Equilibrium Strategy of the Pursuit-Evasion Game in Three-Dimensional Space

The pursuit-evasion game models the strategic interaction among players, attracting attention in many realistic scenarios, such as missile guidance, unmanned aerial vehicles, and target defense. Existing studies mainly concentrate on the cooperative pursuit of multiple players in two-dimensional pursuit-evasion games. However, these approaches can hardly be applied to practical situations where players usually move in three-dimensional space with a three-degree-of-freedom control. In this paper,we make the first attempt to investigate the equilibrium strategy of the realistic pursuit-evasion game, in which the pursuer follows a three-degree-of-freedom control, and the evader moves freely. First, we describe the pursuer's three-degree-of-freedom control and the evader's relative coordinate. We then rigorously derive the equilibrium strategy by solving the retrogressive path equation according to the Hamilton-Jacobi-Bellman-Isaacs(HJBI) method, which divides the pursuit-evasion process into the navigation and acceleration phases. Besides, we analyze the maximum allowable speed for the pursuer to capture the evader successfully and provide the strategy with which the evader can escape when the pursuer's speed exceeds the threshold. We further conduct comparison tests with various unilateral deviations to verify that the proposed strategy forms a Nash equilibrium.

Equilibrium dividend strategies in the dual model with a random time horizon

This paper investigates the dividend problem with non-exponential discounting in a dual model.We assume that the dividends can only be paid at a bounded rate and that the surplus process is killed by an exponential random variable.Since the non-exponential discount function leads to a time inconsistent control problem,we study the equilibrium HJB-equation and give the associated verification theorem.For the case of a mixture of exponential discount functions and exponential gains,we obtain the explicit equilibrium dividend strategy and the corresponding equilibrium value function.Besides,numerical examples are shown to illustrate our results.

Commission Strategy of the Auction House

In a SIPV model, when the commission proportion is not certain, but related with bargain price, generally, it is a linear function of the bargain price, this paper gives bidders＇ equilibrium bidding strategies in the first-and secondprice auctions. We find that the equilibrium strategies in second-price auction are dominant strategies. For seller or auction house, whether the fixed proportion or the unfixed proportion is good is not only related with constant item and the linear coefficient of the linear function, the size of the fixed commission proportion, but also related with the value of the item auctioned. So, in the practical auctions, the seller and the auction house negotiated with each other to decide the commission rules for their own advantage.

Equalizer Zero-Determinant Strategy in Discounted Repeated Stackelberg Asymmetric Game

This paper focuses on the performance of equalizer zero-determinant(ZD)strategies in discounted repeated Stackelberg asymmetric games.In the leader-follower adversarial scenario,the strong Stackelberg equilibrium(SSE)deriving from the opponents’best response(BR),is technically the optimal strategy for the leader.However,computing an SSE strategy may be difficult since it needs to solve a mixed-integer program and has exponential complexity in the number of states.To this end,the authors propose an equalizer ZD strategy,which can unilaterally restrict the opponent’s expected utility.The authors first study the existence of an equalizer ZD strategy with one-to-one situations,and analyze an upper bound of its performance with the baseline SSE strategy.Then the authors turn to multi-player models,where there exists one player adopting an equalizer ZD strategy.The authors give bounds of the weighted sum of opponents’s utilities,and compare it with the SSE strategy.Finally,the authors give simulations on unmanned aerial vehicles(UAVs)and the moving target defense(MTD)to verify the effectiveness of the proposed approach.

On Topics in Quantum Games

This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.

Ensemble feature selection integrating elitist roles and quantum game model

To accelerate the selection process of feature subsets in the rough set theory （RST）, an ensemble elitist roles based quantum game （EERQG） algorithm is proposed for feature selec- tion. Firstly, the multilevel elitist roles based dynamics equilibrium strategy is established, and both immigration and emigration of elitists are able to be self-adaptive to balance between exploration and exploitation for feature selection. Secondly, the utility matrix of trust margins is introduced to the model of multilevel elitist roles to enhance various elitist roles＇ performance of searching the optimal feature subsets, and the win-win utility solutions for feature selec- tion can be attained. Meanwhile, a novel ensemble quantum game strategy is designed as an intriguing exhibiting structure to perfect the dynamics equilibrium of multilevel elitist roles. Finally, the en- semble manner of multilevel elitist roles is employed to achieve the global minimal feature subset, which will greatly improve the fea- sibility and effectiveness. Experiment results show the proposed EERQG algorithm has superiority compared to the existing feature selection algorithms.

Existence of Linear Strategy Equilibrium in Insider Trading with Partial Observations

This paper investigates a linear strategy equilibrium in insider trading in continuous time allowing market makers to know some information on the value of a stock. By the use of filtering theory,the authors prove that in a monopoly market, there exists a unique equilibrium of linear strategy of intensity in a closed form, such that the insider can make positive profits and at which, all of the private information on the value of the stock is released; and the more accurate the information on the value of the stock observed by the market makers, the less the positive profits are made by the insider, and even go to zero. However, there is no Nash equilibrium in a Cournot competition market between two insiders if they both adopt a linear strategy of intensity.

Study on air formation to ground attack-defends decision making in fuzzy condition

With the development of modern military technology, uncertain decision-making problems become more and more exigent to be solved in military command and control. Based on game theory, and taking air formarion to ground attack-defends campaign as the background, this paper established an opposed dynamic decision-making model. As to the problems in military decision-making in fuzzy condition in uncertainty, this paper put forward a fuzzy-influence-factor, which reflects the fuzzy influence on battle units, and establishes a fuzzy opposed decision-making model in anticipant value and in correlative chance way farther to get strategy equilibrium. It can be seen from the simulating results that the model disposes the fuzzy status in battlefield reasonably, analyzes the fighting results objectively, and offers a powerful decision-making support for military operation. The method is practically and effectively.

Time-inconsistent stochastic linear-quadratic control problem with indefinite control weight costs

A time-inconsistent linear-quadratic optimal control problem for stochastic differential equations is studied.We introduce conditions where the control cost weighting matrix is possibly singular.Under such conditions,we obtain a family of closed-loop equilibrium strategies via multi-person differential games.This result extends Yong’s work(2017) in the case of stochastic differential equations,where a unique closed-loop equilibrium strategy can be derived under standard conditions(namely,the control cost weighting matrix is uniformly positive definite,and the other weighting coefficients are positive semidefinite).

On Equilibrium in a Constant Retrial Queuing System with Reserved Time and Vacations

This paper investigates an M/M/1 constant retrial queue with reserved time and vacations.A new arriving customer will take up the server and accept service immediately if the server is idle.Otherwise,if the server is busy or on vacation,customers have to join a retrial orbit and wait for retry.Once a service is completed,the server will reserve a random time to seek a customer from the orbit at a constant retrial rate.If there is no arrivals(from the orbit or outside)during the idle period,to save energy,the server will take a vacation.This paper studies the fully unobservable case.First,the steady-state condition of the system is analyzed by using the Foster’s criterion,and the customers’expected waiting time is obtained based on the generating function technique.And then,by introducing an appropriate revenue structure,the equilibrium strategies of customers and the socially optimal strategy are all derived.Furthermore,a comparison between them is made and the effect of some main system parameters is studied.

Equilibrium Strategies in an Unobservable On-off Fluid Queue

This paper considers an on-off fluid queue model.The on and off states of the system appear alternately,and the sojourn times at these two different states are independent,and each one follows an exponential distribution.The fluid flows into the system buffer with some strategies to wait for the system service under the first-come first-served discipline.Here the system can process the fluid in the buffer only when the system is on state.With given utility functions such as an expected average social profit,and an individual expected profit,the equilibrium strategies are characterized under both the fully unobservable case and the partially observable case.

Worst-Case Investment Strategy with Delay

This paper considers a worst-case investment optimization problem with delay for a fund manager who is in a crash-threatened financial market. Driven by existing of capital inflow/outflow related to history performance, we investigate the optimal investment strategies under the worst-case scenario and the stochastic control framework with delay. The financial market is assumed to be either in a normal state(crash-free) or in a crash state. In the normal state the prices of risky assets behave as geometric Brownian motion, and in the crash state the prices of risky assets suddenly drop by a certain relative amount, which induces to a dropping of the total wealth relative to that of crash-free state. We obtain the ordinary differential equations satisfied by the optimal investment strategies and the optimal value functions under the power and exponential utilities, respectively. Finally, a numerical simulation is provided to illustrate the sensitivity of the optimal strategies with respective to the model parameters.

BUDGET CONSTRAINTS IN HOMOGENOUS MULTI-UNIT AUCTIONS

A homogenous multi-unit auction model is studied in two conditions： One is that the bidders face budget constraints and another is that the bidders do not face budget constraints. Two equilibrium bidding strategies of each bidder are presented in these two conditions. By comparing them, the authors show that the equilibrium unit price bid in the presence of budget constraints will be less than that in the absence of budget constraints. The difference between the two equilibrium bids leads to the reduced revenue of the seller. And this difference will decrease as the number of the bidders increases.