloor \frac{-\gamma }{\alpha }\rfloor =\lfloor \frac{-\delta }{\alpha }\rfloor\). In this case, \(r_*=r^*\), and the number of rounds for \(k\in [x+y+1,x+y+z]\) will be the same as the number of rounds for \(k\in [x+1,x+y]\), and \(WGTC_0=WGTC_+\) for all \(c\). Thus in this case, for any \(c<\alpha\), all voting rules \(k\in [x+1,x+y+z]\) will minimize the total costs incurred by the worst group.

Case 2ii: \(\lfloor \frac{-\gamma }{\alpha }\rfloor <\lfloor \frac{-\delta }{\alpha }\rfloor\). In this case, we solve for \(WGTC_0=WGTC_+\). By (7) and (5), we get \(r^*c= -\delta -(r_*-1)\alpha +r_*c\) hence

$$\begin{aligned} c=c_{*}:=\frac{\delta -(r_*-1)\alpha }{r^*-r_*}=\frac{-\delta -(\lfloor \frac{-\gamma }{\alpha }\rfloor +1)\alpha }{\lfloor \frac{-\delta }{\alpha }\rfloor -\lfloor \frac{-\gamma }{\alpha }\rfloor } \end{aligned}.$$

Thus for \(c\le c_*\), the optimal k-majority rule for the worst group will be any \(k\) in the set \([x+y+1,x+y+z]\). For \(c\in [c_*,\alpha ]\), the optimal k-majority rule for the worst group will be any \(k\) in the set \([x+1,x+y]\).

Case 3: \(\gamma =0\). This case is computed similarly, with the details left to the reader.


  1. 1.

    The numbers \(c_*,c^*\) are critical in the sense that as cost \(c\) crosses these numbers, the optimal voting rule changes. For this population, we are able to fully determine \(c_*,c^*\) if we know the initial gains from passage of the proposal, \(\gamma , \delta\), of members of \(Y,Z\), together with the incremental improvement in the proposals.

  2. 2.

    It is interesting to note that the optimal \(k\) does not depend on the population sizes \(x,y,z\), nor on the initial utility from passage of the proposal, \(\beta\), for the group \(X\).

  3. 3.

    Because \(c^*=\alpha\) in Case 2, one simple conclusion follows. If Group Y (the middling group) is initially opposed to the measure, the optimal voting rule will be \(k\le x\) if and only if per round decision costs exceed the incremental improvement in the proposal (i.e., \(c>\alpha\)).

  4. 4.

    The methods above can be used to determine the optimal voting rule \(k\), as a function of cost \(c\), from the perspective of the \(typical\) voter. However, in this case, the critical cost value \(c^*\) depends on \(y\) and \(z\).

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Dougherty, K.L., Edward, J. & Ragan, R. The value of formalism: re-examining external costs and decision costs with multiple groups. Public Choice 163, 31–52 (2015).

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