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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.

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Can you assume anything about the distributions? All methods I know will assume some parametric model and train the parameters given the data. Some models are more general than others (and some people skip the assumptions and train any model that is known to yield usable results anyway) but without any assumption, I doubt there is much you can do (in a rigorous way). –  Raphael Nov 20 '12 at 9:30
@Raphael you can assume that players will favor the highest value they can get after dividing the object among the people that choose it, can you post an answer with the idea you have? –  Mike G Nov 20 '12 at 23:05
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