The Non-equilibrium in Fixed-Paired Matching Pennies Game

Matching Pennies game (MPG) and and Rock-Paper-Scissors game (RPS) are the two elementary games. These two games are used to illustrate the concept (mixed strategy Nash equilibrium) since its very begin of the modern game theory. Under the traditional multi-person random pairwise-matching experimental protocol, using the definitive measurement, we have found the processes not in equilibrium but in non-equilibrium in these two games. Question left open is whether the equilibrium is violated in general? Target of this paper is to the simplest two-person fixedpaired MPG game. One may argue that, difference from the multi-person protocol, in simple two-person fixed-paired condition, people having full information may engage in more strategic thinking, and the the equilibrium behaviors could be realized. We test this point in experiments. In statistical physics, entropy production rate (EPR) is a real-value observable for non-equilibrium steady state— In the long run, when a system is in equilibrium, its EPR would be zero; otherwise in non-equilibrium, it would be persistently non-zero. In the laboratory experiment data, we observed the persistently non-zero EPR with real value. To confirm this result, we propose a visible and countable graph approach, namely net-loop, to show and count the time irreversibility. We compare the two observable (EPR and net-loop) and find that they are positively and linearly correlated in significant. These results are supported by 10 existed generalized MPG experiments. In Summary, we suggest that — The system of fixed pair two-person MPG, in which the mixed strategy Nash equilibrium is commonly expected, is actually in non-equilibrium. Importantly, the non-equilibrium has its own real-value and clear physical picture — a persistently cyclic motion.