lowers[i] = (1-w) * reputations[i]
uppers[i] = (1-w) * reputations[i] + w
# approximate the scenario as common support, differing
# normal distributions
support = [lowers[0], uppers[-1]]
bajari_params = []
for i in np.arange(n):
location = lowers[i] + w / 2
scale = w / 4
bajari_params.append({'location': location, 'scale': scale})
# compute approximations
dm_bids, dm_costs = dm.solve(w, reputations)
bajari_bids, bajari_costs = bajari.solve(support, bajari_params)
# ensure costs are monotonically increasing
dm_costs, dm_bids = util.ensure_monotonicity(dm_costs, dm_bids)
bajari_costs, bajari_bids = util.ensure_monotonicity(bajari_costs, bajari_bids)
# interpolate bidding functions and their inverses
dm_bid_funcs = []
bajari_bid_funcs = []
dm_inverses = []
bajari_inverses = []
for i in np.arange(n):
# fit
dm_bid_func = util.csplinefit(dm_costs[i], dm_bids)
dm_inverse = util.csplinefit(dm_bids, dm_costs[i])
dm_bid_funcs.append(dm_bid_func)
def compare(w, reputations):
results = {w: {
'reputations': reputations,
'utilities': None,
'prices': None
}}
try:
n = reputations.size
except AttributeError:
reputations = np.array(reputations)
n = reputations.size
# load appropriate version of FSM method
if n == 2:
import dm.fsm.main as dm
else:
import dm.efsm.main as dm
# estimate param
param = estimate_param(w, reputations)
if not param:
return results
# estimate lower and upper extremities
lowers = np.empty(n, dtype=np.float)
uppers = np.empty(n, dtype=np.float)
for i in np.arange(n):
lowers[i] = (1-w) * reputations[i]
uppers[i] = (1-w) * reputations[i] + w
# pick parameters for the common support case
support = [lowers[0], uppers[-1]]
bajari_params = []
for i in np.arange(n):
location = lowers[i] + w / 2
scale = w / 4
bajari_params.append({'location': location, 'scale': scale})
# compute approximations
if n == 2:
dm_bids, dm_costs = dm.solve(w, reputations)
else:
dm_bids, dm_costs = dm.solve(w, reputations, param=param)
bajari_bids, bajari_costs = bajari.solve(support, bajari_params)
# ensure costs are monotonically increasing
dm_costs, dm_bids = util.ensure_monotonicity(dm_costs, dm_bids)
bajari_costs, bajari_bids = util.ensure_monotonicity(bajari_costs, bajari_bids)
if n == 2:
# precondition solutions: remove endpoints where fails
# Lipschitz condition
for c in dm_costs:
diffs = list(map(lambda x,y: abs(x-y), c, dm_bids))
for i in np.arange(len(diffs)):
if diffs[i] <= 5e-5:
index = i
break
try:
dm_costs = np.array([c[:index] for c in dm_costs])
dm_bids = dm_bids[:index]
except NameError:
pass
# interpolate bidding functions and their inverses
dm_bid_funcs = []
bajari_bid_funcs = []
dm_inverses = []
bajari_inverses = []
for i in np.arange(n):
# fit
dm_bid_func = util.csplinefit(dm_costs[i], dm_bids)
dm_inverse = util.csplinefit(dm_bids, dm_costs[i])
dm_bid_funcs.append(dm_bid_func)
dm_inverses.append(dm_inverse)
bajari_bid_func = util.csplinefit(bajari_costs[i], bajari_bids)
bajari_inverse = util.csplinefit(bajari_bids, bajari_costs[i])
bajari_bid_funcs.append(bajari_bid_func)
bajari_inverses.append(bajari_inverse)
# compute ex-ante expected utilities
params = [{'loc': lowers[i], 'scale': w} for i in np.arange(n)]
cdfs = [ss.uniform(**p) for p in params]
dm_utils = []
bajari_utils = []
common_costs = []
for i in np.arange(n):
# common costs
costs = np.linspace(lowers[i], min(dm_costs[i][-1], bajari_costs[i][-1]), 1000)
common_costs.append(costs)
# derive expected utility functions
js = [j for j in np.arange(n) if j != i]
def dm_exp_util(x):
bid = dm_bid_funcs[i](x)
probs = [(1 - cdfs[j].cdf(dm_inverses[j](bid))) for j in js]
return (bid - x) * reduce(lambda x,y: x*y, probs)
def bajari_exp_util(x):
bid = bajari_bid_funcs[i](x)
probs = [(1 - cdfs[j].cdf(bajari_inverses[j](bid))) for j in js]
return (bid - x) * reduce(lambda x,y: x*y, probs)
# compute ex-ante expected utilities
dm_utilities = np.array([dm_exp_util(x) * cdfs[i].pdf(x) for x in costs])
bajari_utilities = np.array([bajari_exp_util(x) * cdfs[i].pdf(x) for x in costs])
dm_util = si.simps(dm_utilities, costs)
bajari_util = si.simps(bajari_utilities, costs)
dm_utils.append(dm_util)
bajari_utils.append(bajari_util)
# sample and generate prices for each auction
size = 1000
sampled_costs = []
params = [{'loc': common_costs[i][0], 'scale': common_costs[i][-1]-common_costs[i][0]} for i in np.arange(n)]
for p in params:
loc = p['loc']
scale = p['scale']
sampled_costs.append(ss.uniform.rvs(loc=loc, scale=scale, size=size))
dm_prices = []
for costs in zip(*sampled_costs):
bids = [dm_bid_funcs[i](costs[i]) for i in np.arange(n)]
dm_prices.append(min(bids))
bajari_prices = []
for costs in zip(*sampled_costs):
bids = [bajari_bid_funcs[i](costs[i]) for i in np.arange(n)]
bajari_prices.append(min(bids))
results[w]['utilities'] = {'dm': dm_utils, 'cp': bajari_utils}
results[w]['prices'] = {'dm': np.mean(dm_prices), 'cp': np.mean(bajari_prices)}
return results