Monday, January 29, 2018

Las Vegas, the Gambler's Ruin, and Index Funds

Las Vegas, the Gambler's Ruin, and Index Funds
Earlier this month, I went to Vegas for my buddy's bachelor party. I had a great time, even if that city was a lot more appealing to 20-something me than it is to 30-something me. Still, I love gambling. It combines two of my favorite things: games, and stakes. Things matter when you have skin in the game.

Don't believe me? Try playing poker without money sometime.

I came home with more money than I left with, somehow. And the bachelor was the big winner: I has a seat next to him Saturday night (well, Sunday morning) as he went on a fantastic run at a blackjack table. As we stumbled home, he was up over five hundred bucks.

Still, thanks to some unbelievably cold dealers and, of course, our stupidity in adding an Andy Reid-led football team into every possible tease and parlay, we ended up losing money as a group. It would be weird if we didn't: Vegas is built on the idea that the house comes out ahead in the end.

But if a lot of the games give odds that approach a 50/50 proposition, why is that? If blackjack or craps have odds that favor the house by only a percent or two, why don't 48% of gamblers come home as winners?

The reason why comes from one of my favorite terms in mathematics: the Gambler's Ruin.

The Gambler's Ruin states that even in a game that is truly a 50/50 split, say, gambling on a coin toss, one gambler is going to eventually lose all his money, if they keep playing long enough. Given enough random outcomes, one side will eventually have a very unlucky streak that costs him all his money. And with no more cash to bet, the gambling is over.

Given the reality of limited funds, the Gambler's Ruin posits that the party with less money has a greater chance going bankrupt: he is less able to survive a bad run. Once he's out of money he has to stop...and right at the worst time, too.

This is a big reason why the casino wins much more than we do: they have far more money on hand than any single gambler. The fact that the odds on any single game leans towards the house just accelerates the likely outcome. But their overwhelming liquidity advantage is why the casino comes out ahead: they won't run out of chips, but you sure can.

So what's an honest gambler to do? The good folks at the University of Massachusetts Amherst Math & Statistics department have some advice:
"As we have seen in previous lectures, in many such games the odds of winning are very close to 1 with p typically around .49. Using our second order difference equations we will show that even though the odds are only very slightly in favor of the casino, this is enough to ensure that in the long run, the casino will make lots of money and the gambler not so much. We also investigate what is the better strategy for a gambler, play small amounts of money (be cautious) or play big amounts of money (be bold). We shall see that being bold is the better strategy if odds are not in your favor (i.e. in casino), while if the odds are in your favor the better strategy is to play small amounts of money"
The math in the paper is well above my head: I haven't solved a math problem or touched a graphing calculator in twenty years. But I love their conclusions. If the odds are not in your favor (and in a casino, they never are), the last thing you want is to place a lot of minimum bets and stretch the game out. Instead, you're better off being bold, both with the size of your bet and the number of hands you play.

If you have $300 and are trying to come home with $500, the best thing to do is to bet $200 on a single hand. If you lose that hand, your best option is to bet your remaining hundred and hope to double up a few times in a row.

The rub is that you're going to have a lot of time to burn for the rest of the weekend after that single minute of gambling. But I appreciate the concept: be bold, not timid. Don't bet the table minimum and try to string the game out.

My buddy who's about to get married employed part of this strategy: betting well above the table minimum, doubling down and splitting to get more of his money on the table, especially if the dealer showed a weak card. In retrospect, I'm sure that's at least part of the reason he was able to capitalize on his good run. He wasn't timid.

The flip side of this betting strategy is that, sometimes, the odds actually are in our favor. While you have a negative expected return at the blackjack table, thankfully we do not spend our entire lives in a casino.

Take the stock market, for example, where we have a positive expected return.

When investing, Mrs. Done by Forty and I want to take the opposite strategy than that of a casino gambler. We don't want to gamble our funds quickly, trying to double our nest egg by the end of the day. We want to stretch the game out, perhaps for decades.

Rather than risking our funds on one bet (say, having our portfolio rest on a single stock, hoping for a ten bagger, or putting everything we have into a cryptocurrency), we want to diversify: we aim for a lot of small bets, spread around.

And sure, investing is not gambling. A decades-long plan for retirement is not a weekend in Vegas.

But when investing for our retirement, maybe we're the house.


*Photo is from Thomas Leuthard at Flickr Creative Commons.

26 comments:

  1. I played poker once with a group of coworkers. I started with $20, lost it all and never made anything. $20 is what I set my limit to.

    I don't really like gambling, especially if it's truly random-win gambling. I prefer their to be some skill involved.

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    1. I prefer poker, too. Like Mike McDermot says, it's a skill game, Jo.

      However, Rounders touched on this same principle of the Gambler's Ruin: "in poker, the size of your stack is as important as the quality of your cards." If the players are roughly equally skilled, the player with the most money has an inherent advantage.

      The flip side is that if you're far less skilled than your opponents, you want to go all-in a lot.

      If you're the shark at the table, grind it out.

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  2. Love this concept. If you're feeling like the underdog, go big. If the scales are tipped in your favor, even slightly, don't let it go to your head and risk everything. Good rules to help keep a level head when you need it, and be bold when it can really pay off!

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    1. That's a good way to summarize it, Emily: wish I'd said it that succinctly.

      I'm trying to think of other areas where I can apply the gambler's ruin: when to swing big (aggressive in your career & investments when young) and when to let your advantages nudge you towards being conservative (making sure our insurance is up to snuff, maybe getting rid of the scooters)...

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    2. I don't think the gambling rule says to be bold when it can really pay off. It says to be bold when the odds are stacked against you, and pray for luck.

      In life and investing in individual stocks, we want to be bold when the odds are in our favor.

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    3. Hmm. I think the UMass paper says to be conservative when the odds are in your favor, and bold when they are not.

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  3. I am not a gambler by nature, so making big bets would be against my natural inclination. More incentive for me to skip all the gambling altogether and go get some food!

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    1. Hey Laurie.

      Agreed, the food in Las Vegas probably gives a better ROI than the average bet. Still, I was surprised how expensive the food was there. While I came home up on my bets, I was down a whole lot on the food & drink.

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  4. I always string out the bets when I play blackjack. Then again, I look at gambling as entertainment -- going in with the assumption that the house will eventually win -- so I want to stretch it out for as long as possible before I lose. If I happen to come out ahead, so much the better. On the other hand, I also usually stick to slots, so you can see just how little winning strategy I have.

    But if I do go back to blackjack again, I'll definitely reconsider the minimum bet!

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    1. Yes! I should have mentioned that the main purpose of the gambling seems to be to have fun. And you're not having a lot of fun placing one bet the whole weekend.

      Still, if you goal is to maximize your chances of winning, it seems like the aggressive, big, single bet is you best shot in blackjack.

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  5. I really like this post! I had never heard of "The Gambler's Ruin" and I'm a math major. Good stuff. It seems to me there ought to be (if there is not already) some research done on combining this Theorem with Game Theory in order to create actionable negotiation and/or strategic rationale.

    At the very least, it beefs up the very real belief that "You have to have money to make money". Increasingly I'm finding that to be the case (real estate is a prime example). It's significantly easier to make gobs of money in real estate, if you have gobs of money to start with.

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    1. Forgot to include another example that happens in our household quite often. We turn down a number of startup job offers that have potentially huge payouts (usually requiring household moves and school disruptions). If we were much more wealthy, we wouldn't care so much about the disruptions and would take on some of these options, willy nilly, and just see which ones pay out, knowing that some would and some wouldn't. Net net, by having more wealth built up, we would be able to increase our wealth at a faster rate than we are presently.

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    2. Hi Tin! I didn't realize you'd majored in math: very cool.

      As you noted, you need money to make money: you ability to remain liquid is a huge consideration in the types of gambles you're willing to make (or at least it ought to be).

      As you noted, there are a lot of opportunities that families don't pursue because of the immediate costs involved. I suspect these limitations are a LOT more restrictive on poorer families: can they really pick up and move to a new city, even if the job available would be great for their long term future?

      One reason that our social safety nets (i.e. - unemployment, welfare, etc.) ought to cover costs of moving in order to help address the immediate liquidity concerns.

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  6. Gambler's ruin works to explain why an *individual* would lose in the long run even in a 50-50 game. It however doesn't explain why a *casino* would win in the long run ;-)

    That's because when you lose your money and get up, another player takes your place. From the casino's POV, they're not playing against an individual player with a finite amount of money; they're playing against an infinite number of players with an infinite supply of money.

    Since each coin toss has the same chance of landing on heads or tails, in the long run (longer than an individual's run, but well within a casino's run), the casino will lose just as many times as they win. It may not be to you, but it'll be to *someone*.

    When you move back from the individual perspective and realize the casino isn't playing against *you* but against an infinite pool of money, you see that the skewed odds don't make the casino win "faster" -- it's literally the only reason they win.

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    1. It's an interesting way of framing it, but I don't know that I agree with your analysis. There are multiple players, but each one is certainly subject to the gambler's ruin.

      The time that each individual player is compelled to quit is likely to be to his detriment and, even when aggregated, drastically in the house's favor.

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    2. I agree that each player is individually subject to the gambler's ruin ... that partly explains why individual players lose. But it doesn't explain why the casino wins, when playing against what (to them) is indistinguishable from an infinite supply of cash.

      In a game that is truly 50-50 (tossing pennies, for instance, but not blackjack), your bad run that causes you to lose and quit doesn't benefit the casino at all ... because from the casino's point of view, the next person to take your place is indistinguishable from you!

      To the casino, every toss of the coin against whoever the opponent is, is part of one long session, and it makes no difference whatsoever if it's against *you* or *me* or anyone else.

      If the odds are truly 50-50, it'll even out, *for them* (though probably not for the individual players they're playing against).

      So *players* lose for a variety of reasons, including their limited bankroll (the Gambler's Ruin), a bad run of cards, etc.

      But the limited bankroll is cancelled out by the infinite bankroll of all the punters taken in aggregate, and your bad run is evened out by other players' good runs.

      So the only way left for the *casino* to win is the skewed odds. Even though in blackjack it's only 0.5% against a player playing optimally, it apparently still adds up :-)

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    3. I hear what you're saying, Pete, but disagree because of one point: when losses or gains are locked in.

      An individual players losses (or gains) are locked in when they leave the table. Due to the gambler's ruin, there's going to be a lot of times the player is locking in losses when he runs out of funds. Locking in gains happens when the player walks out as well, but this is also subject to the liquidity issue noted in the gambler's ruin.

      I do like the framing of seeing gamblers as an infinitely recycling group with an infinite amount of money. Still, they're still a group of individual actors that are playing with a distinct disadvantage even in a true 50/50 game: the disadvantage is seen in the ratio of times the players bust and leave vs. the times the players are up and walk away.

      That's why I don't agree with the conclusion that the only real advantage the casino has is found in the odds of the game itself.

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    4. Of course, this isn't my argument, as I am no mathematician. I'm just summarizing from one of the sources I linked to:

      "Therefore, the player starting out with the smallest number of pennies has the greatest chance of going bankrupt. Even with equal odds, the longer you gamble, the greater the chance that the player starting out with the most pennies wins. Since casinos have more pennies than their individual patrons, this principle allows casinos to always come out ahead in the long run. And the common practice of playing games with odds skewed in favor of the house makes this outcome just that much quicker."

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  7. Very interesting. So basically, we'll win with time in the market. I'm not a gambler so I prefer the odd to be highly in my favor.

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    1. Well put, Joe. Yes, you've got the odds in your favor so your 'betting' should be the opposite of a blackjack players: lots of bets instead of one big one, and a long, long time at the table.

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  8. Now it makes sense why I've always lost when I gambled cautiously :)

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    1. Hey Revanche!

      The gambler's ruin certainly doesn't help. Though we're also not really helped by the inherent lean of the games towards the house. Both of these things point probably to not gambling at all but, if we want to, aggressively.

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  9. Wish I'd had this analysis when I went to Vegas many years ago! I loved playing the cheapest tables, trying to drag out my 'entertainment expense' of gambling...

    Had also never heard of the Gamblers Ruin - I thought this was going to be about that strategy where you continue to double down on every bet you lose, so eventually you get your money back - but there is a very scary, small probability that you'll lose everything you own...

    Cheers,

    Frankie

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    1. Hey Fully Franked Finance,

      Honestly, I think the repeated double down strategy is probably the 'right one' if your goal is to get a really big sum of money (i.e. - I have $100 and want to get to $2,000). It's not like you have good odds, but that sort of strategy at least gives you better odds than trying to get there one $5 bet at a time.

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  10. Texas hold 'em used to be my game back in the day. I'm not a big gambler since the odds are usually against you in slots, blackjack, sports betting, etc. However, I felt like in hold 'em I was more playing against my opponents instead of the house. My college beer money was made playing online poker. There is a lot of analytical skill in poker, which is why I think I liked it so much. Thanks for sharing your story.

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    1. I prefer poker to almost any game in a casino as well. Though the gambler's ruin still applies there, if you can make an assessment of your skill vs. that of your opponents.

      If you're the better poker player, the angle is likely to grind it out.

      If you're overmatched, then quick, aggressive play, with big-but-fewer bets/hands played (with you having the good sense to walk away when you're up) is probably the right move.

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