Minkowski Games

We introduce and study Minkowski games. In these games, two players take turns to choose positions in R^d based on some rules. Variants include boundedness games, where one player wants to keep the positions bounded (while the other wants to escape to infinity), and safety games, where one player wants to stay within a given set (while the other wants to leave it). We provide some general characterizations of which player can win such games, and explore the computational complexity of the associated decision problems. A natural representation of boundedness games yields coNP-completeness, whereas the safety games are undecidable

Perfect Prediction in Minkowski Spacetime: Perfectly Transparent Equilibrium for Dynamic Games with Imperfect Information

The assumptions of necessary rationality and necessary knowledge of strategies, also known as perfect prediction, lead to at most one surviving outcome, immune to the knowledge that the players have of them. Solutions concepts implementing this approach have been defined on both dynamic games with perfect information and no ties, the Perfect Prediction Equilibrium, and strategic games with no ties, the Perfectly Transparent Equilibrium. In this paper, we generalize the Perfectly Transparent Equilibrium to games in extensive form with imperfect information and no ties. Both the Perfect Prediction Equilibrium and the Perfectly Transparent Equilibrium for strategic games become special cases of this generalized equilibrium concept. The generalized equilibrium, if there are no ties in the payoffs, is at most unique, and is Pareto-optimal. We also contribute a special-relativistic interpretation of a subclass of the games in extensive form with imperfect information as a directed acyclic graph of decisions made by any number of agents, each decision being located at a specific position in Minkowski spacetime, and the information sets and game structure being derived from the causal structure. Strategic games correspond to a setup with only spacelike-separated decisions, and dynamic games to one with only timelike-separated decisions. The generalized Perfectly Transparent Equilibrium thus characterizes the outcome and payoffs reached in a general setup where decisions can be located in any generic positions in Minkowski spacetime, under necessary rationality and necessary knowledge of strategies. We also argue that this provides a directly usable mathematical framework for the design of extension theories of quantum physics with a weakened free choice assumption.Comment: 25 pages, updated technical repor

Minimax representation of nonexpansive functions and application to zero-sum recursive games

We show that a real-valued function on a topological vector space is positively homogeneous of degree one and nonexpansive with respect to a weak Minkowski norm if and only if it can be written as a minimax of linear forms that are nonexpansive with respect to the same norm. We derive a representation of monotone, additively and positively homogeneous functions on $L^\infty$ spaces and on $\mathbb{R}^n$, which extend results of Kolokoltsov, Rubinov, Singer, and others. We apply this representation to nonconvex risk measures and to zero-sum games. We derive in particular results of representation and polyhedral approximation for the class of Shapley operators arising from games without instantaneous payments (Everett's recursive games)

Minkowski Games

We introduce and study Minkowski games. In these games, two players take turns to choose positions in ℝd based on some rules. Variants include boundedness games, where one player wants to keep the positions bounded (while the other wants to escape to infinity), and safety games, where one player wants to stay within a given set (while the other wants to leave it). We provide some general characterizations of which player can win such games, and explore the computational complexity of the associated decision problems. A natural representation of boundedness games yields coNP-completeness, whereas the safety games are undecidable.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

Linking of Repeated Games. When Does It Lead to More Cooperation and Pareto Improvements?

Linking of repeated games and exchange of concessions in fields of relative strength may lead to more cooperation and to Pareto improvements relative to the situation where each game is played separately. In this paper we formalize these statements, provide some general results concerning the conditions for more cooperation and Pareto improvements to materialize or not and analyze the relation between both. Special attention is paid to the role of asymmetries.Environmental Policy, Linking, Folk Theorem, Tensor Game, Prsioners' Dilemma, Full Cooperation, Pareto Efficiency, Minkowski Sum, Vector Maximum, Convex Analysis

Noisy Relativistic Quantum Games in Noninertial Frames

The influence of noise and of Unruh effect on quantum Prisoners' dilemma is investigated both for entangled and unentangled initial states. The noise is incorporated through amplitude damping channel. For unentangled initial state, the decoherence compensates for the adverse effect of acceleration of the frame and the effect of acceleration becomes irrelevant provided the game is fully decohered. It is shown that the inertial player always out scores the noninertial player by choosing defection. For maximally entangled initially state, we show that for fully decohered case every strategy profile results in either of the two possible equilibrium outcomes. Two of the four possible strategy profiles become Pareto Optimal and Nash equilibrium and no dilemma is leftover. It is shown that other equilibrium points emerge for different region of values of decoherence parameter that are either Pareto optimal or Pareto inefficient in the quantum strategic spaces. It is shown that the Eisert et al miracle move is a special move that leads always to distinguishable results compare to other moves. We show that the dilemma like situation is resolved in favor of one player or the other.Comment: 14 pages and 6 figure

Relativistic Quantum Games in Noninertial Frames

We study the influence of Unruh effect on quantum non-zero sum games. In particular, we investigate the quantum Prisoners' Dilemma both for entangled and unentangled initial states and show that the acceleration of the noninertial frames disturbs the symmetry of the game. It is shown that for maximally entangled initial state, the classical strategy C (cooperation) becomes the dominant strategy. Our investigation shows that any quantum strategy does no better for any player against the classical strategies. The miracle move of Eisert et al (1999 Phys. Rev. Lett. 83 3077) is no more a superior move. We show that the dilemma like situation is resolved in favor of one player or the other.Comment: 8 Pages, 2 figures, 2 table

Matroids are Immune to Braess Paradox

The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest.Comment: 21 page