Nash-Cournot(mcp) ----> Stackelberg(mpec), still Nash-Cournot output.

32 views
Skip to first unread message

Vegard Skonseng Bjerketvedt

unread,
Apr 22, 2016, 8:37:29 AM4/22/16
to gamsworld
Trying to extend my Nash-Cournot model (mcp) to a Stackelber model (mpec).  
In my Stackelberg model firm 2 act as leader, 1 and 3 are followers. 

The object function of the model is firm 2`s (leaders) protfit function. 
The constraints and parameters are identical to the Nash-Cournot. 

KKT conditions for firm 1 and 3 (followers) are same in both Nash-Cournot
and Stackelberg (mpec) sub-game. 

My output for both models are equal. Do someone know why my "Stackelberg-model" 
acts as a Nash-Cournot? 

x(i,b) in the Nash-Cournot: 
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

wanted output for the Stackelberg model: 
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 



Both codes below.

 
This is the Nash-Cournot model(mcp): 
- 3 firms with production and transmission constraints 


*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/ ;


Stackelberg: 

*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 ;


Reply all
Reply to author
Forward
0 new messages