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.