*4.2. Changing α*1, *α*<sup>2</sup>

The general setting for a potential game involves variables *α*1, *α*2, *β*1, *β*2, *γ*. This suggests mimicking what was done with *β* by carrying out an analysis using *α* = *α*<sup>1</sup> = −*α*2; this is simple, but not necessary. The reason is that most needed information about P's top-ranked cell comes from Equation (5). As this expression shows, with appropriate choices of *α*1, *α*2, and *γ*, any cell can be selected to be P's risk-dominant, top-ranked choice, any admissible pair of cells can be Nash cells, where a designated one is risk dominant, and any cell can be selected to be the sole Nash cell. Finding how the behavioral terms (the *β*1, *β*<sup>2</sup> values) can change which cell is the welfare function's top-ranked cell has been reduced to elementary algebra.

All that is needed to obtain answers is to have a generalized form of Tables 8 and 10, which is given in Table 11. The Table 11a values come from the general form of the potential function in Equation (2). The Table 11b values for the welfare function come from a direct computation of its equation

$$w(t\_1, t\_2) = (a\_1 + \beta\_2)t\_1 + (a\_2 + \beta\_1)t\_2 + 2\gamma t\_1 t\_2. \tag{15}$$


**Table 11.** A potential game's P and w values.

In the manner employed above, Theorem 6 is a sample of results. Here, use Equation (5) to determine the potential function structure, and Theorem 6 to compare social welfare (and *β*) values.

**Theorem 6.** *The social welfare function (Equation (15)) is maximized at TL if and only if α*<sup>1</sup> + *β*<sup>2</sup> + *α*<sup>2</sup> + *β*<sup>1</sup> > 0 *and <sup>β</sup><sup>i</sup>* + *<sup>α</sup>*¬*<sup>i</sup>* > −2*γ, where <sup>i</sup>* = 1, 2 *and* ¬*<sup>i</sup> denotes the agent who is not i. The welfare function is maximized at TR if and only if α*<sup>2</sup> + *β*<sup>1</sup> < −2*γ*, *α*<sup>1</sup> + *β*<sup>2</sup> > 2*γ, and α*<sup>1</sup> + *β*<sup>2</sup> > *α*<sup>2</sup> + *β*1*. The welfare function is maximized at BL if and only if α*<sup>1</sup> + *β*<sup>2</sup> < −2*γ*, *α*<sup>2</sup> + *β*<sup>1</sup> > 2*γ, and α*<sup>2</sup> + *β*<sup>1</sup> > *α*<sup>1</sup> + *β*2*. Finally, the welfare function is maximized at BR if and only if <sup>α</sup>*<sup>1</sup> + *<sup>β</sup>*<sup>2</sup> + *<sup>α</sup>*<sup>2</sup> + *<sup>β</sup>*<sup>1</sup> < <sup>0</sup> *and <sup>β</sup><sup>i</sup>* + *<sup>α</sup>*¬*<sup>i</sup>* < <sup>2</sup>*γ, for i* = 1, 2*.*
