I have a linear model with 6 IVs and would like to analyze the effect of an interaction term applied to all the IVs.

To illustrate, let's say we're predicting the Win/Loose ratio of NBA basketball teams based on a number of players statistics and we want to add the number of spectators coming to the games as an interaction term to all the predictors. The idea is that a higher fans participation in the stadiums will leverage the players skills. Vice-versa if stadiums register low participation (look at the Nets), it will negatively affect the players ability to perform at their best or average levels (side note: we do not want to use the number of spectators as a predictor *per se*).

In MLR terms the model would be:
$$
\hat{Y} = c + b_1X_1 + b_2X_2 + ... + b_nX_n + a_1I_1X_1 + a_2I_2X_2 + ... + a_nI_nX_n$$

Where $X_n$ are the players statistics and $I_n$ is a measure of crowd participation.

If the players skills set (skills IVs) is large, the interaction term will double the model terms, with a higher chance of over-fitting the model data and probably decreasing the predictive ability of the model.

Is there a different methodology that is better designed to adjust the linear coefficients given one or more "background" variables?