Quantum generative adversarial networks with multiple superconducting qubits

Generative adversarial networks are an emerging technique with wide applications in machine learning, which have achieved dramatic success in a number of challenging tasks including image and video generation. When equipped with quantum processors, their quantum counterparts—called quantum generative adversarial networks (QGANs)—may even exhibit exponential advantages in certain machine learning applications. Here, we report an experimental implementation of a QGAN using a programmable superconducting processor, in which both the generator and the discriminator are parameterized via layers of single- and two-qubit quantum gates. The programmed QGAN runs automatically several rounds of adversarial learning with quantum gradients to achieve a Nash equilibrium point, where the generator can replicate data samples that mimic the ones from the training set. Our implementation is promising to scale up to noisy intermediate-scale quantum devices, thus paving the way for experimental explorations of quantum advantages in practical applications with near-term quantum technologies.

Research on Efficient Reinforcement Learning for Adaptive Frequency-Agility Radar

Modern radar jamming scenarios are complex and changeable. In order to improve the adaptability of frequency-agile radar under complex environmental conditions, reinforcement learning (RL) is introduced into the radar anti-jamming research. There are two aspects of the radar system that do not obey with the Markov decision process (MDP), which is the basic theory of RL: Firstly, the radar cannot confirm the interference rules of the jammer in advance, resulting in unclear environmental boundaries; secondly, the radar has frequency-agility characteristics, which does not meet the sequence change requirements of the MDP. As the existing RL algorithm is directly applied to the radar system, there would be problems, such as low sample utilization rate, poor computational efficiency and large error oscillation amplitude. In this paper, an adaptive frequency agile radar anti-jamming efficient RL model is proposed. First, a radar-jammer system model based on Markov game (MG) established, and the Nash equilibrium point determined and set as a dynamic environment boundary. Subsequently, the state and behavioral structure of RL model is improved to be suitable for processing frequency-agile data. Experiments that our proposal effectively the anti-jamming performance and efficiency of frequency-agile radar.

Nash Equilibrium Points for Generalized Matrix Game Model with Interval Payoffs

Game theory-based models are widely used to solve multiple competitive problems such as oligopolistic competitions, marketing of new products, promotion of existing products competitions, and election presage. The payoffs of these competitive models have been conventionally considered as deterministic. However, these payoffs have ambiguity due to the uncertainty in the data sets. Interval analysis-based approaches are found to be efficient to tackle such uncertainty in data sets. In these approaches, the payoffs of the game model lie in some closed interval, which are estimated by previous information. The present paper considers a multiple player game model in which payoffs are uncertain and varies in a closed intervals. The necessary and sufficient conditions are explained to discuss the existence of Nash equilibrium point of such game models. Moreover, Nash equilibrium point of the model is obtained by solving a crisp bi-linear optimization problem. The developed methodology is further applied for obtaining the possible optimal strategy to win the parliament election presage problem.

The Spontaneous Pricing Order in the Noncooperative Game of Monopolistic Manufacturer and Many Retailers

We consider the collective pricing orders in a minimum supply chain that is composed of a monopolistic manufacturer and many retailers that belong to the same chain store firm. The retailers have the freedom to raise or lower the local price. The chain store firm sets up the commercial rules for local retail stores to maximize its total payoff. The monopolistic manufacturer firm controls the total quantity supplied for the market to achieve maximum benefits. We applied the two dimensional Ising model in statistical physics to map the collective distribution of microscopic strategy of local retailers into the macroscopic total payoff of the chain store firm. The local stores choose to raise the price or lower the price based their own mind when the supply in market surpasses the demand. When the supply in market is far less than the demand, the stores synchronously raise prices, even though a local store only have the incomplete information of their nearest neighboring supermarket. We find the critical equation for the balance point between the action of supplier and the action of chain store management based on game theory and statistical physics. The critical equation can identify the Nash equilibrium point of the non-cooperative game between the manufacturer and the chain-store seller, and reveal different levels of collective operations. This statistical physics method also holds for more complicate supply chains and economic systems.

Interrelationships between Prey and Predators and How Predators Choose Their Prey to Maximize Their Utility Functions

In this work, we model the relationship between prey and predators by studying the interactive behavior of this prey-predator model and using the change of prey. The objective is to maximize the profit function of each predator by seeking the strategy provided by each predator to maximize its profit. To do so, we maximize this utility function being constrained by balance equations between biomass and trophic, and we show that this last problem is completely equivalent to finding the generalized Nash equilibrium point. To calculate it, we use the conditions of Karush-Kuhn-Tucker and we show that it is indeed a linear complementarity problem.

A Type of Time-Symmetric Stochastic System and Related Games

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

The effect of price discrimination on dynamic duopoly games with bounded rationality

In a homogenous product market, customers’ different demand elasticities may lead to different prices. This study examined price discrimination’s effect on equilibrium points in Cournot duopoly games by assuming that each firm charges K prices and adjusts its strategies based on bounded rationality. In consideration of price discrimination, two discrete dynamic game systems with 2K variables were introduced for players with homogenous or heterogenous expectations. The stability of the Nash equilibrium point was found to be independent of price discrimination. Given price discrimination, the stability of boundary stationary points for the system with homogenous players is different from that for the system with heterogenous players. Numerical simulations verified the critical point for the system with homogenous players from being stable to its bifurcation.

Local and global analysis of a nonlinear duopoly game with heterogeneous firms

In this paper, we introduce a nonlinear duopoly game whose players are heterogeneous and their inverse demand functions are derived from a more general isoelastic demand. The game is modeled by a discrete time dynamic system whose Nash equilibrium point is unique. The conditions of local stability of Nash point are calculated. It becomes unstable via two types of bifurcations: flip and Neimark–Sacker. Some local and global numerical investigations are performed to show the dynamic behavior of game’s system. We show that the system is noninvertible and belongs to $Z_{2}-Z_{0}$ Z 2 − Z 0 type. We also show some multistability aspects of the system including basins of attraction and regions known as lobes.

SYSTEMIC RISK: THE EFFECT OF MARKET CONFIDENCE

In a crisis, when faced with insolvency, banks can sell stock in a dilutive offering in the stock market and borrow money in order to raise funds. We propose a simple model to find the maximum amount of new funds the banks can raise in these ways. To do this, we incorporate market confidence of the bank together with market confidence of all the other banks in the system into the overnight borrowing rate. Additionally, for a given cash shortfall, we find the optimal mix of borrowing and stock selling strategy. We show the existence and uniqueness of Nash equilibrium point for all these problems. Finally, using this model we investigate if banks have become safer since the crisis. We calibrate this model with market data and conduct an empirical study to assess safety of the financial system before, during after the last financial crisis.

Two-Step Proximal Method for Equilibrium Problems in Hadamard spaces

We propose a novel two-step proximal method for solving equilibrium problems in Hadamard spaces. The equilibrium problem is very general in the sense that it includes as special cases many applied mathematical models such as: variational inequalities, optimization problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithm is the analog of the two-step algorithm for solving the equilibrium problem in Hilbert spaces explored earlier. We prove the weak convergence of the sequence generated by the algorithm for pseudo-monotone bifunctions. Our results extend some known results in the literature for pseudo-monotone equilibrium problems.