Decision theory vs. Game theory?

Question

Game theory is defined (here) as follows:

"Game theory, branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are interdependent. "

And Decision theory is defined (here) as follows:

"Decision theory is concerned with the reasoning underlying an agent’s choices."

Regarding to above definitions, can we say that Decision theory is a subset of Game theory? Or they are completely two different concepts? And in general, what are their common points?

Answer 1

Game theory has people as players, decision theory has probability distributions of decision networks. Game theory is concerned with what the probability and utility need to be to model an observed outcome. The probabilities and utilities are sometimes modelled as piecewise functions. Decision theory uses a prior probability to predict potential outcomes. Such potential outcomes might be a Markov chain. Game Theory is mathematical, decision theory is statistical, if the distinction makes sense. There are overlaps, but that is true with everything. I do not believe that decision theory is a subset of game theory.

Answer 2

Decision theory is a mathematical theory about what rational people choose (or prefer) under conditions of risk or uncertainty about the world, i.e., about what 'nature' will do. Game theory is a mathematical theory about what happens when two or more individuals are making decisions and the outcome depends on what each of them does, and what each of them does is allowed to depend on their beliefs about what the other will do.

One could think of the relationship between them in one of two ways. First, and somewhat awkwardly, decision theory is game theory restricted to the case in which one of the players is 'nature,' and so that 'player's' choices and payoffs are independent of what the other player does. Second, and probably more helpfully, game theory assumes decision theory for individual decision-making. For example, if one player has a probability distribution over the other's actions, she will pick an action which maximizes expected utility with respect to that distribution. (Maximizing expected utility is the orthodox decision theory; but you could in theory plug in other decision theories there.)

Answer 3

I agree with @Trurl's comment on this. Decision theory is more a component of optimal control theory rather than game theory. In control theory you have an environment you are navigating through to maximize some utility, and the decision theory component is in how to design your navigation as best as possible (often due to information and computational constraints). Game theory is about maximizing your utility in a competition against another player who is trying to do the same. It's possible to define your "environment" in control theory as playing against another player, but this leads to inhomogeneities in the environment most people working in optimal control don't normally deal with. Likewise, a game theorist could phrase an optimal control problem as a "game against the environment", but this would lead to payoff matricies that change drastically over time and with large branching factors.