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.