Before you arrive from your home to the gym, you have to arrive halfway. Then you must do half of the remaining way, then half of what remains… Infinitely many happenings must occur before anybody arrives anywhere. Therefore, says Zeno of Elea, nobody arrives anywhere and all motion is an illusion.
This post is about a cute new paper (pdf) by Dov Samet, which uses arguments similar to Zeno’s to study social interaction.

The standard `solution’ to Zeno’s paradox is that as distances diminish, the duration it takes to cover them also diminishes, so that the series of durations converges. Therefore, even though you have to perform infinitely many tasks, the amount of time that is required to do all of them is finite.

I don’t find this explanation fully satisfactory. For me, the mathematical construction is just a re-iteration of the paradox. I still find it counter-intuitive that an infinite amount of happenings are squeezed into a finite time horizon. I have a similar problem with a justification, or excuse, that is sometimes given for studying infinite extensive form games: Instead of talking about players taking actions at stages $n=1,2,3,4,\dots$ and then payoff is given at infinity, the argument says that the players take actions at times $t=1/2,~3/4,~7/8,~15/16,\dots$ and then payoff is given at time $t=1$. Personally, I have no problem with players getting payoff at infinity, but if you do, I can’t see why this mathematical trick solves the problem for you. (I think this argument is attributed to Aumann, so probably there is more to it than meets my eyes)

Anyway, back Dov’s paper. If instead of moving a lazy game theorist from home to the gym you are trying to reach an agreement in a social interaction, it is not even true that when tasks diminish the time it takes to perform them also diminish. This is the essence of Parkinson’s law of triviality:

Time spent on any item of the agenda will be in inverse proportion to the sum involved

Moreover, in a bargaining situation, we usually think of the alternatives as von-Neumann Morgenstern utilities. Since the difference between utilities is defined up to multiplication with a positive constant there is no such a thing as small vs. large tasks.

What it means about reaching agreements in bargaining situations, claims Dov, is that such an agreement can never be reached in finite time. He then axiomatizes a dynamic process of bargaining and shows that under two assumption: 1) restarting — that at every stage of the process the bargaining restarts with the current interim agreemt as the new status quo point — and 2) von Neumann Morgenstern’s scale invariance,  an agreement is never reached in a class of bargaining problems in which a fixed amount of utility is to be divided among the participants. When I read the paper I always had the Israeli-Palestinian negotiations in mind — from one agreement to another, the process continues forever and the bargaining problems that remain never look smaller.

One more point about Zeno’s paradox: There is another way to present the paradox, which for me is even more baffling, even though  the two versions are sometimes claimed to be equivalent: Before you arrive your destination you have to arrive half of the way, and before you arrive half of the way you have to arrive half of the half, and so on. So maybe you not only cannot approach agreement, but in fact you cannot even start moving towards it.