Week 9: Fish

After a series of more algebra bash, I have realized that it is practically impossible to minimize or maximize a fifth degree expression that contains 4 variables using lagrange multipliers.

Though, I have concluded that at least 2 Nash Equilibria must exist in this game of one set fish with trolling. One where neither players troll at all, and ignore the other player’s question, and another one where both players follow the exact same mixed strategy.

On top of that, I tried computing the actual expected values using a C++ program, plugging in actual values for the different constants we assumed (using actual frequencies in games), and obtained expected values that are overall lower with a less number of initial cards, and higher when the opponent has less cards. This fits with my expectations, as you are more likely to get the entire set when you have more cards from the set.

One exception to the trend would be when the 2 players have the entire set, in which case it is almost guaranteed for one player to take it, but other than that, the trend seems fairly consistent. Moreover, after running the recursion with different initial trolling probabilities, the strategies don’t change that much, except for when a player trolls 100% of the time, but that does not make sense anyways.

With this, I concluded my research, and started formulating my thoughts and findings into a final paper.