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Finding Nash Equilibrium in a 2 person game with continuous action space

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92224] Finding Nash Equilibrium in a 2 person game with continuous action space
  • From: "Mauricio Esteban Cuak" <cuak2000 at gmail.com>
  • Date: Tue, 23 Sep 2008 07:34:17 -0400 (EDT)

Hello everyone. Let me tell you something about the model I'm working about.
It's a sort of principal-agent model where each agent (two in this case)
decide only which level of effort to make. They get a fraction ("a"  and
(1-a) ) of
the product they produce, but effort is costly to them. Here are the utility
functions for the players:

U1 = a (x^r + y^r)^(t/r)  - (j*x^2)/2
U2 = (1 - a)*(x^r + y^r)^(t/r)  - (k*x^2)/2


Where x is the effort by agent 1 and "y" the effort by agent  2. "t" is just
a number to give concavity to the function and "r" represents

the elasticity of sustitution between the efforts. "j" and "k" represent the
different costs that each agent faces with their effort. All the parameters
except {a,x,y} can be seen as given.

The Nash Equilibrium would consist in a pair (x*, y*). The efforts MUST be
equal or bigger that zero.

Besides this, I want to maximise this social efficiency function:


op = (x^r + y^r)^(t/r) -   (  (j*x^2)/2 + (k*x^2)/2   )


which must be maximised subject to x  and   y    being a Nash Equilibrium


     What I've been trying to do (but still can't) is to obtain {x*, y*} as
a function of "a" and then maximise the Op. The the Op will be a function

of  x(a)  and y(a).


    I'm not very interested in the value of the parameters, so a numerical
example can work perfectly:



U1 = a (x^0.5 + y^0.5)^(0.6/0.5)  - (0.01*x^2)/2

U2 = (1-a) (x^0.5 + y^0.5)^(0.6/0.5)  - (0.02*x^2)/2


I appreciate any help or comments.

Kind regards!


cd


P.D.: I can solve to get best response functions instead of correspondences,
but I don't want to loose Nash Equilibriums that way.





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