I figured out how Deal or No Deal works (kind of)

Watch out, Banker

This is part three of a ?? part series in which I unravel the method and madness of Deal or No Deal. Want to catch up? Here are parts one and two.

I have reason to believe I’m currently in possession of the largest Deal or No Deal dataset outside of NBC’s studios. I have documented over 100 games, round by round, in meticulous detail, representing almost 800 rounds of play. And using this vast wealth of data, I’m starting to unravel this show.

Let’s see what’s inside

Now that I have all this data, the best thing to do with it is to start visualizing it. I ran pandas-profiling as part of my data validation and it gave me some great statistics and visualizations for each feature, but the most interesting thing I got was this correlation plot showing how each variable affects each other.

Correlation plot using Pearson’s r. Ignore level_0, which is an index and doesn’t exist. Positive values mean variables move together, negative means they move in opposite directions.

We can immediately see a few places where variables are highly correlated. The previous offer is highly correlated to the current offer, bolstering my observation that offers generally don’t vary wildly over the course of a game. The board average (or expected value) is also pretty highly correlated with the offer amount. We can also see that the game round is inversely correlated with the offer. At first glance this feels counterintuitive. But, most games aren’t barn burners, so late game offers reflect people trying to claw back some money after eliminating all of their big values.

So let’s do the next logical thing and just plot the offers in each round for every game (plus some other stuff). When I did that, I got this truly beautiful, unintelligible, chart:

Offers over the course of each game (colored lines), the expected value at each round in those games (faded colored lines), and the average offer in each round (thick blue line). Each colored dot shows where contestants took a deal in each game, so lines extending beyond that show hypothetical offers.

This is very pretty, if only marginally helpful, but there’s more we can do. We suspect that that the Banker’s offers follow some proportion of the expected value of the board, so let’s just plot that.

Offer Percentage of the Expected Value (Y) vs Round (X)

We can see a pretty clear upward trend, but it’s still a bit chaotic seeing every single game overlap. Let’s take a step back and remember what set me on this journey. I want to figure out how the Banker works and reveal their secrets to you, loyal reader. So seeing how every single game progresses isn’t that informative if I want to understand how the Banker operates. I want to strip out the specifics of each game and just look at the aggregate data. So, if I abstract things a little further using a box plot, I get this absolute gem:

@Banker, this you?

This is worth dwelling on for a moment. We can see pretty clearly how the Banker’s offers change over the course of the game. Just as people have theorized, the offers track pretty closely with a proportion of the expected value in each round. This is the smoking gun that I’ve been after.

This chart tells us a lot. It’s clear that there is an underlying pattern in the offers, but not an underlying algorithm, at least in the way I imagined it. Initially, I wondered whether the Banker was just a producer with an Excel sheet, crunching numbers and deterministically spitting out offers, but it’s clear that whether it’s a producer or the actor playing the Banker, someone is pulling the strings and making unique offers (at least during normal play).

This brings me to my second realization. While there is a certain amount of discretion in each offer, it’s driven by all sorts of things beyond the state of the board. The Banker works within a system, and that system has rules, but those rules bend and sometimes break. The Banker can make offers to lowball the contestant at key moments to add some drama to a game, to make a joke, or just out of pure, delicious spite. The Banker has guardrails, which they still sometimes go beyond, but for the most part, the offers fall within certain ranges according to the expected value of the board and the round. This deliberate chaos is something that can’t be done by a dumb computer.

While these realizations mean I can’t reverse engineer a perfect Fake Banker algorithm to run DoND games for me and usurp the Banker altogether, it means I can still create a semi-perfect Banker Approximator (patent pending) with some machine learning. And with that power I might know better than anyone except the Banker themself how the game works.

Luck, timing, and guts

That covers our favorite Certified Financial Analyst for now, so let’s turn to the contestants. Howie often calls DoND a “game of luck, timing, and guts.” A player needs all three of those things to really succeed in a game. I’ve seen so many contestants turn down their highest offer of the game and subsequently just Hindenburg their board, eventually walking away with a fraction of what they could have if they got out at the right time.

So, let’s take a look at what contestants walked away with at the end of their games against their best offers. Just for fun.

Only one person managed to beat their best offer by making a counter-offer late in the game.

In this chart, zero means that someone took the best offer of their game. We can see the majority of people don’t take their best offers and miss out on a lot of potential winnings. In fact, the average theoretical “loss” is over $100,000! That’s skewed by some big outliers here (RIP to the poor fool who missed out on $840,000) but even the median loss is $66,000. Contestants often push their luck just a little too far without regard for the timing.

<caveat>

I should say that this isn’t an evaluation of whether someone made a good deal based on what’s in their case — I’m just considering whether they should have taken a bank offer, not whether they should have held their case to the end. You could argue that those two things are related and you’d probably be right! Considering what’s in your case is a key element of the game. But since I was focused on the Banker’s offers, I often didn’t record what was in someone’s case unless they got to the very end, since that was what they won. So, I’m not able to say with absolute certainty whether these contestants made good or bad decisions. But most of them probably made bad decisions.

</caveat>

To win at DoND means walking away with more than the Banker expects you to just by chance. The fact the show exists means that NBC is willing to accept a hypothetical average payout of the $131,478 for each player (the average actual payout in my dataset is $108,332 so they’re doing just fine). So, if you can get an offer more than that, you’ve already beat the Banker. This is the fundamental misunderstanding of the game that the show exploits. Deal or No Deal isn’t a game about belief, no matter how much it wants to be. It’s definitely not about getting a million dollars in your case and sticking with it to the very end.

This is not how most players approach the game. There’s often a point in a game where everyone — the supporters, the crowd, and even our shiny boy Howie — actively encourage contestants in precarious positions to simply believe that their case holds the largest value on the board, when in fact, at this point they’re just gambling. This play is pure guts. Sometimes it works, and the contestant is rewarded commensurately for their courage, but more often than not, it fails catastrophically.

If you’re willing to approach those decisions as gambling, you can make better decisions during the late game. But if you rely on blind faith alone in the face of unfavorable odds, you’re probably going to implode.

A histogram showing in what round contestants take offers. By round 9, people are often committed to a bad board.

Of course, this is the crux of the show! It’s not very exciting to say, “Nah, don’t go for the million! Just try to maximize your winnings based on probabilities!” That show sucks! I wouldn’t watch that! We want to see someone stick it out against all odds and take home that million!

There are so many different factors in each contestant’s decisions that make the difference between walking away with a few bucks and walking away with a fortune. And while I’ve focused on modeling the Banker’s offers, I can’t pretend that the contestants decisions aren’t also influencing the Banker. For as much as the show tries to convince us, and as much as I’ve tried to prove otherwise, the Banker is just as human as the rest of us.

I can’t come up with a definitive mathematical model for whether someone should take a deal or not. At least not yet. But for now, I think we have to agree with Howie.

It all boils down to luck, timing, and guts.