Monday, April 25, 2016

Is the Road to Hell Paved with Pareto Improvements?

In a large multiplayer prisoner’s dilemma, any change in any one individual’s strategy doesn’t affect anyone else, so a player can know that defection will be a Pareto improvement. We might say that the problem of social evil is that the road to hell is paved with Pareto improvements. -- Ted Poston, “Social Evil,” Oxford Studies in Philosophy of Religion, Volume 5
Poston's "social evil" is what previous authors have called a social trap or, more famously, the tragedy of the commons.

A Pareto improvement is a change that makes at least one person better off without making anyone worse off. According to the standard fable, voluntary exchange results in a Pareto improvement because each party in the exchange gets something they wanted more than what they gave up for it.

A prisoner's dilemma involves a situation where the individual payoff to each player for defection is better, regardless of whether the other player defects or co-operates but the collective payoff is maximized when both players co-operate.

In a large multiplayer prisoner's dilemma game, defection by some players may have no effect on the other players' outcomes, while defection by a large number of players may have catastrophic effects after some vaguely defined tipping point has been reached. Within limits, defections thus appear to result in a Pareto improvement, where some players are made better off and no one is made worse off.

In Fights, Games and Debates, Anatol Rapoport presented a production and exchange model that deserves to be much better known. It is a very elementary model and thus, as Rapoport warns repeatedly, the results should not be taken as a faithful depiction of what is likely to happen in reality. However, it offers some critical insights into "common sense" assumptions and specifically into the idea of Pareto improvement, which is also based on extreme simplification.

Rapoport's production and exchange "society" consists of two people who each produce goods and exchange with each other a uniform, fixed ratio of their products. The individuals derive utility from the goods they produce and, presumably, can increase their utility by exchanging some of the goods they produce for the different goods their counterpart produces.

Effort to produce those goods, however, is a disutility. The utility from goods increases logarithmically as the quantity of goods increases but the disutility of effort increases in proportion to the amount of effort expended.

Agents in this model can only change their utility by increasing or decreasing their own effort and output. Thus, plotted on a graph, X can only move along the x-axis and Y can only move along the y-axis. Under the stipulated conditions, a stable equilibrium can only be achieved when the utility of the proportion retained by each producer is larger than the disutility of effort.That is to say, the proportion retained cannot be too small and the disutility of effort cannot be too large.

In the absence of a stable balance, any relaxation of effort by one of the agents will lead to parasitism by that agent as the other will immediately compensate by increasing effort, the first agent will slack off more to compensate for the increased effort of the other -- and so on.

But even in the presence of a stable equilibrium, the total utility of the two agents, at the balance point, will be less than the total would be without exchange, as long as their production/effort decisions are guided solely by their own utility rather than by some agreement about how to link their production effort to achieve a "social optimum." This outcome is contrary to the "common sense" interpretations of Pareto improvement and Pareto optimality. As Rapoport cites his mentor, Nicolas Rashevsky, it turns out that:
The only 'ethics' which leads to the attainment of maximum joint utility in the model of society we have considered is the 'egalitarian ethic,' in which the concern for self and for other are of equal weight.
It would be easy to dismiss Rapoport's conclusion as pertaining only to very restrictive premises. This is a point that Rapoport reiterates throughout his exposition. But the objection applies equally to Pareto's model.

Vilfredo Pareto is not readily perceived as a proponent of the egalitarian ethic. In his model, though, Rapoport unpacked a tacit premise of Pareto that rational agents would act "as if" guided by some unacknowledged intuition of linkage -- one might even call this invisible intuition "moral sentiments."

Furthermore, the restrictiveness of Rapoport's assumptions may not be as unrealistic as it seems at first. The fixed ratios of exchange can be relaxed to merely widespread similarities in the ratios of exchange. The specification for a stable equilibrium that the proportion of an individual's product exchanged does not exceed the proportion retained can be rationalized by the fact that there is a roughly equal number of hours of unpaid household work performed in the world as there are waged hours of labor. All this is before we move on to the issue of "multiplayer games" -- of a society in which individual actions that ostensively do no harm may accumulate into "social evil."

In Beyond the Invisible Hand, Kaushik Basu examined the issue of outlawing yellow dog contracts, as the Norris-LaGuardia Act did in 1932:
It could he claimed that if one worker prefers to give up the right to join trade unions in order to get a certain job that demands this of workers, then this may be a Pareto improvement. But if such yellow dog contracts are made legal, then lots of firms will offer these contracts, and the terms for jobs without a yellow dog clause may deteriorate so much that those who are strongly averse to giving up the right to join unions will he worse off in this world. 
Basu proceeds to consider labor standards in cases in which there are multiple equilibria. He asks, "Should the law be used to set a limit on the number of hours that a worker is allowed to work?" His answer -- backed by reference to supporting empirical studies -- would earn the scorn of economists who fancy a lump of labor behind every proposal for shorter hours:
A statutory limit on work hours can, by limiting the supply of labor, push up the hourly wage rate, and it is possible that at this higher wage rate people would not want to work that many hours. In other words, the labor market may have two or more equilibria, in which case banning the long work-hours equilibrium is fully compatible with a commitment to the Pareto principle.
Unpacking Pareto optimality and Pareto improvement, as Rapoport's model of production and exchange does, undermines the premise of the road to hell being paved with Pareto improvements. If there is indeed a tacit moral sentiment, a secret egalitarian ethics at the heart of the Paretian idea, then any violation of trust will impose a loss of utility on everyone else -- perhaps even on the violator. Those individuals gains through defecting were only "improvements" assuming an ethical vacuum. What is the point of building a road to a hell when one is already there? In an ethical world, violations of trust are losses of utility.

UPDATE: I have made a pdf copy of the section on the production and exchange model in Rapoport's Fights, Games and Debates. Below is the equation for the utilities of the two members of the society:
In "An Empirical Refutation of Pareto-optimality?" Rupert Read argued that the empirical evidence in Wilkinson and Pickett's The Spirit Level suggests "the remarkable normative conclusion of making the Pareto principle, far from the “conservative” device it is often taken to be, a potential agent of radically-egalitarian/socialist distributive justice." I am arguing here that Rapoport's production and exchange model suggest a mathematical demonstration of the same unexpected conclusion. 


Nick Rowe said...

Poston sounds wrong to me. Let n be the number of players. A defection by one player has an effect 1/(n-1) on each other player, which approaches 0 in the limit, but it affects (n-1) other players, which approaches infinity in the limit.

Rapaport sounds interesting, but I'm not quite getting the game. If a fraction s is shared, what is X's utility function? Is it:

U = log((1-s)x) + log(sy) - x ?

Or is it:

U = log((1-s)x + sy) - x ?

Unknown said...

This is a misleading way of describing the tragedy of the commons. Defection DOES reduce the utility of other by more than the increase to the defector and so it is NOT a Pareto improvement.

Sandwichman said...


Rapoport has it, for x, as U = log(1 + px + qy) - rx , where p is proportion of product kept, q is 1 - p and r is resistance to work. Similarly, for y, the equation is U = log(1 + py + qx) - ry . I've changed the notation slightly because comments doesn;t do subscripts or greek letters. Here is a pdf file of Rapoport's model. said...

When the defection by one player causes the defection of all the others, which presumably happens at some point, then Unknown is right, and the total pie is reduced, the tragedy of the commons hits, or whatever, and the losses to the others outweigh the gains to the defector, with, in fact, the defector in danger of ending up worse off as well.

Sandwichman said...

Well, yes, the tragedy of the commons assumes utility that is individualized and a disutility that is socialized, so that the defector incurs only 1/n of the cost while gaining 100% of the benefit. The ratio of cost to benefit is not specified, however, and could escalate as more and more commoners exploit the commons. Poston is assuming a situation in which there is some threshold before which no damage to the commons occurs so initially there is no cost, 0/n = 0.

My argument contra Poston would be that he is only considering the utility of the physical pasture and disregarding the utility of the social commons. With regard to the latter there is a social cost to the violation of trust regardless of whether any physical damage is done to the pasture.

Peter T said...

The approach here is the standard methodological one where the focus is on some resource subject to individual consumption (production specified as individual effort). It look a bit different if you approach it as an issue of cooperative production. From that standpoint, defection can imperil everyone even at low levels if it tips the cooperative production unit below the threshold of operation - so as production thresholds rise, cooperation rises too, and coercion in some form becomes more essential (so if the tribe depends on successful buffalo hunts to survic=ve the winter, and it takes a minimum of 10 people to hunt, then any number less than 10 dooms everyone, and will not be permitted - but any 10 may do out of, say, 20 available). This is a closer description of our current situation than parsing single defections.

Sandwichman said...

Peter T.,

Rapoport STARTS with a model of individual effort and consumption but uses it to demonstrate the superiority of egalitarian joint-decision making even in the absence of division of labor in the production process. Can you elaborate on why you think this is the "standard model"?

Peter T said...

My issue is that the model does not scale to groups, either morally or practically - so yes it takes as given individual effort and consumption.

Tell it as a story. Some small group undertake a task where there is no division of labour. If one shirks, others must make up the lost effort or suffer the disutility of delayed completion (equivalent to some loss of consumption). Hence the moral obligation on all to do their bit.

Now scale up to two groups. Each has its group task necessary to the overall enterprise. One group (A) has more shirkers. But the other group (B) cannot help (that, after all, is why they are in two groups). B can only exhort A to greater effort. Whether B suffers depends on other factors (maybe the pay-off only comes when the whole project is complete, in which case there's a moral and practical compulsion on all; maybe A and B are contracted just for their group jobs, in which case B walks away paid but grumbling). Scale up again, and the moral element weakens further. Add division of labour, and the link to individual effort and consumption weakens. Add in coordination, division of knowledge and it weakens further. And on up. So it remains a story about individuals and consumption.

Sandwichman said...

Peter T,

I sympathize with your objection. My own method is grounded in "narrative analysis" -- stories. I am normally averse to abstraction ad absurdum and suspicious of mathiness.

But I think you are overlooking the subtlety of Rapoport's argument. He is saying EVEN IF we start with simple two-person exchange, as Paretians do, trust > ratutilmax and the social optimum assumes an egalitarian ethic.

This is not meant to be "the way things are." It is a mathematical inquiry into the logical connection between premises and conclusions. What it demonstrates is that the standard premises of "economic analysis" rely on hidden tacit premises that disavow the standard "common sense" conclusions.

For example, if I say, "competition relies on cooperation," I am making an assertion about reality. It may or may not be true. It may or may not be demonstrable. It may or may not be falsifiable.

However, if I say, "the economist's argument about the economic efficiency of competition embeds a tacit assumption of cooperation," I am making a claim about a logical syllogism that is possible to test using a mathematical model.

My model may not say anything definitive about whether or not competition relies on cooperation. it may not scale to groups, but it still does say something critical about the standard model.

I think the statement "Pareto improvement tacitly assumes an egalitarian ethic" is rather astonishing given the reverence in which a conservative, anti-egalitarian interpretation of Pareto is held. A pseudo-scientific edifice has be erected on that FALSE interpretation. I think it would be nice to tear down that ideological puzzle palace by showing that it depends on assuming the very condition it disdains.

Peter T said...

True. And intellectually interesting, I agree. As someone who sometimes dabble in theology, I'm in no position to throw stones. I just think this particular bit of logic has no effect at all.

Sandwichman said...

"I just think this particular bit of logic has no effect at all."

Does any?

Peter T said...

Well, yes. There are many areas of life where a demonstration of empirical weakness or internal inconsistency would be a major factor in choosing a course of action. It just so happens that this is not one of them.

Sandwichman said...

"It just so happens that this is not one of them."

I am curious as to what makes you so certain.

Peter T said...

Well, I have read a fair number of similar criticisms of economic theorisms - many starting something like "as x pointed out in his devastating article of 1952..." that have similarly made no impression whatever on the mainstream discourse.

It's not a certainty. Just an empirical observation.

Anonymous said...

This is a discussion about two distinct topics, Pareto optimality and "lump of labor".

My usual two important details:

As to «Unpacking Pareto optimality and Pareto improvement», there is no such thing as "Pareto optimality" of *states*. Only *changes* can be properly said to be Pareto optimal or not. Because Pareto optimality is essentially the property of being unanimously agreeable, and only a *change* from which nobody loses can be unanimously agreeable. But in general case there is no unanimous agreement as to a *state*; "right-wing" Economists use the trick of improperly calling *states* Pareto-optimal to block consideration of redistribution. Put another (weaker) way, a *state* cannot be called Pareto-optimal if there are no changes to it that can be unanimously agreed, unless all previous changes were also unanimously agreed, starting from zero for everybody.

Then there is the usual story that the "lump of labour" is indeed *obviously* (consider demand of labour if wages were nugatory) a fallacy, and what Sandwichman talks about is the lump of wages or wage-fund, or more generally the lump of (potential) incomes, which is indeed a lump that only slowly changes.