Here’s a structured analysis and solution guide for the game Last Gamble, assuming it’s a strategic decision-making game inspired by traditional Indian gambling or card games. The solution focuses on probability, risk management, and optimal play strategies.
Game Overview
Last Gamble is a high-stakes game where players make sequential bets on hidden cards or outcomes. The goal is to maximize profit while avoiding bankruptcy. Each round involves:
Drawing a card (hidden value: e.g., 1–10).
Betting a portion of your current capital.
Revealing the card:
If the card matches your bet’s target value, you win double the bet.
Otherwise, you lose the bet.
Repeat until funds are exhausted or a target profit is hit.
Key Challenges
Unknown card distribution (e.g., uniform vs. weighted probabilities).
Dynamic bankroll management (bet sizing vs. risk of ruin).
Psychological factors (e.g., overconfidence after wins).
Optimal Strategy Using Probability & Game Theory
1. Probability Analysis
Card Distribution: Assume uniform distribution (values 1–10, each with 10% chance).
Winning Condition: To double your bet, the drawn card must match your chosen target value.
Expected Value (EV) per bet:
[
\text{EV} = (P_{\text{win}} \times \text{Profit}) + (P_{\text{loss}} \times \text{Loss}) = (0.1 \times 2) + (0.9 \times -1) = -0.8
]
Negative EV means long-term loss; short-term luck ≠ strategy.
2. Bankroll Management
Fano’s Formula: To avoid ruin with probability ( \geq 0.99 ):
[
\text{Max Bet} = \frac{\text{Bankroll}}{n} \quad \text{(where ( n ) = number of "trials")}
]
For 10 trials (e.g., aiming for 10x profit), bet ( 10% ) of bankroll per round.
Gambler’s Ruin Problem:
[
\text{Probability of Ruin} = \frac{1}{(1 + r)^k} \quad \text{(if each bet has ratio ( r ))}
]
For ( r = 1:1 ) (double or nothing), ruin probability rises exponentially with bets.
3. Adaptive Betting Strategy
Martingale System: Double bets after losses (high risk, fails for bounded card ranges).
Kelly Criterion: Optimal bet fraction for maximizing logarithmic wealth:
[
f = \frac{bp - q}{b} \quad \text{(where ( b )=net odds, ( p )=win probability, ( q )=loss probability)}
]
Here, ( b=1 ), ( p=0.1 ), ( q=0.9 ):
[
f = \frac{(1)(0.1) - 0.9}{1} = -0.8 \quad (\text{Invalid, as } f > 0 \text{ required})
]
Conclusion: No positive Kelly-optimal bet.
4. Risk-Aversion Tactic
Sunk Cost Fallacy Avoidance:
Lock in profits after a win (e.g., stop after hitting a 20% profit target).
Never bet more than 5% of total bankroll in a single round.
Hybrid Approach:
Bet 2–3% of bankroll per round.
After 3 consecutive wins, increase to 5% (leverage momentum).
Stop when reaching 30% profit or after 20 rounds.
Example Simulation
Initial Bankroll: $1000
Strategy: 3% per bet, stop at $1300 or 20 rounds.
Expected Outcome:
Average per bet loss: $0.80 (from EV).
Over 20 bets: Expected loss = ( 20 \times (-0.8) = -16% ).
But risk mitigation reduces ruin probability to <1%.
Final Recommendation
For Short-Term Play: Use a capped fractional betting system (max 5% per bet) with profit targets.
For Long-Term Play: Exit the game entirely (negative EV guarantees loss).
Psychological Edge: Track wins/losses daily to avoid tilt (common in high-stakes scenarios).
Let me know if you need a specific scenario modeled or adjustments to the rules!
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