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Partitions.py
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Partitions.py
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# %%
import numpy as np
import itertools as it
# %%
"""
This cell implements the creation of various matrices A
for testing purpose.
"""
def makingOfA(N):
At = np.random.randint(-10,10,(1,N))
A = At.T.dot(At)
return A
# A few test matrix with N=5
def testA(N):
assert N == 5, "Must be a game with 5 players."
A = np.array([
[25, 20, 15, 10, 5],
[24, 19, 14, 9, 4],
[23, 18, 13, 8, 3],
[22, 17, 12, 7, 2],
[21, 16, 11, 6, 1]
])
return A
def testATriangSuppWithZeros(N):
assert N == 5, "Must be a game with 5 players."
A = np.array([
[ 89, -87, -82, -52, -81],
[-96, 18, -70, 7, 0],
[ 62, 71, 61, 0, 0],
[-24, -16, 0, 0, 0],
[ 15, 0, 0, 0, 0],
])
return A
def testATriangSuppWithoutZeros(N):
assert N == 5, "Must be a game with 5 players."
A = np.array([
[ 89, -87, -82, -52, -81],
[-96, 18, -70, 7, 4],
[ 62, 71, 61, 46, 6],
[-24, -16, 78, 23, -80],
[ 15, 2, 54, 6, -15],
])
return A
def testAPositiveExternalities(N):
assert N == 5, "Must be a game with 5 players."
A = np.array([
[ 20, 40, 60, 80, 100],
[ 15, 20, 25, 30, 0],
[ 5, 10, 15, 0, 0],
[ 2, 5, 0, 0, 0],
[ 1, 0, 0, 0, 0],
])
return A
def testANegativeExternalities(N):
assert N == 5, "Must be a game with 5 players."
A = np.array([
[ 6, 5, 4, 3, 2],
[ 7, 6, 5, 4, 0],
[ 8, 7, 6, 0, 0],
[ 9, 8, 0, 0, 0],
[10, 0, 0, 0, 0],
])
return A
lb_A = -100
ub_A = 100
def diagonalA(N):
A = np.diag(np.random.randint(lb_A, ub_A, N))
return A
def identityA(N):
A = np.diag(np.ones(N))
return A
def triangularLowerA(N):
A = np.tril(np.random.randint(lb_A, ub_A, (N,N)))
return A
def mirrorTriangularLowerA(N):
B = np.tril(np.random.randint(lb_A, ub_A, (N,N)))
A = np.zeros((N,N))
for i in range(N):
for j in range(N):
A[i,j] = B[i, N-j-1]
return A
def triangularUpperA(N):
A = np.triu(np.random.randint(lb_A, ub_A, (N,N)))
return A
def mirrorTriangularUpperA(N):
B = np.triu(np.random.randint(lb_A, ub_A, (N,N)))
A = np.zeros((N,N))
for i in range(N):
for j in range(N):
A[i,j] = B[i, N-j-1]
return A
def symmetricA(N):
A = np.random.randint(lb_A, ub_A, (N,N))
for i in range(N):
for j in range(i+1, N):
A[i,j] = A[j,i]
return A
def symmetricANullDiago(N):
A = np.random.randint(lb_A, ub_A, (N,N))
for i in range(N):
A[i,i] = 0
for j in range(i+1, N):
A[i,j] = A[j,i]
return A
def antisymmetricA(N):
A = np.random.randint(lb_A, ub_A, (N,N))
for i in range(N):
A[i,i] = 0
for j in range(i+1, N):
A[i,j] = -A[j,i]
return A
def identicalRowsA(N):
A = np.random.randint(lb_A, ub_A, (N,N))
for i in range(1, N):
for j in range(N):
A[i,j] = A[0,j]
return A
def identicalColumnsA(N):
A = np.random.randint(lb_A, ub_A, (N,N))
for i in range(1, N):
for j in range(N):
A[j,i] = A[j,0]
return A
def identicalElementsA(N):
A = np.random.randint(lb_A, ub_A) * np.ones((N,N))
return A
# %%
def listCopy(l):
new = []
for i in l:
if (type(i)==list):
new.append(listCopy(i))
else:
new.append(i)
return new
def partitions(a):
parts = []
for i in a:
if len(parts) == 0:
parts.append([[i]])
else:
temp = []
for p1 in parts:
for p2 in p1:
t1 = listCopy(p1)
for p3 in t1:
if p3==p2:
p3.append(i)
break
temp.append(t1)
t1 = listCopy(p1)
t1.append([i])
temp.append(t1)
parts = temp
return parts
# %%
def list2Mat(partition):
players = [i for coals in partition for i in coals]
n = len(players)
mat = np.zeros((n,n),dtype=int)
for i, coal in enumerate(partition):
for j in (coal):
mat[i,j] = 1
return mat.T
def Mat2list(mat):
partition = []
n = mat.shape[1]
for j in range(n):
coal = []
for i in range(n):
if mat[i,j] > 0.5:
coal.append(i)
if len(coal) > 0.5:
partition.append(coal)
return partition
# %%
def subcoalitionsForF2(a):
"""
Given a coalition in a list form, return a list of all the ways
the coalition could be split into two subcoalitions.
"""
parts = []
for i in a:
if len(parts) == 0:
parts.append([[i]])
else:
temp = []
for p1 in parts:
t1 = listCopy(p1)
t1[0].append(i)
t2 = listCopy(p1)
if len(p1) < 2:
t2.append([i])
else:
t2[1].append(i)
temp.append(t1)
temp.append(t2)
parts = temp
#print("subcoals", parts[1:])
return parts[1:]
def neighboursForF2(a):
"""
Given a partition in a list form, return a list of all the
partitions neighbour of that partition according to F2 (split).
"""
neighbours = []
for i, coal in enumerate(a):
if len(coal) > 1:
subcoals = subcoalitionsForF2(coal)
for subcoal in subcoals:
t1 = listCopy(a)
t1.pop(i)
t1.insert(i, subcoal[0])
if len(subcoal) > 1:
t1.insert(i+1, subcoal[1])
t1.sort()
neighbours.append(t1)
return neighbours
def neighboursForF4(a):
"""
Given a partition in a list form, return a list of all the
partitions neighbour of that partition according to F4 (joining).
"""
neighbours = []
for i, coal1 in enumerate(a[:-1]):
for j, coal2 in enumerate(a[i+1:]):
#print("Coals", i, coal1, j+i+1, coal2)
t1 = listCopy(a)
t1.pop(j+i+1)
t1[i] += coal2
t1[i].sort()
t1.sort()
neighbours.append(t1)
return neighbours
def compareCoals(a, b):
"""
Given two partitions that are neighbours, find the indexes in the first
partition of the changing coalitions.
"""
diffs = []
for i, coal in enumerate(a):
if coal not in b:
diffs.append(i)
return diffs
def subcoalitionsForCore(coal, n):
"""
Given a coalition in a list form, and the number of parting players from that coal,
return all the resulting partitions of that coal.
"""
parts = []
combinations = it.combinations(coal, n)
for combination in iter(combinations):
temp = []
temp.append(list(combination))
temp.append([x for x in coal if x not in combination])
parts.append((list(combination), temp))
return parts
def neighboursForCore(a, n):
"""
Given a partition in a list form, return a list of all the
partitions neighbour of that partition if there is only one subcoal of
n players splitting from one coal in the original partition.
Returns the index of the new coalition before the new partition, in a tuple.
"""
neighbours = []
if n <= 0: return neighbours
for i, coal in enumerate(a):
if len(coal) > n:
subcoals = subcoalitionsForCore(coal, n)
for subcoal in subcoals:
t1 = listCopy(a)
t1.pop(i)
t1.insert(i, subcoal[1][0])
if len(subcoal) > 1:
t1.insert(i+1, subcoal[1][1])
t1.sort()
t1.sort(key = len, reverse=True)
neighbours.append((t1.index(subcoal[0]), t1))
elif len(coal) == n:
t1 = listCopy(a)
neighbours.append((t1.index(coal), t1))
return neighbours
# %%
if __name__ == "__main__":
N = 5
ps = partitions([i for i in range(N)])
print(len(ps))
A = np.random.randint(1, 10, (N,N))
# v = np.ones((N,1))
v = np.random.randint(1,10,(N,1))
print('A: ',A)
print('v: ',v)
testPns = [12,32,5,16,25,9]
testPartitions = [ps[i] for i in testPns]
for partition in testPartitions:
mat = list2Mat(partition)
# val = v.T.dot(A.dot(mat))
val = v.T.dot(mat)
print('partition: ',partition)
# print(A.dot(mat))
print(val)
print()
print()
for partition in testPartitions:
print("\nPartition", partition)
neighbours2 = neighboursForF2(partition)
print("Neighbours F2", neighbours2)
neighbours4 = neighboursForF4(partition)
print("Neighbours F4", neighbours4)
print("Partition : ", partition, "\n")
neighboursCore = neighboursForCore(partition, 0)
print("Neighbours Core 0", neighboursCore)
neighboursCore = neighboursForCore(partition, 1)
print("Neighbours Core 1", neighboursCore)
neighboursCore = neighboursForCore(partition, 2)
print("Neighbours Core 2", neighboursCore)
neighboursCore = neighboursForCore(partition, 3)
print("Neighbours Core 3", neighboursCore)
neighboursCore = neighboursForCore(partition, 4)
print("Neighbours Core 4", neighboursCore)
neighboursCore = neighboursForCore(partition, 5)
print("Neighbours Core 5", neighboursCore)
#
# for i in testPns:
# pp = ps[i]
# print(pp)
# m = list2Mat(pp)
# print(m)
# print(Mat2list(m))
# print()
def partitionsK(a, k=10):
parts = []
for i in a:
if len(parts) == 0:
parts.append([[i]])
else:
temp = []
for p1 in parts:
for p2 in p1:
t1 = listCopy(p1)
for p3 in t1:
if p3==p2:
if len(p3)<k:
p3.append(i)
break
temp.append(t1)
t1 = listCopy(p1)
t1.append([i])
temp.append(t1)
parts = temp
parts2 = []
for pp in parts:
t2 = [j for k in pp for j in k]
if len(t2)==len(a):
parts2.append(pp)
return parts2