I mathematically proved the best "Guess Who?" strategy [video]

One thing this doesn't take into account (and the paper acknowledges this) is that the characters are assigned by picking cards from a deck. So the two players cannot have the same character.

Taking this into account would make the game much more complicated, because it can introduce an element of bluff.

For a simple example, imagine that there are only 5 characters. On your first turn you know the opponent doesn't have the same card as you, so you've got 4 options remaining. You'd like to ask a question that splits them into 2+2, but if you do this then the card you're holding will make one of the groups into a 3. Your opponent will know that your card is one of the 3, so you've effectively given them a head start. Instead you might sometimes want to split the options 3+2 with your card in the 2, as a bluff.

How often you want to do this must be described by some Nash equilibrium probabilities. It would be interesting to set up a linear programming solver to find these exactly, but so far I haven't had time to set this up. I don't know if it would be practical to solve the full version of the game with 24 characters.

The idea is that if you're winning you can just do a binary search, but if you're losing, it's better to take some risks by making narrower guesses.

For example, let's say it's the last turn and your opponent is about to win. Say you may have 2 options but your opponent has 4 options. Instead of whittling it down to 2 options, it's better to guess one of the four. How outrageous should your guesses be is the content of the result and paper.

Paper is on archive (and linked from the video):

https://arxiv.org/abs/1509.03327

In the video, in the continuous version the game never end and highlight the "loser" strategy.

When you are behind the optimal play is to make a gamble, which most likely will make you even worse. From the naive winning side it seems the loser is just doing a stupid strategy of not following the optimal dichotomy strategy, and therefore that's why they are losing. But in fact they are a "player" doing not only their best, but the best that can be done.

The infinite sum of ever smaller probabilities like in Zeno's paradox, converge towards a finite value. The inevitable is a large fraction of the time, you are playing catch-up and will never escape.

You are losing, playing optimally, but slowly realising the probabilities that you are a loser as evidence by the score which will most likely go down even more next round. Most likely the entire future is an endless sequence of more and more desperate looking losing bets, just hoping to strike it once that will most likely never comes.

In economics such things are called "traps", for example the poverty trap exhibits similar mechanics. Where even though you display incredible ingenuity by playing the optimal game strategy, most of the time you will never escape, and you will need to take even more desperate measures in the future. That's separating the wheat from the chaff from the chaff's perspective or how you make good villains : because like Bane in Batman there are some times (the probability is slim but finite) where the gamble pays off once and you escape the hell hole you were born in and become legend.

If you don't play this optimal strategy you will lose slower but even more surely. The optimal strategy is to bet just enough to go from your current situation to the winning side. It's also important not to overshoot : this is not always taking moonshots, but betting just enough to escape the hole, because once out, the probabilities plays in your favor.