/* 

  Fair coin game in Picat.

  From BL: "Can You Make It A Fair Game? - A Logic Puzzle from xkcd"
  https://medium.com/math-games/can-you-make-it-a-fair-game-bd86dc9b5e22
  """
  Sue and Bob are playing the following game:

  Sue pays Bob $2 to start the game. Then Bob starts tossing coins. If the coin falls 
  on head Bob pays Sue $1 and continues tossing the coin. If however the coin falls 
  on tails then the game is finished.

  Example 1: Sue pays Bob $2. Bob tosses head three times in a row before finally 
  tossing a tails. For each head Bob pays Sue $1 and Sue thus wins $1.

  Example 2: Sue pays Bob $2. Bob starts by tossing tails and Sue thus loses her $2.

  Clearly Sue has a disadvantage here. The question is: what is the correct initial 
  amount, so that neither Sue nor Bob have an advantage.

  Note 1: the difficulty lies in the fact, that Sue could potentially win an infinite 
  amount of money if there is an infinite number of head-tosses.

  Note 2: there is a way to easily get the correct amount without getting into “infinity”.
  """

  Cf my Gamble model gamble_fair_game.rkt

  This program was created by Hakan Kjellerstrand, hakank@gmail.com
  See also my Picat page: http://www.hakank.org/picat/

*/

import ppl_distributions, ppl_utils, ppl_common_utils.
import util.
% import ordset.

main => go.

/*
  With Initial = 1, then the Score is ~ 0, and thus fair.

  initial = -3
  var : score
  Probabilities (truncated):
  -3: 0.4918000000000000
  .........
  -20: 0.0001000000000000
  mean = -4.0076


  initial = -2
  var : score
  Probabilities (truncated):
  -2: 0.5024999999999999
  .........
  -15: 0.0001000000000000
  mean = -3.0026


  initial = -1
  var : score
  Probabilities (truncated):
  -1: 0.4984000000000000
  .........
  -14: 0.0001000000000000
  mean = -2.0149


  initial = 0
  var : score
  Probabilities (truncated):
  0: 0.4996000000000000
  .........
  -13: 0.0002000000000000
  mean = -0.9987


  initial = 1
  var : score
  Probabilities (truncated):
  1: 0.5112000000000000
  .........
  -13: 0.0001000000000000
  mean = 0.0238


  initial = 2
  var : score
  Probabilities (truncated):
  2: 0.4921000000000000
  .........
  -13: 0.0002000000000000
  mean = 0.9775


  initial = 3
  var : score
  Probabilities (truncated):
  3: 0.4960000000000000
  .........
  -8: 0.0001000000000000
  mean = 1.997


*/
go ?=>
  member(Initial,-3..3),
  println(initial=Initial),
  reset_store,
  run_model(10_000,$model(Initial),[show_probs_trunc,mean,truncate_size=1]),
  nl,
  fail,
  nl.
go => true.
  

f(Len,Score) = Res =>
  R = uniform_draw([h,t]),
  if R == h then
    Res = [Len,Score]
  else
    Res = f(Len+1,Score-1)
  end.
  
model(Initial) =>

  % Score is Bob's: At start Sue pays Bob $2
  [_Len,Score] = f(0,Initial),
 
  % add("len",Len),
  add("score",Score).  



/*
  Another approach: Generate initial values (here -3..3) and
  check which value generates a (mean of) 0 score.

  var : score
  Probabilities:
  0.0: 1.0000000000000000
  mean = 0.0

  var : initial
  Probabilities:
  1: 1.0000000000000000
  mean = 1.0

*/
go2 ?=>
  reset_store,
  run_model(100,$model2,[show_probs,mean,min_accepted_samples=1000 % ,show_accepted_samples=true
                        ]),
  nl,
  nl.
go2 => true.
  
check(Initial) = Res =>
  N = 100,
  R = 0,
  foreach(_ in 1..N)
    [_Len,Score] = f(0,Initial),
    R := R + Score
  end,
  Res = R/N.
  
model2() =>

  Initial = discrete_uniform_dist(-3,3),
  Score = check(Initial),
  observe(Score==0.0),
  
  if observed_ok then
    add("score",Score),
    add("initial",Initial)
  end.