I think there are enough grocery apps that it is better to show price data than not. Though I'm certainly baising that off how I shop rather than a general economics / game theory POV.

I regularly use the Ralphs app to cross shop while I'm at Trader Joes. They are only 2 minutes walking apart, so I normally start at TJs. However, sometimes I end up at Ralphs first and now having this data it could lead to an unplanned trip to TJs to save a few bucks.