Markets are efficient if and only if P = NP

I prove that if markets are efficient, meaning current prices fully reflect all information available in past prices, then P = NP , meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can “program” the market to solve NP -complete problems. Since P probably does not equal NP , markets are probably not efficient. Specifically, markets become increasingly inefficient as the time series lengthens or becomes more frequent. An illustration by way of partitioning the excess returns to momentum strategies based on data availability confirms this prediction. Are the concepts of market efficiency in the field of finance and computational efficiency in the field of computer science really the same thing? The efficient market hypothesis claims that all information relevant to future prices is immediately reflected in the current prices of assets. In other words, you cannot consistently make money using publicly available information. Specifically, the weakest form of the EMH states that future prices cannot be predicted by analyzing prices from the past. Therefore, technical analysis cannot work in the long run, though of course any strategy could make money randomly. Most finance academics believe markets are weak form efficient: Doran et al. (2007) survey more than 4,500 finance professors and find that of the nearly 650 usable responses, the majority believe the US stock market is weak form efficient; only 8 percent generally disagree. Stronger forms of market efficiency presume the weak form, so the results here are in full generality. Computational efficiency in computer science distinguishes two kinds of algorithms. One class of algorithms belongs to the set known as P , short for Polynomial, because they can find a solution to an input of length n in a time that is polynomial in n. For example, finding a sorted version of an input list can be done in polynomial time. Another class of algorithms belongs to the set known as NP , short for Nondeterministic Polynomial, because they can verify a proposed solution to an input of length n in a time that is polynomial in n. Verifying a solution means that the algorithm always halts and answers whether the given proposal really does satisfy the input. For example, is it possible to visit each capital city in Europe exactly once and within a week? Verifying a proposed solution can be done in polynomial, indeed linear, time. Does the proposed route visit each capital? Is the total travel time less than a week? (Finding a satisfying solution, however, might only be possible in exponential, not polynomial, time. This problem is known as the traveling salesman problem.) Polynomial time is, in short hand, considered efficient. Obviously P is a subset of NP : any algorithm that can efficiently generate a solution can efficiently verify a proposed solution. The outstanding question in computer science is: does P = NP? In other words, if a solution can be verified efficiently, does that mean it can be computed efficiently? Can the traveling salesman and similar problems be solved efficiently? Consider the problem of satisfiability. Given a Boolean expression composed of variables and negated variables combined through ANDs and ORs, where each variable can be either TRUE or FALSE, it is easy to confirm whether a particular proposed solution in fact satisfies the formula, but in general finding the solution seems to require checking every possible assignment of variables to TRUE or FALSE. For n variables, that requires searching through 2158-5571/11/$27.50 c © 2011 – IOS Press and the authors. All rights reserved 2 Philip Z. Maymin / Markets are efficient if and only if P = NP 2 possibilities, meaning the algorithm works in exponential time, not polynomial time. However, there has been no proof that there does not exist some algorithm that can determine satisfiability in polynomial time. If such an algorithm were found, then we would have that P = NP . If a proof is discovered that no such algorithm exists, then we would have that P 6= NP . Just as most people in the field of finance believe markets are at least weak-form efficient, most computer scientists believe P 6= NP . Gasarch (2002) reports that of 100 respondents to his poll of various theorists who are “people whose opinions can be taken seriously,” the majority thought the ultimate resolution to the question would be that P 6= NP ; only 9 percent thought the ultimate resolution would be that P = NP . The majority of financial academics believe in market efficiency and the majority of computer scientists believe that P 6= NP . The result of this paper is that they cannot both be right: either P = NP and the markets are efficient, or P 6= NP and the markets are not efficient.