Talk:Gambler's ruin

Opening comment
This makes absolutely no sense: This article is full of errors. Admittedly it is easier to complain than to fix. One way to view gambler's ruin is that the typical gambler believes that over time, he/she will eventually recover all of his/her money, because of the law of large numbers. However, what is often misunderstood is that the amount of loss can grow very large (and in particular, larger than the gambler's fortune) before the _ratio_ of wins to games goes to the expected limit.

In particular, a typical example of this is the coin flipping example. In this gambling game, one person will win all the money with probability 1 (if both start with some fix amount of money). Intuitively, the person with more money can withstand losses better, and therefore is more likely to end up the winner. The probability of winning is indeed the ratio of his money to the total money. HOWEVER, the present explanation that "Consider players with 90 and 10 pennies respectively, repeating the game 100 times. The player with 90 pennies is expected to win 90 out of 100 complete games, winning 10 pennies each game." is unfortunately wrong. The player with 90 pennies is still expected to win 1/2 the games. The explanation of why he is more likely to win all the money is more complicated.

I'll delete the obviously incorrect parts. Later I might try to add more, unless someone prefers to do it first. 74.114.213.172 20:01, 18 August 2007 (UTC)

-
 * For example, the rare happening of a coin flip to be heads a dozen times in a row is a gambler's ruin, because it's intuitively well outside the odds, but in reality it's just as likely as any other specific combination of heads or tails.

A coin flip being heads a dozen times in a row is highly unlikely. It is not just as likely as any other combination of heads or tails. Half heads and half tails is the most likely combination, according to statistics.

- Anonymous


 * The piece quoted makes perfect sense (though perhaps it should be reworded); you misunderstood it. It is true that a dozen coin flips coming up all heads is highly unlikely. However the sequence HHHHHHTTTTTT is exactly as likely, as is a more reasonable-looking sequence like HHTTTHTHHTHH. However, there are many, many sequences that allow six heads and six tails, and only one that allows a dozen heads, so of course you're right that a half-dozen heads total is far more likely than a dozen heads total, but the quoted piece in question was discussing individual sequences. The point is that it only happens that 1 time in N, but it will still happen eventually and everything else being equal the tendency will be for the player with the lesser bankroll to lose, when such an improbable sequence clears out his bankroll, whereas the player with the greater bankroll can handle the fluctuations better. See? - furrykef (Talk at me) 17:06, 7 Jan 2005 (UTC)

- Cigor:

Casino has:
 * much more pennies than any participant;
 * Yes, but as noted above, this does not help them get ahead in the long run. They also have to be careful with very high rollers, since the ratio of bankrolls is not so overwhelming, so the probability of a very high roller breaking the casino is non trivial.


 * odds that are skewed in their favor;
 * This is where they make their profit. 5% house edge (e.g. US roulette) means 5% gross profit before taxes, overheads, etc. There is no extra gross profit resulting from the bankroll ratios or other factors. Against reasonable blackjack players, their edge can be less than 1%. Against some clueless blackjack players, it might be over 20%.


 * various risk management techniques that limits their maximum loss;
 * As noted above, they have to be cautious with very rich patrons.

The combination of above allow casino to come ahead in the long run. For a illustration see this Gambler's Ruin simulation


 * I don't see how this link supports your argument. --Mike Van Emmerik 22:41, 17 October 2005 (UTC)
 * Mike,
 * I understand and agree with what you are saying, but I am not convinced why my text needs to be deleted. I never claim that high probability of failure is equivalent to negative expected outcome. However, even with unbiased coin and zero expected reward, 99% failure rate is 99% failure rate even with 100 fold reward. It means that the vast majority will fail even in fair circumstances, and I think this is the essence of Gambler's ruin.
 * As for the link, if you scroll to the bottom there is neat applet simulator where you set probabilities, and starting capital and generates one sequence of outcome based on that. I thought this was pretty educational.
 * So unless you give me stronger reasons, I will revert my changes, at least part of it.--Cigor 00:46, 18 October 2005 (UTC)


 * OK, no problems as long as you make it clear that the bankroll ratio does not affect expected return. That's what I perceived, and what I objected to. Oh, and "many more pennies" rather than "much more pennies" please :-) --Mike Van Emmerik 11:43, 18 October 2005 (UTC)
 * How about now? --Cigor 14:31, 18 October 2005 (UTC)


 * OK, good, though you'll see I could not resist a few edits for clarity. I've only just noticed that this casino related discussion possibly should be integrated more with the next section, titled "Casino games". I'll leave that for someone else. --Mike Van Emmerik 21:48, 18 October 2005 (UTC)

Huh?
None of the lead on this piece matches at all what I understand as the Gambler's Ruin, as that term is used mathematically. Did I miss something entirely?

As I understood it, the lead ought to look something like:


 * The Gambler's Ruin is a reasonably complex theory of statistics which, succinctly stated, says that if you gamble long enough, you will always lose, because the the distribution of random numbers cannot be predicted, and therefore losses will eventually outnumber both wins and the chooser's 'bankroll' (whether that be money in an actual game, or a mathematical equivalent).

I don't see anything in there anything like that. Is that not even close, and I'm a mathematical clod? --Baylink 22:06, 23 March 2006 (UTC)


 * I think the treatment here is best:
 * Eric W. Weisstein. "Gambler's Ruin." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GamblersRuin.html
 * So the main point is: even with fair odds (50% chance of either player winning each individual wager), the player with the greatest bankroll has the greatest chance of ruining the other player in the long term, in the hopefully artificial but interesting case of repeated play until one player is ruined. Having a house edge and (usually!) the greatest bankroll merely accelerates this process for the casino.
 * I was never really happy with this article, either. --Mike Van Emmerik 23:19, 23 March 2006 (UTC)

Removing last part of coin-flip section
I'm removing the following text, and moving the external link to the end of the article:

A casino generally has: The combination of the above ensures that the casino will in the vast majority of cases come out ahead in the long run. For an illustration, see this Gambler's Ruin simulation:
 * many more pennies than any player thus ensuring that the player is much more likely than the casino to experience gambler's ruin;
 * odds that favor the casino resulting in negative expected return for the player; and
 * various risk management techniques that limits their maximum loss.

The first point is misleading; as explained just above this text, the difference in capital does not change the casino's expected return, which is what it needs for long-term profit. The second point is out of place, and is explained in the next section. The third point is vague. --Allen 03:36, 3 May 2007 (UTC)

Typo
I think P1 and P2 exprssions should be exchanged. To see this let p/q -> 1. then the expression for P1 and P2 in fair case are reversed

87.8.225.48 (talk) 21:32, 23 November 2007 (UTC)Mao

Negative expected value?
Can someone please help me understand this passage:

It follows that even with equal odds of winning the player that starts with fewer pennies is most likely to fail. However, the winner does not go unscathed, because in the long-run his expected value is negative. This is because throughout the course of the game, he forfeited more coins than did his counterpart and won merely because he had more coins.

Intuitively, what I think this is saying is that if you have a greater endowment than your opponent, there are more paths by which you could lose more money than they do, and yet still win. But if your forfeit more coins you can still have greater expectation to win them back. What is the "long-run expected value"? [style note: long run is unhyphenated as a noun, hyphenated as an adj/adv] Is it expected wealth at any stage of the game? Or is it expected wealth at the end?

For those familiar with recombining binomial trees, the probability of any path is equal for a 50%/50% repeated trial (e.g., fair coin). The probability of the final nodes is given by the binomial theorem/combinatorial (n-choose-k). Even if the game ends before you reach a final node, because one of the players has gone bust, this doesn't change the probabilities of the individual paths, and thus the expectation at each individual step will always be zero [0.5x(+1) + 0.5x(-1)], and N independent steps times zero is zero. If the long-run expected value refers to the probability-weighted-average over the final nodes, then it would have to be positive because there are more paths by which the gambler with the larger endowment can win than there are by which they could lose, and the magnitudes of the wealth at the final nodes are greater for the person with the higher endowment than for the loser.

To visualise, in an N-step game, if "up" is winning and "down" is losing, at the final nodes the top node will have a value of +N (because the person has won N times in a row), and each node will decrease by one until the bottom node has a value of -N (because the person has lost N times in a row). You can draw out a tree and work through the various paths. With disparate starting endowments, and N steps (N greater than the smaller starting endowment of the two players), some paths will not be possible. The person with a higher starting endowment has a lower floor on the maximum amount they could win, which may make it seem like a negative expected value would follow, but there is a complication in that there is kind of barrier knock-out option quality to the binomial tree where if the person with a smaller endowment loses a net number of times equal to the size of their endowment, they are knocked out of the game, and any path that would allow them to win after that point is disqualified. Oy, vey.

To summarise, I think the original statement above is confusing, adds little insight, and distracts from the flow of the article. Best, Eduardo

Expected value negative
I also take objection to the notion that the man with more pennies has a negative expected value in this game. It is impossible for the winner of this game to be "scathed." He has won back any pennies he lost, plus all of his opponent's. I am striking the line. —Preceding unsigned comment added by 75.65.196.71 (talk) 09:15, 31 May 2008 (UTC)

I added a short section on history
I hope people think it improves the article. AaCBrown (talk) 14:57, 16 September 2013 (UTC)

Brownian Motion
2-D Brownian motion can model G's Ruin for N=2. For the N=3 case, you need an additional axis, the z-axis and it is no longer appropriate to call it Brownian motion (which is 2-D) but is instead referred to as a random walk in 3 dimensional space. Can someone fix this?

Lack of mathematical explanation
It's fine to just give the formulas and show how to use them with examples but there is the beginning of proof showing how to derive the formulas for unfair coins. This looks really promising but it suddenly ends with - look up the rest in a book! 79.75.159.115 (talk) 13:48, 22 August 2018 (UTC)


 * Not suddenly. This is a general encyclopedia, not a math textbook. Solution of a linear homogeneous recurrence relation is outside the scope of this article, but is explained in "Recurrence relation and Linear difference equation". Boris Tsirelson (talk) 17:10, 22 August 2018 (UTC)

But isn't our life a constant gambling?..
A constant gambler's ruin, will be inevitable.

But without the game, there'll be nothing.

— 154.48.253.21 (talk) 02:56, 11 February 2020 (UTC)

On the General Dogmatism on Wikipedia

Why cannot this place be a forum for relevant discussions?

Does freedom of speech really bother you?..

Additionally, wouldn't these discussions in turn help improving the article? (either directly or indirectly)

— Wikipedian Right (talk) 13:04, 16 February 2020 (UTC)
 * Wikipedia is not a place to publish your own thoughts and analyses WP:NOTFORUM. And, this has absolutely nothing to do with freedom of speech. O3000 (talk) 13:10, 16 February 2020 (UTC)

N-dimensional Brownian motion
"For N = 2 and large initial capitals x₁, x₂ the solution can be well approximated by using two-dimensional Brownian motion. (For N ≥ 3 this is not possible.)"

This is completely bogus. First of all, with N players the question is equivalent to a random walk in an N-1-dimensional simplex (see the quoted reference to Ferguson). So it is approximated by 1-dimensional Brownian motion. Second, N-dimensional Brownian motion is a perfectly valid notion — contrary to what an anonymous claimed above. --Ilya-zz (talk) 08:41, 3 April 2021 (UTC)