My question here is... is there any tangible difference between:
10 bets at y for 10% (10x)
100 bets at 1/10y for 10% (10x)?
Well, let's look at a simpler case:
Is there any tangible difference between:
a) betting 1 BTC at 50% (1.98x)
b) betting 0.5 BTC at 50% (1.98x) twice
Both risk 1 BTC, so both have an expected profit of -0.01%.
Let's look at the possible outcomes:
a) has 50% chance of profit 0.98 BTC and 50% chance of profit -1 BTC
b) has 25% chance of profit 0.98 BTC, 25% chance of profit -1 BTC, and 50% chance of profit -0.01
So the maximum and minimum profits are the same, but b) (betting a smaller amount, more times) fills in the middle. The extreme outcomes become less likely the more you subdivide your bet, and the expected outcome becomes more likely. In the extreme, if you bet a single satoshi 100 million times (ignoring rounding), you will be very likely to have a profit very close to the expected profit of -0.01 BTC
For the players, their only hope of winning is to beat the expectation, and so their best strategy (assuming they have a fixed amount to bet, and have to bet at a certain chance of winning) is to make a single large bet.
We could plot the outcomes against the chance of them happening. Both charts would have the same "center of mass" (ie. the expected profit, -1%), but the chart where the bets were divided up more would be more pointy, with the outcomes concentrated around the expected -1% point.
Edit: I missed the end of your question:
Such that let's say it were capped at 100x. Doing this:
If (win) bet wonAmount at 100x
if (lose) returntoBase
Would be the same as just betting straight at 10,000x? Or is there something I'm missing here? In #1, you'd have a 1%*1% chance, or 0.01% = 1/10,000 chance. In the second, you'd still have a 1/10,000 chance.
I think the player is at a disadvantage if you limit them to 100x. They effectively pay 1% of everything they stake.
If they can bet 1 BTC at 10,000x, their expected profit is -0.01 BTC.
But if they can only bet at 100x, they have to bet 1 BTC at 100x, then 100 BTC at 100x. That second bet has an expected profit of -1% of 100 BTC, ie. -1 BTC.
The strategy "bet 1 BTC at 100x, and if I win, bet 100 BTC at 100x" has an expected profit of -0.01 BTC for the first bet, and -0.0099 for the second bet (0.99% chance of having to make the 2nd bet, and an expect profit of -1 BTC if you have to make it, for an overall expectation of -0.0099 BTC), for a total of -0.0199 BTC. That's very nearly twice as bad for the player as if you let them make a single bet.