Conflict destabilizes social interactions and impedes cooperation at multiple scales of biological organization. Of fundamental interest are the causes of turbulent periods of conflict. We analyze conflict dynamics in an monkey society model system. We develop a technique, Inductive Game Theory, to extract directly from time-series data the decision-making strategies used by individuals and groups. This technique uses Monte Carlo simulation to test alternative causal models of conflict dynamics. We find individuals base their decision to fight on memory of social factors, not on short timescale ecological resource competition. Furthermore, the social assessments on which these decisions are based are triadic (self in relation to another pair of individuals), not pairwise. We show that this triadic decision making causes long conflict cascades and that there is a high population cost of the large fights associated with these cascades. These results suggest that individual agency has been over-emphasized in the social evolution of complex aggregates, and that pair-wise formalisms are inadequate. An appreciation of the empirical foundations of the collective dynamics of conflict is a crucial step towards its effective management.

Persistent conflict is one of the most important contemporary challenges to the integrity of society and to individual quality of life. Yet surprisingly little is understood about conflict. Is resource scarcity and competition the major cause of conflict, or are other factors, such as memory for past conflicts, the drivers of turbulent periods? How do individual behaviors and decision-making rules promote conflict? To date, most studies of conflict use simple, elegant models based on game theory to investigate when it pays to fight. Although these models are powerful, they have limitations: they require that both the strategies used by individuals and the costs and benefits, or payoffs, of these strategies are known, and they are tied only weakly to real-world data. Here we develop a new method, Inductive Game Theory, and apply it to a time series gathered from detailed observation of a primate society. We are able to determine which types of behavior are most likely to generate periods of intense conflict, and we find that fights are not explained by single, aggressive individuals, but by complex interactions among groups of three or higher. Understanding how memory and strategy affect conflict dynamics is a crucial step towards designing better methods for prediction, management and control.

Copyright: © 2010 DeDeo et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Introduction

Conflict is dissipative. Organisms, aggregates, and societies must overcome the destabilizing consequences of conflict in order to persist [1]–[5]. Conflict consequently plays a central role in the evolution of social organization. Particularly problematic for social stability is ontogenetic conflict – conflict that finds expression in fights between individuals over the course of their lifetimes. Much is understood about control mechanisms [4]–[11], factors driving escalation of pair-wise contests [12]–[15], the influence of third parties on conflict outcome through coalition formation [16], [17], audience [18], [19] and reputation effects [20], and redirected aggression [21]. Somewhat paradoxically, less is understood about the causes of conflict, and almost nothing is known about the dynamics of multiparty conflict –conflicts that spread to involve more than two individuals to encompass a sizable fraction of a group. Multiparty conflicts are common in many gregarious individual societies [22]. In these systems, it is often difficult to establish why individuals become involved in an ongoing fight or why the fight started.

A standard assumption is that individual strategies are highly tuned to resource competition. Under this assumption the cause of any single conflict is immediate competition for resources over short time intervals. The probability of fighting depends directly on the payoff obtained from acquiring the resource in the present. These resources can include food, mates and dominance status. The latter is thought to improve access to food and mates. However, individual memory for previous interactions can alter the occurrence and course of future conflicts, promoting longer-timescale, competitive dynamics. This is because memory for regular patterns of past conflict facilitates prediction of future conflict, allowing individuals to respond strategically. It is well understood, for example, that competition for dominance between a pair of individuals can be played out over many months and involve alliances and coalitions [23]. Memory can also introduce costs, as it can lead to the amplification of conflict or to the eruption of a sequence of related fights: a “cascade” [24], [25]. Such turbulent periods can increase the probability of injury and stress, both of which are associated with increased mortality [26]. Large conflicts can increase the probability that individuals not involved in an initial dispute will be drawn in, and so can increase the “population cost” of conflict. Thus critical questions include: how do individuals decide to fight, are multiparty conflicts are reducible to pair-wise interactions or do they involve irreducible higher-order interactions. How do alternative decision-making rules, or strategies, effect inter-conflict dynamics and organizational stability, and what role does memory play in amplifying and dampening conflict?

Addressing these questions in multiplayer systems requires models that make few or no assumptions about payoffs, as these are rarely known, and which are tractable when allowing for higher-order interactions (more than pairwise interactions). In standard game theory models – a canonical approach to the study of conflict – payoffs are posited, higher order strategic interactions are typically neglected, and data rarely derive from temporally resolved, natural observations of strategic interactions. The goal is to provide solution concepts for games that find uninvadible strategies, rather than to extract from the data directly those strategies individuals are playing [27].

To complement these standard deductive game theoretic approaches, we introduce Inductive Game Theory, in which the strategies used by individuals, and their consequences for social dynamics, are derived computationally from highly resolved time-series data on competitive processes. Methodologically, this approach borrows from statistical inference methods now standard in genetics. The goal in genetics is typically to reconstruct gene interactions from expression-profile, time-series. The problem is that the number of transcripts is usually far greater than the number of independent observations. Hence priors need to be imposed on permissible solutions. The goal of Inductive Game Theory is to extract decision-making strategies and behavioral time series from known interaction networks. Hence these problems are in some sense inverse of each other. In the Discussion, we return to this issue, expanding the scope of IGT to consider non-conflict time series.

In the body of the paper we develop the inductive game theory approach and apply it to a conflict data set from a pigtailed macaque (Macaca nemestrina) group. The macaque (Macaca) genus and its subset species are natural model systems for studying the role of complex conflict dynamics in social evolution. This is because in macaque societies individual decision making is plastic and guided by learning, conflict is frequent and typically involves multiple, unrelated players, and social dynamics occur over multiple timescales [28], [29]. The particular pigtailed macaque group we study contains 48 socially-mature individuals (84 individuals in total) housed socially in a large compound at the Yerkes National Primate Research Center Field Station in Lawrenceville, Georgia (see Empirical Methods.) Data were collected over a series of four months in which the group was stable (no reversals in dominance status). Conflict events – “fights” – in this group vary in duration, number of participants, and other measures of severity [4], [9]. Because the entire sequence of conflict events was collected, including data on fight duration, participant identity, and participant behavior, we are able to construct a highly-resolved time-series for each observation period. A total of 1,096 fights in 158 hours were observed over the study period; the names of individuals in each fight were recorded.

Time Series Correlations We begin by asking whether fight sizes are correlated in time. An example time-series, from a single eight-hour observation period and showing fight size and duration, is in Fig. 1; one may construct from this various autocorrelation functions. Surprisingly, the sizes of fights are nearly uncorrelated over the course of the day. Larger-than-average fights do not, for example, predict the appearance of larger-than-average fights later. This is discussed in greater detail in the Supporting Information. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 1. Conflict event time-series data from one observation period. Begins at 12:00 hours, and ends just after 20:00 hours. Plotted on the y-axis as “Total Fight Size” is the number of conflict participants per conflict, regardless of whether the participant was an aggressor, recipient, or intervener. The graph gives a sense of the distribution of conflict sizes, and conflict lengths, and the distribution of intervening peaceful periods. Hatched bars indicate periods without data collection. https://doi.org/10.1371/journal.pcbi.1000782.g001 We then ask whether there are correlations across fights in membership – does the the appearance of individual in a fight at time step one predict the appearance of in the next fight? For simplicity, the only within-fight information we use is individual identity data; we do not take into account individual behavior (e.g., aggressor, recipient, intervener, and so forth – see Empirical Methods), nor do we consider which individuals interacted within fights. Given this, the simplest correlations we can observe are correlations in membership across fights separated by one peace bout. We write these , estimated as : the number of fights involving that followed a fight involving , divided by the number of fights involving . Informally, gives the probability of observing in a conflict given that one has just observed a conflict involving . The probabilities will vary for different pairs of individuals. In order to remove time-independent effects on individual participation in fights, we compute ; the difference between the null-expected and that measured from the data: (1)where is the average from a large Monte Carlo set of null models generated by time-shuffling the series but not shuffling identities within fights. Fig. 2 shows some of the strongest correlations of this form found between the 48 individuals, in the form of a directed graph. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 2. The network of the strongest correlations detected in the data set, shown as directed edges between individuals. (as defined in Eq. 1) positive is denoted as a solid line, and negative as a dashed line; arrows denote the forward direction of time. Edges with above 6% (at 95% confidence) are shown; note that detections of different edges are not independent. Node color indicates frequency, with blue meaning rare in fights, and red, frequent. https://doi.org/10.1371/journal.pcbi.1000782.g002 Correlations across fights can also be generated by subgroups deciding to fight in response to other subgroups fighting previously. There are several possible variations in subgroup-generated correlations. We consider only the two computationally simplest correlational structures. Correlations of the form reveal the extent to which the presence or absence of a pair of individuals at one time predicts the appearance of a particular individual at the next step. They can be defined: (2)as can , the extent to which the presence of an individual at the previous step predicts the presence of a pair at the next step: (3)Using this combinatorial Monte Carlo technique, we find significant correlations for both these structures. Plots of the distribution of these three correlations can be found in the Supporting Information. As one measures higher-level correlations, between, for example, triplets and individuals, the effective sample size – number of relevant observations (the conditional and ) – drops, while the number of parameters to estimate rises combinatorically. This leads to a rapid decrease in signal-to-noise on any one observable. Extracting overall significance levels for measurements requires caution. For example, since individuals are correlated within fights, and these correlations are maintained by the null model, the various measurements are not independent of each other. A Monte Carlo simulation of the expected numbers of correlations confirms that the excess of positive values in the observed data is significant at ; these issues are discussed in greater detail in the Supporting Information. A similar analysis can be done for two-step correlations between named individuals and groups; while individual detections can be made, Monte Carlo simulations of the expected noise properties of the measurement suggest that such correlations, should they exist, are too weak to detect even in the full sample of 1096 fights.

The Space of Strategies Given the observed correlations, we now consider the causal mechanisms underlying the detectable individual and subgroup correlations. To do so, we introduce a class of minimal models for social reasoning, or “strategy space.” Full specification of these models is in the Supporting Information section, “Simulation Specification.” We suppose that each individual or subgroup decides whether to join a fight based on composition of the previous fight. The space of possible strategies can then be written as , where is the size of the relevant group in the previous fight, and is the number of individuals making the decision. We allow decisions to be probabilistic (“mixed,” in the game theory terminology), so that a particular fight composition can lead, with some probability distribution, to different kinds of subsequent fights. Each element of a strategy is a number between and , specifying the probability that the appearance of a particular -tuple leads to a recommendation that a particular -tuple join, or avoid, the next fight. These probabilities derive directly from the data, using the equations given in the previous section to determine whether there is a significant identify correlation across fights between two individuals or pairs. A negative value can be interpreted as repulsion or inhibition, and a positive value can be interpreted as attraction or stimulation. In the case that a particular -tuple receive multiple, possibly incompatible, recommendations to join or avoid, which is always possible because each fight has a minimum of two individuals involved, the decision to join or avoid can be resolved by introducing a temperament parameter, which we call a “combinator.” We choose AND and OR to capture the two ends of the spectrum of individual temperaments. Under the conflict averse, or conservative AND combinator, an -tuple must receive recommendations to join from all relevant -tuples. Under the maximally conflict-prone OR combinator, a single recommendation to join is sufficient. We begin with a randomly generated, spontaneous “seed” pair. These seeds can trigger a subsequent series of fights (a “cascade”) that in our simulation build up a time series. At some point, a particular fight may lead to no recommendations to join, or a recommendation that only a single individual join; at this point, the cascade ends, and a new seed pair is chosen. This is a (one-step) Markov model; the restriction to single, as opposed to multi-step models can be justified in part by the absence of detectable correlations at two steps, discussed above, and by the reproduction of this absence in the outputs of the one-step model. Different and combinator choices amount to constraining the transition matrix. The estimation of maximum-likelihood transition probabilities in (hidden) Markov models is often accomplished with a variant of the EM algorithm [30]; however, even in the simplest model, , the number of parameters ( ) to be estimated is larger than the number of events (1096 fights), and so such iterative methods are unlikely to reliably converge. On the other hand, determining the parameters directly by searching the full parameter space is impossible. In this exploratory work, we instead make a convenient Ansatz. Specifically, we take the elements of to be equal to the corresponding measurement of the between the relevant - and -tuple. In the discussion of results (“How Specific are the Strategies”), we consider a number of alterations from this first guess as a way to assess the flatness of the likelihood and thus to suggest, for future investigations, how to reduce the dimensionality of the parameter space. Our choice should be reasonably close to the maximum of the likelihood when fights are small and do not grow or shrink too quickly. We find that some choices of strategy class both “validate” (approximately reproduce the measurements used to specify them) and “predict” (reproduce other features of the data that do not directly influence the values of their parameters.) As shown in Fig. 3 we can define a systematic, discrete space of models that stand in hierarchical relation to one another. Increases in correspond to an increase in the memory capacity of decision makers. Increases in correspond to an increase in coordination among individuals. Hence the space defines an hierarchy of memory and information processing requirements. We confine the space of models we consider to those in which and , as the strategies of higher-order models are unlikely to be within the cognitive capabilities of the individuals. In principle, can be extended systematically to (i) larger values of and , (ii) include a combinator with more complicated functional dependence, and (iii) accommodate longer timescales, for example, by expanding the dimension of the strategy space from to , and even , where and respectively refer to the second and third time steps from the initial fight. Later in the Results section, we consider how factors like power [4] affect strategy use by individuals. Such factors can be incorporated into the IGT framework but caution is warranted as it is nontrivial to do so systematically; we defer this question to future work. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 3. Lattice classification of strategy space. All strategies live in the space of 1-step Markov transition functions. Starting with the simplest model class , we can add individuals to either the first or second fight, systematically building up strategies of increasing complexity based on cognitive, coordination, and computational requirements. https://doi.org/10.1371/journal.pcbi.1000782.g003