Friday, June 10, 2011

The Utility of Gambling

People gamble.

Economists assume people use their money rationally.  If a simple model suggests that a common behavior is a losing strategy, economists seek an extended model which elucidates the behavior.  If the extended model makes predictions, these should be tested against data.  Gambling "violates stochastic dominance," which is the usual argument: "If the expected payoff is 96% of the bet, then playing is irrational."  Gambling suggests that the utility of money is increasing, whereas in many surveys the utility of money is seen to be decreasing.  An economist might predict that when the utility of money is increasing for a rational person, then that person will gamble.

Gambling suggests that ROI increases with wealth.

Warren Buffett makes a better ROI on his investments than I do.  If all of that advantage comes from his brains, then I can't copy him.  But if part of his advantage comes from being rich already, then I should gamble.

ROI model:

Some investments generate an income stream for the owner.  Stocks and bonds pay dividends.  Owning a house near where you work generates an income stream -- you don't have to rent a house, and if you take a long vacation, you can rent it out.  Suppose the investment opportunities I would face after winning a gamble are better than those which I face now.  To take a simple model, suppose that an individual with wealth w can find (by using her spare time to learn about new business ventures, or because she can afford to diversify into risky ventures without risking her daily quality of life, or because large sums of wealth are slightly easier to manage, or because she can put some of it into long-term investments and she doesn't have to keep it in a cash account) earns interest rate 2+log(w) on her wealth.

If the Anderson family invests 1 coin and compounds it 100 times at that rate of interest, they will then have earned 27.71 coins.  The Bakers and the Cooks follow this strategy for 10 days.  At that point they get "anxious" to reach their savings goal.  They take a gamble, to win 0.1 coin or lose 0.1 coin with odds of 51:49 in favor of losing the coin.  I.e., they accept odds slightly worse than 50:50, so that the casino could make a profit.  The Bakers lose and the Cooks win.  They then proceed to compound their money 100 times at the variable rate of interest.

That is... the Andersons apply the following iteration 100 times:

wealth = wealth * (1 +  0.01 * (2 + the natural log of their (wealth) ))

The Bakers and the Cooks do likewise, but after compounding their initial 1 coin 10 times, they then gamble 0.1 coin -- the Cooks add 0.1 coin to their wealth and the Bakers lose 0.1 coin.  The results, after compounding 100 times, is:

A: 27.713150
B: 22.654267
C: 33.364858

The expected wealth of those who follow the Bakers-Cooks strategy is: (B * 0.51 + C * 0.49) since there are 51 losers to every 49 winners in the lottery.  That value exceeds the result realized by A.

Of course, the Cooks and Bakers can do much better if they would bet with each other.

Saving towards a goal.

In real life, people often gamble so as to "make up the difference" between their savings and a desired investment goal -- a house or a business, for example.  When you have a goal in mind, and when you expect that goal to increase your quality of life and generate an income stream with a better ROI than the investments you already own, is it rational to gamble your savings and take a chance on securing the new, desired investment either earlier or later than you would expect?

An example -- brothers buying houses.

Suppose my next purchase will be a house and that I want to buy it with cash so that I will have neither rent nor mortgage payments coming due monthly.  This investment thereby represents an income stream to me.  I can wait for the cash to accumulate and then buy, and then begin to enjoy the utility and income stream from the purchase.  Or, when I have some portion of the money and find a convenient opportunity to buy, I can gamble and stake my savings against the money needed for the investment.  If I win, then I get the income stream early, with its increased ROI.  If I lose, then I get it later.  The average of these two conditions is better than simply waiting.  Suppose that when I start work, my income stream is $50 per day (above expenses).  Suppose that at this rate it will take me 20 years to save the money to buy a house and that I earn no interest on my savings.  Suppose that, having got the house, my income stream will be $150 per day.  After working for 10 years, my brothers gamble.  51% of them lose everything -- their savings are now 0 and they continue to earn income at $50 per day.  On the other hand, 49% of them get their house and now earn income at $150 per day.  My brothers' average income stream is now greater than my own.  The gamble was similar to a pact -- they could have decided to pool their money and get houses, one by one, as they were able to do so.  Gambling and pacts are efficient if ROI is higher for greater sums of wealth.  The Teachers' Credit Union may be a pact allowing teachers to earn ROI available to their total wealth, rather than the ROI available to their individual wealth.

A friendly game of poker:

If you want to buy a house, then join my poker club.  When we have enough savings, between us, then we will play poker (or, if someone is too good at that game, we'll play something more random) until we all lose our small investments and one person has all the money and goes to buy the house.  We could have done some complicated thing where we give him the money and then force him to continue to contribute, but gambling makes this more simple.  If there are only 3 of us and the cost of a house is 6*x, then one of us will save 2*x and then get the house (and the income stream!); another will save 2*x and lose it and then save 3*x and get a house; the third will lose 2*x, lose 3*x, and then save 6*x and get the house.  Together, we saved the price of three houses and some of us got the income stream early.  If ROI is equal for the rich and the poor, then it was all a game.  But if the ROI we get from owning a home is better than the ROI we earned on the savings while we saved it, then it is clever to lose your savings so that another you can close the deal on the good life.


In a lottery, the payoff can be very low.  Considering that the lottery winner pays tax and that the lottery payoff may be only 50%, the payoff might  be 25%.  That is still rational if the ROI on winnings is more than 4 times greater than the ROI the player was making the money which the player gambles.


  1. At I asked whether "moves with negative expected value even below the expected value of paying the ante and always folding" (-EV moves) occur in poker. Since poker players are losing their money constantly due to ante, I supposed that the money can be considered as "invested with a -ROI" and that players who are winning have the luxury of paying a lower ante % and can wait and only play the best moves. Their money is effectively invested with a better ROI. So -EV moves should occur, according to the argument I made above. But -EV moves may make you vulnerable to exploitation.

    The author of that blog mentions that when poker is viewed from the outside, then a -EV move can occur -- if you see a good investment opportunity, you might put your cash on the roulette wheel, since going home without money and going home having played the good opportunity are, on average, better than going home with your cash.

  2. Perhaps variable-ROI can explain how much an individual should gamble, as a function of that person's alternatives -- consumption, investment -- and investment goals.

    This academic article mentions that variable ROI, if it causes increasing marginal utility, makes gambling rational. Maybe the argument was like this: increasing utility makes gambling rational. Bailey et al. (1980) argued that delayed consumption would achieve the same average utility as gambling. (That is, Bailey wrote that rather than gamble 1 coin with 49:51 odds of winning another coin, the consumer might first consume 0, earn another coin, and then consume 2 coins. The consumer's average utility would seem to be the gambler's average utility). The present article argues that the gambler has the advantage of obtaining the 2 coins now. Marginal consumption utility probably diminishes anyway, but marginal investment utility is what increases, and of course it is better to buy an investment now rather than later; the average of buying it now or not buying it is better than the average of not buying it now and then buying it later. The article goes on to discuss how we prefer one time to another; my basic mode for these would be: that we want to smooth out consumption over time, while we want all our investments to be made as soon as possible.

    Of course, this 14-page article from the American Economic Review contains winning strategies, proofs, and commentary at the cutting edge of economic thought, whereas we are presenting what must be an old argument: that increasing ROI makes gambling-then-investment rational.


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