Von Neumann–Morgenstern independence axiom:
Savage independence theorem:
What is the difference between the two? I'm think Von Neumann is talking about the prizes (outcomes) and Savage is talking about the likelihood of acts. Am I correct?
[Math] Von Neumann–Morgenstern independence axiom vs. Savage independence theorm
decision-theoryeconomicsprobability
Best Answer
This post does NOT answer the OPs question. It misleadingly compared Vnm's model with a state-dependent model, as opposed to Savage's model which the OP asks about. See my other more recent answer for a correction.
You are on the right track. It really has to do with the fundamental differences between VnM and Savage's models.
In the VnM model, the alternatives over which agents have preferences are lotteries that assign probabilities to a set of outcomes. If the set of outcomes is $X := \{x_1,\dots,x_n\}$ (e.g. monetary outcomes), then a typical lottery would be
$ L := (p_{x_1}^L, p_{x_2}^L, \dots, p_{x_n}^L),$
assigns probability $p_{x_i}^L$ to any event $x_i \in X$ under the usual restriction that $\sum_{i=1}^n p_{x_i}^L = 1$ (this is for a finite set of outcomes but can be extended to a continuum).
In the Savage model, the alternatives are so-called "compound lotteries" over a set of states of the world $S := \{s_1,\dots,s_m\}$.
(States of the world correspond to what are called "acts" in the statement of the axiom that you quote. This is not exactly Savage's model, but I think it will help to answer your question if we consider this model instead. The model in terms of states of the world with objective probabilities is close enough to the full-fledged subjective probability model of Savage, at least on the point you are trying to understand.)
These compound lotteries are in fact lists of simple lotteries over the outcomes $\{x_1,\dots,x_n\}$. For a given probability distribution over the states $(p_{s_1},\dots,p_{s_m})$, a typical alternative in this setup is
$\boldsymbol{L} := ({L}_{s_1}, {L}_{s_2}, \dots, {L}_{s_m}),$
where $L_{s_i}$ is the lottery over outcomes you would play if you were to play the compound lottery $\boldsymbol{L}$, and state $s_i$ was to realize.
The difference between the two models is somewhat subtle. At first glance, it might seem like they are identical. In effect, for every compound lottery $\boldsymbol{L}$, you can construct a simple lottery $L$. The probability of outcome $x_i$ in this simple lottery is just the sum of the probabilities that $x_i$ occurs in every possible state of the world, times the probability that this state occurs. That is the probability that outcome $x_i$ occurs given compound lottery $\boldsymbol{L}$ and the distribution of probabilities for the states is
$P(x_i|\boldsymbol{L},p_{s_1},\dots,p_{s_m}) = \sum_{j=1}^m [p_{x_i}^{L_{s_k}}*p_{{s_k}}]$
So if the agent's preferences only depend on outcomes, the two models are identical. But if the agent's preferences depend also on the states, they might not be. This is precisely what is allowed in Savage's framework and not in VnM's framework (in that sense Savage is a generalization of VnM). In Savage's framework, two lotteries may yield the same probability distribution over outcomes but not leave the agent indifferent.
Maybe an example would clarify this point. Consider the set of outcomes $X := \{x_1 = $having a monthly wage of 6000USD, $x_2 = $ having a monthly wage of 3000USD$\}$. Suppose you have to choose your course of studies between Construction Engineering and Med school. There are two states of the world.
Suppose that you are more likely to earn 6000 USD per month if you went to med school under state $s_1$, because of the high demand for health practitioners following the pandemic. Assume the payoffs are as follows
$$\begin{array}{c||c|c} P(s_1) = 0.5 & 0.8 & 0.2\\ \hline Med School & 6000 & 3000 \\ Cons. Eng. & 3000 & 6000 \end{array} $$
$$\begin{array}{c||c|c} P(s_2) = 0.5 & 0.8 & 0.2\\ \hline Med School & 3000 & 6000 \\ Cons. Eng. & 6000 & 3000 \end{array} $$
So, in state $s_1$ (which occurs with probability $0.5$), you get a monthly salary of $6000$ USD with a $0.8$ probability if you went to Med School, and with a $0.2$ probability if you did construction engineering (maybe the construction market is down as a consequence of the pandemic).
Notice that if you look at the probability of the outcomes only, the choice of your course of studies is irrelevant. In both case, the probability that you get a high salary is
$$ \begin{align} P(6000 USD | Med School) & = P(s_1) * P(6000|s_1, Med School) + P(s_2)* P(6000|s_2,Med School)\\ & = 0.5 * 0.8 + 0.5 * 0.2\\ & = 0.5 \\ & = 0.5 * 0.2 + 0.5 * 0.8\\ & = P(s_1) * P(6000|s_1, Cons. Eng) + P(s_2)* P(6000|s_2,Cons. Eng.)\\ & = P(6000 | Cons. Eng.) \end{align} $$
Therefore, in the VnM model, you have to be indifferent between the two courses of study. Informally, one may argue that you should prefer Med school, as it pays better in situations where being wealthy is particularly useful (the pandemic situation), whereas construction engineering pays better when everything is just fine (the non-pandemic situation). But under the VnM model, you're not allowed to take this into consideration and you must be indifferent.
In Savage's model, however, your preferences may depend on the state of the world, and you're allowed to prefer a compound lottery over another even when both yield the same probability distribution over outcomes.
To compare the independence axioms, you must first make the two underlying models comparable. Savage model is more general than VnM and boils down to VnM when there is only one state of the world. In this case, Savange independence reduces to VnM Independence.
In the other direction, you could understand VnM independence in a setup with multiple states as applying only to compound lotteries that yield the same simple lottery in every state of the world. With this understanding, in Savage's more general model, VnM independence would be weaker than Savage independence.
For more on the topic, I highly recommend section 6E of Mas-Collel, Whinston and Green, Microeconomic theory, which precisely deals with these issues.