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Duhamel's principle reduces the Cauchy problem for an inhomogeneous partial differential equation to the corresponding homogeneous problem. In the fractional-order setting, the classical principle does not apply directly because fractional…

We analyze the core of a cooperative Cournot game. We assume that when contemplating a deviation, the members of a coalition assign positive probability over all possible coalition structures that the non-members can form. We show that when…

We argue that fractional order (FO) differential equations are more suitable to model complex adaptive systems (CAS). Hence they are applied in replicator equations for non-cooperative game. Rock-Scissors-Paper game is discussed. It is…

In this paper we present numerical methods - finite differences and finite elements - for solution of partial differential equation of fractional order in time for one-dimensional space. This equation describes anomalous diffusion which is…

The properties of the Cournot model based on the most general entanglement operator containing quadratic expressions which is symmetric with respect to the exchange of players are considered. The degree of entanglement of games dependent on…

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by using integrations and…

Using Caputo fractional derivative of order $\alpha,$ $\alpha\in (0,1),$ we consider some chaotic systems of fractional differential equation. We will prove that they can be synchronized and anti-synchronized using suitable nonlinear…

This work proposes a novel distributed approach for computing a Nash equilibrium in convex games with merely monotone and restricted strongly monotone pseudo-gradients. By leveraging the idea of the centralized operator extrapolation method…

We study continuous time Bertrand oligopolies in which a small number of firms producing similar goods compete with one another by setting prices. We first analyze a static version of this game in order to better understand the strategies…

In a variant of communication tasks, players cooperate in choosing their local strategies to compute a given task later, working separately. Utilizing quantum bits for communication and sharing entanglement between parties is a recognized…

This paper presents algorithms for non-zero sum nonlinear constrained dynamic games with full information. Such problems emerge when multiple players with action constraints and differing objectives interact with the same dynamic system.…

Necessary and sufficient stability and instability conditions are obtained for multi-term homogeneous linear fractional differential equations with three Caputo derivatives and constant coefficients. In both cases,…

In this paper we extend the notion of an $\alpha$-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling…

We analyze cooperative Cournot games with boundedly rational firms. Due to cogni- tive constraints, the members of a coalition cannot accurately predict the coalitional structure of the non-members. Thus, they compute their value using…

This paper examines the integration of computational complexity into game theoretic models. The example focused on is the Prisoner's Dilemma, repeated for a finite length of time. We show that a minimal bound on the players' computational…

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.

A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the…

Driven by recent successes in two-player, zero-sum game solving and playing, artificial intelligence work on games has increasingly focused on algorithms that produce equilibrium-based strategies. However, this approach has been less…

For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of the operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the…

We examine sequential equilibrium in the context of computational games, where agents are charged for computation. In such games, an agent can rationally choose to forget, so issues of imperfect recall arise. In this setting, we consider…