How to gamble if you must

Our title comes from a famous paper by Lester Dubbins and Leonard Savage which appeared at the beginning of the Bayesian Theoretical Resurgence, a movement which, I am delighted to report, has now largely infiltrated nearly all of academia.

Import as this movement was, it was slight in gambling because analysis of games of chance were always Bayesian. Prove this by picking up any undergraduate text in statistics and find the chapter on probability (always, bizarrely, hidden in the interior whereas it should come first). You will see examples like this: “A die has probability 1/6 of showing a 6, therefore the probability of two die (or one die thrown twice) showing two 6s is 1/6 * 1/6 = 36.”

Valid answer, but an invalid, or at least incomplete, argument. Close enough, however, because of the implicit recognition that (Bayesian) logic allowed us to deduce the probability of the die showing 1/6. Once that deduction is in hand, we can prove theories of what will happen to that die under scenarios, which come in two flavors.

Simple gambles are those where the physics of the game allow us to deduce the probabilities of events of interest. Examples: “two 6s on two throws”, “a pair of jacks showing from a poker deck”, “00 showing on a roulette wheel”, “three cherries showing on a slot machine”, and so on. Casinos provide simple gambles.

Complex gambles are different from simple ones because we cannot deduce the probability of the events of interest. For example, “The person to my left in this poker game holds a hand superior to mine”, “horse A will win the race or at least come in third”, “stock B will increase in price over the next week”, or “the Detroit Tigers will win tomorrow’s game.”

Unless you are the owner of a casino or a bookie, it is impossible to consistently make money with simple gambles. You might, but probably will not, consistently make money with complex gambles. Here’s why.

Any casino game that does not involve the intelligences of other human beings can be analyzed as simple gambles. This means we can, without error, compute the probability of any outcome of any game, which we can call A, for example A = “the roulette wheel shows red”. We will always know, given the properties and setup of the game, the probability A will be true. For ease, call that probability P(A), which I emphasize we know.

It costs you D dollars to bet on A, and you will win with probability P(A) and will be paid W dollars if A happens. The casino sets the required bet D so that it is more than W * P(A) (alternatively, W is set less than 1/P(A) for every dollar bet). They do this on all simple gambles.

For example, if A = “7 shows on an American roulette wheel” then we can deduce that P(A) = 1/38 (the numbers 0-36 and the symbol “00” are on the wheel). It costs (say) 1 Dollar to play. If you win, you receive 35 Dollars. In this case, W * P(A) = 35/38 which is less than the 1 Dollar it costs to play. Roughly, the casino takes in 8 cents for every dollar bet, meaning you lose 8 cents.1

Meaning you will go hungry if you make gambling on these games a career. The only exception to simple gambles is blackjack, where strategies exist so that D < W * P(A)---you can make money. But because casinos have more money than you, and politicians desire to have that money, casinos are able to buy laws that make these strategies illegal. Just as you go to Walmart to purposely part with your cash, you are meant to go to a casino to lose money.

Money can be made with complex gambles, but it isn’t easy. In simple gambles, everybody has the same information about P(A). This isn’t true in complex gambles where to win, you need to have better information about P(A) than the person or persons betting against you. Those cigarette-wielding guys huddled around the OTB entrance aren’t just dosing themselves with nicotine. They’re trying to gain an advantage in information by subtle probes of their compatriots. Brokerages ponder quarterly reports for the same reason.

Problem is, everybody else is trying to gain an edge the same time you are, which usually means your information is not much better than the next person’s. Plus, in betting horse races and the stock market, there are transaction fees. Tracks skim a percent off the top and set the payouts by the amounts bet, making it extremely difficult to win money. Brokers and banks charge transaction fees which cause the same difficulties.

Besides bar bets and cheating, the only gamble with potential is poker (and its business equivalents, negotiating) which depends on bluffing and the ability to detect it. Being able to read tics and tells is a great skill; but it’s a rare talent and expensive to acquire.

The lesson is: stay away from simple gambles and only take bets where you are sure of your information.

1This figure is “on average” and is, therefore, metaphysical. But we can calculate the exact probabilities of winning for the casino given an assumed number of bets and amounts; the probability the casino wins is nearly 1.


  1. Colinx

    Dr. Briggs,
    Could you please explain the connection/difference between the sample average and the formula W * P(A)?
    Thank you.

  2. Reading tells in poker is certainly useful, but I think it’s equally, if not more important to influence opponents’ perceptions of P(A). And it’s more subtle than bluffing. It’s the ability to employ a flexible strategy to misdirect your opponents.

    Of course, in poker this can consist in both the actual plays made (i.e., playing the same hands in different ways) and in other behavior, like cultivating ‘tells’ and even telling the truth occasionally.

    I’m sure your omission of Baccarat was unintentional.

  3. DAV

    It is possible to make a consistent profit in horse racing but making enough to live on is problematic. One can only lay so many bets a day and the amount per bet has limitations. The trick is to know how good you are and to wait for bets which are potentially profitable. IOW: pay more than break even. This could be a long time if handicapping skills are no better than guessing. At the track and for win bets at least, the payoff after the establishment ‘tax’ is known.

    Stock market purchases and horse bets are quite similar. The difference comes from the nature of the betting field. Most don’t realize that in horse racing one is betting against all of the other bettors and not the track. The largest payoff in horse racing comes from a winning bet that is unpopular (in track parlance, an underlay). In the stock market the largest payoffs usually come from popularity (for long gambles at least). ‘Pump and dump’ schemes don’t work at the track.

    Note that financial institutions, insurance companies and retirement plans all gamble in stocks. I use ‘stock’ to mean all endeavors which rely on price gambling. Stock gambles are easier than horse gambles because the price of a stock is largely irrelevant to its underlying value. One is betting only on human perception.

  4. Doug M

    Your ability to make money is a function of the quality of your information (in excess of transaction costs), your diversification (the number of simultaneous bets you can active at any given time), and your turnover (the interval between placing your bet receiving your payout and placing a new wager).

    As a rule, never wager more money than the quality of your edge. For example, suppose you are betting on the flips of an unbalanced coin with a probability of coming up head 51% of the time. To maximize the growth rate of your endowment, wager 2% of your “nut” on each flip. (2% is E[x]) If you bet less, you will increase the stability of that growth rate. Wager more and you are on the road to ruin.

    Casino games have low diversification but very high turnover. You can only play one hand of poker in a card room at a time. You can play 4 if you play on line. You can play 5 hands of blackjack, but all are dependent on the dealer’s cards. However, you can play 50 hands in an hour. Your edge comes from card counting, shuffle tracking, card manipulation, and “card sense.”

    Sports betting (including the track) has high diversification and high turnover. There are multiple games or races going on at any given time. Each game is independent of the others going on at the time. A game lasts a few hours, and a horse race lasts a few minutes. Your edge may come from analysis of team statistics or the racing form. It may come from a tip from the trainer (not illegal). There may be some “behavioral statistics” to mine, i.e. Atlantic City bookies offer worse odds on New York teams than Vegas bookies.

    Financial markets have moderate diversification and low turnover. There are thousands of securities and derivatives on securities that even an amateur investor has available to him. However, unless you can “go short” all are dependant on economic factors and the supply of money. Diversification has a limit. The holding period of any investment usually ranges from a few days to many years. Unlike casino games or sports betting, there is a reward to the passive investor. As there are always more people who need money than have money, over the long term, the passive investor will be rewarded. The long term, however may take 30 years.

  5. Alan D. McIntire

    Up until the current administration, I always thought of the stock market as a casino rigged in favor of the customer.

  6. I’m no expert but it strikes me that winning at poker involves manipulation of the amount bet, as well as knowing the odds, tells, etc. Manipulating the bet (wide swings in the amount wagered) is the thing casinos hate and legislate against (and will throw you out if you do it). But it is fair play in poker to bet up your good hands, and even make “all in” wagers.

    btw, in our friendly nickel-dime games, my pals and I have maximum bet rules ($1 limit, three-bump limit). Otherwise serious money would change hands and nobody really wants that.

  7. Geckko

    One area that is interesting and proven very difficult for economists to explan behaviour, is gambling with large payoffs at long odds (e.g. dropiing $10 on a lottery). It appears incinsistent with their behaviour when dealing with smaller payoffs at shorter odds (flipping a coin for $10).

    Personally, I appreciate the paradox. I know that buying a lottery ticket is essentially a stupid thing to do based on expected payoff (there are much cheaper ways out there to lose money). But I do it just the same form time to time.

    The reason is in the qualititative rather than quantitative and I rationalise it as follows:

    In the course of my existance I am subjected to unavoidable, but highly improbable, events which extremely high losses associated with them. Getting hit by a piece of falling space debris or a metiorite – a massively negative life changing event. So why not give up a small relative portion of my welfare (say $10 a few times per year) to get myself exposed to the complete opposite – a massively positive life changing event. I can’t really opt out of the adverse, but can leave myself opted out of the latter.

    I know, it still won’t pass any test of rational behaviour, but there you go.

  8. DAV

    Doug M, that’s the essence of the Kelly system. Outside of what Kelly’s paper describes it’s difficult to prove although it sounds intuitive. The unfortunate nature of the Kelly system (if faithfully followed) is that it requires placing bets offering a negative return. I tend to place uniform bets but ratio them when placing multiple bets.

    A problem arises from defining the ‘edge’. It’s easier in casino games because the probability of winning is defined. It’s much harder in horse and sports betting where the probability of winning is unknown. I substitute my long term success average. It’s necessary to know what the other bettors are doing as well , not only to estimate expected payoff but the crowd as a whole performs at the level of a decent handicapper — at least in horse racing.

    Another problem arises in the definition of ‘nut’. Does it mean a maximum wager limit, the starting purse, total purse since the start of the session, or some arbitrary loss limit? I set a session limit and also use the first three races as a gauge. I fully expect those races to fund the majority of my daily session enterprise. If they don’t then it’s time to regroup and discover what went wrong. Doesn’t happen very often. By the end of the daily session I usually end up betting between 5-6 times by session limit.

    I suspect the reason east coast bookies offer less on east coast teams comes from the fact that sports bettors tend to bet on the home team winning. The bookie must counter those bets to make a profit. The net effect is that sports betting is closer to parimutuel pooling than most imagine.

    Gecko, I reason the same way but still wait for the dollar return to exceed the inverse of the probability of winning. For example, greater than $85M for a 1:85M lottery. My reasoning then sounds like yours.

    Here’s an interesting question: what’s the difference between gambling and investing? I expect (and have gotten over the last 15 years) an annual ROI of 20-25%. Last year I bet $70K and got back $88K, a 25.7% return. Am I gambling or investing?

  9. JH

    This reminds me of the classical gambler’s ruin problem (a random walk) in which casinos always come out ahead in the long run. The simplest nontrivial result of the gambler’s ruin problem is that even if the game is perfectly fair, you will lose eventually with a probability of one. Well, I guess, you will need the discipline to stick to your plan (if you have one) and know when to stop.

  10. Doug M

    My comments on the size of the wager are Kelly. Diversification was first discussed by Markowitz. Breadth (or as I called it turnover) I first encoutered in from Griswold.

    Since people may not know what we are talking about — John Kelly wonderd, if he received a transmission from the future with tomorrows race results but the information was somewhat mangled, what would he do? What he won on the first race, he could parley trough the day. If he bet too much and lost, he would have very little to work with on the second race. He worked out a formula to maximise the expectated return at the end of the day.

    Not sure I follow you on following bets with negative return.

    What is the nut? It depends a bit on context. At the race track, my nut would be limited to the ammount of cash in my pocket. For my investment portfolio, all my liquid assets less a several months expenses cash cushion. As a money manager, it is all of my cleints assets.

    It is difficult to quantify the assumptions behind Kelly in a real world situation. However, the rule that an investor should commit more capital to a low return investment than a high return investment is sometimes counterintuitive. For example, if he had the abilitly to borrow freely, a Kelly investor would hold a levered position in high quality bonds and an unlevered position in equities. Bonds are low risk, low return, high Sharpe ratio investments. Equities are high risk, higher return, low sharpe ratio investments.

    Bookmaking a multiple market maker system. A bookie sets a line, and takes on bets untill he feels too exposed. At that point, he either lays-off some of the bet with annother bookie, or he adjusts his prices (or both).

    What is the difference between gambling and investing? Investment losses are tax deductible.

  11. DAV

    “Not sure I follow you on following bets with negative return”

    I have Kelly’s paper around here someplace. I’ll see if I can find it and put it somewhere accessible. Basically, it assumes that all probabilities sum to 1 and all normalized payoffs (called odds at the track) also sum to 1. To faithfully follow his argument it’s necessary to allocate funds regardless of potentially favorable outcome (positive return). IOW: bet on ALL possible outcomes. His proof doesn’t hold if you don’t do this. You need to remember that some of those negative returns will win some of the time. If those bets are skipped, the possible loss in a race is 100% of all other bets

    The summing of the probabilities means that all of the possible events are mutually exclusive. Kelly never covered the situation where all or none of the bets are potentially successful.

    “What is the difference between gambling and investing? Investment losses are tax deductible.”

    LOL! Actually, so are gambling costs but they can only be applied against gambling winnings. I understand that investment losses also have similar limitation.

    My distinction between the two: An investor EXPECTS he will win (with reasonable expectation); the gambler HOPES he will win. If you don;t see the difference substitute driving from point A to B. There’s a big difference between expecting to arrive safely at B and hoping to.

    “the rule that an investor should commit more capital to a low return investment than a high return investment is sometimes counter-intuitive”

    I’ve been at this so long I really find the opposite counter-intuitive. Low risk usually comes with low payoff. The low risk translates to “will succeed more”. At the track it means many are in agreement on the outcome. It’s sometimes hard to bet against the crowd because of the nagging feeling that something obvious has been missed.

  12. OMS

    DAV: If I understand Kelly’s paper correctly, the reasoning is at least a bit more subtle than “bet on all outcomes.” 😉

    Conveniently, the original paper is linked on the Wikipedia entry about the Kelly criterion.

    Briggs: Poker is an interesting situation because such widely varying skill levels can play into a single outcome, whereas blackjack really only admits your own skill (since the dealer has no particular choice in the matter). In poker, you don’t need to beat the whole “game” so long as you beat one person in the game. At the highest levels, yes bluffing and reading and all that is surely important, but I think at lower skill levels is quite possible to gain a major edge against other players simply by understanding some well-known odds better than the other players and having the patience to milk that over a longer period of time than they are willing to wait. The point has been made that poker is one of the few games where complete amateur will play against a professional without the benefit of a handicap. I call that an investment on the part of the pro and gambling on the part of the amateur!

  13. DAV

    OMS, I don’t know about Kelly’s subtlety but there seems to be a necessity for the sum of the probabilities of success to equal one or the return is less than optimum. Omitting bets means that the sum is less than one. If you have a different take I’d like to hear it.

  14. Mike B

    “Here’s an interesting question: what’s the difference between gambling and investing?”

    I suspect that someone who asks this question already knows the answer.

    But I’ll give my answer: both are transactions that can involve “winning” or “losing” money. However, underlying an investment (at some level, even in the case of financial derivatives) is something with intrinsic value, whether it is a piece of real estate, a share of ownership in a corporation, a promissary note (from and individual or a government), a precious metal, a tulip bulb, or a baseball card.

    “Gambling” involves nothing of intrinsic value, unless you define the entertainment as intrinsic value, which essential defines everthing in life down to a gamble.

    My wife and I were trying to describe Las Vegas to our 9-year old son. I explained that you could spend (i.e. lose) $1,000 in a casino, or you could spend $1,000 to see some really good shows. It just depends on how you want to be entertained. But my wife (the gambler in the family) pointed out that if you go to the casino you might win $1,000. And my son (God bless him) responded, “But at the show, you could find a wallet with $1,000 in it under your seat!”

    And if you’ve ever seen my wife gamble, I’ve got a better chance with the wallet under the seat than in the Casino.

  15. Doug M

    In the markets we don’t call it gambling — it is speculating.

    Gambing is a zero sum game. In order for me to win, someone else has to lose.

    Investing is a postive sum game. Someone has money, someone needs money. The person who needs money offers payment in the future for money today. That payment could be fixed or it could be a percentage of profits. The money borrowed, frequently but not necessarily, goes toward a venture that will potentially create some long-term economic gain.

    Derivatives are a zero sum. However, at least one side of the transaction thinks they are buying some form of risk reduction. So, I think this still counts as an investment.

    A day trader, working in his bathroom from home is a pure speculator. An OTC market maker, day trades to provide liquidty. He is neither particularly investing nor speculating.

  16. DAV

    I’m reminded of a TV prograom back in the early 60’s. It was a Playhouse 90 or some such episode called “Big Deal in Laredo”.

    Plot summary: a woman checks into a cattle town hotel and learns of a big cattleman poker game in the back room. The cattlemen surprisingly allow her to watch and even more surprisingly allow her to later join. After receiving requested instuction in the game and many hands later the play is down to her and a very rich cattle baron. She has bet all she has which the rancher promptly sees and raises. She is informed that unless she can get more money she must fold. She turns to the Very Conservative Banker/Spectator and asks to borrow money. When he asks for collateral she shows him her hand. He replies “Sure! As much as you want.” She raises the cattleman who then folds. The cattleman attempts to see her hand but he is stopped by the banker who says the cattleman hadn’t paid for the privilege. Scene shifts and we discover the banker and woman are partners.

    Now that’s bluffing.

  17. DAV

    Doug M: “Investing is a postive sum game. Someone has money, someone needs money. The person who needs money offers payment in the future for money today.”

    Yet if the promise isn’t fulfilled someone has gained at another’s expense. Zero sum, yes? Say the loan is a stock transaction. The person’s net worth is calculated from the price of that stock, is it not? The profit comes mostly from an increase in that price, yes? If so, isn’t that betting on price increase? The ultimate cause of price increase of a stock is demand. Does the stock buyer really care about the cause of that demand or is it just possible evidence when handicapping a stock?

    “A day trader, working in his bathroom from home is a pure speculator. ”

    Your first sentence quoted above implies the only difference between investing and gambling is the possibility of an increase in wealth. So how does that fit with your day trader? You now seem to be saying that the difference lies in the attitude and the positive sum hasn’t anything to do with the difference in terms.

    Which is it?

    FWIW: I think the difference lies in the confidence of expectation of success. And I intend it to be applied to only those who have a rational basis for believing in success. The person who consistently declares “this time for sure!” (alas poor Bullwinkle I knew him well) despite countless failures is really a gambler.

  18. DAV

    Mike B: However, underlying an investment (at some level, even in the case of financial derivatives) is something with intrinsic value, whether it is a piece of real estate, a share of ownership in a corporation, a promissary note (from and individual or a government), a precious metal, a tulip bulb, or a baseball card”

    Baseball card? The value of a baseball card is simply perceived and the demand for a given one is a reflection of it. It’s actually the demand that drive up the price. The same is true for all of the other forms you’ve noted.

    Buying a bet is also ownership. The bet has potential value. That value is inherent in the confluence of future events. The value in owning a piece of real estate is similar unless the ownership is to provide a space for doing something like manufacturing or farming.

  19. Mike B


    In my original post, you’ll note that I mentioned that one could argue (although I didn’t) that the entertainment involved in betting (or investing for that matter) had intrinsic value. I reject that argument as reductio ad absurdem. You’ve taken the opposite extreme ad absurdem: nothing has intrinsic value. I never realize you were a nihilist! (But in a semi-serious vein, I make the same argument to my libertarian friends whenever they start telling me I should be invested in gold).

    Let me try again from an accounting perspective. “Gambling” as I see it, is always a matter of simple cash flows. You place the bet and you end up with either a credit (a win) or a debit (a loss). Investing is more complex, as you are converting assets on the balance sheet from on category to another. For instance, if you make a house payment, the portion of the payment that is principle is converted from a cash asset to equity in your home. The cash flow is zero, you’re just moving stuff around on the balance sheet.

    And yes, you are correct, some bets can have intrinsic value. For instance had you bet $100 on the US to win the Federation Cup at the start of the tournament, you could have gotten some pretty nice odds, maybe 50-1 or better. That ticket is worth a good deal more than $100, right now, and I’d advise anyone holding it to lay off against before the game with Brazil.

  20. DAV

    Mike B, “You’ve taken the opposite extreme ad absurdem: nothing has intrinsic value”

    Well, almost 🙂 If anything has intrinsic value it’s in the form of utility. Food, farm land, whatever. I guess that argument leads to an intrinsic value for entertainment. It also means that money has secondhand utility as it can be traded for things with utility. I guess you could say gambling is just an attempt to increase one’s ability to obtain utility by increasing the amount of available money. Presumably, investing has the same goal. IOW: they don’t differ in any economic sense except that bets on games will likely never increase overall utility.OTOH, neither will ‘investments’ in insurance company stocks.

    I still think the real difference is one of attitude.

    For some reason ‘gambling’ has the stigma of attempting to gain ‘easy money’ so, as pointed out earlier, ‘respectable’ gamblers call themselves ‘speculators’.

  21. OMS

    At that point, doesn’t the distinction between investment and gambling sound sort of made-up?

  22. Doug M

    How about for a economic view of invesment (as opposed to a fincancial view). The economic view isn’t particularly concerned with money. Invisting is the diversion of resources that could be used today to create a benefit in the future.

    Even in the accounting view of investment — the purchase of land or capital equipment does away with concepts of risk and return. It is only in the financial sense, that there are any parallels between gambiling and investing.

    It doesn’t help that most of the theory of probabily was developed to analyse casino games. And that theory was then applied to the world of money and investing. There are a lot of old salts out there who take great offense to the association of random variables and invement returns.

  23. Thank you for sharing! Have a great day. I appreciate the paradox. I know that buying a Lottery ticket is essentially a stupid thing to do based on expected payoff (there are much cheaper ways out there to lose money). But I do it just the same form time to time.

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