January 28, 2001 Eric Rasmusen notes, erasmuse@indiana.edu Http://Php.Indiana.edu/~erasmuse Avner Shaked, ``Existence and Computation of Mixed Strategy Nash Equilibrium for 3-Firms Location Problem,'' Journal of Industrial Economics, 31: 93-96 (September/December 1982). Despite the simplicity of the result--the equilibrium strategy in equation (*) on page 366-- the algebraic manipulations in this paper are unusually difficult. The punctuation of the mathematics is inconsistent in the original, and we have left it that way in the reader. p. 366. Why the "2"s in equation (1)? Because this adds the probabilities of each of the two other firm's being in a particular place, and since they have identical probabilities, this ends up just multiplying the probability of each outcome by two. p. 367. Since R+Q=1, the R's all disappear, replaced by 1-Q. The h transformation allows replacement of the f(z) by f(z) = h'(1/2-z) - (1/2-Q(z))/(1/2-z), since h'= -Q'/(1/2-z) + (1/2-Q)/(1/2-z)^2 and f=Q'. If K=0 in equation (5), then h^3(h-2)=K=0 implies either than h=0 or h=2. Solving the equation h(z) = 2 yields the equation for Q(z) which is the paper's main solution in equation (*). The algebra is (1/2-Q) /(1/2-z) = 2, so 1/2-Q = 1 - 2z, so Q= 2z -1/2. The A(z) equation at the bottom of the page is for when z is definitely to the left of the other two firms. Shaked does not show what happens when z >= 3/4, but this would work out just the same.