Speculative Bubble game

Can someone tell me what I missing as far as the probability of this game? I swear I'm good at probabilities, but I'm missing something basic here.

The game:
1) A base price of 'x' is predetermined.
2) A player buys in to the game for x.
3) Every time someone buys in, there is a .05 chance of a "crash". If there is a crash, the player who just bought-in loses their entire investment, and the cost to buy-in resets to x.
4) If a buy-in does not cause a crash, the cost for the next player to buy-in increases by x1.4 (40% increase).
5) When a player buys-in after you, you are paid (your buy-in x 1.35), for a 35% profit.

Observations:
1) It's obvious for me to see where the house makes its money here, as it keeps the entire first buy-in, and 5% of every future buy-in (actually 5/140, so about 3.57%) of every round. (Arbitrarily calling the time between the first buy-in and crash a 'round')
2) Obviously, everyone makes a little bit of money at the expense of the poor guy at the end who lost more than everyone else won, combined.

However, my basic probability tells me that your individual EV for this game is (.95 x .35y) + (.05 x -1y) = +.2835y where y is your buy-in. Obviously, this game can't both be a 28% player edge and have the house make money.

Also, there would have to be contingencies for if the buy-in gets so high that no one wants to be the next buy-in. I'm not sure how the rules handle that, but even saying theoretically that there will always be a buyer until a crash, where is the hole in my logic that I'm missing? Thanks!

This sounds like the compensation model for a personal direct marketing business :)

Quote: PITdood

However, my basic probability tells me that your individual EV for this game is (.95 x .35y) + (.05 x -1y) = +.2835y where y is your buy-in. Obviously, this game can't both be a 28% player edge and have the house make money.

Your calculation is only valid if there is a next guy standing in line for the entry after you. However y explodes quite rapidly. The entry of the prize doubles every 3 players. If x is $1, after 30 player the prize is ~$1k. And after 100 players it's more than a billion.....

Yeah there would be a pretty negative downside of no one buying in after you. But for argument's sake, say there will be as many buy-ins as necessary until a crash. At the point of crash, the house has always made money (as far as I can tell). I guess my question is, if you have the bankroll to buy-in, and you could be, say, 99% someone will fund after you, is there a mathematical reason to not buy-in? Or is the only negative the chance that there's no-one to follow you? Is this just the Ponzi/Pyramid scheme of games?

It sounds more like a good ponzi scheme.

[deleted - all my math was wrong]

Quote: PITdood

But for argument's sake, say there will be as many buy-ins as necessary until a crash.

Okay, that's a valid assumption to make, at least for academic purpose. You identified a paradox where all players make pure +EV bets, and yet the house is in the money.

The first naive question should be: what is the average length of the game (p=0.95) ?
Calculation would be <n> = sum_n=0^inf n * p^n = p / (1-p)^2 = 380.
Actually we don't much care about this (or any) numbers. Since it is finite, it doesn't help us with the paradox where the money is coming from.

Next, let's compute the average stakes of all players (return r=1.4). For average 380 rounds we could expect it to be large....but how large ? No, it's not r^<n>, it is
<total stake> = sum_n=0^inf r^n p^n = 1 / (1-r*p) for r*p < 1, but it is infinity if r*p > 1.

Since r*p = 1.4 * 0.95 = 1.33 > 1, the average total money put in by all players playing the game is beyond all limits.

If the return of the game would be r*p < 1 then the players money would stay finite - but it would no longer be a +EV bet for any player.

I hope this solves clarifies the paradoxon.

Ah, very nice. Thanks.