MinusGix comments on Unifying Bargaining Notions (2/​2)

[MinusGix]

Utility functions are shift/​scale invariant.

If you have $U_0\left(A\right)=1$ and $U_0\left(B\right)=2$, then if we shift it by some constant $c$ to get a new utility function: $U_c\left(A\right)=1+c$ and $U_c\left(B\right)=2+c$ then we can still get the same result.
If we look at the expected utility, then we get:
Certainty of $A$:

• $1.0\ast U_0\left(A\right)=U_0\left(A\right)=1$

• $1.0\ast U_c\left(A\right)=U_c\left(A\right)=1+c$

50% chance of $B$, 50% chance of nothing:

• $0.5\ast U_0\left(B\right)=0.5\ast 2=1$ (so you are indifferent between certainty of $A$ and a 50% chance of $B$ by $U_0$)

• $0.5\ast U_c\left(B\right)=0.5\ast (2+c)=1+0.5\ast c$

I think this might be where you got confused? Now the expected values are different for any nonzero $c$!
The issue is that it is ignoring the implicit zero. The real second equation is:

• $0.5\ast U_0\left(B\right)+0.5\ast U_0\left(\varnothing \right)=0.5\ast U_0\left(B\right)$ + 0 = 1$

• $0.5\ast U_c\left(B\right)+0.5\ast U_c\left(\varnothing \right)=0.5\ast (2+c)+0.5\ast c=1+0.5c+0.5c=1+c$
  Which results in the same preference ordering.

[MSRayne]

Holy heck I have been enlightened. And by contemplating nothingness too! Thanks for the clarification, it all makes sense now.