The Mystery of the Lottery

I'm not engaged in predicting random number generators. I actually get phone calls from people who want to know what lottery numbers are going to win. I don't have a clue. —Bueno de Mesquita

η ελπίδα πεθαίνει τελευταία [hope dies last] —Pandora

A great deal of human psychology is dedicated to the taming of uncertainty (Krueger & Grüning, 2021). Gaining knowledge is appealing in its own right, and what is more, it enhances feelings of power and control.

Hope springs eternal.

Source: J. Krueger

Yet, people often embrace uncertainty. This, too, can have motivational benefits. We forage for food and adventure in unfamiliar places, enjoy the thrill of suspense, or prefer to sit with uncertainty when we suspect that knowledge would entail bad news (Krueger et al., 2020).

It is often the same people who buy insurance that play the lottery. Doing the former, they accept a negative expected value to avoid catastrophic loss; doing the latter, they accept a negative expected value in hopes of winning big.

These same people both avoid and embrace uncertainty. Buying insurance is consistent with the stylized bias of loss aversion, which is said to pervade much of decision making (Kahneman, 2011), but lottery playing flies in the face of that presumed stylized fact as the losses mount. Why do so many people buy lottery tickets? And why do they keep buying them?

Myopic loss aversion

Consider the phenomenon of myopic loss aversion (Samuelson, 1963). Suppose you are offered a fair bet of winning $200 or losing $100. Most would balk at the prospect of losing $100, even if they were first given $100 in house money. Yet, if they are invited to play the game repeatedly, they jump at the chance. Why?

Take the simplest version of a repeated game: a game with two rounds. The expected value remains the same per round, namely $50 ([$200 - $100]/2). However, there are now three possible outcomes: two wins, one win and one loss [in whatever order], and two losses. The respective probability of these outcomes occurring are .25, .5, and .25.

Compared with the single-round game, two things are different. The worst possible outcome is twice as bad (losing $200 versus losing $100), but the probability of losing anything is cut in half, from .5 to .25. If people accept repeated play but reject single-round play, the reason cannot be a motive to avoid the worst possible scenario: that is, it cannot be the reason that drives insurance buying. Instead, repeated play may be attractive because the prospect of losing anything has become less likely. Decision scientists regard the preference reversal from not playing once to playing twice as a mark of irrationality. Their argument is one of backward induction. If you agree to play the same game N times and also agree to play it N-1 times, there is no justification for switching to refusing to play when N-1 = 1.

Reverse myopic loss aversion

Loss aversion implies that we should not buy lottery tickets. If we buy one ticket, we face a near-certain loss; if we buy many tickets, the cumulative expected value falls fast from the negative to the very negative. Haisley et al. (2008) found that people prefer to buy tickets one (or a few) at a time, a phenomenon they dub reverse myopic loss aversion. For very small amounts of gains and losses, the value function reverses its ordinary concave and convex shape.

People care less about very small gains and very small losses than projections from the rest of the value function would suggest. They dismiss peanuts gained and peanuts lost as if they hardly mattered at all. Hence, the price of today’s lottery ticket feels like almost nothing, and so does the price of tomorrow’s ticket, and so on, but the price paid for, say, a 10-pack feels like something. Perhaps people prefer single-ticket purchases because it allows them to stop buying tickets once they win big—as if it mattered that they bought a few tickets in vain; those wasted tickets would be peanuts.

Consider a numerical comparison between a single-ticket and a double-ticket purchase in a lottery that offers a prize of $10,000 with a probability of .0001 for a ticket that costs $20. The expected value of winning is .0001 x $10,000 = $10. After subtracting the $20 paid for the ticket, the net expected value of this lottery is -$10. The probability of losing $20 is .999.

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Those who buy two tickets face three possible outcomes. They may lose $40 with a probability of .998001; they may win either with the first or with the second ticket with a probability of .001998; or they might win with both tickets with a probability of .000001. The expected values are $19.98 and $.02 respectively for one and two wins, which amounts to $20. Subtracting the $40 paid for the two tickets, we land on an expected value for a two-ticket purchase of -$20, which is proportionately the same as the expected value of one ticket.

The two mysteries of lotteries are: Why do people prefer to buy tickets piecemeal, and why do they buy any tickets at all? The notion of reverse myopic loss aversion helps explain the former question, although other factors likely play a role. As the poor are the most avid lottery players, they may simply not have the money to buy large batches of tickets. If buying a ticket and waiting for the result comes with a small flash of excitement, the spacing out of the excitement over many discrete events is, in fact, the rational choice. Perhaps this fleeting thrill of possessing a possibility, if a remote one, also explains why people play at all. They forage for the big release, and that is exciting. This idea is plausible but hard to nail down with empirical data.

Lotteries and inequality

The empirical data tell a story of immiseration. As the poor play more, they waste precious money they could use to buy more or better food, clothe their children, or give their car the overdue oil change. Money wasted on lottery tickets entails the deferment of necessary upkeep expenses.

Some economists look at state-run lotteries as a form of regressive taxation, where the poor are taxed more than the rich. As lotteries are unlikely to go away, perhaps they can be remodeled. One proposal is to reduce the jackpots to provide more players with sizable wins. This strategy has two downsides. First, the numerical awesomeness of the jackpot adds to the lure of lotteries inasmuch as people pay more attention to large absolute numbers of value than to huge but barely perceptible differences between very small probabilities. Second, very large jackpots ensure that players will never reach a point where they have spent more money than they can recoup with a win.

A rational but politically and psychologically odd proposal would be to make buying lottery tickets an integral part of the taxation system. In such a system, everyone has to buy tickets. If the rich were mandated to buy more than the poor, the lottery would amount to a progressive tax. The poor might object that this is unfair because more rich people—who don’t need the money—would end up winning big than poor people would. Ironically, such a proposal makes it look as if the poor wished to be overtaxed and exploited. From an economic perspective, this may indeed be an instance where the poor suffer from false consciousness (Marx & Engels, 1846/1976). The state or the gaming industry won’t tell them what’s good for them, and academic writing is boring.

Afterburner

There's some question as to whether one would wish to use the word 'poor' or the term 'low socio-economic status.' I turned to my spiritual advisor Hoca Camide to see what he thought. Here is what happened:

I told Hoca Camide “I am not poor; I am of low socio-economic standing.” “Oh, that’s rich!” Hoca replied. “In fact,” I went on, “the term ‘low socio-economic standing’ may be meant to be a euphemism, but it is a put-down because it says there are others with higher standing. A person of low socio-economic standing is branded as a loser. To be called poor might sound harsh, but we could all be poor.” “I think you are one of them sophists,” Hoca snarled, taking a drag from his Murad.