We study a Bayesian persuasion problem with externalities. In this model, a principal sends signals to inform multiple agents about the state of the world. Simultaneously, due to the existence of externalities in the agents' utilities, the principal also acts as a correlation device to correlate the agents' actions. We consider the setting where the agents are categorized into a small number of types. Agents of the same type share identical utility functions and are treated equitably in the utility functions of both other agents and the principal. We study the problem of computing optimal signaling strategies for the principal, under three different types of signaling channels: public, private, and semi-private. Our results include revelation-principle-style characterizations of optimal signaling strategies, linear programming formulations, and analysis of in/tractability of the optimization problems. It is demonstrated that when the maximum number of deviating agents is bounded by a constant, our LP-based formulations compute optimal signaling strategies in polynomial time. Otherwise, the problems are NP-hard.

Principal-agent problems model scenarios where a principal aims at incentivizing an agent to take costly, unobservable actions through the provision of payments. Such interactions are ubiquitous in several real-world applications, ranging from blockchain to the delegation of machine learning tasks. In this paper, we initiate the study of hidden-action principal-agent problems under approximate best responses, in which the agent may select any action that is not too much suboptimal given the principal's payment scheme (a.k.a. contract). Our main result is a polynomial-time algorithm to compute an optimal contract under approximate best responses. This is perhaps surprising, as computing an optimal commitment under approximate best responses is known to be computationally intractable in Stackelberg games. We also investigate the learnability of contracts under approximate best responses, by providing a no-regret learning algorithm for a natural application scenario where the principal does not know anything about the environment.

We study Reinforcement Learning from Human Feedback (RLHF) in settings where multiple labelers may strategically misreport feedback to steer the learned policy toward their own preferences. We show that existing RLHF algorithms, including recent pluralistic methods, are not strategyproof, and that even a single strategic labeler can cause arbitrarily large misalignment with social welfare. Moreover, we prove that, in the worst case, any strategyproof RLHF algorithm must perform $k$-times worse than the optimal policy, where $k$ is the number of labelers. This suggests a fundamental trade-off between incentive alignment (ensuring labelers report truthfully) and policy alignment (maximizing social welfare). To address this, we propose the Pessimistic Median of MLEs algorithm, which, under appropriate policy coverage assumptions, is approximately strategyproof and converges to the optimal policy as the number of labelers and samples increases. Our results apply to both contextual bandits and Markov decision processes.

We study a stochastic principal-agent model. A principal and an agent interact in a stochastic environment, each privy to observations about the state not available to the other. The principal has the power of commitment, both to elicit information from the agent and to signal her own information. The players communicate with each other and then select actions independently. Both players are {\em far-sighted}, aiming to maximize their total payoffs over the entire time horizon. We consider both the computation and learning of the principal's optimal policy. The key challenge lies in enabling {\em history-dependent} policies, which are essential for achieving optimality in this model but difficult to cope with because of the exponential growth of possible histories as the size of the model increases; explicit representation of history-dependent policies is infeasible as a result. To address this challenge, we develop algorithmic techniques based on the concept of {\em inducible value set}. The techniques yield an efficient algorithm that computes an $\epsilon$-approximate optimal policy in time polynomial in $1/\epsilon$. We also present an efficient learning algorithm for an episodic reinforcement learning setting with unknown transition probabilities. The algorithm achieves sublinear regret $\widetilde{\mathcal{O}}(T^{2/3})$ for both players over $T$ episodes.

Stackelberg equilibria, as functions of the players' payoffs, can inversely reveal information about the players' incentives. In this paper, we study to what extent one can learn about the leader's incentives by actively querying the leader's optimal commitments against strategically designed followers. We show that, by using polynomially many queries and operations, one can learn a payoff function that is strategically equivalent to the leader's, in the sense that: 1) it preserves the leader's preference over almost all strategy profiles; and 2) it preserves the set of all possible (strong) Stackelberg equilibria the leader may engage in, considering all possible follower types. As an application, we show that the information acquired by our algorithm is sufficient for a follower to induce the best possible Stackelberg equilibrium by imitating a different follower type. To the best of our knowledge, we are the first to demonstrate that this is possible without knowing the leader's payoffs beforehand.

Canonical models of Markov decision processes (MDPs) usually consider geometric discounting based on a constant discount factor. While this standard modeling approach has led to many elegant results, some recent studies indicate the necessity of modeling time-varying discounting in certain applications. This paper studies a model of infinite-horizon MDPs with time-varying discount factors. We take a game-theoretic perspective – whereby each time step is treated as an independent decision maker with their own (fixed) discount factor – and we study the subgame perfect equilibrium (SPE) of the resulting game as well as the related algorithmic problems. We present a constructive proof of the existence of an SPE and demonstrate the EXPTIME-hardness of computing an SPE. We also turn to the approximate notion of epsilon-SPE and show that an epsilon-SPE exists under milder assumptions. An algorithm is presented to compute an epsilon-SPE, of which an upper bound of the time complexity, as a function of the convergence property of the time-varying discount factor, is provided.

Existing episodic reinforcement algorithms assume that the length of an episode is fixed across time and known a priori. In this paper, we consider a general framework of episodic reinforcement learning when the length of each episode is drawn from a distribution. We first establish that this problem is equivalent to online reinforcement learning with general discounting where the learner is trying to optimize the expected discounted sum of rewards over an infinite horizon, but where the discounting function is not necessarily geometric. We show that minimizing regret with this new general discounting is equivalent to minimizing regret with uncertain episode lengths. We then design a reinforcement learning algorithm that minimizes regret with general discounting but acts for the setting with uncertain episode lengths. We instantiate our general bound for different types of discounting, including geometric and polynomial discounting. We also show that we can obtain similar regret bounds even when the uncertainty over the episode lengths is unknown, by estimating the unknown distribution over time. Finally, we compare our learning algorithms with existing value-iteration based episodic RL algorithms on a grid-world environment.

This paper provides a systematic study of the robust Stackelberg equilibrium (RSE), which naturally generalizes the widely adopted solution concept of the strong Stackelberg equilibrium (SSE). The RSE accounts for any possible up-to-δ suboptimal follower responses in Stackelberg games and is adopted to improve the robustness of the leader's strategy. While a few variants of robust Stackelberg equilibrium have been considered in previous literature, the RSE solution concept we consider is importantly different --- in some sense, it relaxes previously studied robust Stackelberg strategies and is applicable to much broader sources of uncertainties.

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 envyfreeness, 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 envyfreeness cannot be guaranteed. In addition to the existential results, we also investigate the utility guarantees of (Pareto) envy-free assignments in our model.

We study reward design strategies for incentivizing a reinforcement learning agent to adopt a policy from a set of admissible policies. The goal of the reward designer is to modify the underlying reward function cost-efficiently while ensuring that any approximately optimal deterministic policy under the new reward function is admissible and performs well under the original reward function. This problem can be viewed as a dual to the problem of optimal reward poisoning attacks: instead of forcing an agent to adopt a specific policy, the reward designer incentivizes an agent to avoid taking actions that are inadmissible in certain states. Perhaps surprisingly, and in contrast to the problem of optimal reward poisoning attacks, we first show that the reward design problem for admissible policy teaching is computationally challenging, and it is NP-hard to find an approximately optimal reward modification. We then proceed by formulating a surrogate problem whose optimal solution approximates the optimal solution to the reward design problem in our setting, but is more amenable to optimization techniques and analysis. For this surrogate problem, we present characterization results that provide bounds on the value of the optimal solution. Finally, we design a local search algorithm to solve the surrogate problem and showcase its utility using simulation-based experiments.

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

In today's economy, it becomes important for Internet platforms to consider the sequential information design problem to align its long term interest with incentives of the gig service providers. This paper proposes a novel model of sequential information design, namely the Markov persuasion processes (MPPs), where a sender, with informational advantage, seeks to persuade a stream of myopic receivers to take actions that maximizes the sender's cumulative utilities in a finite horizon Markovian environment with varying prior and utility functions. Planning in MPPs thus faces the unique challenge in finding a signaling policy that is simultaneously persuasive to the myopic receivers and inducing the optimal long-term cumulative utilities of the sender. Nevertheless, in the population level where the model is known, it turns out that we can efficiently determine the optimal (resp. $ε$-optimal) policy with finite (resp. infinite) states and outcomes, through a modified formulation of the Bellman equation. Our main technical contribution is to study the MPP under the online reinforcement learning (RL) setting, where the goal is to learn the optimal signaling policy by interacting with with the underlying MPP, without the knowledge of the sender's utility functions, prior distributions, and the Markov transition kernels. We design a provably efficient no-regret learning algorithm, the Optimism-Pessimism Principle for Persuasion Process (OP4), which features a novel combination of both optimism and pessimism principles. Our algorithm enjoys sample efficiency by achieving a sublinear $\sqrt{T}$-regret upper bound. Furthermore, both our algorithm and theory can be applied to MPPs with large space of outcomes and states via function approximation, and we showcase such a success under the linear setting.

We study envy-free policy teaching. A number of agents independently explore a common Markov decision process (MDP), but each with their own reward function and discounting rate. A teacher wants to teach a target policy to this diverse group of agents, by means of modifying the agents' reward functions: providing additional bonuses to certain actions, or penalizing them. When personalized reward modification programs are used, an important question is how to design the programs so that the agents think they are treated fairly. We adopt the notion of envy-freeness (EF) from the literature on fair division to formalize this problem and investigate several fundamental questions about the existence of EF solutions in our setting, the computation of cost-minimizing solutions, as well as the price of fairness (PoF), which measures the increase of cost due to the consideration of fairness. We show that 1) an EF solution may not exist if penalties are not allowed in the modifications, but otherwise always exists. 2) Computing a cost-minimizing EF solution can be formulated as convex optimization and hence solved efficiently. 3) The PoF increases but at most quadratically with the geometric sum of the discount factor, and at most linearly with the size of the MDP and the number of agents involved; we present tight asymptotic bounds on the PoF. These results indicate that fairness can be incorporated in multi-agent teaching without significant computational or PoF burdens.

Real-world security scenarios sometimes involve multiple defenders: security agencies of two or more countries might patrol the same border areas, and domestic security agencies might also operate in the same locations when their areas of jurisdiction overlap. Motivated by these scenarios and the observation that uncoordinated movements of the defenders may lead to an inefficient defense, we introduce a model of multi-defender security games and explore the possibility of improving efficiency by coordinating the defenders — specifically, by pooling the defenders' resources and allocating them jointly. The model generalizes the standard model of Stackelberg security games, where a defender (now a group of defenders) allocates security resources to protect a set of targets, and an attacker picks the best target to attack. In particular, we are interested in the situation with heterogeneous defenders, who may value the same target differently. Our task is twofold. First, we need to develop a good understanding of the uncoordinated situation, as the baseline to be improved. To this end we formulate a new equilibrium concept, and prove that an equilibrium under this concept always exists and can be computed efficiently. Second, to coordinate the heterogeneous defenders we take a mechanism design perspective and aim to find a mechanism to generate joint resource allocation strategies. We seek a mechanism that improves the defenders' utilities upon the uncoordinated baseline, achieves Pareto efficiency, and incentivizes the defenders to report their true incentives and execute the recommended strategies. Our analysis establishes several impossibility results, which indicate the intrinsic difficulties of defense coordination. Specifically, we show that even the basic properties listed above are in conflict with each other: no mechanism can simultaneously satisfy them all, or even some proper subsets of them. In terms of positive results, we present mechanisms that satisfy all combinations of the properties that are not ruled out by our impossibility results, thereby providing a comprehensive profile of the mechanism design problem with respect to the properties considered.

We study Stackelberg security game (SSG) with multiple defenders, where heterogeneous defenders need to allocate security resources to protect a set of targets against a strategic attacker. In such games, coordination and cooperation between the defenders can increase their ability to protect their assets, but the heterogeneous preferences of the self-interested defenders often make such cooperation very difficult. In this paper, we approach the problem from the perspective of cooperative game theory and study coalition formation among the defenders. Our main contribution is a number of algorithmic results for the computation problems that arise in this model. We provide a poly-time algorithm for computing a solution in the core of the game and show that all of the elements in the core are Pareto efficient. We show that the problem of computing the entire core is NP-hard and then delve into a special setting where the size of a coalition is limited up to some threshold. We analyse the parameterized complexity of deciding if a coalition structure is in the core under this special setting, and provide a poly-time algorithm for computing successful deviation strategies for a given coalition.

The Nash social welfare (NSW) is a well-known social welfare measurement that balances individual utilities and the overall efficiency. In the context of fair allocation of indivisible goods, it has been shown by Caragiannis et al. (EC 2016 and TEAC 2019) that an allocation maximizing the NSW is envy-free up to one good (EF1). In this paper, we are interested in the fairness of the NSW in a budget-feasible allocation problem, in which each item has a cost that will be incurred to the agent it is allocated to, and each agent has a budget constraint on the total cost of items she receives. We show that a budget-feasible allocation that maximizes the NSW achieves a 1/4-approximation of EF1 and the approximation ratio is tight. The approximation ratio improves gracefully when the items have small costs compared with the agents' budgets; it converges to 1/2 when the budget-cost ratio approaches infinity.

Complexity of voting manipulation is a prominent topic in computational social choice. In this work, we consider a two-stage voting manipulation scenario. First, a malicious party (an attacker) attempts to manipulate the election outcome in favor of a preferred candidate by changing the vote counts in some of the voting districts. Afterwards, another party (a defender), which cares about the voters' wishes, demands a recount in a subset of the manipulated districts, restoring their vote counts to their original values. We investigate the resulting Stackelberg game for the case where votes are aggregated using two variants of the Plurality rule, and obtain an almost complete picture of the complexity landscape, both from the attacker's and from the defender's perspective.

Recent results have shown that algorithms for learning the optimal commitment in a Stackelberg game are susceptible to manipulation by the follower. These learning algorithms operate by querying the best responses of the follower, who consequently can deceive the algorithm by using fake best responses, typically by responding according to fake payoffs that are different from the actual ones. For this strategic behavior to be successful, the main challenge faced by the follower is to pinpoint the fake payoffs that would make the learning algorithm output a commitment that benefits them the most. While this problem has been considered before, the related literature has only focused on a simple setting where the follower can only choose from a finite set of payoff matrices, thus leaving the general version of the problem unanswered. In this paper, we fill this gap by showing that it is always possible for the follower to efficiently compute (near-)optimal fake payoffs, for various scenarios of learning interaction between the leader and the follower. Our results also establish an interesting connection between the follower’s deception and the leader’s maximin utility: through deception, the follower can induce almost any (fake) Stackelberg equilibrium if and only if the leader obtains at least their maximin utility in this equilibrium.

We study a recently introduced class of strategic games that is motivated by and generalizes Schelling's well-known residential segregation model. These games are played on undirected graphs, with the set of agents partitioned into multiple types; each agent either occupies a node of the graph and never moves away or aims to maximize the fraction of her neighbors who are of her own type. We consider a variant of this model that we call swap Schelling games, where the number of agents is equal to the number of nodes of the graph, and agents may swap positions with other agents to increase their utility. We study the existence, computational complexity and quality of equilibrium assignments in these games, both from a social welfare perspective and from a diversity perspective.

Recent results in the ML community have revealed that learning algorithms used to compute the optimal strategy for the leader to commit to in a Stackelberg game, are susceptible to manipulation by the follower. Such a learning algorithm operates by querying the best responses or the payoffs of the follower, who consequently can deceive the algorithm by responding as if their payoffs were much different than what they actually are. For this strategic behavior to be successful, the main challenge faced by the follower is to pinpoint the payoffs that would make the learning algorithm compute a commitment so that best responding to it maximizes the follower's utility, according to the true payoffs. While this problem has been considered before, the related literature only focused on the simplified scenario in which the payoff space is finite, thus leaving the general version of the problem unanswered. In this paper, we fill this gap by showing that it is always possible for the follower to efficiently compute (near-)optimal payoffs for various scenarios of learning interaction between the leader and the follower.

Strong Stackelberg equilibrium (SSE) is the standard solution concept of Stackelberg security games. As opposed to the weak Stackelberg equilibrium (WSE), the SSE assumes that the follower breaks ties in favor of the leader and this is widely acknowledged and justified by the assertion that the defender can often induce the attacker to choose a preferred action by making an infinitesimal adjustment to her strategy. Unfortunately, in security games with resource assignment constraints, the assertion might not be valid; it is possible that the defender cannot induce the desired outcome. As a result, many results claimed in the literature may be overly optimistic. To remedy, we first formally define the utility guarantee of a defender strategy and provide examples to show that the utility of SSE can be higher than its utility guarantee. Second, inspired by the analysis of leader’s payoff by Von Stengel and Zamir (2004), we provide the solution concept called the inducible Stackelberg equilibrium (ISE), which owns the highest utility guarantee and always exists. Third, we show the conditions when ISE coincides with SSE and the fact that in general case, SSE can be extremely worse with respect to utility guarantee. Moreover, introducing the ISE does not invalidate existing algorithmic results as the problem of computing an ISE polynomially reduces to that of computing an SSE. We also provide an algorithmic implementation for computing ISE, with which our experiments unveil the empirical advantage of the ISE over the SSE.

We consider strategic games that are inspired by Schelling's model of residential segregation. In our model, the agents are partitioned into k types and need to select locations on an undirected graph. Agents can be either stubborn, in which case they will always choose their preferred location, or strategic, in which case they aim to maximize the fraction of agents of their own type in their neighborhood. We investigate the existence of equilibria in these games, study the complexity of finding an equilibrium outcome or an outcome with high social welfare, and also provide upper and lower bounds on the price of anarchy and stability. Some of our results extend to the setting where the preferences of the agents over their neighbors are defined by a social network rather than a partition into types.

In Stackelberg security games when information about the attacker's payoffs is uncertain, algorithms have been proposed to learn the optimal defender commitment by interacting with the attacker and observing their best responses. In this paper, we show that, however, these algorithms can be easily manipulated if the attacker responds untruthfully. As a key finding, attacker manipulation normally leads to the defender learning a maximin strategy, which effectively renders the learning attempt meaningless as to compute a maximin strategy requires no additional information about the other player at all. We then apply a game-theoretic framework at a higher level to counteract such manipulation, in which the defender commits to a policy that specifies her strategy commitment according to the learned information. We provide a polynomial-time algorithm to compute the optimal such policy, and in addition, a heuristic approach that applies even when the attacker's payoff space is infinite or completely unknown. Empirical evaluation shows that our approaches can improve the defender's utility significantly as compared to the situation when attacker manipulation is ignored.

To reduce the air pollution and improve the energy efficiency, many countries and cities (e.g., Singapore) are on the way of introducing electric vehicles (EVs) to replace the vehicles serving in current traffic system. Effective placement of charging stations is essential for the rapid development of EVs, because it is necessary for providing convenience for EVs and ensuring the efficiency of the traffic network. However, existing works mostly concentrate on the mileage anxiety from EV users but ignore their strategic and competitive charging behaviors. To capture the competitive and strategic charging behaviors of the EV users, we consider that an EV user’s charging cost, which is dependent on other EV users’ choices, consists of the travel cost to access the charging station and the queuing cost in charging stations. First, we formulate the Charging Station Placement Problem (CSPP) as a bilevel optimization problem. Then, by exploiting the equilibrium of the EV charging game, we convert the bilevel optimization problem to a single-level one, following which we analyze the properties of CSPP and propose an algorithm Optimizing eleCtric vEhicle chArging statioN (OCEAN) to compute the optimal allocation of charging stations. Due to OCEAN’s scalability issue, we furthermore present a heuristic algorithm OCEAN with Continuous variables to deal with large-scale real-world problems. Finally, we demonstrate and discuss the results of the extensive experiments we did. It is shown that our approach outperform baseline methods significantly.

Most existing models of Stackelberg security games ignore the underlying topology of the space in which targets and defence resources are located. As a result, allocation of resources is restricted to a discrete collection of exogenously defined targets. However, in many practical security settings, defense resources can be located on a continuous plane. Better defense solutions could therefore be potentially achieved by placing resources in a space outside of actual targets (e.g., between targets). To address this limitation, we propose a model called Security Game on a Plane (SGP) in which targets are distributed on a 2-dimensional plane, and security resources, to be allocated on the same plane, protect targets within a certain effective distance. We investigate the algorithmic aspects of SGP. We find that computing a strong Stackelberg equilibrium of an SGP is NP-hard even for zero-sum games, and these are inapproximable in general. On the positive side, we find an exact solution technique for general SGPs based on an existing approach, and develop a PTAS (polynomial-time approximation scheme) for zero-sum SGP to more fundamentally overcome the computational obstacle. Our experiments demonstrate the value of considering SGP and effectiveness of our algorithms.

Preventing crimes or terrorist attacks in urban areas is challenging. Law enforcement officers need to respond quickly to catch the attacker on his escape route, which is subject to time-dependent traffic conditions on transportation networks. The attacker can strategically choose his escape path and driving speed to avoid being captured. Existing work on security resource allocation has not considered such scenarios with time-dependent strategies for both players. Therefore, in this paper, we study the problem of efficiently scheduling security resources for interdicting the escaping attacker. We propose: 1) a new defender-attacker security game model for escape interdiction on transportation networks; and 2) an efficient double oracle algorithm to compute the optimal defender strategy, which combines mixed-integer linear programming formulations for best response problems and effective approximation algorithms for improving the scalability of the algorithms. Experimental evaluation shows that our approach significantly outperforms baselines in solution quality and scales up to realistic-sized transportation networks with hundreds of intersections.

Stackelberg security games have been widely deployed in recent years to schedule security resources. An assumption in most existing security game models is that one security resource assigned to a target only protects that target. However, in many important real-world security scenarios, when a resource is assigned to a target, it exhibits protection externalities: that is, it also protects other “neighbouring” targets. We investigate such Security Games with Protection Externalities (SPEs). First, we demonstrate that computing a strong Stackelberg equilibrium for an SPE is NP-hard, in contrast with traditional Stackelberg security games which can be solved in polynomial time. On the positive side, we propose a novel column generation based approach—CLASPE—to solve SPEs. CLASPE features the following novelties: 1) a novel mixed-integer linear programming formulation for the slave problem; 2) an extended greedy approach with a constant-factor approximation ratio to speed up the slave problem; and 3) a linear-scale linear programming that efficiently calculates the upper bounds of target-defined subproblems for pruning. Our experimental evaluation demonstrates that CLASPE enable us to scale to realistic-sized SPE problem instances.