Ethical considerations on quadratic voting

The range of payments i will potentially make in the voting mechanism is likely to include only payments substantially smaller than \({\hat{u}}_i\). For simplicity, it is natural to assume that \({\hat{U}}_i\) is approximately quasilinear for payments as large as \({\hat{u}}_i\). This would validate our assumption that i is indifferent between \((0,w^e_i)\) and \((1,w^e_i - {\hat{u}}_i)\). However, we do not want to assume quasilinearity for arbitrarily large payments.

Independence is the most favorable assumption for QV to achieve its efficiency benefits.

Of course, this does not mean that valuations are independent of wealth.

If the utility functions \({\hat{U}}_i\) are quasilinear—that is, of the form \({\hat{U}}_i(x,w) = {\hat{u}}_i x + \beta _i w_i\)—and if in addition \(\beta _i=1\) for all i, then one might think that the only ethically reasonable way to calibrate utilities would be to set \(\alpha _i=\alpha\) for some \(\alpha\) and all i, since if \(\alpha _i > \alpha _j\), then the utilitarian would want to make a boundless transfer from j to i. However, by the same token, it is not reasonable to believe to begin with that ethical utility—or decision utility for that matter—is quasilinear for arbitrarily large wealth transfers. Indeed, in Sect. 2.1.1, we only assumed that the \({\hat{U}}_i\) are (approximately) quasilinear when wealth transfers are not too large.

Vickrey (1945) was a precursor to this analysis.

If, instead, we assumed that g is strictly concave, but the slope of g changes sufficiently slowly, then nothing of substance would change.

For example, suppose that the question is whether to create a new public park. Suppose that the enjoyment that voter i would get from the park \(h_i\) does not vary with i’s wealth, but each citizen would have to make a payment of T to finance the park independently of her wealth. Then i’s net utility from the park is \(h_i - b_i T\), where \(b_i=g^{\prime }(w^e_i)\). Since the marginal utility of wealth \(b_i\) does not change over the range of transfers that might occur in the voting mechanism, this example satisfies our assumptions with \(u_i(w_i) =h_i - g^{\prime }(w_i)T\). As another example, the public decision may involve building a facility that the agent may only want to use if she is poor.

In large elections, the election proceeds should be small relative to aggregate willingness to pay. This follows from results of Lalley and Weyl (2016) and Weyl (2017). Taking this into account would itself limit the effect of the refund, and so makes it less important to neutralize its effect, as we do below, although it is still useful to do so, for analytical precision.

Positing non-citizens simplifies some mathematical expressions below, but nothing of substance would change if we assumed that every dollar raised from citizen i is transferred to some other citizen j with the same marginal utility of wealth as i.

In particular, \(g(w^e_i) = u_i + g(w^e_i- {\hat{u}}_i) \Leftrightarrow g(w^e_i) = u_i + g(w^e_i)- b_i{\hat{u}}_i \Leftrightarrow {\hat{u}}_i = \frac{u_i}{b_i}\).

To see this, assume wlog that \(b_i < b_j\), consider the case where \(u_i\,<\,0\,<\,u_j, |u_j|>|u_i|, |\frac{u_i}{b_i}| > |\frac{u_j}{b_j} |\), and \(u_\ell =0, \forall \ell \in N {\setminus } \{i,j\}\).

This departs from the assumption above that g is piecewise linear; to be consistent with this, we may assume that g is a piecewise linear approximation to \(\log\).

See footnote 14.

In Sect. 2.3, we assumed, first, that \(U_i\) measures ethical utility. Second, we assumed that \(U_i\) also represents i’s von Neumann–Morgenstern preferences over lotteries over outcomes. The second assumption (combined with the assumption of a common utility of wealth function g) is what might suggest that risk preferences determine ethical utility. We made the second assumption primarily to make the utilitarian analysis of Sect. 2.3.1 as completely parallel to the preceding analysis of QV in Sects. 2.1–2.2. However, the claim that there should exist a common utility function that simultaneously satisfies the first and second assumptions above, a claim that is integral to Harsanyi’s version of utilitarianism, is a philosophically contentious one. Even if we were to grant that \(U_i\) can play both roles, in reality, different agents will have different risk preferences. If \({\hat{g}}_i\) and \({\hat{g}}_j\) represent i and j’s preferences over wealth gambles, then so do \(\alpha _i{\hat{g}}_i\) and \(\alpha _j{\hat{g}}_j\) for any positive numbers \(\alpha _i\) and \(\alpha _j\), and it will require an ethical judgment to calibrate the utilities—that is, to determine the ethically correct ratio \(\alpha _i{/}\alpha _j\)—before these utilities can be added up to generate an ethically significant quantity in the way required by utilitarianism. These considerations show that the notion that we can empirically infer the marginal utility of wealth from observation without making ethical value judgments is mistaken.

This follows from the analysis of Lalley and Weyl (2016) and Weyl (2017).

For a discussion from a very different perspective, but that echoes the themes raised here, see Buchanan (1959).

With respect to the financing of public decisions, Kaplow (2004) writes, “of the many (consistent) ways that one could adjust the income tax system to achieve budget balance, ideally the intrinsic features of providing a public good or correcting an externality would not become entangled with concerns about the proper extent of income redistribution. Therefore, a distribution-neutral approach to policy analysis is warranted.” Our paper deals with a somewhat different topic, voting institutions, but we agree with the basic sentiment that public decisions should not constantly be entangled with unrelated distributive questions; rather public decisions should be decided on their merits. However, putting aside broader moral considerations that go beyond the utilitarian framework, we think that the utilitarian criterion is superior to the efficiency criterion as a measure of the intrinsic merit of a policy.

We have verified with Weyl that this is the view he was articulating here.

Pareto efficiency is a desirable property when we are near an optimum; if for example, we are far from a utilitarian optimum (e.g., one person has all of the resources), then Pareto efficiency in and of itself has little merit.

A characterization of the boundary between public and private decisions is beyond the scope of this paper.

One may argue that viewed from a broader perspective, the decisions may not be equally efficient. If one considers the entire system of democratic government in contrast with the authoritarian system, with all of their consequences, the democratic system may be more efficient. This is efficiency at the level of the entire political system, not efficiency at the level of the binary decision under consideration. We do not think that this broader notion of efficiency can capture the motivation behind the legitimacy requirement. In any event, the formal efficiency results for QV establish that QV is efficient at the level of binary decisions, and so we focus on this narrower notion of efficiency.

In his contribution to this volume, Ober (2016) argues that QV has trouble with a requirement of democratic legitimacy. While we are friendly to his conclusion, we do not base the legitimacy requirement on the claim that we have equal common interests in the distribution of public goods. On our account, democratic legitimacy is a higher order property that holds when people are consulted and have equal opportunities to influence the outcome of public decisions. On our account, QV has problems with democratic legitimacy even in cases where legislation concerns the distribution of private goods, as well as in cases where individuals have different levels of interest in public goods.

This is related to the issues raised in Sect. 3.6. Agenda-setting is only important against a background of multiple alternatives. That drafting of legislation, not just voting on legislation is important implies that there are more than two alternatives.