A Game of Dice ¹

Hypothetical scenario: a Demon challenges you to a game of dice. He suggests 300 rolls of a six-sided die. At stake is your enormous fortune.

Here are the rules.

1. If the die comes up 1 you pay the Demon 50% of your net wealth.
2. If the die comes up 2, 3, 4, or 5, the Demon pays you 5% of your net wealth.
3. If the die comes up 6 the Demon pays you 50% of your net wealth.
4. You settle up at the end of every roll.

Should you play the Demon's game?

A Comical Calculation ²

In Black Swan Man's illustration of this problem, Mark "Spitz" Spitznagel calculates that playing the Demon's game will increase his fortune by 3.3%. He argues that -50% + 5% +5% + 5% + 5% + 50% all divided by 6 will net him 3.3% of whatever he puts down.

Let's for the sake of argument say that Spitz has $1,000,000,000 dollars laying around to gamble with. A little mad money for a little fun with a supernatural being. 3.3% of 1 billion is $30,300,000. And let's say that 1 round of rolling and settling up takes 1 minute. That's a little more than $6 million an hour. Not bad for 5 hours of work.

But is he right?

The Experiment - Part 1

Here's a Lisp script we can use to calculate Spitz's profit (or loss) after 300 rolls.

(define (dice) (random 6))

(define (wager roll) 
    (cond 
        ((= roll 0) 0.5)
        ((= roll 1) 1.05)
        ((= roll 2) 1.05)
        ((= roll 3) 1.05)
        ((= roll 4) 1.05)
        ((= roll 5) 1.5)))

(define (game fortune)
    (define (game-helper fortune rolls)
        (if (= rolls 0)
            (round fortune)
            (game-helper (* fortune (wager (dice))) (- rolls 1))))
    (game-helper fortune 300))

Our script simulates Spitz rolling the dice 300 times against his fortune and returns however much he won or lost.

1 ]=> (game 1000000000)

;Value: 82247.

A single game of 300 roles returns Spitz's fortune much diminished. So did Spitz make a mistake, or did he just get unlucky?

The Experiment - Part 2: The Multiple Worlds Version

Maybe Spitz was just unlucky on his first go around with the demon. They've just met after all. So what about 100 games of 300 rolls each with Spitz's billion-dollar fortune reset after each game? This will hopefully tell us whether it's ever worth playing the Demon's game, or whether there's something wrong with Spitz's math.

Here's an additional Lisp function so we can play a series of games.

(define (multi-game fortune rounds)
  (define (multi-game-helper rounds-left)
    (if (= rounds-left 0)
        0
        (let ((result (game fortune)))
          (display result)
          (newline)
          (multi-game-helper (- rounds-left 1)))))
  (multi-game-helper rounds))

And here are the results.

43187849.
470.
3359055.
5142377166.
32010346.
134931179284.
279973868.
5602863615.
6855214.
5142377166.
595046500641.
66116277849.
77.
276633.
20320.
622540402.
816250344.
110966.
1426712.
1665817028.
512.
2247493827.
1234684758.
264415401.
3172388.
190458414.
253897.
126.
138844183483.
1027661.
373228.
26204914.
1270490616.
13055.
207513467.
707532679.
604995531.
3765985.
4478.
9911447779.
1599550.
20227444447.
179874423.
329.
466993079406.
182047000020.
21162046.
388690640.
3319.
47055208.
719362.
63486138.
52841.
2312671148.
97190928438.
89612.
66116277849.
3210705468.
56626199.
7778513973.
20565642.
12478530.
179874423.
100009465363.
40787851.
122363553.
7559294434.
4856609412.
14992353254.
1294.
736844746905.
158524.
43187849.
889343431.
2312671148.
1722.
728051126.
57573.
34303246619.
3765985.
141165624.
7139215835.
0.
41970699.
5228356.
12330.
1331.
172719.
9517165.
518157.
261260.
3082982.
14992353254.
740224.
224.
154056.
47623554935.
4663411.
1075.
660240.

That makes an average return of $27,904,897,161.35. Except that Spitz only increased his fortune 30 out of 100 times. 70% of the time he ends up with less than he started with. 31 times out of 100 he ends up with less than $1 million.

Should Spitz Play the Game?

No. So, then what's wrong with his math? Spitz got his 3.3% by saying that - 50% + 5% +5% + 5% + 5% + 50% will get him 3.3% by saying, okay, the +50% and the -50% cancel out and the 5's add up to 20. 20 divided by 6 is 3.3%. Whatever I put at stake times 3.3 is what I should get if I play.

But it doesn't. At least not often enough to make it worth his while. Once in a while it works.

References