In this paper, we study the Facility Location Problem with Scarce Resources (FLPSR) under the assumption that agents' type follows a probability distribution on [0,1]. In the FLPSR, the goal is to identify the optimal locations for one or more capacitated facilities to maximize Social Welfare (SW), defined as the sum of the utilities of all agents. Since the total capacity of the facilities is insufficient to serve all agents, they compete in a First-Come-First-Served game to get accommodated. The main contribution of the paper ties Optimal Transport theory to the problem of selecting a truthful mechanism tailored to the agents' distributions. For the case of a single facility, we show that an optimal mechanism always exists. We examine three classes of probability distributions and characterize the optimal mechanism analytically or provide a routine to numerically compute it. We extend our results to the case in which we have two capacitated facilities to place. Initially, we assume that agents are independent and identically distributed, but our techniques generalize to scenarios where agents are not identically distributed. Finally, we validate our findings through several numerical experiments, including: (i) deriving optimal mechanisms for the class of beta distributions, (ii) assessing the Bayesian approximation ratio of these mechanisms for small numbers of agents, and (iii)assessing how quickly the expected mechanism SW converges to its limit.

Resource competition problems are often modeled using Colonel Blotto games, where players take simultaneous actions. However, many real-world scenarios involve sequential decision-making rather than simultaneous moves. To model these dynamics, we represent the Lottery Colonel Blotto game as a Stackelberg game, in which one player, the leader, commits to a strategy first, and the other player, the follower, responds. We derive the Stackelberg equilibrium for this game, formulating the leader's strategy as a bi-level optimization problem. To solve this, we develop a constructive method based on iterative game reductions, which allows us to efficiently compute the leader’s optimal commitment strategy in polynomial time. Additionally, we identify the conditions under which the Stackelberg equilibrium coincides with the Nash equilibrium. Specifically, this occurs when the budget ratio between the leader and the follower equals a certain threshold, which we can calculate in closed form. In some instances, we observe that when the leader’s budget exceeds this threshold, both players achieve higher utilities in the Stackelberg equilibrium compared to the Nash equilibrium. Lastly, we show that, in the best case, the leader can achieve an infinite utility improvement by making an optimal first move compared to the Nash equilibrium.

In this paper, we explore the Mechanism Design aspects of the m -Capacitated Facility Location Problem ( m -CFLP) on a line, focusing on two frameworks. In the first framework, the number of facilities is arbitrary, all facilities share the same capacity, and the number of agents matches the total capacity of the facilities. In the second framework, we need to locate two facilities, each with a capacity equal to at least half the number of agents. For both frameworks, we propose truthful mechanisms with bounded approximation ratios in terms of Social Cost (SC) and Maximum Cost (MC). When m > 2 , our results stand in contrast to the impossibility results known for the classical m -Facility Location Problem, where capacity constraints are absent. Moreover, all the proposed mechanisms are optimal with respect to MC and either optimal or near-optimal with respect to the SC among anonymous mechanisms. We then establish lower bounds on the approximation ratios that any truthful and deterministic mechanism achieves with respect to SC and MC for both frameworks. Lastly, we run several numerical experiments to empirically evaluate the performances of our mechanisms with respect to the SC or the MC. Our empirical analysis shows that our proposed mechanisms outperform all previously proposed mechanisms applicable in this setting.

Abstract The problem of scheduling unrelated machines has been studied since the inception of algorithmic mechanism design (Nisan and Ronen, Algorithmic mechanism design(extended abstract). In: Proceedings of the Thirty First Annual ACM Symposium on Theory of Computing (STOC), pp. 129–140, 1999. It is a resource allocation problem that entails assigning m tasks to n machines for execution. Machines are regarded as strategic agents who may lie about their execution costs so as to minimize their time cost. To address the situation when monetary payment is not an option to compensate the machines’ costs, Koutsoupias (Theory Comput Syst 54:375–387, 2014) devised two truthful mechanisms, K and P respectively, that achieves an approximation ratio of $$\frac{n+1}{2}$$ n + 1 2 and n, for social cost minimization. In addition, no truthful mechanism can achieve an approximation ratio better than $$\frac{n+1}{2}$$ n + 1 2 . Hence, mechanism K is optimal. While the approximation ratio provides a strong worst-case guarantee, it also limits us to a comprehensive understanding of mechanism performance on various inputs. This paper investigates these two scheduling mechanisms beyond the worst case. We first show that mechanism K achieves a smaller social cost than mechanism P on every input. That is, mechanism K is pointwise better than mechanism P. Next, for each task, when machines’ execution costs are independent and identically drawn from a task-specific distribution, we show that the average-case approximation ratio of mechanism K converges to a constant determined by the task-specific distribution. This bound is tight for mechanism K. For a better understanding of this distribution-dependent constant, on the one hand, we estimate its value by plugging in a few common distributions; on the other, we show that this converging bound improves a known bound (Zhang in Algorithmica 83(6):1638–1652, 2021)) which only captures the single-task setting. Last, we find that the average-case approximation ratio of mechanism P converges to the same constant.

In this article, we explore the Mechanism Design aspects of the Maximum Vertex-Weighted \(b\) -matching (MVbM) problem on bipartite graphs \((A\cup T,E)\) . The set \(A\) comprises agents, while \(T\) represents tasks. The set \(E\) , which connects \(A\) and \(T\) , is the private information of either agents or tasks. In this framework, we investigate three mechanisms— \(\mathbb{M}_{BFS}\) , \(\mathbb{M}_{DFS}\) , and \(\mathbb{M}_{G}\) . We examine scenarios in which either agents or tasks are strategic and report their adjacent edges to one of the three mechanisms. In both cases, we assume that the strategic entities are bounded by their statements: They can hide edges, but they cannot report edges that do not exist. First, we consider the case in which agents can manipulate. In this framework, \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) are optimal but not truthful. By characterizing the Nash Equilibria induced by \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) , we reveal that both mechanisms have a Price of Anarchy ( \(PoA\) ) and Price of Stability ( \(PoS\) ) of \(2\) . These efficiency guarantees are tight; no deterministic mechanism can achieve a lower \(PoA\) or \(PoS\) . In contrast, the third mechanism, \(\mathbb{M}_{G}\) , is not optimal, but truthful and its approximation ratio is \(2\) . We demonstrate that this ratio is optimal; no deterministic and truthful mechanism can outperform it. We then shift our focus to scenarios where tasks can exhibit strategic behavior. In this case, \(\mathbb{M}_{BFS}\) , \(\mathbb{M}_{DFS}\) , and \(\mathbb{M}_{G}\) all maintain truthfulness, making \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) truthful and optimal mechanisms. In conclusion, we investigate the manipulability of \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) through experiments on randomly generated graphs. We observe that (i) \(\mathbb{M}_{BFS}\) is less prone to be manipulated by the first agent than \(\mathbb{M}_{DFS}\) , and (ii) \(\mathbb{M}_{BFS}\) is more manipulable on instances in which the total capacity of the agents is equal to the number of tasks. 1

In this paper, we delve into the problem of using monetary incentives to encourage players to shift from an initial Nash equilibrium to a more favorable one within a game. Our main focus revolves around computing the minimum reward required to facilitate this equilibrium transition. The game involves a single row player who possesses m strategies and k column players, each endowed with n strategies. Our findings reveal that determining whether the minimum reward is zero is NP-complete, and computing the minimum reward becomes APX-hard. Nonetheless, we bring some positive news, as this problem can be efficiently handled if either k or n is a fixed constant. Furthermore, we have devised an approximation algorithm with an additive error that runs in polynomial time. Lastly, we explore a specific case wherein the utility functions exhibit single-peaked characteristics, and we successfully demonstrate that the optimal reward can be computed in polynomial time.

In Federated Learning, a Central Node (CN) coordinates a group of agents to collectively train a shared neural network.However, due to the inherent information asymmetry, some agents may behave as free riders and exploit the system by reaping rewards or by passively benefiting from the common model without contributing to the training process.Proof-of-Training (PoT) effectively allows the CN to verify that an agent has completed training honestly and correctly.However, this method incurs high costs, including proof generation by the agent, communication expenses, and proof verification by the CN.Conducting Proof-of-Training in each FL round is impractical due to these expenses.To enhance verification efficiency, a feasible strategy is to conduct probabilistic verification, where only a subset of agents is sampled for verification in each FL round.This paper aims to design a new incentive mechanism to motivate the agents behave honestly and potentially mitigate free riders.Our model hinges on two parameters: (i) the reward allocated to the local trainers, namely 𝑅, and (ii) a probability vector, denoted as ì 𝑝, indicating the likelihood of subjecting each agent to PoT scrutiny.We show that it is possible to characterize a set of parameters 𝑅 and ì 𝑝 that minimizes the total CN cost and makes the routine Individually Rational and Incentive Compatible, so that every agent will actively train their local model.Finally, we validate our model through extensive experiments.Our findings show that our characterization of the best reward and validation scheme is correct as they minimize the cost of the training routine without compromising the convergence speed.All our experiments are conducted on various datasets, demonstrating the wide applicability of our results.

This paper investigates the Mechanism Design aspects of the $m$-Capacitated Facility Location Problem where the total facility capacity is lower than the number of agents. Following \cite{aziz2020capacity}, the Social Welfare of the facility location is determined through a First-Come-First-Served (FCFS) game where agents compete after the facility positions are established. When the number of facilities is $m>1$, the Nash Equilibrium (NE) of the FCFS game is not unique, thus the utility of the agents and the notion of truthfulness are not well-defined. To address these issues, we consider absolutely truthful mechanisms, i.e. mechanisms able to prevent agents from misreporting regardless of the strategies played during the FCFS game. We pair this more stringent truthfulness requirement with the notion of Equilibrium Stable (ES) mechanism, i.e. mechanisms whose Social Welfare does not depend on the NE of the FCFS game. We show that the class of percentile mechanisms is absolutely truthful and characterize under which conditions they are ES. We then show that the approximation ratio of each ES percentile mechanism is bounded and determine its value. Notably, when all the facilities have the same capacity and the number of agents is large enough, it is possible to achieve an approximation ratio smaller than $1+\frac{1}{2m-1}$. We enhance our findings by empirically evaluating the mechanisms’ performances when agents’ true positions follows a distribution.

In this research, we study the problem that a collector acquires items from the owner based on the item qualities the owner declares and an independent appraiser's assessments. The owner is interested in maximizing the probability that the collector acquires the items and is the only one who knows the items' factual quality. The appraiser performs her duties with impartiality, but her assessment may be subject to random noises, so it may not accurately reflect the factual quality of the items. The main challenge lies in devising mechanisms that prompt the owner to reveal accurate information, thereby optimizing the collector's expected reward. We consider the menu size of mechanisms as a measure of their practicability and study its impact on the attainable expected reward. For the single-item setting, we design optimal mechanisms with a monotone increasing menu size. Although the reward gap between the simplest and optimal mechanisms is bounded, we show that simple mechanisms with a small menu size cannot ensure any positive fraction of the optimal reward of mechanisms with a larger menu size. For the multi-item setting, we show that an ordinal mechanism that only takes the owner's ordering of the items as input is not incentive-compatible. We then propose a set of Union mechanisms that combine single-item mechanisms. Moreover, we run experiments to examine these mechanisms' robustness against the independent appraiser's assessment accuracy and the items' acquiring rate.

Public goods projects, such as open-source technology, are essential for the blockchain ecosystem's growth. However, funding these projects effectively remains a critical issue within the ecosystem. Currently, the funding protocols for blockchain public goods lack professionalism and fail to learn from past experiences. To address this challenge, our research introduces a human oracle protocol involving public goods projects, experts, and funders. In our approach, funders contribute investments to a funding pool, while experts offer investment advice based on their expertise in public goods projects. The oracle's decisions on funding support are influenced by the reputations of the experts. Experts earn or lose reputation based on how well their project implementations align with their advice, with successful investments leading to higher reputations. Our oracle is designed to adapt to changing circumstances, such as experts exiting or entering the decision-making process. We also introduce a regret bound to gauge the oracle's effectiveness. Theoretically, we establish an upper regret bound for both static and dynamic models and demonstrate its closeness to an asymptotically equal lower bound. Empirically, we implement our protocol on a test chain and show that our oracle's investment decisions closely mirror optimal investments in hindsight.

In this paper, we study the Maximum Vertex-weighted b-Matching (MVbM) problem on bipartite graphs in a new game-theoretical environment. In contrast to other game-theoretical settings, we consider the case in which the value of the tasks is public and common to every agent so that the private information of every agent consists of edges connecting them to the set of tasks. In this framework, we study three mechanisms. Two of these mechanisms, namely MBFS and MDFS, are optimal but not truthful, while the third one, MAP, is truthful but sub-optimal. Albeit these mechanisms are induced by known algorithms, we show MBFS and MDFS are the best possible mechanisms in terms of Price of Anarchy and Price of Stability, while MAP is the best truthful mechanism in terms of approximated ratio. Furthermore, we characterize the Nash Equilibria of MBFS and MDFS and retrieve sets of conditions under which MBFS acts as a truthful mechanism, which highlights the differences between MBFS and MDFS. Finally, we extend our results to the case in which agents’ capacity is part of their private information.

Due to the large number of submissions that more and more conferences experience, finding an automatized way to well distribute the submitted papers among reviewers has become necessary. We model the peer-reviewing matching problem as a bilevel programming (BP) formulation. Our model consists of a lower-level problem describing the reviewers’ perspective and an upper-level problem describing the editors’. Every reviewer is interested in minimizing their overall effort, while the editors are interested in finding an allocation that maximizes the quality of the reviews and follows the reviewers’ preferences the most. To the best of our knowledge, the proposed model is the first one that formulates the peer-reviewing matching problem by considering two objective functions, one to describe the reviewers’ viewpoint and the other to describe the editors’ viewpoint. We demonstrate that both the upper-level and lower-level problems are feasible and that our BP model admits a solution under mild assumptions. After studying the properties of the solutions, we propose a heuristic to solve our model and compare its performance with the relevant state-of-the-art methods. Extensive numerical results show that our approach can find fairer solutions with competitive quality and less effort from the reviewers.(Our code website: https://github.com/Galaxy-ZRX/Bilevel-Review.)

Audiences cangain an immersive experience watching videos from multiple angles (a.k.a. viewpoints). Free Viewpoint Video (FVV) is developed to enable users to choose their preferred viewpoints during the play of a video. However, users may experience a delay if video frames of the chosen viewpoint cannot be timely loaded, or synthesized from multiple video streams of neighboring viewpoints. To address this problem, we present Edge-FVV, an edge-assisted FVV system that employs edge caches to reduce the delay in streaming the requested FVV from the server to client users. We first analyze the capacity and delay at edge caches when answering FVV requests. Next, we propose two types of machine learning algorithms that allocate the users’ requests to appropriate edge caches. Our evaluation shows that two types of proposed algorithms outperform benchmarks by 4.2-7.4% and 4.6-6.8%, respectively, in reducing the delay for FVV requests.

In full-knowledge multi-robot adversarial patrolling, a group of robots has to detect an adversary who knows the robots' strategy. The adversary can easily take advantage of any deterministic patrolling strategy, which necessitates the employment of a randomised strategy. While the Markov decision process has been the dominant methodology in computing the penetration detection probabilities on polylines, we apply enumerative combinatorics to characterise the penetration detection probabilities for four penetration configurations. It allows us to provide the closed formulae of these probabilities and facilitates characterising optimal random defence strategies. Comparing to iteratively updating the Markov transition matrices, we empirically show that our method reduces the runtime by up to several hours. This allows us extensive simulations on the two dominant robot movement types for patrolling a perimeter showing that a movement with direction is up to 0.4 more likely to detect an adversary. Therefore, our approach greatly benefits the theoretical and empirical analysis of optimal patrolling strategies with extendability to more complicated attacks and other structured environments.

In a Fisher market, the market maker sells m products to n potential agents. The agents submit their utility functions and money endowments to the market maker, who, upon receiving submitted information, derives market equilibrium prices and allocations of the products. Agents are self-interested entities who wish to maximize their utility, and they may misreport their private information for this purpose. The incentive ratio characterizes the extent to which strategic plays can increase an agent's utility. While agents do benefit by misreporting their private information, we show that the ratio of improvement by a unilateral strategic play is no more than two in markets with gross substitute utilities for the agents. Moreover, it can be pinned down to e1/e≈1.445 in Cobb-Douglas markets. For the Leontief markets in which products are complementary, we show that the incentive ratio is at most two as well.

A mechanism for the random assignment problem takes agents' private preferences over items as input and outputs an allocation. One of the mainstream mechanisms, Random Priority, is asymptotically the best mechanism for welfare maximization. However, its approximation ratio is Θ(n), which implies that there is a large discrepancy between its performance and the optimal social welfare. We evaluate the performance of Random Priority beyond the worst-case analysis in the hope of addressing the inconsistency between the widespread use of the mechanism in practice and the undesired theoretical performance guarantee. We show that with small random noise applied to the worst-case inputs, Random Priority has a constant smoothed approximation ratio. When agents' preference values are independent random variables, Random Priority is nearly optimal evaluated by the average-case approximation ratio. En route to these results, we develop analytics tools to show the insights that the efficiency loss is small on most instances. To our limited knowledge, this is the first work that introduces smoothed analysis to algorithmic mechanism design problems. These results may pave the way for further studies for approximate mechanism design problems beyond the worst-case analysis.

Abstract Apart from the principles and methodologies inherited from Economics and Game Theory, the studies in Algorithmic Mechanism Design typically employ theworst-case analysisand design ofapproximation schemesof Theoretical Computer Science. For instance, theapproximation ratio, which is the canonical measure of evaluating how well an incentive-compatible mechanism approximately optimizes the objective, is defined in the worst-case sense. It compares the performance of the optimal mechanism against the performance of a truthful mechanism, for all possible inputs. In this paper, we take theaverage-case analysisapproach, and tackle one of the primary motivating problems in Algorithmic Mechanism Design—the scheduling problem (Nisan and Ronen, in: Proceedings of the 31st annual ACM symposium on theory of computing (STOC), 1999). One version of this problem, which includes a verification component, is studied by Koutsoupias (Theory Comput Syst 54(3):375–387, 2014). It was shown that the problem has a tight approximation ratio bound of $$(n+1)/2$$ (n+1)/2 for the single-task setting, wherenis the number of machines. We show, however, when the costs of the machines to executing the task followanyindependent and identical distribution, theaverage-case approximation ratioof the mechanism given by Koutsoupias (Theory Comput Syst 54(3):375–387, 2014) is upper bounded by a constant. This positive result asymptotically separates the average-case ratio from the worst-case ratio. It indicates that the optimal mechanism devised for a worst-case guarantee works well on average.

In full-knowledge multi-robot adversarial patrolling, a group of robots have to detect an adversary who knows the robots' strategy. The adversary can easily take advantage of any deterministic patrolling strategy, which necessitates the employment of a randomised strategy. While the Markov decision process has been the dominant methodology in computing the penetration detection probabilities, we apply enumerative combinatorics to characterise the penetration detection probabilities. It allows us to provide the closed formulae of these probabilities and facilitates characterising optimal random defence strategies. Comparing to iteratively updating the Markov transition matrices, our methods significantly reduces the time and space complexity of solving the problem. We use this method to tackle four penetration configurations.

Abstract Often because of limitations in generation capacity of power stations, many developing countries frequently resort to disconnecting large parts of the power grid from supply, a process termed load shedding. This leaves households in disconnected parts without electricity, causing them inconvenience and discomfort. Without fairness being taken into due consideration during load shedding, some households may suffer more than others. In this paper, we solve the fair load shedding problem (FLSP) by creating solutions which connect households to supply based on some fairness criteria (i.e., to fairly connect homes to supply in terms of duration, their electricity needs, and their demand), which we model as their utilities. First, we briefly describe some state-of-art household-level load shedding heuristics which meet the first criteria. Second, we model the FLSP as a resource allocation problem, which we formulate into two Mixed Integer Programming (MIP) problems based on the Multiple Knapsack Problem. In so doing, we use the utilitarian, egalitarian and envy-freeness social welfare metrics to develop objectives and constraints that ensure our FLSP solutions results in fair allocations that consider the utilities of agents. Then, we solve the FLSP and show that our MIP models maximize the groupwise and individual utilities of agents, and minimize the differences between their pairwise utilities under a number of experiments. When taken together, our endeavour establishes a set of benchmarks for fair load shedding schemes, and provide insights for designing fair allocation solutions for other scarce resources.

The fundamental assignment problem is in search of welfare maximization mechanisms to allocate items to agents when the private preferences over indivisible items are provided by self-interested agents. The mainstream mechanism \textit{Random Priority} is asymptotically the best mechanism for this purpose, when comparing its welfare to the optimal social welfare using the canonical \textit{worst-case approximation ratio}. Surprisingly, the efficiency loss indicated by the worst-case ratio does not have a constant bound \cite{FFZ:14}.Recently, \cite{DBLP:conf/mfcs/DengG017} shows that when the agents' preferences are drawn from a uniform distribution, its \textit{average-case approximation ratio} is upper bounded by 3.718. They left it as an open question of whether a constant ratio holds for general scenarios. In this paper, we offer an affirmative answer to this question by showing that the ratio is bounded by $1/\mu$ when the preference values are independent and identically distributed random variables, where $\mu$ is the expectation of the value distribution. This upper bound improves the results in \cite{DBLP:conf/mfcs/DengG017} for the Uniform distribution as well. Moreover, under mild conditions, the ratio has a \textit{constant} bound for any independent random values. En route to these results, we develop powerful tools to show the insights that for most valuation inputs, the efficiency loss is small.

Apart from the principles and methodologies inherited from Economics and Game Theory, the studies in Algorithmic Mechanism Design typically employ the worst-case analysis and approximation schemes of Theoretical Computer Science. For instance, the approximation ratio, which is the canonical measure of evaluating how well an incentive-compatible mechanism approximately optimizes the objective, is defined in the worst-case sense. It compares the performance of the optimal mechanism against the performance of a truthful mechanism, for all possible inputs. In this paper, we take the average-case analysis approach, and tackle one of the primary motivating problems in Algorithmic Mechanism Design -- the scheduling problem [Nisan and Ronen 1999]. One version of this problem which includes a verification component is studied by [Koutsoupias 2014]. It was shown that the problem has a tight approximation ratio bound of (n+1)/2 for the single-task setting, where n is the number of machines. We show, however, when the costs of the machines to executing the task follow any independent and identical distribution, the average-case approximation ratio of the mechanism given in [Koutsoupias 2014] is upper bounded by a constant. This positive result asymptotically separates the average-case ratio from the worst-case ratio, and indicates that the optimal mechanism for the problem actually works well on average, although in the worst-case the expected cost of the mechanism is Theta(n) times that of the optimal cost.

Due to the limited generation capacity of power stations, many developing countries frequently resort to disconnecting large parts of the power grid from supply, a process termed load shedding. During load shedding, many homes are left without electricity, causing them inconvenience and discomfort. In this paper, we present a number of optimization heuristics that focus on pairwise and groupwise fairness, such that households (i.e. agents) are fairly allocated electricity. We evaluate the heuristics against standard fairness metrics in terms of comfort delivered to homes, as well as the number of times they are disconnected from electricity supply. Thus, we establish new benchmarks for fair load shedding schemes.

The Consensus-halving problem is the problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions. We study the epsilon-approximate version, which allows each agent to have an epsilon discrepancy on the values of the portions. It was recently proven in [Filos-Ratsikas and Goldberg, 2018] that the problem of computing an epsilon-approximate Consensus-halving solution (for n agents and n cuts) is PPA-complete when epsilon is inverse-exponential. In this paper, we prove that when epsilon is constant, the problem is PPAD-hard and the problem remains PPAD-hard when we allow a constant number of additional cuts. Additionally, we prove that deciding whether a solution with n-1 cuts exists for the problem is NP-hard.

The approximation ratio has become one of the dominant measures in mechanism design problems. In light of analysis of algorithms, we define the smoothed approximation ratio to compare the performance of the optimal mechanism and a truthful mechanism when the inputs are subject to random perturbations of the worst-case inputs, and define the average-case approximation ratio to compare the performance of these two mechanisms when the inputs follow a distribution. For the one-sided matching problem, Filos-Ratsikas et al. [2014] show that, amongst all truthful mechanisms, random priority achieves the tight approximation ratio bound of Theta(sqrt{n}). We prove that, despite of this worst-case bound, random priority has a constant smoothed approximation ratio. This is, to our limited knowledge, the first work that asymptotically differentiates the smoothed approximation ratio from the worst-case approximation ratio for mechanism design problems. For the average-case, we show that our approximation ratio can be improved to 1+e. These results partially explain why random priority has been successfully used in practice, although in the worst case the optimal social welfare is Theta(sqrt{n}) times of what random priority achieves. These results also pave the way for further studies of smoothed and average-case analysis for approximate mechanism design problems, beyond the worst-case analysis.

We study the problem of locating a single facility on a real line based on the reports of self-interested agents, when agents have double-peaked preferences, with the peaks being on opposite sides of their locations. We observe that double-peaked preferences capture real-life scenarios and thus complement the well-studied notion of single-peaked preferences. As a motivating example, assume that the government plans to build a primary school along a street; an agent with single-peaked preferences would prefer having the school built exactly next to her house. However, while that would make it very easy for her children to go to school, it would also introduce several problems, such as noise or parking congestion in the morning. A 5-min walking distance would be sufficiently far for such problems to no longer be much of a factor and at the same time sufficiently close for the school to be easily accessible by the children on foot. There are two positions (symmetrically) in each direction and those would be the agent’s two peaks of her double-peaked preference. Motivated by natural scenarios like the one described above, we mainly focus on the case where peaks are equidistant from the agents’ locations and discuss how our results extend to more general settings. We show that most of the results for single-peaked preferences do not directly apply to this setting, which makes the problem more challenging. As our main contribution, we present a simple truthful-in-expectation mechanism that achieves an approximation ratio of $$1+b/c$$ for both the social and the maximum cost, where b is the distance of the agent from the peak and c is the minimum cost of an agent. For the latter case, we provide a 3 / 2 lower bound on the approximation ratio of any truthful-in-expectation mechanism. We also study deterministic mechanisms under some natural conditions, proving lower bounds and approximation guarantees. We prove that among a large class of reasonable strategyproof mechanisms, there is no deterministic mechanism that outperforms our truthful-in-expectation mechanism. In order to obtain this result, we first characterize mechanisms for two agents that satisfy two simple properties; we use the same characterization to prove that no mechanism in this class can be group-strategyproof.

We study the problem of locating a single facility on a real line based on the reports of self-interested agents, when agents have double-peaked preferences, with the peaks being on opposite sides of their locations.We observe that double-peaked preferences capture real-life scenarios and thus complement the well-studied notion of single-peaked preferences. We mainly focus on the case where peaks are equidistant from the agents’ locations and discuss how our results extend to more general settings. We show that most of the results for single-peaked preferences do not directly apply to this setting; this makes the problem essentially more challenging. As our main contribution, we present a simple truthful-in-expectation mechanism that achieves an approximation ratio of 1+b/c for both the social and the maximum cost, where b is the distance of the agent from the peak and c is the minimum cost of an agent. For the latter case, we provide a 3/2 lower bound on the approximation ratio of any truthful-in-expectation mechanism. We also study deterministic mechanisms under some natural conditions, proving lower bounds and approximation guarantees. We prove that among a large class of reasonable mechanisms, there is no deterministic mechanism that outpeforms our truthful-in-expectation mechanism.

The Fisher market model is one of the most fundamental resource allocation models in economics. In a Fisher market, the prices and allocations of goods are determined according to the preferences and budgets of buyers to clear the market. In a Fisher market game, however, buyers are strategic and report their preferences over goods; the market-clearing prices and allocations are then determined based on their reported preferences rather than their real preferences. We show that the Fisher market game always has a pure Nash equilibrium, for buyers with linear, Leontief, and Cobb-Douglas utility functions, which are three representative classes of utility functions in the important Constant Elasticity of Substitution (CES) family. Furthermore, to quantify the social efficiency, we prove Price of Anarchy bounds for the game when the utility functions of buyers fall into these three classes respectively.

We study the problem of approximate social welfare maximization (without money) in one-sided matching problems when agents have unrestricted cardinal preferences over a finite set of items. Random priority is a very well-known truthful-in-expectation mechanism for the problem. We prove that the approximation ratio of random priority is Θ( n − 1/2 ) while no truthful-in-expectation mechanism can achieve an approximation ratio better than O ( n − 1/2 ), where n is the number of agents and items. Furthermore, we prove that the approximation ratio of all ordinal (not necessarily truthful-in-expectation) mechanisms is upper bounded by O ( n − 1/2 ), indicating that random priority is asymptotically the best truthful-in-expectation mechanism and the best ordinal mechanism for the problem. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The primary goal of a prediction market is to elicit and aggregate information about some future event of interest. How well this goal is achieved depends on the behavior of self-interested market participants, which are crucially influenced by not only their private information but also their knowledge of others' private information, in other words, the information structure of market participants. In this paper, we model a prediction market using the now-classic logarithmic market scoring rule (LMSR) market maker as an extensive-form Bayesian game and aim to understand and characterize the game-theoretic equilibria of the market for different information structures. Prior work has shown that when participants' information is independent conditioned on the realized outcome of the event, the only type of equilibria in this setting has every participant race to honestly reveal their private information as soon as possible, which is the most desirable outcome for the market's goal of information aggregation. This paper considers the remaining two classes of information structures: participants' information being unconditionally independent (the I game) and participants' information being both conditionally and unconditionally dependent (the D game). We characterize the unique family of equilibria for the I game with finite number of participants and finite stages. At any equilibrium in this family, if player i's last stage of participation in the market is after player j's, player i only reveals his information after player j's last stage of participation. This suggests that players race to delay revealing their information, which is probably the least desirable outcome for the market's goal. We consider a special case of the D game and cast insights on possible equilibria if one exists.