## “What if Achilles and the Tortoise were to Bargain?,” D. Samet (2010)

The Greek Zeno famously claimed that motion was impossible because once half the distance desired has been crossed, we must cross half the remaining distance, then cross half what still remains, and so on infinite times. As Aristotle informally and modern mathematicians formally have discussed, this argument is incorrect, since the problems faced at each step are not precisely the same (the distance remaining becomes shorter), and in particular the infinite sum of times needed to cross is convergent.

However, a similar problem occurs in bargaining games. Consider two agents trying to agree on a policy by reaching the Pareto frontier (for instance, Israelis and Palestinians negotiating on the boundary of their states). In each step of the bargaining process, some proposal is made and accepted such that the agents move closer to the Pareto frontier (for instance, successive rounds of “peace process” negotiations). Given fairly broad assumptions about the nature of the bargaining process, in finite time the agents will never reach the Pareto frontier. The principal difference between bargaining and Zeno’s paradox is that utility is unique only up to affine transformations. That is, when we reach the second stage of the bargaining, we no longer have only “half the utility” still to cover, but instead are at exactly the original problem we faced!