Doubling-on-coin-flip paradox

[Pds314]

Suppose that you given a $1 and flip a coin, if heads, you double it and flip again, if tails, you keep it and nothing else happens.

How much, on average, would this opportunity be worth? In other words, how much, on average, would you make.

Hint: the answer is most-certainly not $2.

[dharak1]

Double as in double the actual 1$ coin and flip two or just double how much you win?

[Pds314]

Double the amount won.

E.g. 100% chance of winning at least $1.

50% chance of winning at least $2.

25% chance of winning at least $4.

12.5% chance of winning at least $8.

100/2^N% chance of winning at least $2^N.

Edited November 12, 2013 by Pds314

[K^2]

Suppose that you given a $1 and flip a coin, if heads, you double it and flip again, if tails, you keep it and nothing else happens.

How much, on average, would this opportunity be worth? In other words, how much, on average, would you make.

Hint: the answer is most-certainly not $2.

My wife was given this as a problem in a probability class. The way it was stated was, "a) Find expectation of the winnings, Should you play this game if it costs you $100 to play?"

I was a bit surprised, to be honest, it kind of rocked everything I thought I understood about statistics. Namely, that an "average" value might be absolutely and totally useless. Basically, yeah, you win on average, given enough time, but unless the opportunity cost is very small, say $10, you just can't physically play enough games to get that infinite tail end on your side.

Ever since I learned of this possibility, I've been wondering if it's possible to play this the other way. In every game of luck in the Casino, house always wins. No matter what strategy you use to bet in, say, roulette, the expectation value is always going to be in favor of the house. In fact, regardless of betting strategy, in a game of luck the expectation value cannot change! But this problem shows that it doesn't necessarily matter. Could there be a betting strategy which, while letting Casino win on average, still lets you get the money over any reasonably finite number of games? It seems that some betting strategies do work on this principle, but they still have rather short fluctuations. So while you can use them to get a bit of extra money in short term, you'll never make any significant amount.

I hope I didn't spoil too much for people still trying to solve it.

[Depth]

http://en.wikipedia.org/wiki/Martingale_%28betting_system%29

This is a good way to get banned from many online casinos and will get you thrown out from many real life ones.

[neamerjell]

This actually sounds a little bit like table top wargaming mechanics:

You roll to see if you hit your target, then if you did hit it, you roll to see how much damage you did. If you failed the to-hit roll, then the damage roll is skipped.

[K^2]

http://en.wikipedia.org/wiki/Martingale_%28betting_system%29

This is a good way to get banned from many online casinos and will get you thrown out from many real life ones.

I don't see why. It's one of the strategies that clearly fails for reasons stated in the article and ones I've mentioned in my post. There are much more clever variations on this that allow you to play at lower risk given table limits, etc, and they still fail in the real world.

[konokono]

The expected payoff is infinite. But if there is even an infinitessimal opportunity cost of time (for every $1 you can earn $(1+r) in the time it takes to make a flip, r>0), then the expected present value becomes finite.

[andrewas]

http://en.wikipedia.org/wiki/Martingale_%28betting_system%29

This is a good way to get banned from many online casinos and will get you thrown out from many real life ones.

I don't see why. The classic martingale, doubling up on losses so that every win brings you back into the black, is a great strategy for casinos - the martingale gambler often comes out slightly ahead, but the day he doesn't will cost him very heavily. The anti-martingale strategy in the OP is also great for casinos since it encourages gamblers to make bigger, more reckless bets. In a game of pure chance, the casino shouldn't care what system you use, in the long run they have their edge and will make lots of money. In truly random games like roulette, all systems are great for casinos because they all encourage the gambler to risk more money, since the gambler 'knows' he will make it back in the end.

Now, games like blackjack only appear to be truly random, after a few hands the player has enough information to win in the long run, which is why the casinos will throw you out if they catch you counting cards.

[Bacterius]

The martingale only works if you have unlimited money. For any finite amount of cash it is a no-win system which gives you no edge in the long run. Use it all you want - you will eventually lose everything you won to that point.

[KerikBalm]

This is why median values are often worth talking about more than mean values.

If you look at the median income, vs mean income of certain countries... you'll see some interesting things

[K^2]

For any finite amount of cash it is a no-win system which gives you no edge in the long run.

The question is how long is the "long run". For martingale specifically, it's not long enough. If you happen to have 2ⁿ minimum bets worth of money, you can absorb n-1 losses. On n-th, game is over. In other words, your expectation for a single round of martingale is winning 1 minimum with probability of 1-2^1-n, or losing it all with probability 2^1-n. Situation changes if you manage to double your starting capital, requiring 2ⁿ consecutive wins. In other words, the odds of you successfully doubling your capital are 1 - (1-2^1-n)ⁿ. For large enough n, that expression goes to zero. So a way of making money it is not.

On the other hand, say you have a million dollars in your savings. And say all you want to do is win enough money to get yourself some drinks. You can pretty much do this whenever you want. Yes, there is a tiny chance that you'll lose your savings, but it's more likely that you die on the way to casino, so it's a reasonable risk. Yeah, collecting interest is more efficient with this sum of money, but we are talking about a principle.

The above basically works for the same reason that the game in OP's post doesn't. Outcome of martingale is a loss in the long term, but it keeps winning in short, despite the game being weighted against you. But what's interesting is that while martingale brakes down pretty fast, given limited resources, the game in OP's post takes an absurdly long time to pay off if you have to pay each time you play. So the question is whether this behavior is something that can depend on the betting tactic. Can there be a betting strategy that allows you to keep winning for a longer time on a significantly smaller budget?

The answer is probably no, but I might be missing something. The idea is that game's average winnings cannot depend on the betting strategy. So in order for you to win small amounts in many games, the strategy has to allow for you to lose all that money in one go. And that means the game will be budget-limited, and you can't really deviate from martingale in any significant way.

[Mr Shifty]

I went on a weekend bender in Reno during college. We used simple Martingale (double after every loss) and played craps. We played the field (2,3,4,9,10,11, or 12 to win), which loses about 56% of the time, but often pays out double on 2 and triple on 12, which reduces the house advantage to about 2.8%. Key to the field bet is that it 1) has nearly even odds and 2) plays on every roll of the dice (unlike most craps bets). We didn't have a large cash balance (about $150 pooled between us to start), so we were playing $2 minimum bet tables and playing the minimum bet. With our initial stake we could lose 6 times in a row without busting (probability 3%). We played this way for 2 hours and built up over $500 of winnings. Then we got greedy and moved to $3 and $4 bets and promptly lost the entire stake on a string of losing rolls when we couldn't double after a loss. We did continue to pursue the strategy with our remaining funds over the course of 6-8 hours, and eventually walked away with about $30 total in winnings over our initial stake (after removing dealer tips), which paid for gas for the trip.

The problem is the even with a larger stake, there are maximum bet limits to prevent exactly this strategy from working. Typically these are $1000 for low minimum bet tables ($2 and $3). If you start with a $2 bet, you're only winning $2 for each winning roll. (Start with $2 and lose. Then bet $4 and win, you've bet $6 and won $8, for a $2 gain.) If you can bet a maximum of $1000, you won't be able to recover your stake if you lose 9 times in a row, which happens about 1 out of 200 runs. Since you're only winning $2 for each run, it's pretty easy to rack up more than 200 runs in a night.

If you're just out to have a good time, though, play the longer bets: pass or come are close to even bets (1.4% house advantage per bet; 0.4% per roll.) You can still pursue martingale, but it'll take longer and you can enjoy the free drinks and rowdy atmosphere and even come away with a few dollars.

Edited November 13, 2013 by Mr Shifty

[3_bit]

http://en.wikipedia.org/wiki/Martingale_%28betting_system%29

This is a good way to get banned from many online casinos and will get you thrown out from many real life ones.

I actually thought of this myself. Start off by taking out a ridiculous sized loan with little to no interest for the first few days, then do what the link stated. It's quite an effective method if you're starting out with small wagers, and no doubt could get you millions if you do it right.