Design of zero-determinant strategies and its application to networked repeated games

Using semi-tensor product (STP) of matrices, the profile evolutionary equation (PEE) for repeated finite games is obtained. By virtue of PEE, the zero-determinant (ZD) strategies are developed for general finite games. A formula is then obtained to design ZD strategies for general finite games with multi-player and asymmetric strategies. A necessary and sufficient condition is obtained to ensure the availability of the designed ZD strategies. It follows that player $i$ is able to unilaterally design $k_i-1$ (one less than the number of her strategies) dominating linear relations about the expected payoffs of all players. Finally, the fictitious opponent player is proposed for networked repeated games (NRGs). A technique is proposed to simplify the model by reducing the number of frontier strategies.Comment: arXiv admin note: substantial text overlap with arXiv:2107.0325

On Universal Eigenvalues and Eigenvectors of Hypermatrices

A cubic hypermatrix of order $d$ can be considered as a structure matrix of a tensor with covariant order $r$ and contra-variant order $s=d-r$. Corresponding to this matrix expression of the hypermatrix, an eigenvector $x$ with respect to an eigenvalue $\lambda$ is proposed, called the universal eigenvector and eigenvalue of the hypermatrix. According to the action of tensors, if $x$ is decomposable, it is called a universal hyper-(UH-)eigenvector. Particularly, if all decomposed components are the same, $x$ is called a universal diagonal hyper (UDH-)eigenvector, which covers most of existing definitions of eigenvalue/eigenvector of hypermatrices. Using Semi-tensor product (STP) of matrices, the properties of universal eigenvalues/eigenvectors are investigated. Algorithms are developed to calculate universal eigenvalues/eigenvectors for hypermatrices. Particular efforts have been put on UDH- eigenvalues/eigenvectors, because they cover most of the existing eigenvalues/eigenvectors for hypermatrices. Some numerical examples are presented to illustrate that the proposed technique is universal and efficient