*4.1. Changing β*1, *β*<sup>2</sup>

The cause of conflict between potential and social welfare rankings now is clear; the first ignores *β* values while the second depends upon them. However, a feature of the previous section is that if TL or BR ended up being the social welfare top-ranked cell, it also was the payoff dominant cell. This property is a direct consequence of the symmetric game structure where the behavioral terms (Table 6b) always favored one of these two cells.

To recognize the many other possibilities, change the *β* structure from *β* = *β*<sup>1</sup> = *β*<sup>2</sup> to *β* = *β*<sup>1</sup> = −*β*2. This affects Table 6b by changing the sign of player 2's entries, so the game's externality features now emphasize either BL or TR. The social welfare function becomes

$$w = 2\gamma t\_1 t\_2 + a(t\_1 + t\_2) + \beta(t\_2 - t\_1) \tag{13}$$

A reason for considering this case is that any (*β*1, *β*2) can be uniquely expressed as

$$h\_1(\beta\_1, \beta\_2) = (b\_1, b\_1) + (b\_2, -b\_2) \text{ where } b\_1 = \frac{\beta\_1 + \beta\_2}{2}, b\_2 = \frac{\beta\_1 - \beta\_2}{2}. \tag{14}$$

Thus, combining Figure 1 with the impact of (*β*, −*β*) captures the general complexity.

Because the decomposition isolates appropriate variables for each measure, Table 8 is the main tool to derive the Figure 3 results. In this new setting, Table 8 is replaced with Table 10, where part a restates the potential function values for each cell and b gives half of the social welfare function's values.

**Table 10.** A quasi-symmetric potential game's P and *<sup>w</sup>* <sup>2</sup> values, with (*β*, −*β*).

**Figure 3.** More conflict and agreement with *α*, *β* values.

As with Figure 1a, if *γ* ≥ 0, then P's top-ranked cell is TL for *α* > 0 and BR for *α* < 0. The same holds for Figure 3a. According to Table 10b, the social welfare function ranks TL over BL if *γ* + *α* > −*γ* + *β*, or if *β* < 2*γ* + *α*. In Fig. 3a, this is the region below the *β* = *α* + 2*γ* slanted line. Similarly, the social welfare function ranks TL over TR if *α* + 2*γ* > −*β*, or *β* > −*α* − 2*γ*, which is the Figure 3a region above the *β* = −*α* − 2*γ* line. P's top ranked cell is BR if *α* < 0, which is the Figure 3a region to the left of the *β* axis. A similar analysis shows that the social welfare function ranks BR above TR if *β* > *α* − 2*γ*, or the region above the Figure 3a *β* = *α* − 2*γ* line. Finally, this function ranks BR above BL if *β* < −*α* + 2*γ*, which is the region below the *β* = −*α* + 2*γ* line.

Consequently, agreement between the two measure's top-ranked cell is in Figure 3a shaded regions, where BR is the common choice to the left of the *β* axis and TL is for the region to the left. Conflict occurs in the unshaded region where BL is the welfare's top-cell on the top and TR is for the region below. Again, these outcomes capture the *β* structure where, now, positive *β* values emphasize the BL cell and negative values enhance the TR entries. Contrary to Figure 1a, the upper unshaded region now is where BL, rather than TL, is the welfare function's top-ranked cell.

Consistent with Figure 1b, the situation becomes more complicated with the anti-coordination *γ* < 0. Again, P's top ranked cell is BR to the left of the vertical strip, *BL* ∼ *TR* in the strip, and TR to the right of the strip. Similar algebraic comparisons show that the social welfare's top-ranked cell is BL in the upper unshaded region of Figure 3b, including the portion of the *β* axis. Similarly, TR is the welfare function's top-ranked cell in the lower unshaded region. Consequently, the two large shaded regions are where agreement occurs (going from left to right, BR, TL).

With the Table 6a Nash structure, outcomes for all possible (*β*1, *β*2) values can be computed from Figures 1 and 3. To illustrate with *γ* = −0.5, *α* = −2, *β*<sup>1</sup> = 11, *β*<sup>2</sup> = 1, it follows from |*γ*| < −*α* that BR is P's top-ranked cell. The information for the welfare function comes from Equation (14) where *b*<sup>1</sup> = 6, *b*<sup>2</sup> = 5. To find half of the welfare functions value, substitute *α* = 2*γ* = −0.5, *β* = 6 in Table 8b, substitute *α* = 2*γ* = −0.5, *β* = 6 in Equation (13b), and add the values. It already follows from plotting these values in Figures 1b and 3b that the outcome is either TL or BL.
