Computationally Efficient Collaborative Communication Via Regularity-Based Coarsening

Our results show that the existence of a short high-utility protocol already suffices for efficient communication. In particular, in a game with n possible observations and m actions: (1) For any achievable target utility α, we give an algorithm with poly(n, m, 1/ε) runtime that designs a protocol achieving utility at least α-ε using only 2^{\mathcal O(CC_α(G))}/ε^2 bits of communication. Here, CC_α(G) is the minimum number of bits used by any protocol, even a computationally inefficient one, to achieve utility α. (2) We prove that this exponential dependence on CC_α(G) is tight up to a constant. That is, unless \mathrm P=NP, no polynomial-time algorithm can in general find optimal protocols using fewer than 2^{CC_α(G) -2} bits. We note that our results strictly weaken the assumptions required by prior work in the multi-agent information aggregation literature, filling a gap that had remained elusive even for games with constant CC_α(G). In particular, prior guarantees for agreement-based information aggregation rely on structural assumptions such as informational substitutes or weak learnability. We show that these assumptions already imply CC_α(G) = O(1) and are therefore more restrictive conditions than required by our protocol to succeed. On a technical level, our results involve a novel strengthening of the Frieze-Kannan weak regularity lemma and yield the following powerful polynomial-time transformation tool: for every communication game G, it constructs a game \hat G that is a coarsening of the agents' observation spaces into constant-size partitions, such that G and \hat G are indistinguishable with respect to every short communication protocol. This coarsening theorem is the engine behind our algorithm and may be of independent interest.