Week 6: Fish

After a whole bunch of calculations, I realized that my recursion formula had been completely wrong. This is mostly due to the fact that changing perspectives in a game that allows for trolling is hard. For example, after you lose a “50-50”, you may think that the opponent will take the entire set from you guaranteed, but the opponent does not know that. They don’t know how many cards you have, as well as whether you trolled or not.

Therefore,  I added a 4th variable in the formula for Expected payoff. Currently, E(a,b,c,d) means you have asked your opponent a questions from this set, your opponent has asked you b questions from this set, c is the number of cards you know your opponent does not have for sure, either because you asked them or because you own those cards, and d is the number of cards your opponent actually has in the set.

The advantage to using a 4th variable that represents the number of cards your opponent actually has is it makes the recursion a relatively simple equation, where you sum all the expected values of what can happen next based on the currrent 4 variables.

Moreover, this fixes the issue of the model potentially having the opponent repeatedly thinking you trolled, and thus asking you for the same card 2 rounds in a row (this happens with probability P^2, if you ask them for a card they previously asked for). This also differentiates the expected values of you before asking a question and you right after trolling on a question.