From c20e9247e329512d1de7e2f8ba2ea36fadd291a7 Mon Sep 17 00:00:00 2001
From: Adrienucl <adrien.payen@student.uclouvain.be>
Date: Fri, 3 May 2024 20:55:35 +0200
Subject: [PATCH] correction tmc et markovDecision
---
ancien/markoVVV.py | 61 +++++-----
ancien/tmcccc.py | 12 ++
markovDecision.py | 27 +++--
plot.py | 4 +-
strategy_comparison.png | Bin 22200 -> 22206 bytes
tmc.py | 241 +++++++++++++++++++---------------------
validation.py | 23 ++--
7 files changed, 192 insertions(+), 176 deletions(-)
diff --git a/ancien/markoVVV.py b/ancien/markoVVV.py
index 8e92921..5cd533d 100644
--- a/ancien/markoVVV.py
+++ b/ancien/markoVVV.py
@@ -1,18 +1,20 @@
import numpy as np
-from ancien.tmc import TransitionMatrixCalculator as tmc
+from tmc import TransitionMatrixCalculator as tmc
class MarkovDecisionSolver:
def __init__(self, layout: list, circle: bool):
- self.Numberk = 15
+ self.nSquares = 15
+ self.precision = 1e-9
+ self.layout = layout
+ self.circle = circle
self.tmc_instance = tmc()
- self.safe_dice = self.tmc_instance._compute_safe_matrix()
- self.normal_dice = self.tmc_instance._compute_normal_matrix(layout, circle)
- self.risky_dice = self.tmc_instance._compute_risky_matrix(layout, circle)
- self.jail = [i for i, x in enumerate(layout) if x == 3]
- self.Dice = np.zeros(self.Numberk)
+ self.matrix_safe = self.tmc_instance._compute_safe_matrix()
+ self.matrix_normal, self.jail_n = self.tmc_instance._compute_normal_matrix(layout, circle)
+ self.matrix_risky, self.jail_r = self.tmc_instance._compute_risky_matrix(layout, circle)
+ self.Dice = np.zeros(self.nSquares, dtype=int)
def solve(self):
- ValueI = np.zeros(self.Numberk)
+ ValueI = np.zeros(self.nSquares)
ValueINew = np.array([8.5, 7.5, 6.5, 7, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 0])
i = 0
@@ -23,36 +25,36 @@ class MarkovDecisionSolver:
np.copyto(ValueI, ValueINew)
# Mettez à jour les valeurs de ValueINew pour chaque état
- for k in range(self.Numberk - 1):
+ for k in range(self.nSquares - 1):
ValueINew[k] = 1 + min(
- np.dot(self.safe_dice[k], ValueI),
- np.dot(self.normal_dice[k], ValueI) + np.sum(self.normal_dice[k][self.jail]),
- np.dot(self.risky_dice[k], ValueI) + np.sum(self.risky_dice[k][self.jail])
+ np.dot(self.matrix_safe[k], ValueI),
+ np.dot(self.matrix_normal[k], ValueI) + np.sum(self.jail_n[k]),
+ np.dot(self.matrix_risky[k], ValueI) + np.sum(self.jail_r[k])
)
- ValueINew[self.Numberk - 1] = min(
- np.dot(self.safe_dice[self.Numberk - 1], ValueI),
- np.dot(self.normal_dice[self.Numberk - 1], ValueI),
- np.dot(self.risky_dice[self.Numberk - 1], ValueI)
+ ValueINew[self.nSquares - 1] = min(
+ np.dot(self.matrix_safe[self.nSquares - 1], ValueI),
+ np.dot(self.matrix_normal[self.nSquares - 1], ValueI),
+ np.dot(self.matrix_risky[self.nSquares - 1], ValueI)
)
# Calculer les actions optimales (indice de l'action + 1)
- for k in range(self.Numberk):
+ for k in range(self.nSquares):
self.Dice[k] = np.argmin([
- np.dot(self.safe_dice[k], ValueI),
- np.dot(self.normal_dice[k], ValueI) + np.sum(self.normal_dice[k][self.jail]),
- np.dot(self.risky_dice[k], ValueI) + np.sum(self.risky_dice[k][self.jail]),
+ np.dot(self.matrix_safe[k], ValueINew),
+ np.dot(self.matrix_normal[k], ValueINew) + np.sum(self.jail_n[k]),
+ np.dot(self.matrix_risky[k], ValueINew) + np.sum(self.jail_r[k]),
]) + 1
# Vérifiez la convergence en utilisant une petite tolérance
- if np.sum(np.abs(ValueINew - ValueI)) < 1e-9:
+ if np.sum(np.abs(ValueINew - ValueI)) < self.precision:
break
# Retourne les valeurs finales de ValueINew et les actions optimales (Dice)
return ValueINew, self.Dice
-def markovDecision(layout : list, circle : bool):
+def markovDecision(layout: list, circle: bool):
solver = MarkovDecisionSolver(layout, circle)
return solver.solve()
@@ -60,12 +62,17 @@ def markovDecision(layout : list, circle : bool):
# Exemple d'utilisation de la fonction markovDecision avec les paramètres layout et circle
layout = [0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 1, 0]
-
# Résolution du problème avec différents modes de jeu
result_false = markovDecision(layout, circle=False)
-print("\nWin as soon as land on or overstep the final square")
-print(result_false)
+print("\nStopping on the square to win")
+print("Expected costs for each square:")
+print(result_false[0])
+print("Dice choices for each square:")
+print(result_false[1])
result_true = markovDecision(layout, circle=True)
-print("\nStopping on the square to win")
-print(result_true)
+print("\nWin as soon as land on or overstep the final square")
+print("Expected costs for each square:")
+print(result_true[0])
+print("Dice choices for each square:")
+print(result_true[1])
diff --git a/ancien/tmcccc.py b/ancien/tmcccc.py
index 388cc13..29086c4 100644
--- a/ancien/tmcccc.py
+++ b/ancien/tmcccc.py
@@ -192,6 +192,18 @@ class TransitionMatrixCalculator:
continue
self.matrix_risky[k,k_prime] += p
return self.matrix_risky
+
+ def print_matrix_with_layout(self, title, matrix):
+ print(f"{title}:")
+ for i in range(matrix.shape[0]):
+ row_str = " | ".join(f"{matrix[i, j]:.3f}" for j in range(matrix.shape[1]))
+ print(row_str)
+ print()
+
+# Example Usage:
+layout_example = [0]*15
+calculator = TransitionMatrixCalculator()
+print(calculator.compute_transition_matrix(layout_example, circle=True))
#tmc = TransitionMatrixCalculator()
#tmc.tst_transition_matrix()
diff --git a/markovDecision.py b/markovDecision.py
index 276043e..5b6e73d 100644
--- a/markovDecision.py
+++ b/markovDecision.py
@@ -1,28 +1,28 @@
import numpy as np
-from tmc_2 import TransitionMatrixCalculator as tmc
+from tmc import TransitionMatrixCalculator as tmc
class MarkovDecisionSolver:
def __init__(self, layout: list, circle: bool):
self.Numberk = 15
self.tmc_instance = tmc()
- self.safe_dice = self.tmc_instance.proba_security_dice()
- self.normal_dice, _ = self.tmc_instance.proba_normal_dice(layout, circle) # Make sure to capture only the normal_dice component
- self.risky_dice, _ = self.tmc_instance.proba_risky_dice(layout, circle) # Make sure to capture only the risky_dice component
+ self.safe_dice = self.tmc_instance._compute_safe_matrix()
+ self.normal_dice, _ = self.tmc_instance._compute_normal_matrix(layout, circle) # Make sure to capture only the normal_dice component
+ self.risky_dice, _ = self.tmc_instance._compute_risky_matrix(layout, circle) # Make sure to capture only the risky_dice component
self.jail = [i for i, x in enumerate(layout) if x == 3]
self.ValueI = np.zeros(self.Numberk)
self.DiceForStates = np.zeros(self.Numberk - 1)
def _compute_vi_safe(self, k):
- return np.sum(self.safe_dice[k] * self.ValueI) + np.sum(self.normal_dice[k][self.jail])
+ return np.dot(self.safe_dice[k], self.ValueI) + np.sum(self.normal_dice[k][self.jail])
def _compute_vi_normal(self, k):
- vi_normal = np.sum(self.normal_dice[k] * self.ValueI) + np.sum(self.normal_dice[k][self.jail])
+ vi_normal = np.dot(self.normal_dice[k], self.ValueI) + np.sum(self.normal_dice[k][self.jail])
return vi_normal
def _compute_vi_risky(self, k):
- vi_risky = np.sum(self.risky_dice[k] * self.ValueI) + np.sum(self.risky_dice[k][self.jail])
+ vi_risky = np.dot(self.risky_dice[k], self.ValueI) + np.sum(self.risky_dice[k][self.jail])
return vi_risky
def solve(self):
@@ -36,15 +36,21 @@ class MarkovDecisionSolver:
vi_normal = self._compute_vi_normal(k)
vi_risky = self._compute_vi_risky(k)
- ValueINew[k] = 1 + min(vi_safe, vi_normal, vi_risky)
+ # Compute the minimum value among vi_safe, vi_normal, and vi_risky
+ min_value = min(vi_safe, vi_normal, vi_risky)
- if ValueINew[k] == 1 + vi_safe:
+ # Find which index (safe, normal, or risky) corresponds to the minimum value
+ if min_value == vi_safe:
+ ValueINew[k] = 1 + vi_safe
self.DiceForStates[k] = 1
- elif ValueINew[k] == 1 + vi_normal:
+ elif min_value == vi_normal:
+ ValueINew[k] = 1 + vi_normal
self.DiceForStates[k] = 2
else:
+ ValueINew[k] = 1 + vi_risky
self.DiceForStates[k] = 3
+
if np.allclose(ValueINew, self.ValueI):
self.ValueI = ValueINew
break
@@ -59,7 +65,6 @@ def markovDecision(layout : list, circle : bool):
return solver.solve()
"""
-
# Exemple d'utilisation de la fonction markovDecision avec les paramètres layout et circle
layout = [0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 1, 0]
diff --git a/plot.py b/plot.py
index 46b023a..cf39b57 100644
--- a/plot.py
+++ b/plot.py
@@ -1,5 +1,5 @@
import matplotlib.pyplot as plt
-from valid import validation
+from validation import validation
import numpy as np
# Example layout and circle settings
@@ -48,7 +48,7 @@ def plot_state_based_turns(save=True):
plt.show()
-def plot_state_based_comparison(validation_instance, num_games=1000):
+def plot_state_based_comparison(validation_instance, num_games=100000):
optimal_turns, empirical_turns = validation_instance.compare_state_based_turns(num_games=num_games)
# Plotting the state-based average turns comparison
diff --git a/strategy_comparison.png b/strategy_comparison.png
index d8b19f2ae884d963e9a1a6f2a97b07cd2f744384..089723dd698d20af173f314912b2e90a01d89143 100644
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diff --git a/tmc.py b/tmc.py
index 04b0392..cb3b32b 100644
--- a/tmc.py
+++ b/tmc.py
@@ -2,156 +2,149 @@ import numpy as np
class TransitionMatrixCalculator:
def __init__(self):
- self.nSquares = 15
- self.matrix_safe = np.zeros((self.nSquares, self.nSquares))
- self.matrix_normal = np.zeros((self.nSquares, self.nSquares))
- self.matrix_risky = np.zeros((self.nSquares, self.nSquares))
+ self.size = 15
+ self.matrix_safe = np.zeros((self.size , self.size ))
+ self.matrix_normal = np.zeros((self.size , self.size ))
+ self.matrix_risky = np.zeros((self.size , self.size ))
- def proba_security_dice(self):
- proba = np.zeros((self.nSquares, self.nSquares))
+ def compute_transition_matrix(self, layout, circle=False):
+ self.matrix_safe = self._compute_safe_matrix()
+ self.matrix_normal, _ = self._compute_normal_matrix(layout, circle)
+ self.matrix_risky, _ = self._compute_risky_matrix(layout, circle)
- for i in range(self.nSquares - 1):
- proba[i][i] = 0.5
- if i == 2:
- proba[i][i + 1] = 0.25 # slow lane
- proba[i][i + 8] = 0.25 # fast lane
- elif i == 9:
- proba[i][i + 5] = 0.5
- else:
- proba[i][i + 1] = 0.5
+ return self.matrix_safe, self.matrix_normal, self.matrix_risky
- proba[self.nSquares - 1][self.nSquares - 1] = 1
- return proba
- def proba_normal_dice(self, layout, circle=False):
- proba = np.zeros((self.nSquares, self.nSquares))
- proba_prison = np.zeros((self.nSquares, self.nSquares))
+ def _compute_safe_matrix(self):
+ p = np.zeros((self.size ,self.size ))
+ for k in range(self.size - 1):
+ if k == 2:
+ p[k,k+1] = 1/4 # slow lane
+ p[k,k+8] = 1/4 # fast lane
+ elif k == 9:
+ p[k,k+5] = 1/2
+ else:
+ p[k,k+1] = 1/2
+ p[k, k] = 1 - np.sum(p[k])
+ p[self.size -1,self.size -1] = 1
+ return p
+
+ def _compute_normal_matrix(self, layout, circle=False):
+ p = np.zeros((self.size ,self.size ))
+ jail = np.zeros((self.size ,self.size ))
- for i in range(self.nSquares - 1):
- proba[i][i] = 1 / 3
- if i == 2:
- proba[i][i + 1] = 1 / 6 # slow lane
- proba[i][i + 2] = 1 / 6 # slow lane
- proba[i][i + 8] = 1 / 6 # fast lane
- proba[i][i + 9] = 1 / 6 # fast lane
- elif i == 8:
- proba[i][i + 1] = 1 / 3
- proba[i][i + 6] = 1 / 3
- elif i == 9:
+ for k in range(self.size - 1):
+ if k == 2:
+ p[k,k+1:k+3] = 1/6 # slow lane # slow lane
+ p[k,k+8:k+10] = 1/6 # fast lane # fast lane
+ elif k == 8:
+ p[k,k+1] = 1/3
+ p[k,k+6] = 1/3
+ elif k == 9:
if circle:
- proba[i][i + 5] = 1 / 3
- proba[i][0] = 1 / 3
+ p[k,k+5] = 1/3
+ p[k,0] = 1/3
else:
- proba[i][i + 5] = 2 / 3
- elif i == 13:
+ p[k,k+5] = 2/3
+ elif k == 13:
if circle:
- proba[i][i + 1] = 1 / 3
- proba[i][0] = 1 / 3
+ p[k,k+1] = 1/3
+ p[k,0] = 1/3
else:
- proba[i][i + 1] = 2 / 3
+ p[k,k+1] = 2/3
else:
- proba[i][i + 1] = 1 / 3
- proba[i][i + 2] = 1 / 3
+ p[k,k+1] = 1/3
+ p[k,k+2] = 1/3
+ p[k, k] = 1 - np.sum(p[k])
- for i in range(self.nSquares - 1):
- for j in range(self.nSquares - 1):
- case_value = layout[j]
- if case_value == 1:
+ for k in range(self.size - 1):
+ for j in range(self.size - 1):
+ s = layout[j]
+ if s == 1:
if j != 0:
- proba[i][0] += proba[i][j] / 2
- proba[i][j] /= 2
- elif case_value == 2:
- proba[i][j - 3 if j - 3 >= 0 else 0] += proba[i][j] / 2
- proba[i][j] /= 2
- elif case_value == 3:
- proba_prison[i][j] = proba[i][j] / 2
- elif case_value == 4:
- proba[i][j] /= 2
- if j != 0:
- proba[i][0] += proba[i][j] / 6
- proba[i][j - 3 if j - 3 >= 0 else 0] += proba[i][j] / 6
- proba_prison[i][j] = proba[i][j] / 6
+ p[k,0] += p[k,j]/2
+ p[k,j] /= 2
+ elif s == 2:
+ p[k,j-3 if j-3 >= 0 else 0] += p[k,j]/2
+ p[k,j] /= 2
+ elif s == 3:
+ jail[k,j] = p[k,j]/2
- proba[self.nSquares - 1][self.nSquares - 1] = 1
- return proba, proba_prison
+ p[self.size -1,self.size -1] = 1
+ return p, jail
- def proba_risky_dice(self, layout, circle=False):
- proba = np.zeros((self.nSquares, self.nSquares))
- proba_prison = np.zeros((self.nSquares, self.nSquares))
+ def _compute_risky_matrix(self, layout, circle=False):
+ p = np.zeros((self.size ,self.size ))
+ jail = np.zeros((self.size ,self.size ))
- for i in range(self.nSquares - 1):
- proba[i][i] = 1 / 4
- if i == 2:
- proba[i][i + 1] = 1 / 8 # slow lane
- proba[i][i + 2] = 1 / 8 # slow lane
- proba[i][i + 3] = 1 / 8 # slow lane
- proba[i][i + 8] = 1 / 8 # fast lane
- proba[i][i + 9] = 1 / 8 # fast lane
- proba[i][i + 10] = 1 / 8 # fast lane
- elif i == 7:
- proba[i][i + 1] = 1 / 4
- proba[i][i + 2] = 1 / 4
- proba[i][i + 7] = 1 / 4
- elif i == 8:
+ for k in range(self.size -1):
+ if k == 2:
+ p[k,k+1:k+4] = 1/8 # slow lane
+ p[k,k+8:k+11] = 1/8 # fast lane
+ elif k == 7:
+ p[k,k+1:k+3] = 1/4
+ p[k,k+7] = 1/4
+ elif k == 8:
if circle:
- proba[i][i + 1] = 1 / 4
- proba[i][i + 6] = 1 / 4
- proba[i][0] = 1 / 4
+ p[k,k+1] = 1/4
+ p[k,k+6] = 1/4
+ p[k,0] = 1/4
else:
- proba[i][i + 1] = 1 / 4
- proba[i][i + 6] = 1 / 2
- elif i == 9:
+ p[k,k+1] = 1/4
+ p[k,k+6] = 1/2
+ elif k == 9:
if circle:
- proba[i][i + 5] = 1 / 4
- proba[i][0] = 1 / 4
- proba[i][1] = 1 / 4
+ p[k,k + 5] = 1/4
+ p[k,0] = 1/4
+ p[k,1] = 1/4
else:
- proba[i][i + 5] = 3 / 4
- elif i == 12:
+ p[k,k+5] = 3/4
+ elif k == 12:
if circle:
- proba[i][i + 1] = 1 / 4
- proba[i][i + 2] = 1 / 4
- proba[i][0] = 1 / 4
+ p[k,k+1:k+3] = 1/4
+ p[k,0] = 1/4
else:
- proba[i][i + 1] = 1 / 4
- proba[i][i + 2] = 1 / 2
- elif i == 13:
+ p[k,k+1] = 1/4
+ p[k,k+2] = 1/2
+ elif k == 13:
if circle:
- proba[i][i + 1] = 1 / 4
- proba[i][0] = 1 / 4
- proba[i][1] = 1 / 4
+ p[k,k+1] = 1/4
+ p[k,0] = 1/4
+ p[k,1] = 1/4
else:
- proba[i][self.nSquares - 1] = 3 / 4
+ p[k,self.size -1] = 3/4
else:
- proba[i][i + 1] = 1 / 4
- proba[i][i + 2] = 1 / 4
- proba[i][i + 3] = 1 / 4
+ p[k,k+1:k+4] = 1/4
+ p[k, k] = 1 - np.sum(p[k])
- for i in range(self.nSquares - 1):
- for j in range(self.nSquares - 1):
- case_value = layout[j]
- if case_value == 1:
- if j != 0:
- proba[i][0] += proba[i][j]
- proba[i][j] = 0
- elif case_value == 2:
- proba[i][j - 3 if j - 3 >= 0 else 0] += proba[i][j]
- proba[i][j] = 0
- elif case_value == 3:
- proba_prison[i][j] = proba[i][j]
- elif case_value == 4:
+ for k in range(self.size - 1):
+ for j in range(self.size - 1):
+ s = layout[j]
+ if s == 1:
if j != 0:
- proba[i][0] += proba[i][j] / 3
- proba[i][j - 3 if j - 3 >= 0 else 0] += proba[i][j] / 3
- proba_prison[i][j] = proba[i][j] / 3
- proba[i][j] /= 3
+ p[k,0] += p[k,j]
+ p[k,j] = 0
+ elif s == 2:
+ p[k,j-3 if j-3 >= 0 else 0] += p[k,j]
+ p[k,j] = 0
+ elif s == 3:
+ jail[k,j] = p[k,j]
- proba[self.nSquares - 1][self.nSquares - 1] = 1
- return proba, proba_prison
+ p[self.size -1,self.size-1] = 1
+ return p, jail
- def compute_transition_matrix(self, layout, circle=False):
- self.matrix_safe = self.proba_security_dice()
- self.matrix_normal, _ = self.proba_normal_dice(layout, circle)
- self.matrix_risky, _ = self.proba_risky_dice(layout, circle)
+"""
+ def display_matrices(self):
+ print("Safe Matrix:")
+ print(self.matrix_safe)
+ print("\nNormal Matrix:")
+ print(self.matrix_normal)
+ print("\nRisky Matrix:")
+ print(self.matrix_risky)
- return self.matrix_safe, self.matrix_normal, self.matrix_risky
+# Example Usage:
+layout_example = [0]*15
+calculator = TransitionMatrixCalculator()
+calculator.compute_transition_matrix(layout_example, circle=True)
+calculator.display_matrices()
+"""
\ No newline at end of file
diff --git a/validation.py b/validation.py
index 16174fc..c86c663 100644
--- a/validation.py
+++ b/validation.py
@@ -1,8 +1,8 @@
import random as rd
import numpy as np
import matplotlib.pyplot as plt
-from tmc_2 import TransitionMatrixCalculator as tmc
-from mdp import MarkovDecisionSolver as mD
+from tmc import TransitionMatrixCalculator as tmc
+from markovDecision import MarkovDecisionSolver as mD
class validation:
def __init__(self, layout, circle=False):
@@ -11,9 +11,9 @@ class validation:
self.layout = layout
self.circle = circle
self.tmc_instance = tmc()
- self.safe_dice = self.tmc_instance.proba_security_dice()
- self.normal_dice, _ = self.tmc_instance.proba_normal_dice(layout, circle) # Make sure to capture only the normal_dice component
- self.risky_dice, _ = self.tmc_instance.proba_risky_dice(layout, circle) # Make sure to capture only the risky_dice component
+ self.safe_dice = self.tmc_instance._compute_safe_matrix()
+ self.normal_dice, _ = self.tmc_instance._compute_normal_matrix(layout, circle) # Make sure to capture only the normal_dice component
+ self.risky_dice, _ = self.tmc_instance._compute_risky_matrix(layout, circle)
solver = mD(self.layout, self.circle)
self.expec, self.optimal_policy = solver.solve()
@@ -74,9 +74,9 @@ class validation:
def simulate_state(self, strategy, layout, circle, n_iterations=10000):
# Compute transition matrices for each dice
- safe_dice = self.tmc_instance.proba_security_dice()
- normal_dice = self.tmc_instance.proba_normal_dice(layout, circle)[0] # Get only the normal dice transition matrix
- risky_dice = self.tmc_instance.proba_risky_dice(layout, circle)[0] # Get only the risky dice transition matrix
+ safe_dice = self.tmc_instance._compute_safe_matrix()
+ normal_dice = self.tmc_instance._compute_normal_matrix(layout, circle)[0] # Make sure to capture only the normal_dice component
+ risky_dice = self.tmc_instance._compute_risky_matrix(layout, circle)[0]
transition_matrices = [safe_dice, normal_dice, risky_dice]
number_turns = []
@@ -162,7 +162,7 @@ class validation:
def compare_empirical_vs_value_iteration(self, num_games=1000):
- value_iteration_turns = self.optimal_policy
+ value_iteration_turns = self.expec
empirical_turns = self.simulate_state(self.optimal_policy, self.layout, self.circle, n_iterations=num_games)
# Calculate the mean turns for each state
@@ -175,7 +175,7 @@ class validation:
- def compare_state_based_turns(self, num_games=1000):
+ def compare_state_based_turns(self, num_games=100000):
value_iteration = self.expec
empirical_turns = self.simulate_state(self.optimal_policy, self.layout, self.circle, n_iterations=num_games)
@@ -183,7 +183,7 @@ class validation:
- def compare_strategies(self, num_games=1000):
+ def compare_strategies(self, num_games=100000):
optimal_cost = self.simulate_game(self.optimal_policy, n_iterations=num_games)
dice1_cost = self.simulate_game(self.safe_strategy, n_iterations=num_games)
dice2_cost = self.simulate_game(self.normal_strategy, n_iterations=num_games)
@@ -265,5 +265,4 @@ random_dice_strategy = validation_instance.random_strategy
mean_turns_random_dice = validation_instance.simulate_state(random_dice_strategy, layout, circle, n_iterations=10000)
print("Mean Turns for Random Dice Strategy:", mean_turns_random_dice)
-
"""
\ No newline at end of file
--
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