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I was trying to write up a justification for the principle of insufficient reason (PIR) in decision theory. I am unable to confirm a bit of mathematical intuition I had.       
         
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Given a formal decision problem, with $n$ states. Let $S$ denote the set of world states. $\#S = n$. $S$ is indexed from $1$.      
       
There is some empiricial (I have since converged on the position that expected utility theory (EUT) is only justified from axioms (non convoluted simple intuitive axioms (this excludes VNM)) if the probabilities used to calculate the expected utility of an act are empirical and not subjective priors)$*$ probability distribution over the states.       
       
Let this probability distribution $= Pr(S)$, the empirical probability of a particular state say $s_i = Pr(s_i)$.

$$\sum_{i = 1}^n (Pr(s_i)) = 1$$

There are an uncountable infinite number of probability distributions over $S$.      
       
For each $s_i$, we can determine the interval $Pr(s_i)$ lies on.      
Let:    
$$a(1) = 0$$
$$a(i) = \sum_{k = 1}^{i-1} {Pr(s_k)}$$
$$b(i) = 1 - a(i)$$
$$\forall i \in [1, n-1]$$
$$Pr(s_i) \in [0, b(i)]$$
$$Pr(s_n) = b(n)$$     
      
For a given $S$, with a given $n$.
We might try to calculate the average probability each $s_i$ would have, and this expected probability of $s_i$ $(E(s_i))$ would be assigned to $s_i$ as the probability of $s_i$, by some transformative decision rule (TDR), let's call it the "principle of expected probability" (PEP). Let the probability that we assign to a particular state using a TDR be $P(s_i)$.    
$$P(s_i) = E(s_i)$$
       
My intuition is:     
$$\forall i \in [1, n] E(s_i) = \frac{1}{n}**$$

For PIR:
$$P(s_i) = 1/n$$

For PEP: 
$$P(s_i) = 1/n$$

As we can see from the above, PEP $\approx$ PIR, and in so much as you accept PEP, you should accept PIP. The justification for PEP is quite similar to the justification for EUT; as such, people who accept EUT should accept PEP.

$*$P.S: Even if you disagree with empirical probability being a requirement for expected utility theory, please just ignore it. Unless otherwise specified, probability in the above refers to empirical probability.

$**$ Determining that would require measure theory and integration; two areas of maths that are currently outside my limited knowledge. 
      
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My question is whether the below is true:      
$$\forall i \in [1, n] E(s_i) = \frac{1}{n}$$?

Fri, 13 Oct 2017 19:33 GMT