Public Choice (2017) 172:109–124 DOI 10.1007/s11127-017-0406-3

Uncertainty, polarization, and proposal incentives under quadratic voting John W. Patty1 • Elizabeth Maggie Penn1

Received: 31 August 2017 / Accepted: 30 September 2017 / Published online: 16 February 2017 Ó Springer Science+Business Media New York 2017

Abstract We consider the quadratic voting mechanism (Lalley and Weyl in Quadratic voting. Working paper, University of Chicago, 2015; Weyl in The robustness of quadratic voting. Working paper, University of Chicago, 2015) and focus on the incentives it provides individuals deciding what proposals or candidates to put up for a vote. The incentive compatibility of quadratic voting rests upon the assumption that individuals value the money used to buy votes, while the budget balance/efficiency of the mechanism requires that the money spent by one voter by redistributed among the other voters. From these assumptions, we show that it follows that strategic proposers will have an incentive to offer proposals with greater uncertainty about individual values. Similarly, we show that, in an electoral setting, quadratic voting provides an incentive to propose candidates with polarized, non-convergent platforms. Keywords Collective choice  Efficiency  Public goods  Mechanism design  Voting

The definition of the alternatives is the supreme instrument of power; the antagonist can rarely agree on what the issues are because power is involved in the definition. He who determines what politics is runs the country, because the definition of alternatives is the choice of conflicts, and the choice of conflicts allocates power.1

1

Schattschneider (1960, p. 68), emphasis in the original.

We thank Sean Ingham and Glen Weyl for their helpful comments. & John W. Patty [email protected] Elizabeth Maggie Penn [email protected] 1

University of Chicago, Chicago, IL, USA

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1 Introduction Lalley and Weyl (2015) propose a mechanism, quadratic voting (QV), that incorporates intensities of individuals’ preferences when making a binary collective decision. Lalley and Weyl show that, in equilibrium, QV is asymptotically ex post efficient when individuals have independently distributed private values. While it is not the only mechanism to achieve such efficiency, QV is an appealing mechanism because the voters’ ‘‘ballots’’ and the mechanism’s calculations of both which outcome is to be implemented and individuals’ charges/rebates (i.e., the ‘‘rules of the mechanism’’) are straightforward. In addition to implementing efficient collective decisions, QV has the virtue of being budget balanced. It achieves incentive compatibility (and hence efficiency) by charging individuals based on the number of votes they decide to cast and, to maintain budget balance, distributes the price paid by any voter to all of the other voters. Our focus in this article is on the incentives the budget balanced nature of QV creates for individuals who can propose items for social consideration in the QV mechanism. We show that QV creates some surprising and interesting incentives. In the setting considered by Lalley and Weyl (2015), we show that QV creates an incentive to offer proposals with higher levels of uncertainty about individuals’ preferences. In addition, this is sometimes, but not always, at odds with majority preferences. Extending that environment, we then show that the proposer can have an incentive to offer proposals offering lower social welfare. Specifically, when individuals’ preferences are independently and Uniformly distributed, the proposer’s monetary incentives lead him or her to prefer distributions that have zero expected value. This is because the symmetry of such distributions implies that the number of votes bought by each voter is strictly increasing in the strength of the voter’s preference for the project for all possible preferences the voter might have. We then extend the application of QV to a setting, more in line with most classical models of politics, in which individuals have probabilistic ‘‘spatial’’ preferences. We demonstrate using a simple example that, because of its budget balanced nature, QV creates an incentive to offer proposals that are more polarized (i.e., are more ‘‘ideologically distant’’ from each other) than they would be if the collective decision were made by majority rule. Specifically, we show that the median voter—the voter whose most-preferred policy (‘‘ideal point’’) should be chosen under majority rule (Black 1948)—has a strict incentive to submit two policies not equal to his or her own ideal point. The structure of the example illustrates that this incentive is entirely due to the monetary incentives created by the QV mechanism’s budget-balancing rebates. We conclude by considering how one might mitigate these proposal incentives. We argue that the only perfect solution is that proposers not be allowed to participate in the QV mechanism. In the absence of side-payments between individuals within the mechanism and the proposer, the incentives we identify are then irrelevant. We discuss a few theoretical and practical challenges one faces when considering implementing such an exclusion.

2 The basics of quadratic voting The basic setting for quadratic voting (see Weyl 2015) is one in which a group of n individuals, N ¼ f1; . . .; ng, is choosing whether to approve some exogenous project. Each

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individual i 2 N has a privately observed net value from approval of the project, which we denote by ui 2 R. If ui \0, individual i would prefer that the project not be approved, while he or she would prefer that the project be approved if ui [ 0. Lalley and Weyl (2015) consider symmetric environments in which each individual’s value, ui , is distributed in an iid fashion according to a continuous probability distribution F with density f and possessing full support on a finite interval ½u; u, with u\0\u. The mean and standard deviation of ui are denoted by l  0 and r, respectively. The assumption that the ui are independent is equivalent to assuming that the individuals’ preferences are ‘‘private values’’: each voter’s preference for or against the project is independent of all other voters’ preferences. If the project fails, each voter receives a payoff of 0. To decide whether to approve the project, QV asks each voter to choose a number of votes, vi 2 R, where vi \0 implies that i is voting against the project and vi [ 0 implies that i is voting in favor of approving the project. We denote P an arbitrary profile of vote choices by v  fvi gi2N . The project is implemented if V  i2N vi  0. Lalley and Weyl (2015) assume that individuals have quasi-linear preferences and if a voter buys P vi votes, v2j

j6¼i that voter is charged v2i for his or her votes and receives a rebate equal to n1 . These assumptions imply that the QV leads to the following payoff for each individual i if the project passes: P 2 j6¼i vj 2 : Ui ðui ; vÞ ¼ ui  vi þ n1

If the project fails, each individual receives: P Ui ð0; vÞ ¼

v2i

þ

2 j6¼i vj

n1

:

2.1 Linear voting and efficiency The QV mechanism willP implement the efficient outcome (the choice that maximizes the sum of voters’ payoffs, i2N ui ) if all players vote according to a rule of the following form: v ðui ; nÞ ¼ bðnÞui ;

ð1Þ

where bðnÞ [ 0 is a scaling factor. To be clear, (1) is not generally an equilibrium strategy profile for arbitrary distributions of ui , but it will be in the cases we examine here. Furthermore, in our discussion of the features of QV more generally, we are always presuming that individuals do use a linear voting strategy of the form described by (1). This allows us to focus on the ‘‘best case’’ for QV in terms of efficiency, which is the principal justification for adopting quadratic voting.2 With this in hand, we now move to consider the incentives QV induces when one is considering what projects or proposals to submit for consideration by the a group using the QV mechanism.

2

Also, we consider it reasonable from a behavioral standpoint that individual choices within a setting QV would be well approximated by a linear voting rule consistent with Eq. (1).

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3 What do we vote on? QV and proposals Viewed broadly, collective decision-making involves at least two separate processes: how to make a choice (a ‘‘decision rule’’), and what to choose between (an ‘‘agenda setting rule’’). QV is a coherent decision rule for binary choices, but it is agnostic about agenda setting. That is, QV takes as given the binary choice that the group must adjudicate. Any practical implementation of QV must address the question of what proposals can be considered, how they are submitted for consideration, and by whom. Unsurprisingly, there is a huge literature on how agendas do, and should, operate in political settings.3 We tackle the problem only in the barest of forms, considering the preferences of a hypothetical, unitary ‘‘proposer’’ with respect to potential binary proposals that could be presented to a group for consideration using QV.

3.1 QV, rebates, and proposers’ incentives The budget-balanced nature of QV affects the incentives of potential proposers of alternatives. Specifically, any individual p 2 N who knows that he or she has a private value of up ¼ 0 for a specific proposal can only gain (and, in fact, will gain with probability 1) from making that proposal and having the group utilize QV to choose whether to implement it. This is in fact true for any individual p whose individual value up is sufficiently close to zero. To keep our presentation as simple as possible, we consider two simple settings of endogenous proposal generation. The first example retains the assumption that the private values are independently and identically distributed and considers the impact of the proposer having private information about his or her own private value (as well as public information about the distribution of other voters’ utilities) when comparing two or more potential proposals. In this setting, we explore both the voters’ and proposer’s preferences with respect to the variance of individual payoffs. We show that the proposer always prefers higher variance, while the other voters sometimes prefer very large variances to smaller variances, but prefer smaller variances among low variance proposals. The second example applies quadratic voting to the canonical setting of two-candidate competition in a unidimensional spatial model. Our assumptions in this example imply that voters’ payoffs are independently, but not identically, distributed. The focus of the example is on the median voter’s incentives with respect to the degree of divergence between the two candidates. That is, the example supposes that the median voter can choose what pair of platforms will subsequently compete for office in the quadratic voting mechanism.

3.2 Maximizing variance We first consider a simple thought experiment involving 3 voters in which one voter, ‘‘voter 3,’’ is comparing proposals about which he or she is indifferent (in the sense that he or she will not buy any votes, regardless of which proposal is put forward) and the payoffs of the other two voters are independently and identically distributed Uniformly between  d2 and d2, where d  0 is chosen by voter 3. We assume that voters 1 and 2 know that voter 3 is indifferent, so that they will play, in equilibrium, as if they are voting ‘‘against’’ only each other. Formally, suppose that N ¼ f1; 2; 3g and that ui is distributed as follows: 3

A small sampling of this literature, spanning various settings and notions of an ‘‘agenda,’’ include Kingdon (1984), Riker (1993), Rasch (2000), Cox and McCubbins (2005), and Perry (2009).

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 8 
for i 2 f1; 2g; for i ¼ 3:

In this setting, our main question is, given equilibrium voting behavior by voters 1 and 2, what are the voters’ preferences with respect to d?

3.3 Majority rule and preferences about d Before considering the QV equilibrium and the voters’ preferences over d, we take a brief detour to consider the baseline case of majority rule. Regardless of the tie-breaking rule, all voters are indifferent about d: the expected payoff of every player is zero, for all values of d: EUiMajority ðdÞ ¼ 0 8d: This calculation is key to understanding the link between individuals’ expected intensities of preference and their induced preferences between QV and majority rule. As we will see below, voters 1 and 2—those with higher expected intensity of preference (i.e., E½ju1 j ¼ E½ju2 j [ E½ju3 j ¼ 0) – will strictly prefer majority rule to quadratic voting when d is small, while voter 3 strictly prefers QV for all values of d [ 0.

3.4 Quadratic voting and preferences about d We first consider the preferences of individual 3 with respect to d. In a symmetric Bayes Nash equilibrium, voters 1 and 2 will use the following strategy: v ðui ; dÞ ¼ bðdÞui ; where b(d) is a constant that depends on d. For either i 2 f1; 2g, ( Pruj ½v [  bðdÞuj ui  v2 if v  0;  EUi ðv; v ; dÞ ¼ Pruj ½v\  bðdÞuj ui  v2 if v 0; In an abuse of notation we have dropped the voter’s rebate from this expression because the voter’s own choice of v doesn’t affect the rebate he or she receives. Focusing on the case of ui [ 0 (and thus v [ 0),4 EUi ðv; v ; dÞ ¼ Pr½v [  bðdÞuj ui  v2 ; uj   v 1 þ ui  v2 ; ¼ dbðdÞ 2 so that the first order condition is

4

The case of ui \0 is symmetric.

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   o o v 1  EUi ðv; v ; dÞ ¼ þ ui  2v ¼ 0; ov ov dbðdÞ 2 ui  2v ¼ 0; ¼ dbðdÞ which is satisfied by v ðui ; dÞ ¼

ui ; 2dbðdÞ

which, after solving for bðdÞ, yields ui 1 ) bðdÞ ¼ ¼ bðdÞui ) bðdÞ ¼ 2dbðdÞ 2dbðdÞ

rffiffiffiffiffi 1 : 2d

Accordingly, the equilibrium strategy for voters 1 and 2 is ui v ðui ; dÞ ¼ pffiffiffiffiffi : 2d Then the expected cost for either voter 1 or 2 (i.e., the expectation of the square of the number of votes bought by either voter) is equal to E½ðv ðui ; dÞÞ2  ¼d1 ¼

Z

d 2

ðv ðx; dÞÞ2 dx;

d2

d : 24

Voter 3 will receive half of each of voters 1’s and 2’s spending, so in expectation he or she d will receive 24 as his or her rebate. Because voter 3 is indifferent about the proposal, his or her expected payoff for any given d is simply equal to his or her conditional expected rebate in equilibrium, given d, which we denote Ri ðdÞ: EU3 ðdÞ ¼ E½R3 ðdÞ ¼

d : 24

Accordingly, voter 3 strictly prefers proposals with larger variances (i.e., larger values of d). The preferences for voters 1 and 2 with respect to d are more complicated to derive. At first blush, one might think that voters 1 and 2 are indifferent about d beyond the effect of d on their expected vosts. However, this is not correct because, in equilibrium, the project will be approved only if the sum of the two voters’ payoffs from it is greater than zero. Thus, the expected sum of the voters’ policy payoffs, given d, is equal to: Z d Z d 2 X 2 2 v ðui ; dÞ [ 0 ¼ ðu1 þ u2 Þ du2 du1 ; E½u1 þ u2 j i¼1

d 2

u1

d3 ¼ : 12 The expected policy payoff for either voter 1 or voter 2 conditional on passage is then, by symmetry, half of this amount:

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E½ui j

2 X i¼1

115

v ðui ; dÞ [ 0 ¼

d3 for both i 2 f1; 2g: 24

Thus, because E½ui jd ¼ 0 for all i 2 f1; 2; 3g and the QV mechanism is budget balanced: E½R1 ðdÞ þ E½R2 ðdÞ þ E½R3 ðdÞ ¼ 0 for all d; the expected payoffs of players 1 and 2, for any given d, are as follows: EU1 ðdÞ ¼ EU2 ðdÞ ¼

d3 d  : 24 48

Voters 1 and 2 strictly prefer not using the QV mechanism whenever the variances are sufficiently small. Specifically, whenever pffiffiffi d3 d 1 d ¼ dðd2  1=2 Þ\ ) d 2 ð0; 2Þ;  16 24 48 24 voters 1 and 2 each derive a negative expected payoff from the QV mechanism relative to simply rejecting the proposal a priori. In terms of their preferences over d, the expected payoffs of voters 1 and 2 are minimized at: rffiffiffi   d d3 d d2 1 1 ¼  ¼0)d¼  ; dd 24 48 6 8 48 so that, for rffiffiffi 1 d\ ; 6 voters 1 and 2 each strictly prefer marginal decreases in d to marginal increases in this uncertainty. Note that—because social welfare is (by assumption) invariant to the QV mechanism’s ex post transfers from voters 1 and 2 to voter 3—the expected social welfare from QV is always greater than that from majority voting (which, assuming the indifferent voter votes against passage, is d8). However, if the group uses majority rule to determine whether to use pffiffiffi QV for a given d, then QV would be rejected whenever d\ 2. This highlights an implementation challenge—when deciding whether to use QV, QV itself would always choose QV (the expected social welfare of QV is always positive—voter 3 would buy more votes than voters 1 and 2 combined when voting over whether to use QV to approve the project), but majoritarian procedures would reject it when uncertainty is small. Figure 1 displays the voters’ net expected payoffs from QV, relative to majority rule, as a function of d. The fact that QV is unanimously preferred for large, but not small, values of uncertainty illustrates the information aggregation value of the QV mechanism—when d is small, the potential gains from the information aggregated through QV is not sufficient to compensate voters 1 and 2 for the amount that they expect to ‘‘pay to’’ voter 3. As d grows, then the conditional expected value of the project to voters 1 and 2 conditional upon it being approved grows and eventually exceeds the amount that they will pay to voter 3 in expectation.

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Voters 1 & 2 QV Voters 1 & 2 Majority Rule Voter 3 QV

δ Fig. 1 Net expected payoffs from QV versus majority voting versus Uncertainty (d)

The conflict between voters 1 and 2 and voter 3 for small values of d is problematic, but it results from the fact that voter 3 is included in the QV mechanism. Under the presumption that it is known that voter 3 is indifferent about the proposal, then obviously he or she should be excluded from the mechanism. Doing so implies that voters 1 and 2 would prefer consider the proposal regardless of the value of d. We return to this question in Sect. 5, but note at this point that the comparisons illustrated in Fig. 1 are robust to allowing for voter 3’s private value to be drawn according to a Uniform ½e; e distribution for small values of e.

3.5 Conflict between social welfare and proposal incentives Suppose now that voter 3 is again indifferent about the proposal and the payoffs of the other two voters are independently and identically distributed Uniformly between l  1=2 and l þ 1=2 , where l 2 ½0; 1=2  is chosen by voter 3.5 We assume that voters 1 and 2 know that voter 3 is indifferent, so that they will play, in equilibrium, as if they are voting ‘‘against’’ only each other. Formally, suppose that N ¼ f1; 2; 3g and that ui is distributed as follows:  U ½l  1 =2 ; l þ 1 =2  for i 2 f1; 2g; ui  0 for i ¼ 3: Again, our main question is, given equilibrium voting behavior by voters 1 and 2, what are voter 3’s preferences with respect to l? Supposing again that voters 1 and 2 are using a linear vote strategy, v ðui ; lÞ ¼ bðlÞui ; the EUi ðv; v ; lÞ for i 2 f1; 2g is defined as follows:

5 The situation is symmetric if one allows for l 2 ½ 1=2 ; 0Þ, and jlj [ 1=2 implies that neither voter 1 nor voter 2 will bid in equilibrium, meaning that voter 3 would receive zero rebate in such situations.

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8 v 2 > < Pruj ½uj [ bðlÞui  v  EUi ðv; v ; lÞ ¼ > Pr ½u \ v u  v2 : uj j i bðlÞ

if v  0; if v 0;

This reduces to  8 v > 1 > u i  v2 = þ l þ > 2 > bðlÞ > <  v EUi ðv; v ; lÞ ¼ 1 > =2  l  u i  v2 > > bðlÞ > > : ui  v2

v 2 ½0; 1=2  l; bðlÞ v 2 ½ðl þ 1=2 Þ; 0; if bðlÞ otherwise; if

so that the first derivative with respect to v is 8 u v i > 2 ½0; 1=2  l; if > > bðlÞ  2v > bðlÞ < o EUi ðv; v ; lÞ ¼ ui  2v if v 2 ½ðl þ 1= Þ; 0; > ov 2 > bðlÞ > bðlÞ > : 2v otherwise; which, after accounting for corner solutions at ui 2 fðl þ 1=2 Þ; 1=2  lg,6 is satisfied by 8 ðl þ 1=2 Þ ui > > if \  ðl þ 1=2 Þ; > > > 2bðlÞ bðlÞ > > < ui ui if 2 ½ðl þ 1=2 Þ; 1=2  l; v ðui ; lÞ ¼ 2bðlÞ bðlÞ > > > > 1=  l > ui > 2 > : if [ 1=2  l; 2bðlÞ bðlÞ which, solving for bðlÞ, yields ui 1 ) bðlÞ ¼ ¼ bðlÞui ) bðlÞ ¼ 2bðlÞ 2bðlÞ

rffiffiffi 1 : 2

Accordingly, the equilibrium strategy for voters 1 and 2 is 8 ðl þ 1=2 Þ ui > > pffiffiffi \  ðl þ 1=2 Þ; if > > > bðlÞ 2 > > < ui ui pffiffiffi if 2 ½ðl þ 1=2 Þ; 1=2  l; v ðui ; lÞ ¼ bðlÞ > 2 > > > 1=  l > ui > 2 > p ffiffiffi : [ 1=2  l: if bðlÞ 2 The only difference between this case and that of d in the setting considered above is that the strategy for voter 1 and voter 2 is truncated: when ui [ 1=2  l for i 2 f1; 2g, the 1 = l number of votes bought by voter i is a constant value ( p2 ffiffi ), because this value guarantees 2

that voter i’s preferred outcome (the project being approved) will occur with certainty. 6

Note that ui ðl þ 1=2 Þ occurs with probability zero for l 6¼ 0.

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With this in hand, the expected cost for either voter 1 or 2 (i.e., the expectation of the square of the number of votes bought by either voter) is equal to Z lþ 1=2 E½ðv ðui ; dÞÞ2  ¼ ðv ðx; lÞÞ2 dx; l 1=2

¼

Z

=2 l 

1

l 1=2

¼

 2 Z lþ 1=2  1 x 2 =2  l pffiffiffi dx þ pffiffiffi dx; 1= l 2 2 2

1 ð1  2lÞ2 ð1 þ 4lÞ: 24

Thus, because EU3 ðlÞ ¼ E½R3 ðlÞ ¼

1 ð1  2lÞ2 ð1 þ 4lÞ; 24

and 1 ð1  2lÞ2 ð1 þ 4lÞ 24 is decreasing on l 2 ½0; 1=2 , voter 3 prefers lower values of l. His or her most-preferred proposal offers zero expected individual welfare from the proposal, l ¼ 0. pffiffiffi We saw in Sect. 3.2 that all three players gain from QV when d  2. Furthermore, it is clear that voters 1 and 2 strictly prefer larger values of l: this simultaneously increases each of the voters’ expected policy payoffs and reduces each of their expected costs of voting. Thus, the three voters have conflicting preferences over l: voters 1 and 2 each most prefer l as large as possible and voter 3 prefers l ¼ 0. It is simple to see that this is a general conclusion for l  0: when l  1=2 , voters 1 and 2 can ensure positive policy payoffs for zero cost, and voter 3 receives nothing.7 Before concluding, it is useful to note that voter 3 is indifferent about l when the project is approved under majority rule. Within this setting, for any given l, there is unanimous agreement among the 3 voters that QV is better than majority rule voting. However, if the group were to choose between majority rule and QV prior to voter 3 setting l, it is possible to construct situations in which voters 1 and 2 would strictly prefer majority rule to QV.8

3.6 QV and proposal incentives in the private values case The two examples illustrate that QV can induce a ‘‘proposer’’ who has little to no direct interest in whether project is approved to have a personal incentive counter to those of society. The first example demonstrates that such a proposer’s preferences are monotonic in the variance of the others’ payoffs, while those voters’ preferences are non-monotonic with respect to that variance among low variance proposals. In that example, there is significant social agreement: if the variance can be made ‘‘large enough,’’ then the voters and proposer essentially have congruent preferences. P This presumes that, when vi ¼ 0, the project is approved. Otherwise, a pure strategy equilibrium will not exist in the QV for l  1=2 . 7

8

For example, suppose that, after majority rule or QV is selected as the collective choice mechanism, voter 3 chooses to whether to pay a cost c [ 0 in order to choose l and, if not, then (say) l is set to some exogenous value, l^ [ 0. If c is small enough relative to l^ and QV is chosen as the collective choice mechanism, then voter 3 would pay the cost c and set l ¼ 0.

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More disturbingly, though, the second example essentially demonstrates a complete divergence of interests with respect to the expected individual payoffs from the proposal. The common link between the examples is a desire by the proposer to maximize the expected number of votes that are bought by the voters. This preference, of course, is induced by the budget balanced nature of QV: if the proposer gets a share of the other voters’ expenditures, then the proposer seeks to maximize the ‘‘expected conflict’’ between the other voters.9 The independent distribution of individual payoffs implied by the private values environment implies a positive link between the variance of individual payoffs and the voters’ ex post expected payoffs. This is a feature of the ex post efficiency of the QV mechanism, but it is dependent on the private values assumption. We now turn to a different, classical environment—probabilistic, ‘‘spatial’’ preferences—that violates this assumption and provides some insight into how QV can affect proposal incentives in environments that arguably better mirror individuals’ political preferences.

4 Applying QV to spatial electoral competition The example in Sect. 3.2 considered how the QV mechanism influences voters’ preferences over uncertainty about their own payoffs. In the private values environment studied by Lalley and Weyl (2015), we showed in Sect. 3.2 that there is unanimous agreement among the voters that, ceteris paribus, really large variances are better than small ones. We consider a setting with three voters, N ¼ f1; 2; 3g, with ideal points given by p ¼ f1  t1 ; 0; 1 þ t3 g, where t1 and t3 are independently and identically distributed on U½0; d for an exogenous and commonly known value d [ 0 and the voters will choose between two candidates, L and R, with corresponding positions, or ‘‘platforms’’ of x 0 (for candidate L) and x  0 (for candidate R). Finally, we assume that the voters’ policy payoffs are ‘‘spatial’’ and represented by the quadratic loss of the distance between the voter’s ideal point and the winning candidate’s platform (x 0 for candidate L and x  0 for candidate R) as follows: ( ð1 þ t1  xÞ2 if W ¼ L; u1 ðW; t1 Þ ¼ ð1 þ t1 þ xÞ2 if W ¼ R;  2 x if W ¼ L; u2 ðWÞ ¼ x2 if W ¼ R; ( ð1 þ t3 þ xÞ2 if W ¼ L; u3 ðW; t3 Þ ¼ ð1 þ t3  xÞ2 if W ¼ R; An example of these payoffs, for an arbitrary pair of realized types, t1 and t3 , is displayed in Fig. 2. The question we are interested in is what voter 2’s preferences over x look like in this setting if the voters will use QV to choose between x and x. As with the previous examples, we construct, for any value x  0, a symmetric voting equilibrium in which voters 1 and 3 are playing the same linear strategy, where

9

This finding is in line with the experimental findings of Goeree and Zhang (2016). In an 11-person private values setting, Goeree and Zhang find that ‘‘moderate’’ voters—those whose values were closer to 0— benefitted from the redistributive aspect of quadratic voting.

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-1-δ

-1

1

0

-1-t1

1+δ

1+t3

u1(x,t1)

u2(x) u3(x,t3)

Fig. 2 A three-voter spatial preferences example

v ðti ; xÞ ¼ bðxÞti denotes, for voter 1, the number of votes bought for candidate L, given t1 2 ½0; d and, for voter 3, the number of votes bought for candidate R, given t3 2 ½0; d. By the assumption that t1 and t3 are iid Uniform[0; d], the probability that candidate 1 wins the election most votes, conditional on buying y maxt2½0;d ½vðtÞ votes and voter 3 following strategy v, is simply pðy; dÞ  Pr½vðt3 Þ\y ¼ Pr½t3 \v1 ðyÞ ¼

v1 ðyÞ y ¼ : d db

Considering voter 1 without loss of generality, his or her expected payoff from buying v maxt2½0;d ½vðtÞ votes is EU1 ðv; x; dÞ ¼ pðv; dÞuðx; 1  t1 Þ þ ð1  pðv; dÞÞuðx; 1  t1 Þ  v2 ; which is maximized by v ðt; xÞ ¼

2xð1 þ t1 Þ : dbðxÞ

After solving for b(x) as in the previous examples, the symmetric voting equilibrium between voters 1 and 3, for any pair fx; xg, is given by rffiffiffiffiffi 2x v ðt; xÞ ¼ ð1 þ tÞ: d The expected sum of the costs of voters 1 and 3 (i.e., the expected sum of the squares of the number of votes bought by voters 1 and 3) is Rd    ðv ðt; xÞÞ2 dt 4x d3 ¼ 2 d þ d2 þ : 2 0 d 3 d In equilibrium, each candidate wins with probability 1=2 (and voter 2’s policy utility is x2 with certainty). Furthermore, voter 2 will buy zero votes with certainty and will receive

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half of the sum of the two other voter’s payments.10 Thus, the median voter (voter 2)’s expected payoff, given x and d, is   2x d3 EU2 ðx; dÞ ¼  x2 þ 2 d þ d2 þ 3 d which is maximized by 1 d x ðdÞ ¼1 þ þ : d 3

ð2Þ

Two points can be drawn from the expression for x ðdÞ in Eq. (2). First, when considering what candidates/platforms to present for consideration using QV, the median voter would choose divergent platforms. This stands in stark contrast to what he or she would choose if the group were going to choose using majority rule. In that case, it is clear that voter 2 would choose x ¼ 0. After all, voter 2’s motivations in that situation are entirely based on his or her policy preferences. Second, Fig. 3 illustrates how the median voter’s most preferred policy, x ðdÞ, varies with d. If one thinks of the magnitude of x ðdÞ as ‘‘polarization,’’ then what is most interesting, beyond the ubiquitousness of the median voter’s preference for polarization, is the nonmonotonicity of his or her most preferred level of polarization with respect to his or her own uncertainty about the other voters’ ideal points. For small values of such uncertainty, the median voter’s optimal level of polarization is decreasing in this uncertainty. For large values of uncertainty, the optimal level is increasing in the level of uncertainty. Because d represents not only voter 2’s uncertainty about the ideal points of voters 1 and 3, but also the uncertainty of voters 1 and 3 with respect to each others’ ideal points, this reflects two competing forces that QV induces between voters 1 and 3 in this setting. The pivotality of an extra vote increases as the level of d decreases, but the probability of one or both voters having an ‘‘extreme’’ type (and thus be willing be to buy a large number of votes) increases as d increases. Voter 2, in pursuit of a large rebate from the cost of the votes bought by voters 1 and 3, is interested in securing large buys of votes with high frequency while also minimizing polarization (i.e., x), due to the policy costs imposed on voter 2 by more polarized policy choices. The non-monotonicity of Fig. 3 reflects voter 2’s attempts to balance these competing forces, for various levels of uncertainty (i.e., d).

4.1 Implications for QV It is important to note that QV is most easily applied to binary choices. When faced with such choices, the nature of individual preferences is relatively unimportant when considering both how one should vote (i.e., incentive compatibility) and, relatedly, how one should elicit individual preferences (i.e., the ‘‘ballot structure’’).11 10 Formally, voter 2 will receive half of each of their payments. This matters only when we allow for more than 3 voters. The case of three voters maximizes the strength of voter 2’s incentive to manipulate x in pursuit of a larger rebate from the ensuing QV election. 11 What is potentially very important—but remains very important when the group must choose between three or more mutually exclusive options—is the nature of individuals’ information about their own preferences. We do not touch upon this issue in any depth in this article, but allowing for correlation between voters’ true preferences in the presence of asymmetric and incomplete information (at the voter level) about these preferences would seem to present a difficult challenge for someone interested in achieving (perhaps even approximate) ex post efficiency.

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Fig. 3 The median voter’s optimal polarization under QV versus uncertainty (d)

Prominent examples of political and social choices that are binary in nature are rarely confronted exogenously. That is, issues like property tax reform, gay marriage, charter schools, legalizing marijuana, and ‘‘right to work’’ laws are brought to the table by individuals and groups within society. As the quoted passage from Schattschneider eloquently describes, the power to offer such alternatives—specifically, the power to secure and direct governmental/social attention at an issue—is a potent one. This potency is unavoidable; it infects any collective decision process. However, the efficiency of QV is based upon a combination of imposition of individual costs and the return of the sum of these costs to the other voters. As such, the use of QV by a group necessarily creates a marginal incentive for a voter with (likely to be) weak preferences about some binary issue to secure consideration of that issue by the group. We now turn to concluding thoughts, including the implications of this incentive and how an institutional designer might mitigate it.

5 Conclusion Quadratic voting is a simple and appealing mechanism for collective choice in large groups making choices in private values environments. Our examples are intended to illustrate the incentives that QV—through its budget balancing transfers—provides to individuals who can make proposals for consideration by a group using the QV mechanism. Viewed broadly, the examples suggest that the use of QV can provide an incentive for individuals to propose policies imbued with greater uncertainty and/or expected conflict between the other voters. While promoting such uncertainty can in some case be aligned with social interests (as is sometimes the case in the example considered in Sect. 3.2), the incentives of a potential proposer can come into direct conflict with maximizing social welfare (as in the examples considered in Sects. 3.3 and 3.3). Those examples provide constructive evidence of a conflict between ex ante majority rule and (known) ex post social efficiency. Before discussing avenues for future work, we briefly consider the cause of this conflict in our examples and possible means of mitigating this tension.

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5.1 Excluding the disinterested As mentioned earlier (page 9), the pathological proposal incentives that emerge in the examples presented in Sects. 3.2 and 3.3 are most easily avoided by excluding the indifferent voter (voter 3) from the QV mechanism. This solution is definitely consonant with the example as presented, but use of this solution is difficult to implement in practice. As we mentioned in Sect. 3.2, the example is robust to allowing the indifferent voter to have strict preferences with probability 1. As long as the variance of voter 3’s private values is smaller than those of voters 1 and 2, voter 3 will receive a positive net rebate from voters 1 and 2 with very high probability, so that there will be proposals that voter 3 will strictly prefer considering but voters 1 and 2 will strictly prefer not considering. This raises the issue of exactly how small an individual’s stakes ‘‘need to be’’ in order to justify excluding him or her from the mechanism. Thinking about more general preference settings (e.g., those with some shared value aspects in the sense of correlation between voters’ individual preferences) immediately suggests more difficulties in this regard, because the stakes that one voter faces might be dependent in various ways on the stakes faced by his or her fellow citizens. Furthermore, and more broadly, regardless of the nature of individual preferences, a key justification for mechanisms such as QV is that they aggregate information that is otherwise difficult or impossible to obtain outside of the mechanism. Thus, presuming that one knows enough to accurately judge which individuals have ‘‘enough at stake’’ from any particular project to warrant their inclusion in the QV mechanism is at least seemingly at odds with a basic justification for using QV in the first place. More generally, any incentive-based mechanism that attempts to discriminate between those who have ‘‘proper’’ (i.e., policy-based) and ‘‘improper’’ (i.e., rebate-seeking) motivations will necessarily have to account for the QV’s rebate. That means that an incentive compatible ‘‘screening’’ mechanism might require outside funds, similar to the conclusions of the Myerson-Satterthwaite theorem’s conclusions about bilateral exchange mechanisms ( Myerson and Satterthwaite 1983).

5.2 Where to go? There are at least two promising paths of future research related to the topics we have addressed in this article; one is normative and the other centers on applications of QV. From a normative standpoint, our analysis suggests that it would be interesting to ask questions such as, when types are non-identically distributed, what determines whether an individual should be excluded ‘‘as a voter’’ from the QV mechanism? That is, we have established that there are situations in which the inclusion of an individual as a voter induces a majority preference that the QV mechanism not be used at all, in spite of the ex post efficiency of the mechanism. Can we say something more general about when such cases will arise, and characterize the types individuals who create such situations? From an applied standpoint, the spatial preferences setting examined in Sect. 4 provides a simple example of how to extend QV to richer, arguably more realistic, applied settings. We think this enterprise is useful not insomuch as QV will actually be used for electoral purposes but, rather, because QV is an elegant framework within which one might study the effects of preference intensity on important phenomena such as proposal generation, electoral campaigns, party formation, and lobbying. In other words, QV might be a useful addition to the applied modeling toolkit in political economy precisely because of its

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tractability. If one drops the budget balanced nature of QV, after all, the mechanism is analogous to an all-pay auction with incomplete information.12 QV provides a parsimonious framework within which to study the effects of ‘‘intensity-based’’ processes of collective choice on political actors and institutional designers.

References Black, D. (1948). On the rationale of group decision-making. Journal of Political Economy, 56(1), 23–34. Cox, G. W., & McCubbins, M. D. (2005). Setting the agenda: Responsible party government in the U.S. House of Representatives. Berkeley, CA: University of California Press. Epstein, G. S., & Nitzan, S. (2006). The politics of randomness. Social Choice and Welfare, 27(2), 423–433. Fang, H. (2002). Lottery versus all-pay auction models of lobbying. Public Choice, 112(3–4), 351–371. Goeree JK, Zhang J (2016) One man, one bid. Games and Economic Behavior. doi:10.1016/j.geb.2016.10. 003. Kingdon, J. W. (1984). Agendas, alternatives, and public policies. Boston: Little, Brown. Lalley, S. P., & Weyl, E. G. (2015). Quadratic voting. Working paper, University of Chicago Myerson, R. B., & Satterthwaite, M. A. (1983). Efficient mechanisms for bilateral trading. Journal of Economic Theory, 29(2), 265–281. Nti, K. O. (1999). Rent-seeking with asymmetric valuations. Public Choice, 98(3–4), 415–430. Perry, H.W. (2009). Deciding to decide: Agenda setting in the United States Supreme Court. Cambridge: Harvard University Press. Rasch, B. E. (2000). Parliamentary floor voting procedures and agenda setting in Europe. Legislative Studies Quarterly, 25(1), 3–23. Riker, W. H. (Ed.). (1993). Agenda formation. Ann Arbor, MI: University of Michigan Press. Schattschneider, E. (1960). The semisovereign people: A realist’s view of democracy in America. Holt, Rinehart and Winston: Austin. Siegel, R. (2009). All-pay contests. Econometrica, 77(1), 71–92. Weyl, E. G. (2015). The robustness of quadratic voting. Working paper, University of Chicago.

12 For more on applying such models to political economy questions, see Nti (1999), Fang (2002), Epstein and Nitzan (2006), and Siegel (2009).

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