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Bruns, B., and C. Kimmich. 2021. Archetypal games generate diverse models of power, conflict, and cooperation. Ecology and Society 26(4):2.
https://doi.org/10.5751/ES-12668-260402
Synthesis

Archetypal games generate diverse models of power, conflict, and cooperation

1Independent Researcher and Consultant, 2Institute for Advanced Studies (IHS), Vienna, Austria, 3Masaryk University, Brno, Czechia

ABSTRACT

Interdependence takes many forms. We show how three patterns of power generate diverse models for understanding dynamics and transformations in social-ecological systems. Archetypal games trace pathways that go beyond a focus on a few social dilemmas to recognize and understand diversity and complexity in a landscape of social situations, including families of coordination and defection problems. We apply the extended topology of two-person two-choice (2 × 2) games to derive simple archetypes of interdependence that generate models with overlapping opportunities and challenges for collective action. Simplifying payoff matrices by equalizing outcome ranks (making ties to show indifference among outcomes) yields three archetypal games that are ordinally equivalent to payoff structures for independence, coordination, and exchange, as identified by interdependence theory in social psychology. These three symmetric patterns of power combine to make an asymmetric archetype for zero-sum conflict and further structures of power and dependence. Differentiating the ranking of outcomes (breaking ties) transforms these primal archetypes into more complex configurations, including intermediate archetypes for synergy, compromise, convention, rivalry, and advantage. Archetypal models of interdependence, and the pathways through which they generate diverse situations, could help to understand institutional diversity and potential transformations in social-ecological systems, to distinguish between convergent and divergent collective action problems for organizations, and to clarify elementary patterns of power in governance.
Key words: asymmetric social situations; coordination games; ecology of games; equilibrium selection; interdependence theory; social dilemmas; system dynamics archetypes

INTRODUCTION

“My mask protects you, your mask protects me,” expresses interdependence in trying to control the spread of COVID-19. Each person has power to determine the other’s outcome but has no control over their own fate. The Buddhist Avatamsaka Sutra describes two people chained in place, each with a spoon too long to feed themself, but each able to feed the other (Aruka 2001). In their Atlas of Interpersonal Situations, Kelley et al. (2003) use the phrase “I’ll scratch your back if you scratch mine” to summarize this kind of reciprocity in an elementary exchange situation.

To practice “social distancing” during a pandemic and reduce the risk of infection when approaching each other in a hallway or along a sidewalk, each of us could move to the left or right. Either direction would be effective, but we are both better off coordinating on the choice, as with driving a vehicle on the left or right. This coordination creates a stable equilibrium outcome in which each person’s action is the best response to what the other does. Lewis (2002) argues that conventions that are used to solve coordination problems are central to social life, including language, culture, and norms. As with the saying about fish not noticing the water, we live within a sea of conventions, usually taken for granted, that enable us to communicate, compete, and cooperate. Norms and their emergence, maintenance, and disappearance play a crucial role in social life (Bicchieri 2005, Legros and Cislaghi 2020), including the governance of natural resources in social-ecological systems (Ostrom 2000, 2005).

I might wear a mask seeking only to protect myself. A mask with an unfiltered exhaust might protect me but not others (Pejó and Biczók 2020). In this case, our actions for self-protection would be independent. “We go our separate ways” (Kelley et al. 2003), regardless of what the other person may do. Each person has a dominant strategy, a strategy that is better whatever the other person does, which, in this (hypothetical) case, still allows each person to get their best outcome.

These three symmetric situations illustrate archetypal games, which are elementary models for interdependence in social-ecological situations. Archetypal patterns can help us to recognize and analyze interactions and dynamics in social-ecological systems (Cullum et al. 2017, Eisenack et al. 2019, Oberlack et al. 2019, Sietz et al. 2019). Social dilemmas such as the “Prisoner’s Dilemma” are not the only forms of interdependence that matter in environmental governance (McAdams 2009, Kimmich 2013, Bisaro and Hinkel 2016). There is a need to better our understanding of the diversity of interdependence in strategic situations and the different challenges to cooperation (Curry et al. 2020). Rules with enforcement by punitive sanctions may help to solve some dilemmas but may be counterproductive in coordination problems. Different situations may require different solutions such as common knowledge, trust building, shared expectations, norms, compromise, reciprocity, or changing rules to improve outcomes.

We apply the extended topology of 2 × 2 games (Robinson and Goforth 2005, Heilig 2011, Hopkins 2014, Bruns 2015; Robinson et al., unpublished manuscript https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.619.4992&rep=rep1&type=pdf) to develop a set of archetypes for interdependence. These archetypes go beyond a few famous games to trace systematic relationships between elementary models of strategic interaction. Rather than a jumble of stories or a scattering of payoff matrices, archetypal games offer landmarks and pathways for navigating a landscape of cooperation and conflict. Archetypal games can aid in understanding the diversity of situations, their dynamics, and their potential transformations. We use a deductive approach to identify the simplest archetypes. We select additional intermediate archetypes based on their relevance for modeling collective action. We find convergence with empirically relevant situations identified in interdependence theory in social psychology (Kelley et al. 2003) and show an efficient way to map relationships between archetypes. These archetypes for interdependence can help to understand a diverse range of situations, including harmony, coordination, exchange, advantage, power, dependence, and conflict.

METHODS

The smiling and frowning faces in Fig. 1A show the relative ranking of different outcomes for wearing a mask that only protects the other person. Numbers from 1 to 4 can indicate the ranks in the normal form payoff matrices used in game theory (Fig. 1B). In this hypothetical case, each person only cares whether the other wears a mask. Either both outcomes are best (4) if the other person wears a mask, or both are worst (1) if the other person does not wear a mask.

Key aspects of the methods can be explained using even simpler preference structures in which each actor prefers a single outcome. Simplification of 2 × 2 games into matrices with ties for the three lowest-ranked outcomes identifies basic archetypes (Fig. 1C and Table 1; Appendix 1). Each actor prefers a single outcome (shown as 4) and is indifferent among the other three outcomes. If the best outcomes are in the same cell, there is agreement on the same win-win outcome. Diagonally opposed preferences create discord, which, in repeated interaction, might be resolved by taking turns.

In symmetric games such as “Win-win” and “Discord,” both actors face the same set of possible outcomes and payoffs. If they switched positions as column and row player, the choices would look the same. Payoffs from symmetric games combine to form asymmetric games. Thus, payoffs from Win-win and Discord combine to form games in which the best outcomes for each actor are either in the same row or in the same column. In this situation, if there is communication or repeated interaction, then one actor has the power to threaten to deny a good outcome for the other unless there is an acceptable agreement such as taking turns to get the best outcome. This potential threat resembles the “Strict Threat” game with four ranked outcomes (Guyer and Rapoport 1970). Switching position as “Row” or “Column” player forms pairs of asymmetric games. One of the two games can be treated as a representative asymmetric archetype, in this case, a “Row Threat” game, to the right (southeast) of the diagonal formed by the two symmetric games.

Asymmetric games can be located and identified using names and abbreviations for the symmetric game payoffs that combine to create their payoff structure (Fig. 1C; Bruns 2015). Interchanging columns or rows (or both) is assumed to represent the same strategic situation, i.e., a variant of the same game (Rapoport et al. 1976). Thus, for example, any game with the two highest ranked outcomes together in the same cell and indifference among the lower ranked outcomes would still be a variant of Win-win. Here, we use a simplified version of Robinson and Goforth’s (2005) Cartesian-style convention, putting Row’s highest payoff in the right-hand column and Column’s highest payoff in the upper row (Row’s 4 right, Column’s 4 up). A convention for consistently displaying payoff matrices makes it easier to compare games. It is particularly helpful for the many asymmetric games that do not have an obvious “cooperative” outcome.

In coordination situations, a social convention or norm helps to select one of the possible equilibria as the preferred solution, for example, driving on the right. In the simplest coordination game, the two alternatives are equally ranked (Fig. 1D). In a second situation, one alternative is preferred and one is ranked lower (Fig. 1D). For games with three ranks, the number 3 is used to show the second-best outcome for simplicity in displaying payoff matrices and for consistency with strict games with four ranks. “Primal Coordination” models an initially arbitrary choice between a convention to keep to the left or to keep to the right. Once agreed, the convention is preferred, and the other outcome is ranked lower. This procedure breaks the ties in payoffs and changes the game. The reverse operation of making ties to show indifference among outcomes simplifies “Convention” into Primal Coordination. The operations of making and breaking ties extend to other archetypal games (see Results and Appendix 1).

Here, we analyze and identify archetypes in game theory payoff matrices using the operations of making ties, breaking ties, and combining payoff patterns. Symmetric payoff patterns combine to make asymmetric games, including asymmetric archetypes. Transforming payoff matrices by equalizing outcome ranks (making ties to show indifference) yields simpler archetypes. Differentiating payoff rankings (breaking ties) transforms simpler archetypes into more complex configurations.

RESULTS

We first show how the three symmetric primal archetypes combine payoff patterns to make asymmetric primal archetypes. We then show how breaking ties in primal archetypes generates families of games linked by making and breaking ties, including intermediate archetypes that exemplify important issues in collective action, such as various types of coordination and defection problems. We present a list of archetypal games that model situations of power, conflict, and cooperation as landmarks for understanding diversity in elementary strategic situations. Selected additional archetypes illustrate overlapping problems of collective action, including trust, externalities, and biased advantage and disadvantage in opportunities and results.

Primal archetypes

Simple games with two “likes” and two “dislikes”, i.e., ties for the two highest and two lowest ranked outcomes, offer interesting examples of elementary interdependence in social or strategic situations, in which each person’s outcomes may depend on what the other person does. Robinson et al. (unpublished manuscript https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.619.4992&rep=rep1&type=pdf) pointed out that this class of payoff structures includes “archetypal games” that exemplify collective action problems, such as the simplest coordination game. Fig. 2 (left side) shows three symmetric primal archetypes matching the situations described in the introduction. In “Primal Independence,” each player can act on their own to achieve their preference (i.e., “we go our separate ways”). Primal Coordination requires joint action to select between alternative equilibria (as in “drive on the left or on the right”). In “Primal Exchange,” each controls the partner’s outcome while having no control over their own payoff (as in “my mask protects you, your mask protects me”).

The payoff patterns for Primal Independence, Primal Coordination, and Primal Exchange combine to form five asymmetric archetypes. These asymmetric archetypes can model patterns of power, dependence, and conflict. In “Primal Help,” one actor’s decision empowers the other actor to have a choice, enabling a result where both obtain their best outcome. Kelley et al. (2003) describe this result as a “helping hand.” In “Primal Gift,” one actor’s choice determines that both reach win-win, regardless of what the other does, benevolently “making us both win.” Primal Gift would also be an archetype for situations in which one person has sufficient incentive to provide a collective good, which then benefits others, what Olson (1971) called a “privileged” group. In “Primal Win-Lose,” one’s choice determines that the first actor gets the best outcome and the other gets the worst, a “best for me, worst for you” result. Finally, interchanging columns (or rows) for one actor in Primal Coordination and combining the resulting payoff pattern with the original pattern for Primal Coordination creates “Primal Conflict.” From each outcome, one person would always prefer to change their move, creating a cyclic game. This situation is often called “Matching Pennies,” based on a simple game with coins that has an equivalent payoff structure. Kelley et al. (2003) summarize this result as “match or mismatch.” It represents a zero-sum conflict of completely opposed interests whereby if one person gains, the other loses.

Each row in Fig. 2 has the same pattern of payoffs for the Row actor and each column has the same pattern for the Column actor. Neighboring games are linked by payoff swaps that switch the ranking of two outcomes. This situation can be visualized as moving one of the 4s into a different cell. Swapping a 4 for Column horizontally, for example, moves to the right, turning Primal Independence into Primal Help, i.e., in the next column of games. Swapping a highest payoff (4) for Row results in moving up into the next row of games, for example, turning Primal Help into Primal Coordination. Thus, this diagram maps possible transformations between one payoff structure and another, “changing the game.”

Families of archetypes

Breaking ties creates more complex games, as in the three families of descendants from primal archetypes (Fig. 3). These families contain games that differ in shared characteristics such as coordination in selecting between multiple equilibria, dominant strategies leading to a single equilibrium with relatively good results, or motivations to defect from a Pareto-optimal outcome.

Primal Coordination differentiates into a Convention game by breaking ties symmetrically for the highest ranked payoffs, so both receive a higher payoff in one of the two equilibria. This situation exemplifies conventions and norms that coordinate on a mutually preferred equilibrium outcome such as driving on the right. Schelling (1960) suggested that such situations with multiple equilibria could be resolved by identifying a prominent focal point based on some salient characteristic. In Lewis’s (2002) discussion of the role of conventions, culture can provide focal points for coordination. The Convention payoff structure is sometimes known as “Hi-lo,” and is used to analyze the problem of coordinating selection of an equilibrium with higher payoff for both (Gold and Colman 2020).

Breaking ties in Primal Coordination so that the equilibria have different payoffs for each actor creates games with rivalry among alternative equilibria whereby one or the other does better. Game theorists often discuss this kind of coordination problem in terms of the “Battle of the Sexes” story about two people who want to do something together but differ about what each would most like for their joint entertainment (Luce and Raiffa 1957). Such models have been used to analyze international relations between rival nations, each seeking its own advantage (Snidal 1985). In an environmental example, fishing sites may yield different production potential; to reduce unproductive conflict, lotteries to assign fishing spots could provide a coordination mechanism for a fair solution (Kaivanto 2018).

Convention and “Rivalry” are intermediate archetypes, with a single pair of ties for each actor, showing indifference between the two lowest ranked outcomes. Breaking ties in the lowest ranked payoffs in Rivalry then creates two strict (no ties) games with four ranked outcomes and rival equilibria (“Hero” and “Leader”). These games differ in the payoff for the actor who changes their move away from a risk-minimizing strategy that avoids the worst outcome for both. In one situation, the Leader does best, whereas in the other situation, the Hero gets second-best (Rapoport 1967).

Breaking low ties in Convention generates two games. In “Safe Choice,” avoiding risk also maximizes payoff. However, in “Assurance,” if both players cooperate, they can both get the best outcome; however, there is a risk of getting the worst outcome if the other player does not cooperate, so cooperation conflicts with caution. This game (and “Stag Hunt,” discussed below) can model issues such as trust and thresholds (tipping points/critical mass) for cooperation in general (Sen 1967, Skyrms 2003), among herders (Runge 1981, Cole and Grossman 2010), for political mobilization (Heckathorn 1996, Oliver and Marwell 2001), and in irrigation technology adoption (Müller et al. 2018).

Breaking high ties in Primal Independence generates another family of games. In “Synergy,” cooperation makes both players better off. Alternatively, breaking ties in Primal Independence generates a situation with a “Second-Best” equilibrium. Breaking low ties in Second-Best generates descendant games that differ in the alignment of the two lowest ranked payoffs: “Deadlock” and “Compromise.” In either case, the equilibrium with second-best payoffs is also the least risky choice. In this case, it is also the choice that offers a chance of getting the best outcome if the other player makes a mistake (“trembling hand”).

Primal Exchange, in which each player controls the partner’s fate, differentiates to form the three most famous and well-studied 2 × 2 models of collective action: Prisoner’s Dilemma, “Chicken,” and Stag Hunt. In these social dilemmas (broadly defined), selfish motives conflict with cooperation (Dawes 1980, Kollock 1998, Van Lange et al. 2014). In Prisoner’s Dilemma, incentives lead away from cooperation to converge on an inferior equilibrium outcome. A narrow definition of social dilemmas would be restricted to situations in which dominant strategies lead to an inferior equilibrium, and further restricted to those in which both players “cooperating” would be better than taking turns “defecting.” However, social dilemmas are often discussed more loosely in terms of a variety of conflicts between individual and collective interests and temptations to defect from cooperation. These dilemmas are often discussed as free-rider problems (Olson 1971). In Chicken, incentives lead away from cooperation to rival equilibria in which one or the other player does best and the other gets second-worst, but with the risk of both getting the worst outcome. Stag Hunt poses a conflict between cooperation to get the outcome that is best for both vs. cautious risk avoidance leading to second-worst for both, similar to the Assurance problem discussed above (Medina 2007). The fourth descendant of Primal Exchange, “Concord,” is called “Max-Diff” by social psychologists. In this situation, dominant strategies could lead harmoniously to a win-win outcome. However, a competitive actor concerned with their own relative advantage and expecting the other actor to follow their dominant strategy might forego win-win, trying to get an unequal outcome in which they do relatively better (Kelley and Thibaut 1978). Such an approach risks getting caught in a spiteful dynamic of “beggar thy neighbor” in which both players end up at the worst outcome.

Starting from the three primal archetypes, breaking ties ultimately generates 12 strict symmetric ordinal games (Fig. 3, right side). Thus, differentiating each primal archetype generates a family of games descending from a common ancestor. The reverse process of simplification, i.e., making ties, creates simpler intermediate games, which can be seen as lying “in between” the strict games. In Appendix 1, we further describe asymmetric descendants for all the primal archetypes and relationships between the resulting families of models within the topology of 2 × 2 games.

Archetypal games

We summarize names and brief descriptions for eight primal archetypes and eight intermediate archetypes in Table 2. This summary combines games derived by simplification or differentiation (Figs. 2 and 3) along with a few additional archetypes (Fig. 4). The 2 × 2 game identifiers use a binomial nomenclature based on how payoff patterns for the 38 symmetric ordinal 2 × 2 games combine to form asymmetric games (Table 2; Bruns 2015). This nomenclature provides a way to identify ordinal games uniquely, including cases in which the same ordinal payoff structure may be discussed using different names, as with Chicken, “Hawk-Dove,” and “Snowdrift.”

Kelley et al. (2003) illustrate the situations (entries) in An Atlas of Interpersonal Situations using payoff matrices with various values. Standardizing payoff ranks to 1–4 and aligning best payoffs (with Row’s 4 right and Column’s 4 up) makes ordinally equivalent matrices that facilitate identifying and comparing games (Robinson and Goforth 2005, Bruns 2015). Appendix 1 (Fig. A10) presents ordinal equivalents for the atlas entries. Standardizing the payoff matrices shows that the examples of the three “single-component” games of independence, coordination, and exchange identified by Kelley et al. (2003) have the same ordinal structure as the simple archetypal games identified by Robinson et al. (unpublished manuscript https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.619.4992&rep=rep1&type=pdf).

More generally, a standardized way of presenting payoffs makes it easier to see the relationships between different games and to compare research from different sources (Robinson and Goforth 2005, Bruns 2015). We put the games identified by Kelley et al. (2003) into context by showing how different games are systematically related through recombining payoff patterns and making and breaking ties. The topology of payoff swaps in 2 × 2 games (Robinson and Goforth 2005), and the “periodic table” display (Appendix 1), offer a visualization that elegantly displays a complex pattern of overlapping relationships. This framework improves on the branching taxonomy suggested by Rapoport et al. (1976) and maps the overlapping properties categorized by Holzinger (2008; see also Holzinger, unpublished manuscript https://doi.org/10.2139/ssrn.399140). Families of archetypal games can offer a somewhat simpler way to see and understand these relationships among different 2 × 2 game theory models. In Appendix 1, we discuss additional ways of analyzing how the potential for cooperative solutions varies between social situations, including dimensions for best response Nash equilibria, coordination, and externalities (Guisasola and Saari 2020).

Several of these archetypal games have also been highlighted by previous researchers. In their synthesis of research on 2 × 2 games, Rapoport et al. (1976) concentrated on strict games (those without ties). However, they also included the payoff structure for Primal Exchange as one of the few non-strict games (games with ties), listed as game #79. They listed Primal Coordination as game #85. Aruka (2001) analyzed Primal Exchange as the Avatamsaka game.

In addition to primal archetypes, we list selected intermediate archetypes that exemplify important situations for collective action (Table 2). These situations include synergy from aligned incentives, compromise on second-best, conventions favoring one of multiple equilibria, and rivalry over alternative equilibria. The payoff structure of the “Defection Dilemma” (between Prisoner’s Dilemma and Chicken in Fig. 3) has occasionally been noted in game theory literature, for example, by Rapoport et al. (1976). However, it does not seem to have been applied much for analyzing collective action. For this reason, Defection Dilemma, and the “Offer” game (between Concord and Stag Hunt) are not listed as intermediate archetypes. However, the topology of 2 × 2 games provides a framework that analysts can use to identify additional archetypes to suit their needs. For example, some descendants of Primal Exchange and Primal Combination are cyclic but have Pareto-optimal outcomes that offer attractive focal points for cooperation (Appendix 1).

When primal archetypes differentiate by breaking ties, their emergent properties converge in some cases, despite starting from different primal archetypes. Primal Exchange differentiates into Stag Hunt, which entails a coordination conflict between risk and trust, similar to Assurance, even though Assurance descends from Primal Coordination. Making middle ties (equalizing the second- and third-ranked outcomes) creates an intermediate game between Assurance and Stag Hunt. In a sense, this game bridges the border between two archetypal families, revealing the process of emergent convergence in the properties of 2 × 2 game theory models. This game (see Fig. 4) represents an intermediate archetype for Assurance and Stag Hunt problems of mutual trust, a “Trust Dilemma.” Rousseau (2004) described a hunter choosing between the certainty of getting a hare or sharing a stag if the others cooperate. With no concern for the other’s outcome, the remaining outcomes would be ranked equally (Rousseau 2004). Hence, this Trust Dilemma (Cronk and Leech 2013) could also be called “Rousseau’s Hunt.”

Primal Exchange also differentiates into Chicken, which shares the problem of coordination between rival equilibria with Leader and Hero, another example of emergent convergence. Forming middle ties between Chicken and Leader makes “Volunteer’s Dilemma” (see Fig. 4), in which all players would like something done but prefer that the other player does it (Diekmann 1985). Volunteer’s Dilemma was also one of the entries in An Atlas of Interpersonal Situations (Kelley et al. 2003). Volunteer’s Dilemma also bridges between families descended from Primal Coordination and Primal Exchange.

Asymmetry, mismatched externalities, and structural bias

Because many situations in life are not symmetric in terms of opportunities or results, asymmetric games deserve more attention for understanding social situations and collective action (Thurow 1975, Ernst 2005, Hauser et al. 2019, Nockur et al. 2020). Asymmetry is relevant to relationships of inequality such as parent-child, teacher-student, supervisor-subordinate, principal-agent, and ruler-ruled, and is also relevant to differences related to factors including, but not limited to, power, knowledge, gender, race, ethnicity, class, education, productivity, and wealth. Examples in social-ecological systems include head- and tail-enders in irrigation (Ostrom and Gardner 1993, Janssen et al. 2011), and unequal access to electricity supply for irrigation pumps (Kimmich 2013).

“Jekyll-Hyde” (see Fig. 4) is a particularly interesting asymmetric game discussed by Kelley et al. (2003) in terms of threat dynamics and has distinctive externalities analyzed by Robinson and Goforth (2005). Robinson and Goforth (2005) examined how the arrangement of incentives and externalities could create common or conflicting interests, systematizing earlier work by Schelling (1960) and building on Greenberg’s (1990) analysis of inducement correspondences. In contrast to situations of pure common interest or complete conflict, Robinson and Goforth (2005) identified “Type” games, such as the payoff structure in Jekyll-Hyde, which exemplify mismatched situations. For each of the other person’s choices, one actor’s incentives always induce moves that give the other a higher payoff, whereas the other person’s incentives always induce moves that give the first person a lower payoff. One person’s choices would always have positive externalities and the other’s choices would always have negative externalities. In a sense, incentives make one cruel and one kind, one harms and one helps.

In Jekyll-Hyde, both players have dominant strategies, and this structure favors one actor who gets their best outcome while the other does worse. Kelley et al. (2003) analyze this situation in terms of the threat that the other actor might be able to make. This situation resembles the Basic Threat game discussed earlier and the Strict Threat game studied by Guyer and Rapoport (1970). In some cases, the actor who does worse might accept the inequality as an act of “loyalty.” However, with communication or repeated interaction, the disadvantaged actor might threaten to deprive the first actor of their best outcome unless given a more equitable result, i.e., “justice.” An agreeable solution might be achieved, for example, by taking turns, using side payments, or changing rules to transform the situation into a win-win game.

In the asymmetric game “Advantage,” only one actor has a dominant strategy and they end up doing best at equilibrium. This is a descendant of Primal Help. Advantage can reflect a crude model of rent-based accumulation in which “the rich get richer.” It is an intermediate archetype for the “Protector” game used by Snyder and Diesing (1978) in their models of international relations. Advantage and Jekyll-Hyde are part of a large set of what we call “bias games.” In this type of asymmetric situation, dominant strategies for one or both actors create a single equilibrium that favors one actor while the other does worse. These have been called “suasion games” (Martin 1992) because the dissatisfied or “aggrieved” (Stein 1982) actor may try to change the game through persuasion or other means. They have also been called “Rambo” games (Zürn 1993, Hasenclever et al. 1997, Holzinger 2008) because one actor can get their way without having to compromise.

Asymmetric situations with an unequal equilibrium outcome can make the disadvantaged actor want to find a way to change the outcome, by persuasion, threats, requesting compensatory side payments, or changing the rules to obtain a better outcome. In the topology of possible 2 × 2 games (Appendix 1), bias games, i.e., those with dominant strategies leading to unequal outcomes at a single equilibrium, are more common than games with equal equilibrium payoffs. In the payoff space of possible 2 × 2 games, the inequality (distributional) problems of bias games, as exemplified by Advantage and Jekyll-Hyde, are much more prevalent than the efficiency problems of Pareto-inferior equilibria in tragic dilemmas and assurance problems. Bias games are another example of how a better menu of models can contribute to understanding institutional diversity in power, conflict, and unequal results.

The 2 × 2 games show cross-cutting problems of collective action (Holzinger 2008): failure or success in achieving Pareto-optimal results; assurance or disagreement problems in choosing among multiple equilibria; and unequal distribution of benefits; as well as instability and zero-sum conflict. The topology of payoff swaps in 2 × 2 games displays similar differences between games according to the presence, Pareto-efficiency, and distribution of equilibrium payoffs (Robinson and Goforth 2005, Bruns 2015). Analysis of archetypes (e.g., Fig. 3; Figs. A1 and A2 in Appendix 1) reveals how families of games descended from primal archetypes display overlapping types of collective action problems:

DISCUSSION

Archetypes in changing systems

Archetypes have been applied to recognize and analyze recurrent patterns in the dynamics of systems (Senge 1990, Kim and Anderson 1998). In systems dynamics, archetypes highlight typical positive and negative feedback loops. Overuse and deterioration of a shared resource, as in a tragedy of the commons, is one archetype; another is competitive escalation of threatening actions with potential collapse, as in the game of Chicken; a third is breakdown in trust, as in Stag Hunt and Assurance problems. Archetypal patterns show how deliberate actions can have unintended consequences. Reinforcing feedbacks can create additional problems, whereas balancing feedbacks can restrict achievement. System archetypes reveal how a more holistic perspective, looking over time and across boundaries, may help in designing suitable solutions (Wolstenholme 2003). System archetypes offer generic models of processes that can be adapted to analyze current conditions or assess planned changes (Braun, unpublished manuscript https://www.albany.edu/faculty/gpr/PAD724/724WebArticles/sys_archetypes.pdf). Transformations between archetypes illustrate changes in the structure and dynamics of systems (Greenwood and Hinings 1993).

As examples for social-ecological systems, analysis using system archetypes can offer a holistic understanding of limits to growth and related tragedy of the commons problems in water resources management, providing insights into options for management and monitoring (Bahaddin et al. 2018). Comparative analysis of pasture social-ecological systems illustrates how archetypes can provide more general insights into the dynamics of problems and potential solutions, including whether archetypes of different system problems are linked or independent (Neudert et al. 2019). As with generic system dynamics archetypes, the archetypal games identified here can provide a menu of models for use in analyzing social-ecological systems.

Archetypal models of interdependence offer insights into incentive structures, their dynamics, and potential transformations. Archetypes can help to go beyond a tendency to concentrate on the (often misdiagnosed and not inevitable) tragedy of the commons and its two-person analog, Prisoner’s Dilemma, and instead consider a broader range of models for situations and the challenges and opportunities they pose for environmental governance, including, but not limited to, various coordination and defection problems. Archetypes can act as building blocks or components to understand forces favoring and hindering cooperation, not only for static equilibrium situations, but also as archetypal models of potential pathways for transformation, such as the following.

Organizational configurations in an ecology of games

Researchers have used archetypes to characterize businesses and other organizations and the relationships between organizations. Such archetypes comprise sets of variables based on typologies deduced from theoretical concepts or from taxonomies inductively based on empirical observation (see reviews by Miller and Friesen 1978, Greenwood and Hinings 1993, Meyer et al. 1993, Short et al. 2008, Misangyi et al. 2017). The difference between coordination and defection problems appears in how relationships within or between firms may depend on convergent or divergent incentive structures (Grandori 1997, Meuer 2014). In situations with convergent incentives (as with descendants of Primal Coordination and Primal Independence), communitarian strategies that emphasize information sharing, teamwork, coordination, and trust may lead to better performance. Other situations have divergent incentives, such as temptations to defect from cooperation (as with descendants of Primal Exchange). These situations might be controlled better through bureaucratic structures and procedures if problems and outcomes are predictable. However, when situations with divergent incentives are more complex or outcomes are more uncertain, an alternative solution may be to create shared incentives through ex-post benefits from property rights, as in a joint venture.

Qualitative comparative analysis provides a way to analyze organizational archetypes in different situations, looking at strategic configurations of organization attributes together with information on contexts and performance. As a means of understanding institutional diversity, qualitative comparative analysis can examine which sets of conditions are causally necessary or sufficient for outcomes and which are complements or substitutes, for example, in assessing the effectiveness of organizational strategies (Ragin 2008, Fiss et al. 2013, Grandori and Furnari 2013, Greckhamer et al. 2018, Villamayor-Tomas et al. 2020). Qualitative comparative analysis and game theory can also be used to study networks of games in social-ecological systems (Kimmich and Villamayor-Tomas 2019).

Differences may exist in the prevalence of coordination or defection problems in an interorganizational ecology of games in water governance, and such problems could affect the choice of institutions for coping with such challenges (Long 1958, Berardo and Scholz 2010, Lubell et al. 2010, Berardo and Lubell 2019). Thus, in networks of organizations, relationships may mainly concern building bridging social capital for mutual understanding and trust between groups or may instead emphasize bonding social capital within groups to overcome temptations to defect from cooperation. When adjacent action situations are interdependent in networks (McGinnis 2011), different situations may face different problems for collective action, and solving coordination problems may also resolve adjacent social dilemmas (Kimmich and Sagebiel 2016, Kimmich and Villamayor-Tomas 2019). Archetypal configurations and their potential prevalence in different contexts may help to explain observed patterns of organizational behavior, illuminate challenges and potential solutions, and contribute to designing improved institutions and monitoring their outcomes.

Elementary patterns of power in governance

The primal archetypes exemplify various forms of power: “power to,” “power with,” and “power over.” Primal Independence exemplifies “power to” as capability or freedom to act independently on one’s own (Sen 2000, Nussbaum 2011). Primal Coordination typifies “power with” to achieve mutual outcomes through cooperation (Follett 1924, Ostrom 1997), as in joint production and coproduction. The symmetrically balanced power in Primal Exchange is technically a reciprocal form of “power over.” However, as long as it stays balanced, it could function as “power with” to obtain mutual gains in an equitable partnership. Maintaining such a potentially precarious balance could depend on mutual adjustment, including voice and exit, as with responsive governance and availability of competitive options (Polanyi 1951, Ostrom et al. 1961, Hirschman 1970). “Power with” could also occur if “power over” is combined with “power with” in an asymmetric but still somewhat balanced way, as when coordination and exchange payoff patterns blend together in Primal Favors. One partner could ensure that the other partner gets a best outcome, but they need to have the favor returned through the other’s choice. Primal Favors could also generate a variety of asymmetric situations with more unequal results (see Appendix 1). Another three primal games offer exemplars of asymmetric “power over”: enabling a choice to win in Primal Help, determining that both actors win in Primal Gift, or determining that one actor gets their best outcome and the other their worst in Primal Win-Lose.

The three symmetric primal archetypes also offer elementary models of principles for organizing social order based on liberty, association, and exchange. Asymmetric power could enable autonomy, ensure a benevolent outcome, or despotically determine who wins and who loses. The primal archetypes for asymmetric power can model governance in ruler-ruled relationships in which the choices of rulers who hold power (executive, governor, collective choice in a legislature, parent, etc.) grant permission, impose a preferred outcome, or prohibit someone from getting their preference. In institutional grammar, deontic rules determine whether someone subject to the rules may, must, or must not do something (Crawford and Ostrom 1995). Thus, archetypal games illustrate elementary relationships that shape social order and governance. These relationships contrast with the cyclic instability and zero-sum opposition of interests in Primal Conflict, as in a “war of all against all” (Hobbes 1651, Ostrom et al. 1992, Ostrom 1997). Archetypal games offer models to help understand and diagnose governance relationships while showing how simple archetypes generate a diversity of more complex situations.

CONCLUSIONS

Analysis of archetypal games affirms the value of distinguishing coordination problems of equilibrium selection from defection problems in social dilemmas and similar situations while tracing how various kinds of coordination or defection problems are related. In turn, these archetypes are part of a larger diversity of situations that pose challenges and opportunities for collective action, including cyclic conflict, asymmetric power, and structural advantage and disadvantage in opportunities and results. Archetypal games show convergence of ideas and potential for further application in understanding transformations in system dynamics; coordination, defection, and other types of problems among networked organizations; best response, coordination, and externality dimensions of behavior in situations of interdependence; and diversity of elementary patterns of power in governance.

Making ties (indifference) between two outcomes for each actor in strict ordinal 2 × 2 games forms archetypes for harmony, social conventions and norms, compromise on second-best, rivalry, Assurance/Stag Hunt tensions between caution and trust in cooperation, and structural bias of advantage and disadvantage. Simplifying payoff rankings to like two outcomes and dislike two outcomes yields three primal archetypes for independence, coordination, and exchange. These symmetric patterns of power combine to create asymmetric models for conflict and structures of power and dependence. Breaking ties differentiates primal archetypes to generate further diversity in interdependence. Half of the primal archetypes generate strict games with relatively good results at equilibrium (best or second-best) while the other half generate a diverse and less stable set of games, most of which yield poor outcomes or severe inequality. Families of games that are descended from primal archetypes display overlapping collective action problems, including risk and rivalry in equilibrium selection, externalities of defection from exchange, partial harmony of good and best, zero-sum opposition of interests, focal points for cooperation, and the prevalence of unequal outcomes at equilibrium.

Simple two-person two-choice games, with ties showing indifference between outcomes, model archetypal situations of interdependence. These archetypes offer insights into similarities, diversity, and potential transformations in social-ecological systems, challenges for cooperation in resource management, and opportunities for improving environmental governance. Archetypes offer starting points and building blocks for understanding more complex situations, networks of action situations, and comparative analysis. Archetypal games can help expand thinking and analysis beyond a few famous games and trace cascading connections in a menu of models for thinking about the diversity and dynamics of interdependent relationships.

RESPONSES TO THIS ARTICLE

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ACKNOWLEDGMENTS

Earlier versions of this paper were presented at the International Association for the Study of Commons, July 1–5, 2019 in Lima, Peru, and the 3rd Workshop on Archetype Analysis in Sustainability Research October 30–November 1, 2019 at Palacký University Olomouc, Czech Republic. The authors express their thanks for comments from Regina Neudert, Matteo Roggero, Santiago Guisasola, Jacob Crandall, and anonymous reviewers, including particularly detailed and helpful comments by one reviewer, as well as comments by Paul A. M. van Lange on a related paper. Open access funding was provided by the Institute for Advanced Studies (IHS), Vienna, Austria.

DATA AVAILABILITY

Appendix 1 and other supplementary information, including spreadsheets with data and figures, are openly available in the osf.io repository at https://doi.org/10.17605/OSF.IO/XTR4N.

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Correspondent author:
Bryan Bruns
bryanbruns@bryanbruns.com
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