Thursday, May 5, 2011

As expected, lottery players are not rational

There is no mystery that under normal circumstances, homo oeconomicus does not play the lottery. Exceptions arise when there is enjoyment in playing the lottery (is this why slot machines a so popular in the US?) or when there are particular reasons. But casual observation indicates people play the lottery, and a lot. Maybe these circumstances mentioned above are met, maybe they are not rational economic agents.

Claus Bjørn Jørgensen, Sigrid Suetens and Jean-Robert Tyran would say lottery players, at least some of them, have a peculiar sense of probabilities. While many change their numbers, among those who change, many avoid numbers that have recently been drawn, as if the lottery were a drawing without replacement. But if a number is on a streak (drawn a few times in a row), then they choose it. If margins were not so high in lotteries, one could possibly make money by arbitraging against these people trying to predict the lottery numbers. But you can actually getting a positive return from lotteries by only buying tickets when large jackpots are at stake. The International Lottery Fund based in Australia is there to prove it.


Milton Recht said...

Actually, lottery players using prior information in a random drawing may be economically rational.

Lotteries allow multiple ticket holders to hold the same numbers and the winnings are divided among this group. A lottery player's winnings increase if there are fewer holders of the same numbers.

While winning numbers are random, the player selection of numbers for tickets is not. It follows certain known behavioral rules. Dates are common, so numbers from 1 to 12 and under 31 are more likely chosen by players than other numbers. 13 is also likely chosen less frequently due to its association with bad luck.

Knowing the previous winning numbers is information to a lottery player to make a prediction about lottery buyers' behavior in selecting numbers in the next drawing.

To determine if a player's selection of lottery numbers is rational or not, one needs to determine if the odds of holding duplicate numbers increases or decreases if one uses the available prior information. This is not the same as trying to change the odds of winning in a random drawing.

The paper as far as I can tell, did not evaluate the odds of holding duplicate numbers with and without using prior information about winners. I suspect different lotteries and different cultures and individuals have predictable behaviors towards the return of a winning number. A subset of lottery buyers could use that predictability rationally to increase their odds of not dividing a winning take, which is not the same as increasing the odds of winning. This behavior of using previous numbers to select new numbers would show up in statistical tests. Yet, it is rational because it will decrease the number of holders of the same numbers.

In other words, using prior winning numbers to select numbers can be rational in a random drawing lottery because it increases the odds of being the sole winner without increasing the odds of winning. The fact that statistical tests show some players select non-randomly and use prior information is not irrational if one attempts to minimize duplicate number winning tickets.

Anonymous said...

Milton Recht is right in that information on how lotto players pick numbers can increase expected winnings. For example, Simon (1999, J. of Risk and Uncertainty) shows that picking the least popular combinations yields a gain that is more than four times higher than picking the most popular combinations in UK lotto. Some popularity patterns tend to be stable. Essentially, some people always pick the same numbers and some of these are more common than others ("lucky numbers", superstition, birthdays etc.) Because of the pari-mutuel structure of lotto (= the prize money in a given category is shared among the winners) it's not a good idea to bet on those numbers.

However, the paper makes a different point. The authors find that some players react to recent RANDOM draws (they are able to establish that finding because they can track the number picking of individual players over time). The authors infer - with reference to the "law of small numbers" - from the observed systematic reactions in number picking that some players think that these numbers are less likely to be drawn subsequently (or more likely if they happen to be drawn several times in a row). Of course, the authors can't be sure about that because they do not directly observe lotto players' inferences but only the numbers they pick. If it is the case, such a belief is not rational. The reason is that subsequent draws are independent and therefore, all numbers are equally likely to be drawn, no matter which numbers have been drawn before.

A related but somewhat more complicated question is whether reacting to recent (random) draws (and by implication, holding such biased beliefs) is actually costly. The authors provide two bits of evidence suggesting that biased beliefs indeed are costly.
First, biased players buy more tickets. Because the payout ratio is set at 45% this means that these players lose more money on average.
The second bit of evidence is that players who are prone to systematically react to previous random draws also win less money if they happen to win. The reason is that the biases are sufficiently common to induce lots of players to coordinate their number picking.

The overall conclusion seems to be that sharing a bias with lots of people is costly (in lotto). Knowing about common biases (and systematically avoiding numbers picked by biased players) is potentially profitable.

Jay Livingston said...

Number that have won in recent drawings get less play. Apparently, bettors think that lightning will not strike twice. But OTOH, stores that sell lottery tickets will put signs in the window telling of recent winning tickets sold at that shop. The bettors' logic must be that numbers don't repeat, but locations do. Go figure.

Archie said...
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