Iyer and Killingback (2020) proved that when individuals in a social dilemma interact with their own type with probability r and otherwise with a random population member, assortativity both raises cooperation levels and - for the specific quadratic (continuous snowdrift) and quadratic-cost/cubic-benefit (continuous tragedy of the commons) functional forms analyzed - inhibits evolutionary branching, the splitting of a population into divergent high- and low-investing types; they conjectured the inhibition is a general phenomenon. Completing this theory raises three linked open questions: (i) establish or refute branching inhibition for general classes of cost and benefit functions under exogenous assortment; (ii) build a well-posed assortativity kernel allowing negative and strategy-dependent assortment in continuous games, together with a strategy-dependent counterpart of the r-replicator dynamics in games with more than two strategies, to determine whether the assortative picture survives either sign of assortment and many strategies; and (iii) settle whether branching in these games on networks is inhibited as mean degree decreases, as the authors' unpublished simulations suggest. The conjecture remains open beyond the cases analyzed in the source paper: relatedness-distribution results cover only the multiplayer snowdrift case, negative assortment has been studied only for two strategies through partner choice, many-strategy games are covered by a different kin-selection formalism, and trait-dependent relatedness has been shown able to promote rather than inhibit diversification.

Background

The evolution of cooperation in social dilemmas - games in which individually advantageous behavior is collectively costly - is a central problem of evolutionary biology and the social sciences. A unifying logic behind the main proposed mechanisms (kin selection, direct and indirect reciprocity, network and group structure) is that they all generate assortative interactions: individuals using the same strategy interact with one another more often than expected at random. Iyer and Killingback (2020) study cooperation under assortativity directly, in the model suggested by the geometric view of relatedness: each individual interacts with another individual of its own type with probability rr and with a uniformly random population member with probability 1−r1-r, where r∈[0,1]r\in[0,1] is a single assortment parameter.

For discrete-strategy games with two pure strategies, they generalize the replicator dynamics (the standard ordinary differential equation describing how strategy frequencies change under selection) to an "r-replicator" dynamics and analyze the donation game (the prisoner's-dilemma exemplar, in which cooperation costs the donor cc and benefits the recipient bb), the snowdrift game (a hawk-dove-class exemplar in which two cooperators share the cost of a joint task), and the sculling game (a coordination-class exemplar, modeled on joint rowing, in which cooperation pays only if enough others cooperate too). They obtain analogs of Hamilton's rule (the classical inclusive-fitness condition that cooperation spreads when relatedness exceeds the cost-to-benefit ratio): for cost-to-benefit ratio ρ=c/b\rho=c/b, complete cooperation prevails in the snowdrift game once r>ρ/2r>\rho/2 and in the sculling game once r>2ρ/3−1/3r>2\rho/3-1/3.

For continuous-strategy games, where the strategy x∈[0,xm]x\in[0,x_m] is a level of investment or consumption, they use adaptive dynamics: the deterministic theory of gradual evolution in a monomorphic population, in which the direction of evolution is set by the selection gradient of the invasion fitness of a rare mutant strategy yy in a resident population of strategy xx. A rest point x⋆x^\star of the selection gradient is a singular strategy; it is an evolutionary branching point (EBP) when it is simultaneously convergence stable (the population evolves toward it) and evolutionarily unstable (mutants on either side of it can invade), in which case the population splits into two diverging phenotypic clusters. Evolutionary branching in snowdrift-type games was first demonstrated by Doebeli, Hauert and Killingback (2004). In the continuous snowdrift (CSD) game with payoff π(x,y)=B(x+y)−C(x)\pi(x,y)=B(x+y)-C(x), quadratic cost C(x)=−c2x2+c1xC(x)=-c_2x^2+c_1x and saturating benefit B(z)=−b2z2+b1zB(z)=-b_2z^2+b_1z, the singular strategy is

x⋆=(1+r)b1−c14(1+r)b2−2c2,x^\star=\frac{(1+r)b_1-c_1}{4(1+r)b_2-2c_2},

it is convergent stable if and only if b2>c2/(2(1+r))b_2>c_2/(2(1+r)), and it is an evolutionarily stable strategy (ESS), rather than a branching point, if and only if b2>c2/(1+3r)b_2>c_2/(1+3r); the branching region therefore shrinks as rr increases. In the continuous tragedy of the commons (CTOC) game with payoff π(x,y)=B(x)−C(x+y)\pi(x,y)=B(x)-C(x+y), convex quadratic cost C(z)=c1z2C(z)=c_1z^2 and sigmoidal cubic benefit B(x)=−b3x3+b2x2+b1xB(x)=-b_3x^3+b_2x^2+b_1x under the further specializations b2=2b1b_2=2b_1 and c1=b1c_1=b_1, the singular strategy is always convergent stable and is an ESS if and only if b1<16r2b12+12b1b3/(2(1−r))b_1<\sqrt{16r^2b_1^2+12b_1b_3}/(2(1-r)), so branching is again inhibited by assortment. Branching in these games produces a "tragedy of the commune" - an unequal and socially inefficient split into high and low investors - so its inhibition means assortativity plays a doubly beneficial role, and Iyer and Killingback conjecture that this inhibition is a general phenomenon holding beyond the specific functional forms and games they analyzed.

Two restrictions frame the open questions. First, the analysis assumes positive assortment r≥0r\ge 0 with a single strategy-independent scalar: negative assortment (interacting with one's own type less often than random) forces the assortment strength to depend on strategy, and for continuous games would require an "assortativity kernel" specifying how much positive or negative assortment each strategy experiences - a model that does not yet exist. Second, although the r-replicator dynamics are formulated for games with any finite number of strategies under a constant scalar rr, the paper's analysis covers only two strategies, and no counterpart of the dynamics exists for strategy-dependent assortment. A complementary framework due to Coder Gylling and Brännström (2018) introduces relatedness through a probability distribution over the number of co-players identical by descent to a focal individual (identical by descent means descended from a recent common ancestor without intervening mutation), and shows for multiplayer nonlinear public-goods (snowdrift-type) games that increased relatedness reduces the possibility of evolutionary branching - a result that contains the two-player CSD result as a special case.

Problem Statement

Within the assortative-interaction framework of Iyer and Killingback (2020) - each individual interacts with its own type with probability rr and with a uniformly random population member with probability 1−r1-r - settle the generality of the framework's central findings, which are currently established only for positive assortment, two strategies, and special functional forms. Three linked open questions arise:

(1) Functional generality of branching inhibition. For the continuous snowdrift game π(x,y)=B(x+y)−C(x)\pi(x,y)=B(x+y)-C(x) and the continuous tragedy of the commons game π(x,y)=B(x)−C(x+y)\pi(x,y)=B(x)-C(x+y) under exogenous assortment rr, with cost and benefit functions in the class considered by Iyer and Killingback - smooth, monotonically increasing functions with C(0)=B(0)=0C(0)=B(0)=0 and B(x)>C(x)B(x)>C(x) for x∈(0,xc)x\in(0,x_c), 0<xc<xm0<x_c<x_m, admitting an interior nondegenerate singular strategy - determine whether the inhibition of evolutionary branching proven for quadratic cost and benefit functions (snowdrift) and for quadratic cost with cubic benefit under the specializations b2=2b1b_2=2b_1, c1=b1c_1=b_1 (tragedy of the commons) extends to this class: prove that the set of admissible cost and benefit functions for which the singular strategy is convergence stable and evolutionarily unstable (a branching point) weakly shrinks, in the sense of set inclusion, as rr increases, or exhibit explicit admissible cost and benefit functions for which increasing rr fails to shrink, or expands, that set.

(2) Sign and strategy-dependence of assortment. Construct a well-posed assortativity kernel for continuous-strategy social dilemmas - a rule specifying, per strategy, the signed degree of positive or negative assortment experienced, with negative values meaning interaction with one's own type less often than expected at random, and reducing to the scalar-rr model when constant - together with the selection dynamics it induces: for continuous games, a generalization of the invasion fitness φxr(y)=rπ(y,y)+(1−r)π(y,x)−π(x,x)\varphi_x^r(y)=r\pi(y,y)+(1-r)\pi(y,x)-\pi(x,x) in which the degree of assortment depends on the strategies, and, for discrete games with more than two strategies, a strategy-dependent counterpart of the r-replicator dynamics (which Iyer and Killingback define for any finite number of strategies under a constant scalar rr). Establish the consistency of the induced dynamics (interaction probabilities well defined, nonnegative, and normalized; the random-matching limit recovered at zero assortment) and characterize how cooperation levels and branching behavior depend on the sign and strength of assortment, determining whether branching inhibition survives, reverses, or breaks down under negative or strategy-dependent assortment.

(3) Population structure. Establish whether evolutionary branching in the continuous snowdrift and tragedy of the commons games on networks is inhibited as the network mean degree decreases - as the authors' unpublished simulations suggest - either analytically or through a systematically specified, replayable simulation study, and determine to what extent the network effect is captured by an effective assortment parameter, as suggested by the identification r=1/(k−1)r=1/(k-1) for discrete games on networks of mean degree kk in the paper's discussion.

The common objective uniting these questions is to determine how far the framework's central findings - assortativity raises cooperation levels and inhibits evolutionary branching - extend beyond the special settings in which they were established: quadratic and cubic functional forms, a positive strategy-independent scalar rr, two strategies, and well-mixed populations. All three questions are jointly required: a complete resolution must address (1), (2), and (3) together, each on its own terms, and settling any one question in isolation does not resolve the task.

References

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