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.