# ex-4-MPEC.mod
# 
# MPEC formulation of Multi-Leader-Follower Game 
# 
# Example 4 from Fukushima and Pang, "Quasi-Variational Inequalities,
# Generalized Nash Equlibria, and Multi-Leader-Follower Games", to
# appear in Computational Management Science.
#
#######################################################################

# ... sets
set I := 1..2;

# ... primal variables
var x{I} >= 0, <= 1;         # ... leader 1 & 2
var y  >= 0;                 # ... follower
var s  >= 0;

# ... multipliers
var chi{I};
var psi{I};
var sigma{I};
var xi{I} >= 0;
var mu{I};

minimize maxmult: sum{i in I} xi[i];

subject to
	# ... first order conditions; player 1
	KKT1x:   0.5 - mu[1] = chi[1];
	KKT1y:   1.0 - mu[1] = psi[1] - xi[1]*s;

	# ... first order conditions; player 2
	KKT2x: - 0.5 - mu[2] = chi[2];
	KKT2y: - 1.0 - mu[2] = psi[2] - xi[2]*s;

	# ... FO conditions wrt slacks s
	KKTs{i in I}: mu[i] - sigma[i] + xi[i]*y = 0;

	# ... definition of slacks
	DefS{i in I}: s = -1 + x[1] + x[2] + y  complements  mu[i];

	# ... complementarity conditions player i
	C1{i in I}: 0 <= x[i] <= 1    complements  chi[i];
	C2{i in I}: 0 <= y            complements  psi[i] >= 0;
	C3{i in I}: 0 <= s            complements  sigma[i] >= 0;
	C4:         0 <= s            complements  y        >= 0;