To me, this is similar to the music industry/pirating example. Find a black market, go to the source of the problem, improve the experience so that users will choose your legal method as the best alternative. In music/movies, it's providing easy-to-access, high quality streams and downloads; here it's providing a way to improve your experience at the ballpark.

Finding the right prices will be key, and take some time, but even less than optimal fee (from the stadium's perspective) will recoup some of the otherwise lost revenue.

How much can stadiums realistically expect to earn a night? Several thousand?