**3. Disagreement in Potential Games**

These coordinates lead to explanations why different behavioral measures can differ about which is the "better" outcome for certain games. This discussion is motivated by innovation diffusion, which is typically modeled by using coordination games with two equilibria, so a key step is to identify the preferred equilibria. A common choice is the risk-dominant pure Nash equilibria. In part, this is because these equilibria have been connected to the long-run behavior of dynamics, such as log-linear learning [18]. Because a coordination game is a potential game, the potential function's global maximum is the risk-dominant equilibrium (Theorem 2).

However, as developed next, there are many games where a maximizing strategy for the potential function differs from the profile that maximizes social welfare. This difference is what allows agents to do "better" by using a profile other than the one leading to the risk-dominant equilibrium. Young [5,6] creates such an example using the utilitarian measure of social welfare, which sums all of the payoffs in a given strategy profile, and Newton and Sercombe [7] discuss similar ideas. A first concern is whether their examples are sufficiently isolated that they can be ignored, or whether they are so prevalent that they must be taken seriously. As we show, the second holds.

An explanation for what causes these conflicting conclusions emerges from the Table 2 decomposition and Theorem 1. A way to illustrate these results is to create any number of new, more complex examples. To do so, start with the fact (Theorem 2.5 in [JS2]) that with 2 × 2 games and two Nash equilibria, a Nash cell is risk dominant over the other Nash cell if and only if the product of the *η* values (see Table 4) of the first is larger than the product of the *η* values of the second. (To handle

<sup>5</sup> This makes sense; "matching pennies" is antithetical (orthogonal) to the cooperative spirit of potential games. The space of matching pennies is the harmonic component discussed in [14].

some of our needs, this result is refined in Theorem 2.) Of significance is that, although the *β* terms obviously affect payoff values, they play no role in this risk-adverse dominance analysis.

To illustrate, let *γ*<sup>1</sup> = *γ*<sup>2</sup> = 4 (Theorems 1-2; to have a potential game) and *α*<sup>1</sup> = *α*<sup>2</sup> = 1 (Theorem 1-3; to have a coordination game). This defines the Table 6a game where the TL Nash cell (with (*η*1,1)(*η*1,2) = 25) is risk dominant over the BR Nash cell (with (−*η*1,1)(−*η*1,2) = 9). There remain simple ways to modify this game that make the payoffs of any desired cell, say BR, more attractive than the risk dominant TL. All that is needed is to select *β<sup>j</sup>* values that increase the payoffs for the appropriate Table 3c cell; for the BR choice, this requires choosing negative *β*1, *β*<sup>2</sup> values (Table 3c). Although these values never affect the risk-dominant analysis, they enhance each player's BR payoff while reducing their TL payoffs. The *β*<sup>1</sup> = *β*<sup>2</sup> = −3 choices define the Table 6b game where each player receives a significantly larger payoff from BR than from the risk dominant TL! In both games, the Nash and risk dominance structures remain unchanged.



These coordinates make it possible to easily create a two-dimensional family of games with such properties. To do so, add Table 3c to the Table 6a game, and then select appropriate *β*1, *β*<sup>2</sup> values to emphasize the payoffs of different cells. For instance, using Young's welfare measure (the sum of a cell's payoffs), no matter which cell is selected, suitable *β* values exist to make that cell preferred to TL. It follows from the Table 3c structure, for instance, that a way to enhance the TR payoffs is to use *β*<sup>1</sup> < 0 and *β*<sup>2</sup> > 0 choices. Adding these values to the Table 6a game defines the TR cell values of −3 + |*β*1| and −5 + *β*<sup>2</sup> while the TL values are 5 − |*β*1| and 5 + *β*2. Thus, the sum of TR cell values dominates the sum of TL values if and only if

$$(-\mathfrak{Z} + |\mathfrak{k}\_1|) + (-\mathfrak{z} + \mathfrak{z}\_2) > (\mathfrak{z} - |\mathfrak{k}\_1|) + (\mathfrak{z} + \mathfrak{z}\_2), \text{ or } \text{iff } -\mathfrak{z}\_1 = |\mathfrak{z}\_1| > \mathfrak{g}.\tag{3}$$

$$
\begin{array}{c|c|c|c|c|c|c|c|c}
\hline
10 & 11 & 2 & 7 \\
\hline
8 & 4 & 8 & 6 \\
\hline
\end{array}
\quad
\begin{array}{c|c|c|c|c|c}
\hline
1 & 2 & -3 & -2 \\
\hline
\hline
\end{array}
\quad
\begin{array}{c|c|c|c|c}
\hline
2 & 2 & -2 & 2 \\
\hline
2 & -2 & -2 & -2 \\
\hline
\end{array}
\quad
\begin{array}{c|c|c|c}
\hline
7 & 7 & 7 & 7 \\
\hline
7 & 7 & 7 & 7 \\
\hline
\end{array}
\quad
\begin{array}{c|c|c|c}
\hline
8 & & & & & & \\
\hline
\end{array}
$$

Conflict among 50-50, payoff, and risk dominance

The coordinates also make it possible to compare other measures by mimicking the above approach. Games with payoff dominant strategies that differ from the risk adverse ones, for example, require appropriate *β* values. To explain, if BR is risk dominant, then, as in the Equation (4) game (from Section 2.6.4 in [JS2]), the product of its *<sup>η</sup>* values from BR in <sup>G</sup>*<sup>N</sup>* (first bimatrix on the right) is larger than the product of the TL <sup>G</sup>*<sup>N</sup>* values. This product condition ensures that the only way the payoff and risk dominant cells can differ is by introducing TL *β* components; this is illustrated with *β*<sup>1</sup> = *β*<sup>2</sup> = 2 in the second bimatrix in Equation (4). More generally and using just elementary algebra as in Equation (3), the regions (in the space of games) where the two concepts differ now can finally be determined.

For another measure, consider the 50–50 choice. This is where, absent any information about an opponent, it seems reasonable to assume there is a 50–50 chance the opponent will select one Nash cell over the other. This assumption suggests using an expected value analysis to identify which strategy a player should select. To discover what coordinate information this measure uses, if TL and BR are the two Nash cells, then for Row and this 50–50 assumption, the expected return from playing T is <sup>1</sup> <sup>2</sup> [|*η*1,1|−|*η*2,1|] + <sup>1</sup> <sup>2</sup> [*β*<sup>1</sup> <sup>−</sup> *<sup>β</sup>*1] + <sup>1</sup> <sup>2</sup> [*κ*<sup>1</sup> <sup>+</sup> *<sup>κ</sup>*1] = <sup>1</sup> <sup>2</sup> [|*η*1,1|−|*η*2,1|] + *κ*1. Similarly, the expected value of playing B is <sup>1</sup> <sup>2</sup> [−|*η*1,1<sup>|</sup> <sup>+</sup> <sup>|</sup>*η*2,1|] + <sup>1</sup> <sup>2</sup> [*β*<sup>1</sup> <sup>−</sup> *<sup>β</sup>*1] + <sup>1</sup> <sup>2</sup> [*κ*<sup>1</sup> <sup>+</sup> *<sup>κ</sup>*1] = <sup>1</sup> <sup>2</sup> [−|*η*1,1| + |*η*2,1|] + *κ*1. Consequently, an agent's larger *η* value completely determines the 50-50 choice. However, if the risk adverse cell of <sup>G</sup>*<sup>N</sup>* is not also <sup>G</sup>*<sup>N</sup>* payoff dominant, as true with the first bimatrix on the right in Equation (4), and if both players adopt the 50-50 measure, they will select a non-Nash outcome. Indeed, in Equation (4), BR is risk dominant, TL is payoff dominant, and BL, which is not a Nash cell (and Pareto inferior to both Nash cells), is the 50–50 choice. Again, elementary algebra of the Equation (3) form identifies the large region of games where this behavior can arise. (By using appropriate *β* values, it is easy to create 50–50 outcomes that are disastrous.)
