Optimal Mixed Strategy Research Articles

Bayesian Stackelberg game model for water supply networks against interdictions with mixed strategies

We address a problem of preventing an interdiction on water supply networks by building a Bayesian Stackelberg game model involving stakeholders of a defender and an interdictor. The defender initiates to allocate resource to network components to make a trade-off between network resilience measured by water satisfaction rate and the defender's cost, whereas the interdictor follows to interdict a component with the objectives of maximising the destruction level on the network structure and minimising the interdictor's cost. Specifically, the defender adopts mixed defence strategies, which implies that the interdictor is uncertain of the defender's resource allocation. Moreover, we propose sufficient conditions for the elimination of the dominated defence and interdiction strategies. A decomposed iterative learning algorithm (DILA) and a smallest-depth binary-partition based hierarchical algorithm (SBHA) are developed to reduce the sizes of the defence and interdiction strategy sets, respectively, thus analysing the optimal mixed defence strategies. Finally, a real case study with private information is conducted, thus providing valuable suggestions for the defender's resource allocation against interdictions.

Dynamic Information Design: A Simple Problem on Optimal Sequential Information Disclosure

We study a dynamic information design problem in a finite-horizon setting consisting of two strategic and long-term optimizing agents, namely a principal (he) and a detector (she). The principal observes the evolution of a Markov chain that has two states, one and one absorbing state, and has to decide how to sequentially disclose information to the detector. The detector's only information consists of the messages she receives from the principal. The detector's objective is to detect as accurately as possible the time of the jump from the good to the bad state. The principal's objective is to delay the detector as much as possible from detective the jump to the bad state. For this setting, we determine the optimal strategies of the principal and the detector. The detector's optimal strategy is described by time-varying thresholds on her posterior belief of the good state. We prove that it is optimal for the principal to give no information to the detector before a time threshold, run a mixed strategy to confuse the detector at the threshold time, and reveal the true state afterwards. We present an algorithm that determines both the optimal time threshold and the optimal mixed strategy that could be employed by the principal. We show, through numerical experiments, that this optimal sequential mechanism significantly outperforms any other information disclosure strategy presented in literature.

Nash Equilibrium Strategy of Port Construction Considering Seawater Erosion

Lu, Q.; Wang, Z., and Wang, R., 2019. Nash equilibrium strategy of port construction considering seawater erosion. In: Li, L.; Wan, X.; and Huang, X. (eds.), Recent Developments in Practices and Research on Coastal Regions: Transportation, Environment and Economy. Journal of Coastal Research Journal of Coastal Research, Special Issue No. 98, pp. 330–334. Coconut Creek (Florida), ISSN 0749-0208.The main construction approaches to ports are based on government investment and contractor construction. This method faces a particular drawback for the implementation of construction: the contractor fails to carry out the work in strict accordance with the plan. By considering the problems of government supervision and contractors' strict adherence to construction plans, as well as to seawater erosion after port construction, this paper uses Nash equilibrium to solve the optimal mixed strategy for both parties. The main influencing factors of the strategy are analyzed. Then this paper finds that the high probability of erosion can stimulate the government and contractor to do their best, and the sum of gains and losses also influence the best strategy. Based on analysis results, the corresponding suggestions are put forward.

CLOSURE: A cloud scientific workflow scheduling algorithm based on attack–defense game model

The multi-tenant coexistence service mode makes the cloud-based scientific workflow encounter the risks of being intruded. For this problem, we propose a CLoud scientific wOrkflow SchedUling algoRithm based on attack–defensE game model (CLOSURE). In the algorithm, attacks based on different operating system vulnerabilities are regarded as different “attack” strategies; and different operating system distributions in a virtual machine cluster executing the workflows are regarded as different “defense” strategies. The information of the attacker and defender is not balanced. In other words, the defender cannot obtain the information about the attacker’s strategies, while the attacker can acquire information about the defender’s strategies through a network scan. Therefore, we propose to dynamically switch the defense strategies during the workflow execution, which can weaken the network scan effects and transform the workflow security problem into an attack–defense game problem. Then, the probability distribution of the optimal mixed defense strategies can be achieved by calculating the Nash Equilibrium in the attack–defense game model. Based on this probability, diverse VMs are provisioned for workflow execution. Furthermore, a task-VM mapping algorithm based on dynamic Heterogeneous Earliest Finish Time (HEFT) is presented to accelerate the defense strategy switching and improve workflow efficiency. The experiments are conducted on both simulation and actual environment, experimental results demonstrate that compared with other algorithms, the proposed algorithm can reduce the attacker’s benefits by around 15.23%, and decrease the time costs of the algorithm by around 7.86%.

On the equivalence of mixed and behavior strategies in finitely additive decision problems

AbstractWe consider decision problems with arbitrary action spaces, deterministic transitions, and infinite time horizon. In the usual setup when probability measures are countably additive, a general version of Kuhn’s theorem implies under fairly general conditions that for every mixed strategy of the decision maker there exists an equivalent behavior strategy. We examine to what extent this remains valid when probability measures are only assumed to be finitely additive. Under the classical approach of Dubins and Savage (2014), we prove the following statements: (1) If the action space is finite, every mixed strategy has an equivalent behavior strategy. (2) Even if the action space is infinite, at least one optimal mixed strategy has an equivalent behavior strategy. The approach by Dubins and Savage turns out to be essentially maximal: these two statements are no longer valid if we take any extension of their approach that considers all singleton plays.

Game-theoretic model of a volunteer computer GRID

In the paper we propose a simple game-theoretic model of a Desktop Grid for volunteer computing. Task replication reduces the risk of accepting wrong answers due to sabotage. Saboteur's attack by intruding multiple computing nodes brings him some profit in case a wrong answer is accepted, while the server suffers some penalty in this case. Nodes are assigned some reputation as a monotone function of the number of produced correct (or not exposed) answers. We obtain the optimal mixed strategies and show that the average gain of the players depends only on the server's penalty, nodes' reputation, and the size of the subgrid of nodes with the same reputation. Also we estimate the server's cost per an answer. Numerical examples show that the average cost of the server is not more than that in the case when the number of intruders is known.

Solution of a Game-Theoretical Model of Resource Allocation

We consider a game-theoretical model of defense in which the opponents use several types of infinitelydivisible attack and defense weapons. The defender (first player) payoff is the probability of destroying each attack weapon by at least one of the defense weapons. It is assumed that defense deploys at least one unit of each type of weapons. The optimal defense strategy is a pure maximin strategy, and the optimal mixed attack strategy involves choosing only one of the available attack weapons with certain probabilities. The search for optimal player strategies is reduced to the solution of linear programs.

Adaptive and Distributed Radio Resource Allocation in Densely Deployed Wireless LANs: A Game-Theoretic Approach

As increasing numbers of access points (APs) are deployed in wireless local area networks (WLANs), they experience severe channel interference. Thus, a well-defined resource management scheme for APs should be devised. Undoubtedly, a centralized and deterministic resource management is not appropriate, owing to the distributed operation and dynamic channel condition of densely deployed WLANs. To address this problem, an adaptive and distributed radio resource allocation (ADRA) algorithm is proposed, in which an optimal mixed strategy for selecting the channel width of APs is derived via a game-theoretic approach. ADRA also fits into the densely deployed WLAN in that it considers partially overlapped channels. Simulation results show that ADRA can provide a comparable throughput and fairness performance to centralized algorithms, even though it functions in a distributed manner. Moreover, it is shown that ADRA works adaptively even when network environments change rapidly.

A Collaborative Learning Automata Team Model for Modeling Multi‐Agent Systems

SUMMARYThe learning automaton (LA) team model has been proposed as one method for modeling multi‐agent systems. It is modeled as a noncooperative game of learning automata. In this model, each LA operates independently from each other, and there exists a Nash equilibrium, that is, the existence of an optimal mixed strategy in the mixed strategy space of the game has been proven. However, for modeling multi‐agent systems more generally, the information exchange among agents and the acquisition of cooperative behaviors such as the formation of autonomous community are required. In this paper, in order to complement the LA team model, we propose a new LA team model with some fully or partially collaborative learning behaviors. In this new model, each automaton performs reinforcement learning process in order to identify random environments exchanging information with its adjacent automata. Several computer simulations indicate the availability of the proposed model.

Competitive target search with multi-agent teams: symmetric and asymmetric communication constraints

We study a search game in which two multi-agent teams compete to find a stationary target at an unknown location. Each team plays a mixed strategy over the set of search sweep-patterns allowed from its respective random starting locations. Assuming that communication enables cooperation we find closed-form expressions for the probability of winning the game as a function of team sizes and the existence or absence of communication within each team. Assuming the target is distributed uniformly at random, an optimal mixed strategy equalizes the expected first-visit time to all points within the search space. The benefits of communication enabled cooperation increase with team size. Simulations and experiments agree well with analytical results.

Disarmament Games

Much recent work in the AI community concerns algorithms for computing optimal mixed strategies to commit to, as well as the deployment of such algorithms in real security applications. Another possibility is to commit not to play certain actions. If only one player makes such a commitment, then this is generally less powerful than completely committing to a single mixed strategy. However, if players can alternatingly commit not to play certain actions and thereby iteratively reduce their strategy spaces, then desirable outcomes can be obtained that would not have been possible with just a single player committing to a mixed strategy. We refer to such a setting as a disarmament game. In this paper, we study disarmament for two-player normal-form games. We show that deciding whether an outcome can be obtained with disarmament is NP-complete (even for a fixed number of rounds), if only pure strategies can be removed. On the other hand, for the case where mixed strategies can be removed, we provide a folk theorem that shows that all desirable utility profiles can be obtained, and give an efficient algorithm for (approximately) obtaining them.

Optimal Mixed Strategy Play: Professionals Can, Students Cannot, But What About the in Between Case?

AbstractUsing penalty kicks in collegiate soccer matches, we test whether kickers choose where to place shots, and whether goalies choose where to dive in a way that is consistent with optimal mixed strategy play. The previous literature, studying professional soccer players, provides evidence of mixed strategy play in penalty kick scenarios. These results contrast with the evidence obtained in a lab, studying subjects who only play a game a few times and have insignificant monetary payoffs. These lab results find no evidence of mixed strategy play. The contrast between the results obtained from these very different environments makes it unclear which result generalizes to other settings. By studying college athletes, we analyze the middle ground, which is where most strategic decisions will be made. We find that college players employ optimal strategic play in some respects, but not in other respects.

Target Evasion from a Missile Performing Multiple Switches in Guidance Law

A planar engagement between a maneuvering target and a homing missile is considered. It is assumed that, during the engagement, the missile carries out multiple switches from one linear guidance law to another, choosing from a prescribed set of guidance laws. This leads to a formulation of a hybrid evasion problem for the target. In the case of complete information on the missile’s strategies and switch moments, the optimal evasion strategy has a bang–bang form. The direction of a maximal acceleration command depends on the sign of a switching function and the sign of the zero-effort miss distance in the switched system. In the case of unknown switch moments, a matrix game is formulated, yielding the maximin evasion strategy, which implies the same bang–bang structure of evasive maneuver and guarantees a miss distance not smaller than the lower game value. Employing an optimal mixed evasion strategy guarantees better results on average. Examples where both the missile and the target have first-order strictly proper dynamics, and the missile (exploiting the proportional navigation guidance law) changes its gain once or twice during the engagement are elaborated upon. Monte Carlo simulation results support the theoretical analysis. The obtained results show that, even if the target acts optimally, the missile benefits from the switch in its guidance gain because the resulting miss distance is smaller when compared to the case when it uses a constant gain.

Consistent Probabilistic Social Choice

Two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives. In the context of traditional non-probabilistic social choice, these axioms are incompatible with each other. We show that in the context of probabilistic social choice, these axioms uniquely characterize a function proposed by Fishburn (1984). Fishburn's function returns so-called maximal lotteries, that is, lotteries that correspond to optimal mixed strategies in the symmetric zero-sum game induced by the pairwise majority margins. Maximal lotteries are guaranteed to exist due to von Neumann's Minimax Theorem, are almost always unique, and can be efficiently computed using linear programming. [web URL: http://onlinelibrary.wiley.com/doi/10.3982/ECTA13337/abstract]

Supply chain coordination with dual procurement sources via real-option contract

In a two-stage supply chain with a supplier and a manufacturer, the manufacturer can purchase a product either from the supplier or in the spot market. The spot market, however, inevitably involves price fluctuation risk and supply risk. Assuming that the manufacturer is the leader in a procurement game and offers a real-option contract to the supplier, we study the manufacturer’s optimal mixed procurement strategy that integrates the use of the real-option contract and the spot market. Moreover, we analyze the effects of the price risk and the supply risk in the spot market on market equilibrium. We show that using the real-option contract mechanism improves the overall expected profit of a supply chain and guarantees supply chain coordination in the presence of the spot market. The results also demonstrate that the price risk and the supply risk in the spot market adversely affect the manufacturer’s expected profit. On the contrary, these two risks bring benefits to the supplier.

Multi-Channel Cognitive Radio Networks: Modeling, Analysis and Synthesis

In this contribution, we establish a model for multi-channel cognitive radio networks (CRNs) using the theory of priority queues. This model enables us to conduct a performance analysis in the most general form by the derivation of the probability mass function (PMF) of queue length at the secondary users (SUs). In the second part, a reverse problem is considered to answer the important top-down question of whether a service requirement can be satisfied in a multi-channel CRN knowing the network parameters and traffic situation with respect to the SUs and the primary users (PUs). Terming this problem as the network synthesis, a precise conservation law is obtained, which relates the packet waiting times of both types of users, and based on which the achievable region of the network is also determined. Lastly, by the introduction of a mixed strategy, the conditions for the existence of an optimal trade-off between the interference onto the PUs and the quality-of-service of the SUs is shown, and the optimal mixed strategy is obtained when those conditions are satisfied.

Individually and socially optimal joining rules for an egalitarian processor-sharing queue under different information scenarios

This paper studies joining behavior of customers into an M/M/1 egalitarian processor-sharing (PS) queue. By constructing a left-multiplication transformation and using its matrix representation, we obtain the expected conditional sojourn time of a tagged customer. Then, in the fully observable case, we first consider the joining strategy in a decentralized manner, that is, arriving customers observe the queue size and then decide whether or not to join the queue based on the net benefit they will obtain upon the completion of service. Secondly, we derive the threshold strategy that will yield the system’s maximal expected profit, to reach the so-called social welfare optimization. Finally, Nash equilibrium and socially optimal mixed strategies are derived in the fully unobservable case. Moreover, some numerical examples are provided to explore the impact of system parameters on customer’s joining behavior.

Does player specialization predict player actions? Evidence from penalty kicks at FIFA World Cup and UEFA Euro Cup

Penalty kicks are analysed in the literature as ‘real life experiments’ for assessing the use of rational mixed strategies by professional players. However, each penalty kick cannot be considered a repetition of the same event because of the varying background conditions, in particular the heterogeneous ability of different players. Consequently, aggregate statistics over data sets composed of a large number of penalty kicks mediate the behaviour of the players in different games, and the properties of optimal mixed strategies cannot be tested directly because of aggregation bias. In this article, we model the heterogeneous ability of players. We then test the hypothesis that differently talented players randomize over different actions. To achieve this aim, we study a data set that collects penalties kicked during shoot-out series in the last editions of FIFA World Cup and UEFA Euro Cup (1994–2012) where kickers are categorized as specialists and non-specialists. The results support our theoretical predictions.

Applications of Games Theory in Analyzing Teaching Process

In the teaching activity, the problem of choosing the best solutions, methods and teaching procedures is usually difficult due to the complexity of pedagogical situations and random nature of individual particularities. This study aims at giving an answer to the question: can we plan the system of educational actions so that each action gives us maximum efficiency, reducing any unwanted results (routine, boredom .). For this purpose we use games theory techniques. Games theory provides models of situations in which each chosen action can give us in different cases, different results with a known probability. The objective is to find the optimal mixed strategy for the professor to ensure the best possible result, giving any mixed response strategies of the students. The analysis is performed at the level of some disciplines highly mathematized: applied mathematics in economics, financial and economic analysis.

Static search games played over graphs and general metric spaces

We define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure. The game involves two players, the searching player and the concealing player, and is played over a metric space. Each player simultaneously chooses to deploy at a point in the space; the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position, the concealing player receiving a payoff of 1 otherwise. The concepts of dominance and equivalence of strategies are examined in the context of this game, before focusing on the more specific case of the game played over a graph. Methods are presented to simplify the analysis of such games, both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph. Lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs.