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Best Strategy

chrisjones

Joined
2/16/12
Messages
3
Points
11
You start with a £100. You can flip a coin 100 times, where if you predict right, you double your stake, if you predict wrong you lose your stake. The stake you gamble with can never be more than what you own.

How can you maximise your profit?
 
You can't do better than zero in expectation. You're playing a game at even odds with a fair coin. You have no advantage.
 
robopool, that only works with an infinite number of flips, you only have 100 games here. To do the 1 penny thing dies indeed almost guarantee that you'll come out ahead eventually, but not quickly enough.

IMO it's the whole risk/reward thing. Sure you can minimize your chance of going bust with penny bets, but you're also going to give up your reward for that. Similarly, you can just gamble all $100 and have a 50% chance of losing everything. Somewhere in there is a happy medium that you'll be content with.
 
Brad Warren: Well actually, there could be a strategy because you have control over:
1) your stake in each round
2) the number of games you choose to play (you can either play a minimum of 1 game of $100 or more than 10,000 games of gambling a penny each time)

I'm thinking that you are forced to play this game anyways. Playing 1 game with a stake of $100 is too risky, you could lose it all with 50% chance. Playing 10,000 games of a stake of $0.01 each time is more favorable to the convergence of the law of large numbers, this way if you are forced to play this game, this is how you can at least guarantee that you leave with your $100 still in pocket.

In thinking about possible "strategies to actually GAIN money"... I do not know what is the best one but I have this idea:
Since the stakes are $0.01 each time, we could do a simple strategy. I used to do this when I was a freshman in university with online casinos. When applied to non-1-1 payouts it worked and I won like $850 from $25 but then I stopped bc I was paranoid they were noticing my strategy and would arbitrarily interfere. For a 1-1 payout it will be a doubling strategy: Always bet on H. If you win, always start over at Step 1. If not, go on to the next step.
Step 1: Bet $0.01.
Step 2: You just lost $0.01. Bet $0.02. If you win, you'll get $0.04 back, subtract the $0.02 you put in, subtract the $0.01 cumulative loss, and you have made a profit of $0.01.
Step 3: You have now lost $0.03. Bet $0.04. If you win, you will get $0.08, subtract the $0.04 you put in, subtract the $0.03 cumulative loss, and you have made a profit of $0.01.
Step 4: You have now lost $0.07. Bet $0.08. If you win, you will get $0.16, subtract the $0.08 you put in, subtract the $0.07 cumulative loss, and you have made a profit of $0.01.
... etc

The chances of this failing correspond to you having lost so much that you are not able to recover by doubling your next bet. For this to be the case, you would have to see 14 consecutive tosses which are tails, which has a probability of (1/2)^14 = 0.00006.

As you win each time, your pot for the gambling becomes bigger. Then you can (slowly) get stronger so that you can withstand 15 consecutive losses, 16 consecutive losses, ... , and the probability of your losing it all will become infinitely small.
 
Oh no! I was not thinking of the 100 flips part.

Well fine, then, I would tailor the doubling strategy to bet a bit more than a penny each time. With a penny each time, I can withstand 14 tails in a row (0.00006 probability). That's not risky enough if we have just 100 flips. So, for differing levels of risk aversion, first we choose a threshold of how many tails in a row we can withstand (ie. 5 consecutive Tails would be 0.03 probability. Then we tailor the bets (make them bigger) such that we go bust if and only if there are 5 consecutive Tails). Effectively, we are taking on more risk but this is good because we only have 100 tosses and cannot be so conservative with only 0.00006 probability of loss.
 
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