Integrating Strategic Properties with Social Perspectives: A Bipartite Classification of Two-by-Two Games

Abstract

Classifying games according to their strategic properties provides meaningful insights into the motivations driving the interacting parties, suggests possible future trajectories, and in some cases also points to potential interventions aiming to influence the interactions’ outcomes. Here, we present a new classification that merges two perspectives: (i) a revised version of Rapoport and Guyer’s taxonomy, which extends beyond the original 78 games they describe by classifying all two-by-two games according to fundamental strategic properties, and (ii) a novel classification grounded in the theory of subjective expected relative similarity, which addresses not only the games’ payoffs but also the players’ strategic perceptions of their opponents. While Rapoport and Guyer’s original taxonomy classifies only strictly-ordinal games, the revised classification addresses all two-by-two games. It comprises eleven categories that are further grouped into five super-categories that focus on the game’s expected outcome and its strategic stability. The second, similarity-based, classification comprises four main categories, specifying whether players’ perceptions of their opponents have the potential to influence strategic decision-making. The merged classification comprises 14 game types, offering a holistic account of the strategic interaction, the players’ underlying motivations, and the expected outcome. It combines the fixed strategic properties with the variable social aspects of the interaction. Moreover, the novel classification points to the potential of social interventions that may influence the game’s outcome by altering strategic similarity perceptions. Therefore, the present work is relevant for both theoretical and experimental research, providing insights into actual choices expected inside and outside of the laboratory.

1. Introduction

A two-by-two game is a basic and useful representation of strategic interactions between two parties (Kilgour & Fraser, 1988; von Neumann & Morgenstern, 2007). In this normal form of games, each of the two players (referred to as the row and the column players) individually and simultaneously chooses between two alternatives. The intersection of both players’ choices determines the outcome of the game and the payoffs obtained by each player (Figure 1). Two-by-two games have been applied to model various social, economic, and political interactions (Axelrod, 1984; Brams, 1993, 2001; Fischer et al., 2022b, 2024, 2025). The classification of games according to their strategic properties provides important insights into the motivations driving the parties, reveals probable trajectories of the interactions, and in some cases also points to potential interventions that may influence the games’ outcomes. For these reasons, several classifications of two-by-two games have been proposed (Bruns, 2015; Harris, 1969, 1972; Huertas-Rosero, 2003; Kilgour & Fraser, 1988; Rapoport & Guyer, 1966; Robinson & Goforth, 2004), focusing on the strategic aspects derived from the games’ payoff structure (for a summary of several classification methods, see Böörs et al., 2022). These aspects include, for example, the payoffs’ impact on players’ choices, the stability of the expected outcomes, or the pressures that may influence choices. Each of these classification methods contributes a unique perspective that sheds light on a specific aspect of the strategic interaction. Here, we aim to develop a classification system that emphasizes the combination of purely strategic (payoff-related) and social (player-related) forces, thereby providing a general and exhaustive account of all two-by-two games. To this end, we propose a bipartite classification that merges strategic and social motivations of the players, thus integrating two fundamental and essential facets of two-by-two games. The proposed classification is of particular value for analyzing real-life conflicts and for planning behavioral interventions aiming to motivate cooperative solutions. We also supplement the theoretical framework presented here with user-friendly tools for classifying two-by-two games (see Supplementary Materials—classification of all two-by-two ordinal games).

The proposed bipartite classification comprises: (i) a revised version of Rapoport and Guyer’s (1966) taxonomy,1 which extends their classification of strictly-ordinal games to account for all two-by-two games; and (ii) a classification based on the theory of Subjective Expected Relative Similarity (SERS; Fischer, 2009, 2012; Fischer & Savranevski, 2023), which integrates the games’ payoffs with the players’ strategic perceptions of each other. The bipartite classification addresses strategic considerations associated mainly with one-shot games and allows for the identification of a single game-type for any two-by-two game, comprising either cardinal or ordinal payoffs.

Game Classifications

While there is an infinite number of two-by-two games, they can be reduced into a finite set by transforming cardinal values into rank-ordered payoffs, where each player’s smallest payoff is assigned the value of 1, the next smallest payoff the value of 2, and so on. Rapoport and Guyer’s (1966) taxonomy addresses only the subset of games that are strictly-ordinal. These are games where the four payoffs of each player have distinct values, ranked as 1, 2, 3, and 4. Rapoport and Guyer’s classification provides a theoretical model that distinguishes between ten different game categories according to a set of strategic properties (described below). However, many interactions cannot be strictly ordered since two or more of a player’s payoffs may have identical values. In fact, out of the 726 rank-ordered two-by-two games,2 only 78 are strictly ordinal, while the remining 648 are non-strictly ordinal (Fraser & Kilgour, 1986; Guyer & Hamburger, 1968). Since Rapoport and Guyer’s taxonomy cannot be applied to most games, the present manuscript proposes a revised classification that accounts for all two-by-two games.

The necessity to classify both strictly and non-strictly ordinal games has been acknowledged by several scholars who proposed other classifications (Bruns, 2015; Harris, 1972; Kilgour & Fraser, 1988; Robinson & Goforth, 2004). These classifications focus mainly on the structure of the game (such as symmetry, preference ordering, or resemblance) and do not directly address the strategic motivations of the players. For example, Kilgour and Fraser explain that their objective is “to classify games according to their inherent properties, rather than according to any specific theory of strategic interaction. [The] taxonomy is thus neutral with respect to theories of strategic choice” (p. 103).

Aiming to classify all two-by-two games while addressing strategic choices, the present work focuses on both strategic and social forces acting on the players. Some forces may coincide and strengthen the motivation to choose a particular alternative, while other forces may contradict each other, driving individual players to simultaneously consider different alternatives. Specifically, we emphasize the combination of purely strategic (i.e., payoff-related) and social (i.e., player-related) forces, each exerting their distinct impact. Therefore, the presented bipartite classification provides a basic, general, and exhaustive account of all two-by-two games, where one dimension focuses on payoff-related characteristics and the other dimension focuses on player-related characteristics.

To account for payoff-related forces, we rely on the fundamental properties described in Rapoport and Guyer’s (1966) classic taxonomy. Their original taxonomy addresses many basic aspects of strategic motivations (e.g., dominance, MaxiMin, Nash equilibrium, Pareto efficiency; described in Section 2) and has been applied in various theoretical and behavioral studies (Colomer, 1995; Fischer et al., 2024, 2025; Guyer & Rapoport, 1970, 1972; Rapoport et al., 1976; Wyer, 1969). However, it is limited to the 78 strictly ranked two-by-two games. Therefore, we revise and extend their classification to include all two-by-two games. To account for player-related forces, we rely on the theory of SERS, which focuses not only on the game’s payoffs but on the interaction of the payoffs with social perceptions of the players, specifically strategic similarity with each other (Fischer, 2009; Fischer et al., 2022a; Fischer & Savranevski, 2023). Importantly, these similarity perceptions are independent of the payoffs and cannot be accounted for by them, therefore providing a separate social yet strategic component of the interaction.

Overall, the suggested classification (i) differentiates between games that are likely to result in non-conflictual and rather satisfactory outcomes for both parties and games that are likely to induce intractable and enduring conflicts, as derived from the revised Rapoport and Guyer’s classification; and (ii) distinguishes between games where the outcomes are dependent on strategic similarity perceptions of the players and games where the outcomes are not dependent on these strategic similarity perceptions, as derived from SERS theory (Fischer, 2009, 2012; Fischer et al., 2024; Fischer et al., 2022a; Fischer & Savranevski, 2023). Hereafter, we describe each part of the bipartite classification separately before presenting the merged classification.

2. Rapoport and Guyer’s Taxonomy of Two-by-Two Games

Rapoport and Guyer (1966) classified all two-by-two strictly-ordinal games into ten categories, grounded in four basic strategic properties. These properties are: dominance (the preference of an alternative that provides larger payoffs than the other alternative, under each and every choice of the opponent), MaxiMin (a strategy that considers only the minimal payoff under each alternative and prefers the alternative that contains the maximum of these minima), the Nash equilibrium (an outcome that none of the players is motivated to abandon unilaterally, assuming the other player does not change his choice; Nash, 1950), and Pareto efficiency (no other outcome of the game can provide better payoffs for both players). Applying these basic criteria in a specific order points to an end-state, referred to as the natural outcome of the game. The natural outcome, together with other considerations (detailed below), defines a specific category for each strictly-ordinal game. Since we rely on Rapoport and Guyer’s taxonomy as a foundation for constructing a revised and comprehensive classification of all two-by-two games, we first summarize their original approach. Readers who are familiar with the taxonomy may skip ahead to Section 2.1, which describes the revised classification.

To derive the natural outcome, Rapoport and Guyer proposed the following algorithm:

• If both players have a dominant alternative, they both choose it, and the resulting outcome constitutes the natural outcome of the game.3
• Else, if only one player has a dominant alternative, he chooses it, while the other player chooses the alternative that maximizes his own payoff under the expectation that the dominant alternative is chosen. The resulting outcome constitutes the natural outcome of the game.
• Else, if the game has a single Pareto efficient outcome, it constitutes the natural outcome of the game.
• Else, if no player has a dominant alternative and there is either none or more than one Pareto efficient outcome, each player chooses according to the MaxiMin strategy. The resulting outcome constitutes the natural outcome of the game.

The natural outcome together with the following properties define the category of the game. Games categorized as Absolutely Stable (also referred to as No Conflict games) are games in which the natural outcome is an optimal Pareto efficient outcome—i.e., an outcome in which both players are completely satisfied, because each of them receives his maximal payoff. Threat Vulnerable games are games in which the natural outcome is the only Nash equilibrium in the matrix, and there is one satisfied player and one aggravated player (i.e., a player that does not receive his maximal payoff). Furthermore, the aggravated player can “induce” the satisfied player to switch alternatives, by threatening to change his initially expected choice. The satisfied player would prefer to switch rather than suffer the consequences of the other player switching. Notice that while this logic can be applied ad infinitum without reaching an end point, Rapoport and Guyer impose a limit of only considering one move ahead. Similarly to threat vulnerable games, Force Vulnerable games’ natural outcome is the only Nash equilibrium in the matrix, and there is one satisfied player and one aggravated player. However, the aggravated player can “force” the satisfied player to switch alternatives, by actually changing his initially expected choice. This means that if the aggravated player decides to switch, the satisfied player is better off switching as well. Notice that while the concepts of inducing and forcing cannot be implemented in one shot games, in later works Rapoport and Guyer suggested that these forces are applicable to repeated games where the players can respond to each other’s choices (Guyer & Rapoport, 1970; Rapoport et al., 1976). This concept is further developed in Brams’ “theory of moves” that considers the possible moves and counter moves made by the players (Brams, 1993; Brams & Ismail, 2022; Brams & Mattli, 1993). Stable games are also games in which the natural outcome is the only Nash equilibrium in the matrix, and there is one satisfied player and one aggravated player. However, the aggravated player is unable to induce or force the other player to switch alternatives. The Strongly Stable category contains games where the natural outcome is the only Nash equilibrium in the matrix, and both players are aggravated players, meaning that neither player receives his maximal payoff. Furthermore, the aggravated players are unable to induce or force each other to switch alternatives. Strongly Stable with Deficient Equilibrium games are strongly stable games in which the natural outcome is not a Pareto efficient outcome. The only game in this category, according to Rapoport and Guyer’s original taxonomy, is the Prisoner’s Dilemma (PD). Unstable games are also games in which the natural outcome is the only Nash equilibrium in the matrix, and there is one satisfied player and one aggravated player; but the aggravated player can both induce and force the other player to switch alternatives (i.e., the game is both threat vulnerable and force vulnerable). Two Equilibria with Equilibrium Outcome games are games in which two cells are both Nash equilibria and Pareto efficient, and one of these cells is the natural outcome of the game. The Two Equilibria without Equilibrium Outcome category (also referred to as Preemption Games) contains games with two cells that are both Nash equilibria and Pareto efficient, but none of these cells are the natural outcome of the game. The final category is Cyclic Games (also referred to as Games without Equilibria); these games have no Nash equilibria (i.e., games with no pure-strategy Nash equilibria; Osborne & Rubinstein, 1994).

2.1. Expanding Rapoport and Guyer’s Taxonomy to Include All Two-by-Two Games

To expand Rapoport and Guyer’s taxonomy to include all two-by-two games, we modify their original algorithm and add a category of games that do not have a natural outcome. The hereby described criteria introduce a concise set of necessary amendments to allow for a comprehensive and strategically meaningful classification of all two-by-two games. Notice that it is not necessary to rank-order the game’s payoffs prior to applying the classification algorithm.

Before presenting the revised natural outcome algorithm, we first describe and explain the required modifications that allow for classifying all two-by-two games. These modifications are as follows:

(1)

If either one or both players obtain identical payoffs in both their alternatives (i.e., two identical rows for the row player or two identical columns for the column player) and therefore no strategic decision can be made by this player, the game has no natural outcome.

(2)

Games may have a weakly dominant alternative that provides a larger payoff under one of the opponent’s choices and an identical payoff under the other choice of the opponent (which is impossible in strictly-ordinal games). Therefore, our definition of dominance includes both weak and strong dominance preferences.

(3)

Since games may have identical minimal payoffs in both rows or both columns (which is impossible in strictly-ordinal games), the players do not necessarily have a preferred alternative that is derived from the MaxiMin strategy (i.e., no unique MaxiMin). Therefore, if only one player has a MaxiMin alternative, he is expected to choose it, and the other player is expected to play under the assumption that his opponent chooses the MaxiMin alternative. This criterion is in line with Rapoport and Guyer’s second criterion (as described above), which states that if only one player has a dominant alternative, the other player chooses under the assumption that this alternative is chosen.

(4)

If neither player has a MaxiMin preferred alternative, the minimization of risk is no longer relevant. At this stage, players who are deterred by risk aversion, and fear the possibility of obtaining one of the minima, may now strive to maximize their gains by using the MaxiMax strategy (a strategy that considers only the maximal payoff under each alternative and chooses the alternative that contains the maximum of these maxima) without being hindered by risk aversion.4

(5)

In line with the third modification, if only one player has a MaxiMax preferred alternative, he is expected to choose accordingly, and the other player is expected to play under the assumption that this choice is made.

(6)

If both players have no dominant alternative, no MaxiMin alternative, and no MaxiMax alternative, the game has no natural outcome.

These modifications allow for the identification of the natural outcome for all two-by-two games, where such an outcome exists, and clearly point to games that do not have a natural outcome. The complete and revised algorithm, which includes all the abovementioned modifications, is depicted in Figure 2.

Alongside the changes to the natural outcome algorithm, the revised classification also contains two modifications to the classification process itself: (i) If the game has no natural outcome, it is classified as a no-natural-outcome game, which does not exist in Rapoport and Guyer’s original taxonomy. (ii) If the game has a natural outcome and has one and only one optimal Pareto efficient outcome (i.e., there is exactly one cell that contains both players’ maximal payoffs) then the game is classified as Absolutely Stable. This change is essential in order to avoid classifying games with two or more optimal Pareto efficient outcomes as absolutely stable, since in these cases the players are not necessarily motivated to choose a specific alternative, and thus such games are not absolutely stable. Apart from these two modifications, games are classified according to the principles proposed by Rapoport and Guyer, as described above. To identify the category of any examined game it is recommended to start by addressing these two criteria before proceeding (if necessary) with the classification process. Note that in the revised classification, which addresses both strictly ordinal and non-strictly ordinal games, both “two equilibria” categories (as defined by Rapoport and Guyer) may in fact have more than two Nash equilibria and are thus better labeled as “multiple equilibria” games (see Supplementary Materials—classification of all two-by-two ordinal games).

2.2. Reducing the Revised Classification into Five Super-Categories

The eleven categories described above may be further reduced into five super-categories in order to better distinguish between games that are likely to result in non-conflictual and rather satisfactory outcomes for both parties, and games that are more likely to develop into intractable and enduring conflicts. We refer to the first category as “Absolutely Stable” games, and to the second category as “Non-Stable” games. We also identify games that do not result in a satisfactory outcome for one or both of the parties but are nonetheless destined to terminate because no party is able to seek a better payoff. These games are referred to as “Stable/Strongly Stable” games. Furthermore, similarly to Rapoport and Guyer’ original taxonomy, we single out the Prisoner’s Dilemma game, since it is widely used as a primary model for the study of cooperation and confrontation (e.g., Axelrod, 1984; Flood, 1958; Rapoport & Chammah, 1965), but expand this category, termed “PD-like” games, to also include games with similar characteristics (as described in the next paragraph). The remaining games, which have no natural outcome and thus do not allow for a clear prediction derived from the payoff structure per se, are assigned to the category of “No Natural Outcome” games.

The reduced classification encompasses the following five super-categories: (i) Absolutely Stable games, where both players obtain their maximal payoff and are thus satisfied with the natural outcome. Such games are regarded as no-conflict games. (ii) Stable/Strongly Stable games, where either one or both players are not satisfied with the natural outcome, but the unsatisfied player/s are not motivated to try to change the outcome of the game, since switching or threatening to switch their initially expected choice neither improves their own expected payoff nor motivates the other player to switch as well. (iii) Non-Stable games, which include unstable, force-vulnerable, threat-vulnerable, multiple equilibria with equilibrium outcome, multiple equilibria without equilibrium outcome, and cyclic games. In these games, either one or both players are not satisfied with the natural outcome, but unlike stable/strongly stable games, the player/s are motivated to try to change the outcome of the game by switching or threatening to switch their initially expected choice, which in turn motivates the other player to threaten or to switch as well. Of course, classifying a game as non-stable does not mean that the unsatisfied players are bound to threaten or to switch, or that their opponents will necessarily respond to the expected threat or switch; but rather that players have strategic motivations that may drive them to deviate from the natural outcome (unlike in the two previous super-categories where departure from the natural outcome is not expected to improve the unsatisfied player’s payoff). (iv) Prisoners’ Dilemma (PD) like games are strongly stable with deficient equilibrium games. In these games both players are not satisfied with the natural outcome but are unable to change the outcome by threat or force. However, unlike stable/strongly stable games, the natural outcome is not Pareto efficient (there is another outcome that is more beneficial for both players). Note that the definition of strongly stable with deficient equilibria games describes more than just the classic PD game when addressing both strictly and non-strictly ordinal games, thus the new super-category is referred to as PD-like games. (v) No Natural Outcome games are games where none of the abovementioned criteria provide strategic guidance for the players, and therefore the game has no clearly expected outcome. Figure 3 depicts a flowchart of the revised and abridged super-categories’ classification process.

3. Similarity-Based Classification of Games

The second perspective proposed here—the similarity-based classification, shifts the focus from the payoff structure per se to the interaction between (i) the game’s payoff structure and (ii) the players’ strategic perceptions of their opponents. Specifically, their prospects of choosing similar (or dissimilar) alternatives, as reflected by their perception of the strategic similarity with the opponent. Similarity has been shown to predict behavioral choices in various mixed-motive interactions (Chierchia & Coricelli, 2015; Clerke & Heerey, 2021; Cornelis et al., 2011; DiLorenzo & Rooney, 2025; Kaufmann, 1967; Krueger, 2013; Lakin & Chartrand, 2003). These considerations have been formalized by the theory of Subjective Expected Relative Similarity (Fischer, 2009, 2012; Fischer & Avrashi, 2024; Fischer & Savranevski, 2023). SERS, as a normative theory, computes Expected Values (EVs) that integrate (i) the payoffs expected under each choice, and (ii) the strategic similarity perceptions, reflecting the prospects of the opponent to choose an identical (or different) alternative to the one selected by oneself. Comparing the EVs associated with each alternative allows for choosing the alternative that maximizes expected payoffs when facing a specific opponent. For example, consider two players interacting in a PD game (with payoff values of T, R, P, and S; see Figure 4). A player that assumes that he and the other player are likely to choose similar alternatives with a probability of p_s (and different alternatives with a probability of 1 – p_s) may compare the EV for the choice of cooperation with the EV for the choice of defection, where:

$$\mathrm{E}\mathrm{V}\left(\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=\mathrm{R}{p}_s+\mathrm{S}\left(1-p_s\right),\mathrm{a}\mathrm{n}\mathrm{d}$$

(1)

$$\mathrm{E}\mathrm{V}\left(\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=\mathrm{P}{p}_s+\mathrm{T}(1-p_s),$$

(2)

and choose the alternative that provides the higher EV.

SERS assumes that the strategic similarity, p_s, between the players is subjectively and individually perceived by each player and is independent of the game’s payoffs. That is, strategic similarity is strictly a behavioral property that is not derived from the payoff matrix, hence providing an additional social aspect that supplements the game’s payoffs. Therefore, two players confronting each other may have identical or different perceptions of their similarity with the opponent. In other words, the row player’s p_s does not necessarily equal the column player’s p_s. Furthermore, different players playing the same game while facing different opponents are likely to assign various p_s values that express their subjective perceptions of the opponents, which in turn may lead to different choices. SERS has been shown to theoretically explain the motivations of players in all two-by-two games (Fischer et al., 2022a; Fischer & Avrashi, 2024), and to provide a descriptive theory of behavioral choice. It has been empirically corroborated by predicting behavioral choices in PD, Chicken, and Battle of the Sexes games (Fischer, 2009, 2012; Fischer & Savranevski, 2023).

SERS provides the opportunity to classify all games into the following categories. (i) Games in which the SERS-based expected choice varies under different perceptions of strategic similarity with the opponent are referred to as Similarity-Sensitive games, whereas (ii) games in which the SERS-based expected choice does not vary under different perceptions of strategic similarity with the opponent are referred to as Non-Similarity-Sensitive games. (iii) In some rare cases, SERS does not point to an expected choice for one or both of the players. Specifically, if the matrix contains a set of identical payoffs for a player in both the matrix’s diagonals (i.e., V(AA) = V(BB) and V(AB) = V(BA) in Figure 1), the game has no similarity-based solution for this player. In these cases, the SERS-based EVs of both alternatives are equal for all values of p_s. Such games are referred to as Similarity-Irrelevant games. In summary, similarity-sensitive games are games in which p_s can influence players’ decisions (Figure 5e); non-similarity-sensitive games are games in which p_s cannot influence players’ decisions (Figure 5f); and similarity-irrelevant games are games in which there is no preferred choice for all p_s levels.

Some games can be similarity-sensitive for one of the players while being non-similarity-sensitive or similarity-irrelevant for the other player. Because the classification addresses each player separately, the game type can be written as a combination of each player’s similarity-based category (e.g., “row player similarity-sensitive and column player non-similarity-sensitive”; or “two-player similarity-sensitive”). Since the roles of the row and column player can be assigned arbitrarily, we propose classifying games into the following four classes: two-player similarity-sensitive games, one-player similarity-sensitive games (where the game is either non-similarity-sensitive or similarity-irrelevant for the other player), two-player non-similarity-sensitive games, and similarity-irrelevant games (which includes “two-player similarity-irrelevant” games as well as “one-player similarity-irrelevant and one player non-similarity-sensitive” games). Figure 5 depicts examples of a two-player similarity-sensitive game (panel a), a two-player non-similarity-sensitive game (panel b), a one-player similarity-sensitive game (panel c), and a similarity-irrelevant game (panel d).

To easily classify games according to their sensitivity to similarity, one may compare the SERS-based EV-maximizing choices under both assumptions of complete strategic similarity (p_s = 1) and complete strategic dissimilarity (p_s = 0) with the opponent. If the EV-maximizing alternative is identical for both cases, the game is non-similarity-sensitive (for the player whose payoffs are examined) since the preferred alternative is the same under both tested boundary conditions, and therefore for all p_s values, and thus not dependent on strategic similarity. If the EV-maximizing alternatives for the cases of p_s = 0 and p_s = 1 are different from each other, the game is similarity-sensitive, since different p_s values may give rise to different preferred alternatives, and thus the choice of alternative is dependent on the exact level of perceived strategic similarity. For example, consider again the PD game depicted in Figure 4. When assuming p_s = 0 in Equations (1) and (2), we obtain $\mathrm{E}\mathrm{V}\left(\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=\mathrm{R}\ast 0+\mathrm{S}\left(1-0\right)=\mathrm{S}$ and $\mathrm{E}\mathrm{V}\left(\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=\mathrm{P}\ast 0+\mathrm{T}\left(1-0\right)=\mathrm{T}$ and hence the preferred choice is defection, since T > S. When assuming p_s = 1 in Equations (1) and (2), we obtain $\mathrm{E}\mathrm{V}\left(\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=\mathrm{R}\ast 1+\mathrm{S}\left(1-1\right)=\mathrm{R}$ and $\mathrm{E}\mathrm{V}\left(\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=\mathrm{P}\ast 1+\mathrm{T}\left(1-1\right)=\mathrm{P}$ and hence the preferred choice is cooperation, since R > P. Since the preferred choices are different in the two boundary conditions, the PD game is indeed similarity-sensitive. Notably, when calculating the EV in the case of p_s = 0 or p_s = 1 some of the components in the equations are canceled out. Therefore, when considering p_s = 0 it is sufficient to compare the payoffs obtained when the players make dissimilar choices (T and S in the case of PD), and when considering p_s = 1 it is sufficient to compare the payoffs obtained when the players make similar choices (R and P in the case of PD). Since each comparison relates to a diagonal in the matrix, it is sufficient to compare the preferred choices under each diagonal. One diagonal may be denoted as the similarity diagonal, which is the diagonal that contains similar choices (e.g., the cells that contain the [R , R] and [P , P] payoffs in the case of PD); and the other diagonal may be denoted as the dissimilarity diagonal, which is the diagonal that contains dissimilar choices (e.g., the cells that contain the [T , S] and [S , T] payoffs in the case of PD). Notice that mislabeling the diagonals, which is more likely to occur in asymmetric games, still results in a preference for one alternative in one diagonal and a preference for the other alternative in the other diagonal, which in turn leads to a correct classification of the game as similarity-sensitive. Therefore, correctly identifying the similarity diagonal is not necessary for classification purposes, though it is necessary for correctly calculating the exact SERS-based EV and the similarity threshold value (described further below).

In the case of a tie in one of the diagonals, that is, the player obtains the same payoff in both cells of one diagonal, the player prefers a specific alternative for all values of p_s except for a single similarity value, either p_s = 0 or p_s = 1, where he has no preference. We consider this game as non-similarity-sensitive, since the probability of p_s = 0 or p_s = 1 is mathematically negligible, and behaviorally quite unlikely. In the case of a tie in both diagonals, the game is similarity-irrelevant, since the player has no preference under all values of p_s. To conclude, comparing the payoffs of both diagonals provides an abridged method for classifying games according to their sensitivity to similarity. Figure 6 depicts the criteria for identifying the similarity-based game type. Notice that identically to the revised Rapoport and Guyer classification, the criteria for classifying games according to SERS does not require knowing the cardinal payoffs and calculating EVs. Therefore, similarity-based classification may be performed on either cardinal or rank-ordered matrices. However, computing the games similarity threshold (as described in the following section) requires using cardinal payoffs.

The Similarity Threshold

Similarity-sensitive games are characterized by their similarity threshold, denoted by p_s*. At this critical value of similarity, the SERS-based EVs of both alternatives are equal. When the perceived similarity with the opponent, p_s, is above the similarity threshold of the game (p_s > p_s*) one of the alternatives provides a higher EV, and when p_s is below the threshold (p_s < p_s*) the other alternative provides a higher EV. For example, calculating the threshold for the PD game is derived while assuming EV(cooperation) = EV(defection), which results in $\mathrm{R}\ast p_s+\mathrm{S}\left(1-p_S\right)=\mathrm{P}\ast p_s+\mathrm{T}(1-p_s)$, and hence

p_s* = (T − S)/(T − S + R − P).

(3)

The lower the value of p_s* in the PD game, the higher the prospect of the players to choose the cooperative alternative, since more values of p_s satisfy p_s > p_s*. This is true for all similarity-sensitive games, though the distinction is not necessarily between cooperation and defection (which are the standard labels of the alternatives in the PD game). For the general case, depicted in Figure 1, the formula for the row player’s threshold value is

$$p_s^{*}={\displaystyle \frac{V{\left(BA\right)}_r-V{\left(AB\right)}_r}{V{\left(BA\right)}_r-V{\left(AB\right)}_r+V{\left(AA\right)}_r-V{\left(BB\right)}_r}},$$

(4)

and the formula for the column player’s threshold value is

$$p_s^{*}={\displaystyle \frac{V{\left(AB\right)}_c-V{\left(BA\right)}_c}{V{\left(AB\right)}_c-V{\left(BA\right)}_c+V{\left(AA\right)}_c-V{\left(BB\right)}_c}},$$

(5)

both assuming that row alternative A is similar to column alternative A and row alternative B is similar to column alternative B (i.e., the similarity diagonal comprises the top left and bottom right cells, as depicted in Figure 1). Importantly, the similarity threshold value, p_s*, can only be calculated using the payoffs’ cardinal values and must not be calculated using the ordinal ranks.

For similarity-sensitive games the threshold value is 0 < p_s* < 1, indicating that the preferred choice is dependent on the perceived strategic similarity with the opponent. For non-similarity-sensitive games the calculated threshold value is either p_s* ≤ 0, p_s* ≥ 1, or undefined (where the denominator is equal to zero but the numerator is different from zero), indicating that the preferred choice does not change within the possible range of strategic similarity values, which spans from complete dissimilarity to complete similarity (0 ≤ p_s ≤ 1). In other words, a non-similarity-sensitive game does not have an actual similarity threshold. In the case of similarity-irrelevant games, the threshold value is undefined (the calculation always results in ${\displaystyle \frac{0}{0}}$).

To calculate the exact value of p_s*, one must first identify the similarity diagonal, which in games with asymmetric payoffs may not necessarily contain identical payoffs for both players. Mislabeling the similarity and dissimilarity diagonals will most likely result in an incorrect p_s* value, therefore leading to erroneous conclusions regarding the players’ motivations in the examined game. However, as explained above, mislabeling the diagonals does not change the game’s classification.

In symmetric two-by-two games, the matrix is structured such that one diagonal comprises two pairs of identical payoffs for both players, and the other diagonal comprises cells that are reflections of each other (see Figure 4 and Figure 5a,b). These games provide each player with the same strategic problem (Kilgour & Fraser, 1988). Therefore, the similarity diagonal for symmetric games is defined as the diagonal that includes the two identical payoff pairs, like in the case of the PD game mentioned above.5 In asymmetric games it is necessary to assess which of the row player’s alternatives is more similar to which column player’s alternative. The similarity of alternatives is conveyed by the extent of similarity of the payoffs. In other words, by choosing similar alternatives in asymmetric games, players obtain payoffs that are closer to each other than the payoffs they obtain by choosing dissimilar alternatives. To correctly identify the cells that constitute the similarity diagonal, one must first test the extent to which the definition of one of the diagonals as the similarity diagonal better represents a game with an identical strategic problem for both players. This may be carried out in various ways, each considering a different approach to defining what constitutes an identical strategic problem. For example, one may consider the similarity of the payoffs, the similarity of the payoff ranks, or the correlation between players’ corresponding payoffs. See Supplementary Materials—similarity diagonal identification methods for a detailed description of several proposed methods for identifying the similarity diagonal (also see Fischer & Avrashi, 2024).

4. Merging the Two Classifications into a Unified Bipartite Classification

Each of the two suggested classifications provides a useful tool for analyzing, understanding, and predicting the expected outcomes of two-by-two games. The revised Rapoport and Guyer’s classification focuses on the game’s payoffs and provides a prediction of the expected trajectory of the interaction as derived solely from the strategic properties of the game; while the similarity-based classification emphasizes the association of the payoffs with the properties of the players, specifically their perceived prospects of choosing similar alternatives. Combining the two classifications merges these two perspectives, therefore offering a holistic understanding of the strategic interaction, the players’ underlying motivations, and the expected outcome. By merging the classifications, we do not intend to show a convergence of strategic motivations. Instead, we aim to portray a more complete depiction of the forces acting upon the players, sometimes pushing in the same direction and sometimes pulling in opposite directions. Therefore, the merged classification reveals both strategic congruencies and inconsistencies. The bipartite classification also points to potential behavioral interventions that aim to manage and possibly resolve social, economic, and political conflicts.

Merging the two classifications should result in 20 categories, derived from all combinations of the five super-categories of the revised Rapoport and Guyer’s classification, and the four categories of the similarity-based classification. However, six of these combinations result in empty sets (i.e., do not contain any games), leaving the bipartite classification with 14 meaningful categories. Figure 7 depicts example ordinal matrices for each category, showing both strictly-ordinal games (if existing) and non-strictly ordinal games (if existing). Also, see Supplementary Materials—Examples of the full classification process—for a detailed description of the bipartite classification process. Notice that both classifications may use either cardinal or ranked-ordered matrices. That is, changing the cardinal payoffs while retaining their ranks does not change the game’s category. However, the similarity threshold (p_s*) may only be calculated using cardinal payoffs.

The following paragraphs outline the main properties of each category. To emphasize the contribution of the added similarity-based classification to the revised Rapoport and Guyer’s classification, we list the super-categories of the revised classification and describe the unique properties derived from the integration of the two perspectives. Importantly, we specifically address cases where the two classifications contradict each other, and point to incongruent outcomes.

Absolutely Stable games: This class of games contains matrices with a single optimal Pareto efficient outcome. As shown in Figure 7, absolutely stable games can be similarity-sensitive for both players, one player, or neither player. The existence of similarity-sensitive games in this class implies that even though the natural outcome of an absolutely stable game offers both players their maximal payoffs, low p_s values may still drive one or both players to choose an alternative that does not result in an optimal Pareto efficient outcome. In other words, if a player perceives the opponent as insufficiently similar to himself, the expected outcome of the game according to SERS is not necessarily the optimal Pareto efficient cell. An example of this type of game is the Stag Hunt game (Skyrms, 2001), shown in matrix 1 in Figure 7.6 In this game, low p_s values may drive the players to choose the non-cooperative alternative (bottom row/right column).

Moreover, the similarity threshold value, p_s*, may take any value from 0 to 1 in many “absolutely stable similarity-sensitive” games (but see matrix 4 in Figure 7 for a game with possible threshold value range restricted to 0 < p_s* < 0.5). For example, in the Stag Hunt game depicted in matrix 1, considering the payoffs as cardinal values and replacing the payoff of 4 with 100 results in $p_s^{*}={\displaystyle \frac{3-1}{3-1+100-2}}=0.02$ (see Equation (4)), whereas replacing the payoff of 1 with −100 results in $p_s^{*}={\displaystyle \frac{3-(-100)}{3-(-100)+4-2}}=0.98$ (see Equation (4)). The former case induces an almost overwhelming preference for the cooperative choice (for all p_s > 0.02), while the latter induces an overwhelming preference for the defective choice (for all p_s < 0.98). In other words, classifying Stag Hunt as an absolutely stable game is insufficient for predicting players’ preferred choices without supplementing the classification with the SERS-driven p_s* value and its comparison with strategic similarity perceptions (p_s) of the players. This again illustrates the necessity to classify games from both the strategic perspective (revised Rapoport and Guyer’s classification) and the similarity-based perspective (SERS), demonstrating the impact of contradictory forces on strategic choices of the players.

Stable/Strongly Stable games: As shown in Figure 7, there are no games in this class that are two-player similarity-sensitive (as empirically observed from classifying all 726 ordinal two-by-two games). This means that in all stable/strongly stable games there is at least one player who has only one possible expected choice according to SERS (the player who plays the non-similarity-sensitive game). This player is not necessarily satisfied with the natural outcome, but his motivation not to switch alternatives is simultaneously driven by three rationales: he can neither force nor induce the other player to switch alternatives, and one of his alternatives has a higher SERS-based EV than the other alternative regardless of the perceived strategic similarity with the opponent (p_s). In the case of “one-player similarity-sensitive stable/strongly stable” games, one should also consider the similarity threshold value (p_s*), since low p_s values may drive the similarity-sensitive player towards a choice that is not predicted by the natural outcome.

Prisoner’s Dilemma like games: All the games in this category are two-player similarity-sensitive. This property alongside the classification of the PD game as a “strongly stable with deficient equilibrium” game in the original Rapoport and Guyer’s taxonomy, suggests that PD-like games can be considered as “two-player similarity-sensitive stable/strongly stable” games. Therefore, PD-like games could be integrated into the stable/strongly stable category of games. However, due to the prominence of the PD game in both theoretic and applied literature, we distinguish it from other games and place it in a category of its own. Note that while the classic PD game, depicted in Figure 4, requires T > R > P > S, the extended set of PD-like games allows for T = R for one (and only one) of the players, since the resulting game exhibits all strategic properties defined for the category of PD-like games, as shown in Figure 3. Therefore, this class contains two rank-ordered games instead of only one (matrices 12 and 13 in Figure 7). This class demonstrates the importance of the bipartite classification by showing how the classic game-theoretic perspective, as reflected by the revised Rapoport and Guyer classification, views PD as a game that motivates defective choices; while the social perspective, as reflected by the similarity-based classification, points to the rationality of both defective and cooperative choices, as corroborated by many experimental studies (Bornstein & Ben-Yossef, 1994; Fischer, 2009, 2012; Fischer & Savranevski, 2023; Gächter et al., 2024; Mengel, 2018; Sally, 1995).

Non-Stable games: This class of games contains matrices that are similarity-sensitive for both players, one player, or neither player. The existence of non-similarity-sensitive games in this category shows that many interactions that have no stable solution according to the revised Rapoport and Guyer’s classification do have a single solution from the perspective of SERS. Moreover, while in some absolutely stable games low similarity perceptions can “destabilize” the interaction and lead to a non-optimal outcome, in some non-stable games high similarity perceptions may “stabilize” the interaction by leading to a solution that is mutually beneficial for both players. Consider for example the Chicken game (Rapoport & Chammah, 1966; matrix 14 in Figure 6)—according to Rapoport and Guyer’s original taxonomy, it is classified as a “two equilibria with no equilibrium outcome” game, since the natural outcome is not one of the two Nash equilibria of the game (each preferred by a different player), and therefore the game is susceptible to deviation from the natural outcome. However, if the players’ perceptions of the others’ strategic similarity (p_s) are sufficiently high (greater than p_s*) the game is likely to result in its natural outcome. Alternatively, if the players’ perceptions of the others’ strategic similarity (p_s) are sufficiently low (smaller than p_s*) the game is likely to result in mutual defection (the bottom right cell in matrix 14), where both players deviate from the natural outcome.

Another noteworthy type of games that is classified as non-stable is coordination games. In these games, both players benefit more from outcomes that are located on one of the diagonals and less from outcomes that are located on the other diagonal (Rapoport, 1967). In many coordination games, the payoffs in each of the cells along the preferred diagonal benefit one of the players more than the other (see Figure 7, matrices 16 and 19, which are typically referred to as the Hero and Battle of the Sexes games, respectively, and see Figure 8 for the Leader game). Hence, each player has a different favorable outcome (both are Nash equilibria), which in turn causes the game to be non-stable. From SERS’s perspective, similarity-sensitive coordination games give rise to a somewhat counterintuitive situation where a pair of players who perceive each other as highly similar are expected to obtain smaller payoffs than a pair of players where one perceives the opponent as highly similar while the other perceives the opponent as dissimilar.7 Consider for example the Leader game depicted in Figure 8. If both players perceive each other as highly similar, the expected outcome is the top left cell in which both obtain their second lowest payoffs. However, if the column player’s p_s is high while the row player’s p_s is low, the expected outcome is the bottom left cell in which the row player obtains his highest payoff and the column player obtains his second highest payoff. That is, an asymmetry in similarity perceptions may lead to a better outcome for both players (as shown in laboratory experiments; see Fischer & Savranevski, 2023). For non-similarity-sensitive coordination games, SERS’s predicted outcome is one of the cells along the non-preferred diagonal.

No Natural Outcome games: This category is added to the Rapoport and Guyer’s classification in order to address some non-strictly ordinal games. Thus, there are no strictly-ordinal games in this category, as shown in Figure 7. Games in this category contain only one or two distinct payoff values for at least one of the players (otherwise, the player has a MaxiMin and/or a MaxiMax preference and the game has a natural outcome). While games where one or both of the players have two distinct payoff values are not necessarily classified as no-natural-outcome games (as for example matrices 6, 7, and 20 in Figure 7), games in which one or both players have only one distinct payoff value (i.e., all four payoffs are identical) are always classified as no-natural-outcome games. As shown in Figure 7, there are no “non-similarity-sensitive no-natural-outcome” games. Yet, there are no-natural-outcome games that are similarity-sensitive for one or both players. In these games, at least one player has no dominant, no MaxiMin, and no MaxiMax preferences, yet a preference may be derived from similarity considerations. Therefore, some no-natural-outcome games may in fact have an expected outcome when analyzed from the perspective of SERS.

No-natural-outcome games include some noteworthy matrices. For example, the game depicted in Figure 7, matrix 21, is a game with complete fate control (Kelley & Thibaut, 1978). In this game, the players are not motivated to choose any alternative over the other, since both alternatives offer them the same payoffs. However, each player, though unable to influence his own payoffs, determines the fate of the other player. From SERS’s perspective, this game is similarity-sensitive for both players, and in fact the similarity threshold is always p_s* = 0.5 regardless of the cardinal payoffs. That is, players that perceive each other as sufficiently similar (p_s > 0.5) are more likely to choose endowing the other player with the higher payoff. Another notable game is the Matching Pennies game (Budescu & Rapoport, 1994), depicted in matrix 23. This game is an example of a “similarity-irrelevant no-natural-outcome” game. In this game, both players have no strategic motivation to prefer any alternative over the other, and may hence be considered as a game of pure chance.

To explore the distribution of game types, we classified all 726 ordinal two-by-two games according to the bipartite classification. Classifying the games according to the revised Rapoport and Guyer classification reveals 236 absolutely stable games, 98 stable/strongly stable games, two PD-like games, 329 non-stable games, and 61 no-natural-outcome games. Notice that games without a stable natural outcome (i.e., non-stable and no-natural-outcome games) constitute about half of all ordinal games (54%), implying that for slightly more than half of all theoretical interactions the outcome of the game cannot be predicted according to the revised Rapoport and Guyer classification. Classifying all ordinal games according to the similarity-based classification results in 91 two-player similarity-sensitive games, 320 one-player similarity-sensitive games, 276 two-player non-similarity-sensitive games, and 39 similarity-irrelevant games. Notice that 411 games are similarity-sensitive for at least one player, meaning that 57% of all theoretic games are dependent on players’ perception of each other. Moreover, these games are susceptible to behavioral interventions aiming to change the players’ perceptions of strategic similarity with their opponent, which in turn is expected to affect the outcome of the interaction.

Table 1. Game type distribution (and their corresponding percentages, rounded to the nearest integer) of all ordinal two-by-two games according to the bipartite classification.

	Absolutely Stable	Stable/Strongly Stable	PD-like	Non-Stable	No Natural Outcome	Total
Two-player similarity-sensitive	23 (3%)	0	2 (<1%)	53 (7%)	13 (2%)	91 (13%)
One-player similarity-sensitive	94 (13%)	38 (5%)	0	156 (21%)	32 (4%)	320 (44%)
Two-player non-similarity-sensitive	107 (15%)	60 (8%)	0	109 (15%)	0	276 (38%)
Similarity irrelevant	12 (2%)	0	0	11 (2%)	16 (2%)	39 (5%)
Total	236 (33%)	98 (13%)	2 (< 1%)	329 (45%)	61 (8%)	726 (100%)

shows the distribution of game types according to the bipartite classification (see also Supplementary Materials—classification of all two-by-two ordinal games for the classes of all 726 ordinal games).

5. Summary and Conclusions

Here, we proposed a novel classification system of all two-by-two games, which combines two perspectives. The first is an expansion of Rapoport and Guyer’s (1966) taxonomy of strictly-ordinal two-by-two games, which emphasizes payoff-related forces of the interaction, and the second is derived from the theory of subjective expected relative similarity (SERS; Fischer, 2009, 2012), focusing on player-related forces. Both perspectives provide distinctive strategic insights into the nature of the examined interaction. The revised Rapoport and Guyer classification comprises 11 categories that are further reduced into 5 super-categories. This classification allows for the identification of interactions that converge on an expected solution, and those that are more likely to develop into intractable and difficult to resolve conflicts. The similarity-based classification integrates the properties of the payoffs with the perceptions of the players, specifically their strategic similarity with the opponent. It identifies whether there exists an opportunity for introducing behavioral interventions that aim to alter opponents’ strategic similarity perceptions of each other, which in turn are expected to change the outcome of the interaction.

The revised Rapoport and Guyer’s classification contributes to the understanding of a wide range of interactions. It allows one to gain a better understanding of why some conflicts are more likely to be peacefully resolved, while others are more likely to develop into intractable and difficult to resolve conflicts. The classification also defines a natural outcome, which is a theoretic end-state of the game. In stable games, this natural outcome may be considered the predicted solution of the interaction. In games that are not stable, the natural outcome is not necessarily the solution of the interaction since several strategic forces may drive the players towards other outcomes.

The similarity-based classification allows for defining games that are dependent on players’ similarity perceptions of each other and games that are independent of these perceptions. Indeed, empirical studies have shown that similarity perception is a crucial determinant of strategic behavior (Fischer, 2009, 2012). This suggests that behavioral interventions that influence similarity perceptions have an impact on strategic interactions, and in turn may help in managing and resolving conflicts. When attempting to resolve intergroup conflicts, conflict management practitioners may rely on historical, cultural, and behavioral properties of the conflicted groups. Those properties that reveal strategic similarity (either directly or indirectly) could be applied to promote cooperation, while those properties that indicate dissimilarity should be played down. Experimental studies have shown that similarity perceptions can be influenced by revealing participants’ similar attitudes, preferences, traits, and reasoning (Chierchia & Coricelli, 2015; Clerke & Heerey, 2021; Fischer, 2012; Fischer & Savranevski, 2023; Kaufmann, 1967). In an applied context, similarity to other individuals has been used as a behavioral tool for changing health-promoting attitudes by invoking individuals’ familial and in-group affiliations (Fischer et al., 2020).

Similarity-sensitive games are also characterized by a similarity threshold, which indicates the switching point between the players’ expected choices. This is more than a mere technical property of the game, since the value of the threshold signals the ease, or difficulty, of changing the preferences of the players. When the threshold is extremely high (close to p_s* = 1) or extremely low (close to p_s* = 0), the expected choices of anonymous players with unknown similarity perceptions could be predicted with rather high confidence, since extreme similarity perceptions are expected to be less common in the general population (Fischer & Savranevski, 2023).8 In these games, influencing similarity perceptions to the level required to affect the parties’ choices is a rather tall order. In similarity-sensitive games with less extreme threshold values, similarity-altering interventions aiming to influence choices are more feasible and should therefore be considered by mediators and conflict management practitioners as means for resolving disputes.

Combining the revised Rapoport and Guyer’s and the similarity-based classifications offers a holistic understanding of the strategic interaction, the players’ underlying motivations, and the expected outcome. The resulting bipartite classification helps elucidate the complexity of many games in which the motivations that stem from the payoffs per se are different from those derived from the interaction between the payoffs and the players’ perceptions. The classification may point to “predictable” games that are in fact unpredictable, and to “unpredictable” games that are in fact predictable. For example, some games that are stable according to the revised Rapoport and Guyer’s classification are similarity-sensitive according to the similarity-based classification. In these games, the payoff structure motivates a specific outcome, whereas the strategic perceptions of the players may suggest another outcome. The opposite case is that of “non-similarity-sensitive non-stable” games, where the revised Rapoport and Guyer’s classification points to an ambiguous solution, while the similarity-based classification suggests that the players are expected to converge on a clearly defined outcome. In such cases where the two perspectives are incongruent, the merit of the merged classification lies in its ability to capture the conflicting payoff-related and player-related forces that drive the players’ decisions.

The bipartite classification also offers a valuable tool for analyzing ecologically-valid interactions. For example, it has been shown that the Israeli-Palestinian conflict is perceived by many individuals in both parties as an absolutely stable game (Fischer et al., 2025). However, the perceptions of whether the interaction is similarity-sensitive or non-similarity-sensitive differ between the parties and within the groups themselves, reflecting the complexity of this decades-long conflict. Similar findings have also been revealed for hypothetical prototypical interactions, encompassing geo-political, socioeconomical, and ideological conflicts (Fischer et al., 2024). These findings demonstrate the applicability of the classification in analyzing game distributions in real-life interactions.

Some final comments are warranted. First, while the proposed classification addresses two-by-two matrices, which capture a wide range of interactions, the revised Rapoport and Guyer’s classification can be easily extended to include all n-by-m two-player matrices, therefore allowing for an analysis of an even wider range of interactions. The same principles that guide the classification process (Figure 2 and Figure 3) do not change when expanding the matrix and thus can be used to classify larger matrices. Clearly, an analysis of an extended set of matrices is beyond the scope of the present manuscript. Additionally, another extension of the classifications may address repeated games. The bipartite classification focuses on one-shot games while incorporating some aspects, such as forcing and inducing, whose manifestation implies a prolonged interaction between the players, either as “cheap talk” or as part of a repeated game. Extending the classification to also account for repeated games requires modifying some of its properties. For example, some non-stable games (e.g., matrices 16, 19, and 23 in Figure 7) may “stabilize” with a mixed-strategies equilibrium, where each player applies an optimal and fixed probability for choosing between alternatives.

The proposed classification describes both basic strategic properties and social aspects, specifically the strategic similarity of the players. Nonetheless, other social perspectives may also motivate players to choose different alternatives and deviate from the purely strategic solution. These perspectives may include, among others, social value orientations, which account for individuals’ rational, cooperative, or competitive nature (Van Lange, 1999); or social preferences motivating fairness, altruism, and even spite (Fehr & Schmidt, 1999, 2005; Levin, 2014). While these perspectives play an important role in shaping players’ choices, they provide an additional psychological utility to each payoff and could be regarded as side payments and ignored in the strategic analysis. Alternatively, they could be integrated into the payoff structure to generate an all-inclusive matrix, which in turn can be classified and analyzed as proposed here. This is not the case for similarity perceptions, since they are independent of the payoffs, and provide additional information on the expected strategic choices of the opponent. Therefore, unlike many other social aspects, strategic similarity does require a dedicated classification.

Finaly, other classification systems (Böörs et al., 2022; Bruns, 2015; Harris, 1969, 1972; Huertas-Rosero, 2003; Kilgour & Fraser, 1988; Robinson & Goforth, 2004) share identical properties with the bipartite classification proposed here, but they also address unique attributes. In line with the proposed concept of forces acting upon the players, these classification systems may further enrich the analyzed array of forces by highlighting additional strategic considerations besides the player-related and payoff-related forces already accounted for by the bipartite classification.