Papers

We revisit the setting of fairly allocating indivisible items when agents have different weights representing their entitlements. First, we propose a parameterized family of relaxations for weighted envy-freeness and the same for weighted proportionality; the parameters indicate whether smaller-weight or larger-weight agents should be given a higher priority. We show that each notion in these families can always be satisfied, but any two cannot necessarily be fulfilled simultaneously. We then introduce an intuitive weighted generalization of maximin share fairness and establish the optimal approximation of it that can be guaranteed. Furthermore, we characterize the implication relations between the various weighted fairness notions introduced in this and prior work, and relate them to the lower and upper quota axioms from apportionment.

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We study a dynamic model of Bayesian persuasion in sequential decision-making settings. An informed principal observes an external parameter of the world and advises an uninformed agent about actions to take over time. The agent takes actions in each time step based on the current state, the principal's advice/signal, and beliefs about the external parameter. The action of the agent updates the state according to a stochastic process. The model arises naturally in many applications, e.g., an app (the principal) can advice the user (the agent) on possible choices between actions based on additional real-time information the app has. We study the problem of designing a signaling strategy from the principal's point of view. We show that the principal has an optimal strategy against a myopic agent, who only optimizes their rewards locally, and the optimal strategy can be computed in polynomial time. In contrast, it is NP-hard to approximate an optimal policy against a far-sighted agent. Further, we show that if the principal has the power to threaten the agent by not providing future signals, then we can efficiently design a threat-based strategy. This strategy guarantees the principal's payoff as if playing against an agent who is far-sighted but myopic to future signals.

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We formalize a voting model for plurality elections that combines Iterative Voting and Calculus of Voting. Each iteration, autonomous agents simultaneously maximize the utility they expect from candidates. Agents are aware of neither other individuals’ preferences or choices, nor of the distribution preferences come from. They know only of candidates’ current expected vote shares (e.g. from a poll, network, etc) and with that calculate expected rewards from each candidate, pondering the probability that voting for each would alter the final outcome. We define the general form of those pivotal probabilities, then we derive both efficient exact calculations as well as heuristics. Lastly, we both prove formally that the model converges with asymptotically large electorates and show empirically (via simulations) that it nearly always converges even with very few agents.

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A growing body of work in game theory extends the traditional Stackelberg game to settings with one leader and multiple followers who play a Nash equilibrium. Standard approaches for computing equilibria in these games reformulate the followers' best response as constraints in the leader's optimization problem. These reformulation approaches can sometimes be effective, but make limiting assumptions on the followers' objectives and the equilibrium reached by followers, e.g., uniqueness, optimism, or pessimism. To overcome these limitations, we run gradient descent to update the leader's strategy by differentiating through the equilibrium reached by followers. Our approach generalizes to any stochastic equilibrium selection procedure that chooses from multiple equilibria, where we compute the stochastic gradient by back-propagating through a sampled Nash equilibrium using the solution to a partial differential equation to establish the unbiasedness of the stochastic gradient. Using the unbiased gradient estimate, we implement the gradient-based approach to solve three Stackelberg problems with multiple followers. Our approach consistently outperforms existing baselines to achieve higher utility for the leader.

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We consider mechanisms for truthfully eliciting probabilistic predictions from a group of experts. The standard approach -- using a proper scoring rule to separately reward each expert -- is not robust to collusion: experts may collude to misreport their beliefs in a way that guarantees them a larger total reward no matter the eventual outcome. Chun and Shachter (2011) termed any such collusion "arbitrage" and asked whether there is any truthful elicitation mechanism that makes arbitrage impossible. We resolve this question positively, exhibiting a class of strictly proper arbitrage-free contract functions. These contract functions have two parts: one ensures that the total reward of a coalition of experts depends only on the average of their reports; the other ensures that changing this average report hurts the experts under at least one outcome.

poster session 5 @ blue 6, poster session 8 @ blue 6, oral session 5 @ blue 6, poster session 5, poster session 8, oral session 5

Despite the many recent practical and theoretical breakthroughs in computational game theory, equilibrium finding in extensive-form team games remains a significant challenge. While NP-hard in the worst case, there are provably efficient algorithms for certain families of team game. In particular, if the game has common external information, also known as A-loss recall---informally, actions played by non-team members (i.e., the opposing team or nature) are either unknown to the entire team, or common knowledge within the team---then polynomial-time algorithms exist (Kaneko and Kline 1995). In this paper, we devise a completely new algorithm for solving team games. It uses a tree decomposition of the constraint system representing each team's strategy to reduce the number and degree of constraints required for correctness (tightness of the mathematical program). Our approach has the bags of the tree decomposition correspond to team-public states---that is, minimal sets of nodes (that is, states of the team) such that, upon reaching the set, it is common knowledge among the players on the team that the set has been reached. Our algorithm reduces the problem of solving team games to a linear program with at most O(NW^{w+1}) nonzero entries in the constraint matrix, where N is the size of the game tree, w is a parameter that depends on the amount of uncommon external information, and W is the treewidth of the tree decomposition. In public-action games, our program size is bounded by the tighter 2^{O(nt)}N for teams of n players with t types each. Our algorithm is based on a new way to write a custom, concise tree decomposition, and its fast run time does not assume that the decomposition has small treewidth. Since our algorithm describes the polytope of correlated strategies directly, we get equilibrium finding in correlated strategies for free---instead of, say, having to run a double oracle algorithm. We show via experiments on a standard suite of games that our algorithm achieves state-of-the-art performance on all benchmark game classes except one. We also present, to our knowledge, the first experiments for this setting where both teams have more than one member.

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The existence of EFX allocations of goods is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and yet, existence results are known only for very restricted settings, such as: (i) agents with identical valuations, (ii) 2 agents, and (iii) 3 agents with additive valuations. It is also known that EFX exists if one can leave n-1 items unallocated, where n is the number of agents. We develop new techniques that allow us to push the boundaries of the enigmatic EFX problem beyond these known results, and (arguably) to simplify proofs of earlier results. Our main result is that every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated. Beyond our main result, we introduce a new class of valuations, termed nice cancelable, which includes additive, unit-demand, budget-additive and multiplicative valuations, among others. Using our new techniques, we show that both our results and previous results for additive valuations extend to nice cancelable valuations.

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We consider a variant of the standard Bayesian mechanism, where players evaluate their outcomes and constraints in an {\em ex-ante} manner. Such a model captures a major form of modern online advertising where an advertiser is concerned with her/his expected utility over a time period and her/his type may change over time. We are interested in the incentive compatibility (IC) problem of such Bayesian mechanism. Under very mild conditions on the mechanism environments, we give a full characterization of IC via the {\em taxation principle} and show, perhaps surprisingly, that such IC mechanisms are fully characterized by the so-called {\em auto-bidding} mechanisms, which are pervasively fielded in the online advertising industry.

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Most prior work on online matching problems has been with the flexibility of keeping some vertices unmatched. We study three related online matching problems with the constraint of matching every vertex, i.e., with no rejections. We adopt a model in which vertices arrive in a uniformly random order and the edge-weights are arbitrary positive numbers. For the capacitated online bipartite matching problem in which the vertices of one side of the graph are offline and those of the other side arrive online, we give a 4.62-competitive algorithm when the capacity of each offline vertex is 2. For the online general (non-bipartite) matching problem, where all vertices arrive online, we give a 3.34-competitive algorithm. We also study the online roommate matching problem (Huzhang et al. 2017) in which each room (offline vertex) holds 2 persons (online vertices). Persons derive non-negative additive utilities from their room as well as roommate. In this model, with the goal of maximizing the sum of utilities of all persons, we give a 7.96-competitive algorithm. This is an improvement over the 24.72 approximation factor in (Huzhang et al. 2017).

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Motivated by fair division applications, we study a fair connected graph partitioning problem, in which an undirected graph with m nodes must be divided between n agents such that each agent receives a connected subgraph and the partition is fair. We study approximate versions of two fairness criteria: \alpha-proportionality requires that each agent receive a subgraph with at least (1/\alpha)*m/n nodes, and \alpha-balancedness requires that the ratio between the sizes of the largest and smallest subgraphs be at most \alpha. Unfortunately, there exist simple examples in which no partition is reasonably proportional or balanced. To circumvent this, we introduce the idea of charity. We show that by "donating" just n-1 nodes, we can guarantee the existence of 2-proportional and almost 2-balanced partitions (and find them in polynomial time), and that this result is almost tight. More generally, we chart the tradeoff between the size of charity and the approximation of proportionality or balancedness we can guarantee.

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We extend the classic regret minimization framework for approximating equilibria in matrix games by greedily weighing iterates based on the regrets observed at runtime. Theoretically, our method retains all previous convergence rate guarantees. Empirically, experiments on large random matrix games and normal-form subgames of the AI benchmark Diplomacy show that greedy weights outperforms previous methods whenever sampling is used, sometimes by several orders of magnitude.

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When aggregating logically interconnected judgments from n agents, the result might be logically inconsistent. This phenomenon is known as the doctrinal paradox, which plays a central role in the field of judgment aggregation. Previous work has mostly focused on the worst-case analysis of the doctrinal paradox, which has lead to many impossibility results. Little is known about its practical occurrence, except for the study on its likelihood under a few i.i.d. distributions. In this paper, we characterize the likelihood of the doctrinal paradox under a much more general and realistic model called smoothed social choice framework introduced in a NeurIPS 2020 paper. In the framework, agents' ground truth judgments can be arbitrarily correlated, while the noises are independent. Our main theorem states that under mild conditions, the smoothed likelihood of the doctrinal paradox is either $0$, $\exp(-\Theta(n))$, $\Theta(n^{-1/2})$ or $\Theta(1)$. This not only answers open questions by List in 2005, but also draws clear lines between situations with frequent and with vanishing paradoxes.

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Making decisions by aggregating the opinions of a collection of agents is a question of interest to a broad array of researchers, from ensemble learning theorists to political scientists designing democratic institutions. Herein, we investigate the optimal number of agents needed to make a decision over a binary issue under a majority rule. We take an \emph{epistemic} view where the group decides on a binary issue with one ground truth ``correct'' outcome. Each one of $n$ voters votes correctly with a fixed probability, known as their competence level, which is drawn from a distribution $\mathcal{D}$. After sampling the competence levels, we choose a number of agents to represent the group. After sampling their votes, the correct outcome is chosen as long as more than half of the representatives chose this option. Assuming that we can identify the best experts, i.e., those with the highest competence, to form an \emph{epistemic congress} we find that the optimal congress size should be linear in the population size. This result is striking because it holds even when allowing the top representatives to be accurate with arbitrarily high probabilities. We then analyze real world data, observing that the actual sizes of congresses are much smaller than the optimal size our theoretical results suggest. We conclude by investigating under what conditions congresses of sub-optimal sizes would still outperform direct democracy, in which all voters vote. We find that a small congress would outperform direct democracy if the rate at which the societal bias towards the ground truth decreases with the population size fast enough, and we quantify the speed needed for constant and polynomial congress sizes.

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We pose and study the problem of planning in Markov decision processes (MDPs), subject to participation constraints as studied in mechanism design. In this problem, a planner must work with a self-interested agent on a given MDP. Each action in the MDP provides an immediate reward to the planner and a (possibly different) reward to the agent. The agent has no control in choosing the actions, but has the option to end the entire process at any time. The goal of the planner is to find a policy that maximizes her cumulative reward, taking into consideration the agent's ability to terminate. Our main result is a fully polynomial-time approximation scheme for this problem. En route, we present polynomial-time algorithms for computing (exact) optimal policies for important special cases of this problem, including when the time horizon is constant, or when the MDP exhibits a "definitive decisions" property. We illustrate our algorithms with two different game-theoretic applications: the problem of assigning rides in ride-sharing and the problem of designing screening policies. Our results imply efficient algorithms for computing (approximately) optimal policies in both these applications.

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While there has been an increasing focus on the use of game theoretic models for autonomous driving, empirical evidence shows that there are still open questions around dealing with the challenges of common knowledge assumptions as well as modeling bounded rationality. To address some of these practical challenges, we develop a framework of generalized dynamic cognitive hierarchy for both modelling naturalistic human driving behavior as well as behavior planning for autonomous vehicles (AV). This framework is built upon a rich model of level-0 behavior through the use of automata strategies, an interpretable notion of bounded rationality through safety and maneuver satisficing, and a robust response for planning. Based on evaluation on two large naturalistic datasets as well as simulation of critical traffic scenarios, we show that i) automata strategies are well suited for level-0 behavior in a dynamic level-k framework, and ii) the proposed robust response to a heterogeneous population of strategic and non-strategic reasoners can be an effective approach for game theoretic planning in AV.

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We study the problem of fair and efficient allocation of a set of indivisible chores to agents with additive cost functions. We consider the popular fairness notion of envy-freeness up to one good (EF1) with the efficiency notion of Pareto-optimality (PO). While it is known that EF1+PO allocations exists and can be computed in pseudo-polynomial time in the case of goods, the same problem is open for chores. Our first result is a strongly polynomial-time algorithm for computing an EF1+PO allocation for bivalued instances, where agents have (at most) two disutility values for the chores. To the best of our knowledge, this is the first non-trivial class of chores to admit an EF1+PO allocation and an efficient algorithm for its computation. We also study the problem of computing an envy-free (EF) and PO allocation for the case of divisible chores. While the existence of EF+PO allocation is known via competitive equilibrium with equal incomes, its efficient computation is open. Our second result shows that for bivalued instances, an EF+PO allocation can be computed in strongly polynomial-time.

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We consider the problem of reforming an envy-free matching when each agent is assigned a single item. Given an envy-free matching, we consider an operation to exchange the item of an agent with an unassigned item preferred by the agent that results in another envy-free matching. We repeat this operation as long as we can. We prove that the resulting envy-free matching is uniquely determined up to the choice of an initial envy-free matching, and can be found in polynomial time. We call the resulting matching a reformist envy-free matching, and then we study a shortest sequence to obtain the reformist envy-free matching from an initial envy-free matching. We prove that a shortest sequence is computationally hard to obtain even when each agent accepts at most four items and each item is accepted by at most three agents. On the other hand, we give polynomial-time algorithms when each agent accepts at most three items or each item is accepted by at most two agents. Inapproximability and fixed-parameter (in)tractability are also discussed.

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We model the societal task of redistricting political districts as a partitioning problem: Given a set of $n$ points in the plane, each belonging to one of two parties, and a parameter $k$, our goal is to compute a partition $\Pi$ of the plane into regions so that each region contains roughly $\sigma = n/k$ points. $\Pi$ should satisfy a notion of ``local'' fairness, which is related to the notion of core, a well-studied concept in cooperative game theory. A region is associated with the majority party in that region, and a point is unhappy in $\Pi$ if it belongs to the minority party. A group $D$ of roughly $\sigma$ contiguous points is called a deviating group with respect to $\Pi$ if majority of points in $D$ are unhappy in $\Pi$. The partition $\Pi$ is locally fair if there is no deviating group with respect to $\Pi$. This paper focuses on a restricted case when points lie in $1$D. The problem is non-trivial even in this case. We consider both adversarial and ``beyond worst-case" settings for this problem. For the former, we characterize the input parameters for which a locally fair partition always exists; we also show that a locally fair partition may not exist for certain parameters. We then consider input models where there are ``runs'' of red and blue points. For such clustered inputs, we show that a locally fair partition may not exist for certain values of $\sigma$, but an approximate locally fair partition exists if we allow some regions to have smaller sizes. We finally present a polynomial-time algorithm for computing a locally fair partition if one exists.

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