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I have a utility function describing the desirability of an outcome state. I weigh the expected utility with the probability of the outcome state occuring. I find the expected utility of an action, a, with $EU(a) = \sum\limits_{s'} P(Result(a) = s' | s)U(s'))$ where Result(a) denotes the outcome state after executing a. There is no global set of actions, the set of actions available to each agent are not identical.

Player1 / Player2 | Action C      | Action D        |
Action A          |  (500,-500)   |  (-1000,1000)   |
Action B          |  (-5,-5)      |  ** (200,20) ** |

Is this a valid approach? All examples of nash equilibriums i can find uses identical action sets for both agents.

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2 Answers

up vote 2 down vote accepted

Short answer: Partly no, and some yes.

First the "yes" bit: There is nothing in the structure of a game that requires each player to be making choices from the same action set. In fact, there is no need for the action sets of the several players even to have the same cardinality. All you need are a pair of payoff functions, one for each player, over the cartesian product of the two action sets.

More generally, let players $i = 1, \ldots, n$ be given, and let $A_i$ denote the action set for player $i$. If $A_i$ contains $m(i)$ possible choices, then we may write $A_i =\{ a_{i1}, \ldots, a_{im(i)} \}$ to itemize player $i$'s set of options. A $n$-player game consists of this list of action sets, together with a set of payoff functions that describe how each player fares as a function of the choices made both by herself and by all other players: let $U_i : A_1 \times A_2 \times A_n \rightarrow \mathbb{R}$ denote agent $i$'s payoff function, where $U_i(a_{1,j_1}, \ldots, a_{n,j_n})$ gives agent $i$'s payoff when the $n$ agents opt for choices $a_{1,j_1}, \ldots, a_{n,j_n}$, respectively.

(Note that you really don't need, for this treatment, to compose the payoff function with a utility function: you can just think of the payoffs as already being described in utility terms, i.e., to be denominated in "utils".)


Now for the "No" part: I believe you are conflating two separate issues: the effect on agent's utility of choices made by the other player, and the effect on utility of random events, a.k.a., "choices made by nature". Nature does not make choices strategically, just randomly. You may of course add a probabilistic dimension to your structure, however. (In that case, you do need to distinguish between payoffs in real terms (e.g., in dollars versus utility terms.)

To summarize: When you ask, "Is this a valid approach?", the answer is, "Not yet. But you could adjust your equation to make it valid: $EU_i(a_i) = \sum_s P(s) \cdot U_i(a \mid s, \mathbb{a_{-i}})$, where $\mathbb{a_{-i}}$ denotes the vector of choices made by all the other players."

To find solutions, e.g., Nash equilibria, you would need to specify utility functions $U_1, \ldots, U_n$ for all the players.

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Set of concepts aimed at decision making in situations of competition and conflict (as well as of cooperation and interdependence) under specified rules. Game theory employs games of strategy (such as chess) but not of chance (such as rolling a dice). A strategic game represents a situation where two or more participants are faced with choices of action, by which each may gain or lose, depending on what others choose to do or not to do. The final outcome of a game, therefore, is determined jointly by the strategies chosen by all participants.


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