**2. Overview of the Coordinate System**

As standard with coordinate systems, the one developed for games in [JS1] can be adjusted to meet other needs. The choice given here [JS2] reflects central features of potential games.

Consider an arbitrary 2 × 2 game G (Table 1), where each agent's strategies are labeled +1 and −1 (cells also will be denoted by TL (top-left), BR (bottom-right), etc.).

**Table 1.** Arbitrary Game G in Normal-Form.


A weakness of this representation is captured by the Table 2 game. Information about which strategy each player prefers, whether they do, or do not want to coordinate with the other player, and where to find group opportunities is packed into the entries. Yet, in general, this structure is not readily available from the Table 2 form.



The coordinate system described here significantly demystifies a game by unpacking its valued behavioral information. This is done by decomposing a game into four orthogonal components, where each captures a specified essential trait: individual preference, individual coordinative pressures, pure externality (or Behavioral), and kernel components (see Table 3) (the orthogonality comment follows by identifying games with vectors). The original game is the sum of the four parts.

**Table 3.** The Four Components of a 2 × 2 Game.


To associate these components with behavioral traits of any 2 × 2 game, the individual preference component identifies an agent's inherent preference for strategy +1 or −1. If *α<sup>i</sup>* > 0, then agent *i* prefers strategy +1 to −1 *independent* of what the other agent plays. In turn, *α<sup>i</sup>* < 0 means that agent *i*'s individual preference is for strategy −1. The Table 2 values will turn out to be *α*<sup>1</sup> = 2, *α*<sup>2</sup> = 3, which indicates that both players prefer +1 (or TL).

The coordinative pressure component *γ<sup>j</sup>* reflects a conforming stress a game imposes on agent *j*. Independent of the *α<sup>i</sup>* sign, a *γ<sup>j</sup>* > 0 value rewards agent *j* a positive payoff by conforming with agent *i*'s preferred *α<sup>i</sup>* choice. Conversely, when *γ<sup>j</sup>* < 0, agent *j*'s payoff is improved by playing a strategy different than what agent *i* wants. With Table 2, *γ*<sup>1</sup> = −3 while *γ*<sup>2</sup> = −1, so neither player is strategically supportive of personally reinforcing the other agent's preferred choice.

The pure externality component represents consequences that an agent's action imposes on the other agent.3 If agent *i* plays +1, for instance, then, independent of what agent *j* does, agent *j* receives an extra *β<sup>j</sup>* payoff! Acting alone, however, agent *j* is powerless to change this portion of the payoff. To see why this statement is true, should Column select L in Table 3c, then no matter what strategy Row chooses, this extra advantage remains *β*1. The sign of *β<sup>j</sup>* indicates which of agent *i*'s strategies contributes to, or detracts from, agent *j*'s payoff. In Table 2, the *β*<sup>1</sup> = −3, *β*<sup>2</sup> = 2 values convert TR into a potential *group* opportunity.

A subtle but important behavioral distinction is reflected by the *γ* and *β* terms. The *γ<sup>j</sup>* values capture whether, in seeking a personally preferred (Nash) outcome, an agent should, or should not, coordinate with the other agent's preferred interests. In contrast, the *β<sup>j</sup>* values identify externalities and opportunities to encourage both to cooperate. For a supporting story, suppose the strategies are to take highway 1 or −1 to drive to a desired location. A *γ*<sup>1</sup> < 0 value indicates the first agent's personal preference to avoid being on the same highway as the second. However, it should it be winter time, then the second agent, who always has a truck with a plow when driving on highway 1, creates a positive externality that can be captured with a *β* value.

The final component is the kernel, which for Table 2 is *κ*<sup>1</sup> = 1, *κ*<sup>2</sup> = 3. This can be treated as an inflationary term that adds the same *κ<sup>i</sup>* value to each of the *ith* agent's payoffs. Methods that compare payoffs cancel the kernel value, so, as in this paper, the kernel can be ignored.

It is important to point out that the individual and coordinative pressure components contain all information from a game that is needed to compute the Nash equilibrium and to analyze related strategic solution concepts [JS1]. To appreciate why this is so, recall that the Nash information relies on payoff comparisons with unilateral deviations. But with the pure externality and kernel components, all unilateral payoff differences equal zero, so they contain no Nash information. This also means that "best response" solutions and methods ignore, and are not affected by the wealth of a game's *β* information (for explicit examples, see [11]).

By involving eight orthogonal directions and independent variables, these components span the eight-dimensional space of all 2 × 2 games. Consequently, any 2 × 2 game can be expressed and analyzed in terms of these eight coordinates. The equations converting a game into this form are

$$\begin{array}{llll} \alpha\_{1} = \frac{1}{4}[(a\_{1} + a\_{3}) - (a\_{2} + a\_{4})], & \alpha\_{2} = \frac{1}{4}[(b\_{1} + b\_{3}) - (b\_{2} + b\_{4})], & \gamma\_{1} = \frac{1}{4}[(a\_{1} + a\_{4}) - (a\_{2} + a\_{3})],\\ \gamma\_{2} = \frac{1}{4}[(b\_{1} + b\_{4}) - (b\_{2} + b\_{3})], & \kappa\_{1} = \frac{1}{4}[a\_{1} + a\_{2} + a\_{3} + a\_{4}], & \kappa\_{2} = \frac{1}{4}[b\_{1} + b\_{2} + b\_{3} + b\_{4}],\\ \beta\_{1} = \frac{1}{2}[a\_{1} + a\_{2}] - \kappa\_{1}, & \beta\_{2} = \frac{1}{2}[b\_{1} + b\_{2}] - \kappa\_{2}. \end{array}$$

For interpretations, *κ<sup>j</sup>* is agent *j*'s average payoff, *β<sup>j</sup>* is agent *j*'s average payoff should the other agent play 1 minus the inflationary *κ<sup>j</sup>* value, *α<sup>j</sup>* is half the difference of agent *j*'s average payoff if the other agent plays 1 and the average if the other agent plays −1, and *γ<sup>j</sup>* is half the difference of the *jth* agent's average TL, BR payoff, and average BL and TR payoff.

To illustrate the derivation of Equation (1), the *α*<sup>1</sup> value of the Table 4a game is computed. All that is needed is a standard vector analysis to find how much of game G is in the *α*<sup>1</sup> coordinate direction, which is denoted by <sup>G</sup>*α*<sup>1</sup> , where *<sup>α</sup>*<sup>1</sup> <sup>=</sup> 1 and *<sup>α</sup>*<sup>2</sup> <sup>=</sup> 0. The sum of the squares of the <sup>G</sup>*α*<sup>1</sup> entries (which in the following notation equals [G*α*<sup>1</sup> , <sup>G</sup>*α*<sup>1</sup> ]) is 12 <sup>+</sup> 02 <sup>+</sup> <sup>1</sup><sup>2</sup> <sup>+</sup> 02 + (−1)<sup>2</sup> <sup>+</sup> 02 + (−1)<sup>2</sup> <sup>+</sup> 02 <sup>=</sup> 4, so, according to vector analysis, *α*<sup>1</sup> = <sup>1</sup> <sup>4</sup> [G, <sup>G</sup>*α*<sup>1</sup> ]. Here, [G, <sup>G</sup>*α*<sup>1</sup> ] is the sum of the products of

<sup>3</sup> This is the behavioral component in [JS1].

corresponding entries from each cell. (Identifying a game's payoffs with components of a vector in R8, [G1, <sup>G</sup>2] is the scalar product of the vectors.) In this example, [G, <sup>G</sup>*α*<sup>1</sup> ]=(12)(1)+(10)(0)+(2)(1) + (2)(0)+(0)(−1)+(4)(0)+(6)(−1)+(0)(0) = 8,, so *<sup>α</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>4</sup> [G, <sup>G</sup>*α*<sup>1</sup> ] = 2.. Similarly, by defining a corresponding <sup>G</sup>*α*<sup>2</sup> , <sup>G</sup>*γ*<sup>1</sup> , ... ,, the remaining values are *<sup>α</sup>*<sup>2</sup> <sup>=</sup> 3, *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> 4, *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> 1, *<sup>β</sup>*<sup>1</sup> <sup>=</sup> 1, *<sup>β</sup>*<sup>2</sup> <sup>=</sup> 2, *<sup>κ</sup>*<sup>1</sup> <sup>=</sup> 5, and *κ*<sup>2</sup> = 4. The Equation (1) expressions can be recovered in this manner.


This decomposition simplifies the analysis by extracting the portion from each payoff that contributes to these different attributes of a game. Illustrating with Table 1, rather than handling each entry separately, behavior can be analyzed with the separated impact of the components. For instance, the *a*<sup>1</sup> entry is *a*<sup>1</sup> = *α*<sup>1</sup> + *γ*<sup>1</sup> + *β*<sup>1</sup> + *κ*1, which, for Table 2, is *a*<sup>1</sup> = −3 = 2 − 3 − 3 + 1.

To connect this notation with [JS1], [JS2], a game's Nash component, denoted by <sup>G</sup>*N*, is the sum of the individual preference and coordinative pressure components as given in Table 5.

**Table 5.** The <sup>G</sup>*<sup>N</sup>* Nash component.


Principal facts about 2 <sup>×</sup> 2 games follow.<sup>4</sup> As a reminder, a game is a potential game if there exists a global payoff function that aggregates the unilateral incentive structure of the game. More precisely, the payoff difference obtained by an agent unilaterally deviating is reflected in the change of the potential function. Potential games are often called "common interest" games. *Coordination games* have pure strategy Nash equilibria precisely where the agents play the same strategy (i.e., the strategy profiles (+1, +1) and (−1, −1)). On the other hand, *anti-coordination games* have pure strategy Nash equilibria when the agents play different strategies (i.e., the strategy profiles (+1, −1) and (−1, +1)).

**Theorem 1.** *Generically (that is, all* <sup>G</sup>*<sup>N</sup> and <sup>β</sup><sup>j</sup> entries are nonzero), the following hold for* <sup>2</sup> <sup>×</sup> <sup>2</sup> *normal form games* G*.*


$$
\pi\_1(t\_1, t\_2) = a\_1 t\_1 + \gamma\_1 t\_1 t\_2 + \beta\_1 t\_2 + \kappa\_1 \text{ and } \pi\_2(t\_1, t\_2) = a\_2 t\_2 + \gamma\_2 t\_1 t\_2 + \beta\_2 t\_1 + \kappa\_2 t\_2
$$

*where t*<sup>1</sup> *and t*<sup>2</sup> *represent the strategy choice (either* +1 *or* −1*) of agents 1 and 2,*

*5. A potential function for a game with components described in Table 3 can be transformed into*

$$P(t\_1, t\_2) = a\_1 t\_1 + a\_2 t\_2 + \gamma t\_1 t\_{2\prime} \text{ where } \gamma = \gamma\_1 = \gamma\_2. \tag{2}$$

<sup>4</sup> All of these results extend to 2 <sup>×</sup> ... <sup>×</sup> 2 games.
