First: Given this definition of the Independence Axiom,

If for all $P$, $P'$, $P''$ in the set of lotteries over outcome space $X$, when:

$P$ preferred to $P'$ $\implies$ $aP + (1-a)P''$ preferred to $aP' + (1-a)P''$ for all $a$ in $(0,1)$."

Can I do what I do here?

Then $P$ preferred to $P'$ $\implies$ $P$ preferred to $aP' + (1-a)P$ for all $a$ in $(0,1)$.

Second: Does satisfaction of the Independence Axiom rely on the statistical independence of the lotteries involved? It doesn't seem to be mentioned anywhere I've looked.

Any direction that can be offered is greatly appreciated.

  • $\begingroup$ I'd say this would be more appropriate on the cross validated.SE $\endgroup$ – Alexis L. May 9 '16 at 2:57
  • 3
    $\begingroup$ @AlexisL. As this is about preferences under uncertainty I would strongly disagree. $\endgroup$ – Giskard May 9 '16 at 6:18

First Question:

Yes, you can obtain $P \succsim P' \Rightarrow P \succsim a P' + (1-a)P \forall a\in (0,1)$ from setting $P''=P$ in the axiom. Thus, your new condition is a special case of the independence axiom as stated.

Second Question:

Statistical independence of $P,P',P''$ is not assumed in the independence axiom. The notion of statistical independence of lotteries is not even meaningful since you can only choose one lottery. Only when you would choose multiple lotteries at the same time their dependence would matter since you may be able to reduce/increase risk by combining lotteries which are not independent.

  • $\begingroup$ Thank you! Regarding your 2nd comment: Suppose we construct a portfolio of lotteries by choosing an optimal mix of lotteries of A and B. While A is a mean preserving spread of B, their returns are correlated. In fact, when the optimal combination of A and B is found, the resultant "portfolio" is a degenerate lottery that yields E[A]=E[B] with certainty. Then this degenerate lottery SOSD B. Is this a violation of independence? Or is there a nuance here to portfolios not existent in compound lotteries that I'm missing? $\endgroup$ – all_thethings May 9 '16 at 13:40
  • $\begingroup$ Yes, in portfolio theory you usually hold several assets ($A$, $B$) in your portfolio and your returns are a sum of the returns of each asset. The probability distribution is derived from their joint probability distribution. If $A$ and $B$ are independent, you may use the convolution (which is still different from the compound). Instead, the compound lottery can be interpreted as lottery $P$ being played with some probability and $P''$ otherwise. This is the nuance that you may be missing? $\endgroup$ – HRSE May 10 '16 at 10:22

The second question of the OP is an important one, because it asks for clarification on a matter that I have never seen explicitly clarified.

The Independence Axiom is defined over simple lotteries. A simple lottery is a set of probabilities that add up to one, and the fixed values ("outcomes") associated with each probability. But these together form a (marginal) statistical distribution. So we can think of a simple lottery as a random variable characterized by/following this distribution.

Let $L'$, $L''$ and $L'''$ be three such random variables. The preferences of a decision maker that obeys the Independence Axiom satisfy

$$L \succsim L' \;\;\text {if and only if}\;\; aL+(1-a)L'' \succsim aL' + (1-a)L'',\;\; a\in (0,1)$$

Note that the Independence Axiom talks about preferences over random variables. $L \succsim L'$ is translated "I prefer random variable $L$ to random variable $L'$", which in more detail means "I prefer that my fortunes are submitted to the uncertainty represented by random variable $L$ than to the uncertainty represented by random variable $L'$." We do not ask why the decision maker prefers that. We do not ask what is the statistical relationship between the two, we just take the preference statement as a primitive given. Then, we say that under the Independence Axiom, "if this is so, then the RHS preference relation also holds, and vice versa".

How are we to translate verbally the expression $aL+(1-a)L''$, or $aL' + (1-a)L''$? The crucial issue here is the interpretation of $a$, as well as to what mathematical operation the expression $aL$ translates into.

What is $a$? Is it a fixed, deterministic "mixture coefficient"? Or is it a probability, and so $aL+(1-a)L''$ is a compound gamble?

The distinction is, again, crucial: if $a$ is a constant and $aL$ represents a multiplication of values with the values (outcomes) $L$ takes, then $aL+(1-a)L''$ is plain convex combination of two random variables, and analogously for $aL' + (1-a)L''$. But then, we cannot rationally defend the Independence Axiom, exactly because there may be stochastic dependence involved between the three simple lotteries such that it may reverse the preference between the two convex combinations. Where moreover $aL$ would mean "multiplication of the random variable $L$ by $a$". But this is not the case (See this post as regards the interpretation of $aL$).

The "optimal portfolio choice" (which from a comment appears to be the concern of the OP), falls into such a framework, and we can see that it cannot be linked to the Independence Axiom.

Continuing, we are necessarily led to view $a$ as something else than a constant. In fact it is treated as a probability, and so the mixture as being a compound gamble. Is a compound gamble also a random variable? It is, but of different nature: if $a$ is meant to reflect a probability, a new source of randomness appears. $a$ here is the "probability of success" of a Bernoulli rv, say $B$, which moreover, is statistically independent from the three simple lotteries. But then ex ante we are essentially looking at the random variable

$$B\cdot L + (1-B)\cdot L'',\;\; B \sim Bern(p=a)$$

and likewise for the other combination (and this is the correct way to write it, in the notation of mathematical statistics). Here the Independence Axiom acquires some intuitive justification:

"Assume that you prefer $L$ to $L'$. Then, if you were asked "what do you prefer: to be subjected to "either $L$ or $L''$" with some probability structure (the $B$), or to be subjected to "either $L'$ or $L''$" with the same probability?" It appears reasonable to say "since I prefer $L$ to $L'$, then I prefer the situation where $L$ is present, rather than the one where $L'$ is present".

Unfortunately, this discussion is not explicitly made in the microeconomic books I know of, and moreover the standard notation is with the $a$'s rather than with the Bernoulli rv.

Sometimes there is some discussion on the matter. Consider for example Mas-Colell, Whinston-Green, p. 172. After stating the Independence Axiom in p. 171, they write (bold my emphasis)

"Assume for example that $L \succsim L'$ and $a=1/2$. Then $aL + (1-a)L'"$ can be thought of as the compound lottery arising from a coin toss in which the decision maker gets $L$ if heads comes up and $L''$ if tails does."

But this description essentially says that $a$ here is the random variable $B$, not some fixed deterministic mixture coefficient.

And a few lines below

"In the theory of consumer demand, for example, there is no reason to believe that a consumer's preferences over various bundles of goods 1 and 2 should be independent of the quantities of the other goods that he will consume.(...) In the present context however (...) this other outcome, $L''$ should be irrelevant to his choice because, in contrast with the consumer context, he does not consume $L$ or $L'$ together with $L''$ but rather, only instead of it (if $L$ or $L'$ is the realized outcome)"

This essentially says again that $a$ in reality is not a mixture constant, but it represents a binary random variable independent of the simple lotteries (our $B$). Unfortunately, the notation used (the $a$) is a notation usually reserved for constants. Moreover the expression $aL$ does not imply multiplication of $L$ by some constant $a$, but it is equivalent to the product random variable $B\cdot L$, with $B$ a Bernoulli random variable and $P(B=1) = a$.

  • $\begingroup$ Thank you so much - what you expound here provides a great exposition of the subtle differences between the random objects that are frequently the subject of analysis in the context of choice under uncertainty. $\endgroup$ – all_thethings May 9 '16 at 21:17

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