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Ralsina.Me — Roberto Alsina's website

The lottery as a rational investment.

There is a prej­u­dice that the poor play lot­ter­ies be­cause they are lazy, can't save and are gen­er­al­ly stupid and are hurt­ing them­selves by chas­ing the fan­ta­sy of win­ning in­stead of sav­ing pen­nies. You know what? It's bull­shit.

When I was in high school (about 13 years old), I once had a plan to make mon­ey: I would play the lot­tery. Here's the mech­a­nism I had in mind.

I would play $1 in the quiniela. Quiniela pays $700 for each $1 you bet, and you have to choose a num­ber be­tween 000 and 999. My idea was: I can bet the $1 my par­ent give me ev­ery day, and there's a chance I make $700. If I had $700 I could buy any­thing a 13-year old kid may wan­t. With $1? Not so much.

Of course you are right now think­ing: What a mo­ron! He has a 0.001 chance of win­ning and it pays 700 to 1, so it's a los­ing bet! Bzzzzzt!

Let's start with some sim­ple sim­u­la­tion code:

import random

n = 476

for tests in range(10000):
    for w in range(1000):
        q = random.randint(0,999)
        if n == q:
            break

    print(w)

Short ex­pla­na­tion: run 10000 sim­u­la­tions of this pro­cess:

  • We play each day for 1000 days.

  • If we win, we stop.

  • If we don't win in 1000 days we stop.

  • We record the num­ber where we stop.

So, I ran it. Here's a graph of the re­sults

histogram

So, how many nev­er won any­thing? In my da­ta set: 3699 play­ers out of 10000 nev­er won any­thing.

How many ac­tu­al­ly lost mon­ey? 5030 play­er­s.

And how many won mon­ey? 4967 play­ers won mon­ey.

2910 play­ers won in less than 350 plays.

3 play­ers got ex­act­ly even mon­ey, win­ning in their 700th play. For them, this was ex­act­ly the same as sav­ing their mon­ey.

So, is it a good idea to play a lot­tery like this? It's a coin toss. Half the time, you end with no mon­ey. Half the time, you end with more mon­ey than if you had saved.

If you are bet­ting dis­pos­able in­come (a suf­fi­cient­ly low amount that "it does­n't hurt"), it works out. You have a fair chance (50%) of a re­ward at least as good as sav­ing the mon­ey, and a de­cent chance (25%) of a re­ward twice as good.

And you have a fair chance (50%) of los­ing mon­ey. But you would lose it very, very slow­ly and pain­less­ly. ¿How well do you think stocks com­pare to that? ¿And what are the bar­ri­ers to en­try on both games?

In short: play­ing the lot­tery is not ir­ra­tional, re­al­ly, it's just a sav­ings plan. It sure was a bet­ter idea than buy­ing can­dy!

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