Optimal Betting Strategy in Coin Toss

Question

Goal:

Participate in a coin-tossing game with the aim to maximize your earnings. A fundamental part of the game is formulating a strategy that optimizes your earnings based on the observation of the coin toss outcomes.

Rules:

1. The Coin:

   • A coin is tossed, landing on "HEADS" with an unknown probability p .

2. Observation and Betting:

   • You have the option to either observe the outcome of the toss without betting or place a fixed bet on an interval where you believe p lies.

3. Placing Bets:

   • Betting involves selecting a confidence interval where you believe the probability p exists.
   • The bet is fixed, meaning you bet the same amount each time.

4. Earnings:

   • If you choose an interval that includes p , you earn a point.
   • If you narrow down the interval size compared to the previous round while p is still within the interval, your earnings are quadratically increased based on the reduction in the interval size.

5. Strategy:

   • The goal of the game is to find the best strategy that maximizes your total earnings.
   • The strategy involves when to observe, when to bet, and how to adjust the betting intervals based on the observed outcomes.

Gameplay:

• In each round, the game simulates a coin toss, revealing whether it lands on "HEADS" or "TAILS".
• Based on the outcomes, you make a decision—either to observe or to bet by choosing a probability interval.
• Implement your strategy based on the observations to optimize the betting intervals and maximize earnings.

Conclusion of the Game:

• The game proceeds for a set number of rounds (e.g., 100 rounds).
• At the end of the game, the actual value of p is disclosed, and your total earnings are calculated, showcasing the effectiveness of your strategy.

Example:

• For instance, you decide to observe for the first five rounds, noting the outcomes.
• In the subsequent round, you may choose to bet with a certain interval based on your observations and strategy.
• Continually adjust your strategy based on the outcomes and the objective to find the most profitable betting intervals.

Remember, the fundamental aim is to develop and apply a strategy that maximally increases your earnings throughout the game, using observation and intelligent betting based on the revealed outcomes.