Greetings to all.

I am currently working in multi-leader-follower games and trying to model them in GAMS using EMP. In fact, in order to create the EPEC’s, I formulate the KKT conditions of the follower and add them as constraints of each leader’s MPEC. Nevertheless, by doing this, follower’s variables are now controlled by both leaders, and I just can’t find a way to model this using EMP/equilibrium, since each variable has to be assigned to exactly one leader.

Does anyone have experience with this? Any thoughts?

I do thank you very much in advance.

Manuel.

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Letâ€™s say for example that there are two leaders and one follower. Leaders control variables x1 and x2 respectively and the follower controls only variable y. In order to form the EPEC, the KKT of the follower are formulated and introduced as constraints on each leader problem. On the new problem, each leader controls his variables (x1 or x2), follower variable y and the resulting Lagrange multipliers On the other hand, it is not possible to formulate this equilibrium in GAMS, since each variable has to be controlled by only one leader. For the multipliers, one can for example has one of them for each leader, but that is not the case of y.

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Manuel,

It sounds like the KKT conditions for the follower (both the equations and variables) could be added to your EMP model not as being owned by a leader but as part of VI side conditions. That how I would first try it. You would have a multi-agent equilibrium model with two optimizing agents (the leaders) and a third VI agent (the follower).

Ideally I could point you to an example in our EMP library, but I don’t recall adding exactly such a model, or seeing one there. Is there a small, complete example you could point me to, ideally something with a known solution that I could try to reproduce with EMP? That could be a nice addition.

-Steve

On Thu, Oct 2, 2014 at 5:16 AM, Manuel Ramos wrote:

Letâ€™s say for example that there are two leaders and one follower. Leaders control variables x1 and x2 respectively and the follower controls only variable y. In order to form the EPEC, the KKT of the follower are formulated and introduced as constraints on each leader problem. On the new problem, each leader controls his variables (x1 or x2), follower variable y and the resulting Lagrange multipliers On the other hand, it is not possible to formulate this equilibrium in GAMS, since each variable has to be controlled by only one leader. For the multipliers, one can for example has one of them for each leader, but that is not the case of y.

Steven Dirkse, Ph.D.

GAMS Development Corp., Washington DC

Voice: (202)342-0180 Fax: (202)342-0181

sdirkse@gams.com

http://www.gams.com

Hello Steven,

Thank you for your answer and interest. I propose the following simple example: 2 leaders, 1 follower.

-Leader 1 problem:

min(x1) f1=1/2x1+y1

s.t. x1>=0

-Leader 2 problem:

min(x2) f2= -1/2x2+y2

s.t. x2>=0

-Follower’s KKT:

-1+x1+x2+lam-mu1+mu2=0

y2+lam-mu3+mu4=0

y1+y2=1

y1>=0 perpendicular to mu1

y1=0 perpendicular to mu3

y3 wrote:

Letâ€™s say for example that there are two leaders and one follower. Leaders control variables x1 and x2 respectively and the follower controls only variable y. In order to form the EPEC, the KKT of the follower are formulated and introduced as constraints on each leader problem. On the new problem, each leader controls his variables (x1 or x2), follower variable y and the resulting Lagrange multipliers On the other hand, it is not possible to formulate this equilibrium in GAMS, since each variable has to be controlled by only one leader. For the multipliers, one can for example has one of them for each leader, but that is not the case of y.

Steven Dirkse, Ph.D.

GAMS Development Corp., Washington DC

Voice: (202)342-0180 Fax: (202)342-0181

sdi...@gams.com

http://www.gams.com

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Hello Steven,

Thank you for your answer and interest. I propose the following simple example: 2 leaders, 1 follower.

-Leader 1 problem:

min(x1) f1=1/2x1+y1

s.t. x1>=0

-Leader 2 problem:

min(x2) f2= -1/2x2+y2

s.t. x2>=0

-Follower’s KKT:

-1+x1+x2+lam-mu1+mu2=0

y2+lam-mu3+mu4=0

y1+y2=1

y1>=0 perpendicular to mu1

y1=0 perpendicular to mu3

y3 wrote:

Letâ€™s say for example that there are two leaders and one follower. Leaders control variables x1 and x2 respectively and the follower controls only variable y. In order to form the EPEC, the KKT of the follower are formulated and introduced as constraints on each leader problem. On the new problem, each leader controls his variables (x1 or x2), follower variable y and the resulting Lagrange multipliers On the other hand, it is not possible to formulate this equilibrium in GAMS, since each variable has to be controlled by only one leader. For the multipliers, one can for example has one of them for each leader, but that is not the case of y.

Steven Dirkse, Ph.D.

GAMS Development Corp., Washington DC

Voice: (202)342-0180 Fax: (202)342-0181

sdi...@gams.com

http://www.gams.com

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Manuel,

I quite agree with Steve in his first statement.

In addition, it seems that your follower KKT contains redundant information – you could quite effectively replace y1 and y2 by w and (1-w), and hence write only two complementarity constraints instead of four :

w >= 0, perpendicular to mu1

w wrote:

Letâ€™s say for example that there are two leaders and one follower. Leaders control variables x1 and x2 respectively and the follower controls only variable y. In order to form the EPEC, the KKT of the follower are formulated and introduced as constraints on each leader problem. On the new problem, each leader controls his variables (x1 or x2), follower variable y and the resulting Lagrange multipliers On the other hand, it is not possible to formulate this equilibrium in GAMS, since each variable has to be controlled by only one leader. For the multipliers, one can for example has one of them for each leader, but that is not the case of y.

Steven Dirkse, Ph.D.

GAMS Development Corp., Washington DC

Voice: (202)342-0180 Fax: (202)342-0181

sdi...@gams.com

http://www.gams.com

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Dax,

I don’t see how you can express the follower’s problem as a VI, since in order to solve the multi-leader-follower problem. In fact, by expressing it as a VI, its variables are not going to be seen by the leaders as variables and therefore, the gradient of each leader’s Lagrangian respect to the follower’s variables is not going to be computed. As long as I know (note that my knowledge in Nash games and multi-leader-follower games is not very extensive, since my expertise is traditional optimization), in order to solve the multi-leader-follower problem and the equivalent EPEC is made of (in part at least) of these equations.

Manuel

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