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 > 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. > > > > -- > Por favor eviten enviarme archivos adjuntos de Word o Powerpoint ( > http://www.gnu.org/philosophy/no-word-attachments.es.html ) >