code: #######################################################################
  # Copyright (C)                                                       #
  # 2016-2018 Shangtong Zhang(zhangshangtong.cpp@gmail.com)             #
  # 2016 Kenta Shimada(hyperkentakun@gmail.com)                         #
  # 2017 Nicky van Foreest(vanforeest@gmail.com)                        #
  # Permission given to modify the code as long as you keep this        #
  # declaration at the top                                              #
  #######################################################################
  
  import numpy as np
  import matplotlib
  matplotlib.use('Agg')
  import matplotlib.pyplot as plt
  import seaborn as sns
  from tqdm import tqdm
  
  # actions: hit or stand
  ACTION_HIT = 0
  ACTION_STAND = 1  #  "strike" in the book
  ACTIONS = [ACTION_HIT, ACTION_STAND]
  
  # policy for player
  POLICY_PLAYER = np.zeros(22, dtype=np.int)
  for i in range(12, 20):
      POLICY_PLAYER[i] = ACTION_HIT
  POLICY_PLAYER[20] = ACTION_STAND
  POLICY_PLAYER[21] = ACTION_STAND
  
  # function form of target policy of player
  def target_policy_player(usable_ace_player, player_sum, dealer_card):
      return POLICY_PLAYER[player_sum]
  
  # function form of behavior policy of player
  def behavior_policy_player(usable_ace_player, player_sum, dealer_card):
      if np.random.binomial(1, 0.5) == 1:
          return ACTION_STAND
      return ACTION_HIT
  
  # policy for dealer
  POLICY_DEALER = np.zeros(22)
  for i in range(12, 17):
      POLICY_DEALER[i] = ACTION_HIT
  for i in range(17, 22):
      POLICY_DEALER[i] = ACTION_STAND
  
  # get a new card
  def get_card():
      card = np.random.randint(1, 14)
      card = min(card, 10)
      return card
  
  # get the value of a card (11 for ace).
  def card_value(card_id):
      return 11 if card_id == 1 else card_id
  
  # play a game
  # @policy_player: specify policy for player
  # @initial_state: [whether player has a usable Ace, sum of player's cards, one card of dealer]
  # @initial_action: the initial action
  def play(policy_player, initial_state=None, initial_action=None):
      # player status
  
      # sum of player
      player_sum = 0
  
      # trajectory of player
      player_trajectory = []
  
      # whether player uses Ace as 11
      usable_ace_player = False
  
      # dealer status
      dealer_card1 = 0
      dealer_card2 = 0
      usable_ace_dealer = False
  
      if initial_state is None:
          # generate a random initial state
  
          while player_sum < 12:
              # if sum of player is less than 12, always hit
              card = get_card()
              player_sum += card_value(card)
  
              # If the player's sum is larger than 21, he may hold one or two aces.
              if player_sum > 21:
                  assert player_sum == 22
                  # last card must be ace
                  player_sum -= 10
              else:
                  usable_ace_player |= (1 == card)
  
          # initialize cards of dealer, suppose dealer will show the first card he gets
          dealer_card1 = get_card()
          dealer_card2 = get_card()
  
      else:
          # use specified initial state
          usable_ace_player, player_sum, dealer_card1 = initial_state
          dealer_card2 = get_card()
  
      # initial state of the game
      state = [usable_ace_player, player_sum, dealer_card1]
  
      # initialize dealer's sum
      dealer_sum = card_value(dealer_card1) + card_value(dealer_card2)
      usable_ace_dealer = 1 in (dealer_card1, dealer_card2)
      # if the dealer's sum is larger than 21, he must hold two aces.
      if dealer_sum > 21:
          assert dealer_sum == 22
          # use one Ace as 1 rather than 11
          dealer_sum -= 10
      assert dealer_sum <= 21
      assert player_sum <= 21
  
      # game starts!
  
      # player's turn
      while True:
          if initial_action is not None:
              action = initial_action
              initial_action = None
          else:
              # get action based on current sum
              action = policy_player(usable_ace_player, player_sum, dealer_card1)
  
          # track player's trajectory for importance sampling
          player_trajectory.append([(usable_ace_player, player_sum, dealer_card1), action])
  
          if action == ACTION_STAND:
              break
          # if hit, get new card
          card = get_card()
          # Keep track of the ace count. the usable_ace_player flag is insufficient alone as it cannot
          # distinguish between having one ace or two.
          ace_count = int(usable_ace_player)
          if card == 1:
              ace_count += 1
          player_sum += card_value(card)
          # If the player has a usable ace, use it as 1 to avoid busting and continue.
          while player_sum > 21 and ace_count:
              player_sum -= 10
              ace_count -= 1
          # player busts
          if player_sum > 21:
              return state, -1, player_trajectory
          assert player_sum <= 21
          usable_ace_player = (ace_count == 1)
  
      # dealer's turn
      while True:
          # get action based on current sum
          action = POLICY_DEALER[dealer_sum]
          if action == ACTION_STAND:
              break
          # if hit, get a new card
          new_card = get_card()
          ace_count = int(usable_ace_dealer)
          if new_card == 1:
              ace_count += 1
          dealer_sum += card_value(new_card)
          # If the dealer has a usable ace, use it as 1 to avoid busting and continue.
          while dealer_sum > 21 and ace_count:
              dealer_sum -= 10
              ace_count -= 1
          # dealer busts
          if dealer_sum > 21:
              return state, 1, player_trajectory
          usable_ace_dealer = (ace_count == 1)
  
      # compare the sum between player and dealer
      assert player_sum <= 21 and dealer_sum <= 21
      if player_sum > dealer_sum:
          return state, 1, player_trajectory
      elif player_sum == dealer_sum:
          return state, 0, player_trajectory
      else:
          return state, -1, player_trajectory
  
  # Monte Carlo Sample with On-Policy
  def monte_carlo_on_policy(episodes):
      states_usable_ace = np.zeros((10, 10))
      # initialze counts to 1 to avoid 0 being divided
      states_usable_ace_count = np.ones((10, 10))
      states_no_usable_ace = np.zeros((10, 10))
      # initialze counts to 1 to avoid 0 being divided
      states_no_usable_ace_count = np.ones((10, 10))
      for i in tqdm(range(0, episodes)):
          _, reward, player_trajectory = play(target_policy_player)
          for (usable_ace, player_sum, dealer_card), _ in player_trajectory:
              player_sum -= 12
              dealer_card -= 1
              if usable_ace:
                  states_usable_ace_count[player_sum, dealer_card] += 1
                  states_usable_ace[player_sum, dealer_card] += reward
              else:
                  states_no_usable_ace_count[player_sum, dealer_card] += 1
                  states_no_usable_ace[player_sum, dealer_card] += reward
      return states_usable_ace / states_usable_ace_count, states_no_usable_ace / states_no_usable_ace_count
  
  # Monte Carlo with Exploring Starts
  def monte_carlo_es(episodes):
      # (playerSum, dealerCard, usableAce, action)
      state_action_values = np.zeros((10, 10, 2, 2))
      # initialze counts to 1 to avoid division by 0
      state_action_pair_count = np.ones((10, 10, 2, 2))
  
      # behavior policy is greedy
      def behavior_policy(usable_ace, player_sum, dealer_card):
          usable_ace = int(usable_ace)
          player_sum -= 12
          dealer_card -= 1
          # get argmax of the average returns(s, a)
          values_ = state_action_values[player_sum, dealer_card, usable_ace, :] / \
                    state_action_pair_count[player_sum, dealer_card, usable_ace, :]
          return np.random.choice([action_ for action_, value_ in enumerate(values_) if value_ == np.max(values_)])
  
      # play for several episodes
      for episode in tqdm(range(episodes)):
          # for each episode, use a randomly initialized state and action
          initial_state = [bool(np.random.choice([0, 1])),
                         np.random.choice(range(12, 22)),
                         np.random.choice(range(1, 11))]
          initial_action = np.random.choice(ACTIONS)
          current_policy = behavior_policy if episode else target_policy_player
          _, reward, trajectory = play(current_policy, initial_state, initial_action)
          first_visit_check = set()
          for (usable_ace, player_sum, dealer_card), action in trajectory:
              usable_ace = int(usable_ace)
              player_sum -= 12
              dealer_card -= 1
              state_action = (usable_ace, player_sum, dealer_card, action)
              if state_action in first_visit_check:
                  continue
              first_visit_check.add(state_action)
              # update values of state-action pairs
              state_action_values[player_sum, dealer_card, usable_ace, action] += reward
              state_action_pair_count[player_sum, dealer_card, usable_ace, action] += 1
  
      return state_action_values / state_action_pair_count
  
  # Monte Carlo Sample with Off-Policy
  def monte_carlo_off_policy(episodes):
      initial_state = [True, 13, 2]
  
      rhos = []
      returns = []
  
      for i in range(0, episodes):
          _, reward, player_trajectory = play(behavior_policy_player, initial_state=initial_state)
  
          # get the importance ratio
          numerator = 1.0
          denominator = 1.0
          for (usable_ace, player_sum, dealer_card), action in player_trajectory:
              if action == target_policy_player(usable_ace, player_sum, dealer_card):
                  denominator *= 0.5
              else:
                  numerator = 0.0
                  break
          rho = numerator / denominator
          rhos.append(rho)
          returns.append(reward)
  
      rhos = np.asarray(rhos)
      returns = np.asarray(returns)
      weighted_returns = rhos * returns
  
      weighted_returns = np.add.accumulate(weighted_returns)
      rhos = np.add.accumulate(rhos)
  
      ordinary_sampling = weighted_returns / np.arange(1, episodes + 1)
  
      with np.errstate(divide='ignore',invalid='ignore'):
          weighted_sampling = np.where(rhos != 0, weighted_returns / rhos, 0)
  
      return ordinary_sampling, weighted_sampling
  
  def figure_5_1():
      states_usable_ace_1, states_no_usable_ace_1 = monte_carlo_on_policy(10000)
      states_usable_ace_2, states_no_usable_ace_2 = monte_carlo_on_policy(500000)
  
      states = [states_usable_ace_1,
                states_usable_ace_2,
                states_no_usable_ace_1,
                states_no_usable_ace_2]
  
      titles = ['Usable Ace, 10000 Episodes',
                'Usable Ace, 500000 Episodes',
                'No Usable Ace, 10000 Episodes',
                'No Usable Ace, 500000 Episodes']
  
      _, axes = plt.subplots(2, 2, figsize=(40, 30))
      plt.subplots_adjust(wspace=0.1, hspace=0.2)
      axes = axes.flatten()
  
      for state, title, axis in zip(states, titles, axes):
          fig = sns.heatmap(np.flipud(state), cmap="YlGnBu", ax=axis, xticklabels=range(1, 11),
                            yticklabels=list(reversed(range(12, 22))))
          fig.set_ylabel('player sum', fontsize=30)
          fig.set_xlabel('dealer showing', fontsize=30)
          fig.set_title(title, fontsize=30)
  
      plt.savefig('../images/figure_5_1.png')
      plt.close()
  
  def figure_5_2():
      state_action_values = monte_carlo_es(500000)
  
      state_value_no_usable_ace = np.max(state_action_values[:, :, 0, :], axis=-1)
      state_value_usable_ace = np.max(state_action_values[:, :, 1, :], axis=-1)
  
      # get the optimal policy
      action_no_usable_ace = np.argmax(state_action_values[:, :, 0, :], axis=-1)
      action_usable_ace = np.argmax(state_action_values[:, :, 1, :], axis=-1)
  
      images = [action_usable_ace,
                state_value_usable_ace,
                action_no_usable_ace,
                state_value_no_usable_ace]
  
      titles = ['Optimal policy with usable Ace',
                'Optimal value with usable Ace',
                'Optimal policy without usable Ace',
                'Optimal value without usable Ace']
  
      _, axes = plt.subplots(2, 2, figsize=(40, 30))
      plt.subplots_adjust(wspace=0.1, hspace=0.2)
      axes = axes.flatten()
  
      for image, title, axis in zip(images, titles, axes):
          fig = sns.heatmap(np.flipud(image), cmap="YlGnBu", ax=axis, xticklabels=range(1, 11),
                            yticklabels=list(reversed(range(12, 22))))
          fig.set_ylabel('player sum', fontsize=30)
          fig.set_xlabel('dealer showing', fontsize=30)
          fig.set_title(title, fontsize=30)
  
      plt.savefig('../images/figure_5_2.png')
      plt.close()
  
  def figure_5_3():
      true_value = -0.27726
      episodes = 10000
      runs = 100
      error_ordinary = np.zeros(episodes)
      error_weighted = np.zeros(episodes)
      for i in tqdm(range(0, runs)):
          ordinary_sampling_, weighted_sampling_ = monte_carlo_off_policy(episodes)
          # get the squared error
          error_ordinary += np.power(ordinary_sampling_ - true_value, 2)
          error_weighted += np.power(weighted_sampling_ - true_value, 2)
      error_ordinary /= runs
      error_weighted /= runs
  
      plt.plot(np.arange(1, episodes + 1), error_ordinary, color='green', label='Ordinary Importance Sampling')
      plt.plot(np.arange(1, episodes + 1), error_weighted, color='red', label='Weighted Importance Sampling')
      plt.ylim(-0.1, 5)
      plt.xlabel('Episodes (log scale)')
      plt.ylabel(f'Mean square error\n(average over {runs} runs)')
      plt.xscale('log')
      plt.legend()
  
      plt.savefig('../images/figure_5_3.png')
      plt.close()
  
  
  if __name__ == '__main__':
      figure_5_1()
      figure_5_2()
      figure_5_3()
  
apis: numpy.ones numpy.asarray matplotlib.pyplot.subplots_adjust matplotlib.pyplot.ylabel matplotlib.pyplot.savefig numpy.random.choice matplotlib.pyplot.xscale matplotlib.use numpy.where numpy.zeros matplotlib.pyplot.subplots numpy.argmax numpy.arange numpy.max numpy.power matplotlib.pyplot.close matplotlib.pyplot.ylim numpy.random.binomial matplotlib.pyplot.legend numpy.flipud numpy.add.accumulate numpy.errstate numpy.random.randint matplotlib.pyplot.xlabel