*3.2. A Basic "Risk Dominant" Theorem*

The coordinates explicitly display a tension between what individuals can achieve on their own (Nash behavior) and with cooperative forces. With a focus on individualistic forces, the potential function is useful because its local maxima are pure Nash equilibria. Even more, as known, the potential's global maximum is the risk dominant equilibrium. This fact is re-derived for 2 × 2 potential games in a manner that now highlights the roles of a potential game's coordinates.

**Theorem 2.** *For a* 2 × 2 *potential game* G*, its potential function has a global maximum at the strategy profile* (*t* , *t* ) *if and only if* (*t* , *t* ) *is* G*'s risk-dominant Nash equilibrium. With two Nash equilibria where one is risk dominant,* (*t* , *t* ) *is the risk dominant strategy if and only if the following inequalities hold*

$$\mathbf{a}. \left(\mathfrak{a}\_1 t' + \mathfrak{a}\_2 t''\right) > 0, \quad \mathbf{b}. \left|\gamma\right| > \left|\mathfrak{a}\_1\right|, \left|\mathfrak{a}\_2\right|, \quad \mathbf{c}. \left|t' t'' \gamma > 0. \tag{5}$$

As shown in the proof, inequality c identifies the Nash cells; e.g., if *γ* < 0, then *t t* = −1, so the Nash cells are BL and TR. With two Nash cells, the inequality a identifies which one is a global maximum of the potential function. Similarly, inequality b requires *γ* to have a sufficiently large value to create two Nash cells. Of importance, Equation (5) does not include *β* values!

To illustrate these inequalities, let *α*<sup>1</sup> = 1, *α*<sup>2</sup> = −2, and *γ* = 3. By satisfying Equation (5b), there are two Nash cells. According to Equation (5c), *t t* = 1, so *t* = *t* , which positions the Nash cells at TL and BR. From Equation (5a), *t* − 2*t* > 0, or *t* < 0, so the risk dominant strategy is the *t* = *t* = −1 BR cell. Conversely, to create an example where a desired cell, say TR, is risk dominant, the *t* = 1, *t* = −1 values require (Equation (5c)) *γ* < 0 and (Equation (5a)) *α*<sup>1</sup> > *α*2. Finally, select *γ* < 0 that satisfies Equation (5b); e.g., *α*<sup>1</sup> = 1, *α*<sup>2</sup> = −1 and *γ* = −2 suffice.

The Equation (5) inequalities lead to the following conclusion.

**Corollary 1.** *If a* 2 × 2 *potential game* G *has two pure Nash equilibria where one is risk dominant, then* G *is a coordination game. If γ* > 0*, the Nash cells are at TL and BR, where* G *is a coordination game. If γ* < 0*, the Nash cells are at BL and TR, where* <sup>G</sup> *is an anti-coordination game.*<sup>6</sup>

**Proof of Corollary 1.** According to Theorem 2, the Corollary 1 hypothesis ensures that Equation (5) hold. With the b inequality, it follows from Theorem 1-3 that G is a coordination game. In a 2 × 2 game, pure Nash cells are diagonally opposite. If *γ* > 0, it follows from Equation (5), c that the Nash strategies satisfy *t t* = 1, so the Nash cells are at TL (for *t* = *t* = 1) and BR (for *t* = *t* = −1), and that this is a coordination game. Similarly, if *γ* < 0, then *t t* = −1, so the Nash cells are at BL (for *t* = −1, *t* = 1) and TR (for *t* = 1, *t* = −1) to define an anti-coordination game.

<sup>6</sup> As pointed out by a referee, the case where *γ* < 0 appears to be related to the notion of self-defeating externalities, making the potential game in this case a stable game, as defined in [19].

**Proof of Theorem 2.** In a non-degenerate case (i.e., *P*(*t*1, *t*2) is not a constant function), *P* has a maximum, so there exists at least one pure Nash cell. If a game has a unique pure strategy Nash equilibrium, then, by default, it is risk-dominant and *P*'s unique maximum.

Assume there are two Nash cells; properties that the potential, *P*, must satisfy at a global maximum are derived. Pure Nash equilibria must be diametrically opposite in a normal form representation, so if G has two pure strategy Nash equilibria where one is (*t* , *t* ), then the other one is at (−*t* , −*t* ). Consequently, if *P* has a global maximum at (*t* , *t* ),, then *P* has a local maximum at (−*t* , −*t* ), so *P*(*t* , *t* ) > *P*(−*t* , −*t* ). According to Equation (2), this inequality holds iff [*α*1*t* + *α*2*t* + *γ*(*t* )(*t* )] − [*α*1(−*t* ) + *α*2(−*t* ) + *γ*(−*t* )(−*t* )] = 2[*α*1*t* + *α*2*t* ] > 0, which is inequality Equation (5)a.

The local maximum structure of *P*(−*t* , −*t* ) requires that *P*(−*t* , −*t* ) > *P*(*t* , −*t* ) and *P*(−*t* , −*t* ) > *P*(*t* , −*t* ). Again, according to Equation (2), the first inequality is true if

$$
\alpha\_1(-t') + \alpha\_2(-t'') + \gamma(-t')(-t'') > \alpha\_1 t' + \alpha\_2(-t'') + \gamma(t')(-t''),
$$

or *γt t* > *α*1*t* . Similarly, the second inequality is true iff *γt t* > *α*2*t* . Thus, for a potential game with two Nash cells, *P* has a global maximum at (*t* , *t* ) iff

$$
\alpha \left[ a\_1 t' + a\_2 t'' \right] > 0, \quad \gamma t' t'' > a\_1 t', \quad \gamma t' t'' > a\_2 t'', \quad \gamma t' t'' > 0. \tag{6}
$$

The last inequality follows from the first one, which requires at least one of *α*1*t* , *α*2*t* to be positive. Thus, the *γt t* > 0 inequality follows from either the second or third inequality.

All that is needed to establish the equivalence of the Equation (6) inequalities and those of Equation (5) is that Equation (5b) is equivalent to the two middle inequalities of Equation (6). Equation (5b) implies the two middle inequalities of Equation (6) is immediate. In the opposite direction, the first Equation (6) inequality requires at least one of *α*1*t* or *α*2*t* to be positive. If it is *αjt*, then because |*γ*| = *γt t* , this positive term requires the appropriate middle inequality of Equation (6) to be |*γ*| > |*αj*|. If it holds for both terms, the proof is completed. If it holds for only one term, say *α*1*t* > 0, but *α*2*t* < 0, then the first Equation (6) inequality requires that |*α*1| > |*α*2|, which completes the proof.

The second step requires showing that (*t* , *t* ) is a risk-dominant Nash equilibrium if Equation (5) holds. According to Harsanyi and Selten (1988), (*t* , *t* ) is a game's risk-dominant Nash equilibrium if

$$\left( \left( P(-t',t'') - P(t',t'') \right) \left( P(t',-t'') - P(t',t'') \right) \right) > \left( P(t',-t'') - P(-t',-t'') \right) \left( P(-t',t'') - P(-t',-t'') \right), \tag{7}$$

which is (−2*α*1*t* − 2*γt t* )(−2*α*2*t* − 2*γt t* ) > (2*α*1*t* − 2*γt t* )(2*α*2*t* − 2*γt t* ). This inequality reduces to

$$
\gamma t' t'' (a\_1 t' + a\_2 t'') \, > 0. \tag{8}
$$

If *t t* = 1, the Nash cells are at TL and BR, so both entries of these two Table 4 cells must be positive (Theorem 1-1). For the TL cell, this means that *γ* > −*α*1, −*α*2,, while for the BR cell it requires *γ* > *α*1, *α*2. Consequently, *γ* > |*α*1|, |*α*2| and *γ* > 0: inequalities b and c of Equation (5) are satisfied. That inequality a of Equation (5) holds follows from *γt t* > 0 and Equation (8).

Similarly, if *t t* = −1, then BL and TR are the Nash cells. Again, each entry of each of these Table 4 cells must be positive: from BL, we have that −*α*1, *α*<sup>2</sup> > *γ*,, while from TR we have that *α*1, −*α*<sup>2</sup> > *γ*. Consequently, *γ* < 0, *γt t* > 0 and |*γ*| > |*α*1|, |*α*2|; these are inequalities b and c of Equation (5). That inequality a holds again follows from Equation (8).

A consequence of Theorem 2 is that the potential function can serve as a comparison measure of Nash outcomes. Other natural measures reflect the overall well-being of all agents, such as the utilitarian social welfare function that sums each strategy profile's payoffs. To obtain precise conclusions, our results use this social welfare function. However, as indicated later, everything extends to several other measures.
