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MathGroup Archive 1998

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What kind of math problem is this?



I know this is a little off topic, but I am hoping there are enough
mathematicians with tolerance on this list that I can get some help for
a problem relevant to the game theoretic analysis of legal rules.

Also, my apologies for the one-dimensional Mathematica notation here.
Until newgroups better understands Mathematica notebooks or until
MathML, this is the best I can do.

Suppose I have two functions f[x,y] and g[x,y]. Both map two real
numbers onto a real number.  Suppose that f[x,y]+g[x,y] is uniquely
maximized at some point { x*,y*}.  What relationships and qualities
must f and g have such that there exists a transfer function t[x,y] so
that f[x,y]-t[x,y] is maximized with respect to x at x* for all values
of y and so that g[x,y]+t[x,y] is maximized with respect to y at y* for
all values of x.

By way of example:

Define functions f and g as follows:

In[82]:=
f[x_,y_]=y-(x-2)^2;

In[83]:=
g[x_,y_]=x-(y-3)^2;

We can determine (somewhat sloppily) the maximum of these functions as
follows:

In[6]:=
Solve[{D[f[x,y]+g[x,y],x]==0 ,D[f[x,y]+g[x,y],y]==0},{x,y}]

Out[6]=
\!\({{x \[Rule] 5\/2, y \[Rule] 7\/2}}\)

There exists a function t[x_,y_]=y-x such that my requirements are met

In[84]:=
t[x_,y_]=y-x;

In[19]:=
Solve[D[f[x,y]-t[x,y],x]==0,x]

Out[19]=
\!\({{x \[Rule] 5\/2}}\)

In[20]:=
Solve[D[g[x,y]+t[x,y],y]==0,y]

Out[20]=
\!\({{y \[Rule] 7\/2}}\)

On the other hand for f1 and g1 I don't believe a function t exists.

In[85]:=
f1[x_,y_]= x y - y ^2 -(x-2)^2;

In[86]:=
g1[x_,y_]=2 x y - x^2 - (y-3)^2;

In[62]:=
Solve[{D[f1[x,y]+g1[x,y],x]==0,D[f1[x,y]+g1[x,y],y]==0},{x,y}]

Out[62]=
\!\({{x \[Rule] 34\/7, y \[Rule] 36\/7}}\)

Here is a function t1 that makes part of the requirements come true

In[87]:=
t1[x_,y_]=x y - 40 x /7;

In[70]:=
Solve[D[f1[x,y]-t1[x,y],x]==0,x]//Simplify

Out[70]=
\!\({{x \[Rule] 34\/7}}\)

But not the other part. (the optimal y value ends up depending on x)

In[68]:=
Solve[D[g1[x,y]+t1[x,y],y]==0,y]//Simplify

Out[68]=
\!\({{y \[Rule] \(3\ \((2 + x)\)\)\/2}}\)

My problem is, I can't even figure out what kind of a problem this is.
Could someone out there help? If they wanted to use Mathematica to
illuminate the problem, that would be fabulous.




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