Abstract
We propose a novel model to explain the mechanisms underlying dominance hierarchical structures. Guided by a predetermined social convention, agents with limited cognitive abilities optimize their strategies in a Hawk-Dove game. We find that several commonly observed hierarchical structures in nature such as linear hierarchy and despotism, emerge as the total fitness-maximizing social structures given different levels of cognitive abilities.
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Notes
See also Nakamaru and Sasaki (2003) for a study on the evolution of transitive inference.
It is still possible for the agents to infer the relative ranks between themselves and their opponents without having the identities of their opponents in memory. For example, the top ranked agent can infer that all its opponents have lower ranks; the bottom ranked agent can infer that all its opponents have higher ranks; the second top ranked agent who has the top ranked agent in its memory can infer that all other opponents have lower ranks. Nevertheless, inferring about the ranks of those agents that an agent has no memory of arguably requires strong cognitive ability, which cannot simply be modeled as memory. Hence, we do not consider such a possibility in this paper.
\(\lceil x \rceil\) is the ceiling function, which gives the least integer greater than or equal to x.
See the appendix for a description of the computation method. The code can be found at https://github.com/harrisonhhy/optimal_social_structure.
Symmetrically, a social system in which one agent is dominated by all others is also an equilibrium social structure that follows social convention and maximizes the total fitness.
Symmetrically, a social system in which the bottom ranked agent is dominated by the second bottom ranked and the second top ranked agent is dominated by all agents beside the bottomed ranked agent is also an equilibrium social structure that follows social convention and maximizes the total fitness.
We thank an anonymous reviewer for the suggestion.
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Acknowledgements
We sincerely thank the editor F.J.A. Jacobs and two anonymous reviewers for their suggestions that help to greatly improve the paper. We also thank Jonathan Newton for his comments.
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Appendix: Computational method to find the optimal social structure
Appendix: Computational method to find the optimal social structure
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1.
Set N = given population, m = given memory ability.
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2.
Create the memory profile matrix base and the strategy profile matrix base, they are N-by-N null matrices.
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3.
Digitize the strategies (we use: hawk = 3, dove = 1, mix = 2), to be used in the strategy profile matrix.
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4.
Generate an arbitrary memory profile matrix under the given memory ability.
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5.
Generate an arbitrary strategy profile matrix at equilibrium (the off-diagonal pairs must sum to 4, meaning players must play \(H-D\), \(D-H\), or \(M-M\) at equilibrium).
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6.
Check whether the strategy profile can be supported by the memory profile. The strategy profile can be supported by the memory profile if the agents play the same strategy against all non-memorized opponents:
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(a)
Locate all 0-value entries in the first row in the memory profile matrix. This reflects all non-memorized opponents for the first agent.
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(b)
Find all corresponding entries in the strategy profile matrix to the entries found in (a).
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(c)
If all entries found in (b) have the same value, then this agent’s strategy profile can be supported by its memory profile.
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(d)
Repeat (a) to (c) for every agents. The strategy profile matrix is supported by the memory profile matrix if all agents’ strategy profiles can be supported by their memory profiles.
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(a)
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7.
If the strategy profile can be supported by the memory profile, then it is an equilibrium. Count the number of strategy M by counting entries with value=2 in the strategy profile matrix. Divide by 2 will give us the number of pairs that play \(M-M\).
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8.
Repeat 5-7 for all possible strategy profile matrices at equilibrium.
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9.
Repeat 4-8 for all possible memory profile matrices under the given memory ability.
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10.
The social structure that induces the least number of \(M-M\) pairs is the optimal social structure. If there are multiple, compare their memory profile matrices. The one with more zeroes (fewer ones) is the superior one in terms of fitness, because of lower memory usage.
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Huang, H., Wu, J. Limited Cognitive Abilities and Dominance Hierarchies. Acta Biotheor 70, 17 (2022). https://doi.org/10.1007/s10441-022-09442-6
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DOI: https://doi.org/10.1007/s10441-022-09442-6