## Winning against a bet-dependent coin

For the discussion of math. Duh.

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### Winning against a bet-dependent coin

You are playing a gambling game on a casino machine. It plays as follows:
You bet a number, R, chosen from the set of integers [1, N]. R is proportional to the amount of money at stake. With probability PR you will win back 2R. With probability 1- PR you win back nothing (so you lose R). PR is dependent on R but you do not know what it is.

You know the following:
- The maximum and minimum values in the set of all P are 0.9 and 0, respectively.
- For all k != j, |Pk - Pj | > 1/(2N)
- After X bets, the probabilities are shifted cyclically such that P’k = Pk-1, and P’1 = PN.
- After 10N bets, the entire system resets, and a new set of probabilities is generated that satisfies the above conditions.

Suppose N = 100, X = 10, and you have a lot of money. Can you game the system? What is the best strategy?

My current strategy (not sure if optimal):
1. Set some confidence counter to 0. Set current bet to 1.
2. Bet 10 times.
3. If those 10 bets yielded net profit, increase confidence by 1. If they yielded net loss, decrease confidence by 1.
4. If confidence is negative, set it to 0. If confidence is positive, increase bet by 1.
5. Go back to Step 2.
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quantropy
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### Re: Winning against a bet-dependent coin

So you have 1000 goes. The probabilities cycle every 10 goes but you can keep track of that. I can see that you should start with low bets, so that you're not using up too much money, and towards the end of the game when you have a better idea of the probabilities these will be high bets. I would try to vary my bets more than you suggest to sample a wider range. Say choose N with probability DN, where DN decreases with N according to some rule. As the game proceeds modify DN according to how often that slot has won in the past. I'm not sure how to make this more precise though. DN=2-N is a possibility, although maybe that drops off too quickly.

>-)
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### Re: Winning against a bet-dependent coin

First make 10 bets of 1, then when p_n goes over to p_1, bet on the new 1 a few times. Now you have a really good sense of the probabilities on the high end of the range and can decide whether or not to bet N

Also, I'm not sure this will help, but are the probabilities selected at random from the acceptable range, or maliciously assigned?

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### Re: Winning against a bet-dependent coin

Probabilities are randomly generated but must fit the constraints. The permutation, however, can be maliciously selected.
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