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I am referring to the definition in Proposition 6.B.3 on Page 176 of Mas Colell. I follow the formal proof and the application of the Independence axiom at various steps (mathematical application of the definition), but I am having a hard time understanding the intuition behind the Expected Utility Theorem and the use of Independence axiom as a central assumption.

My basic understanding is that we use Expected Utility theorem to help make decisions when difference in probabilities between two lotteries are very low. Are we using the independence axiom to mix our two lotteries with the third one, to help make a decision on lottery choice since the preference relationship over lotteries would still be preserved because it is independent of any third lottery? Are there other uses?

Overall, I am just trying to get a better understanding of this concept so I am able to replicate this proof naturally with ease.

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  • $\begingroup$ Hint: The expectation is a linear operator. Can you see the similarity between the independence axiom and the linearity of a operator? $\endgroup$
    – user141240
    Nov 19 at 23:40
  • $\begingroup$ Do you mean how the independence axiom allows us to express preferences in an expected utility form because the indifference curves are straight lines and parallel? $\endgroup$
    – Rumi
    Nov 21 at 3:54
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The expected utility theorem (EUT), first and foremost, establishes a utility representation of the preference over lotteries. This is akin to establishing utility representation of preference over deterministic consumption bundles in consumer theory. The representation (in both cases) is valuable because it gives us tools like algebra and calculus to do further analysis.

Secondly, the EUT provides a very specific functional form, i.e. the expected utility form (Def 6.B.5), that is linear in probabilities. This result is due mainly to the independence axiom (see Step 5 in MWG's proof).

In terms of the use of the EUT, rather than thinking about how each axiom (e.g. independence) is used in evaluating lotteries, I would view all the axioms as a package. When given two lotteries $L$ and $L'$, I would put them through the expected utility function $U(\cdot)$ that is given by the EUT and rank them based on the output of $U(\cdot)$. This is the same as using ordinary utility functions to assess the desirability of different consumption bundles in consumer theory.


Despite the preceding paragraph, it is true that the independence axiom plays a more important role both in delivering the EUT result and in limiting its applicability.

As you noted in the comments, independence restricts the set of indifference curves to be both linear and parallel. This has been shown to be inconsistent with a lot of empirically observed behavior, most notably the Allais paradox. Much of what we know as behavioral economics started as attempts to address inconsistencies of this sort. I would point you to Burghart (2020), which decomposes independence into a part that gives linearity (which the author calls "homotheticity") and another part that gives parallelism ("betweenness"), and shows that these two parts are necessary and sufficient for independence.

Coming back to its roots, the significance of the independence axiom lies in its implications for game theory. For example, Crawford (1990) shows that, without independence as a property of the underlying preference, Nash equilibrium may fail to exist.

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We don't use the EUT for comparing lotteries with close probabilities, except if we are solving an exercise involving an Allais paradox type of question. We don't actually use it at all if we are not theorists trying to prove something.

The EUT tells you that if you have preferences over lotteries satisfying the basic axioms, including independence, then there exists a (Bernoulli) utility function on prizes such that you choose between lotteries in a way as if you were evaluating them by their expected utility.

The reason is that independence guarantees that, geometrically speaking, your indifference curves in the space of lotteries are parallel lines (for the illustrative case of three prizes). This in turn implies that the graph of your utility-over-lotteries function (your vNM utility function) is a plane, meaning that this utility function is linear in probabilities. It is then easy to show that linearity in probabilities implies that it can be written as an expected-utility-over-prizes function.

The EUT thus justifies the common assumption in economic models that agents facing risk maximize expected utility, which would otherwise be just a questionable ad-hoc assumptions.

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