To play with this, I got data for 1 season (2006) of the US show from a paper on player behavior. For some obvious questions you might have...

#### Are the offers consistent?

An immediate question I wanted to answer is 'does a player with a better board get a better offer?' The answer is 'usually', but there were many instances where this was not the case. Some specific examples...

- round 8, player 1 had $50, $200, and $1,000,000 cases remaining and was offered $267,000
- round 8, player 2 had $400, $1,000, and $1,000,000 cases remaining and was offered $215,000

Player 2 clearly had the better board but received a much lower offer.

- round 8, player 1 had $0.01, $400,000, and $750,000 cases remaining and was offered $375,000
- round 8, player 2 had $25, $500,000, and $750,000 cases remaining and was offered $359,000

Again, player 2 clearly had the better board but received a lower offer.

Interestingly, if I restrict it to offers made in rounds 7 or 8 that were worth more than $100,000, 9 of 26 fit this pattern (player received an offer better than someone who had a better board). I'll cover another topic briefly then come back to this for speculation.

#### Does the bank ever offer more than the board is worth?

What is the board 'worth'? The most obvious answer is just the expectation value of the cases on the board. If you have 3 cases with $100, $500, and $1000 in them, the board's expectation value is ($100 + $500 + $1000)/3, or ~$533. You'd think it never makes sense for the banker to offer more than $533 for that board right?

Turns out, for this season, 16 of the 62 round 7 and round 8 offers were for more than the board was worth using the definition above. Why would this ever make sense?

#### Speculating so far

There could just be a random noise generator weighting each offer to keep things interesting. There are some legit things that might explain the two questions above though that I can't answer confidently without more data. An idea that came to mind is basically...people get sad watching someone lose horribly, and the show might lose interest if that happens too often.

In one instance of this pattern, a player had the following cases: $5, $75, and $400,000. That board is 'worth' ~$133,000, but the banker offered $137,000. It's sad if people in that situation commonly end up with $75, so the banker might offer more than it's worth just to keep that from happening.

A sample safer board that got an offer below the board's worth had cases $200, $50,000, and $75,000 remaining. The offer was $35,000 for a board that's 'worth' ~$42,000. If the next case opened was $50,000, the player would receive an offer of $37,500 so there's no catastrophe. Thus, I think there's likely a random component added to each offer that can be tuned up when they want to encourage the player to accept the offer.

This might not be what's happening, but I can't think of a better reason for why the banker sometimes offers more than the expectation value of the board.

#### Modeling the offers

Now to answer the title question...can we reverse-engineer the model? Because of the above, I think it's impossible to get exactly. Further, I have no idea what type of model they're using. It could be a giant decision tree. It could be some random multiplier on the expectation value. It could be regression on, say, 5 parameters. I can get pretty close to the results though using the available information.

To get it out of the way, here's what you get if you simply use the board's 'worth' from above:

That's too simple to be exciting, so it's wrong.

The most intuitive simple model to me thinking about this problem is:

"Offer = constant * expectation value of board" where 'constant' is fixed based on the round

"Offer = constant * expectation value of board" where 'constant' is fixed based on the round

Fitting the data to that, I get the following constants for each round (round # is list #):

- 0.11
- 0.24
- 0.38
- 0.49
- 0.6
- 0.72
- 0.85
- 0.85
- 0.99

This makes sense. In round 1, they don't want you to stop and there is a huge spread of outcomes left so the offer is so low that no one would accept it (11% of the board's value). By round 9, you have 2 cases left, so they just offer the average value of those two cases. In-between they build drama and steadily make the offers more attractive.

How well does that predict the actual offers?

That actually has an r^2 of ~95%, so we likely won't do much better with any sort of linear regression.

Another approach is to try something like a random forest regressor. Using that with that weighted average, standard deviation of remaining case values, largest remaining case, smallest remaining case, and round number as features, I get:

Another approach is to try something like a random forest regressor. Using that with that weighted average, standard deviation of remaining case values, largest remaining case, smallest remaining case, and round number as features, I get:

That's a bit better, but it's likely overfit and I don't have enough data to split into large training and test sets. The simple weighted expectation value above works much better than I'd expected, so that's a decent model I think for this.

One cool thing about the random forest regression is that you can get the importance of each feature. Those importances are:

- round-weighted expectation value: 0.96
- standard deviation of remaining cases: 0.02

and all the rest are less than 0.01. Running linear regression with spread included (so model is 'offer = C1*round-weighted average + C2*standard deviation of remaining cases) yields basically the same as just the round-weighted average:

#### Conclusion

It looks like a simple model of 'offer some % of the average of the remaining cases where that % depends on current round' works well enough, and in reality they likely add some noise and probably alter it a bit as needed to keep interest/ratings up.

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