poker

#!/usr/bin/env python

from __future__ import division
from numpy import *
import pylab

def limit_donkey_equity(k):
  """In the case of infinite stack size, Alice's optimal strategy is simply to wait for AA.  The resulting
  win probability function satisfies
    p[t] = 0, t <= 0
    p[t] = 1, t >= 1
    p[t] = a p[2t] + (1-a) p[2t-1], 0 < t < 1
  where a is the probability of winning with AA (splitting ties).  Important properties:
    1. p is continuous.
    2. If t is a dyadic rational, p[t] is a function of simpler dyadic rationals.
  This provides an easy, superfast way of computing p.

  Bug: actually, splitting ties doesn't produce the right answer.  To be correct, "a" below should
  become b/(1-c), where a is the win probability and c is the tie probability.  Still a fractal, though.
  """
  a = 893604787/1048786200
  p = zeros(1)
  for k in xrange(k):
    p = hstack([a*p,a+(1-a)*p])
  return hstack([p,1])

k = 20
t = arange(2**k+1)/2**k
p = limit_donkey_equity(k)

pylab.plot(t,p)
pylab.show()