Optimal Strategy for Chosing Lottery Tickets You have 2 types of lottery tickets: one that costs $c_1$ and has a probability of winning of $p_1$, and the other costs $c_2$ and has a probability of winning of $p_2$. The goal, as you might expect, is to win by spending the least money. You play until you win it.
Here the optimal strategy is to pick whichever ticket has $c/p$ lower. The reasoning behind is that $c/p$ is the expected cost or each ticket (geometric distribution).
Now consider the following twist: every time you make a pick (1 or 2), two new types of tickets are created, with arbitrary $c$ and $p$.
Intuitively, I would say the optimal strategy remains the same: pick whichever has lower $c/p$. But I fail to find a convincing mathematical argument.
 A: First, your problem is ambiguous, every go, can you only buy the new ticket type or can you buy old or new?
Anyway, this does not matter.
Your conujecture is wrong:
without knowing the distribution of future $c_i$ and $p_i$, it is unclear what the optimal solution is. here is an example
Let $c_1= 100$ $p_1=1$ and $c_2 = 1$ and $p=0.00001$, then according to your strategy,  then you buy ticket 1.
However, if you know two new tickets are created, let one of ticket created will be $c_3 = 1$ $p_3=1$, the optimal strategy might be to buy ticket 2 followed by 3, the maximum cost possible is 2, comparing to 100 if you buy ticket 1 on turn 1.
A problem with similar favour:
https://mathoverflow.net/questions/153006/partially-observable-markov-decision-process-finding-a-hidden-object-with-some
It is very hard to solve this type of problem explicitly. In my example, the optimal policy is unknown though simulation agrees with the conjecture. In your case, i do not think the conjecture is 'not usually' correct because we often want to 'wait for a better deal' if the cost is very very low.
