How can something be in Nash equilibrium without being Pareto optimal?

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.

Game theory isn’t my area of expertise, but just going from the definitions, it seems like a Nash equilibrium could be Pareto suboptimal if anybody can improve someone else’s situation without making anyone worse off, and there’s a sufficient lack of information so that you can’t reasonably expect the person you’re helping to be aware of it and feel obligated to help you out later on. But this is just a guess, and I know we have people here who know this better than I do, so I’d wait for one of them to answer.

The canonical example of a Nash equilibrium which is not Pareto-optimal is in the Prisoner’s Dilemma. Suppose you have two arrested suspects, each of which gets to choose whether to rat out the other or not. If they both stay silent, they each do 1 year in prison. If only one talks, then he goes free while the other does 10 years in prison. And if they both talk, then they both do 5 years. Consider the situation where they both talk: this is a Nash equilibrium, because every player has a strong disincentive to unilaterally changing their strategy: their sentence would go up from 5 years to 10. But it’s definitely not Pareto optimal, as both players would be better off if they both kept their mouths shut.

Yes, it hinges on the fact that players make their decisions independently, instead of contracting a cooperative strategy, but it’s not really that awkward, and it has nothing to do with being indifferent about choices that accrue benefits to others.

To put it another way, it’s a question of “non-local” optimization: a sequence of player strategy changes may end up improving things for everybody, even though some of the individual steps along the way, in isolation, hurt the player enacting the change. To be a Nash equilibrium is a “local” property, while Pareto-optimality is a “global” one (in more ways than one, I suppose, this characterizes the difference between the two).

Indistinguishable, thanks. For some reason I have been conceptually stuck thinking about making pareto improvements by forcing one player to change their strategy, not sets of players.