Abstract
An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, states of general interaction systems are seen to arise as linear superpositions of pure quantum states and Fourier transformation to become meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context.
Highlights
In an interaction system, economic agents interact pairwise, but do not necessarily cooperate towards a common goal
The relation with quantum models is immediate: The states of an n-decision system are described by complex n-dimensional vectors of unit length in exactly the same way as such vectors are assumed to describe the states of an n-dimensional quantum system in the Schrödinger picture of quantum theory
We present a linear model for the evolution of interaction systems and discuss its
Summary
Economic (or general physical) agents interact pairwise, but do not necessarily cooperate towards a common goal. The relation with quantum models is immediate: The states of an n-decision system are described by complex n-dimensional vectors of unit length in exactly the same way as such vectors are assumed to describe the states of an n-dimensional quantum system in the Schrödinger picture of quantum theory The latter yields a very intuitive interpretation of quantum games: the players select their strategies as to move some decision makers into accepting an alternative that offers financial rewards to the individual players, where the financial reward is given as a linear functional on the space to decision states (Section 5.4). Interaction systems provide a general model for interaction and decision-making Their analysis with vector space methods yields isomorphic representations in real and complex space, which suggests quantum theoretic interpretations. The dual interpretation suggests novel concepts of “Markov evolution” of cooperation
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