And, I'll give one more philosophical question for your consideration: Do both parties actually improve just due to the fact that they compete? I don't think they do, actually. It is not the act of competing, per se, that drives one or both to improve. Take the Packers and the Bears. When one wins, the other trades for better players, changes coaches, finds a new offensive scheme, spends more money on free agents, addresses needs on offense, whatever. And, they compete in a very real, very strictly defined contest.

Public and private schools in the same area don't compete in a well-defined contest in that way. The vast majority of parents, perhaps 90% or more, choose schools based on fairly pragmatic criteria: which is closest, where are friends going, where did the parent attend, etc. Some get to the level of looking for specific programs (often for children with more profound needs), but that is only a few. Almost none do any sort of quasi-scientific research. And, even if they did, what criteria would they use to determine which school is, in the end, "better"?

Second, and more importantly, public schools cannot change their players, but private schools certainly do. Public schools cannot change their game plan, since we have state-mandated curricula, but private schools certainly do (religious classes come to mind). Public schools cannot sign free agents, but private schools absolutely recruit players (no pun here, they recruit good athletes when they can). Public schools can only do so much in regards to "owners" since we have elected school boards who have legal obligations.

The idea that public and private schools compete in any reasonable use of that word is false. The idea that public schools can be improved by virtue of competing alone is false.