Guessing the best choice to maximize returns There are $N$ number of people and $X$ amount of objects with different values. Each person will choose an object and will obtain that objects value. If multiple people choose the same object then the value of the object is shared among the people. For example, If there are 2 objects and 3 people, one of the objects will be chosen by at least 2 people, thus the value of the object is divided over 2. 

Each round the people make their decisions with new objects and new values, how can a person maximize their expected values accumulated from their choices in Q number of rounds, in order to win(i.e accumulated most value)? ($Q$ belongs to [1, +infinite).

Note that keeping track of other player's decisions and their accumulated values can help in future decision making.

Note: An approximation would be great as well, so far my strategy is to choose a random object at each round, I am looking for ways to maximize this strategy.
One simple way, better than random choice, is to use modification of one step ExpectiMax. Each round you will give each Objects Value some new estimated value V(obj).

V(obj)= $\frac{ObjectValue(obj)}{ ( 1+ \sum\limits_{i=1}^n P_{i}(obj) )}$

where $ P_{i}(obj)$ = probability that person i will chose Object obj

And every round you will chose the one Object with greatest estimated value V.

You can choose at the beginning  $ P_{i}(obj)$ to be some function over the Object Value- the bigger the value the greatest probability that it will be picked.

Learning:

From your question "Note that keeping track of other player's decisions and their accumulated values can help in future decision making." and your comment  "(learning opponent strategy under uncertainty) " I guess that you need to learn the strategies of your opponents.

One way is after each game to see which Object Value was picked by each of your opponents and how the picked value differs from your expectations ($ P_{i}(obj)$). Using this information you can slightly modify the probabilities $ P_{i}(obj)$ (decrease or increase them a little for specific Object Values) and use the improved probabilities for each of the opponent in the next rounds s.