*3.1. The Potential and Welfare Functions*

Moving beyond examples, these coordinates can fully identify those games for which different measures agree or disagree, which is one of the objectives of this paper. The importance of this analysis is that it underscores our earlier comment that this conflict between the conclusions of measures is a fundamental concern that is suffered by a surprisingly large percentage of all 2 × 2 games.

Our outcomes are described using maps of the space of 2 × 2 games. The maps show where the potential and social welfare functions (e.g., the ones used by Young [5,6] and Newton and Sercombe [7]) agree, or disagree, on which is the "better" choice of two equilibria. Not only do these diagrams demonstrate the preponderance of this conflict, but they identify which behavior a specific game will experience. As an illustration, the dark regions of Figure 1 single out those potential games (so *γ*<sup>1</sup> = *γ*2), where *α*<sup>1</sup> = *α*<sup>2</sup> = *α* and *β*<sup>1</sup> = *β*<sup>2</sup> = *β*, which are without conflict; for games in the unshaded regions, different measures support different outcomes.

**Figure 1.** Conflict and agreement with *α*, *β* values.

Thanks to the coordinate system for games, the game theoretic analysis is surprisingly simple; it merely uses a slightly more abstract version of the Equation (3) analysis. To illustrate with the above 50-50 discussion, if the Nash cells are TL and BR, then *η*1,1, −*η*2,1, *η*1,2, −*η*2,2 are all positive (if the Nash cells are BL and TR, all of these entries are negative). Consequently, the two surfaces *η*1,1 + *η*2,1 = 0 and *η*1,2 + *η*2,2 = 0 separate which one of an agent's *η* values is larger. Even though the discussion applies to the four dimensional space of *η* values, one can envision the huge wedges these surfaces define where the *η* values force the 50–50 approach to select a non-Nash outcome.

A similar approach applies to all of the maps derived here. In Figure 1a, the dividing surface separating which Nash cell is selected by potential function outcomes is *α* = 0; if *α* > 0, the game's top Nash choice for the potential function is TL; if *α* < 0, the top Nash choice is BR. However, the social welfare conclusion is influenced by *β* values, so it will turn out that the separating line between a social welfare function selecting TL or BR is the *α* + *β* = 0 line. Above this line, TR is the preferred Nash choice; below it is BR. Given this legend, Figure 1a demonstrates those games for which the different measures agree or disagree about the top choices, and the magnitude of the problem. Stated simply, regions that emphasize behavioral terms place emphasis on payoff and social welfare dominant measures; regions that emphasize Nash strategic terms emphasize risk dominant measures.

Stated differently, difficulties in what follows do not reflect the game theory; the coordinate system handles all of these problems. Instead, all of the complications (there are some) reflect the geometric intricacy of the seven-dimensional space of 2 × 2 potential games. Consequently, readers that are interested in applying this material to specific games should emphasize the maps and their legends (given in the associated theorems). Readers that are interested in the geometry of the space of potential functions will find the following technical analysis of value. However, first, a fundamental conclusion about potential games is derived.
