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

• To: mathgroup at smc.vnet.net
• Subject: [mg92229] Re: 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 07:35:15 -0400 (EDT)
• References: <d451304b0809221444p33468e2eqb72336678113c060@mail.gmail.com>

```ups, sorry, I should have said I'm only interested in equilibriums with pure
strategies.

2008/9/22 Mauricio Esteban Cuak <cuak2000 at gmail.com>

> Hello everyone. Let me tell you something about the model I'm working
> 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.
>
>
>
> --
> Por favor eviten enviarme archivos adjuntos de Word o Powerpoint (
> http://www.gnu.org/philosophy/no-word-attachments.es.html )
>

```

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