Eficient Capital Markets 385 in terms of expected returns and that equilibrium expected returns are formed on the basis of (and thus" fully reflect " the information set t have a major empirical implication-they rule out the possibility of trading systems based only on information in t that have expected profits or returns in excess of equilibrium expected profits or returns. Thus let E(P3+1|重) Then E(xt+1|)=0 (3) which, by definition, says that the sequence(x3t) is a fair game"with respect to the information sequence (%t]. Or, equivalently, let t+1=rt+1-E(ft+1④), (4) E(2t+1D) so that the sequence (Zjt) is also a " fair game "with respect to the information sequence [) In economic terms, x, t +1 is the excess market value of security j at time t+1: it is the difference between the observed price and the expected value of the price that was projected at t on the basis of the information t. And similarly, zj, t+1 is the return at t+ 1 in excess of the equilibrium expected return projected at t. Let a(4)=[a1(t),a2(t),…,an(t) be any trading system based on which tells the investor the amounts a @t) of funds available at t that are to be invested in each of the n available secu- rities. The total excess market value at t+ 1 that will be generated by such a V+1=2吗(4)[r…+1-E(E+1)], game"property of(5)has expectatic E(W+1厘)=2a(亚)E(动t+1)=0 The expected return or "fair game efficient markets model has other important testable implications, but these are better saved for the later dis cussion of the empirical work. Now we turn to two special cases of the model the submartingale and the random walk, that (as we shall see later)play an important role in the empirical literature 2. Though we shall sometimes refer to the model summarized by(1)as the "fair game" model, keep in mind that the "fair game properties of the e implications of the assumptions that (i) the conditions of market equilibrium can be stated in terms of expected returns, and (ii)the nformation pt is fully utilized by the market in equilibrium expected returns and thus current he role of fair game models in the theory of efficient markets was first recognized and ously by Mandelbrot [27] and Samuelson [38]. Their work will be discussed in more detail late