Update app.py

app.py

@@ -126,8 +126,47 @@ Step 4: State Final Recommendation.
 Base your analysis strictly on the provided frequencies and the stated RPS rules.
 """
 
-# Empty Markov system prompt as requested
-DEFAULT_SYSTEM_PROMPT_MARKOV = ""
+# Updated Markov system prompt with the working version
+DEFAULT_SYSTEM_PROMPT_MARKOV = """You are analyzing a Rock-Paper-Scissors (RPS) game using a Markov transition matrix.
+
+### TRANSITION MATRIX:
+[
+  [0.20, 0.60, 0.20],  # Row 0 (After Rock)
+  [0.30, 0.10, 0.60],  # Row 1 (After Paper)
+  [0.50, 0.30, 0.20]   # Row 2 (After Scissors)
+]
+
+### EXPLANATION:
+- This matrix shows P(Next Move | Previous Move)
+- Each row represents the previous move (0=Rock, 1=Paper, 2=Scissors)
+- Each column represents the next move (0=Rock, 1=Paper, 2=Scissors)
+- For example, entry [0,1]=0.60 means: after playing Rock, 60% chance of playing Paper next
+
+### PLAYER INFORMATION:
+- The player's last move was: Paper
+- Our goal is to predict their most likely next move and determine our choice that counters the predicted move
+
+### YOUR TASK:
+1. Find the row in the matrix corresponding to the player's last move
+2. From that row, identify which move has the highest probability value
+3. That highest probability move is the player's predicted next move
+4. Determine the optimal counter move using RPS rules:
+   * Rock beats Scissors
+   * Scissors beats Paper
+   * Paper beats Rock
+
+### SHOW YOUR MATHEMATICAL WORK:
+- Identify the correct row number for the player's last move
+- Extract all probability values from that row
+- Compare the numerical values to find the maximum
+- Apply game rules to determine the counter move
+
+### OUTPUT FORMAT:
+Player's Last Move: [Move]
+Probabilities: [List the probabilities]
+Predicted Next Move: [Move with highest probability]
+Optimal Counter: [Move that beats the predicted move]
+"""
 
 # --- Default Input Values ---
 DEFAULT_PLAYER_STATS = "Rock: 40%\nPaper: 30%\nScissors: 30%"
@@ -209,7 +248,7 @@ with gr.Blocks(theme=gr.themes.Soft()) as demo:
             return {
                 frequency_inputs: gr.update(visible=False),
                 markov_inputs: gr.update(visible=True),
-                system_prompt_input: gr.update(value=DEFAULT_SYSTEM_PROMPT_MARKOV) # Load empty Markov prompt
+                system_prompt_input: gr.update(value=DEFAULT_SYSTEM_PROMPT_MARKOV) # Load Markov prompt
             }
         else: # Default case
              return {