So you're telling me there's a chance?

January 10, 2009 in Math

Reader's Digest has written a brief piece looking at state lottery games and noting that an increased proportion of recession spending is allocated toward these games of chance.

Lotteries are excellent laboratories for testing the theories of Israeli psychologist Daniel Kahneman, who won the Nobel prize in economics in 2002.  Kahneman's focus was on the cognitive errors that people make - basically, he rooted out the shortcuts which the brain uses in lieu of complex calculations, particularly when dealing with probabilities.   Along with Amos Tversky, Kahneman developed prospect theory, which describes why people may behave irrationally when faced with a set of potentially risky choices, such as buying insurance or entering a lottery.

Our brains are exceptionally good at dealing with things of equal probability - betting on a coin toss, for example.  However, the mathematical shortcuts we employ break down as probabilities become very small.  According to prospect theory, we overweight the impact of small probabilitie and - perhaps because we subconsciously demand probabilities sum to 1 - we underweight events which are nearly certain.  This helps explain risk aversion.  For example, the fair price to insure a $100,000 home against the 1% chance it burns down is $1,000.  However, most people would be willing to pay considerably more than $1,000 for the insurance.  This is because of the dual impact of 1) perceiving that something which happens 1% of the time actually happens much more frequently and 2) considering a one time loss of $100,000 to be much more painful than 100 losses of $1000, despite the total loss being the same.  Essentially, prospect theory suggests that not only is the utility curve convex (this is universally accepted), but so is our perception of the probability curve.

So what does this have to do with lotteries?  The chance of winning the powerball jackpot is infinitesimally small, about 1 in 150 million.  This number is so small it does not have meaning.  As you read this, you might be thinking "well I know that's extremely unlikely", but you do not actually comprehend how unlikely it is.  Reader's Digest points out that the chance of hitting not one but two holes in one in a round of golf is roughly 1 in 70 million.  While it may be interesting to know that the lottery is twice as hard to win as a double hole in one, at this scale that comparison is meaningless.  It is like when people say that if an atom's diameter were as large as the Empire State Building is tall, its nucleus would be the size of a tennis ball -- that's great, I get that the nucleus is small, but I'm not closer to understanding how much closer to nothingness it actually is, because I just can't comprehend that scale.  

Fortunately, my brain overweights small probabilities.  1 in 150 million may be so unlikely it is incomprehensible, but all my brain thinks is, "So you're telling me there's a chance?"  And it goes on to associate this with the smallest probability it can safely comprehend, which, unfortunately, happens to be literally millions of times more likely than actually winning the lottery.  Let's suppose my brain is capable of accurately assessing 1% chances (which may in itself be a stretch).  If that's the smallest thing I can deal with, then the lottery just got bumped up to 1% likelihood and I'd have to be crazy not to pay $1 for a 1% chance at millions, right??

If this seems idiotic, it's because it is.  And yet no less than a nobel prize went to the man who demonstrated that heuristics such as these are how our brains deal with, categorize, and make decisions regarding the otherwise incomprehensible number of datapoints which constantly barrage it from our senses.

As we get older, or more educated, we develop tools that either help us internally perceive probabilities more accurately, or help us perform rational calculations which can inform our perceptions (I'm not sure which).  Some supporting evidence comes from the Reader's Digest piece, which notes that college-educated people spend roughly $8.50 each month on Texas lotteries.  A typical player without a high school degree spends almost twice as much, $16.  I should note that a potential confounding variable is that college-educated players will be typically wealthier, and therefore have less motive to play the lottery.  However, this remains a returns-to-education question.

Is it fair, then, to call lotteries a "stupidity tax"? An "irrationality tax" would seem a more appropriate term.  Certainly, people with less education will be more cognitively susceptible to the lure of the low-probability, high-stakes event.  In their minds, the loss of $1 is more than offset by the probability-weighted utility of gaining millions.  But do they recognize the marginal loss of that dollar against their future income stream?  

Another theory of Kahneman and Tversky may help explain people's willingness to throw away money -- the representativeness heuristic.  This refers to people's tendency to associate their perceived frequency of something occuring with how easily they can recall an example.  Media coverage of lottery winners enforces the idea that the lottery is frequently won.  Indeed, every single night someone wins the lottery. But there is never any coverage of the millions of lottery losers (It's not much of a story to report on Joe, who lost $1.50).  The representativeness heuristic suggests that when people think about lottery outcomes, they think about winners even more than they would otherwise, since the winners are publicized.  This increases the perceived probability of winning.

In the end, the billions of dollars raised from lotteries do a lot of good, like padding state education budgets.  So one could argue the lottery is not all bad.  However, it serves as a regressive tax on the people with the least discretionary income.  Moreover, gambling addictions can arise from the seemingly tame behavior of spending a couple dollars a week on an outcome which never occurs.  Perhaps a limit on the number of times any one person can play the lottery would be helpful, or a greater focus on the winners of lesser prizes - or even better - the losers themselves, would make it easier for people to stay away.  In the end, however, any policy must recognize that the irrational decision to play the lottery is in fact one which has been hardwired into the brain by years of evolution -- the byproduct of a series of mathematical shortcuts which developed when the only lottery was the reproductive one, and it was beneficial to overreact to otherwise unlikely but dangerous outcomes.

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