Suppose each player is randomly and secretly assigned a number between $1$ and $6$ and then a communal dice is rolled four times. After each roll, you can place a bet or fold. After the fourth bet is placed, the person who has the most die facing their number wins.
As each game progresses, the game can move in your favour or against you (e.g., 1/1, 1/2, 2/3, 3/4). If you bet too high at the beginning, the progression of the game could move against you. If you bet too low, it would be difficult to increase the pot at the end because the other players could fold without losing anything. There are infinitely many games available and a finite buy-in amount but an infinite amount of money to replenish the buy-in.
How do you calculate the optimum betting amount?
This relates to one aspect of Texas Hold 'Em poker betting odds, in abstract terms. The other aspects of the betting odds can be ignored.
Any guidelines will be appreciated. I think the question could be simplified to:
Does an initial bet of $10\%$ of capital on $10\%$ odds with a $20\times$ return approach $0$ or $\infty$? What are the limits in this case?
If you continually bet $10\%$ of capital with $0.1$ chance of returning $20x$ you will win almost surely. Your position after $n$ wins and $m$ losses only depends on $n$ and $m$, not on the order of wins and losses. Each win triples your bankroll, while each loss multiplies it by $0.9$. You will have $3^n\cdot 0.9^m$ times your original stake. The law of large numbers is on your side. After one win and nine losses your capital has multiplied by $3 \cdot 0.9^9\approx 1.16$. You will never run out of money because you never bet your whole bankroll. After a large number of games the winning fraction will be close to $10\%$ and you will be way ahead. The bet is in your favor-take it!
You might be interested to look up the Kelly criterion for how much to bet in a favorable case. If every dollar is equally useful to you, the highest expectation is to bet all your money each time. If twice as much money is not twice as good, you don't necessarily want to bet all your money.