Strategic Group Formation in Agent-based Simulation

Andrew J. Collins and Erika FrydenlundOld Dominion University

Suffolk, VA{ajcollin, efrydenl}@odu.edu

AbstractMost often, researchers model crowds as individuals rather than groups with social cohesion. This approach diminishes the impact of altruism and other group-supporting behaviors. For example, in real evacuation scenarios, some people will move counterintuitively towards danger to help friends and loved ones. Current modeling approaches to capture group formation and dynamics lack strategic elements required to model the complexity of human decision-making. Game Theory provides a mechanism to introduce this strategic behavior. This paper investigates strategic group formation through the introduction of Cooperative Game Theory techniques into an agent-based model (ABM). This approach requires looking at the Core instead of the more common Nash Equilibrium. This paper shows some empirical results from introducing the cooperative group formation into a simple agent-based model. In the model, implementation of a simple game results in a large dominant group, much like real-world mobs. The model and results are relevant to policymakers trying to understand how humans navigate an environment through strategic social interactions.

Author KeywordsAgent-based modeling; agent-based simulation; cooperative game theory; group formation.

INTRODUCTIONWhen modeling large numbers of individuals, not taking into account group structure can lead to erroneous conclusions. For example, in an evacuation model, one might assume that individuals will flee from danger if possible; however, this is not always true (1). Counterintuitive movement toward danger could be motivated by an individual’s desire to retrieve a love one from another location, such as school or another residence. Misunderstanding social group dynamics such as these can lead to conclusions and policy decisions that adversely affect the entire population. In this case, contraflow lane closures could block travelers driving toward danger and cause delays across the entire transportation system.

Agent-based modeling (ABM), as well as other techniques like social network analysis (SNA), provide a mechanism to model and analyze large groups of diverse individuals with independent decision-making abilities. However, the formation of groups within these methods tends to be simplistic and focused on homophily or popularity (2). Analytical techniques, like Game Theory, provide a method to investigate more sophisticated group behavior but are, realistically, limited to only a small number of agents. These agents tend to be homogenous, particularly when considering more than a dozen agents using Cooperative Game Theory.

Our research intends to bridge the gap between ABM and Game Theory by incorporating cooperative game theoretic methods into a agent-based model. In recent years, our research group has been working to capture group behavior in pedestrian evacuation models and simulations (3-5). This paper represents a move to generalize this work for broader application in social science modeling. The approach takes a dynamic group interaction scenario and searches for generalizable empirical insights. This research evaluated both homogeneous and heterogeneous agents, though we only present the homogeneous findings in this paper. This paper is an extension of our 2016 Spring Simulation Conference presentation (6) adapted for this special issue of Simulation Transactions through inclusion of empirical results.

The next section gives some background of the research followed by a description of the scenario and the ABM. Empirical results precede the conclusions.

GROUP FORMATIONStrategic group formation relies on the assumption that agents maximize their individual utilities by deciding to join or leave a social group. Here, ‘utility’ is purposefully vague to support extension of the theory developed in this study to a variety of social contexts including international relations, political science, anthropology, and business. In general, maximizing utility in the model translates to acquiring more resources. The current state-of-the-art in group formation and dynamics studies focuses on two selection preferences: grouping with similar individuals (homophily) or joining the most popular groups. While strategic in a very limited sense, these approaches to the study of groups do not allow for more complex decision-making. Choosing the most popular groups does replicate similar scale-free social networks found in the real world, as seen in the Barabási–Albert models (7). The generalizability of this approach, however, is limited (2). Cooperative Game Theory provides some additional insights by adding a mechanism to capture strategic group/coalition formation (8). This approach is computationally limited as only a few dozen agents can be considered at once. Growing a population composed of groups using simple group formation strategies and utility maximizing preferences, this study proposes to

advance the understanding of social dynamics in conditions more analogous to real-world scenarios. Our approach takes the solution concepts of imputations and the Core (9), which are key concepts from Cooperative Game Theory, and implements these key concepts in a iterative manner in an ABM.Prior work at the intersection of ABM and Game Theory is relatively restricted in that most approaches allow only a very small number of agents to interact. Schelling (10) and Axelrod (11) provide classic examples of work that marry ABM and Game Theory. Joshua Epstein also contributed to this field by incorporating a two-player prisoner’s dilemma game into an ABM (12). Cellular automata (13) and multi-agent systems (14, 15) approaches have also delved into ways to model game theoretic principles, but these applications of game theory are add-ons to an ABM as opposed to a fundamental incorporation, as we propose here.

Many agent-based modelers come from a Game Theory background--for example, Thomas Schelling and Robert Axelrod--but there has been little more than a superficial bridging of the subjects. For example, Joshua Epstein incorporated the prisoner’s dilemma game into an ABM (12); however, the simulated game was played exclusively by two, and only two, agents when they interacted. Though the researchers used a single game scenario, this research will contribute to the broader conversation of Game Theory, group formation and social dynamics, and extensibility of theory derived from modeling and simulation to the social sciences. Additionally, this work bridges two modeling paradigms—ABM and Game Theory—to support development of generalizable group dynamics theories with broader application across the social sciences.

Cooperative Game Theory

The formation of coalitions and groups has been of interest to the social science community because of the diverse areas in which it is found. From coalitions of countries, e.g., the North Atlantic Treaty Organization (NATO), to ad hoc formations of teams in organizations (16-18), coalitions have an impact on our everyday lives. There are many ways to analyze group formation, from modern social network analysis (7) to Cooperative Game Theory (8). Cooperative Game Theory provides mechanisms to investigate sophisticated (strategic) group formation through concepts like the Core (19), Nucleolus (20), and Shapley value (21). These concepts afford a means to find solutions to cooperative games; each is quite complex and we will only define the Core concept below. Cooperative Game Theory, however, has its limitations, such as the feasible number of agents that can be considered at once and the sophistication of the agents’ characteristics.

In response to the many limitations of Game Theory, many researchers have turned to other approaches such as Bayesian networks (22) and ABM (11). For example, Miller and Page (13) used an ABM approach to investigate the outcome of possibilities of repeated 2 x 2 normal-form games using adaptive agents. What was interesting from their results was that humans “play” many games simultaneously and/or sequentially. The agents in our scenario also face a changing game environment as their group and groups around them change. The approach by Miller and Page was an application of ABM to solve a Game Theory problem, whereas our approach is the reverse, with the incorporation of Game Theory techniques into an ABM. The Miller and Page approach did not take into account agent heterogeneity. Heterogeneity in our model is a critical means to reflect the differences between human decision-makers.

With researchers moving from Game Theory to the more versatile ABM, the ability to model sophisticated group formation was lost because the agents follow simple action selection mechanisms.

Researchers like Joshua Epstein have tried to compensate for this lack of sophistication by introducing new concepts like Agent_Zero (23), however, these approaches do not capture the truly strategic nature of Game Theory because each agent does not take into account other agents’ actions. Thus, we argue, there is a need to incorporate more Game Theoretic elements back into ABM and this incorporation must go beyond the simple implementation of games to govern the pair-wise interaction of agents such as those found in Epstein (12). Pair-wise game playing is inadequate because different, potentially more beneficial results can occur if agents act in a group as opposed to individually. Arguably, this observation has been used to explain the formation of civilizations (24). Thus when modeling more than two agents, a researcher must consider the formation of coalitions if strategic play is to be incorporated.

Game Theory has a whole sub-field dedicated to analyzing coalition formation known as Cooperative Game Theory (pre-1990 it was known as n-person Game Theory (25)). Due to the nature of Cooperative Game Theory, whole new solution concepts and terminology were needed for its analysis, namely imputation, Core (19), Nucleolus (20), and Shapley Value (21). This new approach is needed because of the computational complexity of solving games with three or more players using the Nash Equilibrium (8).

Cooperative Game Theory gained popularity due to the Nobel Prize winning economist Lloyd Shapley. Cooperative Game Theory is concerned with determining which coalitions form and who cooperates with whom. The heart of Cooperative Game Theory is the concept of an “imputation,” which expresses some criteria about how coalitions should form and distribute a reasonable share of the rewards. Equation (1) shows the criteria for an imputation:

1) ∑i=1

n

x i=v (N ) (Efficient) (1)

2) x i≥v ({i })∀ i∈ N (Individually rational)

v(S) is the value function, which gives the total worst-case return (in terms of resources) of any coalition S {1, 2, …, n} = N. “Worst-case” here represents what coalition S can guarantee itself given that N\S forms a coalition to minimize S’s return; this is equivalent to the maximin concept of normal-form Game Theory and effectively reduces the game to only two players for each coalition (them and us). Each agent i will receive a return of xi. If this spread of rewards is an imputation then the two criteria in Equation 1 must be satisfied. The first criteria is global efficiency, which states that the total return observed must be equal to the return observed if they all cooperated, thus satisfying Pareto optimality. The reason for this criterion is that if the total return was less, then the agents could all form a coalition of size N and all be better off by some amount (assuming side-payments1). Note that total return can never be greater than v(N) because the agents could cooperate to get that return if they are all working together. The second criterion of an imputation is individual rationality. This means that an agent will never accept a return that is less than if it went off on its own (as they could always do this).

Not all imputations are the same and some dominate others. By dominate, we mean that a subgroup of agents gains a greater reward in one imputation than the dominated one. Obviously, that subgroup would prefer the dominating imputation to the dominated one, and it will focus on achieving it. Thus for an imputation to be satisfactory for all sub-groups, it must not be dominated by any other imputation. The 1 Side-payments is game mechanism that allows of agents to redistribute part of their payoff, to other players, after they have received them.

group of imputations that satisfy this criteria is called the Core (19), which was popularized by Lloyd Shapley in 1960s (9). Determining whether an imputation should be in the Core can be problematic, as all imputations require consideration. Luckily, a theorem exists for determining the Core:

Theorem: An allocation {x1, x2,…,xn} is in the Core if and only if

1) ∑i=1

n

x i=v (N ) (Efficient) (2)

2)∀S :∑

i∈Sxi≥v (S ) (Subgroup power)

The first criterion of the theorem is the same as the imputation criterion (as the Core is a subset imputations if it exists). The second criterion ensures that no coalition has the bargaining power to leave the current imputation.

This research uses the Core concept to introduce strategic group formation in an ABM. Though group formation exists within extant ABMs, like the famous Boids model (26) or our own pedestrian evacuation models (3-5), they do not include any strategic behavior. Because of the differences between Game Theory and ABM, and a wish to avoid vast computational complexities, the researchers cannot directly inject the Core concept into an ABM environment. Thus, the researchers developed a testing scenario to investigate the ideas of integration.

SCENARIOThe project team developed a scenario to investigate the integration of the Core concept into an ABM to enable strategic group formation behavior. It was determined that the key goals of the scenario are (1) maintain simplicity, (2) encourage cooperative behavior (group formation), and (3) encourage sub-group formation. The third goal is required to ensure that they agents do not just form one large single group, which would lead to trivial results and not reflect real-world group formation.

In the experimental scenario, agents are arranged in a static grid configuration. Figure 1 shows a screenshot of the layout in the prototype model. An agent only interacts with the four agents in its Von Neumann neighborhood. This interaction determines the allocation of some shared resource, which is given a value of one for simplicity. Thus, an agent has the potential to obtain up to four resources. Ability to acquire the resource is dependent on the agent’s relative “strength.” The neighbor with the highest strength gets the resource; if there is a tie then the resource is split evenly. Heterogeneity enters the model through agents’ varying strengths to acquire resources. If all agents remain in their singleton group then each agent would get two resources, 0.5 for each of its neighbors (assuming a torus shaped world).

Resource Allocation with Groups

Agents can join groups only with those sharing direct contact (i.e. agents in their Moore neighborhood). Neighboring group members evenly split resource gains, but also benefit from additional ‘strength’ in resource exchanges. The image on the left in Figure 2 helps to clarify this dynamic. In Figure 2, four separate groups are denoted by shades of gray: black (3members), dark gray (2 members), light gray (1 member), and white (2 members). If agents B and C were not in groups, they would be evenly matched because they are, individually, of equal strength. However, agents B and C are in different groups. Agent B has a fellow group member (D) also adjacent to agent C. Likewise, agents C and A are both adjacent to B. In the resource exchange negotiations, agent B draws upon both its own strength and the additional reinforcement of its fellow group member, D. Note that agent B’s other group member to the south cannot affect the negotiations since it is not in C’s Von Neumann neighborhood. Similarly, agent C utilizes its own strength and the reinforcement of its group member, A, in negotiations with B. Having come to the negotiation table with equal strength and group reinforcements, C and B would evenly split the resources.

Trade agreement negotiations provide a real-world example of this model’s potential application. Country leaders come to the negotiation table with varying political strength (e.g. national GDP, military size, global or regional political influence) as well as the backing of allied countries. These influences exert pressure on the terms of the agreement until the resources (the trade agreements) are eventually allocated. Spatially, then, the neighborhoods and proximity to the agents engaging in negotiations represent influence on opponents. Not all members of a coalition have influence on the opposing party at a negotiation, as with the agent to the south of B in regards to agent C.

Figure 2. Diagram showing a network of agents of different colored groups (right-hand side) and how this network is interrupted when determining the value function of the central black agent (left-hand side).

CBA

D

Figure 1. Screenshot from the NetLogo model with heterogeneous agents.

Group and Sub-group Dynamics

Benefits of negotiating influence (i.e. “strength”) incentivize agents to join groups and cooperate. There is little incentive to form one mega-group composed of all agents as that world has no opposition from whom to gather resources. The preponderance of strength in groups motivates singleton agents to seek out groups to join in order to defend against resource loss. This interaction of individual interests with group benefits allows the formation of subgroups.

The value function from our scenario is different from the actual allocation because it focuses on the maximin criteria. That is, the value function for determining resource allocation assumes that every agent not in the agent’s group is assisting the opposing group. To explain this, consider the scenario given in Figure 2 again. When agent B determines the value of the interaction with agent C, it assumes that all non-black agents are in agent C group (white). This interpretation is shown on the right side of Figure 2. In this case, agent B would determine that agent C would win the negotiation (two black agents against three white agents).

Simplification of the value function implies that total allocations based on these rules may deviate from those computed under the complete cooperation scenario. The allocation actually realized and the value function being different implies that the total value function-based allocations will not necessarily added up to the value function from complete cooperation (2n is the total allociation of all agents). Thus, in our scenario, it is possible to find allocations that are not imputations and thus not in the Core. It is even possible that the Core does not exist. Note that v({i}) = 0. The scenario assumes no reallocation (side-payments) of resources after distribution to the agents.

The scenario provides a basis for exploring strategic group formation in ABM using Cooperative Game Theory. Cooperative Game Theory has been used in a multi-agent environment for software task allocation (27) and also in networks (28). However, the focuses of both of these research projects was not group/coalition formation and neither environment considered would be considered an ABM environment.

AGENT-BASED SIMULATIONA simulation was developed by the researchers in the NetLogo agent-based modeling environment (29), which has been used to generate several of the graphics found in this paper. Critically, this simulation focused on dynamic group formation as opposed to determining the Core of the different scenarios. There currently does not exist any software package that can solve Cooperative Game Theory problems and find the Core, though there is software available to solve normal-form games, e.g., Gambit (30).The data provided by this dynamic situation provides insight into the Core of the game by using agents that follows the Core’s principles. The critical aspect of a Core imputation is that, as shown in equation 2, an agent will not be satisfied with its current group (coalition) if there exists another group that could provide more resources. Technically, it is the group that is dissatisfied with the current allocation, not the individual agent; however, an ABM works at the agent level and thus our dynamic decision-making will occur at that level. It could be possible to consider decision-making at the group level, but with 2n possible subgroups to consider at each iteration, this would be computationally challenging.

The iterative mechanic for our simulation focuses on the agent’s decision to either leave its current group (coalition) with a subgroup of that group or merge its current group with a neighbor’s group (thus

acting as a catalyst for unification in this case). We also check to see if the rest of the group would benefit from kicking that agent out. Thus, the pseudo-algorithm for this iterative process is as follows:

1. Select random agent.2. Randomly determine a new subgroup containing selected agent and determine value of subgroup

if it was independent from its group.3. If the subgroup value is greater than the current group value, then subgroup detaches from the

main group and move to step 9.4. Determine value of group without the selected agent.5. If group value is higher without the selected agent, kick selected agent out of the group and move

to step 9.6. Select another random local (Moore neighborhood of a group agent) group. Determine if selected

agent’s group benefits from merging with this random local group. 7. Determine if the random local group benefits from joining selected agent’s group.8. If both benefit, group merge. 9. Randomly select a previously unselected agent and repeat from Step 210. If all agents have been selected, move to the next time step.

To make the visualization clearer, the algorithm checks all groups to see if they are internally connected (using a Moore neighborhood) and, if not, splits the groups accordingly. The researchers could have developed other variations of the model just described, e.g., inclusion of a mechanism to allow side-payments, which we leave for further research. Since we considered geographically static agents, we do not need to develop other algorithms, like movement patterns and evolution, for this simulation.

SIMULATION RUNSThe simulation runs were conducted for different numbers of agents, ranging from 4 (2 x 2) to 36 (6 x 6). We could have considered a much larger group size but we limit ourselves to this number to understand the effects of our group formation mechanism. There was no guarantee that a simulation run would reach steady state, so we imposed stopping conditions. The computer code allowed each simulation to run until 100n steps had occurred without any changes. This extreme stopping condition built in the necessary redundancy to account for the fact that the simulation only considered randomly one of 2n subsets each step. Even with this redundancy, we could not guarantee that every subgroup would be checked (for 36 agents there are 69 billion possible subgroups). We repeated the simulation runs 100 times for a 2,500 total runs that an eight-core computer with 8GB of RAM completed in under one day.

Initially, all the agents in the simulation start in their own singleton groups. By the pseudo-algorithm given above, over time some will probably join to form a new group, thus we expected the number of groups to decrease generally over time. The data collected focused on the number and size of the groups for each run as presented in the results below.

EMPIRICAL RESULTSThe complexity of simulations often makes it difficult to use to standard statistical measures to represent the results. This is due to the number of agents and configurations that could occur. Social Network Analysis does have standard measures for completed graphs, e.g., diameter or compactness, (7) but these

measures are inadequate to portray the insights found in the simulation runs. As such, we used visual examples and a novel approach of transition matrix representation to represent the outputs from the simulation runs. Given the novelty of the research, i.e., an investigation into dynamic strategic group formation, it is not surprising that novel representations are required for the results.

All of the simulation runs where completed within 10,000 time-steps (which was another mandatory stopping condition). A general result from the simulation runs was that some agents formed a super-group with the remaining forming smaller groups. Examples of this can been seen in the screenshots shown in Figure 3 and Figure 4. Figure 3 (a) shows the only outcome, given rotational, reflective, or wrapping equivalences, of the 2 x 2 game. The outcome was a group of three agents and a singleton group. Figures 3 (b) and 3 (c) show the only two final configurations for a 2 x 3 game, which was a group of four agents and a group of two agents. Those games that had symmetrical setups, e.g., 2 x 3 and 3 x 2, had the also identical outcomes.

(a) (b) (c)Figure 3: Final group formation configurations from the simulation runs for the 2 x 2 (a) and 2 x 3 (b & c) games

As the size of the game increased, so did the number of possible final group configurations. However, the numbers in each group tended to be the same, implying convergence in group sizes but not group location. For example, Figure 4 shows some of the group configurations from the 4 x 4 game. All three results have ten agents in the larger group and six other agents. Notice that Figure 4 (c) has two smaller groups which are unconnected (so the algorithm would not allow them to form a group).

(a) (b) (c)Figure 4: Possible final group formation configurations from the simulation runs for the 4 x 4 game

The split of the groups into multiple groups with one larger group is unsurprising. To understand why this is, consider the case when all agents join to form a single group of n agents. The payoff observed by each agent will be the same. However if any single agent is kicked out the group, then all the agents around that kicked-out agent would benefit from an increase payoff, thus the main group (not the new singleton) would benefit overall. This means, in our game setup, that the formation of a single group is not stable and there will always be more than one group. Our next concern was what proportion of agents formed largest group.

Table 1 shows the average proportion of agents in the largest group. The cells of the table represent the results from game (row number) x (column number). The games that contained a single row or column (i.e., r x 1 or 1 x c) were determined by hand due to their simplicity. The table has three noticeable features beyond the obvious (i.e., the table is symmetric). Firstly, the changes in proportions are not monotone when keeping a row or column static, i.e., they do not uniformly increase or decrease. This result may be due to the discrete nature of the group size (i.e., you cannot have half an agent in a group). The second result is that the games containing five rows or columns have 60% of agents in the larger group. We are not sure the cause of this consistent proportion. The third feature is that, with the exception of the very small games, the larger group generally has approximately 50-60% of the total agents in it. We believe that proportion range is due to the balancing act the larger group must play between (a) dominating as many other agents as possible and (b) having enough agents to dominant them in the first place.

Table 1: Average proportion of the agents in largest group final configurations.

1 2 3 4 5 61 1.000 1.000 0.667 0.750 0.600 0.6672 1.000 0.750 0.667 0.625 0.600 0.5833 0.667 0.667 0.556 0.583 0.600 0.5564 0.750 0.625 0.583 0.625 0.600 0.5835 0.600 0.600 0.600 0.600 0.600 0.6006 0.667 0.583 0.556 0.583 0.600 0.583

Another result that was not observed was an even split of the agents into two groups. This was surprising because stability can be achieved with an even split as shown in Figure 7. The diagram shows two groups in an equilibrium with no agent having an incentive to join the other side or be kicked out by its own group. In other words, each agent is essential is obtaining a certain amount of resources that would be lost if it got kicked out. However, this equilibrium state is unlikely (if not impossible) from the initial conditions because of the delicate balance required for it to be a feasible choice: one alliance change from any agent would lead to a complete collapse of the structure. In Game Theory terminology, though an equilibrium has been reached in Figure 3, it is not a trembling hand perfect equilibrium (31), that is, it is an unstable equilibrium. As expected, the researchers never observed this equilibrium in the simulation runs. To verify stability, the researchers manually inputted this agent setup.

Figure 5. A stable result of two groups from initial runs.

As the discussion about Figure 5 shows, the way that groups are formed affects the outcome. Several questions immediately come to mind when thinking about dynamic group formation: (1) is there a monotone decline in the number of groups? (2) does the number of groups cycle? and (3) does the game always end in the same results? In the simulation results, there was neither a monotone decline nor any evidence of cycling. As Figure 4 shows, the same game results were not achieved for different simulation runs. Thus, we are left with the more difficult question of how the groups form over time. This is a difficult question because of the challenge of concisely representing this group formation. For example, Figure 6 shows the distribution of group size for the 6 x 6 game but does not show how these groups change from n singletons to the final grouping of sizes of 21 and 15.

Figure 6: A graph showing normalized (by largest) number of times that contain a group of given size for the 6x6 game, based on results from 100 runs.

Thus to understand group formation, we must first understand how it evolves. As Joshua Epstein said, “If you didn’t grow it, you didn’t explain it” (12, 32). Transition matrices provide researchers an understanding of a dynamic environment using a static picture (33) and, as such, form the basis of our dynamic analysis. Since the agents store no previous group information, the system is memory-less (only updating based on current conditions).

Transition Matrices

Transition matrices shows the probably of moving from one state to another. We created a transition matrix for changes in the group size, for which we derived probabilities from our simulation results. Note that the system will have multiple group changes happening concurrently, which we amalgamate into a single transition matrix. Transition matrices are traditionally associated with Markov chains; however, our resultant transition matrix is not, strictly speaking, a Markov chain as other factors that could influence transition are not represented in the transition matrix (i.e., the group configuration). Exclusion of these could violate the memory-less requirement of a Markov chain.

A transition matrix can be difficult to read for any formed patterns or potential generalizations due to be being a list of number. Thus, there is benefit from deriving graphical representations like the one shown in Figure 7. It shows an approximation of the state chain associated with the transition matrix, with the probability values and self-transitions removed. The nodes are grey-scaled by the number of time-steps sent in each. A transparent node indicates an unobserved a group size in any of that game’s runs. The dark arrows indicate the dominant transition from that state and the light arrows indicate a possible transition. Self-transitions have not been included in the diagrams.

Figure 7: An approximate transition matrix representation of the group transition of a 2 x 2 game.

Figure 7 shows the transitions for the 2 x 2 game. We see that the singleton group is most likely to change into a pair and a pair is most likely to change to a triplet of agents. This results in a triplet of agents and a singleton. What is interesting from this diagram is that there is no disaggregation of agent groups, e.g., a group of two agents does not devolve to two singletons. Thus, there is never the case when two pairs form. This occurs because once a pair is formed there is no increase in payoff from the other two agents joining. This occurs because there is an anomaly in the 2 x 2 game due to the wrapping of the agent-space. The torus shape of the agent’s environment results in each neighbor being used multiple times in the payoff calculations.

Figure 8: An approximate transition matrix representation of the group transition of a 2 x 3 game.

Figure 8 shows the results from a 2 x 3 game. Here we see the same increase in group size as the 2 x 2 game, but we also experience some regression in the group sizes from four to five. This regression occurs because the agents in a group of size five realize they can increase their overall payoff by kicking one agent out. Again, the 2 x 3 game does suffer from the same wrapping issue as the 2 x 2 game. It is only once there are at least three rows and three columns that this wrapping issue no longer has an effect on the outcomes.

Figure 9: An approximate transition matrix representation of the group transition of a 4 x 4 game.

The final game considered in this paper is the 4 x 4 game. As Figure 9 shows, this is a much more complicated game with many more transitions. The game experiences only expansion of groups of size ten or less, with groups larger than ten disaggregating. This implies a steady progression toward larger groups of ten. These results from the 4 x 4 game are typical of the results from the 3 x 3 game or larger. That is a regimented transition to a particular group size for the game with few or no chaotic transitions along the way. Given the complexity of the games, this may be a surprising result to the reader but you might expect a chaotic transition between group at the start of the simulation.

Heterogeneous Impact

The final task of the project was to introduce agent heterogeneity into the model by varying the agents’ strengths and determining the impact on the results found from the homogeneous case. Impacts could be quite severe, with weaker agents dominated by the stronger ones, though group formation is still likely to happen as shown in Figure 1. We have left the results of this analysis to a follow-on paper due to the extreme complexity and depth of explanation required. We leave it to future work to determine adequate mechanism to explain the heterogeneous results.

Future research from this work will simulate games of much larger size to see if there is consistent convergence to one large group containing approximately 60% of population. We will also complete

another set of runs with the agents starting in a single large group as opposed to all being in singletons, to see if this affects our results.

CONCLUSIONSThis paper considers an adaption of Cooperative Game Theory into an ABM environment. This is achieved by using a Core-inspired approach for exploring strategic group formation. The paper considers a simple game setting, where static agents compete for resources with their neighbors. Researchers used a simulation approach and analyzed results produced through 2,500 runs. The insights gained from this study have the potential to change the way that researchers analyze group formation processes. Empirical results from this study support a methodology for better exploring strategic group formation.

The results from the game indicated that a large group would form to dominate the resource consumption over the remaining agents. The results also indicate a regimented change in the group sizes before this dominant group forms completely. However, this regimented fashion of group changes might be due to the simplicity of the underlying game (homogeneous agent are only competing for resources with their direct neighbors). As such, there is a demand for further research into more complicated game scenarios before any general conclusions about group formation can be drawn.

Cooperative Game Theory has a wide range of potential social science and business applications. By placing strategic group selection behavior into an ABM environment, researchers gain the benefit of empirical results without the limitation of standard Cooperative Game Theory, such as computational complexity. More importantly, this approach will allow ABM to move away from the physics-based models of agent interaction and group formation, particularly in the study of pedestrian movement, thus potentially initiating a new era of its use.

ACKNOWLEDGEMENTThis paper was adapted from a 2016 Spring Simulation Multi-Conference paper of the same name. The researchers greatly appreciate critical feedback provided at the Spring Simulation Conference, much of which went into the development of this version.

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Author BiographyANDREW J. COLLINS is a Research Assistant Professor at the Virginia Modeling, Analysis and Simulation Center (VMASC) at Old Dominion University. He holds a Ph.D. (2009) and an MSc (2001) from the University of Southampton in Operations Research, and a bachelor’s degree in Mathematics from the University of Oxford. Dr. Collins’ current research focus is agent-based modeling and simulation and he is the principle analyst on an award winning investigation applying agent-based modeling to the foreclosure crisis. Dr. Collins’ research interests include investigating the practical problem of applying simulation to the real world. He has spent the last 10 years, while working on his Ph.D. and as an analyst for the UK’s Ministry of Defense, applying game theory to a variety of practical Operations Research problems. His email and web address are ajcollin@odu.edu and www.drandrewjcollins.com.

ERIKA FRYDENLUND is a Research Assistant Professor at the Virginia Modeling, Analysis and Simulation Center (VMASC) at Old Dominion University. She holds a Ph.D. in International Studies with a focus on refugees and forced migration. Her research interests include agent-based modeling, computational social science, mobility and social justice, and forced migration. Her email address is efrydenl@odu.edu.