Fig 1.
Game play and updating mechanism.
In each round agents must (1) select an interaction partner, (2) play a game of conflict, and (3) update both their network weight and strategy weights based on the payoff π.
Fig 2.
In games of conflict with power asymmetry and random interaction, individuals with different ranks effectively face different games.
A, Analytical analysis of our model reveals that, under random interaction, individuals play different games, depending on their rank and the degree of the power asymmetry f (see S1 Text). For f < dh, power asymmetry has no effect and individuals of all ranks engage in a mean game of conflict (the orange area). For f ≥ dh, the game is dominant-solvable for the top ranked individual(s). B, Numerical results from evolutionary simulations show that conventions prevail when f is below the critical value dh; individuals of all ranks play correlated conventions (either all ownership or host-guest; each appear with equal frequency in simulation seeds; ownership shown). At f ≥ dh a transition occurs as ranking individuals break from the convention and adopt pure hawk behavior both home and away. The correlated convention is resilient among most individuals until high values of f make it unsustainable. A hierarchy forms in which the top-half of individuals are pure hawks while the bottom-half are pure doves. Note how the analytic result on the left matches the numeric result on the right. (Both panels use n = 20; dh = 0.4; dd = 0.6).
Fig 3.
Partner choice restores the correlated convention and increases cooperation in the presence of bullies.
A, With partner choice and low f-value, individuals of all ranks play correlated conventions (predominantly host-guest as first shown in [12]). A transition occurs at f ≥ dh, identical to the random interaction case, where top ranked individual(s) break from the convention and become pure hawks. The correlated convention is preserved among outranked individuals. However, contrary to the random interaction case, partner choice allows the convention to remain sustainable among a majority of individuals even for high f values. This is infeasible under random interaction. B, The pattern in which some top ranked individual(s) break from the convention is consistent across population sizes. Notice how the shape of the curve in B matches the boundary where strategies change in A. Pure hawks are defined conservatively as agents with a likelihood of at least 0.8 for playing hawk both at home and away. C, The network structure that emerges resembles a hierarchy. Ranking individuals play pure hawk strategies but receive few or no visitors at all. Outranked individuals adopt the correlated convention and attract many visitors. The individual who is outranked by all others is visited by everyone. The graph shows average in-weights across seeds grouped by rank. Node size is scaled by incoming edge weight (dh = 0.4;dd = 0.6; f = 0.6). D, The highest degree of network centralization (most hub-like) is reached at f = 0.2. Nodes with disproportionate (too many) connections stop emerging entirely at f = 0.8 (color by mean network centralization). E, Network centralization over time, averaged across seeds. Networks with heterogeneous node weights emerge for a period of time. Network centralization of random Erdős–Rényi networks of the same size and density are shown as reference (the dashed line shows the median, the grey area is 95% CI).
Fig 4.
Cycles of interacting rank and strategy changes.
Allowing individuals’ ranks to change based on cumulative payoffs (dynamic ranks) leads to coupled cycles of aggressive-cooperative strategy, shifting network position, and rising and falling rank.
Fig 5.
Cycles have equalizing force that reduce wealth inequality.
A, Two example cycles of coupled changes in network positions, strategies, and ranks (dh = 0.4; dd = 0.6; f = 0.4). B, Below the critical value f < dh, ranks remain static and no cycles emerge (the grey shaded area). At f ≥ dh, a phase transition occurs and coupled cycles emerge. The cycle length varies as a function of f. Cycles are shorter when power asymmetry is higher. C, Static networks lead to low total payoff due to inefficient hawk-hawk conflict while partner choice stabilizes the convention and leads to high payoffs. D, Static networks lead to high inequality as bullies earn higher payoffs. Partner choice with static ranks reverses the pattern with low-ranking individuals earning high payoffs as they attract more visitors. Partner choice and dynamic ranks preserving perfect equality among individuals. (dh = 0.4; dd = 0.6; f = 0.6).