-Rank: Practically Scaling -Rank through Stochastic Optimisation

Recently, -Rank, a graph-based algorithm, has been proposed as a solution to ranking joint policy profiles in large scale multi-agent systems. -Rank claimed tractability through a polynomial time implementation with respect to the total number of pure strategy profiles. Here, we note that inputs to the algorithm were not clearly specified in the original presentation; as such, we deem complexity claims as not grounded, and conjecture solving -Rank is NP-hard. The authors of -Rank suggested that the input to -Rank can be an exponentially-sized payoff matrix; a claim promised to be clarified in subsequent manuscripts. Even though -Rank exhibits a polynomial-time solution with respect to such an input, we further reflect additional critical problems. We demonstrate that due to the need of constructing an exponentially large Markov chain, -Rank is infeasible beyond a small finite number of agents. We ground these claims by adopting amount of dollars spent as a non-refutable evaluation metric. Realising such scalability issue, we present a stochastic implementation of -Rank with a double oracle mechanism allowing for reductions in joint strategy spaces. Our method, -Rank, does not need to save exponentially-large transition matrix, and can terminate early under required precision. Although theoretically our method exhibits similar worst-case complexity guarantees compared to -Rank, it allows us, for the first time, to practically conduct large-scale multi-agent evaluations. On random matrices, we achieve speed reduction. Furthermore, we also show successful results on large joint strategy profiles with a maximum size in the order of ( million joint strategies) -- a setting not evaluable using -Rank with reasonable computational budget.
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