# How are payoff values in game theories calculated / estimated?

In game theories such as Nash Equilibrium, exactly how are each of the players' payoff values for each of their potential strategies created? In the common 2x2 matrices I've seen in academic papers, the various payoff values seem to just "appear" in the matrices without an explanation of how they were derived / calculated / estimated, whether or not those values have (or need) a confidence level associated with them, etc. Is it possible, for example to have a range of payoff values rather than a single number for a given player/strategy choice?

## 1 Answer

Though I'd ask for a bit more clarification which papers you are referring to for a more specific answer, overall the answer depends a bit on what the purpose of the paper/game and payoffs being proposed is.

Papers like Selten's 1978 Chain Store Paradox use payoffs that are important in their relative magnitude (not their absolute value) as a means of creating a perceived disconnect between Industrial Organization and Game Theory. Similarly, in the classic Prisoner's Dilemma game, the actual values for "fink/tattle" and "mum/don't" aren't as important as the fact that the dominant strategy is for both players to tattle. That payoff can be (0,0) or (-10,-10), so long as that payoff makes tattling the dominant strategy, it works for the game.

One of the larger reasons why "exact" values are given instead of confidence intervals is because of how strategies are calculated. Indifference is the key to defining mixed strategies, which is a much simpler feat if exact values are given.

That being said, there is a variety of games of imperfect information and, perhaps most important for your question, games with imperfect monitoring which do exactly as you suggest- allow payoffs to be dependent on some random variable not observed by one (or more) of the players.

I'm sorry if that wasn't very direct, or didn't really fully answer your question- please let me know if I can improve the answer. Confidence intervals can be difficult to use because they are so dependent on the statistical results as opposed to the theoretical model. However ranges of values are a relatively common phenomena in certain types of game theory models (for example, see auction theory). Usually it is assumed that players know the relative likelihoods of different payoffs, but even that can be relaxed in certain circumstances.

• Thanks for the clarification. I'll do more research on auction theory and imperfect monitoring games. Isn't it weird though that in Prisoner's Dilemma you need to choose payoffs such that they cause a certain outcome--i.e. making tattling the dominant strategy? Seems to this layman that that's fudging the numbers to make the theory work. – charlie K3 May 30 '17 at 14:13