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We study a single-agent contracting environment where the agent has misspecified beliefs about the outcome distributions for each chosen action. First, we show that for a myopic Bayesian learning agent with only two possible actions, the empirical frequency of the chosen actions converges to a Berk-Nash equilibrium. However, through a constructed example, we illustrate that this convergence in action frequencies fails when the agent has three or more actions. Furthermore, with multiple actions, even computing an $\varepsilon$-Berk-Nash equilibrium requires at least quasi-polynomial time under the Exponential Time Hypothesis (ETH) for the PPAD-class. This finding poses a significant challenge to the existence of simple learning dynamics that converge in action frequencies. Motivated by this challenge, we focus on the contract design problems for an agent with misspecified beliefs and two possible actions. We show that the revenue-optimal contract, under a Berk-Nash equilibrium, can be computed in polynomial time. Perhaps surprisingly, we show that even a minor degree of misspecification can result in a significant reduction in optimal revenue.

A principal can propose a project to an agent, who then decides whether to accept. Their payoffs from launching the project depend on an unknown binary state. The principal can obtain more precise information about the state through a test at no cost, but crucially, it is common knowledge that she can falsify the test result. In the most interesting case where players have conflicted interests, the optimal test is a binary lemon-detecting test. We also find that coordination is possible when the principal is pessimistic but not when the agent is pessimistic. Moreover, when the agent has private information about the state, a single binary lemon-detecting test remains optimal even though the principal has the option to screen the agent by providing a menu of tests. Our finding is consistent with observed tests in real practice.

This paper studies a joint design problem where a seller can design both the signal structures for the agents to learn their values, and the allocation and payment rules for selling the item. In his seminal work, Myerson (1981) shows how to design the optimal auction with exogenous signals. We show that the problem becomes NP-hard when the seller also has the ability to design the signal structures. Our main result is a polynomial-time approximation scheme (PTAS) for computing the optimal joint design with at most an $\epsilon$ multiplicative loss in expected revenue. Moreover, we show that in our joint design problem, the seller can significantly reduce the information rent of the agents by providing partial information, which ensures a revenue that is at least $1 - \frac{1}{e}$ of the optimal welfare for all valuation distributions.

An expert with no inherent interest in an unknown binary state can exert effort to acquire a piece of falsifiable evidence informative of it. A designer can incentivize learning using a mechanism that provides state-dependent rewards within fixed bounds. We show that eliciting a single report maximizes information acquisition if the evidence is revealing or its content predictable. This conclusion fails when the evidence is sufficiently imprecise, the failure to find it is informative, and its contents could support either state. Our findings shed light on incentive design for consultation and forecasting by showing how learning dynamics qualitatively shape effort-maximizing contracts., Comment: A preliminary version of this paper has been accepted in the Twenty-Fifth ACM Conference on Economics and Computation (EC'24) as a one-page abstract with the title "Optimal Scoring for Dynamic Information Acquisition."

We study a sender-receiver model where the receiver can commit to a decision rule before the sender determines the information policy. The decision rule can depend on the signal structure and the signal realization that the sender adopts. This framework captures applications where a decision-maker (the receiver) solicit advice from an interested party (sender). In these applications, the receiver faces uncertainty regarding the sender's preferences and the set of feasible signal structures. Consequently, we adopt a unified robust analysis framework that includes max-min utility, min-max regret, and min-max approximation ratio as special cases. We show that it is optimal for the receiver to sacrifice ex-post optimality to perfectly align the sender's incentive. The optimal decision rule is a quota rule, i.e., the decision rule maximizes the receiver's ex-ante payoff subject to the constraint that the marginal distribution over actions adheres to a consistent quota, regardless of the sender's chosen signal structure.

We study the design of screening mechanisms subject to competition and manipulation. A social planner has limited resources to allocate to multiple agents using only signals manipulable through unproductive effort. We show that the welfare-maximizing mechanism takes the form of a contest and characterize the optimal contest. We apply our results to two settings: either the planner has one item or a number of items proportional to the number of agents. We show that in both settings, with sufficiently many agents, a winner-takes-all contest is never optimal. In particular, the planner always benefits from randomizing the allocation to some agents.

We provide a justification for the prevalence of linear (commission-based) contracts in practice under the Bayesian framework. We consider a hidden-action principal-agent model, in which actions require different amounts of effort, and the agent's cost per-unit-of-effort is private. We show that linear contracts are near-optimal whenever there is sufficient uncertainty in the principal-agent setting.

This paper develops a framework for the design of scoring rules to optimally incentivize an agent to exert a multi-dimensional effort. This framework is a generalization to strategic agents of the classical knapsack problem (cf. Briest, Krysta, and V\"ocking, 2005, Singer, 2010) and it is foundational to applying algorithmic mechanism design to the classroom. The paper identifies two simple families of scoring rules that guarantee constant approximations to the optimal scoring rule. The truncated separate scoring rule is the sum of single dimensional scoring rules that is truncated to the bounded range of feasible scores. The threshold scoring rule gives the maximum score if reports exceed a threshold and zero otherwise. Approximate optimality of one or the other of these rules is similar to the bundling or selling separately result of Babaioff, Immorlica, Lucier, and Weinberg (2014). Finally, we show that the approximate optimality of the best of those two simple scoring rules is robust when the agent's choice of effort is made sequentially.

We study the aggregate welfare and individual regret guarantees of dynamic \emph{pacing algorithms} in the context of repeated auctions with budgets. Such algorithms are commonly used as bidding agents in Internet advertising platforms, adaptively learning to shade bids by a tunable linear multiplier in order to match a specified budget. We show that when agents simultaneously apply a natural form of gradient-based pacing, the liquid welfare obtained over the course of the learning dynamics is at least half the optimal expected liquid welfare obtainable by any allocation rule. Crucially, this result holds \emph{without requiring convergence of the dynamics}, allowing us to circumvent known complexity-theoretic obstacles of finding equilibria. This result is also robust to the correlation structure between agent valuations and holds for any \emph{core auction}, a broad class of auctions that includes first-price, second-price, and generalized second-price auctions as special cases. For individual guarantees, we further show such pacing algorithms enjoy \emph{dynamic regret} bounds for individual value maximization, with respect to the sequence of budget-pacing bids, for any auction satisfying a monotone bang-for-buck property. To complement our theoretical findings, we provide semi-synthetic numerical simulations based on auction data from the Bing Advertising platform., Comment: Preliminary version has appeared in ITCS 2023: the 14th Conference on Innovations in Theoretical Computer Science. Compared to the initial version, the current version features revised presentation and expanded discussions, adds numerical experiments (since Aug'24), and fixes an inaccuracy in the definition of the MBB property (since Sep'22)

Participation incentives a well-known issue inhibiting randomized clinical trials (RCTs). We frame this issue as a non-standard exploration-exploitation tradeoff: an RCT would like to explore as uniformly as possible, whereas each patient prefers "exploitation", i.e., treatments that seem best. We incentivize participation by leveraging information asymmetry between the trial and the patients. We measure statistical performance via worst-case estimation error under adversarially generated outcomes, a standard objective for RCTs. We obtain a near-optimal solution in terms of this objective: an incentive-compatible mechanism with a particular guarantee, and a nearly matching impossibility result for any incentive-compatible mechanism. We consider three model variants: homogeneous patients (of the same "type" comprising preferences and medical histories), heterogeneous agents, and an extension with estimated type frequencies.

A prevalent assumption in auction theory is that the auctioneer has full control over the market and that the allocation she dictates is final. In practice, however, agents might be able to resell acquired items in an aftermarket. A prominent example is the market for carbon emission allowances. These allowances are commonly allocated by the government using uniform-price auctions, and firms can typically trade these allowances among themselves in an aftermarket that may not be fully under the auctioneer's control. While the uniform-price auction is approximately efficient in isolation, we show that speculation and resale in aftermarkets might result in a significant welfare loss. Motivated by this issue, we consider three approaches, each ensuring high equilibrium welfare in the combined market. The first approach is to adopt smooth auctions such as discriminatory auctions. This approach is robust to correlated valuations and to participants acquiring information about others' types. However, discriminatory auctions have several downsides, notably that of charging bidders different prices for identical items, resulting in fairness concerns that make the format unpopular. Two other approaches we suggest are either using posted-pricing mechanisms, or using uniform-price auctions with anonymous reserves. We show that when using balanced prices, both these approaches ensure high equilibrium welfare in the combined market. The latter also inherits many of the benefits from uniform-price auctions such as price discovery, and can be introduced with a minor modification to auctions currently in use to sell carbon emission allowances.

In this paper, we study how to fairly allocate m indivisible chores to n (asymmetric) agents. We consider (weighted) proportionality up to any item (PROPX) and show that a (weighted) PROPX allocation always exists and can be computed efficiently. For chores, we argue that PROPX might be a more reliable relaxation for proportionality by the facts that any PROPX allocation ensures 2-approximation of maximin share (MMS) fairness [Budish, 2011] for symmetric agents and of anyprice share (APS) fairness [Babaioff et al, 2021] for asymmetric agents. APS allocations for chores have not been studied before the current work, and our result implies a 2-approximation algorithm. Another by-product result is that an EFX and a weighted EF1 allocation for indivisible chores exist if all agents have the same ordinal preference, which might be of independent interest. We then consider the partial information setting and design algorithms that only use agents' ordinal preferences to compute approximately PROPX allocations. Our algorithm achieves a 2-approximation for both symmetric and asymmetric agents, and the approximation ratio is optimal. Finally, we study the price of fairness (PoF), i.e., the loss in social welfare by enforcing allocations to be (weighted) PROPX. We prove that the tight ratio for PoF is Theta(n) for symmetric agents and unbounded for asymmetric agents.

We consider a model of a data broker selling information to a single agent to maximize his revenue. The agent has a private valuation of the additional information, and upon receiving the signal from the data broker, the agent can conduct her own experiment to refine her posterior belief on the states with additional costs. To maximize expected revenue, only offering full information in general is suboptimal, and the optimal mechanism may contain a continuum of menu options with partial information to prevent the agent from having incentives to acquire additional information from other sources. However, our main result shows that the additional benefit from price discrimination is limited, i.e., posting a deterministic price for revealing full information obtains at least half of the optimal revenue for arbitrary prior and cost functions., Comment: accepted in EC 2022

We study mechanisms for selling a single item when buyers have private costs for participating in the mechanism. An agent's participation cost can also be interpreted as an outside option value that she must forego to participate. This substantially changes the revenue maximization problem, which becomes non-convex in the presence of participation costs. For multiple buyers, we show how to construct a $(2+\epsilon)$-approximately revenue-optimal mechanism in polynomial time. Our approach makes use of a many-buyers-to-single-buyer reduction, and in the single-buyer case our mechanism improves to an FPTAS. We also bound the menu size and the sample complexity for the optimal single-buyer mechanism. Moreover, we show that posting a single price in the single-buyer case is in fact optimal under the assumption that either (1) the participation cost is independent of the value, and the value distribution has decreasing marginal revenue or monotone hazard rate; or (2) the participation cost is a concave function of the value. When there are multiple buyers, we show that sequential posted pricing guarantees a large fraction of the optimal revenue under similar conditions., Comment: accepted at SODA 2024

This paper considers prior-independent mechanism design, in which a single mechanism is designed to achieve approximately optimal performance on every prior distribution from a given class. Most results in this literature focus on mechanisms with truthtelling equilibria, a.k.a., truthful mechanisms. Feng and Hartline (2018) introduce the revelation gap to quantify the loss of the restriction to truthful mechanisms. We solve a main open question left in Feng and Hartline (2018); namely, we identify a non-trivial revelation gap for revenue maximization. Our analysis focuses on the canonical problem of selling a single item to a single agent with only access to a single sample from the agent's valuation distribution. We identify the sample-bid mechanism (a simple non-truthful mechanism) and upper-bound its prior-independent approximation ratio by 1.835 (resp. 1.296) for regular (resp. MHR) distributions. We further prove that no truthful mechanism can achieve prior-independent approximation ratio better than 1.957 (resp. 1.543) for regular (resp. MHR) distributions. Thus, a non-trivial revelation gap is shown as the sample-bid mechanism outperforms the optimal prior-independent truthful mechanism. On the hardness side, we prove that no (possibly non-truthful) mechanism can achieve prior-independent approximation ratio better than 1.073 even for uniform distributions., Comment: This paper will appear in Proceedings of the ACM Symposium on Theory of Computing 2021 (STOC'21)

We examine the long-term behavior of a Bayesian agent who has a misspecified belief about the time lag between actions and feedback, and learns about the payoff consequences of his actions over time. Misspecified beliefs about time lags result in attribution errors, which have no long-term effect when the agent's action converges, but can lead to arbitrarily large long-term inefficiencies when his action cycles. Our proof uses concentration inequalities to bound the frequency of action switches, which are useful to study learning problems with history dependence. We apply our methods to study a policy choice game between a policy-maker who has a correctly specified belief about the time lag and the public who has a misspecified belief.

We study a fair resource sharing problem, where a set of resources are to be shared among a group of agents. Each agent demands one resource and each resource can serve a limited number of agents. An agent cares about what resource they get as well as the externalities imposed by their mates, who share the same resource with them. Clearly, the strong notion of envy-freeness, where no agent envies another for their resource or mates, cannot always be achieved and we show that even deciding the existence of such a strongly envy-free assignment is an intractable problem. Hence, a more interesting question is whether (and in what situations) a relaxed notion of envy-freeness, the Pareto envy-freeness, can be achieved. Under this relaxed notion, an agent envies another only when they envy both the resource and the mates of the other agent. In particular, we are interested in a dorm assignment problem, where students are to be assigned to dorms with the same capacity and they have dichotomous preference over their dormmates. We show that when the capacity of each dorm is 2, a Pareto envy-free assignment always exists and we present a polynomial-time algorithm to compute such an assignment. Nevertheless, the result breaks immediately when the capacity increases to 3, in which case even Pareto envy-freeness cannot be guaranteed. In addition to the existential results, we also investigate the utility guarantees of (Pareto) envy-free assignments in our model., Comment: accepted in JAIR 2023

We examine a patient player's behavior when he can build reputations in front of a sequence of myopic opponents. With positive probability, the patient player is a commitment type who plays his Stackelberg action in every period. We characterize the patient player's action frequencies in equilibrium. Our results clarify the extent to which reputations can refine the patient player's behavior and provide new insights to entry deterrence, business transactions, and capital taxation. Our proof makes a methodological contribution by establishing a new concentration inequality., Comment: accepted at Journal of Economic Theory

We study multinomial logit bandit with limited adaptivity, where the algorithms change their exploration actions as infrequently as possible when achieving almost optimal minimax regret. We propose two measures of adaptivity: the assortment switching cost and the more fine-grained item switching cost. We present an anytime algorithm (AT-DUCB) with $O(N \log T)$ assortment switches, almost matching the lower bound $\Omega(\frac{N \log T}{ \log \log T})$. In the fixed-horizon setting, our algorithm FH-DUCB incurs $O(N \log \log T)$ assortment switches, matching the asymptotic lower bound. We also present the ESUCB algorithm with item switching cost $O(N \log^2 T)$., Comment: Accepted for presentation at the International Conference on Machine Learning (ICML) 2020

This paper introduces an objective for optimizing proper scoring rules. The objective is to maximize the increase in payoff of a forecaster who exerts a binary level of effort to refine a posterior belief from a prior belief. In this framework we characterize optimal scoring rules in simple settings, give efficient algorithms for computing optimal scoring rules in complex settings, and identify simple scoring rules that are approximately optimal. In comparison, standard scoring rules in theory and practice -- for example the quadratic rule, scoring rules for the expectation, and scoring rules for multiple tasks that are averages of single-task scoring rules -- can be very far from optimal.

We show that economic conclusions derived from Bulow and Roberts (1989) for linear utility models approximately extend to non-linear utility models. Specifically, we quantify the extent to which agents with non-linear utilities resemble agents with linear utilities, and we show that the approximation of mechanisms for agents with linear utilities approximately extend for agents with non-linear utilities. We illustrate the framework for the objectives of revenue and welfare on non-linear models that include agents with budget constraints, agents with risk aversion, and agents with endogenous valuations. We derive bounds on how much these models resemble the linear utility model and combine these bounds with well-studied approximation results for linear utility models. We conclude that simple mechanisms are approximately optimal for these non-linear agent models.

This paper compares two leading approaches for robust optimization in the models of online algorithms and mechanism design. Competitive analysis compares the performance of an online algorithm to an offline benchmark in worst-case over inputs, and prior-independent mechanism design compares the expected performance of a mechanism on an unknown distribution (of inputs, i.e., agent values) to the optimal mechanism for the distribution in worst case over distributions. For competitive analysis, a critical concern is the choice of benchmark. This paper gives a method for selecting a good benchmark. We show that optimal algorithm/mechanism for the optimal benchmark are equal to the prior-independent optimal algorithm/mechanism. We solve a central open question in prior-independent mechanism design, namely we identify the prior-independent revenue-optimal mechanism for selling a single item to two agents with i.i.d. and regularly distributed values. Via this solution and the above equivalence of prior-independent mechanism design and competitive analysis (a.k.a. prior-free mechanism design) we show that the standard method for lower bounds of prior-free mechanisms is not generally tight for the benchmark design program.

We extend the model of stochastic bandits with adversarial corruption (Lykouriset al., 2018) to the stochastic linear optimization problem (Dani et al., 2008). Our algorithm is agnostic to the amount of corruption chosen by the adaptive adversary. The regret of the algorithm only increases linearly in the amount of corruption. Our algorithm involves using L\"owner-John's ellipsoid for exploration and dividing time horizon into epochs with exponentially increasing size to limit the influence of corruption.

We study how to maximize the broker's (expected) profit in a two-sided market, where she buys items from a set of sellers and resells them to a set of buyers. Each seller has a single item to sell and holds a private value on her item, and each buyer has a valuation function over the bundles of the sellers' items. We consider the Bayesian setting where the agents' values are independently drawn from prior distributions, and aim at designing dominant-strategy incentive-compatible (DSIC) mechanisms that are approximately optimal. Production-cost markets, where each item has a publicly-known cost to be produced, provide a platform for us to study two-sided markets. Briefly, we show how to covert a mechanism for production-cost markets into a mechanism for the broker, whenever the former satisfies cost-monotonicity. This reduction holds even when buyers have general combinatorial valuation functions. When the buyers' valuations are additive, we generalize an existing mechanism to production-cost markets in an approximation-preserving way. We then show that the resulting mechanism is cost-monotone and thus can be converted into an 8-approximation mechanism for two-sided markets., Comment: Appeared in IJCAI 19

The revenue optimal mechanism for selling a single item to agents with independent but non-identically distributed values is complex for agents with linear utility (Myerson,1981) and has no closed-form characterization for agents with non-linear utility (cf. Alaei et al., 2012). Nonetheless, for linear utility agents satisfying a natural regularity property, Alaei et al. (2018) showed that simply posting an anonymous price is an e-approximation. We give a parameterization of the regularity property that extends to agents with non-linear utility and show that the approximation bound of anonymous pricing for regular agents approximately extends to agents that satisfy this approximate regularity property. We apply this approximation framework to prove that anonymous pricing is a constant approximation to the revenue optimal single-item auction for agents with public-budget utility, private-budget utility, and (a special case of) risk-averse utility., Comment: Appeared at EC 2019

Linear contextual bandit is an important class of sequential decision making problems with a wide range of applications to recommender systems, online advertising, healthcare, and many other machine learning related tasks. While there is a lot of prior research, tight regret bounds of linear contextual bandit with infinite action sets remain open. In this paper, we address this open problem by considering the linear contextual bandit with (changing) infinite action sets. We prove a regret upper bound on the order of $O(\sqrt{d^2T\log T})\times \text{poly}(\log\log T)$ where $d$ is the domain dimension and $T$ is the time horizon. Our upper bound matches the previous lower bound of $\Omega(\sqrt{d^2 T\log T})$ in [Li et al., 2019] up to iterated logarithmic terms., Comment: 10 pages, accepted for presentation at AISTATS 2021

We study the linear contextual bandit problem with finite action sets. When the problem dimension is $d$, the time horizon is $T$, and there are $n \leq 2^{d/2}$ candidate actions per time period, we (1) show that the minimax expected regret is $\Omega(\sqrt{dT (\log T) (\log n)})$ for every algorithm, and (2) introduce a Variable-Confidence-Level (VCL) SupLinUCB algorithm whose regret matches the lower bound up to iterated logarithmic factors. Our algorithmic result saves two $\sqrt{\log T}$ factors from previous analysis, and our information-theoretical lower bound also improves previous results by one $\sqrt{\log T}$ factor, revealing a regret scaling quite different from classical multi-armed bandits in which no logarithmic $T$ term is present in minimax regret. Our proof techniques include variable confidence levels and a careful analysis of layer sizes of SupLinUCB on the upper bound side, and delicately constructed adversarial sequences showing the tightness of elliptical potential lemmas on the lower bound side.

We study the revenue maximization problem with an imprecisely estimated distribution of a single buyer or several independent and identically distributed buyers given that this estimation is not far away from the true distribution. We use the earth mover's distance to capture the estimation error between those two distributions in terms of both values and their probabilities, i.e., the error in value space given a quantile, and the error in quantile space given a value. We give explicit characterization of the optimal mechanisms for the single buyer setting. For the multi-buyer case, we provide an algorithm that finds an approximately optimal mechanism (FPTAS) among the family of second price auctions with a fixed reserve.

In this note we prove bounds on the upper and lower probability tails of sums of independent geometric or exponentially distributed random variables. We also prove negative results showing that our established tail bounds are asymptotically tight., Comment: 9 pages

In this paper, we implement the Stochastic Damped LBFGS (SdLBFGS) for stochastic non-convex optimization. We make two important modifications to the original SdLBFGS algorithm. First, by initializing the Hessian at each step using an identity matrix, the algorithm converges better than original algorithm. Second, by performing direction normalization we could gain stable optimization procedure without line search. Experiments on minimizing a 2D non-convex function shows that our improved algorithm converges better than original algorithm, and experiments on the CIFAR10 and MNIST datasets show that our improved algorithm works stably and gives comparable or even better testing accuracies than first order optimizers SGD, Adagrad, and second order optimizers LBFGS in PyTorch.

Generating good revenue is one of the most important problems in Bayesian auction design, and many (approximately) optimal dominant-strategy incentive compatible (DSIC) Bayesian mechanisms have been constructed for various auction settings. However, most existing studies do not consider the complexity for the seller to carry out the mechanism. It is assumed that the seller knows "each single bit" of the distributions and is able to optimize perfectly based on the entire distributions. Unfortunately, this is a strong assumption and may not hold in reality: for example, when the value distributions have exponentially large supports or do not have succinct representations. In this work we consider, for the first time, the query complexity of Bayesian mechanisms. We only allow the seller to have limited oracle accesses to the players' value distributions, via quantile queries and value queries. For a large class of auction settings, we prove logarithmic lower-bounds for the query complexity for any DSIC Bayesian mechanism to be of any constant approximation to the optimal revenue. For single-item auctions and multi-item auctions with unit-demand or additive valuation functions, we prove tight upper-bounds via efficient query schemes, without requiring the distributions to be regular or have monotone hazard rate. Thus, in those auction settings the seller needs to access much less than the full distributions in order to achieve approximately optimal revenue.

In this paper, we focus on how to dynamically allocate a divisible resource fairly among n players who arrive and depart over time. The players may have general heterogeneous valuations over the resource. It is known that the exact envy-free and proportional allocations may not exist in the dynamic setting [Walsh, 2011]. Thus, we will study to what extent we can guarantee the fairness in the dynamic setting. We first design two algorithms which are O(log n)-proportional and O(n)-envy-free for the setting with general valuations, and by constructing the adversary instances such that all dynamic algorithms must be at least Omega(1)-proportional and Omega(n/log n)-envy-free, we show that the bounds are tight up to a logarithmic factor. Moreover, we introduce the setting where the players' valuations are uniform on the resource but with different demands, which generalize the setting of [Friedman et al., 2015]. We prove an O(log n) upper bound and a tight lower bound for this case.