*6. A potential game's potential function is invariant to the pure externality and kernel components.*

To explain certain comments, recall that a potential game has a potential function; if an agent changes a strategy, the change in the agent's payoff equals the change in the potential function. To illustrate, suppose the first agent changes from strategy *t*<sup>1</sup> = 1 to *t*<sup>1</sup> = −1, while agent 2 remains at *t*<sup>2</sup> = 1. According to Table 2, the change in the first agent's payoff is

$$[-\kappa\_1 - \gamma\_1 + \beta\_1 + \kappa\_1] - [\kappa\_1 + \gamma\_1 + \beta\_1 + \kappa\_1] = -2[\kappa\_1 + \gamma\_1]\_{\mathcal{I}}$$

or −2[*α*<sup>1</sup> + *γ*] for potential games. According to Equation (2), the change in the potential is the same value

$$P(-1,1) - P(1,1) = [-a\_1 + a\_2 + (1)(-1)\gamma] - [a\_1 + a\_2 + (1)(1)\gamma] = -2[a\_1 + \gamma].$$

Statement 1 is proved in [JS1]. To prove the second assertion, in (Chap. 2 of [JS2]) it is shown that game <sup>G</sup> is a potential game if and only if it is orthogonal to the 2 <sup>×</sup> 2 matching pennies game <sup>G</sup>*pennies*; 5this orthogonality condition is [G, <sup>G</sup>*pennies*] = 0. A direct computation shows that the individual preference, pure externality, and kernel components always satisfy this condition. The coordinative pressure component satisfies the condition if and only if *γ*<sup>1</sup> = *γ*2.

Statement 4 is a direct computation. Statement 5 is a direct computation showing that changes in an agent's strategy have the same change in the potential function as in the player's payoff. Statement 6 follows, because *P*(*t*1, *t*2) (Equation (2)) does not have *β* or *κ* values. A proof of the remaining statement 3 is in [JS2]. For intuition, the players in a coordination game coordinate their strategies, which is the defining feature of the coordinative pressure component. Thus, for G to be a coordination game, the *γ* values must dominate the game's Nash structure.
