How to pick your football team

Abstract

Team captains Alice and Bob divide up $2m$ footballers, each reduced to a real-valued score, into two teams of $m$ footballers each. On each turn, one captain plays picker, and the other chooser: the picker names a footballer yet to be selected, and the chooser decides which captain's team receives that footballer. Alice starts as picker, Bob as chooser, and roles alternate. The game ends as soon as either captain has a full team of $m$ footballers, at which point the other team receives all remaining footballers. The team with the larger sum of scores wins. Settling a problem raised by Eccles in 2015, we show that Alice cannot win.