**1. Introduction**

When Schelling (1960) wrote *Strategy of Conflict*, it pivoted attention from zero-sum games to the more general behavior allowed by games with mutually beneficial outcomes (which was appropriate during this Cold War period) [1]. Specifically, Schelling made a case for coordination games, which Lewis (1969) used to discuss culture and convention [2]. This behavioral notion of mutually beneficial outcomes was further explored by Rosenthal (1973) with his development of the congestion game [3]. Monderer and Shapley (1996) built on the congestion game with "common interest" games; namely the potential games (which include coordination games) [4]. More recently, Young (2006, 2011) and Newton and Sercombe (2020) took this analysis a step further by modeling, with potential games, how populations on networks evolve from one convention to another [5–7]. The natural question in this work is to discover whether the status quo or an innovation will be accepted.

This issue of finding the "better outcome" (e.g., an innovation or the status quo), which is a theme of this paper, is a fundamental and general concern for game theory; answers require selecting a measure of comparison. A natural choice is to prefer those outcomes where the players receive larger payoffs. Rather than payoff dominance, another refinement of Nash equilibria offered by Harsanyi and Selton (1988) is risk dominance [8].<sup>1</sup> The choice used by Young (2006, 2011) and later by Newton and Sercombe (2020) is to maximize social welfare. Still, other measures can be constructed [5–7].

With the social welfare measure, Young constructed a clever ad-hoc example where, although it is seemingly profitable to adopt the innovation, the innovation is worse than the status quo [5]. Young's observation underscores the important need to understand when and why a model's conclusions can change. This includes his concerns of identifying when and why a new cultural convention is "better". Is there a boundary between the quality of innovations?

<sup>1</sup> A payoff dominant Nash cell is where each agent does at least as well as in any other Nash cell, and at least one does better. A risk-dominant Nash cell is less costly should coordination be mistakenly expected; see Section 3.1.

Conflicting conclusions must be anticipated because different measures emphasize different traits of games. Thus, answering the "better" question requires determining which aspects of a game a given measure ignores or accentuates. The approach used here to address this concern is new; it appeals to a recent decomposition (or coordinate system) created for the space of games [9–12] (in what follows, the [9–11] papers are designated by [JS1], and the book [12] by [JS2]). There are many ways to decompose games, where the emphases reflect different objectives. An early approach was to express a game in terms of its zero sum and identical play components, which plays a role in the more recent Kalai and Kalai bargaining solution [13]. Others include examining harmonic and potential games [14] and strategic behavior such as in [15,16]. While some overlap must be expected, the material in [JS1] and [JS2] appears to be the first to strictly separate and emphasize Nash and non-Nash structures.

Indeed, in [JS1], [JS2], and this paper, appropriate aspects of a game are used to extract all information, and only this information, needed for the game's Nash structures; this is the game's "Nash" component. Other coordinate directions (orthogonal to, and hence independent of the Nash structures) identify those features of a game that require interaction among the players, such as coordination, cooperation, externalities, and so forth. By isolating the attributes that induce behavior among players, these terms define the game's "Behavioral" component. The final component, the kernel, is akin to adding the same value to each of a player's payoffs. While this is a valuable variable with transferable assets, or to avoid having games with negative payoffs, it plays no role in the analysis of most settings. In [JS2], the [JS1] decomposition is extended to handle more player and strategies.2

One objective of this current paper is to develop a coordinate system that is more convenient to use with a wide variety of choices that include potential games (a later paper extends this to more players and strategies). An advantage of using these coordinates is that they intuitively organize the strategic and payoff structures of all games. This is achieved by extracting from each payoff entry the behavioral portions that capture ways in which players agree or disagree (e.g., in accepting or rejecting an innovation) and affect payoff values. Of interest is how this structure applies to all 2 × 2 normal form games. When placing the emphasis on potential games, these coordinates cover their full seven-dimensional space, so they subsume the lower dimensional models in the literature.

By being a change of basis of the original decomposition, this system still highlights the unexpected facts that Nash equilibria and similar solution concepts (e.g., solution notions based on "best response" such as standard Quantal Response Equilibria) ignore nontrivial aspects of a game's payoff structure; see [11]. In fact, this is the precise feature that answers certain concerns in the innovation diffusion literature. Young's example [5], for instance, turns out to combine disagreement between two natural measures of "good" outcomes: one measure depends on unilateral deviations; the other aggregates the collective payoff. Newton and Sercombe re-parametrize Young's model to further explore this disagreement [7]. As we show, Young's example and the Newton and Sercombe arguments stem from a game's tension between group cooperative behavior and individualistic forces.

Other contributions of this current paper are to


<sup>2</sup> Experimental work has been done by Jessie and Kendall [17] by building on the decomposition in [JS1]. More precisely and as given in this paper, the separation aspect of the decomposition permits constructing large classes of games with an identical Nash component (or, the strategic component), but with wildly different externalities components (or, the behavioral component). As they showed, the choice of the behavioral term influenced an agent's selection. Section 2 discusses these components.

The paper begins with an overview of the coordinate system for all 2 × 2 normal form games. After identifying the source of all conflict with symmetric potential games, the full seven-dimensional class is described.
