The Core of the Participatory Budgeting Problem

In participatory budgeting, communities collectively decide on the allocation of public tax dollars for local public projects. In this work, we consider the question of fairly aggregating the preferences of community members to determine an allocation of funds to projects. This problem is different from standard fair resource allocation because of public goods: The allocated goods benefit all users simultaneously. Fairness is crucial in participatory decision making, since generating equitable outcomes is an important goal of these processes. We argue that the classic game theoretic notion of core captures fairness in the setting. To compute the core, we first develop a novel characterization of a public goods market equilibrium called the Lindahl equilibrium, which is always a core solution. We then provide the first (to our knowledge) polynomial time algorithm for computing such an equilibrium for a broad set of utility functions; our algorithm also generalizes (in a non-trivial way) the well-known concept of proportional fairness. We use our theoretical insights to perform experiments on real participatory budgeting voting data. We empirically show that the core can be efficiently computed for utility functions that naturally model our practical setting, and examine the relation of the core with the familiar welfare objective. Finally, we address concerns of incentives and mechanism design by developing a randomized approximately dominant-strategy truthful mechanism building on the exponential mechanism from differential privacy

Voting with Coarse Beliefs

The classic Gibbard-Satterthwaite theorem says that every strategy-proof voting rule with at least three possible candidates must be dictatorial. Similar impossibility results hold even if we consider a weaker notion of strategy-proofness where voters believe that the other voters' preferences are i.i.d.~(independent and identically distributed). In this paper, we take a bounded-rationality approach to this problem and consider a setting where voters have "coarse" beliefs (a notion that has gained popularity in the behavioral economics literature). In particular, we construct good voting rules that satisfy a notion of strategy-proofness with respect to coarse i.i.d.~beliefs, thus circumventing the above impossibility results

Federal mandates by popular demand

This paper proposes a new framework for studying federal mandates regarding public policies in areas such as environmental quality, public health, highway safety, and the provision of local public goods. Voters have single-peaked preferences along a single policy dimension. There are two levels of government, federal and local. The federal level can constrain local policy by mandating a minimum (or maximum) policy. Localities are free to adopt any policy satisfying the constraint imposed by the federal mandate. We show that voters choose federal mandates that are too strict, which leads to excessively severe mandates. We show that similar results can obtain when federal provision of the public-provided good is more efficient than local provision

Strategic voting and nomination

Using computer simulations based on three separate data generating processes, I estimate the fraction of elections in which sincere voting will be a core equilibrium given each of eight single-winner voting rules. Additionally, I determine how often each voting rule is vulnerable to simple voting strategies such as 'burying' and 'compromising', and how often each voting rule gives an incentive for non-winning candidates to enter or leave races. I find that Hare is least vulnerable to strategic voting in general, whereas Borda, Coombs, approval, and range are most vulnerable. I find that plurality is most vulnerable to compromising and strategic exit (which can both reinforce two-party systems), and that Borda is most vulnerable to strategic entry. I support my key results with analytical proofs.strategic voting; tactical voting; strategic nomination; Condorcet; alternative vote; Borda count; approval voting

Mixture Selection, Mechanism Design, and Signaling

We pose and study a fundamental algorithmic problem which we term mixture selection, arising as a building block in a number of game-theoretic applications: Given a function $g$ from the $n$-dimensional hypercube to the bounded interval $[-1,1]$, and an $n \times m$ matrix $A$ with bounded entries, maximize $g(Ax)$ over $x$ in the $m$-dimensional simplex. This problem arises naturally when one seeks to design a lottery over items for sale in an auction, or craft the posterior beliefs for agents in a Bayesian game through the provision of information (a.k.a. signaling). We present an approximation algorithm for this problem when $g$ simultaneously satisfies two smoothness properties: Lipschitz continuity with respect to the $L^\infty$ norm, and noise stability. The latter notion, which we define and cater to our setting, controls the degree to which low-probability errors in the inputs of $g$ can impact its output. When $g$ is both $O(1)$-Lipschitz continuous and $O(1)$-stable, we obtain an (additive) PTAS for mixture selection. We also show that neither assumption suffices by itself for an additive PTAS, and both assumptions together do not suffice for an additive FPTAS. We apply our algorithm to different game-theoretic applications from mechanism design and optimal signaling. We make progress on a number of open problems suggested in prior work by easily reducing them to mixture selection: we resolve an important special case of the small-menu lottery design problem posed by Dughmi, Han, and Nisan; we resolve the problem of revenue-maximizing signaling in Bayesian second-price auctions posed by Emek et al. and Miltersen and Sheffet; we design a quasipolynomial-time approximation scheme for the optimal signaling problem in normal form games suggested by Dughmi; and we design an approximation algorithm for the optimal signaling problem in the voting model of Alonso and C\^{a}mara