I am currently working on a game-theoretic paper and trying to match two complex best response functions. The equation system includes two decision variables x
and y
, for which I want to find all possible solutions, and two static parameters b
and t
. The best response functions are given by
x == 1/18 (-6 b - t + 12 y + Sqrt[t] Sqrt[12 b + t + 12 y]
and
y == 1/18 (-6 b + t + 12 x - Sqrt[t] Sqrt[-12 b + t - 12 x])
Using
Solve[x == 1/18 (-6 b - t + 12 y + Sqrt[t] Sqrt[12 b + t + 12 y]) &&
y == 1/18 (-6 b + t + 12 x - Sqrt[t] Sqrt[-12 b + t - 12 x]), {x, y}]
yields:
{{x -> -b, y -> -b}, {x -> 1/25 (-25 b - 2 t), y -> 1/225 (-225 b - 17 t)}}
I also tried the Reduce
command with the same result. Working on, two problems occurred.
Problem 1: Checking results
I wanted to double check my results and therefore replaced x
and y
in both equations above with
x = 1/25 (-25 b - 2 t)
y = 1/225 (-225 b - 17 t)
and used the Simplify
command, to see whether the results hold true. While this was the case for the first equation, simplifying the second equation yielded
t==0.
Thus, the solution seems valid only for t = 0
, though I expected Solve
to give me generic solutions for any value of t
(in fact, in my model t > 0
).
Problem 2: Further solutions
I also inverted the second equation using
Solve[y == 1/18 (-6 b + t + 12 x - Sqrt[t] Sqrt[-12 b + t - 12 x]), x]
which yielded
x == 1/8 (4 b - t + 12 y - Sqrt[t] Sqrt[-8 b + t - 8 y])
and
x == 1/8 (4 b - t + 12 y + Sqrt[t] Sqrt[-8 b + t - 8 y])
Solving the equation system as before but using the inverted form of the second equation yielded different results
{{x -> -b, y -> -b}, {x -> 1/225 (-225 b + 17 t), y -> 1/25 (-25 b + 2 t)}
Again, I tried to check these results by inserting them into both equations, with the same result as before: True
for the first equation but t==0
for the second one.
Therefore my overall question is:
How can I get all possible and reliable solutions for the equation system above?