Is there a surefire strategy in a fair coin toss game?

Niti998

**Strategy in Coin Toss Game**

Suppose you and Little Mei are playing a coin toss game multiple times. Before each round, you can declare any amount, denoted as x. If the coin lands heads up, Little Mei gives you x dollars; if it lands tails up, you give Little Mei x dollars. Assume: 1) the game is entirely fair; 2) both you and Little Mei have enough money and time; 3) as soon as both sides settle their money, you can end the entire game.

Can you guarantee winning money?

Most people might think like this: since the probability of winning each round is 1/2, in each game round, the mathematical expectation of my income and expenses is 0. Therefore, neither side has a guaranteed winning strategy. However, it can be proven that I do have a guaranteed winning strategy.

I can adopt the following strategy: In the first round, I declare $1. If I win, I end the game, and I earn $1. If I lose the first round, in the second round, I declare $2. If I win the second round, I end the game, and I ultimately win $1. If I lose the second round as well, in the third round, I declare $4. If I win the third round, I end the game, and I ultimately win $1. In other words, the next round's bet is double the previous round's bet. As soon as I win, I end the game.

Next, let's prove that this strategy can ensure a stable gain of $1. First, it's impossible for you to keep losing indefinitely. If you continuously toss the coin multiple times, heads will eventually come up. Furthermore, it can be known that in this strategy, in the nth round, I bet x dollars. Suppose I win for the first time in the n+1 round, which means I had been losing continuously for n rounds. In the n+1 round, I win 2^n dollars. So, my final profit is 2^n - (1 + 2 + 4 + ... + 2^(n-1)) = 1 dollar.

In fact, you don't have to settle for"guaranteed to win $1". The above strategy is "if you lose, then double the next round's bet." If the strategy becomes "if you lose, then triple the next round's bet," what would happen? The same analysis process applies, and the final profit is 3^n dollars, which is exponential!

However, don't celebrate too soon! The above analysis is based on certain conditions. In real life, your funds are limited. If you follow the "doubling bet strategy," betting $1 in the first round, the bet escalates to $1.05 million by the 20th round. If your assets are only $1 million, you will go bankrupt or won't have enough funds to start the next round. Furthermore, continuously tossing a coin and getting tails 20 times in a row is still a probability. Additionally, the game assumes that Little Mei can keep playing with you indefinitely. If she wins $20,000 from you and leaves, you won't have a chance to recover your investment. Moreover, strictly speaking, this is not a fair game because you can win money and leave, while Little Mei cannot declare the bet; she has no say in it.

At the end of the article, I would like to propose two questions: Apart from the "doubling strategy" mentioned above, are there other strategies that guarantee a winning strategy? If the probability of the coin landing heads is 0.3, are there still guaranteed winning strategies? Feel free to share your thoughts in the comments.

Shri333

"You'll only know if it works by trying it out."

Jatin255

"Having a guaranteed win is also quite impressive."