There are N players and M objects, each of the objects has a value. Each player has a strategy in choosing an object. Each round a player will choose an object, many players can choose the same object. However the value of each object is divided evenly among every player that has chosen it. There will be 9000 rounds(choices) per game. Our goal is to maximize the values that we accumulate at the end of the game.
Question: how can I build a probability distribution function for each playing assuming that their decisions are random variables?
Current Approach: My current approach is to count the frequency of a player choosing a specific object and dividing by the total number of rounds, that would give a probability a player is likely to choose that specific object.
Problem: With each player playing aggressively trying to be unpredictable as possible(noise), with my current approach the probability distribution functions are not accurate(9000 rounds doesn't seem to be enough data). Is there a better way to build these distribution functions?
Note: I've read somewhere that (Bayes model and HMM) are more superior than frequency counts, but I am not sure how to adapt it to this situation.