Nash-Cournot(mcp) to Stackelber(mpec), same output.

[Vegard Skonseng Bjerketvedt]

Hey. 

My Nash-Cournot (mcp) and Stackelberg (mpec) gives the same output. This is definitely wrong. 
Both models have the same constraints and parameters.  

In the Stackelberg model the profit-function is the object-function. 
The KKT-conditons from the Nash-Cournot model is adapted to the Stackelberg model 

for the follwers, firm 1 and 3. Is this correct?   
Why do my Stackelberg model end up as the Nash-Cournot?

The Nash-Cournot model should and gives the output: 

x(i,b): produced by i with technology b 

x(1,1) = 21 000 x(1,2) = 12 200 

x(2,1) = 17 000 x(2,2) = 0 

x(3,1) = 18 000 x(3,2) = 0  

The Stackelberg model gives the same, but should give:

x(1,1) = 21 000 x(1,2) = 16 000 

x(2,1) = 17 000 x(2,2) = 0 

x(3,1) = 18 000 x(3,2) = 0  

Nash-Cournot

*Linear inverse demand function with 3 price-making firms

*Transmission contraint

Sets

i firms /i1, i2, i3 /

n node /n1, n2, n3 /

b production tech /b1, b2 /

alias(i,j)

TABLE C(i,b) variable costs

         b1      b2

i1       0.55    0.81

i2       0.62    1.25

i3       0.78    1.35

TABLE K(i,b) peocess capacities

          b1      b2

i1       21000    16000

i2       17000    22000

i3       18000    14000

TABLE T(i,n) peocess capacities

          n1      n2       n3

i1       1000000  1000000  1000000

i2       1000000  1000000  1000000

i3       1000000  1000000  1000000

PARAMETERS alpha(n) inverse demand intercept node n

/n1 2.5, n2 2.5, n3 2.5/ ;

PARAMETERS beta(n) inverse demand slope node n

/n1 0.00005, n2 0.00005, n3 0.00005/ ;

POSITiVE VARIABLES

x(i,b) production by i with tech b

z(i,n) sold fom firm i to node n

q(n) demand quant in node n

gamma(i,b) dual of capacity constraint

omega(i,n) dual til transmisjons constraint

theta(i) dual

;

EQUATIONS

Cap(i,b) Production capacity

Trans(i,n) Transsmision capacity

Salg(i) Sold

KKTx(i,b) KKTcondition (MR=MC)

KKTz(i,n) KKTcondition

Demand(n) demand in node n

;

Cap(i,b).. - x(i,b) + K(i,b) =g= 0 ;

Trans(i,n).. - z(i,n) + T(i,n) =g= 0 ;

Salg(i).. - sum(n,z(i,n)) + sum(b,x(i,b)) =g= 0 ;

KKTx(i,b).. C(i,b) + gamma(i,b) - theta(i) =e= 0 ;

KKTz(i,n).. - alpha(n) + beta(n)*q(n) + beta(n)*z(i,n) + omega(i,n) + theta(i) =e= 0 ;

Demand(n).. q(n) - sum(i,z(i,n)) =e= 0 ;

MODEL

Nash /Cap.gamma, Trans.omega, Salg.theta, KKTx.x, KKTz.z, Demand.q/ ;

Solve

Nash using mcp ;

Stackelberg

Sets

i firms /i1, i2, i3 /

ifol(i) following firms /i1, i3/

b production tech /b1, b2/

n node /n1, n2, n3 /

TABLE C(i,b) variable costs

         b1      b2

i1       0.55    0.81

i2       0.62    1.25

i3       0.78    1.35

TABLE K(i,b) peocess capacities

          b1      b2

i1       21000    16000

i2       17000    22000

i3       18000    14000

TABLE T(i,n) peocess capacities

          n1      n2       n3

i1       1000000  1000000  1000000

i2       1000000  1000000  1000000

i3       1000000  1000000  1000000

PARAMETERS alpha(n) inverse demand intercept node n

/n1 2.5, n2 2.5, n3 2.5/ ;

PARAMETERS beta(n) inverse demand slope node n

/n1 0.00005, n2 0.00005, n3 0.00005/ ;

POSITIVE VARIABLES

x2(b) production by leader from b

z2(n) sold by leder i n

x(ifol, b) production by i from tech b

z(ifol, n) sold av ifol i n

q demanded quantity

gamma(ifol, b) dual of capacity cont for followers

omega(ifol, n) dual of tran cont for followers

theta(ifol) dual til kan ikke selge mer en produsert

;

VARIABLES

NegStackObj negative of Stack objtive

;

EQUATIONS

StackObjDef Stackelberg objective defination

Capacity2(b) upper limmint cap generation for b

Trans2(n) upper limint trans

Salg2 kan ikke selge mer en produsert

Capacity(ifol, b) upper limmit

Trans(ifol, n) upper lim trans

Salg(ifol) kan ikke selge mer en produsert

MarkClear market clearing

KKTx(ifol,b) kktx followers

KKTz(ifol,n) kktz followers

;

StackObjDef.. sum(b, C('i2', b)*x2(b)) - sum(n,(alpha(n) - beta(n)*q(n))*z2(n))

              - NegStackObj =e= 0 ;

Capacity2(b).. x2(b) - K('i2', b) =l= 0 ;

Trans2(n).. z2(n) - T('i2', n) =l= 0;

Salg2.. sum(n,z2(n)) - sum(b,x2(b)) =l= 0;

Capacity(ifol, b).. - x(ifol, b) + K(ifol, b) =g= 0 ;

Trans(ifol, n).. - z(ifol, n) + T(ifol, n) =g= 0 ;

Salg(ifol).. - sum(n, z(ifol,n)) + sum(b,x(ifol,b)) =g= 0 ;

MarkClear(n).. q(n) - sum(ifol, z(ifol, n)) - z2(n) =e= 0 ;

KKTx(ifol,b).. C(ifol,b) + gamma(ifol,b) - theta(ifol) =e= 0 ;

KKTz(ifol,n).. - alpha(n) + beta(n)*q(n) + beta(n)*z(ifol,n) + omega(ifol,n)

               + theta(ifol) =e= 0 ;

MODEL Stackelberg

/StackObjDef, Capacity2, Trans2, Salg2, Capacity.gamma, Trans.omega, Salg.theta,

MarkClear, KKTx.x, KKTz.z/ ;

Stackelberg.optfile = 1 ;

SOLVE Stackelberg minimizing NegStackObj using mpec ;

[Michael Ferris]

Attached is a modified version of your model that uses the GAMS EMP framework for solving

these models.   I believe this makes the model much easier to understand.

The only issue is the definition of revenue which is the integral of the inverse demand.

That gives the slightly awkward correction term in objective function.

Cheers, Michael

[Vegard Skonseng Bjerketvedt]

Thanks Michal.