Some Results on Adjusted Winner 1 Eric Pacuit Rohit Parikh Samer Salame Abstract We study the Adjusted Winner procedure of Brams and Taylor for di- viding 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 geo- metric 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. 1 Introduction In this paper we study one particular algorithm, or procedure, for settling a dispute between two players over a finite set of goods. The algorithm we are interested in is called Adjusted Winner (AW ) and due to Steven Brams and Alan Taylor [2]. See also [1] for a relevant discussion. Suppose there are two players, called Ann (A) and Bob (B), and n (divisible 2 ) goods (G 1 ,...,G n ) which must be distributed to Ann and Bob. The goal of the Adjusted Winner algorithm is to fairly distribute the n goods between Ann and Bob. We begin by discussing an example which illustrates the Adjusted Winner algorithm. Suppose Ann and Bob are dividing three goods: G 1 ,G 2 , and G 3 . Adjusted Winner begins by giving both Ann and Bob 100 points to divide among the three goods. Suppose that Ann and Bob assign these points according to the following table. Item Ann Bob G 1 10 7 G 2 65 43 G 3 25 50 Total 100 100 1 Working paper which has been presented at the Stony Brook International Game Theory Conference, June 2005 and Multiagent Resource Allocation Workshop (MARA), September 2005. 2 Actually all we need to assume is that one good is divisible. However, since we do not know before the algorithm begins which good will be divided, we assume all goods are divisible. See [2, 3] for a discussion of this fact.