Improving Unlinkability of Attribute-based Authentication through Game Theory

Paul Erickson. <i>The World the Game Theorists Made</i>. Chicago: University of Chicago Press, 2015. Pp. 384. $35.00 (paper).

We begin with an exposition on the theory of noncooperative games. The strategic form game representation is introduced first, followed by selected equilibrium concepts based on the Nash equilibrium and some of its refinements like its undominated variant, the Coalition-Proof and the Strong Nash equilibrium. The second part on noncooperative games covers their representation in extensive form, where we introduce another variant, the subgame perfect Nash equilibrium Next comes the consideration of cooperative games. We first define the notion of a cooperative game as compared to a noncooperative one and introduce the characteristic function which assigns the gains from cooperation to each possible coalition of players. Depending on whether there even exist overall gains from cooperation and how these gains specifically arise and change within and over different coalitions, we can classify cooperative games and assign certain properties to them. The last part of this chapter is devoted to some of the most prominent solution concepts for cooperative games. These concepts determine how the gains from cooperation are or can be distributed among the players and are generally either set- or point-valued. We begin with the classic von Neumann Morgenstern solution to which we relate the concept of the core. There, we show different conditions under which, regarding the underlying game, a solution in terms of the core exists. The section on point-valued solutions, also called allocation rules, starts with a general definition and some common properties of these rules and how the latter respond to changes to the game they are based on. The allocation rule we cover in detail is the Shapley value, followed by its weighted variant, which again is related to the core. The chapter concludes with a brief treatment of the so-called bargaining problem and corresponding bargaining solutions of Nash and Kalai–Smorodinsky. With this we include an alternative approach to allocate the gains from cooperation among a number of players.

One contribution of 18 to a theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

An equilibrium in group decision and its association with the Nash equilibrium in game theory

In this paper, a novel algorithm based on Nash equilibrium and memory neural networks has been suggested for the path selection of autonomous vehicles in highly dynamic and complex environments. The proposed algorithm has been investigated in a racing game containing two vehicles: ego and opponent. However, the proposed method can be easily generalized to a race game with more than two cars. The suggested method, called GT-LSTM, is comprised of three primary parts. First, memory neural networks for learning and predicting the opponent’s vehicle behavior. Second, game theory and payoff matrices which agents use to select the action that leads to the maximum payoff, and finally, PID controllers in order to make lane changing and path following smoother. The motivation behind combining game theory and memory neural networks was to facilitate and make the decision-making process for the ego vehicles more accurate by learning the opponent vehicle’s behavior through memory neural networks. In the designed scenario for investigating the effectiveness of the proposed method in the CARLA simulator, the opponent vehicle performs extreme maneuvers to block the ego’s route. On the other hand, the ego vehicle has the privilege of using the GT-LSTM method for overtaking the opponent and winning the race. For that, it uses the predicted action of the opponent vehicle by means of memory neural networks to update its own payoff matrices and opt for the best action, knowing the next move of its rival. For better performance comparison of the proposed algorithm, two other planning methods, including game theory without having knowledge about the opponent vehicle and also game theory and payoff matrices along with conventional neural networks, have been implemented. With respect to the yielding results from the simulation, the proposed method is superior to the two other methods, having a success rate of 55 percent, compared to 15 and 32 percent, respectively, for the game theory without having knowledge about the opponent vehicle and also game theory and payoff matrices next to conventional neural networks. Moreover, it has been shown that the obtained response from the suggested algorithm matches with Nash equilibrium in 90.2 percent of the situation during the simulation experiments.

Game Theory and its Relation to Bayesian Theory

Dyadic two-person mixed strategy games form the simplest case for which we can determine straightforwardly Nash equilibrium sets. As in the precedent chapters, the set of Nash equilibria in a particular game is determined as an intersection of graphs of optimal reaction mappings of the first and the second players. In contrast to other games, we obtain not only an algorithm, but a set-valued/multi-valued Nash equilibrium set function (Nash function) \({ NES}(A,B)\) that gives directly as its values the Nash equilibrium sets corresponding to the values of payoff matrix instances. The function is piecewise and it has totally 36 distinct pieces, i.e. the domain of the Nash function is divided into 36 distinct subsets for which corresponding Nash sets have a similar pattern. To give an expedient form to such a function definition and to its formula, we use a code written in the Wolfram language (Wolfram, An elementary introduction to the Wolfram language, Wolfram Media, Inc., Champaign, XV+324 pp, 2016, [1]; Hastings et al., Hands-on start to Wolfram mathematica and programming with Wolfram language, Wolfram Media, Inc., Champaign, X+470 pp, 2015, [2]) that constitutes a specific feature of this chapter in comparison with other chapters. To prove the main theoretic result of this chapter, we apply the Wolfram language code too. The Nash function \({ NES}(A,B)\) is a multi-valued/set-valued function that has in the quality of its domain the Cartesian product \(\mathbb {R}^{2\times 2}\times \mathbb {R}^{2\times 2}\) of two real spaces of two \(2\times 2\) matrices and in the quality of a Nash function image all possible sets of Nash equilibria in dyadic bimatrix mixed-strategy games. These types of games where considered earlier in a series of works, e.g. Vorob’ev (Foundations of game theory: noncooperative games, Nauka, Moscow (in Russian), 1984, [3]; Game theory: lectures for economists and systems scientists, Nauka, Moscow (in Russian), 1985, [4]), Gonzalez-Diaz et al. (An introductory course on mathematical game theory, American Mathematical Society, XIV+324 pp, 2010, [5]), Sagaidac and Ungureanu (Operational research, CEP USM, Chisinau, 296 pp (in Romanian), 2004, [6]), Ungureanu (Set of nash equilibria in \(2\times 2\) mixed extended games, from the Wolfram demonstrations project, 2007, [7]), Stahl (A gentle introduction to game theory, American Mathematical Society, XII+176 pp, 1999, [8]), Barron (Game theory: an introduction, 2nd ed, Wiley, Hoboken, XVIII+555 pp, 2013, [9]), Gintis (Game theory evolving: a problem-centered introduction to modeling strategic interaction, 2nd ed, Princeton University Press, Princeton and Oxford, XVIII+390 pp, 2009, [10]). Recently, we found the paper by John Dickhaut and Todd Kaplan that describes a program written in Wolfram Mathematica for finding Nash equilibria (Dickhaut and Kaplan, Economic and financial modeling with mathematica®, TELOS and Springer, New York, pp 148–166, 1993, [11]). The program is based on the game theory works by Rapoport (Two-person game theory: the essential ideas, University of Michigan Press, 229 pp, 1966, [12]; N-person game theory: concepts and applications, Dover Publications, Mineola, 331 pp, 1970, [13]), Friedman (Game theory with applications to economics, 2nd ed, Oxford University Press, Oxford, XIX+322 pp, 1990, [14]), and Kreps (A course in microeconomic theory, Princeton University Press, Princeton, XVIII+839 pp, 1990, [15]). Some examples from Harsanyi and Selten book (Harsanyi and Selten, General theory of equilibrium selection in games, The MIT Press, Cambridge, XVI+378 pp, 1988, [16]) are selected for tests. Unfortunately, the program and package need to be updated to recent versions of the Wolfram Mathematica and the Wolfram Language.

Game theory plays an important role in applied mathematics, economics and decision theory. There are many works devoted to game theory. Most of them deals with a Nash equilibrium. A global search algorithm for finding a Nash equilibrium was proposed in [13]. Also, the extraproximal and extragradient algorithms for the Nash equilibrium have been discussed in [3]. Berge equilibrium is a model of cooperation in social dilemmas, including the Prisoner’s Dilemma games [15]. The Berge equilibrium concept was introduced by the French mathematician Claude Berge [5] for coalition games. The first research works of Berge equilibrium were conducted by Vaisman and Zhukovskiy [18; 19]. A method for constructing a Berge equilibrium which is Pareto-maximal with respect to all other Berge equilibriums has been examined in Zhukovskiy [10]. Also, the equilibrium was studied in [16] from a view point of differential games. Abalo and Kostreva [1; 2] proved the existence theorems for pure-strategy Berge equilibrium in strategic-form games of differential games. Nessah [11] and Larbani, Tazdait [12] provided with a new existence theorem. Applications of Berge equilibrium in social science have been discussed in [6; 17]. Also, the work [7] deals with an application of Berge equilibrium in economics. Connection of Nash and Berge equilibriums has been shown in [17]. Most recently, the Berge equilibrium was examined in Enkhbat and Batbileg [14] for Bimatrix game with its nonconvex optimization reduction. In this paper, inspired by Nash and Berge equilibriums, we introduce a new notion of equilibrium so-called Anti-Berge equilibrium. The main goal of this paper is to examine Anti-Berge equilibrium for bimatrix game. The work is organized as follows. Section 2 is devoted to the existence of Anti-Berge equilibrium in a bimatrix game for mixed strategies. In Section 3, an optimization formulation of Anti-Berge equilibrium has been formulated.

In this paper, we have introduced two models for spectrum management (spectrum trading and spectrum competition) in cognitive radio network. The first model is without game theory and the second one is with game theory. The first model for spectrum management without game theory, called SMWG, which has been designed to provide an efficient and dynamic equations to enhance the network performance. SMWG provides a novel function, called QoS function, to support three levels of QoS. In addition, it provides a novel factor, called Competition Factor, to control the behaviors among spectrum owners (primary users). In the second model, we have applied the game theory concept into spectrum management (SMG) to compare the network performance in both cases (SMWG and SMG). SMG provides two games, the first one is to model the dynamic behavior of spectrum competition among primary users, a Bertrand game is formulated where the Nash equilibrium is considered as the solution. The second game is to model the spectrum trading between the primary user and the secondary user, a Stackelberg game is formulated where the Nash equilibrium is again considered as the solution. Basically, the solution is found in terms of the size of the offered spectrum to the secondary users with regards to the offered spectrum price. We compare SMWG with conventional scheme, also compare SMG with conventional scheme, and finally compare SMWG with SMG in terms of network performance.

Attribute-Based Authentication (ABA) is becoming more prevalent in everyday interactions. In this paper, we propose the Decision-Making Method for Authentication (DMMA) to address the privacy concerns in ABA. The need for DMMA is supported through multiple observations. First, in practice, the indistinguishability of crypto-proof-based assertions (that are posessed by different users) fails with non-zero probability. This explains why cryptographic means alone are insufficient to provide a substantial level of unlinkability in ABA systems with n users. Second, each user in ABA possesses multiple credentials: they can be used interchangeably to get access to the service(s) which is provided by a relying party (RP). DMMA addresses the challenge of interchangeable usage. As an initial step, we synthesized the criterion of unlinkability: it is based on the definitions of international standard ISO 27551 as well as the information theoretic measure of conditional entropy. We then use that criterion to formalize the task of authentication as a non-cooperative coordination game. In this game, players (targets of the attack) maximize their utilities by using their assertions interchangeably. The experiment demonstrates that a number of equilibria with substantially higher unlinkability can be achieved. Unlinkability vary depending on: i) the information (and its trustworthiness) about the moves of the other players in the game; ii) the statistical distribution of user attributes. DMMA demonstrates how users may be provided recommendations over the optimal selection of assertions for ABA. These recommendations can have a practical impact if DMMA is implemented as a feature within Digital Credential Wallets (DCWs).

Non-cooperative behavior in wireless networks

Game Theory and Exercises introduces the main concepts of game theory, along with interactive exercises to aid readers’ learning and understanding. Game theory is used to help players understand decision-making, risk-taking and strategy and the impact that the choices they make have on other players; and how the choices of those players, in turn, influence their own behaviour. So, it is not surprising that game theory is used in politics, economics, law and management. This book covers classic topics of game theory including dominance, Nash equilibrium, backward induction, repeated games, perturbed strategie s, beliefs, perfect equilibrium, Perfect Bayesian equilibrium and replicator dynamics. It also covers recent topics in game theory such as level-k reasoning, best reply matching, regret minimization and quantal responses. This textbook provides many economic applications, namely on auctions and negotiations. It studies original games that are not usually found in other textbooks, including Nim games and traveller’s dilemma. The many exercises and the inserts for students throughout the chapters aid the reader’s understanding of the concepts. With more than 20 years’ teaching experience, Umbhauer’s expertise and classroom experience helps students understand what game theory is and how it can be applied to real life examples. This textbook is suitable for both undergraduate and postgraduate students who study game theory, behavioural economics and microeconomics.

Game theory is a theory that has been widely used and analyzed in real life circumstances such as trading or investing. It is the study of strategic decision making. This theory can be modeled and applied to mathematics or economy to help people, societies, and governments make better decisions. This paper will discuss the basic concepts of game theory, such as what is game theory, the basic elements of game theory and the practical application of game theory. This paper will elaborate the principles of game theory through the example of rock paper scissors. This paper will also explain the concept of Nash equilibrium, which is a special equilibrium in game theory. This paper will use another example to show under which circumstances Nash equilibrium will be achieved. Through Nash equilibrium theory and case analysis, it is found that rational people will maximize their own interests by looking for a Nash equilibrium in their daily behavior decisions. This indicates that when making some decisions, society and the government need to consider and look for Nash equilibrium as much as possible, so that the whole country can obtain the maximum benefit.

Much of the economic value of electronic commerce comes from the automation of interactions between businesses and individuals. Game theory is a useful set of tools that can be used by designers of electronic-commerce applications in analyzing and engineering of automated agents and communication protocols. The central theoretical concept used in game theory is the Nash equilibrium. In this article, I show how the outcomes supported by a Nash equilibrium can positively be enlarged using automated negotiations.

Publications on mathematical game theory with many (not less than 2) players one can conditionally distribute in four directions: noncooperative, hierarchical, cooperative and coalition games. The two last in its turn are divided in the games with side and nonside payments and respectively in the games with transferable and nontransferable payoffs. If the first ones are being actively investigated (St. Petersburg State Faculty of Applied Mathematics and Control Processes, St. Petersburg Economics and Mathematics Institute, Institute of Applied Mathematical Research of Karelian Research Centre RAS), then the games with nontransferable payoffs are not covered. Here we suggest to base on conception of objections and counterobjections. The initial investigations were published in two monographies of E.\;I.\;Vilkas, the Lithuanian mathematician (the pupil of N.\;N.\;Vorobjev, the professor of St. Petersburg University). For the differential games this conception was first applied by E.\;M.\;Waisbord in 1974, then it was continued by the first author of the present article combined with E.\;M.\;Waisbord in the book &lt;&lt;Introduction in the theory of differential games of n-persons and its application&gt;&gt; M.: Sovetskoye Radio, 1980, and in monography of Zhukovskiy &lt;&lt;Equilibrium of objections and counterobjections&gt;&gt;, M.: KRASAND, 2010. However before formulating tasks of coalition games, to which this article is devoted, we return to noncoalition variant of many persons game. Namely we consider noncooperation game in normal form, defined by ordered triple: $$G_N=\langle\mathbb{N}, \{X_i\}_{i \in \mathbb{N}}, \{f_i (x)\}_{i \in \mathbb{N}}\rangle.$$ Here $\mathbb{N}=\{1,2,\ldots , N\ge2\}$ "--- set of ordinal numbers of players, each of them (see later) chooses its strategy $x_i\in X_i\subseteq \mathbb{R}^{n_i}$ (where by the symbol $\R^k$, $k\ge 1$, as usual, is denoted $k$-dimensional real Euclidean space, its elements are ordered sets of $k$-dimensional numbers, as well Euclidean norm $\parallel \cdot \parallel$ is used); as a result situation $x=(x_1,x_2,\ldots,x_N)\in X=\prod \limits_{i\in \mathbb{N}}X_i\subseteq \mathbb{R}^{\sum \limits_{i\in \mathbb{N}}n_i}$ form in the game. Payoff functions $f_i (x)$ are defined on set $X$ of situations $x$ for each players: \begin{gather*} f_1(x)=\sum_{j=1}^{N}x_{j}^{'} D_{1j}x_{j}+2d_{11}^{'} x_1,\\ f_2(x)=\sum_{j=1}^{N}x_{j}^{'} D_{2j}x_{j}+2d_{22}^{'} x_2,\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ f_N(x)=\sum_{j=1}^{N}x_{j}^{'} D_{Nj}x_{j}+2d_{NN}^{'} x_N. \end{gather*} In the opinion of luminaries of game theory to equilibrium as acceptable solution of differential game has to be inherent the property of stability: the deviation from it of individual player cannot increase the payoff of deviated player. The solution suggested in 1949 (at that time by the 21 years old postgraduate student of Prinston University John Forbs Nash (jun.) and later named Nash equilibrium "--- NE) meets entirely this requirement. The NE gained certainly &lt;&lt;the reigning position&gt;&gt; in economics, sociology, military sciences. In 1994 J.\;F.\;Nash was awarded Nobel Prize in economics (in a common effort with John Harsanyi and R.\;Selten) &lt;&lt;for fundamental analysis of equilibria in noncooperative game theory&gt;&gt;. Actually Nash developed the foundation of scientific method that played the great role in the development of world economy. If we open any scientific journal in economics, operation research, system analysis or game theory we certainly find publications concerned NE. However &lt;&lt;And in the sun there are the spots&gt;&gt;, situations sets of Nash equilibrium must be internally and externally unstable. Thus, in the simplest noncoalition game of 2 persons in the normal form $$ \langle\{1,2\},~\{X_i=[-1,1]\}_{i=1,2},~ \{f_i(x_1,x_2)=2x_1x_2-x_i^2\}_{i=1,2}\rangle$$ set os Nash equilibrium situations will be $$X^{e}=\{x^{e}=(x_1^{e},~x_2^{e})=(\alpha, \alpha)~|~\forall \alpha=const\in [-1,1]\},~f_i(x^e)=\alpha^2~(i\in 1,2).$$ For elements of this set (the segment of bisectrix of the first and the third quarter of coordinate angle), firstly, for $x^{(1)}=(0,0)\in X^{(e)}$ and $x^{(2)}=(1,1)\in X^{e}$ we have $f_i(x^{(1)})=0&lt;f_i(x^{(2)})=1~(i=1,2)$ and therefore the set $X^e$ is internally unstable, secondly, $f_i(x^{(1)})=0&lt;f_i(\frac{1}{4},\frac{1}{3})~(i=1,2)$ and therefore the set $X^e$ is externally unstable. The external just as the internal instability of set of Nash equilibrium is negative for its practical use. In the first case there exists situation which dominates NE (for all players), in the second case this situation is Nash equilibrium. Pareto maximality of Nash equilibrium situation would allow to avoid consequences of external and internal instability. However such coincidence is an exotic phenomenon. Thus to avoid trouble connected with external and internal instability then we add the requirement of Pareto maximality to the notion of equilibrium of objections and counterobjections offered below. However we first of all reduce generally accepted solution concepts "--- NE and BE for the game $G_N$. It is proved in the article that in mathematical model both NE and BE are absent but there exist equilibria of objections and conterobjections as well as sanctions and countersanctions and simultaneously Pareto maximality.