Applied — page 5
A core is a rare thing to find at random
A game has a core when the whole can be shared so that no group would do better on its own, and Bondareva and Shapley turned that into a finite list of checks. For three players there are five checks; for four, forty-one; for five, 1,291. Draw the coalitions' values at random and the list wins: a third of three-player games fail, nineteen four-player games in twenty, and not one five-player game in two thousand passes. The cores that theorems promise come from structure, and structure is exactly what a random game lacks.
The element the judges are worst at
A court can vote element by element or on the verdict, and the comparison so far assumed judges equally good at every element. Real judges are surer about the law than about a contested fact. Make one element hard and both procedures sink towards the majority on that element, but they sink differently: the verdict vote sees only the product of the judges' competences and cannot tell an even court from a lopsided one, while the premise vote is ruled by its weakest majority. And when a judge's errors on two elements come from the same misreading of the facts, the gap between the procedures closes.