optimal_auction_correlated.py

import random

def sample():
  phase = random.random()
  if phase < 3/8.0:
    return random.random(), random.random()
  elif phase < 6/8.0:
    return random.random() + 1, random.random() + 1
  elif phase < 7/8.0:
    return random.random() + 1, random.random()
  else:
    return random.random(), random.random() + 1

def main():
  N = 1000 * 1000

  # r = 2/3 should be optimal with the max expected revenue = 89/108 ~= 0.824074.
  #
  # For comparison, r = 1 should only get an expected revenue = 0.75. Note that
  # r = 1 is optimal if two bidders are independently and uniformly distributed
  # between [0, 2). The marginal PDF of each bidder in our setup is indeed
  # uniform between [0, 2). (However, our bidders are not independent.)
  #
  # Please feel free to change r and see how the revenue changes.
  #
  # BTW, see the calculation of 89/108 on
  # https://www.wolframcloud.com/obj/21f990d3-95bb-4895-b7ae-59379cf9ce11
  r = 2/3.0

  sum = 0.0
  for i in range(N):
    x, y = sample()
    first, second = max(x, y), min(x, y)
    if second >= r:
      sum += second;
    elif first >= r:
      sum += r
  print( 'r = %s has expected revenue %s' % (r, sum / N) )

main()