Addendum to ‘Deriving Equality from Ignorance’ Suppose that a government must divide a fixed quantity of resources R > 0 between n individuals. Write x i for the amount given to individual i . We make the following assumptions: 1. Feasible allocations. Resources are perfectly divisible, and any allocation satisfying x i ≥ 0 and ∑ n i =1 x i = R is feasible. 2. Utility. Each individual’s utility depends only on their own allocation, through a function u i ( x i ) 3. Possible utility functions. The government believes that each individual might have one of m possible utility functions, denoted f 1 , . . . , f m , each mapping [0 , R ] to R and each strictly increasing and strictly concave. 4. Equal ignorance. For every individual i , the government believes that they have utility function j = 1 , ..., m with probability p j ; thus, beliefs are symmetric with respect to individuals. 5. Weak prioritarianism. An allocation is optimal if it maximises E [ n ∑ i =1 g ( u i ( x i )) ] , where g : R → R is strictly increasing and weakly concave. This includes utilitarianism when g is linear and prioritarianism when g is strictly concave. Theorem. Under these assumptions, the uniquely optimal allocation divides resources equally. Proof. Under these assumptions, an optimal allocation maximises E [ n ∑ i =1 g ( u i ( x i )) ] = n ∑ i =1 E [ g ( u i ( x i ))] = n ∑ i =1 m ∑ j =1 p j g ( f j ( x i )) = n ∑ i =1 F ( x i ) , where we have defined F ( x ) := ∑ m j =1 p j g ( f j ( x )) . Since each f j is strictly concave and g is strictly increasing and weakly concave, each composition g ◦ f j is strictly concave. Therefore, F is the non-negative weighted sum of strictly concave functions – and thus itself strictly concave. The problem is thus to maximise the sum of identical strictly concave functions. Using Jensen’s inequality, we conclude that x ∗ i = R/n 1