Independence axiom of lottery when $\alpha \ge 1$

Question

When studying preference over lotteries we learned the independence axiom which goes like this:

The preference relation $\succsim$ on the space of simple lotteries $\mathscr{L}$ satisfies the independence axiom if for all $L$，$L^\prime$，$L^{\prime \prime} \in \mathscr{L}$ and $\alpha \in (0,1)$ we have

$$L \succsim L^\prime \iff \alpha L + (1 - \alpha) L^{\prime \prime} \succsim \alpha L^{\prime} + (1 - \alpha) L^{\prime \prime} $$

What if $\alpha \ge 1$? I guess that will be a quick extension for the case of $\alpha \in (0,1)$ but I don't know how to show it.

Answer 1

To understand why $\alpha$ must be constrained in $(0,1)$, one has to contemplate the meaning of the expression

$$\alpha L$$

when $L$ is a "lottery". How is a lottery denoted mathematically? Authors do not agree on that: for example, the way Jahle and Reny define a lottery (a "gamble" in their terminology), a lottery can be written as a vector whose elements are bivariate vectors themselves:

$$L=\{(p_1,w_1),...,(p_n, w_n)\}$$

where $p_i$ are probabilities, and $w_i$ are quantitative outcomes.

But MasColell, Whinston and Green, define a lottery as a vector containing only the probabilities:

$$L=\{p_1,...,p_n\}$$

and so the "Lottery Space" is a vector space, including vectors each containing only probabilities.

But in both cases, the authors make clear that an expression like $\alpha L$, when the time comes to translate it into a mathematical operation, denotes a multiplication of only the probabilities linked with $L$ by another probability, $\alpha$. This is implementing the reduction of compound lotteries to simple ones. JR describe it as "the decision maker cares only about effective probabilities of each outcome". MWG call it the "consequentialist premise", that allows them to work only with simple lotteries.

So it is not valid to consider $\alpha$ outside $(0,1)$, because it is defined as a probability (the open bounds are used in order to avoid triviality in the statement of the Independence axiom).

Also, the above imply that $\alpha$ does not directly interact with the quantitative outcomes of the lottery/gamble...

...which points to a (perhaps interesting, perhaps not) research direction: What can we say (if anything), if we start tweaking the outcomes linked with a lottery? Will "attitude towards risk" get in the way and prevent us from drawing any general conclusions? Or not?