We study the approval-based multi-winner election problem where \(n\) voters jointly decide a committee of \(k\) winners from \(m\) candidates. We focus on the axiom average justified representation (AJR) proposed by Fernández, Elkind, Lackner, García, Arias-Fisteus, Basanta-Val, and Skowron (2017). AJR postulates that every group of voters with a common preference should be sufficiently represented in that their average satisfaction should be no less than their Hare quota. Formally, for every group of \(\lceil \ell \cdot \tfrac{n}{k} \rceil\) voters with \(\ell\) common approved candidates, the average number of approved winners for this group should be at least \(\ell\). It is well-known that a winning committee satisfying AJR is not guaranteed to exist for all multi-winner election instances. In this paper, we study the likelihood of the existence of AJR under the Erdos–Rényi model. We consider the Erdos–Rényi model parameterized by \(p \in [0,1]\) that samples multi-winner election instances from the distribution where each voter approves each candidate with probability \(p\) (and the events that voters approve candidates are independent), and we provide a clean and complete characterization of the existence of AJR committees in the case where \(m\) is a constant and \(n\) tends to infinity. We show that there are two phase transition points \(p_1\) and \(p_2\) (with \(p_1 \le p_2\)) for the parameter \(p\) such that: 1) when \(p \lt p_1\) or \(p \gt p_2\), an AJR committee exists with probability \(1 - o(1)\), 2) when \(p_1 \lt p \lt p_2\), an AJR committee exists with probability \(o(1)\), and 3) when \(p = p_1\) or \(p = p_2\), the probability that an AJR committee exists is bounded away from both \(0\) and \(1\).

We investigate majority voting where agents possess private information about an unobservable ground truth that determines their preferences. In such settings, agents may hold different preferences and (collectively) engage in counterintuitive strategic behaviors. Previous work either assumes strategic behavior occurs without coordination or with unlimited coordination, yielding overly inclusive or exclusive predictions about voting outcomes. We incorporate coordination ability—the largest coalition size at which agents could strategically coordinate—into the analysis. Under the ex-ante Bayesian k-strong equilibrium framework, where no group of at most k agents can benefit from deviation, we provide closed-form characterizations of when informed majority decisions, the decision favored by the majority if the ground truth is common knowledge, are achievable. Specifically, we determine (1) when all k-strong equilibria reach the informed majority decision and (2) when at least one such equilibrium exists. These conditions depend on three factors: coordination ability, fraction of majority agents, and information structure. The boundary for the second question exhibits surprising complexity--non-continuous, non-linear, and segmental. Our results reveal the complicated landscape and provide refined predictions for strategic behavior across different coordination levels.

We study the voting problem with two alternatives where voters' preferences depend on a not-directly-observable state variable. While equilibria in the one-round voting mechanisms lead to a good decision, they are usually hard to compute and follow. We consider the two-round voting mechanism where the first round serves as a polling stage and the winning alternative only depends on the outcome of the second round. We show that the two-round voting mechanism is a powerful tool for making collective decisions. Firstly, every (approximated) equilibrium in the two-round voting mechanisms (asymptotically) leads to the decision preferred by the majority as if the state of the world were revealed to the voters. Moreover, there exist natural equilibria in the two-round game following intuitive behaviors such as informative voting, sincere voting, and surprisingly popular strategies. This sharply contrasts with the one-round voting mechanisms in the previous literature, where no simple equilibrium is known. Finally, we show that every equilibrium in the standard one-round majority vote mechanism gives an equilibrium in the two-round mechanisms that is not more complicated. Therefore, the two-round voting mechanism provides a natural equilibrium in every instance, including those where one-round voting fails, and it can reach an informed majority decision whenever one-round voting can. Our experiments on LLM voters also imply that two-round voting leads to the correct outcome more often than one-round voting under some circumstances.

We study cascades in social networks with the independent cascade (IC) model and the Susceptible-Infected-recovered (SIR) model. The well-studied IC model fails to capture the feature of node recovery, and the SIR model is a variant of the IC model with the node recovery feature. In the SIR model, by computing the probability that a node successfully infects another before its recovery and viewing this probability as the corresponding IC parameter, an equivalence between the two models is established, except that the events of the infections along different out-going edges of a node become dependent in the SIR model, whereas these events are independent in the IC model. In this paper, we thoroughly compare the two models and examine the effect of this extra dependency in the SIR model. By a carefully designed coupling argument, we show that the seeds in the IC model have a stronger influence spread than their counterparts in the SIR model, and sometimes it can be significantly stronger. Specifically, we prove that, given the same network, the same seed sets, and the parameters of the two models being set based on the above-mentioned equivalence, the expected number of infected nodes at the end of the cascade for the IC model is weakly larger than that for the SIR model, and there are instances where this dominance is significant. We also study the influence maximization problem (the optimization problem of selecting a set of nodes as initial seeds in a social network to maximize their influence) with the SIR model. We show that the above-mentioned difference in the two models yields different seed-selection strategies, which motivates the design of influence maximization algorithms specifically for the SIR model. We design efficient approximation algorithms with theoretical guarantees by adapting the reverse-reachable-set-based algorithms, commonly used for the IC model, to the SIR model.

We study the problem of fairly and truthfully allocating m indivisible items to n agents with additive preferences. Specifically, we consider truthful mechanisms outputting allocations that satisfy $\mathrm{EF}_{-v}^{+u}$, where, in an $\mathrm{EF}_{-v}^{+u}$ allocation, for any pair of agents i and j, agent i will not envy agent j if u items were added to i ‘s bundle and v items were removed from j ‘s bundle. Previous work easily indicates that, when restricted to deterministic mechanisms, truthfulness will lead to a poor guarantee of fairness: even with two agents, for any u and $v, \mathrm{EF}_{-v}^{+u}$ cannot be guaranteed by truthful mechanisms when the number of items is large enough. In this work, we focus on randomized mechanisms, where we consider ex-ante truthfulness and ex-post fairness. For two agents, we present a truthful mechanism that achieves $\mathrm{EF}_{-1}^{+0}$ (i.e., the well-studied fairness notion EF1). For three agents, we present a truthful mechanism that achieves $\mathrm{EF}_{-1}^{+1}$. For n agents in general, we show that there exists a truthful mechanism that achieves $\mathrm{EF}_{-O(\sqrt{n})}^{+0}$. We further consider fair and truthful mechanisms that also satisfy the standard efficiency guarantee: Pareto-optimality. We provide a mechanism that simultaneously achieves truthfulness, EF1, and Pareto-optimality for bi-valued utilities (where agents’ valuation on each item is either p or q for some $p\gt q \geq 0$). For tri-valued utilities (where agents’ valuations on each item belong to $\{p, q, r\}$ for some $p\gt q\gt r \geq 0$) and any $u, v$, we show that truthfulness is incompatible with $\mathbf{E F}_{-v}^{+u}$ and Pareto-optimality even for two agents.

The problem of fair allocation of indivisible goods studies allocating a set of m goods among n agents in a fair manner. While fairness is a fundamental requirement in many real-world applications, it often conflicts with (economic) efficiency. This raises a natural and important question: How can we identify the most welfare-efficient allocation among all fair allocations? This paper gives an answer from the perspective of computational complexity. Specifically, we study the problem of maximizing utilitarian social welfare (the sum of agents' utilities) under two widely studied fairness criteria: envy-freeness up to any item (EFX) and envy-freeness up to one item (EF1). We examine both normalized and unnormalized valuations, where normalized valuations require that each agent's total utility for all items is identical.

The classic notion of truthfulness requires that no agent has a profitable manipulation — an untruthful report that, for some combination of reports of the other agents, increases her utility. This strong notion implicitly assumes that the manipulating agent either knows what all other agents are going to report, or is willing to take the risk and act as-if she knows their reports.

We study the group strategic behaviors in Bayesian games. Equilibria in previous work do not consider group strategic behaviors with bounded sizes and are too ''strong'' to exist in many scenarios. We propose the ex-ante Bayesian k-strong equilibrium and the Bayesian k-strong equilibrium, where no group of at most k agents can benefit from deviation. The two solution concepts differ in how agents calculate their utilities when contemplating whether a deviation is beneficial. Intuitively, agents are more conservative in the Bayesian k-strong equilibrium than in the ex-ante Bayesian k-strong equilibrium. With our solution concepts, we study collusion in the peer prediction mechanisms, as a representative of the Bayesian games with group strategic behaviors. We characterize the thresholds of the group size k so that truthful reporting in the peer prediction mechanism is an equilibrium for each solution concept, respectively. Our solution concepts can serve as criteria to evaluate the robustness of a peer prediction mechanism against collusion. Besides the peer prediction problem, we also discuss two other potential applications of our new solution concepts, voting and Blotto games, where introducing bounded group sizes provides more fine-grained insights into the behavior of strategic agents.

We initiate the study of fair allocation with the set of divisible or indivisible items distributed in multiple regions. The key requirement is that each agent can only obtain items from one region. In this work, we consider two kinds of fairness concepts: envy-based notions including envy-freeness (EF) and envy-freeness up to one/any item (EF1/EFX), and share-based notions including proportionality (PROP) and proportionality up to one/any item (PROP1/PROPX). On the negative side, we show NP-hardness and inapproximability results about the aforementioned fairness notions. On the positive side, we propose several algorithms to compute the partial allocations that satisfy envy-based notions and allocations that approximate the above fairness notions.

We consider the fair division problem of indivisible items. It is well-known that an envy-free allocation may not exist, and a relaxed version of envy-freeness, envy-freeness up to one item (EF1), has been widely considered. In an EF1 allocation, an agent may envy others' allocated shares, but only up to one item. In many applications, we may wish to specify a subset of prioritized agents where strict envy-freeness needs to be guaranteed from these agents to the remaining agents, while ensuring the whole allocation is still EF1. Prioritized agents may be those agents who are envious in a previous EF1 allocation, those agents who belong to underrepresented groups, etc. Motivated by this, we propose a new fairness notion named envy-freeness with prioritized agents EFprior, and study the existence and the algorithmic aspects for the problem of computing an EFprior allocation. With additive valuations, the simple round-robin algorithm is able to compute an EFprior allocation. In this paper, we mainly focus on general valuations. In particular, we present a polynomial-time algorithm that outputs an EFprior allocation with most of the items allocated. When all the items need to be allocated, we also present polynomial-time algorithms for some well-motivated special cases.

We study the problem of designing truthful and fair mechanisms when allocating a mixture of divisible and indivisible goods. We first show that there does not exist an EFM (envy-free for mixed goods) and truthful mechanism in general. This impossibility result holds even if there is only one indivisible good and one divisible good and there are only two agents. Thus, we focus on some more restricted settings. Under the setting where agents have binary valuations on indivisible goods and identical valuations on a single divisible good (e.g., money), we design an EFM and truthful mechanism. When agents have binary valuations over both divisible and indivisible goods, we first show there exist EFM and truthful mechanisms when there are only two agents or when there is a single divisible good. On the other hand, we show that the mechanism maximizing Nash welfare cannot ensure EFM and truthfulness simultaneously.

We study the voting game where agents' preferences are endogenously decided by the information they receive, and they can collaborate in a group. We show that strategic voting behaviors have a positive impact on leading to the "correct" decision, outperforming the common non-strategic behavior of informative voting and sincere voting. Our results give merit to strategic voting for making good decisions.

We study the fair division problem on divisible heterogeneous resources (the cake cutting problem) with strategic agents, where each agent can manipulate his/her private valuation to receive a better allocation. A (direct-revelation) mechanism takes agents' reported valuations as input and outputs an allocation that satisfies a given fairness requirement. A natural and fundamental open problem, first raised by Chen, Lai, Parkes, and Procaccia [1] and subsequently raised in reference [2] , [3] , [4] , [5] , [6] , [7] , etc., is whether there exists a deterministic, truthful, and envy-free (or even proportional) cake cutting mechanism. In this paper, we resolve this open problem by proving that there does not exist a deterministic, truthful and proportional cake cutting mechanism, even in the special case where all of the following hold: • there are only two agents; • each agent's valuation is a piecewise-constant function; • each agent is hungry: each agent has a strictly positive value on any part of the cake. The impossibility result extends to the case where the mechanism is allowed to leave some part of the cake unallocated. We also present a truthful and envy-free mechanism when each agent's valuation is piecewise-constant and monotone. However, if we require Pareto-optimality, we show that truthful is incompatible with approximate proportionality for any positive approximation ratio even for piecewise-constant and monotone value density functions. To circumvent the main impossibility result, we aim to design mechanisms that possess a certain degree of truthfulness. Motivated by the kind of truthfulness possessed by the classical I-cut-you-choose protocol, we propose a weaker notion of truthfulness, the proportional risk-averse truthfulness . We show that the well-known moving-knife (Dubins-Spanier) procedure and Even-Paz algorithm do not have this truthful property. We propose a mechanism that is proportionally risk-averse truthful and envy-free, and a mechanism that is proportionally risk-averse truthful that always outputs allocations with connected pieces.

We study the fair division problem on divisible heterogeneous resources (the cake cutting problem) with strategic agents, where each agent can manipulate his/her private valuation in order to receive a better allocation. A (direct-revelation) mechanism takes agents' reported valuations as input, and outputs an allocation that satisfies a given fairness requirement. A natural and fundamental open problem, first raised by Chen, Lai, Parkes, and Procaccia [22] and subsequently raised in reference [9 , 11 , 12 , 19, 33, 35], etc., is whether there exists a deterministic, truthful and envy-free (or even proportional) cake cutting mechanism. In this paper, we resolve this open problem by proving that there does not exist a deterministic, truthful and proportional cake cutting mechanism, even in the special case where all of the following hold: there are only two agents; each agent's valuation is a piecewise-constant function; each agent is hungry: each agent has a strictly positive value on any part of the cake. The impossibility result extends to the case where the mechanism is allowed to leave some part of the cake unallocated.

In the influence maximization problem, one chooses a fixed number of initial seeds in a social network to maximize the spread of their influence. We study this problem with the r-complex contagion model, where each uninfected vertex in the network becomes infected if it has at least r infected neighbors. We focus on a random graph model called the stochastic hierarchical blockmodel. When the graph is not exceptionally sparse, under certain mild assumptions, we prove the optimal seeding strategy puts all the seeds in a single community. This matches the intuition that, in a nonsubmodular cascade model, placing seeds near each other creates synergy. However, it sharply contrasts with the intuition for submodular cascade models (e.g., the independent cascade model) in which nearby seeds tend to erode each others' effects. We use this observation to design a polynomial-time dynamic programming algorithm for a slightly more general setting.

We consider the adaptive influence maximization problem: given a network and a budget k, iteratively select k seeds in the network to maximize the expected number of adopters. In the full-adoption feedback model, after selecting each seed, the seed-picker observes all the resulting adoptions. In the myopic feedback model, the seed-picker only observes whether each neighbor of the chosen seed adopts. Motivated by the extreme success of greedy-based algorithms/heuristics for influence maximization, we propose the concept of greedy adaptivity gap, which compares the performance of the adaptive greedy algorithm to its non-adaptive counterpart. Our first result shows that, for submodular influence maximization, the adaptive greedy algorithm can perform up to a (1 − 1/e)-fraction worse than the non-adaptive greedy algorithm, and that this ratio is tight. More specifically, on one side we provide examples where the performance of the adaptive greedy algorithm is only a (1−1/e) fraction of the performance of the non-adaptive greedy algorithm in four settings: for both feedback models and both the independent cascade model and the linear threshold model. On the other side, we prove that in any submodular cascade, the adaptive greedy algorithm always outputs a (1 − 1/e)-approximation to the expected number of adoptions in the optimal non-adaptive seed choice. Our second result shows that, for the general submodular diffusion model with full-adoption feedback, the adaptive greedy algorithm can outperform the non-adaptive greedy algorithm by an unbounded factor. Finally, we propose a risk-free variant of the adaptive greedy algorithm that always performs no worse than the non-adaptive greedy algorithm.

When can cooperation arise from self-interested decisions in public goods games? And how can we help agents to act cooperatively? We examine these classical questions in a pivotal participation game, a variant of public good games, where heterogeneous agents make binary participation decisions on contributing their endowments, and the public project succeeds when it has enough contributions. We prove it is NP-complete to decide the existence of a cooperative Nash equilibrium such that the project succeeds. We demonstrate that the decision problem becomes easy if agents are homogeneous enough. We then propose two algorithms to help cooperation in the game. Our first algorithm adds an external investment to the public project, and our second algorithm uses matching funds. We show the cost to induce a cooperative Nash equilibrium is near-optimal for both algorithms. Finally, the cost of matching funds can always be smaller than the cost of adding an external investment. Intuitively, matching funds provide a greater incentive for cooperation than adding an external investment does.

We consider two-alternative elections where voters' preferences depend on a state variable that is not directly observable. Each voter receives a private signal that is correlated to the state variable. As a special case, our model captures the common scenario where voters can be categorized into three types: those who always prefer one alternative, those who always prefer the other, and those contingent voters whose preferences depends on the state. In this setting, even if every voter is a contingent voter, agents voting according to their private information need not result in the adoption of the universally preferred alternative, because the signals can be systematically biased.We present a mechanism that elicits and aggregates the private signals from the voters, and outputs the alternative that is favored by the majority. In particular, voters truthfully reporting their signals forms a strong Bayes Nash equilibrium (where no coalition of voters can deviate and receive a better outcome).

Financial options are contracts that specify the right to buy or sell an underlying asset at a strike price by an expiration date. Standard exchanges offer options of predetermined strike values and trade options of different strikes independently, even for those written on the same underlying asset. Such independent market design can introduce arbitrage opportunities and lead to the thin market problem. The paper first proposes a mechanism that consolidates and matches orders on standard options related to the same underlying asset, while providing agents the flexibility to specify any custom strike value. The mechanism generalizes the classic double auction, runs in time polynomial to the number of orders, and poses no risk to the exchange, regardless of the value of the underlying asset at expiration. Empirical analysis on real-market options data shows that the mechanism can find new matches for options of different strike prices and reduce bid-ask spreads. Extending standard options written on a single asset, we propose and define a new derivative instrument ---combinatorial financial options that offer contract holders the right to buy or sell any linear combination of multiple underlying assets. We generalize our single-asset mechanism to match options written on different combinations of assets, and prove that optimal clearing of combinatorial financial options is coNP-hard. To facilitate market operations, we propose an algorithm that finds the exact optimal match through iterative constraint generation, and evaluate its performance on synthetically generated combinatorial options markets of different scales. As option prices reveal the market's collective belief of an underlying asset's future value, a combinatorial options market enables the expression of aggregate belief about future correlations among assets.

We consider the adaptive influence maximization problem: given a network and a budget k, iteratively select k seeds in the network to maximize the expected number of adopters. In the full-adoption feedback model, after selecting each seed, the seed-picker observes all the resulting adoptions. In the myopic feedback model, the seed-picker only observes whether each neighbor of the chosen seed adopts. Motivated by the extreme success of greedy-based algorithms/heuristics for influence maximization, we propose the concept of greedy adaptivity gap, which compares the performance of the adaptive greedy algorithm to its non-adaptive counterpart. Our first result shows that, for submodular influence maximization, the adaptive greedy algorithm can perform up to a (1-1/e)-fraction worse than the non-adaptive greedy algorithm, and that this ratio is tight. More specifically, on one side we provide examples where the performance of the adaptive greedy algorithm is only a (1-1/e) fraction of the performance of the non-adaptive greedy algorithm in four settings: for both feedback models and both the independent cascade model and the linear threshold model. On the other side, we prove that in any submodular cascade, the adaptive greedy algorithm always outputs a (1-1/e)-approximation to the expected number of adoptions in the optimal non-adaptive seed choice. Our second result shows that, for the general submodular cascade model with full-adoption feedback, the adaptive greedy algorithm can outperform the non-adaptive greedy algorithm by an unbounded factor. Finally, we propose a risk-free variant of the adaptive greedy algorithm that always performs no worse than the non-adaptive greedy algorithm.

We study learning statistical properties from strategic agents with private information. In this problem, agents must be incentivized to truthfully reveal their information even when it cannot be directly verified. Moreover, the information reported by the agents must be aggregated into a statistical estimate. We study two fundamental statistical properties: estimating the mean of an unknown Gaussian, and linear regression with Gaussian error. The information of each agent is one point in a Euclidean space.Our main results are two mechanisms for each of these problems which optimally aggregate the information of agents in the truth-telling equilibrium:• A minimal (non-revelation) mechanism for large populations — agents only need to report one value, but that value need not be their point.• A mechanism for small populations that is non-minimal — agents need to answer more than one question.These mechanisms are “informed truthful” mechanisms where reporting unaltered data (truth-telling) 1) forms a strict Bayesian Nash equilibrium and 2) has strictly higher welfare than any oblivious equilibrium where agents' strategies are independent of their private signals. We also show a minimal revelation mechanism (each agent only reports her signal) for a restricted setting and use an impossibility result to prove the necessity of this restriction.We build upon the peer prediction literature in the single-question setting; however, most previous work in this area focuses on discrete signals, whereas our setting is inherently continuous, and we further simplify the agents' reports.

We study envy-free cake cutting with strategic agents, where each agent may manipulate his private information in order to receive a better allocation. We focus on piecewise constant utility functions and consider two scenarios: the general setting without any restriction on the allocations and the restricted setting where each agent has to receive a connected piece. We show that no deterministic truthful envy-free mechanism exists in the connected piece scenario, and the same impossibility result for the general setting with some additional mild assumptions on the allocations. Finally, we study a large market model where the economy is replicated and demonstrate that truth-telling converges to a Nash equilibrium.

We consider the classic cake cutting problem where one allocates a divisible cake to n participating agents. Among all valid divisions, fairness and efficiency (a.k.a. ~social welfare) are the most critical criteria to satisfy and optimize, respectively. We study computational complexity of computing an efficiency optimal division given the conditions that the allocation satisfies proportional fairness and assigns each agent a connected piece. For linear valuation functions, we give a polynomial time approximation scheme to compute an efficiency optimal allocation. On the other hand, we show that the problem is NP-hard to approximate within a factor of Ω 1/√n for general piecewise constant functions, and is NP-hard to compute for normalized functions.