The last examination of the rock-paper-scissors (RPS) model added spatial considerations to the replicator equation, used as the basis for agent logic. Here, the outputs and responsiveness to geographic considerations are examined using the modified replicator equation discussed earlier.
The modified version of the replicator equation transitioned the agents in the system from a featureless soup, where the agents were perfectly mixed, into a spatial configuration where each agent owned sets of territory or cells on a lattice. This change allowed agents to modify the likelihood or weighting of interacting with other agents, and therefore their respective influence on their adaptive strategic allocations based on geographic proximity and, if neighbors, border lengths.
Two simple parameter sweeps show the characteristic ways in which geography, at least as implemented in this version of the replicator equation in an RPS game, affects the dynamics of the system and population of states. The first parameter sweep examined geographic weighting, but did not consider border length. The second sweep only examined border length, which then implicitly favored neighboring states because that variable is naturally set to 0 in all other cases. In order to simplify the analysis, all experiments were conducted with an initial random seed of 1234 and twenty agents or states. By locking down the configuration of the agent population, both with respect to spatial configurations and their initial strategies, all other sources of variations across cases were eliminated. Because the model is deterministic, the only sources of variation in results were the changing of the distance weighting or border length weighting parameters.
The landscape and initial population generated by the selected random seed of 1234 is shown below:
Likewise, State 0 is shown below:
Other basic parameters in the model were held constant, such as employing the standard RPS payoff structure where the payoffs of a win was 2.0, a loss was -2.0, and a draw was 0.0. The initial fitness of each state was set at 7.0 (in code, this is an unexposed parameter in the model interface).
Geography Distance Weighting
The effects of making geography increasingly important in agents’ calculations was examined over the interval from [0.0 3.0] with a step-size of 0.1. The result was 31 individual experiments, with the model run one time for each point in the interval. A simple analysis of the results presented below examines two measures of the system: the average “percentage rocks” played by the population as a whole, and the “percentage rocks” played by State 0. These two simple measures show how the population’s and individual agents change as a result of increasing the importance of geography in agents’ decision-making.
At the population level, qualitatively different behaviors emerge as the significance of geography increases. The graphic below shows the average percentage rocks played by the entire population of states in the system for each distance weighting examined. While this graphic is hard to see, there is a clear distinction between those values where geography was relatively unimportant, mirroring the familiar strategic oscillations that increase in amplitude and then duration once the corners have been reached. Alternatively, as geographic distance increases in importance, the behavior of the system changes, with strategies staying in a more balanced, but less structured mode. The secondary graphic shows the only the runs from the interval [2.5 3.0] in order highlight the effects of changing the distance weighting parameter.
While the population as a whole avoids the corner regions of competing strategies with the growing importance of geography, the same is not true for the agents in the system. Just tracking the percentage of rocks employed in State 0’s strategy for each setting of the geographic weighting parameter reveals how the experience of the population is not representative of the experiences of its individual members. Again, the percentage of rocks played by State 0 is depicted below for all runs, and for a subset of runs with higher geographic weighting.
What is clear is that the percentage of rocks played by State 0 does approach the corners of 0.0 or 1.0 percent despite the fact the system itself remains in a narrower, more balanced set of strategies with rocks, papers, and scissors represented more equally. These differences are suggestive, and may indicate that spatial or strategic niches exist when the interactions between states deviate from the pure “soup” of perfect mixing assumed in the mathematical model. Given the extreme strategic adaptations made by State 0, whose movements that are not reflected in the larger population level statistics, it is likely that other states are making similar counter moves, thus canceling out one another’s strategic choices to create a more balanced set of strategies when aggregated.
To further capture the differences between the population as a whole, and the strategic choices of State 0, a simple assessment was made that summed the absolute value of the difference in the percentage rocks played by State 0 and the population as whole over the course of the simulation. What emerges from this comparison is a general tendency that shows increasing differences between the individual state and the population as whole as geographic weight increases. While this pattern is not absolute, the general trend is nevertheless clear—increasing localization of strategic concerns corresponds with greater differences between agents in the local environments and their mean values of the population as a whole.
A simpler test of geography in the RPS model simply accounts for the effects of border lengths between neighboring states. The way in which this parameter was implemented varied significantly than with the geographic distance parameter. Rather than affect how each agent assessed its view of the entire population, the borer length parameter only modified the weights of those agents that were directly on its borders. Thus, all agents that were not immediate neighbors were treated equally regardless of their spatial position. As a result, agent decision-making was highly localized and did not consider the extended geography of the system, whereas the geographic distance weighting parameter was applied continuously, the border length parameter was discrete. As before, 31 experiments were conducted, adjusting the border length weighting parameter on the interval [0.0 3.0] with a step size of 0.1.
The border length parameter adjusts the speed of the system’s dynamics but not its qualitative properties. The strategic cycles exhibited by the RPS game persist, with the system entering into corners regardless of the border length weighting applied. However, as the border length weighting increased the speed at which the system moved through its strategic cycles intensified. The population level percentage rocks are shown below for all values and then for a smaller subset to make the trend more apparent.
What is clear from these outputs is that the higher the border length weighting, the more quickly the system enters into strategic transitions that are shown by the percentage of rocks being played moving from 0.0 to 1.0 percent and back.
Finally, a look at the difference between State 0’s percentage rocks and the overall population shows a reversed trend when compared with the geographic distance weighting discussed earlier. Rather than intensifying differences as the border length weighting increased, the differences between the population as a whole and the strategy of the individual agents diminishes as the border lengths become increasingly important.
The results of the modified RPS model have some interesting implications for real world arms races. However, given the limitations of the model, e.g. the differences between social/organizational learning vs. biological evolution or the real-world problem of determining the differences between and payoffs for wins, loses, and draws, these model outputs should not be over interpreted or taken literally.
In most cases, it would seem that the international relations would more likely resemble the case of the geographic weighting, where each state pays close attention to the capabilities of rivals and adjusts it threat assessments based on factors such as distance, the character of military technology and logistics, etc. To this equation, border length is a consideration, but it is unlikely that states view all others equally after securing or considering their borders except in special cases, such as being unable (or unwilling) to differentiate between rival groups beyond one’s sovereign domain. This may be the case in frontier states that border on nomadic or tribal regions whose political composition and distinctions are simply ignored or not known by the state.
Therefore, it is more likely that changes in technology, doctrine, organization, and other military innovations will cause qualitative changes in the dynamics shown by the RPS model than simply accelerating or slowing its core dynamics. The addition of geography is a notable start, but greater exploration of the model is needed, particularly with respect to agent logic, and the structure of payoffs in order to account for the internal variations of states, i.e. their domestic politics, and asymmetries between offense and defense. Future postings will continue to explore the evolution of this model, making it increasingly more complex.
Data from these experiments can be found in these two spreadsheets.