Some Results on Adjusted Winner

We study the Adjusted Winner procedure of Brams and Taylor for dividing goods fairly between two individuals, and prove several results. In particular we show rigorously that as the differences between the two individuals become more acute they both benefit. We introduce a geometric approach which allows us to give alternate proofs of some of the Brams-Taylor results and which gives some hope for understanding the many-agent case also. We also point out that while honesty may not always be the best policy, it is as Parikh and Pacuit [4] point out in the context of voting, the only safe one. Finally, we show that provided that the assignments of valuation points are allowed to be real numbers, the final result is a continuous function of the valuations given by the two agents and suggest a generalization of the adjusted winner function to take into account nonlinear utility functions.