Several recent studies in experimental economics have tried to measure beliefs of subjects engaged in strategic games with other subjects. Using data from one such study (Nyarko-Schotter, 2002) we conduct an experiment where our experienced subjects observe early rounds of strategy choices from that study and are given monetary incentives to report forecasts of choices in later rounds. We elicit beliefs using three different scoring rules: linear, logarithmic, and quadratic. There are differences between the elicited beliefs under quadratic and logarithmic scoring rules in spite of both being proper scoring rules. The (improper) linear scoring rule frequently elicits boundary forecasts as theory predicts, and is poorly calibrated. We compare the forecasts of our trained observers to forecasts of the actual players in the Nyarko-Schotter experiment and identify several differences. There was a significant positive correlation between observer forecasts and the choice behavior in the game under both proper scoring rules, but no significant correlation between the players' own forecasts and the actual play. This raises doubts about whether beliefs can be reliably elicited from players who simultaneously have a stake in the target of their forecast, in this case the opponent's choice. The distribution of player forecasts also tended to be more extreme than the observer forecasts using either of the proper scoring rules. We also find evidence of belief convergence when beliefs are elicited iteratively from a group.