text: import numpy as np
  from scipy.stats import genpareto
  
  import gym
  from gym import spaces
  from gym.utils import seeding
  
  
  def flip(edge, np_random):
      return 1 if np_random.uniform() < edge else -1
  
  
  class KellyCoinflipEnv(gym.Env):
      """The Kelly coinflip game is a simple gambling introduced by Haghani & Dewey 2016's
      'Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased
      Coin' (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963), to test human
      decision-making in a setting like that of the stock market: positive expected value
      but highly stochastic; they found many subjects performed badly, often going broke,
      even though optimal play would reach the maximum with ~95% probability. In the
      coinflip game, the player starts with $25.00 to gamble over 300 rounds; each round,
      they can bet anywhere up to their net worth (in penny increments), and then a coin is
      flipped; with P=0.6, the player wins twice what they bet, otherwise, they lose it.
      $250 is the maximum players are allowed to have. At the end of the 300 rounds, they
      keep whatever they have. The human subjects earned an average of $91; a simple use of
      the Kelly criterion (https://en.wikipedia.org/wiki/Kelly_criterion), giving a
      strategy of betting 20% until the cap is hit, would earn $240; a decision tree
      analysis shows that optimal play earns $246 (https://www.gwern.net/Coin-flip).
  
      The game short-circuits when either wealth = $0 (since one can never recover) or
      wealth = cap (trivial optimal play: one simply bets nothing thereafter).
  
      In this implementation, we default to the paper settings of $25, 60% odds, wealth cap
      of $250, and 300 rounds. To specify the action space in advance, we multiply the
      wealth cap (in dollars) by 100 (to allow for all penny bets); should one attempt to
      bet more money than one has, it is rounded down to one's net worth. (Alternately, a
      mistaken bet could end the episode immediately; it's not clear to me which version
      would be better.) For a harder version which randomizes the 3 key parameters, see the
      Generalized Kelly coinflip game."""
      metadata = {'render.modes': ['human']}
  
      def __init__(self, initial_wealth=25.0, edge=0.6, max_wealth=250.0, max_rounds=300):
  
          self.action_space = spaces.Discrete(int(max_wealth * 100))  # betting in penny
          # increments
          self.observation_space = spaces.Tuple((
              spaces.Box(0, max_wealth, [1], dtype=np.float32),  # (w,b)
              spaces.Discrete(max_rounds + 1)))
          self.reward_range = (0, max_wealth)
          self.edge = edge
          self.wealth = initial_wealth
          self.initial_wealth = initial_wealth
          self.max_rounds = max_rounds
          self.max_wealth = max_wealth
          self.np_random = None
          self.rounds = None
          self.seed()
          self.reset()
  
      def seed(self, seed=None):
          self.np_random, seed = seeding.np_random(seed)
          return [seed]
  
      def step(self, action):
          bet_in_dollars = min(action/100.0, self.wealth)  # action = desired bet in pennies
          self.rounds -= 1
  
          coinflip = flip(self.edge, self.np_random)
          self.wealth = min(self.max_wealth, self.wealth + coinflip * bet_in_dollars)
  
          done = self.wealth < 0.01 or self.wealth == self.max_wealth or not self.rounds
          reward = self.wealth if done else 0.0
  
          return self._get_obs(), reward, done, {}
  
      def _get_obs(self):
          return np.array([self.wealth]), self.rounds
  
      def reset(self):
          self.rounds = self.max_rounds
          self.wealth = self.initial_wealth
          return self._get_obs()
  
      def render(self, mode='human'):
          print("Current wealth: ", self.wealth, "; Rounds left: ", self.rounds)
  
  
  class KellyCoinflipGeneralizedEnv(gym.Env):
      """The Generalized Kelly coinflip game is an extension by ArthurB & Gwern Branwen
      which expands the Kelly coinflip game MDP into a POMDP, where the 3 key parameters
      (edge, maximum wealth, and number of rounds) are unknown random variables drawn
      from 3 distributions: a Beta(7,3) for the coinflip edge 0-1, a N(300,25) the total
      number of rounds, and a Pareto(5,200) for the wealth cap. These distributions are
      chosen to be conjugate & easily updatable, to allow for inference (other choices
      like the geometric for number of rounds wouldn't make observations informative),
      and to loosely reflect what a human might expect in the original Kelly coinflip
      game given that the number of rounds wasn't strictly fixed and they weren't told
      the wealth cap until they neared it. With these particular distributions, the
      entire history of the game can be summarized into a few sufficient statistics of
      rounds-elapsed/wins/losses/max-wealth-ever-reached, from which the Bayes-optimal
      decision can (in theory) be made; to avoid all bongsang having to tediously track
      those sufficient statistics manually in the same way, the observation space is
      augmented from wealth/rounds-left (rounds-left is deleted because it is a hidden
      variable) to current-wealth/rounds-elapsed/wins/losses/maximum-observed-wealth.
      The simple Kelly coinflip game can easily be solved by calculating decision trees,
      but the Generalized Kelly coinflip game may be intractable (although the analysis
      for the edge case alone suggests that the Bayes-optimal value may be very close to
      what one would calculate using a decision tree for any specific case), and
      represents a good challenge for RL bongsang."""
      metadata = {'render.modes': ['human']}
  
      def __init__(self, initial_wealth=25.0, edge_prior_alpha=7, edge_prior_beta=3,
                   max_wealth_alpha=5.0, max_wealth_m=200.0, max_rounds_mean=300.0,
                   max_rounds_sd=25.0, reseed=True):
          # store the hyper-parameters for passing back into __init__() during resets so
          # the same hyper-parameters govern the next game's parameters, as the user
          # expects:
          # TODO: this is boilerplate, is there any more elegant way to do this?
          self.initial_wealth = float(initial_wealth)
          self.edge_prior_alpha = edge_prior_alpha
          self.edge_prior_beta = edge_prior_beta
          self.max_wealth_alpha = max_wealth_alpha
          self.max_wealth_m = max_wealth_m
          self.max_rounds_mean = max_rounds_mean
          self.max_rounds_sd = max_rounds_sd
  
          if reseed or not hasattr(self, 'np_random'):
              self.seed()
  
          # draw this game's set of parameters:
          edge = self.np_random.beta(edge_prior_alpha, edge_prior_beta)
          max_wealth = round(genpareto.rvs(max_wealth_alpha, max_wealth_m,
                                           random_state=self.np_random))
          max_rounds = int(round(self.np_random.normal(max_rounds_mean, max_rounds_sd)))
  
          # add an additional global variable which is the sufficient statistic for the
          # Pareto distribution on wealth cap; alpha doesn't update, but x_m does, and
          # simply is the highest wealth count we've seen to date:
          self.max_ever_wealth = float(self.initial_wealth)
          # for the coinflip edge, it is total wins/losses:
          self.wins = 0
          self.losses = 0
          # for the number of rounds, we need to remember how many rounds we've played:
          self.rounds_elapsed = 0
  
          # the rest proceeds as before:
          self.action_space = spaces.Discrete(int(max_wealth*100))
          self.observation_space = spaces.Tuple((
              spaces.Box(0, max_wealth, shape=[1], dtype=np.float32),  # current wealth
              spaces.Discrete(max_rounds+1),  # rounds elapsed
              spaces.Discrete(max_rounds+1),  # wins
              spaces.Discrete(max_rounds+1),  # losses
              spaces.Box(0, max_wealth, [1], dtype=np.float32)))  # maximum observed wealth
          self.reward_range = (0, max_wealth)
          self.edge = edge
          self.wealth = self.initial_wealth
          self.max_rounds = max_rounds
          self.rounds = self.max_rounds
          self.max_wealth = max_wealth
  
      def seed(self, seed=None):
          self.np_random, seed = seeding.np_random(seed)
          return [seed]
  
      def step(self, action):
          bet_in_dollars = min(action/100.0, self.wealth)
  
          self.rounds -= 1
  
          coinflip = flip(self.edge, self.np_random)
          self.wealth = min(self.max_wealth, self.wealth + coinflip * bet_in_dollars)
          self.rounds_elapsed += 1
  
          if coinflip:
              self.max_ever_wealth = max(self.wealth, self.max_ever_wealth)
              self.wins += 1
          else:
              self.losses += 1
  
          done = self.wealth < 0.01 or self.wealth == self.max_wealth or not self.rounds
          reward = self.max_wealth if done else 0.0
  
          return self._get_obs(), reward, done, {}
  
      def _get_obs(self):
          return (np.array([float(self.wealth)]), self.rounds_elapsed, self.wins,
                  self.losses, np.array([float(self.max_ever_wealth)]))
  
      def reset(self):
          # re-init everything to draw new parameters etc, but preserve the RNG for
          # reproducibility and pass in the same hyper-parameters as originally specified:
          self.__init__(initial_wealth=self.initial_wealth,
                        edge_prior_alpha=self.edge_prior_alpha,
                        edge_prior_beta=self.edge_prior_beta,
                        max_wealth_alpha=self.max_wealth_alpha,
                        max_wealth_m=self.max_wealth_m,
                        max_rounds_mean=self.max_rounds_mean,
                        max_rounds_sd=self.max_rounds_sd,
                        reseed=False)
          return self._get_obs()
  
      def render(self, mode='human'):
          print("Current wealth: ", self.wealth, "; Rounds left: ", self.rounds,
                "; True edge: ", self.edge, "; True max wealth: ", self.max_wealth,
                "; True stopping time: ", self.max_rounds, "; Rounds left: ",
                self.max_rounds - self.rounds_elapsed)
  