def compute_optphragmen(profile, committeesize,
algorithm="gurobi", resolute=False, verbose=0):
enough_approved_candidates(profile, committeesize)
# optional output
if verbose:
print(header(rules["optphrag"].longname))
if resolute:
print("Computing only one winning committee (resolute=True)\n")
if verbose >= 3:
if algorithm == "gurobi":
print("Using the Gurobi ILP solver")
# end of optional output
if algorithm != "gurobi":
raise NotImplementedError("Algorithm " + str(algorithm)
+ " not specified for compute_optphragmen")
committees = abcrules_gurobi.__gurobi_optphragmen(
profile, committeesize, resolute=resolute, verbose=verbose)
committees = sort_committees(committees)
# optional output
if verbose:
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
# end of optional output
return committees
def __revseq_thiele_irresolute(profile, committeesize, scorefct_str):
"""Compute an *irresolute* sequential Thiele method
Consider all possible ways to break ties between candidates
(aka parallel universe tiebreaking)
"""
scorefct = scores.get_scorefct(scorefct_str, committeesize)
allcandcomm = tuple(range(profile.num_cand))
comm_scores = {allcandcomm: scores.thiele_score(
scorefct_str, profile, allcandcomm)}
for _ in range(profile.num_cand - committeesize):
comm_scores_next = {}
for committee, score in comm_scores.items():
marg_util_cand = scores.marginal_thiele_scores_remove(
scorefct, profile, committee)
score_reduction = min(marg_util_cand)
# find smallest elements in marg_util_cand and return indices
cands_to_remove = [cand for cand in range(profile.num_cand)
if marg_util_cand[cand] == min(marg_util_cand)]
for c in cands_to_remove:
next_comm = tuple(set(committee) - set([c]))
comm_scores_next[next_comm] = score - score_reduction
comm_scores = comm_scores_next
return sort_committees(list(comm_scores.keys()))
def compute_lexmav(profile, committeesize, algorithm="brute-force",
resolute=False, verbose=0):
"""Lexicographic Minimax AV"""
enough_approved_candidates(profile, committeesize)
if not profile.has_unit_weights():
raise ValueError(rules["lexmav"].shortname +
" is only defined for unit weights (weight=1)")
if algorithm != "brute-force":
raise NotImplementedError(
"Algorithm " + str(algorithm)
+ " not specified for compute_lexmav")
opt_committees = []
opt_distances = [profile.num_cand + 1] * len(profile)
for comm in combinations(list(range(profile.num_cand)), committeesize):
distances = sorted([hamming(pref, comm)
for pref in profile],
reverse=True)
for i in range(len(distances)):
if opt_distances[i] < distances[i]:
break
if opt_distances[i] > distances[i]:
opt_distances = distances
opt_committees = [comm]
break
else:
opt_committees.append(comm)
committees = sort_committees(opt_committees)
if resolute:
committees = [committees[0]]
# optional output
if verbose:
print(header(rules["lexmav"].longname))
if resolute:
print("Computing only one winning committee (resolute=True)\n")
print("Minimum maximal distance: " + str(max(opt_distances)))
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
print("Corresponding distances to voters:")
for comm in committees:
print([hamming(pref, comm) for pref in profile])
print()
# end of optional output
return committees
def __minimaxav_bruteforce(profile, committeesize):
"""Brute-force algorithm for computing Minimax AV (MAV)"""
opt_committees = []
opt_mavscore = profile.num_cand + 1
for comm in combinations(list(range(profile.num_cand)), committeesize):
score = scores.mavscore(profile, comm)
if score < opt_mavscore:
opt_committees = [comm]
opt_mavscore = score
elif scores.mavscore(profile, comm) == opt_mavscore:
opt_committees.append(comm)
committees = sort_committees(opt_committees)
return committees
def __seqphragmen_irresolute(profile, committeesize,
start_load=None, partial_committee=None):
"""Algorithm for computing irresolute seq-Phragmen (>=1 winning committees)
"""
approvers_weight = {}
for c in range(profile.num_cand):
approvers_weight[c] = sum(pref.weight for pref in profile if c in pref)
load = start_load
if load is None:
load = {v: 0 for v, _ in enumerate(profile)}
if partial_committee is None:
partial_committee = [] # build committees starting with the empty set
comm_loads = {tuple(partial_committee): load}
for _ in range(len(partial_committee), committeesize):
comm_loads_next = {}
for committee, load in comm_loads.items():
approvers_load = {}
for c in range(profile.num_cand):
approvers_load[c] = sum(pref.weight * load[v]
for v, pref in enumerate(profile)
if c in pref)
new_maxload = [
Fraction(approvers_load[c] + 1, approvers_weight[c])
if approvers_weight[c] > 0 else committeesize + 1
for c in range(profile.num_cand)]
# exclude committees already in the committee
for c in range(profile.num_cand):
if c in committee:
new_maxload[c] = sys.maxsize
# compute new loads
# and add new committees
for c in range(profile.num_cand):
if new_maxload[c] <= min(new_maxload):
new_load = {}
for v, pref in enumerate(profile):
if c in pref:
new_load[v] = new_maxload[c]
else:
new_load[v] = load[v]
new_comm = tuple(sorted(committee + (c,)))
comm_loads_next[new_comm] = new_load
comm_loads = comm_loads_next
committees = sort_committees(list(comm_loads.keys()))
return committees, comm_loads
def __monroe_bruteforce(profile, committeesize, resolute):
"""Brute-force computation of Monroe's rule"""
opt_committees = []
opt_monroescore = -1
for comm in combinations(list(range(profile.num_cand)), committeesize):
score = scores.monroescore(profile, comm)
if score > opt_monroescore:
opt_committees = [comm]
opt_monroescore = score
elif scores.monroescore(profile, comm) == opt_monroescore:
opt_committees.append(comm)
committees = sort_committees(opt_committees)
if resolute:
committees = [committees[0]]
return committees
def compute_thiele_method(scorefct_str, profile, committeesize,
algorithm="gurobi", resolute=False, verbose=0):
"""Thiele methods
Compute winning committees of the Thiele method specified
by the score function (scorefct_str)
"""
enough_approved_candidates(profile, committeesize)
scorefct = scores.get_scorefct(scorefct_str, committeesize)
# optional output
if verbose:
print(header(rules[scorefct_str].longname))
if resolute:
print("Computing only one winning committee (resolute=True)\n")
if verbose >= 3:
if algorithm == "gurobi":
print("Using the Gurobi ILP solver\n")
if algorithm == "branch-and-bound":
print("Using a branch-and-bound algorithm\n")
# end of optional output
if algorithm == "gurobi":
committees = abcrules_gurobi.__gurobi_thiele_methods(
profile, committeesize, scorefct, resolute)
committees = sort_committees(committees)
elif algorithm == "branch-and-bound":
committees = __thiele_methods_branchandbound(
profile, committeesize, scorefct_str, resolute)
else:
raise NotImplementedError(
"Algorithm " + str(algorithm)
+ " not specified for compute_thiele_method")
# optional output
if verbose >= 2:
print("Optimal " + scorefct_str.upper() + "-score: "
+ str(scores.thiele_score(scorefct_str, profile, committees[0])))
print()
if verbose:
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
# end of optional output
return committees
def compute_mav(profile, committeesize, algorithm="brute-force",
resolute=False, verbose=0):
"""Minimax AV (MAV)"""
enough_approved_candidates(profile, committeesize)
# optional output
if verbose:
print(header(rules["mav"].longname))
if resolute:
print("Computing only one winning committee (resolute=True)\n")
if verbose >= 3:
if algorithm == "gurobi":
print("Using the Gurobi ILP solver\n")
if algorithm == "brute-force":
print("Using a brute-force algorithm\n")
# end of optional output
if algorithm == "gurobi":
committees = abcrules_gurobi.__gurobi_minimaxav(
profile, committeesize, resolute)
committees = sort_committees(committees)
elif algorithm == "brute-force":
committees = __minimaxav_bruteforce(profile, committeesize)
if resolute:
committees = [committees[0]]
else:
raise NotImplementedError("Algorithm " + str(algorithm)
+ " not specified for compute_mav")
opt_mavscore = scores.mavscore(profile, committees[0])
# optional output
if verbose:
print("Minimum maximal distance: " + str(opt_mavscore))
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
print("Corresponding distances to voters:")
for comm in committees:
print([hamming(pref, comm) for pref in profile])
print()
# end of optional output
return committees
def __thiele_methods_branchandbound(profile, committeesize,
scorefct_str, resolute):
"""Branch-and-bound algorithm to compute winning committees
for Thiele methods"""
enough_approved_candidates(profile, committeesize)
scorefct = scores.get_scorefct(scorefct_str, committeesize)
best_committees = []
init_com = compute_seq_thiele_method(
profile, committeesize, scorefct_str, resolute=True)[0]
best_score = scores.thiele_score(scorefct_str, profile, init_com)
part_coms = [[]]
while part_coms:
part_com = part_coms.pop(0)
# potential committee, check if at least as good
# as previous best committee
if len(part_com) == committeesize:
score = scores.thiele_score(scorefct_str, profile, part_com)
if score == best_score:
best_committees.append(part_com)
elif score > best_score:
best_committees = [part_com]
best_score = score
else:
if len(part_com) > 0:
largest_cand = part_com[-1]
else:
largest_cand = -1
missing = committeesize - len(part_com)
marg_util_cand = scores.marginal_thiele_scores_add(
scorefct, profile, part_com)
upper_bound = (
sum(sorted(marg_util_cand[largest_cand + 1:])[-missing:])
+ scores.thiele_score(scorefct_str, profile, part_com))
if upper_bound >= best_score:
for c in range(largest_cand + 1,
profile.num_cand - missing + 1):
part_coms.insert(0, part_com + [c])
committees = sort_committees(best_committees)
if resolute:
committees = [committees[0]]
return committees
def compute_monroe(profile, committeesize, algorithm="brute-force",
resolute=False, verbose=0):
"""Monroe's rule"""
enough_approved_candidates(profile, committeesize)
# optional output
if verbose:
print(header(rules["monroe"].longname))
if resolute:
print("Computing only one winning committee (resolute=True)\n")
if verbose >= 3:
if algorithm == "gurobi":
print("Using the Gurobi ILP solver\n")
if algorithm == "brute-force":
print("Using a brute-force algorithm\n")
# end of optional output
if not profile.has_unit_weights():
raise ValueError(rules["monroe"].shortname +
" is only defined for unit weights (weight=1)")
if algorithm == "gurobi":
committees = abcrules_gurobi.__gurobi_monroe(
profile, committeesize, resolute)
committees = sort_committees(committees)
elif algorithm == "brute-force":
committees = __monroe_bruteforce(
profile, committeesize, resolute)
else:
raise NotImplementedError(
"Algorithm " + str(algorithm)
+ " not specified for compute_monroe")
# optional output
if verbose:
print("Optimal Monroe score: "
+ str(scores.monroescore(profile, committees[0])) + "\n")
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
# end of optional output
return committees
def __seq_thiele_irresolute(profile, committeesize, scorefct_str):
"""Compute an *irresolute* reverse sequential Thiele method
Consider all possible ways to break ties between candidates
(aka parallel universe tiebreaking)
"""
scorefct = scores.get_scorefct(scorefct_str, committeesize)
comm_scores = {(): 0}
# build committees starting with the empty set
for _ in range(committeesize):
comm_scores_next = {}
for committee, score in comm_scores.items():
# marginal utility gained by adding candidate to the committee
additional_score_cand = scores.marginal_thiele_scores_add(
scorefct, profile, committee)
for c in range(profile.num_cand):
if additional_score_cand[c] >= max(additional_score_cand):
next_comm = tuple(sorted(committee + (c,)))
comm_scores_next[next_comm] = (
score + additional_score_cand[c])
comm_scores = comm_scores_next
return sort_committees(list(comm_scores.keys()))
def compute_greedy_monroe(profile, committeesize,
algorithm="standard", resolute=True, verbose=0):
""""Greedy Monroe"""
enough_approved_candidates(profile, committeesize)
if not profile.has_unit_weights():
raise ValueError(rules["greedy-monroe"].shortname +
" is only defined for unit weights (weight=1)")
if not resolute:
raise NotImplementedError(
"compute_greedy_monroe does not support resolute=False.")
if algorithm != "standard":
raise NotImplementedError(
"Algorithm " + str(algorithm)
+ " not specified for compute_greedy_monroe")
num_voters = len(profile)
committee = []
# remaining voters
remaining_voters = list(range(num_voters))
remaining_cands = set(range(profile.num_cand))
assignment = []
for t in range(committeesize):
maxapprovals = -1
selected = None
for c in remaining_cands:
approvals = len([i for i in remaining_voters
if c in profile[i]])
if approvals > maxapprovals:
maxapprovals = approvals
selected = c
# determine how many voters are removed (at most)
if t < num_voters - committeesize * (num_voters // committeesize):
num_remove = num_voters // committeesize + 1
else:
num_remove = num_voters // committeesize
# only voters that approve the chosen candidate
# are removed
to_remove = [i for i in remaining_voters
if selected in profile[i]]
if len(to_remove) > num_remove:
to_remove = to_remove[:num_remove]
assignment.append((selected, to_remove))
remaining_voters = [i for i in remaining_voters
if i not in to_remove]
committee.append(selected)
remaining_cands.remove(selected)
committees = sort_committees([committee])
# optional output
if verbose:
print(header(rules["greedy-monroe"].longname))
if verbose >= 2:
score1 = scores.monroescore(profile, committees[0])
score2 = len(profile) - len(remaining_voters)
print("The Monroe assignment computed by Greedy Monroe")
print("has a Monroe score of " + str(score2) + ".")
if score1 > score2:
print("Monroe assignment found by Greedy Monroe is not "
+ "optimal for the winning committee,")
print("i.e., by redistributing voters to candidates a higher "
+ "satisfaction is possible "
+ "(without changing the committee).")
print("Optimal Monroe score of the winning committee is "
+ str(score1) + ".")
# build actual Monroe assignment for winning committee
for t, district in enumerate(assignment):
cand, voters = district
if t < num_voters - committeesize * (num_voters // committeesize):
missing = num_voters // committeesize + 1 - len(voters)
else:
missing = num_voters // committeesize - len(voters)
for _ in range(missing):
v = remaining_voters.pop()
voters.append(v)
print("Assignment (unsatisfatied voters marked with *):\n")
for cand, voters in assignment:
print(" candidate " + profile.names[cand]
+ " assigned to: ", end="")
output = ""
for v in sorted(voters):
output += str(v)
if cand not in profile[v].approved:
output += "*"
output += ", "
print(output[:-2])
print()
if verbose:
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
# end of optional output
return committees
def __separable(rule_id, profile, committeesize, resolute, verbose):
enough_approved_candidates(profile, committeesize)
appr_scores = [0] * profile.num_cand
for pref in profile:
for cand in pref:
if rule_id == "sav":
# Satisfaction Approval Voting
appr_scores[cand] += Fraction(pref.weight, len(pref))
elif rule_id == "av":
# (Classic) Approval Voting
appr_scores[cand] += pref.weight
else:
raise UnknownRuleIDError(rule_id)
# smallest score to be in the committee
cutoff = sorted(appr_scores)[-committeesize]
certain_cands = [c for c in range(profile.num_cand)
if appr_scores[c] > cutoff]
possible_cands = [c for c in range(profile.num_cand)
if appr_scores[c] == cutoff]
missing = committeesize - len(certain_cands)
if len(possible_cands) == missing:
# candidates with appr_scores[c] == cutoff
# are also certain candidates because all these candidates
# are required to fill the committee
certain_cands = sorted(certain_cands + possible_cands)
possible_cands = []
missing = 0
if resolute:
committees = sort_committees(
[(certain_cands + possible_cands[:missing])])
else:
committees = sort_committees(
[(certain_cands + list(selection))
for selection
in combinations(possible_cands, missing)])
# optional output
if verbose:
print(header(rules[rule_id].longname))
if resolute:
print("Computing only one winning committee (resolute=True)\n")
if verbose >= 2:
print("Scores of candidates:")
for c in range(profile.num_cand):
print(profile.names[c] + ": " + str(appr_scores[c]))
print("\nCandidates are contained in winning committees")
print("if their score is >= " + str(cutoff) + ".")
if len(certain_cands) > 0:
print("\nThe following candidates are contained in")
print("every winning committee:")
namedset = [profile.names[c] for c in certain_cands]
print(" " + ", ".join(map(str, namedset)))
print()
if len(possible_cands) > 0:
print("The following candidates are contained in")
print("some of the winning committees:")
namedset = [profile.names[c] for c in possible_cands]
print(" " + ", ".join(map(str, namedset)))
print("(" + str(missing) + " of those candidates is contained\n"
+ " in every winning committee.)\n")
if verbose:
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
# end of optional output
return committees
def compute_phragmen_enestroem(profile, committeesize, algorithm="standard",
resolute=True, verbose=0):
""""Phragmen-Enestroem (aka Phragmen's first method, Enestroem's method)
In every round the candidate with the highest combined budget of
their supporters is put in the committee.
Method described in:
https://arxiv.org/pdf/1611.08826.pdf (Section 18.5, Page 59)
"""
enough_approved_candidates(profile, committeesize)
if not profile.has_unit_weights():
raise ValueError(rules["phrag-enestr"].shortname +
" is only defined for unit weights (weight=1)")
if algorithm != "standard":
raise NotImplementedError(
"Algorithm " + str(algorithm)
+ " not specified for compute_phragmen_enestroem")
num_voters = len(profile)
start_budget = {i: Fraction(profile[i].weight)
for i in range(num_voters)}
price = Fraction(sum(start_budget.values()), committeesize)
cands = range(profile.num_cand)
committees = [(start_budget, set())]
for _ in range(committeesize):
# here the committees with i+1 candidates are
# stored (together with budget)
next_committees = []
# loop in case multiple possible committees
# with i filled candidates
for committee in committees:
budget, comm = committee
curr_cands = set(cands) - comm
support = {c: 0 for c in curr_cands}
for nr, pref in enumerate(profile):
voting_power = budget[nr]
if voting_power <= 0:
continue
for cand in pref:
if cand in curr_cands:
support[cand] += voting_power
max_support = max(support.values())
winners = [c for c, s in support.items()
if s == max_support]
for cand in winners:
b = dict(budget) # copy of budget
if max_support > price: # supporters can afford it
# (voting_power - price) / voting_power
multiplier = Fraction(max_support - price,
max_support)
else: # set supporters to 0
multiplier = 0
for nr, pref in enumerate(profile):
if cand in pref:
b[nr] *= multiplier
c = comm.union([cand]) # new committee with candidate
next_committees.append((b, c))
if resolute:
committees = [next_committees[0]]
else:
committees = next_committees
committees = [comm for b, comm in committees]
committees = sort_committees(committees)
if resolute:
committees = [committees[0]]
# optional output
if verbose:
print(header(rules["phrag-enestr"].longname))
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
# end of optional output
return committees
def compute_rule_x(profile, committeesize, algorithm="standard",
resolute=True, verbose=0):
"""Rule X
See https://arxiv.org/pdf/1911.11747.pdf, page 7
"""
enough_approved_candidates(profile, committeesize)
if not profile.has_unit_weights():
raise ValueError(rules["rule-x"].shortname +
" is only defined for unit weights (weight=1)")
if algorithm != "standard":
raise NotImplementedError(
"Algorithm " + str(algorithm)
+ " not specified for compute_rule_x")
# optional output
if verbose:
print(header(rules["rule-x"].longname))
if resolute:
print("Computing only one winning committee (resolute=True)\n")
# end of optional output
start_budget = {v: Fraction(committeesize, len(profile))
for v, _ in enumerate(profile)}
cands = range(profile.num_cand)
commbugdets = [(set(), start_budget)]
final_committees = set()
# optional output
if resolute and verbose >= 2:
print("Phase 1:\n")
print("starting budget:")
output = " ("
for v, _ in enumerate(profile):
output += str(start_budget[v]) + ", "
print(output[:-2] + ")\n")
# end of optional output
for _ in range(committeesize):
next_commbudgets = []
for committee, budget in commbugdets:
curr_cands = set(cands) - committee
min_q = {}
for c in curr_cands:
q = __rule_x_get_min_q(profile, budget, c)
if q is not None:
min_q[c] = q
if len(min_q) > 0: # one or more candidates are affordable
next_cands = [c for c in min_q.keys()
if min_q[c] == min(min_q.values())]
for next_cand in next_cands:
new_budget = dict(budget)
for v, pref in enumerate(profile):
if next_cand in pref:
new_budget[v] -= min(budget[v], min_q[next_cand])
new_comm = set(committee)
new_comm.add(next_cand)
next_commbudgets.append((new_comm, new_budget))
# optional output
if resolute and verbose >= 2:
output = "adding candidate number "
output += str(len(committee)) + ": "
output += profile.names[next_cand] + "\n"
output += " with maxmimum cost per voter q = "
output += str(min(min_q.values()))
print(output)
print(" remaining budget:")
output = " ("
for v, _ in enumerate(profile):
output += str(new_budget[v]) + ", "
print(output[:-2] + ")")
if len(next_cands) > 1:
output = " tie broken in favor of "
output += profile.names[next_cand] + ","
output += "\n candidates "
output += str_candset(next_cands[1:])
output += " are tied"
print(output)
print()
# end of optional output
if resolute:
break
else: # no affordable candidates remain
# fill committee via seq-Phragmen
# optional output
if resolute and verbose >= 2:
print("Phase 2 (seq-Phragmén):\n")
# end of optional output
start_load = {}
# translate budget to loads
for v in range(len(profile)):
start_load[v] = (Fraction(committeesize, len(profile))
- budget[v])
# optional output
if resolute and verbose >= 2:
print("starting loads (= budget spent):")
output = " ("
for v, _ in enumerate(profile):
output += str(start_load[v]) + ", "
print(output[:-2] + ")\n")
# end of optional output
if resolute:
committees, _ = __seqphragmen_resolute(
profile, committeesize, verbose=verbose,
partial_committee=list(committee),
start_load=start_load)
else:
committees, _ = __seqphragmen_irresolute(
profile, committeesize,
partial_committee=list(committee),
start_load=start_load)
final_committees.update([tuple(comm) for comm in committees])
# after filling the remaining spots these committees
# have size committeesize
commbugdets = next_commbudgets
final_committees.update([tuple(comm) for comm, _ in commbugdets])
committees = sort_committees(final_committees)
if resolute:
committees = committees[:1]
# optional output
if verbose:
print(str_committees_header(committees, winning=True))
print(str_candsets(committees, names=profile.names))
# end of optional output
return committees
