3.3.1. A Map of Games and Symmetries

A portion of a map that describes the structure of all 2 × 2 games (Chapter 3, [JS2]) is expanded here to concentrate on the symmetric games. While variables *α*, *γ*, *β* require a three-dimensional space, the potential function does not depend on *β*, so Figure 2a highlights the *α*-*γ* plane. Treat the positive *β* axis as coming out orthogonal to the page.

**Figure 2.** The *α*, *β*, *γ* structures.

Changes in the potential game and *P* (Equation (9)) depend on *α*, *γ*, *γ* − *α*, and *γ* + *α* values, which suggests dividing the *α*-*γ* plane into sectors where these terms have different signs. That is, divide the plane into eight regions (Figure 2a) with the lines *α* = 0, *γ* = 0, *α* = *γ*, and −*α* = *γ*. The first two lines represent changes in a game's structure by varying the *α* and *γ* signs. For instance, reversing the sign of *α* changes which strategy the agents prefer; swapping the *γ* sign exchanges a game's coordination, anti-coordination features. The other two lines are where certain payoffs (Table 6a) change sign, which affects the game's Nash structure. The labelling of the regions follows:

$$\begin{array}{ccccccccc}(1) & \gamma > a > 0 & \text{(2)} & a > \gamma > 0 & \text{(3)} & a > -\gamma > 0 & \text{(4)} & -\gamma > a > 0\\(5) & \gamma > -a > 0 & \text{(6)} & -a > \gamma > 0 & \text{(7)} & -a > -\gamma > 0 & \text{(8)} & -\gamma > -a > 0\end{array}$$

A natural symmetry simplifies the analysis. In Table 6, interchanging each matrix's rows and then columns creates an equivalent game, where the *t* , *t* cell in the original becomes the −*t* , −*t* cell in the image. This equivalent game is identified in Figure 2a with the mapping

$$F(a, \gamma, \beta) = (-a, \gamma, -\beta). \tag{12}$$

Geometrically, *F* flips a point on the right (a symmetric potential game) about the *γ* axis to a point on the left (which corresponds to the described changing of rows and columns of the original game); e.g., in Figure 2a, the bullet in region 2 is flipped to the bullet in region 6. Similarly, the original *β* value is flipped to the other side of the *α*-*γ* plane. Consequently, anything stated about a (*t* , *t* ) strategy or cell for a game in region *k* holds for a (−*t* , −*t* ) strategy or cell of the corresponding game in region (*k* + 4). Thanks to this symmetry, by determining the *P* ranking of region *k* to the right of the *γ* axis, the *P* ranking of region *k* + 4 is also known.

The following theorem describes each region's *P* ranking. The reason the decomposition structure simplifies the analysis is that all of the comments about Nash cells follow directly from Table 6a. If, for instance, *γ* > *α* > 0 (region 1), then only cells TL and BR have all positive entries (Table 6a), so they are the only Nash cells. Similarly, if −*γ* > *α* > 0 (region 4), only cells BL and TR have all positive entries, so they are the Nash cells (Theorem 1-1). Each cell's *P* value is specified in matrix Table 8a, so the P ranking of the cells follows immediately. Region 2, for instance, has *α* > *γ* > 0, so TL is P's top-ranked cell. Whether BL is P-ranked over BR holds (Table 8a) iff −*γ* > *γ* − 2*α*, or *α* > *γ*, which is the case. This leads to P's ranking of *TL* (*BL* ∼ *TR*) *BR* for all region 2 games (here, *A B* means *<sup>A</sup>* is ranked over *<sup>B</sup>* and *<sup>A</sup>* <sup>∼</sup> *<sup>B</sup>* means they are tied). Each cell's <sup>1</sup> <sup>2</sup>*w*'s value (half the social welfare function), which comes from Equation (11), is given in Table 8b.

**Table 8.** A symmetric potential game's P and *<sup>w</sup>* <sup>2</sup> values.


**Theorem 4.** *The following hold for a* 2 × 2 *symmetric potential game.*


The content of this theorem is displayed in Figure 2b. Notice, with *α* > 0, the potential function has BR bottom ranked unless BR is a Nash cell. This makes sense; *α* > 0 means (Table 3a) that both agents prefer a "+1" strategy, so they prefer T and L. In fact, with *α* > 0, the only way TL loses its top-ranked P status is with a sufficiently strong negative *γ* value (region 4 of Figure 2a). This also makes sense; a negative *γ* value (Table 3b) captures the game's anti-coordination flavor, which, if strong enough, can crown BL and TR as Nash cells.

Similar comments hold for *α* < 0; this is because TL and BR reverse roles in the P rankings (properties of F (Equation (12)). Thus, the P ranking of region 1 is *TL BR* (*BL* ∼ *TR*), so the P ranking of region 5 is *BR TL* (*BL* ∼ *TR*). Accordingly, for *α* < 0, P always bottom-ranks TL unless TL is a Nash cell, which reflects that *α* < 0 is where the players have a preference for B and R.

The next theorem describes where the potential and social welfare function rankings agree or disagree. As its proof relies on Table 8b values, it is carried out in the same manner as for Theorem 4. Namely, to determine whether TL is ranked above BL or TR, it must be determined (Table 8b) whether *γ* + (*α* + *β*) > −*γ*, or whether 2*γ* > *α* + *β* > 0. Next, according to (Table 8b), the social welfare function ranks BR above BL (or TR) iff *γ* − (*α* + *β*) > −*γ*, or 2*γ* > *α* + *β* > 0. Thus, if 2*γ* > *α* + *β* > 0, the social welfare ranking is *TL BR* (*BL* ∼ *TR*).

**Theorem 5.** *For a* 2 × 2 *symmetric potential game with a coordinative flavor of γ* > 0*, the social welfare function (Equation (10)) ranks the cells in the following manner:*


*For games with an anti-coordinative flavor of γ* < 0*, the social welfare rankings are*


As with Theorem 4, this theorem ignores certain equalities, such as *α* + *β* = 2*γ* > 0, but the social welfare ranking is the obvious choice. This equality captures the transition between parts 1 and 2, so its associated ranking is *TL BR* ∼ (*BL* ∼ *TR*).

A message of these theorems is to anticipate strong differences between the potential and social welfare rankings. Each game in region 1 of Figure 2a, for instance, has the unique P ranking of *TL BR* (*BL* ∼ *TR*), so TL is P's "best" cell. In contrast, each game in each region has four different social welfare rankings<sup>7</sup> where most of them involve ranking conflicts! In region 1 of Figure 2a, for instance, an admissible social welfare ranking (Theorem 5-4) is *BR* (*BL* ∼ *TR*) *TL*, where the payoff dominant BR is treated as being significantly better than P's top choice of TL.

The coordinates explicitly identify why these differences arise: The potential function ignores *β*, while the *β* values contribute to the social welfare rankings. By influencing a game's payoffs and identifying (positive or negative) externalities that players can impose on each other, the *β* values constitute important information about the game. To illustrate, Table 5 has two different symmetric games; they differ only in that the first game has *β* = 0 with no externalities while the second has *β* = −3, which is a sizable externality favoring BR payoffs (Table 6b). Both of the games are in region 1 of Figure 2, so both have the same P ranking of *TL BR* (*BL* ∼ *TR*) (Theorem 4-1), where TL is judged the better of the two Nash cells. However, the social welfare ranking for the Table 5b game is *BR TL* (*BL* ∼ *TR*), which disagrees with the P ranking by crowning BR as the superior cell. By examining this Table 5b game, which includes externality information, it would seem to be difficult to argue otherwise.

Viewed from this externality perspective, Theorem 5 makes excellent sense. It asserts that, with sufficiently large positive *β* values, the social welfare function favors TL, which must be expected. The decomposition (Table 6b) requires *β* > 0 to favor TL payoffs. Conversely, *β* < 0 enhances the BR payoffs.



<sup>7</sup> If ties, such as *TL* <sup>∼</sup> *BR* or *BR* <sup>∼</sup> (*BL* <sup>∼</sup> *TR*) are included, there are seven distinct social welfare rankings for each game in each Figure 2a region.

The potential and social welfare rankings can even reverse each other. According to Theorem 4-2, all games in region 2 of Figure 2a have a single Nash cell with the P ranking of *TL* (*BR* ∼ *TR*) *BR*. This region requires *α* > *γ* > 0, so the Table 9a example is constructed with *α* = 3, *γ* = 1. For this game, where *β* = 0, the P and social welfare rankings agree. To modify the game to obtain the reversed social welfare ranking of *BR* (*BL* ∼ *TR*) *TL*, where the non-Nash cell BR will be the social welfare function's best choice, Theorem 5 describes precisely what to do; select *β* values that satisfy (*α* + *β*) < −2*γ*. For Table 9, this means that *β* < −2(1) − 3 = −5. The *β* = −6 choice leads to the Table 9b game, where the social welfare ranking reverses that of the potential function. Again, it is difficult to argue against this game's social welfare ranking.
