Expected Value Gambling Qn
Solution 1:
Your question is ill-posed. It has a couple of hidden issues with it.
Your problem has multiple components, some of which are not addressed in your post. They cannot be answered without knowing them.
The first element is that your game appears to have a stopping rule of some kind. People are permitted to stop playing. You take it for granted, it seems. If early stopping is permitted, then that opens up other issues. Do you want to create a Bayes action, a decision rule, or choose the combination of the two by minimizing integrated risk? You must have a utility function of wealth for any of this. You may also need to specify the utility of an estimation rule.
Without considering a stopping rule, consider the following utilities of wealth, $$\mathcal{U}(w)=\log(w),$$ and $$\mathcal{U}(w)=\begin{cases}w&\text{if } w\le{E}(w)\\ E(w)&\text{otherwise}\end{cases}$$
In the first utility function, additional wealth is always valued. In the second utility function, wealth past a threshold, in this case, the long-run expected value, has no added utility. Both utilities are real-world functions, and there are countless others (well, maybe they could be countable as they are rules).
The next issue is about whether there are other costs. Can you get tired? Are you willing to accept an interior solution for a reason not mentioned? For example, if your spouse wants you to go swimming with them, have you specified it in your utility function. The example is meant to be fanciful and get the point across.
Your question currently reads that a highly unspecified video poker machine exists. A strategy has been discovered that gives an edge. In the case where there are a large number of initial winnings, should that fact be considered?
It appears to assume that an expected value is relevant in the decision making, but it may not be. Remember that $$\mathcal{U}(E(x))\ne{E}(\mathcal{U}(x))$$ in the general case.
The stated edge is ill-specified. Does that edge exist in all configurations of observed cards? Are the cards shuffled, or is there a memory? Are there different edges under differing circumstances? Is the edge sometimes negative or zero?
If the “edge” is estimated, it creates another host of issues. Only Bayesian tools are coherent. A decision would be coherent if a competitor could not create a game of “heads I win, tails you lose” by using your probabilities against you.
I do believe, as do other commentators, that you felt that winning early would imply a reversal later. In systems without memory, that is not true.
Nonetheless, you also lack a rule of how much money to bet. That should also optimize utility and any constraints. Are bets in discrete units or real ones? This is math; mathematicians let you get away with real-valued bets. If the bets are discrete and there are no loans available, then you can run out of money in many typical situations. If you made that much extra up, it is likely you could collapse down too.
Does your decision rule consider total failure in the allocation or stopping decisions? What about in hand specific decisions? Would you change your bets if you had a flush showing in seven card stud?
How does this machine handle bets? What are the rules?
As shown in the comments, if the game is memoryless, you have won a fixed sum already, and you will continue to have a positive expected value, then the prior win should not impact the future patterns of wins.
However, that does not imply that you should continue playing.
Finally, I would argue that statistics, at least, is not a branch of mathematics, but as is often pointed out by Cosmo Shalizi, a branch of rhetoric. Decisions, of course, are the property of statistical decision theory, psychology and/or microeconomics. If this video poker is an MMOG, then also add in game theory.
I hope this post did not come across as snarky. You left things out, but that is okay too. Placing money at risk in a state of uncertainty is a profession in itself. It is a gigantic field. I wanted to be certain that you were thinking carefully.