Intermediate flexibility in cooperation games prevents free riding,polarization,and societal disintegration

Abstract

We explored the properties of a computational model, “APEM,” inspired by empirical work on human group behavior in a coordination game called “The Slider Game.” Players in a network manipulate sliders on computer screens in an effort to achieve across-the-board aggregation on the same side of the slider. The model is an agent-based dynamical system. We studied the model’s behavior under manipulation of a control parameter called “flexibility,” which determines the willingness of players to adjust their behavior as a function of their agreement with their neighbors. Optimal coordination occurs only in a narrow range near the middle of the parameter space (a “Principle of Intermediacy”). Additionally, the model exhibits two types of non-successful asymptotic states, called “fraught” and “obdurate,” as well as a condition of prohibitively slow convergence where its behavior is very noisy. A review of the anthropological literature provides evidence that each regime of failure is a simple version of an empirically observed condition of flummoxed coordination in human societies: polarization, free-riding, and infrastructure disintegration under societal demise, respectively. Although each of these phenomena has been individually studied at some length, we are aware of no prior integrative treatment.

Significance statement

Successful group coordination often depends upon individuals being flexible enough to compromise, yet not so flexible that they fail to achieve any structured goals. We introduce a formal model of consensus formation that reveals conditions of optimal flexibility for group members interacting via a network. In the model, failure to achieve consensus can occur due to inflexibility—leading to polarization or free-riding-like behavior—or pervasive caprice, which causes severe communication breakdown. The model places polarization, free-riding, and social disintegration in a single explanatory framework. Simulations indicate that intermediate flexibility yields optimal coordination across a range of sparse connectivity patterns.

Introduction

A key feature of social systems is their ability to achieve coordination of individually motivated members. In this paper, we investigate the role of flexibility, where members adjust individual goals in service of attaining coordination. Intuitively, social systems with members that are inflexible will coordinate poorly. Individuals will prioritize their own well-being above that of the group, and coordination efforts may result in polarization of the group (Cinelli et al., 2021; DiFonzo et al., 2013; Eschert and Simon, 2019; Hewstone et al., 2002; McPherson et al., 2001; Neal, 2020; Sieber and Ziegler, 2019; Yahani et al., 2022). On the other hand, if members are too flexible, such that they change course at the slightest suggestion, then coordination can also fail because there is not enough consistency to achieve any structured goals.

We suggest that effective coordination requires striking a balance between these extremes—community members must get the level of flexibility “just right.” These qualitative arguments identify a commonsense insight (“compromise judiciously”) that many real-world negotiators might endorse (Gutmann and Thompson, 2012; Wolak, 2022). However, because it is a qualitative insight about what is arguably, a quantitative question, its practical usefulness is limited. Here, we report on a formal model of group coordination that suggests a way of exploring conflict resolution via judicious compromise with mathematical precision.

The model is based on experiments with humans playing a simple coordination game called the Slider Game, where players attempt to reach consensus with limited knowledge of other players’ actions (Tabor et al., 2024). In the Slider Game, players often experience conflict when clusters of contradictory opinions result in a polarized state we refer to as fraughtness. To resolve fraught states, the players must compromise by temporarily worsening their agreement. To explore the conditions under which successful compromise occurs, Tabor et al. (2024) proposed a computational model. The model, called the Agreement-Position Emigration (or APEM) model is an agent-based model that implements a form of binary-opinion formation among nodes in a connected network. It has a real-valued parameter, a _θ , which we call the flexibility parameter, that specifies agents’ willingness to change their opinions. a _θ ranges in the real interval [0, 1] where 0 corresponds to maximal rigidity and 1 corresponds to maximal flexibility. For simplicity, we focus here on cases where all agents have the same flexibility setting.

APEM is a continuous dynamical system whose behavior is organized around various attractors. ¹ This paper reports simulation evidence for three points which connect to matters of anthropological and sociological interest:

1. Fraught states can act as attractors in the model. As such they constitute one type of impediment to successful coordination and provide a formal analog of societal polarization.

2. In addition to fraught attractors, there is another kind of attractor in the model, which we dub an obdurate attractor. Obdurate attractors are states in which one or more players could improve agreement with their neighbors by switching sides, but are unwilling to do so. Tabor et al. (2024) did not observe obdurate attractors because they focused on parameter settings where these do not occur. Here, we show that, for many values of the flexibility parameter, a _θ , obdurate attractors are common. We suggest that certain obdurate attractors provide a natural model of free-riding in human societies.

3. a _θ , has a critical value, a _c (0 < a _c < 1), such that performance is optimal at a _c because the model achieves 100% successful coordination in the shortest possible amount of time. As a _θ grows linearly above a _c , the amount of time it takes to reach solution grows at least exponentially. The high convergence times for a _θ ≫ a _c stem from erratic direction changes by the players which hamper effective coordination. We refer to this phase as the long-transient regime and suggest that it provides a natural analogue of societal demise and collapse.

The optimal setting, a _θ = a _c has some properties in common with criticality in self-organizing systems and the related notion, Edge of Chaos (Teuscher, 2022). It would be valuable to know if this setting is a critical state, but at present, we do not know its status in this regard. We discuss this point further in General Discussion.

In sum, we suggest that three behaviors of the model provide simple formal analogs of polarization, free-riding, and societal collapse. A strength of the model is that it unifies these three kinds of challenges to social coordination, each of which has been independently observed. Because we empirically observe optimal behavior at or near a _θ = a _c and performance deteriorates rapidly on either side of the optimal region, we propose a Principle of Intermediacy for optimal coordination. Understanding why there is a principle of intermediacy and where the optimum lies could have practical value for groups navigating complex coordination.

The Slider Game

Players (typically 7 in runs with humans) are seated at separate computers, all of which are connected to the same network. Each player’s monitor displays a slider that can be moved incrementally to the left or right by pressing arrow keys on the keyboard. The players are told that the goal of the game is for everyone’s slider to reach the same side. Let $\vec{s}\in {[-\mathrm{1,1}]}^n$ denote the set of player positions, where −1 corresponds to the left endpoint of the slider, 1 corresponds to the right endpoint, and the other points are linearly interpolated. In other words, the player positions lie in a hypercube in n dimensions. To win the game, a group of players must arrive at one of the two consensus corners (also called “success states”), $\vec{s}={\{1\}}^n$ and $\vec{s}={\{-1\}}^n$ . The orthants containing these corners, called the consensus orthants, are denoted ${\mathcal{O}}_{-}$ and ${\mathcal{O}}_{+}$ , respectively.

In a given game, the network has a fixed topology. Figure 1 shows two topologies used in the human trials. ² Let N _i be the neighborhood of player i. |N _i | denotes the cardinality of player i’s neighborhood. Equation (1) defines a _i , the agreement of player i.

$\underline{\text{Def.}}(agreement\text{of\hspace{0.17em}player}i):a_i=\frac{|{\sum }_{jϵN_i}\hspace{0.17em}{s}_j|}{|N_i|}$

(1)

Figure 1.

Examples of network topologies used in human experiments. (a) Hierarchy with apex player (player 1). (b) Full topology.

Note that a _i lies in [0, 1]. Agreement will be low if all players in a neighborhood adopt positions near the middle of the slider. It will also be low if roughly half the players are near the left end and half are near the right end. Agreement will be high if all players are near the same side, but it can also be high if the neighborhood is large with most players near one side, and just a few near the other. In addition to showing their slider, each player’s screen shows their agreement value as a percentage of 1. This value is updated in real time. The players are told that each of them is linked to a subset of the other players and that a value called Agreement displayed on their screen will continuously indicate their degree of coordination with their neighbors. They are told that if all group members achieved 100% agreement, then they win the game. A time limit of 5 minutes or 3 minutes was imposed. Subsequent runs with players solicited via a cloud worker platform (Amazon Mechanical Turk) produced similar results (Tabor et al., 2024).

In sum, the only relevant information that the interface allows players of the Slider Game to observe is the position of their slider and their current Agreement value. The game has commonalities with other social coordination games in that the players do well if they align their actions, but unlike typical games studied in the Rational Choice paradigm, there is no explicit utility function that the players can reason about. Instead, the players have only local information about their state and this information is updated in real time. In particular, the players do not have information about the network topology or their position in the topology. The game dynamics are thus effectively focused on a continuous-time negotiation among the players to decide what form their coordination will take (either all-to-the-left or all-to-the-right).

Fraughtness

As Tabor et al. (2024) discuss, despite the lack of an explicit utility function, there are some network topologies of the Slider Game that, like some versions of the Prisoner’s Dilemma, exhibit non-Pareto-optimal Nash equilibria. These are called fraught states, and they arise when different clique-like subparts of the network are on opposite sides of the slider.

Formally, let $\vec{s}\in R^n$ be a vector of player positions with s _i different from 0 for all i. For i ∈ {1, …, n}, let friends(i) be the indices of the neighbors of i, not including i, whose position has the same sign as i, and let enemies(i) be the indices of i’s neighbors with opposite sign. Let F _i = |Σ_{j∈friends(i)}s _j | and E _i = |Σ_{j∈enemies(i)}s _j |.

Def. (fraughtness). If s _i ≠ 0 for i = 1, …, n, $\vec{s}\notin {\mathcal{O}}_{+}$ , $\vec{s}\notin {\mathcal{O}}_{-}$ , and, for each i, F _i − E _i ≥ 0, then the network is in a fraught state. For any state, the value y _i = F _i − E _i is called the alignment asymmetry of i.

Figure 2 shows the Hierarchy network of Figure 1(a) in a fraught state.

Figure 2.

Example of a fraught state. Players {2, 4, 5, 6, 7} have no enemies and at least one friend so they all have positive alignment asymmetry. Moreover, F₃ − E₃ = 2 − 1 = 1 and F₁ − E₁ = 1 − 1 = 0. Since all players have non-negative alignment asymmetry, the state is fraught. The numbers in square brackets below each node give the corresponding player’s agreement value.

Prior human experiment results and model

Tabor et al. (2024) report several results with human participants that highlight the interest of studying fraught states in the Slider Game. Overall, the groups were quite successful, but they were significantly faster with Hierarchy than with Full, even though Full has no fraught states and the Hierarchy groups spent on average 11.6% of their time in fraughtness. Tabor et al. (2024) suggest that the problem for Full is loss of the signal in the noise: if all the players are moving most of the time (as the human players on the Full topology are prone to do), and each player is continuously receiving input from all other players, then it is very difficult for an individual player to determine the effect that their own actions are having on their agreement. By contrast, in Hierarchy, the players interact in smaller groups. The smaller groups incur the possibility of fraughtness which impedes success, but they also make it much easier for players to detect their effects on agreement and mutually work out differences—on average, the benefits of easier local coordination appear to outweigh the challenges of fraughtness.

Simulation experiments

Tabor et al. (2024) defined the “Agreement-Position Emigration” (APEM) model, an agent-based model of Slider Game play. They found that the profile of APEM trajectories involving high fraughtness in the Hierarchy Topology closely resembled the profiles of the human groups that experienced high fraughtness in that topology.

Model definition

An APEM model, M, is a stochastic dynamical system defined by

$M=(R,B,G)$

(2)

where R is the model’s state space, B are the model’s scalar parameters (μ, maxtime, a _θ , s _θ —explained below), and G is an undirected, connected network. ³ R, the state space, is defined by three vectors: player positions, $\vec{s}\in \mathit{S}={[-\mathrm{1,1}]}^n$ , player directions, $\vec{\varphi }\in \mathit{\varphi }={\{-\mathrm{1,1}\}}^n$ , and player timer states, $\vec{\tau }\in \mathit{\tau }={\left[0\right.,\left.\infty \right)}^n$ :

$R=(\mathit{S},\mathit{\varphi },\mathit{\tau })$

(3)

The model players generally move at a constant speed to the left or right on the slider. The directions indicate “heading left” (−1) and “heading right” (1). At the start of the simulation, the timer states are drawn from a negative exponential distribution (specified by a parameter, μ, the mean of the distribution). Going forward, the timer states decrease at a constant rate until the minimum timer value reaches 0. At this point, the player whose timer reached zero evaluates its current state and decides whether to maintain or change its current direction. This player will flip its direction precisely in case two conditions are met: (i) its agreement is less-than-or-equal to the flexibility threshold, a _θ , and (ii) the absolute value of its position is greater-than-or-equal-to a value, s _θ , called the position threshold. Then, the timer process repeats. Each game proceeds until the players arrive at a consensus state or the time of play exceeds a pre-specified time limit. (See Supplemental Materials: APEM Model Specification). The flexibility threshold and position threshold criteria are designed to implement a plausible human strategy: ⁴ precisely in case one has made a strong commitment to a side (high s _i ) but agreement is low (low a _i ), a switch in preference is warranted (Tabor et al., 2024), ⁵ With a high value of a _θ , players are relatively flexible—they are willing to switch directions even if their agreement value is quite good, while with a low value of a _θ , their agreement has to be rock-bottom for them to be willing to switch.

Distribution of consensus states and fraught states across topologies

Our models’ network topologies, (G), were randomly generated using a variant of the Erdös–Renyi–Gilbert (ERG) method. For a network of N nodes, this method considers every pair of nodes in turn, and places an edge between the pair with probability, P(edge). Since we were only interested in connected topologies, we ran the ERG method repeatedly until a connected network was produced. Figure 4 shows some topologies generated by this modified ERG method. Table 1 summarizes the model parameter values used in the paper.

Table 1.

Unless otherwise specified, our models used these parameters. Note 50 values of a _θ are explored, ranging from 0 to 0.98, in 0.02 increments—we did not simulate the special case a _θ = 1.

Model parameter	Symbol	Value
Flexibility threshold	a θ	[0.00, 0.02, 0.04, 0.06, …, 1]
Position threshold	s θ	1/2
Mean of negative exponential	μ	1/10
Network	G	ERG sampling until connected
Edge probability in ERG	P(edge)	[0.05, 0.10, 0.15, …, 1]
Inherent player speed (when direction is not changing)	r	1
Maximum run time	maxtime	300
Number of runs per a θ value	nruns	200

Due to our interest in games where players experience and resolve fraughtness, we systematically explored performance across networks generated with a range of edge probabilities. We wanted to identify edge probabilities such that networks are (1) prone to fraughtness at some a _θ value, and (2) capable of high success rates at that a _θ value so that we could study how modulation of a _θ adjudicates between fraughtness and success. For each P(edge) value in the set [0.05, 0.10, 0.15, …, 1], we generated 20 connected topologies. For each topology, we performed 100 model runs for a wide range of a _θ values. We tracked how many of the 100 games ended in a consensus state (success rate) and the portion of time that the players spent in fraught states (fraughtness rate) across the 100 games. In Figure 3, we report the highest success and fraughtness rates for each topology tested. In line with the difference between human performance on Hierarchy (12% edges included) and Full (100% of edges included), the topologies with fewer connections show more success and experience fraughtness more often. Since maximal success is essentially 100% for values of a _θ below 0.5 and fraughtness rises to its maximum as a _θ approaches 0, we can find networks that support both high fraughtness and high success by picking a _θ near 0. Therefore, we fixed the values of the other model parameters (s _θ = 1/2, μ = 0.1, maxtime = 300) and studied success and fraughtness of the final state as P(edge) ranged from 0.05 to 0.2 and a _θ ranged over its possible values (a _θ ∈ [0, 1]).

Figure 3.

(a) Reports the success rate (portion of games that ended in a consensus state) recorded for each topology’s best performing a _θ value. (b) Reports the average fraughtness rate recorded for each topology’s a _θ value that resulted in the highest fraughtness. All topologies included 10 players. Similar results were found for topologies with 7, 15, and 20 players. The box and whiskers illustrate standard quantile ranges, with outliers presented as circles.

Effects of flexibility

Figure 4 shows success and fraughtness of the final game state as a function of the flexibility parameter, a _θ , for an illustrative sample of topologies. Each data point on each graph shows an average over 200 runs. The first two examples (P(edge) = 0.1, P(edge) = 0.15) are in the class of cases where maximal success and maximal fraughtness are both relatively high. The graphs reveal several simultaneous discontinuities in the success and fraughtness rates. Such discontinuities suggest bifurcations. Also, for the P(edge) = 0.15 example, there is a section in the middle, between two discontinuities, where the fraughtness and success rates sum almost exactly to 1. We refer to this pattern as mirroring. Mirroring of fraughtness and success indicates that fraughtness is the main impediment to success. The third example (P(edge) = 0.45) illustrates relatively low fraughtness and success for a highly connected topology.

Figure 4.

Three example topologies and their performance in APEM. (a) (P(edge) = 0.1) Maximal success is 100% over a wide range of a _θ values; abrupt change in model performance at a _θ = 1/3. (b) (P(edge) = 0.15) Maximal success is 100% over a narrower range of a _θ values; abrupt changes at a _θ = 1/3 and a _θ = 1/2; mirroring of success and fraughtness for a _θ ∈ [1/3, 1/2]. (c) (P(edge) = 0.45) Much lower maximal success (73%); very little fraughtness; smaller discontinuities. (In the topology graphs on the right, the self-connections on all nodes are not shown.)

Pursuing our hypothesized Principle of Intermediacy, we studied, as a function of the flexibility parameter, a _θ , the success profiles of 20 topologies generated with each of the P(edge) values in {0.05, 0.1, 0.15, 0.2}. Figure 5 shows a sample of results. Every topology shows a peak of success in the interval, a _θ ∈ [1/3, ≈2/3] and the average success curve for each P(edge) value falls away on either side of the peak nearly monotonically, consistent with our hypothesis of a unimodal optimum.

Figure 5.

For each P(edge) in the set [0.05, 0.1, 0.15, 0.2] we randomly generated 30 topologies and tested their performance with 100 games over a full range of agreement values. Thick lines illustrate average success rate for each P (edge) value. Thin lines illustrate individual profiles (10 randomly selected from each set of 30).

To further explore the observation about mirroring, we made a mirroring assessment of 200 games for each of 20 topologies generated with P(edge) = 0.1 (Figure 6). Each panel shows the sum of the success and fraughtness rates at game-end across the range of a _θ values. In each case, the left-most point of pure success (i.e., the left-most point of each green curve) is directly adjacent to a region of high fraughtness. Indeed, in 71 out of 80 (89%) of the topologies we tested, fraughtness made an abrupt descent (to 0%) within 0.02 a _θ units of the first high-success value. ⁶ These observations suggest that fraughtness, though undesirable, is closely linked to success.

Figure 6.

Evidence that fraughtness is the primary impediment to success for intermediate values of a _θ and that perfect success generally abuts the region of success-fraughtness trade-off. Each curve shows the sum of the portion of runs that ended in fraughtness and the portion that ended in success (a “mirroring analysis”). The curve is plotted in shades of red if the fraughtness rate is nonzero. It is plotted in shades of green if, among success and fraughtness, only success is present. To facilitate interpretation, the shading shows the same information as the curve height.

This section has provided evidence that success is maximal at intermediate values of a _θ and that success and fraughtness account for the bulk of the outcomes in the middle range, but not at the extremes. We would like to understand what causes the intermediacy effect, how to determine which values of a _θ are best, and why fraughtness and success dominate the outcomes for middle values but not elsewhere. In an effort to answer these questions, we undertake a deeper analysis of the model.

Model analysis

Standard dynamical systems are defined on complete metric spaces. A closed set, T, in the metric space is called an attractor if, from most points close to T, the system converges to T. To make sense of the notion, “close,” it is convenient to apply the Taxicab Metric on ${\mathcal{R}}^n$ . With these assumptions in place, many states of the APEM model are attractors. For example, the consensus states are attractors. (Supplemental Materials: Consensus States are Attractors). ⁷

The consensus attractors are singleton attractors in the sense that they consist of a single point in S crossed with a single point in ϕ crossed with the simplex, τ . There are also manifold attractors whose position subspaces are higher (than 0) dimensional simplexes in R ⁿ . An example occurs in the 7-dimensional hierarchy topology shown in Figure 1(a) which, for 1/3 ≤ a _θ < 1/2, includes two attractors whose position-space components are 1-dimensional. As an illustration, consider the positional state shown in Figure 2, let the directional states be equal to the illustrated positional values and let a _θ = 5/12 = mean ([1/3, 1/2]) and s _θ = 1/2. APEM’s position will then stay indefinitely on the set ([−1, 1], −1, 1, −1, −1, 1, 1) (a 1-dimensional manifold) and it will return to this manifold if moved slightly away from it. (See Supplemental Materials: 1-dimensional attractor).

Based on the mirroring analysis reported above, as well as the example just mentioned, we suspected that fraught states, when they exist, might be attractors. In order to efficiently explore this hypothesis, we defined a Simple Model whose dynamics approximate the dynamics of the Full Model. The Simple Model works like the Full Model except that instead of moving continuously across the slider, if a player qualifies to move, it jumps all the way to the other side of the Slider in one step. In Supplemental Materials we show empirically that even though the dynamics of the Simple Model only approximate the dynamics of the Full Model, in the cases we consider here, there seems to be a close relationship between the two models: all attractors of both models contain corner states, and for every attractor of the Simple Model, there is a corresponding attractor of the Full Model that has exactly the same corner states, and vice versa. This situation is convenient because, while we are not aware of an efficient way to discover all attractors for a Full Model, there is a simple, inexpensive procedure for discovering all attractors in the corresponding Simple Model. This situation supports discovering attractors in the Full Model by searching for attractors in the Simple Model. The main findings from implementing this method are as follows (See https://osf.io/w5rgx for code):

1. When a _θ is set to 0, the number of attractors is finite, positive and ≫ 2. As a _θ increases above 0, the number of attractors decreases monotonically until it reaches 2. The a _θ value at which this occurs is always strictly less than 1, is labeled a _c , and is called the resolution bifurcation point (rbp).

2. When the number of attractors is 2, the only attractors are the consensus states.

3. For a _θ ∈ [a _c , 1), the average convergence time grows at least exponentially in a _θ . (See Supplemental Materials: Post-resolution convergence time).

4. The non-consensus attractors are often singleton fraught states or manifolds of fraught states, but there is also another kind of attractor, which we call an obdurate attractor, and to which we now turn.

Obdurate attractors

Def. (obdurateness) An APEM positional state that is not a fraught state, not in a consensus orthant, and in which no player has their slider at 0 is called an obdurate state. An obdurate state that is also an attractor is called an obdurate attractor.

In an obdurate state, there exists an i ∈ {1, …, n} such that E _i > F _i —in other words, the alignment asymmetry, F _i − E _i is negative. For example, in Figure 8(a), the outlier player has more enemies than friends. This makes the whole state of the game obdurate. If an attractor consists entirely of obdurate states, we call it an obdurate attractor. (See Supplemental Materials: Obdurate Attractors). Further, upon investigating the networks we have studied, we find that many obdurate attractors exist, especially at low values of a _θ . Unlike fraught attractors, where no single player can improve their agreement by changing sides, obdurate attractors do contain such individuals. In such cases, it is natural to say the “blame” for being stuck in an obdurate attractor falls on individual players. This property of obdurate attractors lends itself to a comparison to free-riding behavior—see General Discussion.

An analysis of the Simple Model reveals the existence of consensus, fraught, and obdurate attractors in APEM. In a small subset of networks, there exist a fourth type of attractor that contains both fraught and obdurate states. We refer to this type as a mixed fraught/obdurate attractor (See Supplemental Materials: Mixed Fraught/Obdurate Attractor). With the discovery of obdurate attractors (and mixed attractors), we can extend the Mirroring Analysis of Figure 6 to a comprehensive characterization of the impediments to coordination for our sample of high-fraughtness, high-success topologies. Figure 7 shows end-state proportions as a function of a _θ for each of the 20 topologies generated with P(edge) = 0.1. Similar patterns were observed for P(edge) = 0.05, 0.15, and 0.2. In the bulk of cases, there is (i) an early phase where obdurateness dominates, followed by (ii) the previously observed “middle” phase where fraughtness is prominent, followed by (iii) a phase of only consensus attractors and rapidly increasing convergence time. The resolution bifurcation point, a _c , is at the boundary between phase (ii) and phase (iii). The large green swath in the middle of each plot reflects the Principle of Intermediacy.

Figure 7.

Portions of end-state types for each of the 20 topologies generated with P (edge) = 0.1. For each topology, a _c is the lowest a _θ value for which consensus is achieved 100% of the time. In the long-transient regime (a _θ > a _c ), transient final states become more and more prevalent as a _θ grows because the model increasingly fails to discover an attractor state before the time limit (maxtime = 300) is reached.

General discussion

In this section, we summarize our primary findings and then review results from the anthropological and sociological literature to address the question, “Is APEM a plausible model of human behavior?”

Summary

Via simulations, we studied APEM, an agent-based model of the Slider Game (Tabor et al., 2024) across a range of network topologies by manipulating its flexibility parameter, a _θ . We observed the following:

1. A Principle of Intermediacy: coordination was always very close to optimal at one particular value of a _θ , called a _c , which was strictly between the minimum value (a _θ = 0, corresponding to maximal stubbornness) and the maximum value (a _θ = 1, corresponding to maximal flexibility).

2. For values of a _θ below a _c , failure of coordination occurred because the model got stuck in attractors. Attractors exhibited two profiles—obdurate attractors, where one or more players could change sides to improve their agreement but refused to do so, and fraught attractors, where the network was divisible into two pools of opposite polarity, such that no player could improve agreement by switching sides.

3. For values of a _θ above a _c , the network would always achieve full coordination eventually (i.e., the consensus states were the only attractors), but the average convergence time grew, eventually very rapidly, as a _θ approached 1. We dubbed this parameter region the “long-transient” regime.

APEM and human behavior

APEM’s behavior is very simple relative to humans’. Nevertheless, the variety of APEM’s regimes encouraged us to investigate whether impediments to coordination in APEM correspond to those in humans in a systematic way. In Figure 8, we illustrate the four regimes of APEM, and their proposed societal analogs. We suggest that the obdurate players in point attractors that occur when a _θ is just below a _c are like free-riders—they benefit from the coordination of the players around them in the sense that they achieve a high agreement value on account of their neighbors’ coordination, but they themselves don’t contribute to that coordination; fraught attractors resemble entrenched polarization in the sense that two or more subgroups of players are stuck in states of opposing agreement; the long-transient regime resembles societal degradation in the sense that coordination persistently falters; and being at the critical value, a _θ = a _c , produces maximally successful coordination, a state of unusual synergy which human societies appear to achieve only rarely.

Figure 8.

Illustrations of APEM’s four regimes that we compare to common impediments in human coordination. (a) A network stuck in an obdurate attractor. All agents are in agreement except one. This agent could improve overall network agreement and individual agreement by changing its sign, but it won’t do so even after a small perturbation. (b) A network that is stuck in a fraught attractor. Agents get high agreement from neighbors but the network fails to escape fraughtness, even if slightly perturbed. (c) A network with a _θ far above a _c . The agents switch directions frequently in a way that generally flummoxes coordination, resulting in a long-transient. (d) A network with a _θ = a _c . This network was previously in a fraught state but is now very likely to converge to a consensus state. (The self-connections on all nodes are not shown.)

There is an immense literature on social coordination phenomena. Our comments here only address a few aspects of it. Nevertheless, we suggest that an integrative framework is worth considering: prior treatments have rarely, if ever, made a link between polarization, free-riding and societal degradation. Such a framework, if vindicated, may aid our understanding of the challenges of social change. In particular, such a unified framework highlights the way adjustments aimed at correcting one societal ill may, if carried too far, engender another one. Additionally, the framework points to the value of accomplishing precise quantitative modeling of social dynamics, because, it suggests, optimal behavior requires precise parameter tuning.

APEM, as we have defined here, does not have any parameter dynamics. To specify its relevance to human societies, we thus add some additional assumptions to the core framework provided by the model:

One assumption is that humans pay a cost whenever they alter their position—whether physically or mentally.

Our second assumption is that humans generally try to minimize cost, and thus would, if left to their own devices, set their flexibility parameter to 0. However, societies composed of individuals who are completely inflexible do not function very efficiently. Therefore, many types of behaviors have emerged to aid in increasing flexibility—some of which manifest in the culture of a group.

A third assumption is that flexibility tends to be noisy—societies have to work hard to maintain a steady value. Therefore, most societies, most of the time, reside in a region where flexibility is in the region just below a _c : coordination in this region is not perfect, but it is relatively strong, and there is no great risk of falling into societal collapse. We refer to this region as $a_c^{-}$ .

Based on the parameter space layout of APEM combined with the assumptions just described, we expect: (1) there should be cultural mechanisms that discourage selfish behavior in the service of coordination (these drive a _θ upward from 0); (2) under normal circumstances, societies should reside in the region $a_c^{-}$ and thus exhibit moderate degrees of free-riding and polarization; (3) special cultural mechanisms may drive a _θ from $a_c^{-}$ toward a _c nearly eliminating free-riding and polarization, but this will occur rarely; (4) societies can exhibit societal degradation and collapse; this should also be infrequently observed, because if it occurs, either the high flexibility gets quickly lowered or the society disappears.

We now review some anthropological evidence to assess these claims.

Obdurateness

First, we review evidence of cultural mechanisms that discourage selfishness. Food sharing is widespread among hunter-gatherers (Crittenden and Zes, 2015; Kishigami, 2021). In some groups—for example, the Ache of Paraguay—cultural norms discourage hunters from consuming any of their own kills (Kaplan et al., 1984, 1985). Among the !Kung of the Kalahari Desert, a successful day of hunting is met with playful insults rather than praise (Lee, 1976). These practices ensure that others in the tribe besides the hunters get enough to eat. Similar selfishness-prevention strategies have been observed in agricultural communities. Abramitzky (2018) reports that Jewish kibbutzim in the earlier 20th century pooled money earned outside the kibbutz by the labor of individual members and distributed it for the common good—individuals who violated the sharing requirement underwent severe censure.

Of course, the impulse to behave selfishly is not always successfully curbed. Resource pooling is vulnerable to cheating, and supervision of all members may be so expensive that the cost of completely eliminating free-riders outweighs the cost imposed by tolerating a small number of them (Maynard Smith and Price, 1973; Parker and Maynard Smith, 1990; Madgwick and Wolf, 2020). APEM provides a model of such tolerated free-riding with topologies that have obdurate point attractors in the region $a_c^{-}$ (Figure 7). The obdurate players in such states receive benefit (in the sense that they receive high agreement, specifically agreement $>a_{\theta }$ ) from coordination among their opposite-signed neighbors, but they do not contribute to this coordination themselves. Moreover, the fact that the states are attractors means that, even though the obdurate players hinder coordination, their effect is not so detrimental as to cause a shift in the status quo. In this sense, their free-riding behavior is “tolerated.”

However, there are less common cultural practices that come very close to achieving the ideal of complete participation. Anthropologists have often observed norms and rituals that impose great costs to the group and produce very little tangible benefits (Bird et al., 2001; Smith et al., 2003; Sosis and Alcorta, 2003). Celibacy, fasting, self-inflicted pain, and bans on certain food items are all examples of what are called, “costly signals.” Such behaviors seem, at first glance, to be maladaptive, but they make sense if we recognize their coordination-enhancing properties. Indeed, costly signals are especially prevalent among religious communities and are quantitatively associated with greater cooperation and longevity of communes (Ruffle and Sosis, 2007; Sosis and Bressler, 2003). In the framework we are describing, costly signaling is hypothesized to be a way to drive coordination above $a_c^{-}$ toward a _c .

Fraughtness

In the Slider Game, both human (Tabor et al., 2024) and APEM groups encounter fraughtness, where players become split into two or more subgroups with high local agreement and contrasting opinions. Similarly, real-world social networks often have clusters based on shared features (Apicella et al., 2012; McPherson et al., 2001; Redhead and Power, 2022), and such clusters often compete with one another for dominance and resources (Bissonnette et al., 2015; Hewstone et al., 2002).

As with obdurateness, a fraught group’s overall coordination suffers due to inflexibility of group members. However, the form of inflexibility differs from obdurateness because every member of the group is in high agreement with their local group members. Hyenas provide a clear illustration: they hunt in packs with great coordination, but once a kill is made, they divide into subgroups that compete over the kill (Smith et al., 2010).

APEM models in the $a_c^{-}$ range, which are prone to get stuck in fraughtness, can nevertheless achieve high average agreement since most players are surrounded by friends. Similarly, human groups may be characterized as fraught, yet be stable and high functioning. In traditional Melanesian society, intragroup alliances form around influential individuals called “big men” (Sahlins, 1963). The title of “big man” does not come with any authority over the tribe as a whole. Rather, a big man is supported locally by relatives and neighbors who, in exchange, receive protection and resources. Within a tribe, several big men will compete with each other by displaying their wealth with donations to other alliances. While such a system promotes resource pooling, Sahlins observed that big men pose problems to wider group cooperation by collecting resources that would otherwise serve the whole group. Such semi-effective social structuring like these thus aligns well with $a_{\theta }=a_c^{-}$ .

In APEM games where players are stuck in a fraught state, an increase in player flexibility is needed to resolve the fraughtness—a _θ must match or exceed a _c . But sustaining a _θ = a _c is costly. Because of this, we predict groups may resolve fraughtness by temporarily increasing flexibility. Indeed, anthropologists have noted that many rituals are characterized by a temporary change of ordinary social structure called liminality (Gennep, 1960; Turner, 1969).

Victor Turner’s account of the Ndembu of Zambia highlights the role of liminality in aiding a smooth transition of power, a potentially polarizing period (Turner, 1969). When a new chief is chosen, they only assume power after a long and intense ritual where they are sleep deprived, forced into physical labor, and berated by fellow tribe members for any previous wrongdoing. The soon-to-be-chief is told by the ritual officiate to surrender selfish intent and become an impartial servant of the peoples’ will. As is typical of such liminal phases, the normal strictures on behavior, in this case, behavior toward the chief-to-be, are temporarily relaxed. This process evidently helps the chief and the tribe transition to a new relationship which serves the tribe effectively once the chief takes on his leadership role.

Some liminal rituals are more frequent and provide regular opportunities for groups to address internal conflicts or important decisions. Consider the ritualistic consumption of kava, a mild sedative and euphoria-inducing beverage, among Fijians (Shaver, 2015; Shaver and Sosis, 2014; Turner, 1992). Kava is typically consumed in a group setting involving community members of different ranks whose interactions are normally strongly constrained. During the ceremony, social barriers are relaxed and the more relaxed interactions facilitate resolution of conflict and social mobility.

Such liminality cycles align with our prediction that groups might resolve intragroup conflict with practices that raise a _θ temporarily above its default $a_c^{-}$ value, and then lower it. Moreover, rituals tend to be either temporally brief or physiologically mild (Whitehouse, 2004), such that they soften, but do not fully dismantle restrictive social constraints; this property corresponds to the APEM Principle of Intermediacy.

In sum, fraughtness in APEM and intragroup conflict in human groups both result from group members who are unwilling to compromise on account of having relatively high local agreement. Additionally, temporary elevations of a _θ in APEM which resolve fraughtness bear a resemblance to liminality cycles by which human societies navigate challenging transitions.

The long-transient regime

As we noted in Introduction, for a _θ ≥ a _c , APEM has only consensus attractors and therefore will always eventually converge to a solution, but the convergence time grows very large as a _θ approaches 1. Practically, this amounts to another kind of coordination failure which we called “long transient” because it is brought about by players responding incoherently to the signals they receive from one another and therefore taking a very long time to reach consensus. We suggest that the long-transient regime models societal dysfunction and collapse.

Several authors define societal collapse as a condition in which once-vital complex societal mechanisms are lost in favor of simpler ones. Simplification is typically accompanied by human suffering and demise (Diamond, 2005; Tainter, 1988). A number of historical cases have been studied (Brosović, 2023; Tainter, 1988). As we noted in Introduction, basic APEM does not posit any mechanism driving the change of its parameters in a particular way, but the long-transient regime has the property that the communication that supports coordination at lower a _θ values is increasingly disrupted as a _θ grows above a _c . This variation across settings corresponds to a complexity difference (Bardi et al., 2019; Brunk, 2002; Livni, 2019; Tainter, 1988, 1995) in that the capacity of the model for meaningful information processing is reduced.

One subset of complexity-based models (Bardi et al., 2019; Tainter, 1988, 1995) attributes collapse to a complexity overrun—building complexity requires increasing energy and resource inputs; eventually the growth outruns the resources needed to sustain it. Bardi et al. (2019) argue for overshoot on the basis of a hysteresis effect that shows up in cases ranging from the fall of the Roman Empire to collapse of fisheries (Perissi et al., 2017; Sverdrup et al., 2012). Similarly, augmented APEM models societal dysfunction as the long transients that result from overshoot of mechanisms that drive a _θ from lower values toward a _c .

Explaining the regimes

We noted at the outset that prior treatments of challenges to collective behavior have focused independently on polarization, free-riding, and societal collapse. We argued that APEM is interesting because it unites the three. In closing, we take stock of how it does this. Self-organization can be defined intuitively as the situation in which local feedback interactions of many small parts give rise to organized structure at the scale of the group (Haken, 2008). With regard to polarization, when a self-organizing system has symmetries, then the system must break symmetry to arrive at solution. Since the system proceeds from the bottom up, if different parts of it break symmetry in different ways, polarization is a possible outcome. Relatedly, with regard to free-riding, each particular emergent structure in such a system is stable: therefore, if there are just a few players that prefer dissent—not enough to overcome the stability—then the system will persist in the near but not completely optimal condition of supporting free-riders. In other words, both polarization and free-riding stem from the positive local feedback loops which drive self-organization. The case of the long-transient regime is more esoteric. In the Slider Game, its possibility is related to the system geometry: there are two consensus corners on opposite ends of a high-dimensional space. Between the two is a 1-dimensional manifold which constitutes the shortest path between them. Self-organization chooses between the two consensus states via a random walk. In general, a random walk is capable of going anywhere in the space. However, it is most efficient if it stays on the shortest path between the consensus corners. Putting the system at the Resolution Bifurcation Point ensures that it does this.

We noted at the outset, that the Resolution Bifurcation Point has some properties in common with edge of chaos (EOC), or “criticality” phenomena (Teuscher, 2022). In particular, diverse parameterizable complex systems exhibit an intermediate sweet spot in their parameter space where system behavior is arguably optimal. EOC situations often involve networks with the following properties:

1. The systems consist of cells that communicate locally with one another to make global decisions.

2. The connectivity is sparse.

3. The sweet spot is at a boundary between order and disorder.

4. The sweet spot is a tiny region, often a single point.

5. The sweet spot is a phase transition point.

6. Behavior at the sweet spot is optimal in the sense that the system performs effective computation there.

7. If the system is adaptive, it gravitates to the sweet spot.

We have observed that (1)-(5) hold of APEM at the Resolution Bifurcation Point. Additionally, we have shown via simulation that (6) holds in the sense that APEM efficiently solves the Slider Game coordination problem at RBP. However, we have not demonstrated any other complex computation abilities. Regarding point (7), vanilla APEM is not adaptive, but in the beginning of General Discussion, we argued that an adaptive version may provide a helpful model of societal patterns.

Criticality is also distinguished by a number of technical properties:

1. Perturbations persist without blowing up or dying out.

2. The maximal Lyapunov exponent is 0.

3. The system exhibits long-range correlations.

4. The system exhibits power law distribution of damage spreading (Rohlf and Bornholdt, 2002).

It would be informative to determine whether APEM at rbp exhibits these technical properties. The technical properties point to ways of evaluating regime status in experimental and natural settings. If APEM at rbp is not critical, the commonalities may support useful generalization of the concept of EOC. We leave this evaluation to future research.

Conclusion

With a very simple set of assumptions, APEM derives a rich variety of collective behavior patterns which resemble documented human social phenomena. Prominent among these is the Principle of Intermediacy which posits optimal group behavior at an intermediate flexibility value, thus giving precise formal substance to the notion that compromise is a delicate balancing act. The model links polarization, free-riding, and societal disintegration phenomena, relating them all to core properties of self-organization. We have argued that the three patterns, obdurateness, fraughtness, and long transients, correspond to common collective organization impediments among humans. Future research should ask whether APEM covers the bulk of coordination challenges that human societies encounter.

Limitations

The model has some notable shortcomings. First, we have only stipulated, not derived the dynamics on the parameter space—it is desirable to develop a causal formal model of APEM’s parameter evolution. Also, APEM resides in an opinion-space—everything it says depends on the dynamics of opinion-formation. It seems evident that real human negotiation outcomes are affected by emotional states like trust, respect, embarrassment, fear, etc. We suggest that it will be helpful to think about how these properties can either be added to the model or understood as collective properties of a system like APEM. In another vein, the Slider Game stipulates that universal agreement is the goal, and we have evaluated APEM with respect to its success at achieving that goal. However, it is arguable that, at least in some contexts, diversity of opinion is more desirable than universal agreement. This suggests exploring an alternate model in which concrete problems must be solved (e.g., acquiring nourishment, navigating terrain, sustaining healthy relationships) and the collective effort is mediated by opinion dynamics—in some such cases, we might expect opinion diversity to be favored. Finally, we note that our treatment of free-riding as a feature of sub-critical obdurate attractors only seems to capture some properties of the way free-riding is currently understood. For example, free-riders are often understood as failing to pay costs that other players shoulder—this suggests identifying a property of players’ behavior—for example, their rate of direction change, or rate of travel—as costly. Also, non-free-riding players tend to discourage free-riding if it is detected. This suggests adding a detection element as well as a cost-imposition element into the model. We identify these issues as directions for future research.

Finally, when we introduced cultural considerations above (at the beginning of General Discussion) we adopted a standard, essentially Hobbesian view of the role of culture: left to their own devices, humans will not behave in particularly cooperative ways; the role of culture is to cajole them into cooperation so the benefits of complex coordination can be achieved. While our account does not deny the rough validity of this portrayal, in retrospect, it suggests a subtly different view: cultures are mechanisms for tuning the control parameters of a dynamical system—they do not simply curtail native tendencies; rather they modulate their expression so that groups go through productive episodes (like liminality cycles). APEM supports this novel conception by mapping out a landscape of coordination regimes on which such migrations may occur.

Supplemental Material

Supplemental Material - Intermediate flexibility in cooperation games prevents free riding, polarization, and societal disintegration

Supplemental Material for Intermediate flexibility in cooperation games prevents free riding, polarization, and societal disintegration by Matthew Conrad and Whitney Tabor in Collective Intelligence.