**4. Conflict and Agreement**

These negative conclusions, where potential and social welfare rankings disagree, can be overly refined for many purposes. Similar to an election, the interest may be in the winner rather than who is in second, third, or fourth place. Thus, an effective but cruder measure is to determine where potential and social welfare functions have the same top-ranked cell.

All of the conflict in potential and social welfare rankings are strictly caused by *β* values, which suggests identifying those *β* values that allow the same potential and social welfare preferred cell. It is encouraging how answers follow from *α* and *β* comparisons.

**Corollary 2.** *For symmetric* 2 × 2 *games, the following hold for γ* ≥ 0*:*


The content of this corollary serves as a legend for the Figure 1a map; the shaded regions are where both measures have the same top-ranked cell. A simple way to interpret this figure is that for all games to the right of the *β* axis (*α* > 0), P's top-ranked cell is TL, while to the left it is BR. In contrast, above the *α* + *β* = 0 slanted line, the social welfare's top-ranked cell is TL, while below it is BR. Thus, in the unshaded regions, one measure has BR top-ranked, while the other has TL.

This corollary and Figure 1a show that if the *α* value (indicating a preference of the agents for T and L or B and R) is not overly hindered by the externality forces (e.g., if *α* > 0 and *β* > −*α*) then the potential and social welfare functions share the same top ranked cell. But should conflict arise between these two fundamental variables, where the *α* and *β* values favor cells in opposite directions, disagreement arises between the choices of the top ranked P and social welfare cells.

**Proof.** The proof follows directly from Table 8. With *γ* ≥ 0, P's top-ranked cell is TL for *α* > 0 (to the right of the Figure 1a *β* axis), and BR for *α* < 0 (Table 8a). According to Table 8b, the social welfare's top-ranked cell is TL iff *γ* + [*α* + *β*] > *γ* − [*α* + *β*], or iff *α* + *β* > 0; this is the region above the Figure 1a slanted line. The same computation shows that the social welfare's top-ranked cell is BR for the region below the slanted line. This completes the proof.

Everything becomes slightly more complicated with *γ* < 0. The reason is that this *γ* < 0 anti-coordination factor permits BL and TR to become Nash cells. This characteristic is manifested in Figure 1b, where the Figure 1a *α* = 0 and *α* + *β* = 0 lines are separated into strips.

The content of the next corollary is captured by Figure 1b, where the potential and social welfare's top-ranked cells agree in the three shaded regions. To interpret Figure 1b, P's top-ranked cell is BR for all games to the left of the vertical strip (*α* < −|*γ*|), cells BL and TR (or *BL* ∼ *TR*) in the vertical strip (−|*γ*| < *α* < |*γ*|), and cell TL to the right of the vertical strip (*α* > |*γ*|). Similarly, the social welfare's top-ranked cell is BR below the slanted strip, *BL* ∼ *TR* in the slanted strip, and TL above the slanted strip. As *γ* → 0, the width of the strips shrink and Figure 1b merges into Figure 1a.

**Corollary 3.** *For symmetric* 2 × 2 *potential games, the following hold for γ* < 0*:*


**Proof.** The proof follows directly from Table 8. With *γ* < 0, it follows from Table 8a that P's top-ranked cell is TL if it is preferred to either BL or TR, which is if 2*α* + *γ* > −*γ* or if *α* > |*γ*|. This is the Figure 1b region to the right of the *α* = |*γ*| vertical line. Similarly, P's top-ranked cell is BR if *γ* − 2*α* > −*γ*, or if −*α* > |*γ*|; this is the region to the left of the *α* = −|*γ*| vertical line. The same computation shows that in the vertical strip −|*γ*| < *α* < |*γ*|, P's top-ranked cells are the two Nash cells BL and TR, where P's ranking is *BL* ∼ *TR*.

Using the same approach with Table 8b, it follows that the social welfare's top-ranked cell is TL if it has a higher score than BL or TR, which is if −|*γ*| + (*α* + *β*) > |*γ*|, or if *α* + *β* > 2|*γ*|. This is the region above the *α* + *β* = |*γ*| slanted line. Similarly, the social welfare top ranked cells are *BL* ∼ *TR* for −2|*γ*| < (*α* + *β*) < 2|*γ*|, which is the slanted strip (which expanded the Figure 1a slanting line), and BR for (*α* + *β*) < −2|*γ*|, which is the region below the slanting strip. This completes the proof.
