How to Win Credit Card Roulette
Note: All graphs are for a bill of 50 people, each with a $20 share
Picture this: 20 people at a brewery, each ordering an entree, a couple beers, maybe even some fries to split. Eventually the bill comes. How do you split it up? You could have everyone go through it, figure out what they ordered, what % of the shared fries they’re responsible for, and what percent to tack on top of that to take care of tax and tip.
Or you could put everyone’s credit card into a hat and pick one poor sucker to pay for it all. That is the essence of credit card roulette, or CCR.
The way that it’s typically played is that everyone puts a credit card into a hat, and then cards are pulled out one at a time until only one remains. That card has to pay for the meal. One additional rule, which makes the game a little more interesting, is that those who don’t want to participate can be “bought” out by somebody. What that means is that in return for $20 (or whatever their share of the bill is), I put an extra card of mine in the hat instead of theirs. That way they just have to pay for their own bill, and I have a chance of actually making money if I don’t end up having to pay the whole check.
It’s fairly obvious that if everyone’s contribution to the bill is equal, then the expected value of playing CCR is the same as just paying your own bill.
And, of course, no matter how many people you buy out, that won’t ever change.
But expected value doesn’t tell the whole story. Intuitively, you might think that buying out increases your risk. After all, you are putting more cards into the mix, which means you have a higher chance of losing. But all those losses are somewhat mitigated by the guaranteed payout from the people you are buying out! The safest move in CCR, in fact, is to buy out everybody, which guarantees that you only pay for own meal. So depending on your appetite for risk, you can select how many people to buy out. Also note that by participating, you are basically buying out yourself (since you make $20 or whatever in the form of your food).
You can see in the graph above that the “riskiest” move, the one with the highest variance, is to buy out half the people. That means you have a 50% chance of losing half the check and a 50% chance of winning half the check. Also note that buying out everybody is the same as not playing.
You might also consider whether you care more about losing a lot and winning a little, or losing a little and winning a lot. The more people that you buy out, the less money you stand to lose and the more you stand to gain (until you’ve bought out everyone, at which point you can’t make money). You could even win the entire cost of the check, minus one person’s bill!
The far right of that graph sure looks good, doesn’t it? But bear in mind you have an extremely high chance of losing a little money in that case. Since all strategies have an equal expected value, which one you choose in this case is really just a question of how loss averse you are. Frankly, looking at that graph makes me want to start buying people out more. It seems more fun to occasionally win $480 at the cost of frequently paying $20 then it does to avoid paying $20 at the occasional cost of $480.
However, there is a situation where CCR’s expected value is not level! Rarely are peoples’ actual shares of the bill equivalent, and in those cases, the people who spend more have a better expected value.
Also note that it lowers your expected value to buy people out who spent less than the average amount.
“Ah, so I should spend the most then, and not buy anyone out”, think you, the clever reader. Yes, but of course there comes a point where you don’t want more food or even (perhaps) more beer. Additionally, if you spend more for the sake of improving your position in CCR, the only gains come during that meal. It doesn’t increase the actual amount of cash you could win, and increases the amount of cash you could lose. What you really want to do is get the benefits of a higher bill, without being forced to put all your money into food.
Now is where we get to the winning move in CCR. Buy out anyone who spent more than the average on the check! If you do this enough you are guaranteed, statistically speaking, to win money playing CCR.
This introduces a little bit of game theory into CCR. Ignoring the battle over buying out anyone who spent more than the average, you can increase your EV by ordering more. But so can everybody else.
Looking at this from a game theory perspective, there is no Nash Equilibrium here. No matter what strategy your opponent is deploying, it is always in your interest (in an expected value sense), to order more food.
Unfortunately, there is no easy solution to this, so you’ll have to be a little honorable when playing CCR. And be careful trying to max out your expected value, because a high expected value won’t help you when you can’t get up from the table!
P.S. You can solve all the “problems” with CCR by adjusting everybody’s odds to match the % of their contribution to the check and picking somebody with an random number generator. This is what the developers at Everlaw do, although it’s certainly less exciting than pulling cards out of a hat.