Improved Metric Distortion for Deterministic Social Choice Rules

In this paper, we study the metric distortion of deterministic social choice rules that choose a winning candidate from a set of candidates based on voter preferences. Voters and candidates are located in an underlying metric space. A voter has cost equal to her distance to the winning candidate. Ordinal social choice rules only have access to the ordinal preferences of the voters that are assumed to be consistent with the metric distances. Our goal is to design an ordinal social choice rule with minimum distortion, which is the worst-case ratio, over all consistent metrics, between the social cost of the rule and that of the optimal omniscient rule with knowledge of the underlying metric space. The distortion of the best deterministic social choice rule was known to be between $3$ and $5$. It had been conjectured that any rule that only looks at the weighted tournament graph on the candidates cannot have distortion better than $5$. In our paper, we disprove it by presenting a weighted tournament rule with distortion of $4.236$. We design this rule by generalizing the classic notion of uncovered sets, and further show that this class of rules cannot have distortion better than $4.236$. We then propose a new voting rule, via an alternative generalization of uncovered sets. We show that if a candidate satisfying the criterion of this voting rule exists, then choosing such a candidate yields a distortion bound of $3$, matching the lower bound. We present a combinatorial conjecture that implies distortion of $3$, and verify it for small numbers of candidates and voters by computer experiments. Using our framework, we also show that selecting any candidate guarantees distortion of at most $3$ when the weighted tournament graph is cyclically symmetric.Comment: EC 201

What (If Anything) Can Economics Say About Equity?

Does economics have anything to teach us about the meaning of fairness? The leading practitioners of law and economics disagree. Judge Richard Posner argues that economics is largely irrelevant to distributive issues. Posner maintains that the most useful economic measure of social welfare is cost-benefit analysis (which he calls wealth maximization). But, he observes, this economic measure ratifies and perfects an essentially arbitrary distribution of wealth. Given an ethically acceptable initial assignment of wealth, rules based on economic efficiency may have some claim to be considered fair. On the critical issue of distributional equity, however, Posner apparently believes that economics has little to say. In contrast, Professors Louis Kaplow and Steven Shavell believe that economics can teach us nearly everything about equity. In Fairness Versus Welfare, they argue that there is only one viable notion of equity: resources should be distributed so as to maximize overall social welfare. As we will see, the full import of this argument is ambiguous. On the one hand, Kaplow and Shaven expressly concede that multiple ways of calculating social welfare might exist, so Wy might have to look beyond economics to determine the right one. On the other hand, much of their argument is implicitly predicated on a specific social welfare function, and in a footnote they give the argument for adopting this function universally. If a unique social welfare function is given, economic analysis would completely resolve all equity issues under their approach. Thus, although they do not say so explicitly, the book can be read to endorse a singl� definition of equity based on economic analysis. This reading would completely eliminate any independent role for judgments about equity. At the least, however, Kaplow and Shaven believe that economics can restrict value judgments about equity to a single, sharply defined place in policy analysis: the choice of an appropriate social welfare function. Posner, in turn, sees only modest merit in the Kaplow and Shaven theory of social welfare, except to the extent that it reduces in practice to his own theory of wealth maximization

The metric distortion of multiwinner voting

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of k alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the q-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of k and q: The distortion is unbounded when q≤k/3, asymptotically linear in the number of agents when k/3k/

Discounting Dollars, Discounting Lives: Intergenerational Distributive Justice and Efficiency

The view that intergenerational distributive justice and efficiency should be treated separately is familiar, yet controversial. This article elaborates the often-implicit justifications for separate treatment and provides a more express statement of how and when such treatment is appropriate. Substantial attention is devoted to an approach that holds constant the intra- and intergenerational distribution of well-being, which proves to be a valuable analytical device even for intergenerational policies that are not distribution neutral. Also explored are possible interrelationships between intergenerational distributive justice and efficiency, the choice of interest rate for discounting dollars, and how the present approach relates to those that would employ direct social weights to dollars at different points in time.

Bias In, Bias Out

Police, prosecutors, judges, and other criminal justice actors increasingly use algorithmic risk assessment to estimate the likelihood that a person will commit future crime. As many scholars have noted, these algorithms tend to have disparate racial impact. In response, critics advocate three strategies of resistance: (1) the exclusion of input factors that correlate closely with race, (2) adjustments to algorithmic design to equalize predictions across racial lines, and (3) rejection of algorithmic methods altogether. This Article’s central claim is that these strategies are at best superficial and at worst counterproductive, because the source of racial inequality in risk assessment lies neither in the input data, nor in a particular algorithm, nor in algorithmic methodology. The deep problem is the nature of prediction itself. All prediction looks to the past to make guesses about future events. In a racially stratified world, any method of prediction will project the inequalities of the past into the future. This is as true of the subjective prediction that has long pervaded criminal justice as of the algorithmic tools now replacing it. What algorithmic risk assessment has done is reveal the inequality inherent in all prediction, forcing us to confront a much larger problem than the challenges of a new technology. Algorithms shed new light on an old problem. Ultimately, the Article contends, redressing racial disparity in prediction will require more fundamental changes in the way the criminal justice system conceives of and responds to risk. The Article argues that criminal law and policy should, first, more clearly delineate the risks that matter, and, second, acknowledge that some kinds of risk may be beyond our ability to measure without racial distortion—in which case they cannot justify state coercion. To the extent that we can reliably assess risk, on the other hand, criminal system actors should strive to respond to risk with support rather than restraint whenever possible. Counterintuitively, algorithmic risk assessment could be a valuable tool in a system that targets the risky for support