Lottoptimization

The good: waking up at 5:30 am full of energy due to the wonderful westbound reverse jet lag after returning from my vacation.

The bad: wasting the reverse jet lag the very next day by reading dumb stuff online until 1 am.

The ugly: blobfish. Also not pretty: the rigor level of this post. This is intended as lighter fare for my western readers recovering from the anti-voting post and my Russian readers recovering from a two week holiday drinking binge.

blobfish 2
How some Russians are feeling January 11th.

Walking into the office where I pretend to work while actually spending all day thinking of dumb puns to use on the blog, I was surrounded by colleagues cajoling me to put $10 into their lottery pool for the $1.5 billion Powerball drawing tonight. You can probably guess my take on lotteries from my last post, and also from the simple fact that I write a blog about statistics. This morning I happily indulged in a delicious anti-lottery rant, but at least my coworkers enjoyed a big box of chocolates I brought from overseas while I patronized them (I’ll promptly explain why I did that).

Since WordPress doesn’t support candy sharing (yet?), I’ll spare you the ordeal; instead of sermonizing on “lotteries are dumb” I’ll explore some ways to make the most of lotteries, whether you play them or not.


The saddest thing about lotteries is that my colleagues don’t even play for the hope of being rich, they each play because of the fear of being the only one left out when their coworkers quit and sail to Aruba. If I had one question to ask an employer about company culture I’d ask whether there’s an office pool for lotteries. I doubt that a tight-knit group of rational, savvy people who are driven by optimism instead of jealousy is going to extort each other for Powerball.

What to do in the face of office pressure? The reason I am happy to pontificate on the stupidity of lotteries at work is to credibly commit to never playing. My coworkers aren’t winning the lottery, but even if they did I would feel no regret that I didn’t participate because I know for certain that there was no way they could have talked me into it, especially after I publicly established myself as the anti-lottery guy. If I walk into the office tomorrow and half the people have quit I imagine I’ll get at least a double salary and a corner office, what’s not to like? By pledging to never play I align my hopes with my friends and root for them without even having to buy a ticket!

This optimistic thinking also protects me from getting personally outraged about lotteries. I used to get angry at the government for stealing money from poor, uneducated, desperate people. But the government also gives some money to poor, uneducated desperate people so maybe it’s a wash?  Also, politics are weather and more importantly no one asked me so there no sense in wasting outrage on it.

It’s hard to get worked up about my middle class peers wasting money on lotteries either. Is $10 spent on a movie ticket any better? The only truth regarding disposable income is that it will, in fact, be disposed of. Are dead bats a nobler purchase than a lotto ticket?

ooh_bat
Actually, that’s exactly the kind of rich person I’d be.

The argument that I’m most sympathetic to is that lotteries are a waste of hope, draining emotional energy away from “good” fantasies like shepherding the next billion dollar start up. Being the next Zuck is a more fun fantasy, (slightly) more realistic, it doesn’t expire twice a week and it’s completely free! It seems foolish to buy a fantasy for $10 instead of spending 5 minutes conjuring a better one in your imagination. But what about buying comic books to fantasize about being a super hero? We all have wild fantasies, we all function reasonably well despite indulging in them, and ranking these fantasies from admirable to contemptible seems pointless.

If you want to pay a few bucks to buy a couple of days of yacht-dreaming, I won’t judge. I’ll just tell you how to do it better, with numbers!


The reverie of lottery riches works because our brains can’t accurately process tiny or huge numbers. The pleasure of imagining yourself rich doesn’t diminish a thousandfold when the odds of winning go from 1 in a million to 1 in a billion, and it doesn’t increase a thousandfold when the jackpot grows similarly. It makes sense to buy the cheapest ticket possible for the smallest jackpot that’s still life-altering*. The latter is because larger jackpots attract exponentially more participants, decreasing the expected value of a ticket by raising the chances that you’ll have to split the prize with several others. The winners actually win more money when the jackpots are smaller, and who wants to fantasize about sharing money with strangers?

* life altering = making you richer than anyone you are friends with on Facebook. That’s really what you’re dreaming about, aren’t you?

Another reason to play smaller jackpots is the decreasing marginal utility of money: the more money you have the less each additional dollar makes you happy. $40,000 is a lot of money to most people, but not when it’s difference between having $4,501,328,000 and $4,501,288,000. The only thing that increases on the margin with more money is taxes.

The fantasy bubble survives only until the draw itself, so you should buy the lottery ticket as far in advance as possible to maximize fantasizing time. Of course, the government wants you spending as much as possible so draws are frequent and buying tickets in advance is usually disallowed. A tobacco company would never sell you a cigarette that you can enjoy for a whole year, will it?

You can get a discount on the cost of dreaming by committing in advance to flip a coin right before the lottery draw and buying a ticket only if it lands heads. This saves you half the money while not affecting your enjoyment much. Who cares if it’s half of an already astronomically tiny chance?  If the coin lands tails, it is only a few minutes before draw takes place and the relief of not having wasted the money replaces the anxiety. A professional fantasizer can buy almost a full year of Powerball dreaming for a mere $50.

If the above examples seem a bit contrived, the important point is that even silly behavior can be optimized. Everyone spends money on fantasies, you can still make smarter purchases. Thinking that a sports car will make women want to sleep with you is no saner than thinking lotteries will make you rich. If you can’t help yourself, you can still find a $9,000 sports car that will make you feel almost as sexy as the $100,000 version.

chicks car
Hot chicks in a sports car

Two last tips for those who play the lottery tonight:

  1. Tell everyone you know that you pinky-swear to donate half your winnings to charity. For example, a single Powerball winner might get close to eradicating malaria by herself! Since you’re not actually winning anything, you get free virtue signalling. The upside for humanity is that now you also have a self-image as a philanthropist and are likelier to actually go and donate a (more modest sum) to charity.
  2. When you come to work tomorrow not having won, consider yourself very lucky. If you make it to the office you probably weren’t hit by lightning twice either, which was likelier to happen than the jackpot. Avoiding both the riches and the voltage makes you decidedly luckier than average!

I’m not ready for a regular posting schedule but Putanumonit will be back before the end of next week. This time, we’re going to Mars!

19 thoughts on “Lottoptimization

    1. Now that you made me think of that, it sounds like a great idea for a future post or ten! I definitely have some thoughts on the matter and some reading I could do. Have you put any thought into it yourself?

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      1. I recall having read that one should
        1, buy experiences, not things (and as a sort of corollary, try to avoid upkeep costs (e.g a bigger house not only costs you more now))
        2, spend money on others, and not on oneself (for the fuzzy feelings)
        And in general, one should prioritize things based on the frequency of interaction with the things (e.g buying a 4×4 vehicle for offroad use is an inefficent use of money, not only since it costs more than your regular car, and has a larger associated upkeep costs, but also since you most likely overestimate the frequency of your utilizing it’s offroad capabilities. OTOH, as a personal example, I tend to brush my teeth with too much force, sometimes causing it to bleed (especially with my wisdom teeth still going for the last push to become full members of my oral cavity), and apart from the obvious solution (go gentler, you dummy!), an electronic toothbrush may ease this issue.)

        As it currently stands, I am not optimizing for hedonic pleasure, or only remotely.
        Having recently finished high school, my current goal is to get some sort of land tenure (I am not above even squatting, if need dictates, but the uncertainty of long-term ownership precludes enthusiastic investment into the land. I also sort of require (or really prioritize) having a largish area to build my projects on. (e.g living in an apartment does not lend itself as easily to some kinds of amateur radio stuff) (of course, largish is relative here. Larger than a 40 sqm apartment, yes. But even the smallest plots available around here are more than sufficient for my needs)

        As such, my current goal is more like getting enough money for a piece of land as soon as possible (one can occasionally find 1000 sqm lands for ~3500 USD around here. (which still can be more than one year’s wages.) but at around 20K USD one can reliably find something larger, for a similar price/sqm. After having a secure piece of land, I am confident in my abilities to provide for my needs (hell, after not freezing to death is provided for, I have not got much of a care for whether I live in a tent or the royal palace. Of course, one is easier for founding a family, which I DO care about.) but if some gentlemen came with a bulldozer, inquiring as to what business do I have here, there is not much I could do.)
        ‘Geo-arbitrage’ or (temporary) ‘economic migration’ is kind of easy in my case, but that still leaves me without knowing the amount of time to my return, the money accrued, and these sorts of things.

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        1. Re – first part: I completely agree with the experience > stuff paradigm, but I include in experience non-durable things. For example, I have the cheapest car in my neighborhood but I buy the most expensive bar soap I could find and it’s fantastic. It can be an interesting project to analyze things by amount paid per time used, i.e. $15 soap for 4 hours of showering or $15,000 car for 2,000 hours of driving. I’ll look into that.

          Re – second part: your life sounds fascinating, and it doesn’t seem like you need any help analyzing your own situation. I guess I’m an economic immigrant too, but I’m paying a huge rent for a tiny apartment in NYC so our land use goals are quite different 🙂

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  1. I am annoyed by this ‘lottery is irrational’ stance not because I have a problem with not playing (I never do) but because it exhibits a fundamental flaw I see too often in ‘rationalist’ arguments: the implicit assumption of a linear utility function. It is perfectly possible to model a selfish rational actor playing the lottery as long as his utility function is nonlinear wrt currency. It isn’t irrational to follow such a function because choosing a utility function isn’t a rational act to begin with. Rationality only enters into play when you choose actions with the aim of maximizing utility wrt a given function. You can use observed lottery participation to reconstruct human utility (or to explain the notion of money in anthropological terms) but you don’t get to call it irrational.

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    1. Leaving aside the fact that I explicitly talked about the non-linearity of utility on money, I don’t even see what your argument can possibly prove. If you assume ahead of time that anything people do is following their utility AND define rationality as “following utility” then of course not one person ever did anything that wasn’t perfectly rational. Except arguing semantics online 🙂

      Anyway, assuming that utility is whatever actions people already do doesn’t give you anything. I tried to figure out what people are actually trying to achieve: pleasant fantasies, social conformity, reduction of regret etc. Maybe I’m right and maybe I’m wrong, but only once utility is defined on outcomes and not actions it’s meaningful to talk about what actions can increase utility.

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      1. Exactly, you did mention non-linearity in “decreasing marginal utility of money”. If anything this makes the oversight worse, because it will follow almost as a corollary that if the utility of amounts far smaller than your net worth are disproportionally uninteresting, perhaps the utility of amounts far larger than your net worth (i.e. what you wish to gain from participating in a lottery) are disproportionately interesting to you. Which is my entire point.

        You are not wrong concerning possible additional benefits like “pleasant fantasies” etc. I am not disputing that. My point is that it is conceivable that you could play the lottery even outside of these, just for monetary utility. Let me construct an artificial and therefore glaringly obvious example: You have a disease and need a $100k operation or you will die in two months. You have $1k. There is no way you will gain $99k in two months in any way other than playing the lottery. Say investing $1 gives you a 1E-7 chance of winning $1E5. So investing your $1k in the lottery raises your chance of survival (in this contrived example) from zero to 1E-4, i.e. it is clearly in your interest to play if you are a rational agent trying to optimize your chance of survival. That’s because your situation has a built-in discontinuity for the utility of money at $100k. More realistic examples will not have strict discontinuities but you get the idea.

        Your criticism that the idea that you can deduce utility functions based on people’s behavior results in everyone behaving rationally, all the time is worth thinking about for a minute. It’s good to spend a minute worrying that your entire rationalist edifice is a null statement.
        But I think you will find that if you do think about it you will come to a different conclusion, i.e. that rational behaviour can be defined as self-consistent (no matter how crazy or unorthodox the underlying utility function), and that irrational behaviour can be defined as behaviour that is self-contradictory, i.e no utility function exists that is being optimized by the observed behaviour.
        In my book, that’s a perfectly satisfactory definition of rational vs. irrational behaviour and it certainly isn’t a pure game of semantics.

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        1. Dear heavens, were all 635,103,137 tickets for Wednesday’s Powerball sold to people who are dead today for lack of a medical operation?

          I didn’t write about these outlandish scenarios because I’m concerned with actual people (my coworkers and my readers) making actual decisions. The conclusion from your example isn’t that every decision can be understood as rational under a model of pure utility of money, but that it’s possible to construct a contrived mathematical example to justify any observed behavior. The constructed example will bear little resemblance to the context of people’s actual thoughts when playing the lottery.

          I believe that utility exists outside observed action (i.e. in your thoughts), so consistency is just one of the requirements for rationality. Even if all you care about is consistency, what do you make of the majority of people making inconsistent gambles in Allais’ paradox or Ellsberg’s Paradox situations?

          Finally, here’s a tip for you good health: roulettes pay out 97 or 94 cents on the dollar, lotteries pay 60 at best even before taxes. If you need $100,000, go to Vegas and put $77 on a number twice in a row. If you need $100 million, take the $100,000 from Vegas and buy an extreme-value call option on a volatile stock. If I was going to die next week for lack of a nine figure sum, I would at least do my research on maximizing my odds.

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          1. look: I took care to signal that I am aware the example was contrived. I was illustrating a principle. I could construct a more realistic example, but it would be lengthy and not necessarily worth the effort. I do think there is an interesting discussion in this, and I am interested in having this discussion with somebody who is coming from where you are, but I cannot make you want to have the discussion.

            I guess here are my two main points:
            a) it is a difference between evaluating rationality when the utility function is known and when it is unknown
            b) when judging other people’s behaviour, it is reasonable to start out by assuming it is unknown
            c) even when evaluating YOUR OWN actions, you should think long and hard about whether your utility function is in fact known, because you are HUMAN, meaning you have lots of unconscious incentives.
            d) (extra credit) rationality is orthogonal to ethics

            I am perfectly willing to agree that people taking inconsistent actions are acting irrationally, this is part of my point (my actual point is that people not acting inconsistently cannot be claimed to act irrationally unless their utility function is known). It you who concluded that according to me any behaviour whatsoever can be made out to be rational, and I then disputed that this was the case.

            Ok, so if you think the above points are adding anything of interest, you are welcome. If you do not think so, well, it’s your blog and I don’t have a problem to find myself in disagreement with people on the internet.

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          2. fab, let me start by expressing my gratitude that you are taking the time to engage me on these questions. Mi blog es su blug, etc.

            In my last two responses, my main point was that lottery can’t be seen as rational if we only look at the value of money. By “can’t” I don’t mean a hard mathematical “can’t ever”, I mean it anthropologically: if you ask a lot of people how happy different amount of money make them their preferences would be consistent with the lottery being a poor choice. People play the lottery because they are poor intuitive reasoners about money and other considerations (social pressure, fantasizing) easily override their judgment. Even if you had constructed a more realistic example of utility of money alone, would it really apply to more than a fraction of the hundreds of millions who played?

            One quibble about your point “c”: if my utility is unknown even to me, then where does it exist? I use “rationality” to mean “acting optimally to achieve one’s goals”, with the goals meant to be the goals of your conscious self. The goals themselves don’t have to be “rational” – for example I have a goal minimize possible regret of not joining a Powerball pool. Whether regret is “rational” or now, I know I’m susceptible to it and want to avoid it for example by precommitting. Inasmuch as one’s conscious self is a poor planner for the future, or inasmuch as it cedes control to the instant gratification monkey and other distractions, one is acting irrationally.

            If rationality means something else to you, then we need to establish new terms. I don’t start from an assumption of ignorance: I know what I want, have a decent idea what other people want, and the occasional glimpse into what women want 🙂 To me, rationality is about making better decisions in my own personal life and the lives of humans whose brains work similarly, not about an abstract game theory concept.

            And regarding point “d”, most if not all people’s actual ethics as implemented are inconsistent and self-contradictory. Rationality won’t prove that an ethical position is right or wrong, but it can straighten out a muddled system of morals.

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  2. The value I got from putting money into the lottery is that I got to talk about what I would buy with my parents, friends, and I got to daydream about being rich; dreaming about being rich and luxuriating in my expansive villa with multiple pools and hot tubs is almost as pleasurable as the thing itself.

    In no way is the lottery illogical because it is a stimulant to the imagination.

    By being the “anti-lottery” guy in the office, you get a chance to signal how much smarter you presume yourself to be, which gives you pleasure, and in a way, you are a free-riding parasite on the lottery because you are extracting value at no cost to yourself. You even got material for a blog post!

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    1. I agree with everything you wrote, I just have one question: how many pools do you need in your villa? Wouldn’t you just hang out in the best pool all the time? Couldn’t the other pools be converted into something else? Do you need a quiet pool for when someone is shooting a dance music video in your main pool?

      My fantasies of riches include a strictly limited number of pools, maybe I need the lottery to stimulate my imagination!

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      1. Infinity pool outside because it looks cool. Sauna adjacent.
        Indoor pool for winter.

        Jacuzzi, one in master bathroom , one outdoors adjacent to infinity pool.

        “Endless” (workout) pool for exercise in the gym area. This will be full of salt water rather than chlorine because I will use it most often.

        I see no reason not to do this if I am very wealthy.

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