We present a polynomial-time algorithm for exactly computing second-price pacing equilibria (SPPE) in auction markets with a constant number of buyers. SPPE plays a central role in modern advertising auctions; however, computing or even approximating it is PPAD-hard in general. To overcome this computational barrier in the restricted setting, we adopt the cell-decomposition method. Specifically, we partition the solution space into polynomially many cells, each defined by hyperplanes corresponding to a fixed ordering of buyers’ scaled valuations across goods. Within each cell, the equilibrium computation reduces to solving a constant number of linear programs. Notably, our algorithm can also efficiently identify equilibria that optimize key objectives such as revenue or social welfare. To the best of our knowledge, this is the first algorithm that efficiently computes an exact SPPE for a simple and natural class of second-price pacing games.

Effective urban planning is crucial for enhancing residents' quality of life and ensuring societal stability, playing a pivotal role in the sustainable development of cities. Current planning methods heavily rely on human experts, which are time-consuming and labor-intensive, or utilize deep learning algorithms, often limiting stakeholder involvement. To bridge these gaps, we propose Intelli-Planner, a novel framework integrating Deep Reinforcement Learning (DRL) with large language models (LLMs) to facilitate participatory and customized planning scheme generation. Intelli-Planner utilizes demographic, geographic data, and planning preferences to determine high-level planning requirements and demands for each functional type. During training, a knowledge enhancement module is employed to enhance the decision-making capability of the policy network. Additionally, we establish a multi-dimensional evaluation system and employ LLM-based stakeholders for satisfaction scoring. Experimental validation across diverse urban settings shows that Intelli-Planner surpasses traditional baselines and achieves comparable performance to state-of-the-art DRL-based methods in objective metrics, while enhancing stakeholder satisfaction and convergence speed. These findings underscore the effectiveness and superiority of our framework, highlighting the potential for integrating the latest advancements in LLMs with DRL approaches to revolutionize tasks related to functional areas planning.

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.

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.

One primary topic of multimodal learning is to jointly incorporate heterogeneous information from different modalities. However, most models often suffer from unsatisfactory multimodal cooperation, which cannot jointly utilize all modalities well. Some methods are proposed to identify and enhance the worse learnt modality, but they are often hard to provide the fine-grained observation of multimodal cooperation at sample-level with theoretical support. Hence, it is essential to reasonably observe and improve the fine-grained cooperation between modalities, especially when facing realistic scenarios where the modality discrepancy could vary across different samples. To this end, we introduce a sample-level modality valuation metric to evaluate the contribution of each modality for each sample. Via modality valuation, we observe that modality discrepancy indeed could be different at sample-level, beyond the global contribution discrepancy at dataset-level. We further analyze this issue and improve cooperation between modalities at sample-level by enhancing the discriminative ability of low-contributing modalities in a targeted manner. Overall, our methods reasonably observe the fine-grained uni-modal contribution and achieve considerable improvement. The source code and dataset are available at https://github.com/GeWu-Lab/Valuate-and-Enhance-Multimodal-Cooperation.

We consider a competitive information design game in which there are multiple senders vie for the selection of a risk-neutral receiver by disclosing information about their individual state realizations. The receiver aims to select the sender who has the highest expected value of the state. We consider a general setting where the senders may be ex-ante heterogeneous, namely, while the senders' state realizations are independently distributed, they do not necessarily follow the same prior distributions. Each sender can only control the disclosure of information regarding his own state realizations, but there is no structural restriction on the set of the feasible information disclosing strategies.

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.

Time or money? That is a question! In this paper, we consider this dilemma in the pricing regime, in which we try to find the optimal pricing scheme for identical items with heterogenous time-sensitive buyers. We characterize the revenue-optimal solution and propose an efficient algorithm to find it in a Bayesian setting. Our results also demonstrate the tight ratio between the value of wasted time and the seller's revenue, as well as that of two common-used pricing schemes, the k-step function and the fixed pricing. To explore the nature of the optimal scheme in the general setting, we present the closed forms over the product distribution and show by examples that positive correlation between the valuation of the item and the cost per unit time could help increase revenue. To the best of our knowledge, it is the first step towards understanding the impact of the time factor as a part of the buyer cost in pricing problems, in the computational view.

We continue the study of the performance for fixed-price mechanisms in the bilateral trade problem, and improve approximation ratios of welfare-optimal mechanisms in several settings. Specifically, in the case where only the buyer distribution is known, we prove that there exists a distribution over different fixed-price mechanisms, such that the approximation ratio lies within the interval of [0.71, 0.7381]. Furthermore, we show that the same approximation ratio holds for the optimal fixed-price mechanism, when both buyer and seller distributions are known. As a result, the previously best-known (1 − 1/e+0.0001)-approximation can be improved to 0.71. Additionally, we examine randomized fixed-price mechanisms when we receive just one single sample from the seller distribution, for both symmetric and asymmetric settings. Our findings reveal that posting the single sample as the price remains optimal among all randomized fixed-price mechanisms.

We consider designing reward schemes that incentivize agents to create high-quality content (e.g., videos, images, text, ideas). The problem is at the center of a real-world application where the goal is to optimize the overall quality of generated content on user-generated content platforms. We focus on anonymous independent reward schemes (AIRS) that only take the quality of an agent's content as input. We prove the general problem is NP-hard. If the cost function is convex, we show the optimal AIRS can be formulated as a convex optimization problem and propose an efficient algorithm to solve it. Next, we explore the optimal linear reward scheme and prove it has a 1/2-approximation ratio, and the ratio is tight. Lastly, we show the proportional scheme can be arbitrarily bad compared to AIRS.

Motivated by a new generation of Internet advertising that has emerged in the live streaming e-commerce markets (e.g., Tiktok) over the past five years, we study a variant of online bipartite matching problem: advertisers send ad requests to influencers (aka, key opinion leaders) on a social media platform. Each influencer has a maximum number of ad requests she can accommodate. We assign a fixed number of influencers to an advertiser when she enters the platform. The advertiser then matches with each of the assigned influencers with a probability, which can be thought of as a set of negotiations between the advertiser and the set of assigned influencers. Unlike the standard online assignment problems, the outcome of any of these matches is not revealed throughout the session (negotiations take time). Our goal is to maximize the expected number of matches between advertisers and influencers. We put forward a new deterministic algorithm with a competitive ratio of $1/2$ and prove that no deterministic algorithm can achieve a better competitive ratio. We also show that the competitive ratio can be improved when randomness is allowed. We then study a setting where a match is successful with either probability 0 or a fixed $p$. We present an optimal randomized algorithm that achieves a competitive ratio of $1-1/e$ in this setting.

Vehicular CrowdSensing (VCS) has become a powerful sensing paradigm by selecting users driving vehicles to perform tasks. In most existing research, the platform centrally allocates tasks according to the collected user information. We argue that the information collection process results in user privacy leakage, and the centralized allocation leads to a heavy computation complexity. We propose to apply a distributed task allocation method in the widely-used route navigation system. The navigation system recommends several routes to a user and each route may cover some tasks. Then, the user distributively selects a route according to the route profit (task reward minus route cost). Since the task reward is shared by users, the route selections of users may influence each other. Hence, it remains unclear how to design a distributed route navigation approach to reach an equilibrium state (i.e., each user is satisfied with the selected route), while guaranteeing a good total profit. To this end, we formulate the problem as a multi-user potential game and propose a distributed route navigation algorithm. The trace-based simulation results verify that the proposed algorithm achieves a Nash equilibrium, while achieving a total user profit performance close to that of the optimal solution.

Mobile CrowdSensing (MCS) has recently become a powerful paradigm that recruits users to cooperatively perform various tasks. In many realistic settings, users participate in real time and we have to recruit them in an online manner. The existing works usually formulate the online recruitment problem as a budgeted optimal stopping problem with submodular user utility, while we first argue that not only the budget but also the time constraints can jointly influence the recruitment performance. For example, if we have less budget but plenty of time, we should recruit users with more patience. Second, considering the user’s cooperative willingness, its contribution may be diminishing or even irregular. Hence, we also need to address not only submodular cases but also their non-submodular utility. In this paper, we study the online user recruitment problem with (non-)submodular utility under the budget and time constraints. To deal with the two constraints, we first estimate the number of users to be recruited and then recruit them in segments. Moreover, we extend the segmented strategy with a non-submodular utility, which has the submodularity ratio$\gamma $and the competitive ratio$\gamma ^{2}(1-e^{-1})/7$. Furthermore, to correct estimation errors and utilize newly obtained information, we dynamically re-adjust the segmented strategy and also prove that the dynamic strategy achieves a competitive ratio of${\gamma ^{2}(1-e^{-1})(1-e^{-\gamma /2})/7}$. Finally, a reverse auction-based online pricing mechanism is lightly built into the proposed user recruitment strategy, which achieves truthfulness and individual rationality. Extensive experiments on three real-world data sets validate the proposed online user recruitment strategy under the (non-) submodular utility and two constraints.

We consider the principal-agent problem with heterogeneous agents. Previous works assume that the principal signs independent incentive contracts with every agent to make them invest more efforts on the tasks. However, in many circumstances, these contracts need to be identical for the sake of fairness. We investigate the optimal common contract problem. To our knowledge, this is the first attempt to consider this natural and important generalization. We first show this problem is NP-complete. Then we provide a dynamic programming algorithm to compute the optimal contract in O(n2m) time, where n,m are the number of agents and actions, under the assumption that the agents' cost functions obey increasing difference property. At last, we generalize the setting such that each agent can choose to directly produce a reward in [0,1]. We provide an O(log n)-approximate algorithm for this generalization.

Film release time play an important part in box office revenues due to obvious seasonality demand in the film industry and severe competition among films shown at the same time. In this paper, we study how film studios choose release time for movies they produce to maximize their box offices. We first formalize this problem as an attraction competition game where players (film studios) consider both potential profits and competitors' choices when deciding the release time. Then we prove that there always exists a pure Nash equilibrium and give the sufficient condition of the uniqueness of the Nash equilibrium. Our model can be generalized to an extensive game and we compute the subgame-perfect equilibrium for homogeneous players. For the case that one film studio could have multiple movies to release, we prove that finding a player's best response is NP-hard and it does not guarantee the existence of a pure Nash equilibrium. Experiments are provided to support the soundness of our model. In the final state, most of the film studios, accounting for 84 percent of the market, would not change their release time. The behaviors of film studios imply they are following some strategies to reach a Nash equilibrium.

Econometric inference allows an analyst to back out the values of agents in a mechanism from the rules of the mechanism and bids of the agents. This paper gives an algorithm to solve the problem of inferring the values of agents in a dominant-strategy mechanism from: the social choice function implemented by the mechanism and the per-unit prices paid by the agents (the agent bids are not observed). For single-dimensional agents, this inference problem is a multi-dimensional inversion of the payment identity and is feasible only if the payment identity is uniquely invertible. The inversion is unique for single-unit proportional weights social choice functions (common, for example, in bandwidth allocation); and its inverse can be found efficiently. This inversion is not unique for social choice functions that exhibit complementarities. Of independent interest, we extend a result of Rosen (1965), that the Nash equilbria of “concave games” are unique and pure, to an alternative notion of concavity based on Gale and Nikaido (1965).

Motivated by online ad auctions, we consider a repeated auction between one seller and many buyers, where each buyer only has an estimation of her value in each period until she actually receives the item in that period. The seller is allowed to conduct a dynamic auction but must guarantee ex-post individual rationality. In this paper, we use a structure that we call credit accounts to enable a general reduction from any incentive compatible and ex-ante individual rational dynamic auction to an approximate incentive compatible and ex-post individually rational dynamic auction with credit accounts. Our reduction obtains stronger individual rationality guarantees at the cost of weaker incentive compatibility. Surprisingly, our reduction works without any common knowledge assumption. Finally, as a complement to our reduction, we prove that there is no non-trivial auction that is exactly incentive compatible and ex-post individually rational under this setting.

This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1−1/e≈.63, derived using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the worst known example, the first price auction achieves a ≈.869-fraction of the optimal welfare, far better than the theoretical guarantee. Despite this large gap, it was unclear whether the 1−1/e bound was tight. We prove that it is not. Our analysis eschews smoothness, and instead uses the independence assumption on agents’ value distributions to give a more careful accounting of the welfare contribution of agents who win despite not having the highest value.

Time-inconsistency refers to a paradox in decision making where agents exhibit inconsistent behaviors over time. Examples are procrastination where agents tend to postpone easy tasks, and abandonments where agents start a plan and quit in the middle. To capture such behaviors and to quantify inefficiency caused by such behaviors, Kleinberg and Oren (2014) propose a graph model with a certain cost structure and initiate the study of several interesting computation problems: 1) cost ratio: the worst ratio between the actual cost of the agent and the optimal cost, over all the graph instances; 2) motivating subgraph: how to motivate the agent to reach the goal by deleting nodes and edges; 3) Intermediate rewards: how to incentivize agents to reach the goal by placing intermediate rewards. Kleinberg and Oren give partial answers to these questions, but the main problems are open. In this paper, we give answers to all three open problems. First, we show a tight upper bound of cost ratio for graphs, and confirm the conjecture by Kleinberg and Oren that Akerlof’s structure is indeed the worst case for cost ratio. Second, we prove that finding a motivating subgraph is NP-hard, showing that it is generally inefficient to motivate agents by deleting nodes and edges in the graph. Last but not least, we show that computing a strategy to place minimum amount of total reward is also NP-hard and we provide a 2n- approximation algorithm.

We consider the problem of designing revenue-optimal auctions for selling two items and bidders' valuations are independent among bidders but negatively correlated among items. Abstractly, this setting can be thought of as an instance with single-dimensional type space but multi-dimensional allocation space. Such setting has been extensively studied in the literature, but all under the assumption that the items are positively correlated. Under the positive correlation assumption, the optimal allocation rules are simple, avoiding difficulties brought by ensuring Bayesian incentive compatibility (BIC) under multi-dimensional feasibility constraints and by ensuring interim individual rationality (IIR) given the possibility that the lowest utility point may no longer be at the boundary of the type domain. However, the nice properties no longer hold when there is negative correlation among items.

We solve the Bayesian sequential equilibrium of a general class of single-item first-price or all-pay auctions of incomplete information. Our main contribution is a general methodology for solving the optimal commitment problem, in closed form, for asymmetric continuous-type distributions.

We consider revenue-optimal mechanism design for the case with one buyer and two items. The buyer's valuations towards the two items are independent and additive. In this setting, optimal mechanism is unknown for general valuation distributions. We obtain two categories of structural results that shed light on the optimal mechanisms. These results can be summarized into one conclusion: under certain conditions, the optimal mechanisms have simple menus.