*3.3. Symmetric Games*

Without question, it is difficult to understand the structures of a four-dimensional object, leave alone the seven-dimensions of the space of 2 × 2 potential games. Thus, to underscore the ideas, we start with the simpler (but important) symmetric games (all symmetric games are potential games); doing so reduces the dimension of the space of games from seven to four (with the kernel). This is where *α*<sup>1</sup> = *α*<sup>2</sup> = *α*, *β*<sup>1</sup> = *β*<sup>2</sup> = *β*, *γ*<sup>1</sup> = *γ*<sup>2</sup> = *γ*, and *κ*<sup>1</sup> = *κ*<sup>2</sup> = *κ*. Ignoring the kernel term, the coordinates are given in Table 7.


**Table 7.** Decomposition for a symmetric potential game.

To interpret Table 7, the Nash entries combine terms, where each agent prefers a particular strategy independent of what the other agent does (if *α* > 0, each agent prefers to play 1; if *α* < 0 each agent prefers −1) with terms that impose coordination pressures. That is, if *γ* > 0, the game's Nash structure inflicts a pressure on both agents to coordinate; if *γ* < 0, the game's joint pressure is for anti-coordination. With symmetric games, the payoffs for both agents agree at TL and at BR, so if one of these cells is social welfare top-ranked (the sum of the entries is larger than the sum of entries of any other cell), the cell also has the stronger property of being payoff dominant.

All 2 × 2 symmetric games can be expressed with these four *α*, *β*, *γ*, *κ* variables (Equation (1)). Ignoring *κ*, the remaining variables define a three-dimensional system. To tie all of this in with commonly used symmetric games, if BR is the sole Nash point, then the above defines a Prisoner's Dilemma game if −*α* + *γ* > 0 and *α* + *γ* + *β* > −*α* + *γ* − *β*, the second of which reduces to *β* + *α* > 0. Similarly, the Hawk–Dove game with Nash points at BL and TR, and "hawk" strategy as +1, has −*α* − *γ* > 0 and *α* − *γ* > 0 for the Nash points, and *β* < 0 to enhance the BR payoffs, so *γ* < 0, *α* < −|*γ*|, and *β* < 0. A coordination game simply has *γ* > |*α*| (these inequalities allow the different games to easily be located in Figure 1 and elsewhere). As stated, because *γ*<sup>1</sup> = *γ*2, it follows (Theorem 1-2) that all 2 × 2 symmetric games are potential games with the potential function (Equation (2))

$$P(t\_1, t\_2) = a(t\_1 + t\_2) + \gamma t\_1 t\_2 \tag{9}$$

A computation (Table 7) proves that the social welfare function (the sum of a cell's entries) is

$$w(t\_1, t\_2) = (\alpha + \beta)(t\_1 + t\_2) + 2\gamma t\_1 t\_2. \tag{10}$$

Our goal is to identify those games for which the potential and social functions agree, or disagree, on the ordering of strategies. Here, the following results are useful.

**Theorem 3.** *A* 2 × 2 *symmetric potential game* G *satisfies*

$$w(t\_1, t\_2) = 2P(t\_1, t\_2) + (\beta - a)(t\_1 + t\_2). \tag{11}$$

*If α* = *β, the potential function and the welfare function rankings of* G*'s four cells agree.*

*Both the potential and welfare functions are indifferent about the ranking of the BL and TR cells, denoted as BL* ∼ *TR. If one of these cells is a Nash cell for* G*, then so is the other one, but neither is risk dominant.*

Equation (11) explicitly demonstrates that *β* values—the game's behavioral or externality values—are solely responsible for all of the differences between how the potential and social welfare functions rank G's cells.

**Proof.** Adding and subtracting *α*(*t*<sup>1</sup> + *t*2) to Equation (10) leads to Equation (11). Thus, if *α* = *β*, then <sup>1</sup> <sup>2</sup>*w*(*t*1, *t*2) = *P*(*t*1, *t*2), so both functions have the same ranking of G's four cells.

The BL and TR cells correspond, respectively, to (*t*<sup>1</sup> = −1, *t*<sup>2</sup> = 1) and (*t*<sup>1</sup> = 1, *t*<sup>2</sup> = −1), where *t*<sup>1</sup> + *t*<sup>2</sup> = 0. Thus, (Equation (11)), the <sup>1</sup> <sup>2</sup>*w*, and *P* values for each of these cells is *γ*. As both measures have the same value for each cell, both have a tie ranking for the two cells denoted by *BL* ∼ *TR*.

The Nash entries of BL and TR are the same but in different order (Table 6a). Thus, if both entries of one of these cells are positive, then so are both entries of the other. This means that both are Nash cells (Theorem 1-1). The risk dominant assertion follows because inequality a of Equation (5) is not satisfied; it equals zero. Equivalently, *P* does not have a unique global maximum in this case.
