# Jackpots, Complexity, and Difficulty

The Problem:  In considering decisions that do not have a transparent answer, many business leaders and strategists fail to distinguish between problems that are hard (i.e., those that require specialized skills or information that is not reliably or readily accessible) from those that are complex in that they require many steps to solve, but each step is not in itself particularly difficult.  This would not be a problem, except that when even bright people believe that they cannot solve a problem, they often fall back on conventional wisdom and platitudes, leading to missed opportunities and wasted resources.

Jackpot lotteries provide a prime example of this fallacy.  State lotteries are big business, generating billions of dollars in profits for the states that run them.  Despite the number of possible outcomes and prizes, calculating the expected result of a jackpot lottery requires nothing more than multiplication, division, and some basic algebra.  Since jackpot lotteries have fixed odds of winning but a variable top prize, if the top prize gets large enough, it can, theoretically, have an expected payoff greater than the cost of the ticket.

Although solving for this “break-even” value requires no more than solid arithmetic, few people present arguments about lottery prizes based on solving for their expected value.  Instead, thinking about lotteries tends to devolve into two categories, the positive camp of “they’re fun, if you can’t win you can’t play, etc.” and the anti- camp, whose position can be summarized as “a tax on those who can’t do math/immoral/throwing money away/worse odds than Vegas”.  A strictly rationalist view, however, says that one should play the lottery only when the expected value of a ticket exceeds the cost of that ticket.  And so while I can’t tell you what numbers to play, I can tell you which games offer the best expected value, and when you should play it.  The most common expected value is the probability times the payout.

For jackpot games, probabilities and payouts are publicized by legal requirement, and so a simple equation identifies the breakeven point at which the expected value exceeds the cost of the ticket.  There are four jackpot games available in many states: a daily drawing with a minimum jackpot of around \$ 100,000, a larger lotto with a minimum jackpot around \$ 1 Million, and the multi-state Mega Millions and Powerball games.  For the purpose of this exercise I will refer to the odds and payouts associated with the Mega Millions game, but the same logic can (and should) be applied to any jackpot-structured game.

Prizes and Probabilities

From the above calculations, in order for the purchase of a Mega Millions ticket to be a break-even proposition, the expected value of the ticket must be 1-(Expected value from Non-Jackpot Prizes), or \$ 0.825.  With a probability of winning the jackpot of 1 in 258,890,850, the present value of the Jackpot must reach at least a level of \$ 213,482,528.

Time-Value of Money:  The advertised Mega Millions jackpot is the total payout to a winner over a period of time.  If you win, you have the option of taking the jackpot amount distributed in the form of an immediate payment of approximately 1.5% of the total stated prize, followed by 29 subsequent payments that increase by 5% each year.  Assuming a 3% inflation rate over 25 years, the present-value of those winnings is only 60.3% of the nominal value; as a result, the stated jackpot must be 65.8% greater (1/0.603) than it would otherwise, implying that in order for the present value to exceed \$213.5 Million, the nominal jackpot must exceed \$ 354 M.  The other option is to take a lump-sum payment of ~ 58% of the stated jackpot, which is less than the value of the annuity.  (Note: if the jackpot were significantly smaller, tax treatment might also favor the annuity, as the winner would be taxed at a lower rate for having lower income in each year).

Taxes:  Speaking of taxes, the dollar that you choose to spend on the lottery ticket which will make your fortune is one that you have already earned and could spend without paying additional taxes on it.  Gambling winnings (including from the lottery) net of losses are taxed at your ordinary income tax rate.  This means that the tax impact of the lottery depends in part on how you play: if you truly restrict your playing to those instances when the jackpot exceeds the payout point and bought every possible combination (assuming you have \$ 259 Million to spend on tickets and are very patient in filling out purchase orders: no Quick Picks, or you might miss the perfect combination), you would only pay on your net winnings.  For the sake of argument, however, I will assume that you will not be able to purchase every possible combination, and will choose to purchase a more modest number of tickets.  In that case, your winnings (in excess of losses) will be taxed at differing marginal tax rates depending on your income.

For 2014, the IRS has published the income tax rates below for income in excess of the standard deduction:
Income Tax Rates

For simplicity’s sake, I will assume that you are single and earn \$ 50,000/year; your actual circumstances may vary, and you should identify your marginal tax rate accordingly.  With a standard deduction of \$ 6,200, your income is already \$ 43,800 in excess of the standard deduction, putting you in the 25% marginal tax bracket.  If you win more than \$ 45,550 from the lottery (net), then you will pay marginal taxes in the 28% bracket, and so on.  This means that only the top two prizes (the jackpot and the \$ 1,000,000 prize) trigger any other taxation.  I assume that you buy only one ticket, and therefore can deduct only the cost of that ticket.

Taxes’ Effect on Prizes

This reduction in the payouts of your non-Jackpot prizes requires that your NPV increase to \$221,681,108.  Since that number represents the value after 39.6% of the jackpot is withheld in taxes, your required pre-tax jackpot NPV increases to \$ 367,000,000.  As noted above, with 3% inflation, your nominal Mega Millions jackpot must be a whopping \$ 608,659,143.

Split Jackpots:  For better or for worse, other people play the lottery (even when the jackpot is below the critical threshold, but especially when the jackpots get large enough to attract media attention).  These other players change your expected jackpot payout, because there is a chance that, even if you win the jackpot, you will have to share it with another winner.

Figuring out the impact of a split jackpot requires an estimation exercise; you won’t know in advance how many tickets will be sold, but Mega Millions does post the number of winning tickets sold.  On May 20, 2014, there were a total of 2,056,000 winning tickets sold.  From the above probabilities of winning, the odds of a ticket winning any prize are one in 14.5, implying that 29,800,000 tickets were sold.  Since the stated jackpot was < \$ 150 Million, and as jackpots get very large many more tickets will be sold, I will assume that a quadrupling of the jackpot leads to a quadrupling in ticket sales as well (in fact, you would likely get a more accurate estimation by calculating the tickets sold with a > \$ 400 Million jackpot that approaches the critical threshold).

Note: The probability that at least one other player wins at the same time that you do is 1 minus the odds of each ticket NOT winning the jackpot.  With 120 million tickets sold, therefore, the probability that if you win, you will have to share the jackpot is: 1-(1/258,000,000)^(120,000,000), or 37%.
Effects of Split Jackpots

Note that the probability of each split is calculated as the probability of one fewer winner than the number of the split; we assume that you, dear reader, are the first winner and not a part of the population who buys tickets willy-nilly, therefore the odds that you will have to share your prize are the odds that any one of the horde purchasing tickets wins.  Be warned, however: in point of fact, as jackpots rise, more people choose to play (some rational like you, others out of simple fascination with/fantasy over staggeringly large numbers), and so the odds of a split jackpot increase.  In the most extreme examples, however, the odds of a split remain low, and the odds of a more-than-3-way split are slim, even given your victory.

Considering all of these factors, the jackpot required to make playing Mega Millions a positive-ROI proposition is nearly triple the 1 in 258 Million odds of winning the jackpot, making it a poor investment.  The Mega Millions jackpot has never exceeded \$ 750 Million.  But it has exceeded \$ 600 Million, and could plausibly reach higher levels.

Perhaps more interestingly, jackpot games that pay out as a lump-sum rather than an annuitized payment (such as many daily Lotto games) become ROI-positive at much lower levels- as noted above, the present value of the Mega Millions jackpot is decreased to only 60% of the advertised value due to the annuity factor alone.

Conclusion: There are two key lessons to learn from this analysis:
1.    Though the problem is complex (i.e., it requires many steps), it is not particularly hard (except for possibly the probability of splitting, which can be looked up or derived, depending on your inclinations),
2.    The required levels of lottery jackpots are reachable, particularly if a few of the variables are changed (as they are in different states/games).

Business forecasts and analyses are often analogous to this problem.  Consultants and employees alike are prone to presenting a “directional” answer and making simplifying assumptions.  The problem is that numbers, once presented, have a tendency of being remembered.  It is fair to say “we realized that the jackpot would have to be greater than \$ 400 million and thought that was so large as to be unlikely- we recommend you never play,” but it would be disingenuous to say “based on our analysis, you should not play the lottery if the jackpot is under \$ 1 Billion.”  While the latter statement is true, it is misleading, because it implies a more rigorous analysis that accurately identified \$ 1 Billion as the break-even threshold.  Having worked as both a consultant and a client of consultants, I can assure you that such misinterpretations are frequently made by busy leaders who have learned to remember numbers that seem important, but may not remember the full context or precise limitations on those numbers.

Postscript: if you’ve said that the lottery is “a tax on people who can’t do math” solely because the odds are only 1 in 200 million-plus, you’ve missed the analysis.  Jackpots can (albeit rarely) reach levels at which a pure expected-value analysis exceeds the cost of the ticket.     It is obviously possible to spend too much on lottery tickets.  But if you play only at the rational break-even threshold, it substantially limits your losses.   If (in an optimistic scenario) one jackpot game in your state reaches its critical threshold every other month and you decided to save and invest your dollar instead of “throwing it away” on a lottery ticket, you would save \$ 6 per year.  Doing so each year from age 18-67, even with a solid post-inflation compound interest rate of 8%, would only total \$ 3,000 (in present dollars), which would likely make no difference in your life, while one multimillion payout, however unlikely, would provide substantially new opportunities for most people.  The only truly risk-neutral approach is to perform the calculations and identify the levels at which it makes sense to play.   From a purely probabilistic perspective, insurance is also “a tax on people who can’t do math”: if an insurance company paid out on average more than the premiums it received, it would go out of business.  The reasons most people don’t describe insurance with the same derisive tone that they apply to lottery tickets are twofold: first, the law of diminishing marginal utility applies, meaning that we derive less utility (or joy) from a large gain than we lose from a large loss.  The second, less benign explanation, has to do with class-ism.  Wealthy people are likely to have substantial insurance policies, while the poor are more likely to take their chances (and to play the lottery).  It therefore makes for better cocktail-party “wisdom” to deride lottery-players than to mock purchasers of insurance.

Post-Postscript:  If you do choose to play the lottery at irrational levels, but are embarrassed by this lapse in your judgment, please contact me if you find yourself in the awkward position of having won a jackpot sufficiently large that you fear your friends and neighbors would find out.   I will purchase the offending ticket from you for the price that you paid for it, and keep your secret safe.