# how to reach Continuous Expected Utility (EU)?

Consider EU on monetary outcomes. Say we have a utility function $$u:\mathbb R\to\mathbb R$$

The common axioms of EU are continuity, independence and weak order.

These axioms do not imply that the utility function is continuous.

My question: how to reach a continuous utility?

Edit: the usual continuity conditions for EU are mixture continuity and Archimedean.

• Expected Utility is linear is probabilities and therefore, it is continuous.
– Amit
Commented Jul 12 at 9:03
• This is a fairly technical issue. There is a nice discussion in the "Notes on Theory of Choice" by Kreps. Commented Jul 12 at 11:25
• @Amit We are talking about the continuity of small u function.
– dodo
Commented Jul 12 at 16:33
• @dodo Thanks. Yes, I misinterpreted.
– Amit
Commented Jul 12 at 22:02

I believe we get it from the axioms you have provided with a few technical conditions on the spaces we are working with.

Let $$X$$ be a separable, metrizable space (like your $$\mathbb{R}$$) and let $$\Delta(X)$$ be the set of countably additive probability distributions, endowed with the weak convergence topology.

The complete preorder $$\succeq$$ on $$\Delta(X)$$ satisfies Continuity and Independence if and only if it has an expected utility representation:

$$p \succeq q \quad \text{ if and only if } \quad \int_X u(x) dp \geq \int_X u(x)dq$$

where $$u : X \rightarrow \mathbb{R}$$ is continuous and bounded.

• Hi, the usual continuity conditions for EU are mixture continuity and Archimedean. Could you please explain how is your continuity stronger than the mixture continuity?
– dodo
Commented Jul 12 at 16:38
• Its stronger in the sense that we require the preference to be continuous with respect to the weak convergence topology. i.e. $(p_n) \rightarrow p$ if for every bounded and continuous $f$, $\int f(x) dp_n \rightarrow \int f(x) dp$. Commented Jul 12 at 17:07