Guessing the best choice to maximize returns
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.