  In a situation where each player has control over the transition
probabilities of each subsystem, we game-theoretically analyze the optimization
problem of minimizing both the partial entropy production of each subsystem and
a penalty for failing to achieve a given state transition. In the regime of
linear irreversible thermodynamics, we obtain the Nash equilibrium solution of
the probability flow and calculate each partial entropy production for this
solution. We find a trade-off such that a partial entropy production should be
larger if we want the other partial entropy production to be smaller. The total
entropy production can be minimized if each subsystem equally shares the
penalty. We identify that this trade-off is due to the interacting contribution
of the probability flow and discuss a possible biological validity for
Escherichia coli chemotaxis.
