Someone told me something interesting today. They said they were going to take their bonus check from work, to the casino because they have "better odds" of paying off more debt then if they would just apply it to the principle balance of their outstanding credit card. Talking some more with the person they indicated they had about $20,000 in debt at about 20% interest.
Lets assume that their bonus check is $1000, they pay 500 a month towards their credit card, and they play a game with a 'better' chance of winning at the casino (craps, baccarat, etc) and have a 0.48 chance to win on the dollar.
We know their expected return is -0.04 per game, but since their owed interest and balance is so high; how would one go about calculating if their statement is true?
Clearly, this is not a wise decision in general, assuming constant utility per dollar. But let's explore it in brief, anyway.
We'll make some more assumptions. They put the entire grand on a single even-money bet, and win another grand with probability $0.48$, and lose their bet with probability $0.52$. That's it; then they walk home. (Obviously, this is not the way real people bet, but there's a limit to how much we can model. We just want to see if there's a non-linear effect one can take advantage of.)
On the debt side, they have debt at $20$ percent interest, and we'll assume zero inflation. (Now we're really fictional!) With those assumptions, then a debt of $D$ will get paid off in $n$ months according to
$$ \frac{500(1-r^n)}{1-r} = D $$ $$ 500(1-r^n) = D(1-r) $$ $$ 500r^n = 500-D(1-r) $$ $$ r^n = 1-\frac{D(1-r)}{500} $$
with $r = 60/61$. For $D = 20000$, this yields $n \doteq 64.513$; for $D = 19000$, $n \doteq 61.699$; for $D = 18000$, $n \doteq 59.009$. Paying down a thousand dollars of debt immediately saves your co-worker about $1357$ dollars; paying another thousand of winnings saves them only another $1345$ dollars. So there's a non-linearity, but it goes the wrong way. (Not surprising, really, since the further in debt you are, the more additional debt hurts you.)