CONVERGENCE OF EXPECTED UTILITY FOR UNIVERSAL AI 5 By equation 4, for each j ∈ N there exists uj ∈ N such that uj ≤ j and θuj**(0) =**
B (j).

Now we define a map on function indices G : N → N **such that:**
φG(n) (γ) = H(θn**(0))**|γ| So G takes the θ-index of an N → N function (say, g), and returns the φ**index of an environment which is compatible with all the data so far, and which**
is guaranteed to produce utility greater than g(0). We can assume that G **is a** computable function.

So our sequence of environments will be φG(uj )
	∞
j=1.

Then UWp(φG(uj ))≥ B(j**). Now let**
(5) ¯ρ(j) = ⌈max

## K≤J 1 Ρ(G(K)) ⌉

Then ¯ρ is a computable, nondecreasing function. Since ¯ρ **is computable,** B(j) ≥ ρ¯(j)
infinitely often. Since uj ≤ j**, then by definition, ¯**ρ(j) ≥1 ρ(G(uj )) ≥1 P ({φG(uj
)})
.

P(ΓQ(y n 1) = x n 1|Q = φG(uj )**) = 1, so by Bayes' Rule,** P({φG(uj )}|ΓQ(y n 1) =
x n 1) ≥ P({φG(uj )}**). Since both sides are positive, we take the reciprocal to get**
1 P ({φG(uj
)}|y n 1
,xn 1
) ≤1 P ({φG(uj
)})
. By transitivity, UWp(φG(uj ))≥1 P ({φG(uj
)}|ΓQ(y n 1
)=x n 1
)
infinitely often, so UWp(φG(uj ))P({φG(uj )}|ΓQ(y n 1
) = x n 1
) ≥ **1 infinitely often.**
Since the series contains infinitely many terms ≥ 1, its limit is either +∞ **or nonexistent.** 
Corollary 1. If U is computably unbounded from below on D*, then* E (U (x n 1Ψ (**Q, p, y**n 1, xn 1))|ΓQ(y n 1) = x n 1) is either undefined or −∞.

Proof. By definition, −U **is computably unbounded from above. Thus, by theorem 1,** E (−U (x n 1 Ψ (**Q, p, y**n 1
, xn 1
))|ΓQ(y n 1
) = x n 1
) is either undefined or +∞**. So**
E (U (x n 1Ψ (**Q, p, y**n 1
, xn 1
))|ΓQ(y n 1
)) = −E (−U (x n 1 Ψ (**Q, p, y**n 1
, xn 1
))|ΓQ(y n 1
) = x n 1
) is either undefined or −∞. 

Corollary 2. If U is computably unbounded from both below and above on D*, then* E (U (x n 1Ψ (**Q, p, y**n 1
, xn 1
))|ΓQ(y n 1
) = x n 1
) *is undefined.*
Proof. **By theorem 1,** E (U (x n 1 Ψ (**Q, p, y**n 1
, xn 1
))|ΓQ(y n 1
) = x n 1
) is either undefined or
+∞. By corollary 1, it is either undefined or −∞**. Thus it is undefined.** 

## 6. Discussion

Our main result implies that if you have an unbounded, perception determined, computable utility function, and you use a Solomonoff-like prior (Solomonoff, 1964), then you have no way to choose between policies using expected utility. So which of these things should we change?

We could use a non-perception determined utility function. Then our **main result**
would not apply. In this case, the existence of bounded expected utility will depend on the utility function. It may be possible to generalize our argument **to some larger** class of utility functions which have a different domain.

We could use an uncomputable utility function. For instance, if the utility of any perception-sequence is defined as equal to its Kolmogorov complexity, then the utility function is unbounded but the expected utility of any policy is finite.