Negative Expected Value

 

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Negative Expected Value:   The Ultimate Triumph of Loss.

Summary: Though  only one form of predatory gambling,  lottery  illustrates how and why predatory gambling drains not only the gambler  but also everyone around each gambler from whom he or she can winkle or steal a dollar to pursue “the chase.”   Using a simple model of lottery, this essay distinguishes “wins” from net gain, showing that unless a compulsive  gambler quits, “extinction” is inevitable.  Every rational person knows this. Sad to say,  however, most people deny  how badly  trusting persons around the gambler will be hurt as the gambler fends off   the inevitable.  “The gambler’s ruin” is not confined to  him or her.   This is the central evil of predatory gambling.

Adam  Smith wrote about  lottery  “That the chance of gain is naturally overvalued we may learn from the universal success of lotteries.  The world neither ever saw, nor ever will see, a perfectly fair lottery; or one in which the whole gain compensated the whole loss; because the undertaker could make nothing by it . . .  There is not, however, a more certain proposition in mathematics than that the more tickets you adventure upon, the more likely you are to be a loser.  Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets the nearer you approach to that certainty.”   Wealth of Nations book I Chapter 10 p 153

On first reading  this, in an article by Grinols and Omorow*,  I needed a concrete example to  reckon “how much is the chance of gain overvalued?”  A second query, answered from the first, is why the more tickets you buy the more likely you are “to be a loser.”  Imagine a simple lottery with one prize, $100 in cash.   Each ticket costs one dollar.  The market worth of a ticket pre-drawing, however, is not its purchase price,

Ticket price           Prize value                   Tickets sold     Ticket value pre-draw

   $1    $100     50  $2.00
   $1    $100     90  $1.11
   $1    $100   100  $1.00
   $1    $100   110  $0.91
   $1    $100   120  $0.83
   $1    $100   200  $0.50
   $1    $100   400  $0.25

 

but its statistical expected value, the probability of a given payoff multiplied by the amount of that payoff.  The probability of a payoff in this one-prize example depends on the number of tickets sold up to the time of drawing.   If that number is pre-set and met,  the value of the ticket is as shown in the table.   If exactly 100 tickets are sold @ $1 , the probability of a win is 1/100 = 0.01 .  If the prize is $ 100, the expected value of a ticket is (0.01 times $100 = $1.00)  .   Post-drawing,  all tickets but one are worthless; one is worth $100.  The never-to-be-seen perfectly  fair lottery is shown in line 3 of the table in larger type. .

Very few lotteries would have a drawing if the prize value is greater than the total ticket sales, unless the prize (as for charity, often)  has been donated.  All aim to have gross ticket sales exceed the cost of the prize to the operators.  The larger the difference,  the greater the return to the operators ( Adam Smith’s “undertakers”).   On the other hand, buyers know that if the prize value is fixed for a scheduled drawing but sales are unusually brisk the value of a ticket will go down. Ticket buyers in most lotteries do not know at the time of drawing how deeply discounted their tickets are, but understand they are worth less than purchase price.  They accept that discount in the dream of a big payout or, as with many charity raffles, a non-deductible contribution to a “good cause.”

The discount (purchase price less market value)  can be learned from the table above for the very simple one-prize lottery.  When 120 tickets have been sold @ $1,  a single ticket  has been discounted by 20/120 =  0.1666 .  This rounds off to 17 cents; the market value is then 83 cents pre-draw.   The more tickets are sold, the deeper the discount.

Not remembering that tickets in a big (say state-run) lottery are statistically worth less than their purchase price, many people puzzle at Smith’s apparent paradox: the more tickets you buy,  the more money you are likely to lose.  Even less intuitive is that  if you buy them all you are certain to lose.  This is an important facet of gambling, that one can lose while “winning”  and “win” while losing.  It emerges in Smith’s description of lottery.  Here, buying every ticket guarantees their holder the prize but also, when the tickets are discounted,   guarantees the holder will have laid out more than the prize is worth.

If the venturer buys 55 tickets of the 110, she or he has a 50% chance to win $100.  If one of the 55 wins, the venturer nets $45.  If all 55 lose, she or he loses $55.  The expected value (predraw market value) of the $55 investment is $50.  A venturer who plays this lottery exactly the same way a thousand times will on the average lose $5 per drawing though scoring $ 100 payoffs on about half the drawings.

The amount of money the venturer can expect to lose is a linear function of the proportion of tickets she or he buys, as shown in this table for a lottery  with always exactly  110 tickets sold @ $1 and the single prize $100

Number tickets           Total Cost                  Investment                Investment  bought                                                               value                   less  cost ( = loss)

1 $1 $   0.91 $  0.09
10 $10 $   9.10 $  0.90
40 $40 $  36.36 $  3.64
55 $55 $  50.00 $  5.00
110 $110 $ 100.00 $10.00

 

I’m probably belaboring the point, but once more:  the more tickets someone buys, the better his or her chance of winning $100 on that drawing but the greater the expected cash loss.  All real-world lotteries have  a “negative expected value.” This leads to another term,  “gambler’s ruin,”   which applies not just to lotteries.  Wiki: “The most common use of the term [gambler’s ruin]  today is for the unsurprising idea that a gambler playing a negative expected value game will eventually go broke, regardless of betting system.”

In the simple example above, if 110 tickets are sold to the same person,  he or she will have spent $110 to “win” $100, a net loss of $10.

Smith’s comments  apply  to statistical expectations, not to single events.  The chance of a jackpot does go up in proportion to the number of tickets bought.  It is possible to buy a large number of tickets for one drawing,  one of which does hit the jackpot.    It is certain that in an interstate lottery where the winning ticket paid $500 million pre-tax, that ticket holder’s lifetime investment in lottery tickets to then was  nowhere near the payout.   In the long run, however, the  more an individual or a sector of society “plays,” the more that party loses.  When half  of the loss is claimed by a government “to benefit education,” that noble phrase seeks to justify an exchange that draws ever more from gamblers AND FROM THOSE AROUND THEM when their “own” resources are gone.

The same principle, losing while “winning,” came up recently in another way.  An executive for a company that makes gambling machines was asked in an interview for the secret to winning on a slot.  He replied “stay on the machine.”   An opponent of predatory gambling took issue on twitter, saying that the longer someone is on device the more he or she will lose.  Both are right.  If “win” means getting a big payout, that event is more likely the longer one plays.   “Win” here, however,  does not mean net win.  Playing long enough against a house with much greater  resources,  a gambler will always lose more than she or he wins.

The adage “quitters never win” does not apply to gambling.   For someone in “the chase” that’s the only way to win in the long run, to stop losing.  Sad to say, quitting is nearly impossible without the conjunction of catastrophe and great timely help from outside.

* Earl Grinols and J.D. Omorow J. Law and Commerce vol 16 1997 p. 49 ff

The opinions in this essay are those of the author, Stephen Q. Shafer,  and do not necessarily reflect those of any or all members of CAGNY.  Permission to quote or reproduce in any length is hereby granted on condition that the url above is cited.

The photograph shows avifauna of  New York State and boxes that house the state bird.

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