Briefcase Banker

The briefcase game show is one of those games you can explain in a single breath and then argue about for an hour. There are twenty-six sealed cases, each holding a different amount of money, from one cent up to a million dollars. You pick one case to keep without looking inside, and then you open the others one at a time. Every amount you reveal is an amount that is not in your case. So the game is a long, nerve-shredding process of working out what you might be holding by ruling out what you're not. Between rounds a Banker offers to buy your case for a guaranteed sum, and over and over you have to decide whether the sure money in front of you is worth more than the gamble still sealed in your hand.

We built this version because underneath all the noise the show is really a probability puzzle, and that made it an interesting thing to program. There's no skill in opening a case. That part is pure luck. But every Banker offer is a real decision that you can learn to make well, and most of our work went into getting the Banker to behave believably.

How the Banker actually thinks

Our Banker isn't reading your mind and he isn't cheating, even though at times it will feel like he's doing both. His offer comes from two numbers. One is the average of every amount still in play, including your own case. The other is how far into the game you are. Early on he lowballs you hard. His first offer is usually around a tenth of the true average, because he knows the urge to keep playing is strongest at the start, and a cheap offer is unlikely to be taken then anyway. As the board empties and the tension rises, his factor goes up, and by the final offers he's paying close to the real mathematical average of what's left. We deliberately modelled that curve on how the banker on the real show behaves, and it's why a disciplined player can sometimes get a very good deal in the late game.

What follows from this is that the expected value of your case and the Banker's offer are two different numbers, and they chase each other around the board. Say you have a bad round and knock out three of the top four amounts. Watch the next offer collapse, because the average just fell off a cliff. Have a great round where you only eliminate small amounts, and the Banker suddenly gets very generous. Most of the strategy comes down to reading the gap between the offer and the average.

So when should you take the deal?

It depends on how much risk you can stomach, and that's exactly why the show works. There are some real principles, though. If the Banker offers more than the average of the remaining cases, that's a mathematically good deal and you should think very hard about taking it. It doesn't happen often, and when it does it usually means he's nervous about a big amount still on the board. If the offer is well below the average and several large amounts are still in play, the odds favour holding on. Once you're down to a few cases the swings get enormous. With two cases left, one holding a dollar and one holding half a million, no formula can tell you the right move, and that's the point.

The trap most people fall into is fixating on the biggest number left on the board and refusing every offer because a million dollars is technically still possible. It stays technically possible right up until you open your own case at the end and find eleven dollars in it. The players who do well treat every offer as a fresh, cold decision and ask one question. Given only what's on the board right now, is this money worth more than the gamble? Whatever you were hoping for a few rounds ago doesn't come into it.

Notes from building it

The money board down both sides is the main information display, and we spent longer on it than on anything else. As soon as a case is opened, its amount greys out and gets struck through, so you can tell at a glance whether the top end of the board is still alive. The cases are the classic gold briefcases. Offers are rounded to numbers a real banker would actually say out loud instead of raw decimals, and each offer has a little deliberate randomness in it so the game never feels like it's just reciting a formula at you.