Layman trying to grasp nuances here. Be gentle, please! Apologies in advance if this post is very disjointed; the question in the subject is the important one.
In my musings on all things thingy, I stumbled across wikipedia’s entry on Holmstrom’s theorem. After locating the paper and reading through it about a zillion times I understand the main thrust of the argument (and would like to take issue with the phrase “budget-balancing” which is not what I initially thought it meant), which is basically exactly as wikipedia lays it out.
But here’s the part where my stunning lack of technical knowledge bites me in the ass. How could you ever have a situation where it was not Pareto optimal but still was in Nash equilibrium? As I understand the latter, it is that people have no incentive to alter their strategy. As I understand the former, it is that someone could be made better off without making anyone else worse off. But if someone could be made better off without making anyone else worse off, how could this be a stable solution to a payoff matrix? Why wouldn’t this individual choose the alternate strategy?
If it helps, which it may not, depending on where my understanding is failing, the paper in question expresses it thus:
s is the sharing rule, x is the monetary outcome to be shared, a is the action taken, and v is the private (nonmonetary) cost.
How does this particular a* “satisfy the condition for Pareto optimality”? What would violate it, but still be a Nash equilibrium? As stated, that’s exactly what I would expect the equilibrium to be. What is the extra bit that somehow satisfied the condition for Pareto optimality?
I think I have the answer to the question in the OP, but…
Conceptually, is it possible that Pareto optimality is violated because one person couldn’t benefit themselves by changing strategy, but could benefit someone else without bearing a cost; but, since this is a non-cooperative game, this won’t happen. That is, person 2 (say) could alter their strategy, receive the same payoff under the sharing rules, but benefit person 3 (they got a bigger payoff)? In my head it seems plausible that this is how we could get into a situation where there was a Nash equilibrium but no Pareto optimality, but then it seems that the non-cooperation angle is an artificial constraint on the situation, as bargaining by (in this case) person 2 to alter their strategy to share in the benefits 3 gets would violate non-cooperation, even if it seems like the most natural result. Yes, sure, this sounds plausible, but it can’t be the case, as then it could just happen that person 2 chose this action which had this unintended consequence of bettering person 3, and the theorem suggests that there’s no way to unite these three conditions (budget-balancing, Nash equilibrium, and Pareto optimality), one of the three must be violated. Unless it means that the non-3 decision and the pro-3 decision are a matter of indifference for person 2, meaning that person 2’s optimal long-term strategy is a coin toss strategy (or a six-sided die strategy, or…), meaning person 3 could be better off if 2 would just express a damn preference to help 3 and cooperate (which returns to the artificiality of “non-cooperation”). So therefore this theorem is really asserting that there must exist for at least one player a set of choices over which the player is indifferent, but for which the benefits accrue to another player? The whole theorem hinges critically on non-cooperation?
Whenever I go off on a thinking spree like that paragraph, I feel I’ve either settled the matter or headed off in completely the wrong direction. So, any help, fellow dopers? Thanks.