## Natural Games: Game Theory as a Thermodynamic Process

As game theory for the mathematical modeling of human behavior was inspired by the behavior of thermodynamic systems, it is only fitting that behavior in game theory can be viewed directly from the perspective of thermodynamic processes, as Anttila and Annila discuss in their article “Natural games”. In game theory in general (as we have discussed extensively in this course), behavior is modeled as a game where all players are trying to maximize their payoff. The universality of this basic structure of games which appears in all sorts of areas such as economics and biology implies that there exists some sort of universal payoff function (i.e. the payoff for each player given a set of strategies corresponding to each player), which takes into account all incentives of each player in competition with the incentives of all other players. Looking at economics and biology as specific examples, it can be seen that the payoff functions are quite different. In economics, the payoff can usually be described as money, or when other objects are valued, some sort of utility. In biology, specifically ecological systems, the payoff can be roughly equated with fitness of the species, which is similar to utility. In general, it is very difficult to quantify utility in any specific scenario.

In “Natural games”, the authors map game theory concepts to their physical counterparts in thermodynamics. In the thermodynamics picture, everything can be expressed in terms of energy, where the overall system follows the 2nd law of thermodynamics in that the system tends toward greater overall entropy. The assets of players can be viewed as a pool of energy, and information or asset exchange is simply energy flow. (Note that information theory says that information and energy are essentially interchangeable, in the sense that gaining information costs energy, and energy can be gained by “spending” information.) The “thermodynamics” game can then be viewed as a generalized scenario where all players are trying to increase their free energy as fast as possible (i.e. maximize the rate of free energy increase). As a result of the fact that the rate of change of free energy (of a “player”) directly corresponds to the rate of change of entropy (of the system as a whole), the payoff function can be expressed as the rate of increase of entropy in the system.

The expression of the payoff function as the rate of increase of entropy reveals the subjective nature of decision making in choosing a strategy. In almost any scenario, the payoff for a specific player is a function of the possessions and options that are available for that player. Similarly, the rate of entropy increase as a result of a specific “player” is dependent on the current energy and methods of increasing the gain rate of free energy of that “player”. Additionally, a few simple scenarios find themselves quite easily expressed in thermodynamic terms. A zero-sum game, where any gain or loss by one player corresponds directly to a loss or gain respectively by another player or players, can be viewed as a thermodynamic system where the total energy is conserved. A non-zero-sum game on the other hand, where players can also compete for external resources, can be viewed as a thermodynamic system where external forces or factors can add or remove energy from the system.

The formulation of game theory as simply the enforcement of a natural law is illuminating in that it clarifies why game theory can be applied so ubiquitously. By reminding ourselves of the origins of game theory and why it works so well in describing behavior in so many different scenarios, we can see that this behavior is simply obeying the laws of the thermodynamics and the universe. From this analysis, we can gain a better overall understanding of game theory and why it is so powerful.

Source:

“Natural games” by Jani Anttila & Arto Annila (Mar. 2011) https://arxiv.org/abs/1103.1656