First, and I don’t mean this disrespectfully, why should we care? Trying to generate pareto optimality through a series of games played by randomly but fully paired-off players, with no potential for trade, doesn’t resemble anything I can think of. This is particularly true because players of games have little interest in pareto optimality; the very concept of “game” implies “competition,” and the concept of “pareto optimality” implies “cooperation.” Indeed, this assumes that trade is impossible, or else making game outcomes pareto optimal is redundant. While I may in some sense prefer pareto optimal outcomes for games, what most players are assumed to care about is maximizing their own utility, which is largely unrelated to pareto optimality. Perhaps my imagination is insufficient, but this doesn’t seem applicable to anything.
Second, the =1 seems arbitrary, because there are no units. So long as none of the factors are zero, the product will be some number that could be scaled to 1. I don’t think undermines your point; it just seems odd. FUrthermore, it’s interesting to note that there is always a consistent scale in which the product equals zero, even if there is another where it equals 1. I’m not sure if this is significant; again it just seems odd..
Who said there is no potential for trade? Each game could be exactly that—a trade—where the “game” is simply dividing up the gain from trade.
The product of the three theta cannot be scaled. First of all, the theta are unitless (as a ratio between two utility functions), so you can’t scale them by changing their units. You can change individual thetas by scaling the u1, u2 or u3, but this will not change their product: scaling u1, for instance, will scale θ12 one way and θ31 the opposite way, cancelling out. The =1 is well defined.
If trade is permitted, and there are no transaction costs, all outcomes are pareto optimal. That’s what I was objecting to.
I think you might be worried more about maximizing expected (or average) utility than pareto optimization. The two are largely unrelated.
Again, I may be misunderstanding something, but the requirement that there be a universal scale for utility seems to require that individuals be interchangeable, if the scale is dependent on observable characteristics (e.g. someone who has $200 has 2 utils of happiness). Unless this scale is somehow contingent on measuring subjective utility, you’re saying that pareto optimality should be possible only when players have an agreed-upon set of preferences.
Even if I’m confused on that point, I still fail to see how this hypothetical relates to anything in reality.
If people are Pareto optimal in a trade, then yes, that trade is Pareto optimal.
However if there are three trades between three people (12, 23 and 31) and each trade is Pareto-optimal, that does not mean that the set of three trades as a whole is Pareto-optimal. It’s generally possible to change these trades and get a better outcome for everyone (which is kinda what a market with price signals tries to do).
The only cases where you cannot do this is precisely where there is an agreed upon weighting of everyone’s utility function etc...
The individuals need not be interchangeable, the weighting need not be fair. But Pareto-optimality is only possible in this situation (when the outcome set is smooth).
Two concerns.
First, and I don’t mean this disrespectfully, why should we care? Trying to generate pareto optimality through a series of games played by randomly but fully paired-off players, with no potential for trade, doesn’t resemble anything I can think of. This is particularly true because players of games have little interest in pareto optimality; the very concept of “game” implies “competition,” and the concept of “pareto optimality” implies “cooperation.” Indeed, this assumes that trade is impossible, or else making game outcomes pareto optimal is redundant. While I may in some sense prefer pareto optimal outcomes for games, what most players are assumed to care about is maximizing their own utility, which is largely unrelated to pareto optimality. Perhaps my imagination is insufficient, but this doesn’t seem applicable to anything.
Second, the =1 seems arbitrary, because there are no units. So long as none of the factors are zero, the product will be some number that could be scaled to 1. I don’t think undermines your point; it just seems odd. FUrthermore, it’s interesting to note that there is always a consistent scale in which the product equals zero, even if there is another where it equals 1. I’m not sure if this is significant; again it just seems odd..
Who said there is no potential for trade? Each game could be exactly that—a trade—where the “game” is simply dividing up the gain from trade.
The product of the three theta cannot be scaled. First of all, the theta are unitless (as a ratio between two utility functions), so you can’t scale them by changing their units. You can change individual thetas by scaling the u1, u2 or u3, but this will not change their product: scaling u1, for instance, will scale θ12 one way and θ31 the opposite way, cancelling out. The =1 is well defined.
If trade is permitted, and there are no transaction costs, all outcomes are pareto optimal. That’s what I was objecting to.
I think you might be worried more about maximizing expected (or average) utility than pareto optimization. The two are largely unrelated.
Again, I may be misunderstanding something, but the requirement that there be a universal scale for utility seems to require that individuals be interchangeable, if the scale is dependent on observable characteristics (e.g. someone who has $200 has 2 utils of happiness). Unless this scale is somehow contingent on measuring subjective utility, you’re saying that pareto optimality should be possible only when players have an agreed-upon set of preferences.
Even if I’m confused on that point, I still fail to see how this hypothetical relates to anything in reality.
If people are Pareto optimal in a trade, then yes, that trade is Pareto optimal.
However if there are three trades between three people (12, 23 and 31) and each trade is Pareto-optimal, that does not mean that the set of three trades as a whole is Pareto-optimal. It’s generally possible to change these trades and get a better outcome for everyone (which is kinda what a market with price signals tries to do).
The only cases where you cannot do this is precisely where there is an agreed upon weighting of everyone’s utility function etc...
The individuals need not be interchangeable, the weighting need not be fair. But Pareto-optimality is only possible in this situation (when the outcome set is smooth).