This paper studies game-theoretic decision-making for autonomous vehicles (AVs). A receding horizon multi-player game is formulated to model the AV decision-making problem. Two classes of games, including Nash game and Stackelber games, are developed respectively. For each of the two games, two solution settings, including pairwise games and multi-player games, are introduced, respectively, to solve the game in multi-agent scenarios. Comparative studies are conducted via statistical simulations to gain understandings of the performance of the two classes of games and of the two solution settings, respectively. The simulations are conducted in intersection-crossing scenarios, and the game performance is quantified by three metrics: safety, travel efficiency, and computational time.
This paper investigates inventory management in a multi channel distribution system consisting of one manufacturer and an arbitrary number of retailers that face stochastic demand. Existence of the pure Nash equilibrium is proved and parameter restriction which implies uniqueness of it is derived. Also the Stackelberg game where the manufacturer plays a roll as a leader is discussed. Under specified parameter restrictions which guarantee profitability, sufficient condition for uniqueness of Stackelberg equilibrium is obtained. In addition comparison with simultaneous move game is made. The res…
There has been significant recent interest in game-theoretic approaches to security, with much of the recent research focused on utilizing the leader-follower Stackelberg game model. Among the major applications are the ARMOR program deployed at LAX Airport and the IRIS program in use by the US Federal Air Marshals (FAMS). The foundational assumption for using Stackelberg games is that security forces (leaders), acting first, commit to a randomized strategy; while their adversaries (followers) choose their best response after surveillance of this randomized strategy. Yet, in many situations, a…
We study the computational complexity of finding Stackelberg Equilibria in general-sum games, where the set of pure strategies of the leader and the followers are exponentially large in a natrual representation of the problem. In \emph{zero-sum} games, the notion of a Stackelberg equilibrium coincides with the notion of a \emph{Nash Equilibrium}~\cite{korzhyk2011stackelberg}. Finding these equilibrium concepts in zero-sum games can be efficiently done when the players have polynomially many pure strategies or when (in additional to some structural properties) a best-response oracle is availabl…
In this paper, we discuss a class of two-stage hierarchical games with multiple leaders and followers, which is called Nash-Stackelberg-Nash (N-S-N) games. Particularly, we consider N-S-N games under decision-dependent uncertainties (DDUs). DDUs refer to the uncertainties that are affected by the strategies of decision-makers and have been rarely addressed in game equilibrium analysis. In this paper, we first formulate the N-S-N games with DDUs of complete ignorance, where the interactions between the players and DDUs are characterized by uncertainty sets that depend parametrically on the play…
Interactions among selfish users sharing a common transmission channel can be modeled as a non-cooperative game using the game theory framework. When selfish users choose their transmission probabilities independently without any coordination mechanism, Nash equilibria usually result in a network collapse. We propose a methodology that transforms the non-cooperative game into a Stackelberg game. Stackelberg equilibria of the Stackelberg game can overcome the deficiency of the Nash equilibria of the original game. A particular type of Stackelberg intervention is constructed to show that any pos…
In this paper, we employ a hypergame framework to analyze the single-leader-multiple-followers (SLMF) Stackelberg security game with two typical misinformed situations: misperception and deception. We provide a stability criterion with the help of hyper Nash equilibrium (HNE) to investigate both strategic stability and cognitive stability of equilibria in SLMF games with misinformation. In fact, we find mild stable conditions such that the equilibria with misperception and deception can become HNE. Moreover, we discuss the robustness of the equilibria to reveal whether players have the ability…
There has been significant recent interest in leader-follower security games, where the leader dominates the decision process with the Stackelberg equilibrium (SE) strategy. However, such a leader-follower scheme may become invalid in practice due to subjective or objective factors, and then the Nash equilibrium (NE) strategy may be an alternative option. In this case, the leader may face a dilemma of choosing an SE strategy or an NE strategy. In this paper, we focus on a unified three-player leader-follower security game and study the coincidence between SE and NE. We first explore a necessar…
We study multi-player general-sum Markov games with one of the players designated as the leader and the other players regarded as followers. In particular, we focus on the class of games where the followers are myopic, i.e., they aim to maximize their instantaneous rewards. For such a game, our goal is to find a Stackelberg-Nash equilibrium (SNE), which is a policy pair $(π^*, ν^*)$ such that (i) $π^*$ is the optimal policy for the leader when the followers always play their best response, and (ii) $ν^*$ is the best response policy of the followers, which is a Nash equilibrium of the followers…
Equilibrium refinements are important in extensive-form (i.e., tree-form) games, where they amend weaknesses of the Nash equilibrium concept by requiring sequential rationality and other beneficial properties. One of the most attractive refinement concepts is quasi-perfect equilibrium. While quasi-perfection has been studied in extensive-form games, it is poorly understood in Stackelberg settings---that is, settings where a leader can commit to a strategy---which are important for modeling, for example, security games. In this paper, we introduce the axiomatic definition of quasi-perfect Stack…
A growing body of work in game theory extends the traditional Stackelberg game to settings with one leader and multiple followers who play a Nash equilibrium. Standard approaches for computing equilibria in these games reformulate the followers' best response as constraints in the leader's optimization problem. These reformulation approaches can sometimes be effective, but often get trapped in low-quality solutions when followers' objectives are non-linear or non-quadratic. Moreover, these approaches assume a unique equilibrium or a specific equilibrium concept, e.g., optimistic or pessimistic…
This paper studies a multi-period demand response management problem in the smart grid where multiple utility companies compete among themselves. The user-utility interactions are modeled by a noncooperative game of a Stackelberg type where the interactions among the utility companies are captured through a Nash equilibrium. It is shown that this game has a unique Stackelberg equilibrium at which the utility companies set prices to maximize their revenues (within a Nash game) while the users respond accordingly to maximize their utilities subject to their budget constraints. Closed-form expres…
Zero-sum stochastic games have found important applications in a variety of fields, from machine learning to economics. Work on this model has primarily focused on the computation of Nash equilibrium due to its effectiveness in solving adversarial board and video games. Unfortunately, a Nash equilibrium is not guaranteed to exist in zero-sum stochastic games when the payoffs at each state are not convex-concave in the players' actions. A Stackelberg equilibrium, however, is guaranteed to exist. Consequently, in this paper, we study zero-sum stochastic Stackelberg games. Going beyond known exis…
It is sometimes the case that one solution concept in game theory is equivalent to applying another solution concept to a modified version of the game. In such cases, does it make sense to study the former separately (as it applies to the original representation of the game), or should we entirely subordinate it to the latter? The answer probably depends on the particular circumstances, and indeed the literature takes different approaches in different cases. In this article, I consider the specific example of Stackelberg mixed strategies. I argue that, even though a Stackelberg mixed strategy …
We introduce and study incentive equilibria for multi-player meanpayoff games. Incentive equilibria generalise well-studied solution concepts such as Nash equilibria and leader equilibria (also known as Stackelberg equilibria). Recall that a strategy profile is a Nash equilibrium if no player can improve his payoff by changing his strategy unilaterally. In the setting of incentive and leader equilibria, there is a distinguished player called the leader who can assign strategies to all other players, referred to as her followers. A strategy profile is a leader strategy profile if no player, exc…
We study a class of games in which a finite number of agents each controls a quantity of flow to be routed through a network, and are able to split their own flow between multiple paths through the network. Recent work on this model has contrasted the social cost of Nash equilibria with the best possible social cost. Here we show that additional costs are incurred in situations where a selfish ``leader'' agent allocates his flow, and then commits to that choice so that other agents are compelled to minimise their own cost based on the first agent's choice. We find that even in simple networks,…
Real world applications such as economics and policy making often involve solving multi-agent games with two unique features: (1) The agents are inherently asymmetric and partitioned into leaders and followers; (2) The agents have different reward functions, thus the game is general-sum. The majority of existing results in this field focuses on either symmetric solution concepts (e.g. Nash equilibrium) or zero-sum games. It remains open how to learn the Stackelberg equilibrium -- an asymmetric analog of the Nash equilibrium -- in general-sum games efficiently from noisy samples. This paper ini…
We consider multiple access games in which primary users are interested in maximizing their confidential data rate at the minimum possible transmission power and secondary users employ eavesdropping as a leverage to maximize their data rate to a common destination at minimum transmission energy. For the non-cooperative static game, Nash equilibria in pure and mixed strategies are derived and shown to be Pareto inefficient in general, when channel gains are common knowledge. For the two-player Stackelberg game where the primary user is the leader, it is shown that the secondary user is forced t…
Robust Reinforcement Learning (RL) focuses on improving performances under model errors or adversarial attacks, which facilitates the real-life deployment of RL agents. Robust Adversarial Reinforcement Learning (RARL) is one of the most popular frameworks for robust RL. However, most of the existing literature models RARL as a zero-sum simultaneous game with Nash equilibrium as the solution concept, which could overlook the sequential nature of RL deployments, produce overly conservative agents, and induce training instability. In this paper, we introduce a novel hierarchical formulation of ro…
A noncooperative game theoretical approach for analysing opportunistic spectrum access (OSA) in cognitive radio (CR) environments is proposed. New concepts from game theory are applied to spectrum access analysis in order to extract rules of behaviour for an emerging environment. In order to assess OSA scenarios of CRs, two oligopoly game models are reformulated in terms of resource access: Cournot and Stackelberg games. Five CR scenes are analysed: simultaneous access of unlicensed users (commons regime) with symmetric and asymmetric costs, with and without bandwidth constraints and sequentia…
Min-max optimization problems (i.e., min-max games) have attracted a great deal of attention recently as their applicability to a wide range of machine learning problems has become evident. In this paper, we study min-max games with dependent strategy sets, where the strategy of the first player constrains the behavior of the second. Such games are best understood as sequential, i.e., Stackelberg, games, for which the relevant solution concept is Stackelberg equilibrium, a generalization of Nash. One of the most popular algorithms for solving min-max games is gradient descent ascent (GDA). We …
In this paper, we study the transmission strategy adaptation problem in an RF-powered cognitive radio network, in which hybrid secondary users are able to switch between the harvest-then-transmit mode and the ambient backscatter mode for their communication with the secondary gateway. In the network, a monetary incentive is introduced for managing the interference caused by the secondary transmission with imperfect channel sensing. The sensing-pricing-transmitting process of the secondary gateway and the transmitters is modeled as a single-leader-multi-follower Stackelberg game. Furthermore, t…
We explore the computational complexity of computing pure Nash equilibria for a new class of strategic games called integer programming games with difference of piecewise linear convex payoffs. Integer programming games are games where players' action sets are integer points inside of polytopes. Using recent results from the study of short rational generating functions for encoding sets of integer points pioneered by Alexander Barvinok, we present efficient algorithms for enumerating all pure Nash equilibria, and other computations of interest, such as the pure price of anarchy, and pure threa…
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
Adversarial deep learning is to train robust DNNs against adversarial attacks, which is one of the major research focuses of deep learning. Game theory has been used to answer some of the basic questions about adversarial deep learning such as the existence of a classifier with optimal robustness and the existence of optimal adversarial samples for a given class of classifiers. In most previous work, adversarial deep learning was formulated as a simultaneous game and the strategy spaces are assumed to be certain probability distributions in order for the Nash equilibrium to exist. But, this as…
In this study, we analyze an advertising competition in a duopoly. We consider two different notions of equilibrium. We model the companies in the duopoly as major players, and the consumers as minor players. In our first game model we identify Nash Equilibria (NE) between all the players. Next we frame the model to lead to the search for Multi-Leader-Follower Nash Equilibria (MLF-NE). This approach is reminiscent of Stackelberg games in the sense that the major players design their advertisement policies assuming that the minor players are rational and settle in a Nash Equilibrium among thems…
This paper investigates effects of realistic, non-ideal, decisions of energy users as to whether to participate in an energy trading system proposed for demand-side management of a residential community. The energy trading system adopts a non-cooperative Stackelberg game between a community energy storage (CES) device and users with rooftop photovoltaic panels where the CES operator is the leader and the users are the followers. Participating users determine their optimal energy trading starting time to minimize their personal daily energy costs while subjectively viewing their opponents' acti…
Algorithms for computing game-theoretic solutions have recently been applied to a number of security domains. However, many of the techniques developed for compact representations of security games do not extend to {\em Bayesian} security games, which allow us to model uncertainty about the attacker's type. In this paper, we introduce a general framework of {\em catcher-evader} games that can capture Bayesian security games as well as other game families of interest. We show that computing Stackelberg strategies is NP-hard, but give an algorithm for computing a Nash equilibrium that performs w…
In order to assess TV whitespace access scenarios, three oligopoly game models are considered and reformulated in terms of radio access: Cournot, Stackelberg, and Bertrand. Besides revisiting the relevance of Nash and Pareto equilibria, a new equilibrium concept, the joint Nash-Pareto equilibrium, is considered. An evolutionary game equilibria detection method is used. The analysis of the simulation results brings relevant insights on the issue of autonomy vs. regulation in emerging cognitive radio environments.
In this paper, an incentive proactive cache mechanism in cache-enabled small cell networks (SCNs) is proposed, in order to motivate the content providers (CPs) to participate in the caching procedure. A network composed of a single mobile network operator (MNO) and multiple CPs is considered. The MNO aims to define the price it charges the CPs to maximize its revenue while the CPs compete to determine the number of files they cache at the MNO's small base stations (SBSs) to improve the quality of service (QoS) of their users. This problem is formulated as a Stackelberg game where a single MNO …
We model the coexistence of DSRC and WiFi networks as a strategic form game with the networks as the players. Nodes in a DSRC network must support messaging of status updates that are time sensitive. Such nodes would like to achieve a small age of information of status updates. In contrast, nodes in a WiFi network would like to achieve large throughputs. Each network chooses a medium access probability to be used by all its nodes. We investigate Nash and Stackelberg equilibrium strategies.
As the communication network is in transition towards a commercial one controlled by service providers (SP), the present paper considers a pricing game in a communication market covered by several wireless access points sharing the same spectrum and analyzes two business models: monopoly (APs controlled by one SP) and oligopoly (APs controlled by different SPs). We use a Stackelberg game to model the problem: SPs are the leader(s) and end users are the followers. We prove, under certain conditions, the existence and uniqueness of Nash equilibrium for both models and derive their expressions. I…
In this paper, we study an interference avoidance scenario in the presence of a smart interferer which can rapidly observe the transmit power of a backscatter wireless sensor network (WSN) and effectively interrupt backscatter signals. We consider a power control with a sub-channel allocation to avoid interference attacks and a time-switching ratio for backscattering and RF energy harvesting in backscatter WSNs. We formulate the problem based on a Stackelberg game theory and compute the optimal transmit power, time-switching ratio, and sub-channel allocation parameter to maximize a utility fun…
In the framework of finite games in extensive form with perfect information and strict preferences, this paper introduces a new equilibrium concept: the Perfect Prediction Equilibrium (PPE). In the Nash paradigm, rational players consider that the opponent's strategy is fixed while maximizing their payoff. The PPE, on the other hand, models the behavior of agents with an alternate form of rationality that involves a Stackelberg competition with the past. Agents with this form of rationality integrate in their reasoning that they have such accurate logical and predictive skills, that the world …
It is known that there are uncoupled learning heuristics leading to Nash equilibrium in all finite games. Why should players use such learning heuristics and where could they come from? We show that there is no uncoupled learning heuristic leading to Nash equilibrium in all finite games that a player has an incentive to adopt, that would be evolutionary stable or that could "learn itself". Rather, a player has an incentive to strategically teach such a learning opponent in order secure at least the Stackelberg leader payoff. The impossibility result remains intact when restricted to the classe…
In this paper, a Stackelberg game is built to model the hierarchic power allocation of primary user (PU) network and secondary user (SU) network in OFDM-based cognitive radio (CR) networks. We formulate the PU and the SUs as the leader and the followers, respectively. We consider two constraints: the total power constraint and the interference-to-signal ratio (ISR) constraint, in which the ratio between the accumulated interference and the received signal power at each PU should not exceed certain threshold. Firstly, we focus on the single-PU and multi-SU scenario. Based on the analysis of the…
Autonomous vehicles (AVs) are inevitably entering our lives with potential benefits for improved traffic safety, mobility, and accessibility. However, AVs' benefits also introduce a serious potential challenge, in the form of complex interactions with human-driven vehicles (HVs). The emergence of AVs introduces uncertainty in the behavior of human actors and in the impact of the AV manufacturer on autonomous driving design. This paper thus aims to investigate how AVs affect road safety and to design socially optimal liability rules for AVs and human drivers. A unified game is developed, includ…
Equilibrium analysis of three-player General Lotto game with leader-follower framework
Shanghai Research Institute for Intelligent Autonomous System, Tongji University（同济大学上海智能无人系统研究院）；Department of Control Science and Engineering, College of Electronics and Information Engineering, Tongji University（同济大学电子与信息工程学院控制科学与工程系）
The Robinson-Goforth topology of swaps in adjoining payoffs elegantly arranges 2x2 ordinal games in accordance with important properties including symmetry, number of dominant strategies and Nash Equilibria, and alignment of interests. Adding payoff families based on Nash Equilibria illustrates an additional aspect of this order and aids visualization of the topology. Making ties through half-swaps not only creates simpler games within the topology, but, in reverse, breaking ties shows the evolution of preferences, yielding a natural ordering for the topology of 2x2 games with ties. An ordinal…
