Computing Effective Mixed Strategies for Protecting Targets in Large-Scale Critical Infrastructure Networks

Most critical infrastructure networks often suffer malicious attacks, which may result in network failures. Therefore, how to design more robust defense measures to minimize the loss is a great challenge. In recent years, defense strategies for enhancing the robustness of the networks are developed based on the game theory. However, the aforementioned method cannot effectively solve the defending problem on large-scale networks with a full strategy space. In this study, we achieve the purpose of protecting the infrastructure networks by allocating limited resources to monitor the targets. Based on the existing two-person zero-sum game model and the Double Oracle framework, we propose the EMSL algorithm which is an approximation algorithm based on a greedy search to compute effective mixed strategies for protecting large-scale networks. The improvement of our approximation algorithm to other algorithms is discussed. Experimental results show that our approximation algorithm can efficiently compute the mixed strategies on actual large-scale networks with a full strategy space, and the mixed defense strategies bring the highest utility to a defender on different networks when dealing with different attacks.

Game Self-organization of Hamiltonian Cycle of the Graph

This paper proposes a new application of the stochastic game model to solve the problem of self- organization of the Hamiltonian cycle of a graph. To do this, at the vertices of the undirected graph are placed game agents, whose pure strategies are options for choosing one of the incident edges. A random selection of strategies by all agents forms a set of local paths that begin at each vertex of the graph. Current player payments are defined as loss functions that depend on the strategies of neighboring players that control adjacent vertices of the graph. These functions are formed from a penalty for the choice of opposing strategies by neighboring players and a penalty for strategies that have reduced the length of the local path. Random selection of players’ pure strategies is aimed at minimizing their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution built on the basis of dynamic vectors of mixed strategies. The elements of the vectors of mixed strategies are the probabilities of choosing the appropriate pure strategies that adaptively take into account the values of current losses. The formation of vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which the gradient method of stochastic approximation is used. During the game, the method increases the value of the probabilities of choosing those pure strategies that lead to a decrease in the functions of average losses. For given methods of forming current payments, the result of the stochastic game is the formation of patterns of self-organization in the form of cyclically oriented strategies of game agents. The conditions of convergence of the recurrent method to collectively optimal solutions are ensured by observance of the fundamental conditions of stochastic approximation. The game task is extended to random graphs. To do this, the vertices are assigned the probabilities of recovery failures, which cause a change in the structure of the graph at each step of the game. Realizations of a random graph are adaptively taken into account when searching for Hamiltonian cycles. Increasing the probability of failure slows down the convergence of the stochastic game. Computer simulation of the stochastic game provided patterns of self-organization of agents’ strategies in the form of several local cycles or a global Hamiltonian cycle of the graph, depending on the ways of forming the current losses of players. The reliability of experimental studies is confirmed by the repetition of implementations of self-organization patterns for different sequences of random variables. The results of the study can be used in practice for game-solving NP-complex problems, transport and communication problems, for building authentication protocols in distributed information systems, for collective decision-making in conditions of uncertainty.

Competition Strategies for Location-Based Mobile Coupon Promotion

Location-based service heightens consumers’ shopping convenience. By utilizing spatial flexibility of consumers, retailers can target consumers via location-based mobile coupons (LBMCs) to enhance market performance. Considering the strategies for LBMC promotion for two competing retailers, we find that under different market intensities, only no adoption and symmetric adoption of LBMC promotion are the possible equilibria for the competing retailers at a low marginal targeting cost. Then, we extend our model to consider vertical (quality) differentiation and analyze the implications of adopting LBMC promotion for a superior-quality firm (with higher product valuation) and an inferior-quality firm (with lower product valuation). Mixed strategies for LBMC promotion emerge when firms’ products have different qualities. Our research findings provide useful guidance for managers and marketing practitioners to formulate strategies for targeted LBMC promotion.

Dynamic tax competition between symmetric countries with multiple strategic investors

Two symmetric countries compete over two-period under a non-preferential taxation regime to attract multiple investors where investors are strategic and investments are sunk once invested. Contrary to the existing results, we find that tax holidays do not arise during the initial period. Equilibria in mixed strategies arise in both periods where competing countries set strictly positive tax rates during the initial period. Strategic interaction between large investors reduces competition and increases tax rates during the initial period. We provide full characterization and uniqueness of equilibria in mixed strategies.JEL classification: F21, H21, H25, H87

THE CORRELATION DEPENDENCE OF PAYOFFS FOR MIXED STRATEGIES IN GAMES WITH NATURE

Цель исследования состоит в развитии и применении к задачам инвестирования методов принятия решений в играх с природой, учитывающих корреляцию случайных значений выигрышей для каждой пары чистых стратегий. При этом рассматриваются два критерия: математическое ожидание выигрыша и среднеквадратическое отклонение как оценка риска. Двухкритериальная модель принятия решений формализована путем перевода оценки риска в ограничение. Для такой обобщенной задачи квадратичного программирования получены аналитические методы решения. Приведен пример применения предложенного метода на реальных статистических данных. The aim of the research is to develop and apply to investment problems the methods of decision-making in games with nature, considering the correlation of random values of payoffs for each pair of pure strategies. In this case, two criteria are considered: the mathematical expectation of a payoff and the standard deviation as a risk assessment. The two-criteria decision-making model is formalized by translating the risk assessment into a constraint. For such a generalized quadratic programming problem, analytical solution methods are obtained. An example of applying the proposed method to real statistical data is given.

A new method to solve intuitionistic fuzzy matrix game based on the evaluation of different experts

In the word of uncertain competitive situations everything is in the state of flux. Under such situations knowing the exact outcomes of mixed strategies adopted by a player is nearly impossible. It is highly rational to assume that no two experts will project the similar fuzzy payoffs for mix of strategies used. Aggregation of expert’s judgement becomes utmost important before solving such competitive situations. Considering this the present paper proposes a method to solve intuitionistic fuzzy game problems by using aggregation operators on payoff judgments of more than one expert. The proposed method significantly adds to the existing literature by overcoming the limitation of Li’s existing method that considers only one expert’s opinion for solving intuitionistic fuzzy game problems. Illustrative example has been given for showing the superiority of the proposed method.

Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems

In this paper we address game theory problems arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problems addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of a Nash game. The resulting problems are nonconvex QP ones with linear constraints and turn out to be very challenging. We will show that the most recent approaches for the minimization of nonconvex QP functions over polytopes, including commercial solvers such as and , are unable to solve to optimality even test instances with $$n=50$$ n = 50 variables. For this reason, we propose to extend with them the current benchmark set of test instances for QP problems. We also present a spatial branch-and-bound approach for the solution of these problems, where a predominant role is played by an optimality-based domain reduction, with multiple solutions of LP problems at each node of the branch-and-bound tree. Of course, domain reductions are standard tools in spatial branch-and-bound approaches. However, our contribution lies in the observation that, from the computational point of view, a rather aggressive application of these tools appears to be the best way to tackle the proposed instances. Indeed, according to our experiments, while they make the computational cost per node high, this is largely compensated by the rather slow growth of the number of nodes in the branch-and-bound tree, so that the proposed approach strongly outperforms the existing solvers for QP problems.

Equilibrium payoffs in two-player discounted OLG games

This paper studies payoffs in subgame perfect equilibria of two-player discounted overlapping generations games with perfect monitoring. Assuming that mixed strategies are observable and a public randomization device is available, it is shown that sufficiently patient players can obtain any payoffs in the interior of the smallest rectangle containing the feasible and strictly individually rational payoffs of the stage game, when we first choose the rate of discount and then choose the players’ lifespan. Unlike repeated games without overlapping generations, obtaining payoffs outside the feasible set of the stage game does not require unequal discounting.