**5. Discussion**

Questions regarding the various measures for game theory have proved to be difficult to analyze. There is an excellent reason for this complexity; answers must depend upon the particular payoffs of a game, but it was not clear what portions of each payoff contribute to which aspects of a game. As such, a surprising and welcomed property of the coordinate system is how it identifies how all of a game's entries interact; the coordinates precisely dissect and extract from each payoff entry its contribution to the different attributes of a game.

Support for these comments come from equations such as Equation (9) for the potential function and Equation (10) for the welfare function. The different signs of *t*1*t*<sup>2</sup> and *ti* coefficients, for instance, nicely capture the complexity of a standard approach; it indicates there exists a twisting of certain portions of the payoff entries that are needed to carry out an analysis. The decomposition's separation of which parts of a payoff entry affect Nash structures and which affect payoff and externality factors explain why different measures of a game can have different conclusions. What illustrates the power of doing so is how the discovery and proofs of many subtle results now reduce to elementary algebraic computations.

Our analysis described how and why differences can arise among potential function, payoff dominance, and social welfare conclusions about games. Everything extends more generally. As the decomposition demonstrates, expect methods, learning approaches, and measures that emphasize "best response", comparisons of individual payoff differences, and obtaining Nash equilibria to ignore behavioral terms. Should the objective be to identify properties of Nash structures, doing so simplifies the analysis by eliminating the redundant (for a Nash analysis) *β<sup>j</sup>* variables. However, by not including *β* terms, it must be expected that answers from these approaches about games will differ from those measures that capture the value of payoffs, such as the social welfare function and payoff dominance. They must; the two different classes of measures depend upon different information about the games.

**Author Contributions:** Conceptualization, all authors; investigation, all authors; methodology, all authors; supervision, all authors; validation, all authors; writing-original draft, all authors; writing-review and editing, all authors. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** Our thanks for useful comments from two referees and an editor.

**Conflicts of Interest:** The authors declare no conflict of interest.
