Yao's Principle

In computational complexity theory, Yao's principle or Yao's minimax principle states that the expected cost of any randomized algorithm for solving a given problem, on the worst case input for that algorithm, can be no better than the expected cost, for a worst-case random probability distribution on the inputs, of the deterministic algorithm that performs best against that distribution. Thus, to establish a lower bound on the performance of randomized algorithms, it suffices to find an appropriate distribution of difficult inputs, and to prove that no deterministic algorithm can perform well against that distribution. This principle is named after Andrew Yao, who first proposed it.

Yao's principle may be interpreted in game theoretic terms, via a two-player zero sum game in which one player, Alice, selects a deterministic algorithm, the other player, Bob, selects an input, and the payoff is the cost of the selected algorithm on the selected input. Any randomized algorithm R may be interpreted as a randomized choice among deterministic algorithms, and thus as a strategy for Alice. By von Neumann's minimax theorem, Bob has a randomized strategy that performs at least as well against R as it does against the best pure strategy Alice might choose; that is, Bob's strategy defines a distribution on the inputs such that the expected cost of R on that distribution (and therefore also the worst case expected cost of R) is no better than the expected cost of any single deterministic algorithm against the same distribution.