/*

  Anne Obborn:
  https://twitter.com/AnneOgborn/status/1560681513548759042
  """
  I offer a game-  flip a coin. Heads, I pay $1 and the game ends. Tails, we flip again- 
  if Heads, I pay $2 and game ends, Tails, we flip again, doubling each time.
  What would you pay me to play this game?
  """

  Comment by Richard Barrell:
  """
  Roughly 0.5*log2(your bankroll)
  Each round has an EV of $0.50 and the number of rounds is log2(the maximum possible 
  amount of money you can pay out)
  """
  
  * Approx 0.1

    [program = 0.5 * log2(N * 4),res = 5.982892142331043,count = 1]

    gen = 32
    [program = log2(N * 4.161105861031034) / 2,res = 6.011375638269768,count = 1]

    gen = 53
    [program = (log2(N) + 2) * (N / N * 0.5),res = 5.982892142331043,count = 1]

  * Approx 0.01
    gen 17
    [program = 0.5 * log2(N * 4),res = 5.982892142331043,count = 1]

    gen = 19
    [program = log2(N) / 2 - log2(0.5),res = 5.982892142331043,count = 1]
         
    gen = 50
    [program = 0.5 * log2(3.970501084798247 * N),res = 5.977552687190866,count = 1]

    With num_gens=100
    [program = log2(2) + (log2(N) / 2 + (N - N)),res = 5.982892142331043,count = 39]
    [program = log2(N) / 2 + (N - N) + log2(2),res = 5.982892142331043,count = 32]
    [program = N / N + (log2(N) / 2 + (N - N)),res = 5.982892142331043,count = 31]
    [program = log2(N) / 2 + (N - N) + N / N,res = 5.982892142331043,count = 31]
    [program = N / (N + N) * 2 + (log2(N) / 2 + (N - N)),res = 5.982892142331043,count = 30]
    [program = log2(N) / 2 + (N - N) + N * (0 + N) / N / N,res = 5.982892142331043,count = 29]
    [program = log2(N) / 2 + (N - N) + N / (N + N) * 2,res = 5.982892142331043,count = 28]
    [program = log2(N) / 2 + (N - N) - -1,res = 5.982892142331043,count = 19]
    [program = 0 * 0 - -1 + (log2(N) / 2 + (N - N)),res = 5.982892142331043,count = 19]
    [program = log2(N) / 2 + (N - N) + (N - N - -1),res = 5.982892142331043,count = 12]
    [program = N - N - -1 + (log2(N) / 2 + (N - N)),res = 5.982892142331043,count = 10]
    [program = N * (0 + N) / N / N + (log2(N) / 2 + (N - N)),res = 5.982892142331043,count = 5]
    [program = log2(N) / 2 + (N - N) - -1 * N / N,res = 5.982892142331043,count = 3]

    resultMap = [5.982892142331043 = 13]


  See my WebPPL model coin_flip_game.wppl

*/
import util.

data(coin_flip_game,Data,Vars,Unknown,Ops,Constants,MaxSize,Params) :-
  games(Games),
  Data = [ [[N],Est] : [N,Est] in Games],
  Ops = [+,/,-,*,log2,log],
  Constants = [0.5,2,4] ++ [random(-10,10) : _ in 1..10],
  Vars = ['N'],
  Unknown = [1000],
  MaxSize = 15,
  Params = new_map([approx=0.01,num_gens=100]).


% n,expectation
games(Data) =>
  Data = chunks_of([
1,1,
2,1.5,
4,2,
8,2.5000000000000004,
16,3.000000000000001,
32,3.5000000000000004,
64,4.000000000000001,
128,4.500000000000001,
256,5.000000000000001,
512,5.5,
1024,6.000000000000001,
2048,6.500000000000002,
4096,7.000000000000002,
8192,7.500000000000002,
16384,8.000000000000002,
32768,8.500000000000002,
65536,9.000000000000002,
131072,9.500000000000002,
262144,10.000000000000002,
524288,10.5
],2).