Eficient Capital Markets 389 1. Random Walks and Fair Games: A Little Historical Background As noted earlier, all of the empirical work on efficient markets can be con- sidered within the context of the general expected return or "fair game del, and much of the evidence bears directly on the special submartingale expected return model of (6). Indeed, in the early literature, discussions of the efficient markets model were phrased in terms of the even more special andom walk model, though we shall argue that most of the early authors were rned with more general versions of the "fair game'model Some of the confusion in the early random walk writings is understandable. Research on security prices did ent of a theory of price formation which was then subjected to empirical tests. Rather, the mpetus for the development of a theory came from the accumulation of ev- idence in the middle 1950s and early 1960 s that the behavior of common stock and other speculative prices could be well approximated by a random walk. Faced with the evidence, economists felt compelled to offer some ratio nalization. What resulted was a theory of efficient markets stated in terms of random walks, but usually implying some more general"fair game ' model It was not until the work of Samuelson [38] and Mandelbrot [27] in 1965 and 1966 that the role of "fair game"expected return models in the theory of efficient markets and the relationships between these models and the theory of random walks were rigorously studied. And these papers came somewhat after the major empirical work on random walks. In the earlier work,"theo- retical""discussions, though usually intuitively appealing, were always lacking in rigor and often either vague or ad hoc. In short, until the Mandelbrot- Samuelson models appeared, there existed a large body of empirical results in search of a rigorous theor Thus, though his contributions were ignored for sixty years, the first state- ment and test of the random walk model was that of Bachelier [3] in 1900 But his "fundamental principle for the behavior of prices was that specula tion should be a"fair game; in particular, the expected profits to the specu- lator should be zero. With the benefit of the modern theory of stochastic processes, we know now that the process implied by this fundamental principle is a martingale. After Bachelier, research on the behavior of security prices lagged until the 6. Basing their analyses on futures contracts in commodity markets, Mandelbrot and Samuelson show that if the price of such a contract at time t is the expected value at t(given information ) of the spot price at the termination of the contract, then the futures price will follow a artingale with respect to the information sequence (p,; that is, the expected price change from riod to period will be zero, and the price changes will be a"fair game. If the equilibrium ex- pected return is not assumed to be zero, our more general"fair game model, summarized by (1) informationΦ, :出= f the assumptions the returns and that the