This paper considers a game-theoretic framework for distributed machine learning problems over networks where the information acquisition at a node is modeled as a rational choice of a player. In the proposed game, players decide both the learning parameters and the network structure. The Nash equilibrium characterizes the tradeoff between the local performance and the global agreement of the learned classifiers. We first introduce a commutative approach which features a joint learning process that integrates the iterative learning at each node and the network formation. We show that our game is equivalent to a generalized potential game in the setting of undirected networks. We study the convergence of the proposed commutative algorithm, analyze the network structures determined by our game, and show the improvement of the social welfare in comparison with standard distributed learning over fixed networks. To adapt our framework to streaming data, we derive a distributed Kalman filter. A concurrent algorithm based on the online mirror descent algorithm is also introduced for solving for Nash equilibria in a holistic manner. In the case study, we use telemonitoring of Parkinson's disease to corroborate the results.
We study new classes of games, called zero-sum equivalent games and zero-sum equivalent potential games, and prove decomposition theorems involving these classes of games. We say that two games are "strategically equivalent" if, for every player, the payoff differences between two strategies (holding other players' strategies fixed) are identical. A zero-sum equivalent game is a game that is strategically equivalent to a zero-sum game; a zero-sum equivalent potential game is a zero-sum equivalent game that is strategically equivalent to a common interest game. We also call a game "normalized" …
This is a short introduction to the subject of strategic games. We focus on the concepts of best response, Nash equilibrium, strict and weak dominance, and mixed strategies, and study the relation between these concepts in the context of the iterated elimination of strategies. Also, we discuss some variants of the original definition of a strategic game. Finally, we introduce the basics of mechanism design and use pre-Bayesian games to explain it.
We introduce the concept of budget games. Players choose a set of tasks and each task has a certain demand on every resource in the game. Each resource has a budget. If the budget is not enough to satisfy the sum of all demands, it has to be shared between the tasks. We study strategic budget games, where the budget is shared proportionally. We also consider a variant in which the order of the strategic decisions influences the distribution of the budgets. The complexity of the optimal solution as well as existence, complexity and quality of equilibria are analyzed. Finally, we show that the t…
In [Van Benthem 2007] the concept of a public announcement is used to study the effect of the iterated elimination of strictly dominated strategies. We offer a simple generalisation of this approach to cover arbitrary strategic games and many optimality notions. We distinguish between announcements of optimality and announcements of rationality.
We provide a compositional coalgebraic semantics for strategic games. In our framework, like in the semantics of functional programming languages, coalgebras represent the observable behaviour of systems derived from the behaviour of the parts over an unobservable state space. We use coalgebras to describe and program stage games, finitely and potentially infinitely repeated hierarchical or parallel games with imperfect and incomplete information based on deterministic, non-deterministic or probabilistic decisions of learning agents in possibly endogenous networks. Our framework is composition…
This paper studies strategic decentralization in binary choice composite network congestion games. A player decentralizes if she lets some autonomous agents to decide respectively how to send different parts of her stock from the origin to the destination. This paper shows that, with convex, strictly increasing and differentiable arc cost functions, an atomic splittable player always has an optimal unilateral decentralization strategy. Besides, unilateral decentralization gives her the same advantage as being the leader in a Stackelberg congestion game. Finally, unilateral decentralization of …
A Machine Learning Approach to Detect Strategic Behavior from Large-Population Observational Data Applied to Game Mode Prediction on a Team-Based Video Game
This paper considers a class of reinforcement-learning that belongs to the family of Learning Automata and provides a stochastic-stability analysis in strategic-form games. For this class of dynamics, convergence to pure Nash equilibria has been demonstrated only for the fine class of potential games. Prior work primarily provides convergence properties of the dynamics through stochastic approximations, where the asymptotic behavior can be associated with the limit points of an ordinary-differential equation (ODE). However, analyzing global convergence through the ODE-approximation requires th…
Private blockchain is driving the creation of business networks, resulting in the creation of new value or new business models to the enterprises participating in the network. Such business networks form when enterprises come together to derive value through a network which is greater than the value that can be derived solely by any single company. This results in a setting that combines both competitive and cooperative behavior, and which we call strategic coopetition. Traditionally, cooperative and competitive behavior have been analyzed separately in game theory. In this article, we provide…
In the context of strategic games, we provide an axiomatic proof of the statement Common knowledge of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies. Rationality here means playing only strategies one believes to be best responses. This involves looking at two formal languages. One is first-order, and is used to formalise optimality conditions, like avoiding strictly dominated strategies, or playing a best response. The other is a modal fixpoint language with expressions for optimality, rationality and bel…
In recent work, semantic games of independence-friendly logic were studied in strategic form in terms of (mixed strategy) Nash equilibria. The class of strategic games of independence-friendly logic is contained in the class of win-loss, zero-sum two-player games. In this note we draw on the theory of linear programming to develop tools to analyze the value of such games. We give two applications of these tools to independence-friendly logic under the so-called equilibrium semantics.
Candogan et al. (2011) provide an orthogonal direct-sum decomposition of finite games into potential, harmonic and nonstrategic components. In this paper we study the issue of decomposing games that are strategically equivalent from a game-theoretical point of view, for instance games obtained via transformations such as duplications of strategies or positive affine mappings of of payoffs. We show the need to define classes of decompositions to achieve commutativity of game transformations and decompositions.
Several notions of game enjoy a Nash-like notion of equilibrium without guarantee of existence. There are different ways of weakening a definition of Nash-like equilibrium in order to guarantee the existence of a weakened equilibrium. Nash's approach to the problem for strategic games is probabilistic, \textit{i.e.} continuous, and static. CP and BR approaches for CP and BR games are discrete and dynamic. This paper proposes an approach that lies between those two different approaches: a discrete and static approach. multi strategic games are introduced as a formalism that is able to express b…
This paper presents the research on the interdisciplinary research infrastructure for understanding human reasoning in game-theoretic terms. Strategic reasoning is considered to impact human decision making in social, economical and competitive interactions. The provided introduction explains and connects concepts from AI, game theory and psychology. First result is a concept of interdisciplinary game description language as a part of the focused interdisciplinary research infrastructure. The need of this domain-specific language is motivated and is aimed to accelerate the current developments…
Using Sperner's lemma for modified partition of a simplex we will constructively prove the existence of a Nash equilibrium in a finite strategic game with sequentially locally non-constant payoff functions. We follow the Bishop style constructive mathematics.
We introduce a new measure of the discrepancy in strategic games between the social welfare in a Nash equilibrium and in a social optimum, that we call selfishness level. It is the smallest fraction of the social welfare that needs to be offered to each player to achieve that a social optimum is realized in a pure Nash equilibrium. The selfishness level is unrelated to the price of stability and the price of anarchy and is invariant under positive linear transformations of the payoff functions. Also, it naturally applies to other solution concepts and other forms of games. We study the selfish…
In this paper we survey various notions of anonymity and symmetry for finite strategic-form games present in relevant literature, and discuss notions of fairness; show that game bijections and game isomorphisms form groupoids; introduce matchings as a convenient characterisation of strategy triviality; and outline how to construct and partially order parameterised (symmetric) games with examples that range all combinations of surveyed symmetry notions, which when combined with other results in this paper gives the precise relationship between the various symmetry notions.
We provide an epistemic analysis of arbitrary strategic games based on possibility correspondences. We first establish a generic result that links true common beliefs (and, respectively, common knowledge) of players' rationality defined by means of `monotonic' properties, with the iterated elimination of strategies that do not satisfy these properties. It allows us to deduce the customary results concerned with true common beliefs of rationality and iterated elimination of strictly dominated strategies as simple corollaries. This approach relies on Tarski's Fixpoint Theorem. We also provide an…
We consider the task of computing an approximation of a trembling hand perfect equilibrium for an n-player game in strategic form, n >= 3. We show that this task is complete for the complexity class FIXP_a. In particular, the task is polynomial time equivalent to the task of computing an approximation of a Nash equilibrium in strategic form games with three (or more) players.
In this paper, we first devise two algorithms to determine whether or not a bimatrix game has a strategically equivalent zero-sum game. If so, we propose an algorithm that computes the strategically equivalent zero-sum game. If a given bimatrix game is not strategically equivalent to a zero-sum game, we then propose an approach to compute a zero-sum game whose saddle-point equilibrium can be mapped to a well-supported approximate Nash equilibrium of the original game. We conduct extensive numerical simulation to establish the efficacy of the two algorithms.
The aim of the strategic analysis is to (simply) carry out the game between the implementing body and possible links to the existing market situation. We are therefore playing a strategic game between us and the outside world. This situation is most often associated with existing and potential threats, such as competition, fashion trends, cost situation, marketing effectiveness, market demand, etc. However, this does not exhaust all aspects resulting from the situation, as conditions conducive to the implementation may also be taken into account, such as the possibility of cooperation, favoura…
The paper studies properties of functional dependencies between strategies of players in Nash equilibria of multi-player strategic games. The main focus is on the properties of functional dependencies in the context of a fixed dependency graph for pay-off functions. A logical system describing properties of functional dependence for any given graph is proposed and is proven to be complete.
It is well-known that in finite strategic games true common belief (or common knowledge) of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies. We establish a general theorem that deals with monotonic rationality notions and arbitrary strategic games and allows to strengthen the above result to arbitrary games, other rationality notions, and transfinite iterations of the elimination process. We also clarify what conclusions one can draw for the customary dominance notions that are not monotonic. The main tool …
We give a strict mathematical description for a refinement of the Marinatto-Weber quantum game scheme. The model allows the players to choose projector operators that determine the state on which they perform their local operators. The game induced by the scheme generalizes finite strategic form game. In particular, it covers normal representations of extensive games, i.e., strategic games generated by extensive ones. We illustrate our idea with an example of extensive game and prove that rational choices in the classical game and its quantum counterpart may lead to significantly different out…
We introduce a game-theoretic approach to the study of recommendation systems with strategic content providers. Such systems should be fair and stable. Showing that traditional approaches fail to satisfy these requirements, we propose the Shapley mediator. We show that the Shapley mediator fulfills the fairness and stability requirements, runs in linear time, and is the only economically efficient mechanism satisfying these properties.
This paper presents the "Game Theory Explorer" software tool to create and analyze games as models of strategic interaction. A game in extensive or strategic form is created and nicely displayed with a graphical user interface in a web browser. State-of-the-art algorithms then compute all Nash equilibria of the game after a mouseclick. In tutorial fashion, we present how the program is used, and the ideas behind its main algorithms. We report on experiences with the architecture of the software and its development as an open-source project.
CWI（荷兰国家数学与计算机科学研究所）；University of Amsterdam（阿姆斯特丹大学）
We provide here an epistemic analysis of arbitrary strategic games based on the possibility correspondences. Such an analysis calls for the use of transfinite iterations of the corresponding operators. Our approach is based on Tarski's Fixpoint Theorem and applies both to the notions of rationalizability and the iterated elimination of strictly dominated strategies.
We establish a network formation game for the Internet's Autonomous System (AS) interconnection topology. The game includes different types of players, accounting for the heterogeneity of ASs in the Internet. In this network formation game, the utility of a player depends on the network structure, e.g., the distances between nodes and the cost of links. We also consider the case where utility (or monetary) transfers are allowed between the players. We incorporate reliability considerations in the player's utility function, and analyze static properties of the game as well as its dynamic evolut…
The COVID-19 lockdowns have created a significant socioeconomic impact on our society. In this paper, we propose a population vaccination game framework, called EPROACH, to design policies for reopenings that guarantee post-opening public health safety. In our framework, a population of players decides whether to vaccinate or not based on the public and private information they receive. The reopening is captured by the switching of the game state. The insights obtained from our framework include the appropriate vaccination coverage threshold for safe-reopening and information-based methods to …
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…
We formalize the current practice of strategic mining in multi-cryptocurrency markets as a game, and prove that any better-response learning in such games converges to equilibrium. We then offer a reward design scheme that moves the system configuration from any initial equilibrium to a desired one for any better-response learning of the miners. Our work introduces the first multi-coin strategic attack for adaptive and learning miners, as well as the study of reward design in a multi-agent system of learning agents.
We present a new class of strategic games, mixed capability games, as a foundation for studying how different player capabilities impact the dynamics and outcomes of strategic games. We analyze the impact of different player capabilities via a capability transfer function that characterizes the payoff of each player at equilibrium given capabilities for all players in the game. In this paper, we model a player's capability as the size of the strategy space available to that player. We analyze a mixed capability variant of the Gold and Mines Game recently proposed by Yang et al. and derive its …
Strategic interactions between competitive entities are generally considered from the perspective of complete revelation of benefits achieved from those interactions, in the form of public payoff functions and/or beliefs, in the announced games. However, there exist strategic interplays between competitors where the players have a choice to strategise under the availability of private payoffs, in similar competitive settings. In this contribution, we propose a formal framework for a competitive ecosystem where each player is permitted to defect from publicly optimal strategies under certain pr…
In this paper we describe an approach to resolve strategic games in which players can assume different types along the game. Our goal is to infer which type the opponent is adopting at each moment so that we can increase the player's odds. To achieve that we use Markov games combined with hidden Markov model. We discuss a hypothetical example of a tennis game whose solution can be applied to any game with similar characteristics.
Among the strategic choices made by today's economic actors are choices about algorithms and computational resources. Different access to computational resources may result in a kind of economic asymmetry analogous to information asymmetry. In order to represent strategic computational choices within a game theoretic framework, we propose a new game specification, Strategic Bayesian Networks (SBN). In an SBN, random variables are represented as nodes in a graph, with edges indicating probabilistic dependence. For some nodes, players can choose conditional probability distributions as a strateg…
We introduce a new solution concept, called periodicity, for selecting optimal strategies in strategic form games. This periodicity solution concept yields new insight into non-trivial games. In mixed strategy strategic form games, periodic solutions yield values for the utility function of each player that are equal to the Nash equilibrium ones. In contrast to the Nash strategies, here the payoffs of each player are robust against what the opponent plays. Sometimes, periodicity strategies yield higher utilities, and sometimes the Nash strategies do, but often the utilities of these two strate…
The paper studies properties of functional dependencies between strategies of players in Nash equilibria of multi-player strategic games. The main focus is on the properties of functional dependencies in the context of a fixed dependency graph for pay-off functions. A logical system describing properties of functional dependence for any given graph is proposed and is proven to be complete.
We introduce a new class of games where each player's aim is to randomise her strategic choices in order to affect the other players' expectations aside from her own. The way each player intends to exert this influence is expressed through a Boolean combination of polynomial equalities and inequalities with rational coefficients. We offer a logical representation of these games as well as a computational study of the existence of equilibria.
We consider a partially asymmetric multi-players zero-sum game with two strategic variables. All but one players have the same payoff functions, and one player (Player $n$) does not. Two strategic variables are $t_i$'s and $s_i$'s for each player $i$. Mainly we will show the following results. 1) The equilibrium when all players choose $t_i$'s is equivalent to the equilibrium when all but one players choose $t_i$'s and Player $n$ chooses $s_n$ as their strategic variables. 2) The equilibrium when all players choose $s_i$'s is equivalent to the equilibrium when all but one players choose $s_i$'…
CWI Amsterdam（阿姆斯特丹数学与计算机科学研究院）；University of Amsterdam（阿姆斯特丹大学）；University of Padova（帕多瓦大学）
The notion of optimality naturally arises in many areas of applied mathematics and computer science concerned with decision making. Here we consider this notion in the context of three formalisms used for different purposes in reasoning about multi-agent systems: strategic games, CP-nets, and soft constraints. To relate the notions of optimality in these formalisms we introduce a natural qualitative modification of the notion of a strategic game. We show then that the optimal outcomes of a CP-net are exactly the Nash equilibria of such games. This allows us to use the techniques of game theory…
The article considers strategies of coalitions that are based on intelligence information about moves of some of the other agents. The main technical result is a sound and complete logical system that describes the interplay between coalition power modality with intelligence and distributed knowledge modality in games with imperfect information.
An agent, or a coalition of agents, faces an ethical dilemma between several statements if she is forced to make a conscious choice between which of these statements will be true. This paper proposes to capture ethical dilemmas as a modality in strategic game settings with and without limit on sacrifice and for perfect and imperfect information games. The authors show that the dilemma modality cannot be defined through the earlier proposed blameworthiness modality. The main technical result is a sound and complete axiomatization of the properties of this modality with sacrifice in games with p…
Given the importance of line-of-sight in mmWave communications, a strategic adversary can harm a transmission by obstructing the receiver, which in turn can react by trying to move around this hurdle. To expand on this point, we study one such scenario from the perspective of game theory, considering a mobile mmWave receiver and an adversary interacting strategically as players in a zero-sum game, where they want to maximize, or respectively minimize, the spectral efficiency of the communication. To do so, the adversary attempts at screening the receiver's line of sight as an obstacle, while t…
Decision making in modern large-scale and complex systems such as communication networks, smart electricity grids, and cyber-physical systems motivate novel game-theoretic approaches. This paper investigates big strategic (non-cooperative) games where a finite number of individual players each have a large number of continuous decision variables and input data points. Such high-dimensional decision spaces and big data sets lead to computational challenges, relating to efforts in non-linear optimization scaling up to large systems of variables. In addition to these computational challenges, rea…
Many real-world domains contain multiple agents behaving strategically with probabilistic transitions and uncertain (potentially infinite) duration. Such settings can be modeled as stochastic games. While algorithms have been developed for solving (i.e., computing a game-theoretic solution concept such as Nash equilibrium) two-player zero-sum stochastic games, research on algorithms for non-zero-sum and multiplayer stochastic games is limited. We present a new algorithm for these settings, which constitutes the first parallel algorithm for multiplayer stochastic games. We present experimental …
We consider the decentralized power allocation and spectrum sharing problem in multi-user, multi-channel systems with strategic users. We present a mechanism/game form that has the following desirable features. (1) It is individually rational. (2) It is budget balanced at every Nash equilibrium of the game induced by the game form as well as off equilibrium. (3) The allocation corresponding to every Nash equilibrium (NE) of the game induced by the mechanism is a Lindahl allocation, that is a weakly Pareto optimal allocation. Our proposed game form/mechanism achieves all the above desirable pro…
