#### Supplement to Epistemic Utility Arguments for Probabilism

## Proof of Theorem 6

We wish to prove the following theorem (Selten 1984):

Theorem 6.Perspective Indifference,World Indifference, andWeakly Non-Trivialentail

G(p,v) = λΣ|v(v_{i}) −p(v_{i})|²for all

pinPandvinV, wherev_{1}, …,v_{n}are the atoms ofF.

First, by **World Indifference**, every world
*v* is exactly the same distance from every other world
*v*′. Thus, let *G*(*v*, *v*′)
= *c*, for all *v* ≠ *v*′. Now suppose
*p* ∈ **P** and *v*_{i} ∈
**V**. Then, by **Perspective
Indifference**,

GExp

_{G}(p|p) − GExp_{G}(v_{i}|p) = GExp_{G}(v_{i}|v_{i}) − GExp_{G}(p|v_{i})

That is, *p* is the same distance from *v _{i}*
as

*v*is from

_{i}*p*, where the distance of one probability function from another is given by the expected inaccuracy of the first by the lights of the second corrected so that every probability function is at distance 0 from itself.

Spelling out the definition of GExp_{G} we
derive

GExp_{G}(p|p) − Σp(v)G(v_{i},v) = Σv_{i}(v)G(v_{i},v) − Σv_{i}(v)G(p,v)

Thus, by **World Indifference**, we have

GExp_{G}(p|p) −c(1 −p(v_{i})) = -G(p,v_{i})

which gives

G(p|v_{i}) =c−cp(v_{i}) − GExp_{G}(p|p)

From this point on, we are simply manipulating formulae: there is no
deep explanation for what is going on. Substituting this into the
definition of GExp_{G}, we have

GExp _{G}(p|p)= Σ p(v)[c−cp(v) − GExp_{G}(p|p)]= c− GExp_{G}(p|p) −cΣp(v)²

So

GExp_{G}= ½c− ½cΣp(v)²

Substituting this into the expression for *G* given above,
we have

G(p,v_{i})= c−cp(v_{i}) − ½c+ ½cΣp(v)²= ½ c− ½c[2p(v_{i}) − Σp(v)²]= ½ c||p−v_{i}||².

as required.