Asymptotic growth of the expected number of internal equilibria in general d-player n-strategy random evolutionary games (Duong–Han conjecture)

In a random evolutionary game, the payoff entries are treated as independent Gaussian random variables and the number of internal equilibria of the replicator dynamics becomes a random variable whose expectation, $E(n,d)$, measures the expected complexity and behavioral diversity of a $d$-player game with $n$ strategies. Duong and Han proved computable formulas and bounds, and conjectured on numerical evidence that $E(n,d)$ grows like $(d-1)^{(n-1)/2}$ as the group size $d$ grows, with the expected density $f_{n,d}$ of internal equilibria growing on the same scale. The two-strategy case ($n=2$) was later proven with sharp constant, and the asymmetric-game analogue was solved exactly (under iid Gaussian induced coefficients) through an identification with the Kostlan–Shub–Smale random polynomial system. The conjecture for symmetric games with $n\geq 3$ strategies remains unproven: prove or refute that $\lim_{d\to\infty}\ln E(n,d)/\ln(d-1)=(n-1)/2$ and that $f_{n,d}=\Theta((d-1)^{(n-1)/2})$.

Background

Evolutionary game theory (EGT) studies how frequencies of competing strategies in a large population evolve under selection, typically via the replicator dynamics: the share $x_i$ of strategy $i$ grows when its expected fitness (average payoff) exceeds the population average. In a $d$-player game, individuals interact in randomly formed groups of $d$ participants, so the payoff to a strategist depends on the multiset of strategies of the other $d-1$ group members. An internal equilibrium is a rest point of the replicator dynamics in the interior of the strategy simplex, i.e. a state where all $n$ strategies coexist; the number of internal equilibria is a standard proxy for the behavioral, cultural, or biological diversity a game can support. In random evolutionary games the payoff entries are random variables (here independent identically distributed Gaussians), and the number of internal equilibria becomes a random variable. Writing $E(n,d)$ for its expectation and $f_{n,d}(t)$ for the expected density of internal equilibria in the coordinates $t_i=x_i/x_n$ (so that integrating $f_{n,d}$ over a region gives the expected number of equilibria there), the equilibrium conditions reduce to a system of $n-1$ random polynomial equations of degree $d-1$ in $t_1,\dots,t_{n-1}$, which places the problem inside the theory of real zeros of random polynomial systems (via the Kac–Rice formula, the standard tool that expresses the expected number of zeros of a random function through the joint distribution of the function and its derivatives). Duong and Han established computable formulas for $E(n,d)$ and $f_{n,d}$, the exact value $E(n,2)=2^{-(n-1)}$ for two-player games, and bounds $\sqrt{d-1}\lesssim E(2,d)\lesssim\sqrt{d-1}\ln(d-1)$ for two-strategy games. Motivated by the elliptic (Kostlan–Shub–Smale) random polynomial system — a Gaussian system whose covariance kernel is the generating function $(1+x\cdot y)^{d-1}$ and whose expected number of real zeros is exactly $(d-1)^{(n-1)/2}$ — they conjectured on extensive numerical evidence (Conjecture 2 of Duong & Han 2016) that the same power law holds for general symmetric games: $f_{n,d}\sim(d-1)^{(n-1)/2}$ and $\lim_{d\to\infty}\ln E(n,d)/\ln(d-1)=(n-1)/2$, envisaging the stronger form $E(n,d)\sim(d-1)^{(n-1)/2}$. Two later results sharpen the picture without settling it: Can, Duong, and Pham proved sharp asymptotics for the univariate (two-strategy) case, giving $E(2,d)\sim\sqrt{(d-1)/2}$; and Duong and Han showed that for asymmetric games (where a player's payoff depends on the ordering of group members) the equilibrium system is exactly the Kostlan–Shub–Smale system, yielding $E(N_{d,n})=2^{-(n-1)}(d-1)^{(n-1)/2}$ for all $d,n$ (under the assumption that the induced polynomial coefficients are iid Gaussian), together with a central limit theorem and universality, and proving for two-strategy games that symmetry increases the expected number of internal equilibria. The obstruction in the symmetric multi-strategy case is structural: for $n>2$ the Gaussian coefficient vectors of the $n-1$ equilibrium polynomials are not independent, and the covariance kernel involves squared multinomial coefficients rather than the generating function, so the classical multivariate Kac–Rice computation for elliptic systems does not apply.

Problem Statement

For symmetric $d$-player $n$-strategy random evolutionary games whose payoff entries are independent identically distributed Gaussian random variables, prove or refute the asymptotic growth law conjectured by Duong and Han (2016, Conjecture 2) for the general case $n\geq 3$, which is currently supported only by numerical evidence. The conjecture is the conjunction of two assertions; both are asked for, and a submission that settles only one of them resolves that part but not the whole conjecture:

$\lim_{d\to\infty}\frac{\ln E(n,d)}{\ln(d-1)}=\frac{n-1}{2},$

where $E(n,d)$ is the expected number of internal equilibria of the replicator dynamics (equivalently, the expected number of solutions in the positive orthant of the induced system of $n-1$ random polynomial equations of degree $d-1$ in the coordinates $t_i=x_i/x_n$); and, for the expected density of internal equilibria, the pointwise order statement that for each fixed $t=(t_1,\dots,t_{n-1})$ in the positive orthant there are positive constants $0<c(t)\leq C(t)$ with

$c(t)\,(d-1)^{(n-1)/2}\ \leq\ f_{n,d}(t)\ \leq\ C(t)\,(d-1)^{(n-1)/2}$

for all sufficiently large $d$ — the density analogue of the elliptic (Kostlan–Shub–Smale) system, whose expected density is of the same pointwise order and whose expected number of real zeros over $\mathbb{R}^{n-1}$ is exactly $E_{EP}=(d-1)^{(n-1)/2}$; by sign-flip symmetry only $2^{-(n-1)}(d-1)^{(n-1)/2}$ of those zeros lie in the positive orthant, the value quoted below for the asymmetric case.

The two settled boundary cases anchor the conjecture: $E(2,d)\sim\sqrt{(d-1)/2}$ for symmetric two-strategy games (leading constant $1/\sqrt{2}$) and $E(N_{d,n})=2^{-(n-1)}(d-1)^{(n-1)/2}$ for the asymmetric counterpart (the latter proved under iid Gaussian induced coefficients); for two-strategy games it is also proved that symmetry increases the expected number of internal equilibria, $E^{\mathrm{sym}}(N_{d,2})\ge E^{\mathrm{asym}}(N_{d,2})=\sqrt{d-1}/2$.

References

1. The Anh Han (2023). Evolutionary game theory: the mathematics of evolution and collective behaviours. arXiv:2311.14480, DOI: 10.48550/arXiv.2311.14480. https://arxiv.org/abs/2311.14480
2. Manh Hong Duong and The Anh Han (2016). Analysis of the expected density of internal equilibria in random evolutionary multi-player multi-strategy games. Journal of Mathematical Biology 73(6): 1727–1760, DOI: 10.1007/s00285-016-1010-8. https://doi.org/10.1007/s00285-016-1010-8 (arXiv:1505.04676)
3. Manh Hong Duong and The Anh Han (2016). On the Expected Number of Equilibria in a Multi-player Multi-strategy Evolutionary Game. Dynamic Games and Applications 6(3): 324–346, DOI: 10.1007/s13235-015-0148-0. https://doi.org/10.1007/s13235-015-0148-0 (arXiv:1408.3850; published online 25 March 2015)
4. Van Hao Can, Manh Hong Duong, and Viet Hung Pham (2022). On the expected number of real roots of random polynomials arising from evolutionary game theory. Communications in Mathematical Sciences 20(6): 1613–1636, DOI: 10.4310/cms.2022.v20.n6.a7. https://doi.org/10.4310/cms.2022.v20.n6.a7 (arXiv:2010.14198)
5. Manh Hong Duong and The Anh Han (2025). Random evolutionary games and random polynomials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 481(2319): 20240911, DOI: 10.1098/rspa.2024.0911. https://doi.org/10.1098/rspa.2024.0911 (arXiv:2304.13831)
6. Manh Hong Duong and The Anh Han (2025). Evolutionary dynamics with random payoff matrices. Europhysics Letters 149(3): 32001, DOI: 10.1209/0295-5075/ada0d8. https://doi.org/10.1209/0295-5075/ada0d8 (arXiv:2410.22010)
7. Manh Hong Duong, Hoang Minh Tran, and The Anh Han (2019). On the Expected Number of Internal Equilibria in Random Evolutionary Games with Correlated Payoff Matrix. Dynamic Games and Applications 9(2): 458–485, DOI: 10.1007/s13235-018-0276-4. https://doi.org/10.1007/s13235-018-0276-4 (arXiv:1708.01672; published online 28 July 2018)
8. Manh Hong Duong, Hoang Minh Tran, and The Anh Han (2019). On the distribution of the number of internal equilibria in random evolutionary games. Journal of Mathematical Biology 78(1–2): 331–371, DOI: 10.1007/s00285-018-1276-0. https://doi.org/10.1007/s00285-018-1276-0 (arXiv:1711.03848; published online 1 August 2018)