Conflict and Convention
in Dynamic Networks

Key Results

We study the hawk dove game to explore the dynamics of conflict and cooperation, where hawks act aggressively, and doves cooperatively. We use a coevolutionary model where both network ties and behavioural strategies evolve by reinforcement learning. In a dynamic network environment, host-guest norms (paradoxical behavior) account for the vast majority of the final state solutions, while ownership norms (bourgeois behavior) are quite rare. Only when payoffs are very strongly biased in favor of playing dove when visiting, or against dove when hosting, do we find the bourgeois solution more often than the paradoxical solution. If there is no bias or any bias in favor of dove when hosting, or against dove when visiting, the paradoxical solution is far more prominent. The paradoxical solution has a large basin of attraction because it can be reached through numerous network configurations, while the bourgeois solution can typically only be reached when the network maintains a high level of homogeneity throughout its evolution.

A common view on the origin of territoriality or norms of ownership takes the evolution and stability of the bourgeois correlated equilibrium to provide a rationale for aggressively defending territories or resolving conflicts over resources in favor of hosts. In stark contrast, our results show that a host-guest norm, where the host concedes resources to a visitor, evolves far more readily than any norm of territoriality. More work is needed to unveil the evolutionary dynamics underlying ownership norms.

Evolutionary Trajectories

Click on the buttons to learn more about the most common evolutionary trajectories in the model. In the network plots here and in the interactive visualization, node size corresponds to total incoming edge weight (expected number of visits). The outer edge color of the nodes corresponds to the visitor strategy, and the node center color corresponds to host strategy. Darker red is more prominent hawk strategy, darker blue is more prominent dove strategy, and light gray is neutral strategy (equal chance of playing hawk or dove). Arrow size on the edges corresponds to the proportion of outgoing edge weight for a single node (i.e. a very large arrow means the source agent almost exclusively visits the target agent).

Paradoxical Solution

Bourgeois Solution

Network Solution

Mixed Network Solution

All agents start with neutral strategy and uniform outgoing edge weights.

By time step 100, the network is still homogeneous, but a few dove hosts have started attracting visitors at a higher than average rate. Most agents have already started to lean towards playing hawk when visiting other agents.

At time step 1000, the network is starting to develop a hub-spoke structure with multiple dove hubs. Every agent has evolved a strong hawk visiting strategy, and all agents that attract a significant amount of visitors have learned a strong dove host strategy.

The hawk-visiting spokes from time step 1000 have attracted hawk visitors and subsequently learned to play dove as hosts to maximize their payoffs. This strategy evolution, along with discounting and the presence of errors, has facilitated the network evolution away from the heterogeneous degree distribution and back towards the initial homogeneous state. This final configurationx is called the paradoxical correlated equilibrium.

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All agents start with neutral strategy with equal outgoing edge weights.

By time step 100, the network is still mostly homogeneous, but a few hawk-leaning hosts have started attracting visitors at a higher than average rate. Most agents have already started to lean towards playing dove when visiting other agents. When payoffs are symmetric, random fluctuations will occasionally, but not often, allow for this early time evolution.

By time step 500 the network is almost entirely bourgeois, with just a couple agents that are not yet pure dove visitors or pure dove hawks. With discounting and errors present, it is all but guaranteed that agents will continue to work towards the pure bourgeois state.

By time step 1000, the network has already evolved to a steady state - the bourgeois correlated equilibrium. When payoffs are symmetric, all evolutionary trajectories towards the bourgeois state will resemble this example - the network must maintain relative homogeneity for the bourgeois solution to survive.

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All agents start with neutral strategy and uniform outgoing edge weights. In this simulation, the agents do not commit errors.

Quickly, by time step 100, many agents learn to play hawk away, and a single dove host emerges.

By time step 500, the dove host hub has attracted the vast majority of visits from other agents, and has found another dove host to play hawk against. A few of of the "spokes" are still playing dove as a visitor, not yet maximizing their potential payoffs. Due to the lack of errors, some may get trapped in this suboptimal strategy regime.

By time step 10000, the network has evolved to its final state. The dove hub now attracts nearly every possible available visit, and exclusively visits the other paradoxical agent. One agent was trapped into playing dove as visitor.

When errors and discounting are present, groups that reach the network solution here very often evolve out of it and into the homogeneous paradoxical solution, given enough time.

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All agents start with neutral strategy and uniform outgoing edge weights.

In the early time steps, a few different trajectories are still possible. Multiple dove hosts have begun evolving, and a few hawk hosts have as well. Similarly, dove and hawk visitors are dispersed throughout the network.

Three dove hubs have started to evolve, with two playing hawk away and one playing dove as visitor. Despite this, many less popular agents still are playing dove as visitor.

Four dove hosts have now emerged, and nearly all other agents have taken on the pure hawk spoke role. With so many dove hubs, there is not enough selection pressure to force agents to visit any one agent, and thus they maintain more possible links than in the network solution.

When errors and discounting are present, groups that reach the network solution here very often evolve out of it and into the homogeneous paradoxical solution, given enough time.

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Interactive Visualization

We created an interactive visualization to accompany the paper that allows you to explore select aspects of our raw data.

First, choose model parameters you want to explore using the dropdown menus. Then, scroll over the line chart to view the proportion of interaction types over time, and the network will update correspondingly. You can also click on the legend markers to toggle them - the default state shows Bourgeois and Paradoxical interactions, while hiding Cooperative and Aggressive interactions. Click while mousing over the chart to "freeze" the network visualization at one particular time step. You can hover over the nodes to see their visitor and host strategy states, and you can drag them around as well.

In the payoff dropbox menu, DD is the dove-dove payoff, and DH is the dove payoff in a dove-hawk interaction. Visitor Bias implies that dove visitor payoffs are larger than dove host payoffs. Static simulations do not allow agents to update their network connections. In all simulations, hawk-hawk payoffs are fixed at 0, and hawk payoffs in hawk-dove interactions are fixed at 1. See Figure 1a in the paper for more payoff details.