New Algorithms for Solving Tropical Linear Systems

The problem of solving tropical linear systems, a natural problem of tropical mathematics, has already proven to be very interesting from the algorithmic point of view: it is known to be in $NP\cap coNP$ but no polynomial time algorithm is known, although counterexamples for existing pseudopolynomial algorithms are (and have to be) very complex. In this work, we continue the study of algorithms for solving tropical linear systems. First, we present a new reformulation of Grigoriev's algorithm that brings it closer to the algorithm of Akian, Gaubert, and Guterman; this lets us formulate a whole family of new algorithms, and we present algorithms from this family for which no known superpolynomial counterexamples work. Second, we present a family of algorithms for solving overdetermined tropical systems. We show that for weakly overdetermined systems, there are polynomial algorithms in this family. We also present a concrete algorithm from this family that can solve a tropical linear system defined by an $m\times n$ matrix with maximal element $M$ in time $\Theta\left({m \choose n} \mathrm{poly}\left(m, n, \log M\right)\right)$, and this time matches the complexity of the best of previously known algorithms for feasibility testing.Comment: 17 page

Matrix analysis for associated consistency in cooperative game theory

Hamiache's recent axiomatization of the well-known Shapley value for TU games states that the Shapley value is the unique solution verifying the following three axioms: the inessential game property, continuity and associated consistency. Driessen extended Hamiache's axiomatization to the enlarged class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. In this paper, we introduce the notion of row (resp. column)-coalitional matrix in the framework of cooperative game theory. Particularly, both the Shapley value and the associated game are represented algebraically by their coalitional matrices called the Shapley standard matrix $M^{Sh}$ and the associated transformation matrix $M_\lambda,$ respectively. We develop a matrix approach for Hamiache's axiomatization of the Shapley value. The associated consistency for the Shapley value is formulated as the matrix equality $M^{Sh}=M^{Sh}·M_\lambda.$ The diagonalization procedure of $M_\lambda$ and the inessential property for coalitional matrices are fundamental tools to prove the convergence of the sequence of repeated associated games as well as its limit game to be inessential. In addition, a similar matrix approach is applicable to study Driessen's axiomatization of a certain class of linear values. Matrix analysis is adopted throughout both the mathematical developments and the proofs. In summary, it is illustrated that matrix analysis is a new and powerful technique for research in the field of cooperative game theory

Navigating the Topology of 2x2 Games: An Introductory Note on Payoff Families, Normalization, and Natural Order

The Robinson-Goforth topology of swaps in adjoining payoffs elegantly arranges 2x2 ordinal games in accordance with important properties including symmetry, number of dominant strategies and Nash Equilibria, and alignment of interests. Adding payoff families based on Nash Equilibria illustrates an additional aspect of this order and aids visualization of the topology. Making ties through half-swaps not only creates simpler games within the topology, but, in reverse, breaking ties shows the evolution of preferences, yielding a natural ordering for the topology of 2x2 games with ties. An ordinal game not only represents an equivalence class of games with real values, but also a discrete equivalent of the normalized version of those games. The topology provides coordinates which could be used to identify related games in a semantic web ontology and facilitate comparative analysis of agent-based simulations and other research in game theory, as well as charting relationships and potential moves between games as a tool for institutional analysis and design.Comment: 8 pages including 4 figures in text and 4 plate

Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.Comment: 27 pages, 6 figures, revised versio

On the optimality of the uniform random strategy

The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H o}s, is a central topic in the field of positional games, with deep connections to the theory of random structures. For any given hypergraph ${\cal H}$ the main questions is to determine the smallest bias $q({\cal H})$ that allows Breaker to force that Maker ends up with an independent set of ${\cal H}$. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies a couple of natural `container-type' regularity conditions about the degree of subsets of its vertices. This will enable us to derive a hypergraph generalization of the $H$-building games, studied for graphs by Bednarska and {\L}uczak. Furthermore, we investigate the biased version of generalizations of the van der Waerden games introduced by Beck. We refer to these generalizations as Rado games and determine their threshold bias up to constant factors by applying our general criteria. We find it quite remarkable that a purely game theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, when the generalizations to hypergraphs in the analogous setup of sparse random discrete structures are usually quite challenging.Comment: 26 page