This blog is primarily derived from Multiplicative Updates in Coordination Games and the Theory of Evolution (2012) (arXiv:1208.3160) authored by E. Chastain, A. Livnat, C. Papadimitriou & U. Vazirani. The paper takes a crack on the famous enigma in evolutionary biology.

The longstanding puzzle of ‘Why sex?’ remains one of the widely debated upon questions in evolutionary biology. Sexual reproduction is everywhere in nature — yet, from a cost–benefit point of view, it looks almost irrational.

Consider the drawbacks:

• It’s complex, risky, and energy-hungry.
• It breaks apart well-adapted gene combinations.
• It halves/dilutes a parent’s genetic contribution to offspring.

Classic explanations — like faster adaptation, removal of deleterious mutations, or Red Queen–style arms races — have been insightful, but none have perfectly nailed down why sex consistently wins out despite these costs.

A Different Angle: Mixability

The authors propose reframing the problem. Under sexual reproduction, natural selection doesn’t chase the single “fittest” combination of alleles. Instead, it rewards alleles that play well with others.

Mixability of an Allele: An allele’s ability to work reasonably well with a wide range of genetic partners.

From this angle, what used to be called sex’s biggest flaw — constantly breaking apart “lucky” genetic combinations — becomes its central feature. The shuffling keeps the focus on robustness and flexibility, not on a brittle, perfect match.

The central result is proved in the weak selection regime, where fitness differences are small (think fitness values in \([1-s,\, 1+s]\) with small \(s\)).

Key points:

1. Quick drift to the Wright manifold
   The system rapidly settles into a state with no linkage disequilibrium — meaning genes behave as if inherited independently. Genotype distributions can then be written as products of allele frequencies, which simplifies the analysis.

2. Evolution ≡ Multiplicative Updates
   In weak selection, the population genetics dynamics are exactly the same as the multiplicative update rule from online learning (classes of learning algorithm that Netflix might be using for optimizing the recommendations) and game theory.

   • Genes are the players.
   • Alleles are strategies.
   • Allele frequencies are strategy probabilities.
   • The payoff is mixability.

   The paper is very emphatic:

   “We are not saying the equilibrium can be found through multiplicative updates — we are saying something stronger: Evolution is multiplicative updates.”

3. Robust convergence
   This process converges to an equilibrium and remains stable even under small random perturbations to fitness values.

The Diversity Question

But simply convergence to equilibrium is not good enough for a population. One of the most appealing outcomes of this framework is that it ensures that diversity can persist at equilibrium (opposed to the case where a population of a bacterium would become uniform eventually). It’s not obvious — multiplicative updates often converge to narrow supports in other contexts — so the authors go further:

• They show that for two genes, there are exponentially many equilibria where the support (the set of alleles with nonzero frequency) contains a significant fraction of the alleles for each gene.
• This “exponential support” result means the loss of diversity is not inevitable — the dynamics can sustain large, mixed populations.

The proof uses a neat trick:

• Represent the differential fitness between allele combinations as a random matrix.
• Show (via a potential function argument) that, with high probability, the inverse of this matrix has non-negative row and column sums.
• This property guarantees the existence of large-support equilibria.

The potential function argument is simple but clever: Flip the sign of any row or column whose sum is negative, track a global measure or potential (here the sum of all the elements of the matrix), and observe that it always increases. Eventually, one end up in a state where all row/column sums are non-negative.

Limitations & Open Questions

The framework is elegant but comes with boundaries:

• The proof relies on small fitness differences (weak-selection). It’s unclear which other fitness landscapes would yield the same multiplicative-update dynamics.

• The results assume that linkage disequilibrium disappears quickly (Wright manifold assumption). A characterization of landscapes that stay on or return to this manifold still remains open.

• Once an allele is lost, it’s gone (unless mutation reintroduces it). Real evolution, of course, never stabilizes — mutations are a slow but constant background force.

• The equilibrium reached may not be a Nash equilibrium of the full game, but of a smaller ‘subgame’ containing only the alleles that survived.

• The formal diversity result is proved for the two-gene case. Extending it to three or more genes is mathematically tough beacuse it would involve tensors of higher order, though simulations hint that large-support equilibria still exist.

A Computational Detour: The PLS Question

Here’s where things get unexpectedly fun for anyone who likes computational complexity theory. The row/column-flipping step above isn’t just a neat trick — it’s also a search problem in disguise:

Given a nonsingular square matrix, can one flip rows and columns so that its inverse has non-negative row and column sums?

The authors point out that this problem lives in the complexity class PLS — Polynomial Local Search. PLS problems have three key traits:

1. A solution always exists.
2. One can check the quality of a solution efficiently (in polynomial time).
3. One can move to a better solution by a small local change.

The authors ask:

Is this matrix-flipping problem PLS-complete?

• PLS-complete problems are the ‘hardest’ in PLS. If one can solve a single such problem efficiently, one solve all of them.
• But if this matrix problem were PLS-complete, it would mean the computational challenge of predicting diversity-maintaining equilibria is as hard as any local search problem we know.

In other words: If this is PLS-complete, the challenge of predicting/determining evolution’s equilibrium won’t be just due to messy biology and bad data— it’d be a fundamentally (or more aptly computationally) hard..

Closing Thoughts

This work reframes sexual mode of reproduction as a coordination game between genes, where multiplicative updates naturally push the system toward alleles with strong mixability. It’s a clean, learning-theoretic way to capture the why of sex — at least in the weak selection regime — and it opens the window into other questions.

Me and my friend Pritipriya redid the calculations of the paper as a part of project in our ‘Theoretical ML’ coursework. The blog was one of the attempts at commentary on the article. For technical details on calculations/proof one can take a look at the preliminary project report we wrote.

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