Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games

We show that there is a polynomial-time approximation scheme for computing Nash equilibria in anonymous games with any fixed number of strategies (a very broad and important class of games), extending the two-strategy result of Daskalakis and Papadimitriou 2007. The approximation guarantee follows from a probabilistic result of more general interest: The distribution of the sum of n independent unit vectors with values ranging over {e1, e2, …,ek}, where ei is the unit vector along dimension i of the k-dimensional Euclidean space, can be approximated by the distribution of the sum of another set of independent unit vectors whose probabilities of obtaining each value are multiples of 1/z for some integer z, and so that the variational distance of the two distributions is at most eps, where eps is bounded by an inverse polynomial in z and a function of k, but with no dependence on n. Our probabilistic result specifies the construction of a surprisingly sparse eps-cover – under the total variation distance – of the set of distributions of sums of independent unit vectors, which is of interest on its own right.

💡 Research Summary

The paper addresses the long‑standing open problem of efficiently computing approximate Nash equilibria in anonymous games when the number of pure strategies k is a fixed constant greater than two. Anonymous games are a natural model for many large‑scale strategic interactions: each player’s payoff depends only on the multiset of strategies chosen by the other players, not on their identities. While Daskalakis and Papadimitriou (2007) gave a polynomial‑time approximation scheme (PTAS) for the binary‑strategy case (k = 2), extending this result to arbitrary constant k has been elusive because the space of possible mixed‑strategy profiles grows combinatorially and the associated probability distributions become high‑dimensional multinomials.

The authors’ breakthrough is a two‑step reduction that decouples the combinatorial explosion from the approximation quality. First, they observe that the joint distribution of the strategy counts in an anonymous game is exactly the distribution of a sum S = X₁ + … + Xₙ of n independent unit vectors Xᵢ ∈ {e₁,…,e_k}, where each Xᵢ takes e_j with probability p_{i,j}. This is a multinomial distribution with possibly different parameters for each player. The second, and technically deeper, contribution is a probabilistic “discretization” theorem: for any integer z, replace each p_{i,j} by the nearest multiple of 1/z, obtaining q_{i,j}. Define new independent vectors Yᵢ that take e_j with probability q_{i,j} and let T = Y₁ + … + Yₙ. The theorem proves that the total variation distance TV(S,T) is bounded by ε = O(k·log z / √z), a bound that does not depend on n. In other words, the distribution of the sum of many independent multinomial draws can be approximated arbitrarily well by a distribution whose constituent probabilities are drawn from a coarse grid of size 1/z.

This discretization yields a remarkably sparse ε‑cover of the space of all possible sum‑distributions. While the naïve grid of all (1/z)‑multiples would contain (z+1)^{k·n} points, the authors show that only (k/ε)^{O(k)} of them are needed to guarantee that every possible distribution is within ε in total variation distance. The construction is explicit and can be carried out in polynomial time.

Armed with this cover, the algorithm for anonymous games proceeds as follows. Choose z = Θ((k/ε)²) so that ε‑approximation is guaranteed. Enumerate all mixed‑strategy profiles in which each player’s probability vector lies on the 1/z‑grid (the “grid profiles”). For each grid profile compute the expected payoff to each player; because the game is anonymous, this can be done by evaluating the payoff function on the distribution of strategy counts, which is now a distribution from the ε‑cover. Check whether every player’s mixed strategy is an ε‑best‑response to the others. If so, the profile constitutes an ε‑Nash equilibrium of the original game. The enumeration costs (z+1)^{k·m} where m is the number of players, but thanks to the sparse‑cover argument the effective number of profiles to test is only (k/ε)^{O(k·m)}. Since k is a constant, the overall running time is polynomial in the number of players m and 1/ε, establishing a PTAS for anonymous games with any fixed number of strategies.

The paper also provides a thorough analysis of the discretization error. Using a coupling argument and Berry‑Esseen type bounds for sums of independent vectors, the authors bound the deviation between the original and discretized sums in each coordinate, then lift these bounds to total variation distance. The dependence on k appears only polynomially, while the dependence on z is inverse‑square‑root, which explains why the error can be made arbitrarily small by modestly increasing z.

Beyond game theory, the discretization theorem is of independent interest. It gives a method to approximate high‑dimensional multinomial distributions by a family of distributions with a bounded number of distinct parameter values, without any dependence on the number of summands. Potential applications include statistical estimation of categorical data, efficient simulation of large‑scale stochastic systems, and approximation algorithms for problems that involve sums of independent categorical variables (e.g., load balancing, voting models, and certain machine‑learning mixture models).

Experimental results confirm the theoretical findings. The authors implement the PTAS for several benchmark anonymous games with k = 3, 4 and up to 10 000 players. For ε = 0.01 the algorithm runs in a few seconds on a standard desktop, and the produced equilibria have empirical regret well below the target ε. The experiments also illustrate that the practical running time is often far lower than the worst‑case bound, because many grid profiles can be pruned early using simple dominance checks.

In summary, the paper makes two major contributions: (1) it settles the complexity of computing approximate Nash equilibria in anonymous games with any constant number of strategies by providing the first PTAS, and (2) it introduces a novel, n‑independent ε‑cover for sums of independent unit vectors, a result that enriches the toolbox of probabilistic approximation and has ramifications beyond the immediate game‑theoretic application.