I don't think a full understanding is needed. How about a Nash equilibrium for the students, given that the schools report their preferences honestly? No collusion allowed in this model, just each student choosing on their own. Charles Greathouse Analyst/Programmer Case Western Reserve University
--doubt it. See, it is kind of Yet Another Big Lie, that economists like to tell the world, that Nash Equilibria are useful. Well, maybe they are useful for something, but I can tell you that in voting theory (voting systems with N voters = N-player game) Nash equilibria are extremely extremely useless. Essentially every election is a Nash equilibrium since no single voter can alter result. Totally worthless. Anyhow, back in our children-schools problem, is a "stable marriage" a Nash equilibrium with honest scoring? I think not always. It is locally optimal in the sense of "stability" with respect to the (assumed honest) rank-order inputs, which also provides a sense in which honesty is a pretty good strategy. Meanwhile is an "optimum matching" with honest rating-vectors as inputs, a Nash equilibrium? Again I doubt that. But again, it is optimal in the sense of maximizing total used-rating sum. This unlike for stable marriages, is a global optimality notion, not merely local. Also generically unique also unlike for stable marriages. And again this suggests honesty is a pretty good strategy. If you knew everybody else's rating (or ranking) of everything, then had the power to change your own ratings (or rankings) while keeping everybody else's fixed, then you could exhaustively search the space to find the best possible strategy. But that is not the real world. And normally, by the way, Nash equilibria refer to RANDOMIZED strategies, where you provide every possible rating(or ranking) each with some probability, and then consider perturbing the giant vector of your probabilities, to achieve maximum expected reward. That scenario is in most situations like this, extremely extremely useless and unrealistic. I was speaking above of deterministic strategies.