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.
- In state $s_1$, there is a severe pandemic due to some bacterias becoming increasingly resistant to antibiotics.
- In state $s_2$, some new cheap drug to limit the growth of bacteria has been discovered and the pandemic does not occur.
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.
First off, econometrics does not assume that relationships of $x$ and $y$ in economics are linear [in general]. Rather, it says that if a relationship is linear, then you can use OLS to estimate these effects. Moreover, OLS is the most efficient estimator in this case, as you undoubtedly have learned in econometrics.
Second, fair enough, if you look at the applied economics literature, then researchers often do not bother to discuss whether the linearity assumption is plausible in their case. And very often it will not be. Still, sometimes the linearity assumption is innocuous. For example, if all your $x$ are dummy variables denoting group-membership, then you just estimate conditional group-means. Or if the relationship is quadratic, then you can still include higher order terms in your OLS regression and everything is fine, or in fact any other higher order polynomial.
More generally, I like to view OLS not as a linear model but one that assumes additive separability. Because $y=\alpha+\beta \log(x_1)+\gamma x_2+\epsilon$ can still be estimated with OLS even though it is clearly not linear in $x_1$. And OLS is very often used to estimate such nonlinear relationships (e.g., logs when estimating elasticities).
Third, if you are looking at a relationship where you believe that additive separability is not plausible, then there are other tools and you should use them. For example nonlinear least squares. Or, if the dependent variable is binary (zero-one), then applied economists tend to use the nonlinear logit regression more than OLS.
Fourth, there is a different approach in economics called "structural estimation". This is probably closer to what you are used to from physics. The idea here is that you write down an economic theory model and then estimate the parameters of this model empirically. This is very popular in the field of industrial organization. The relationship of two economic variables in such structural models will be linear only if they are also linear in the economic model based on the assumed utility functions, error distributions, etc.
Overall, I agree that economists tend to use OLS a lot, and sometimes in cases where they shouldn't. A main reason is probably indoctrination: OLS is covered in grad school while Poisson regression or other nonlinear models aren't. Another reason is that OLS is amazingly simple to interpret. Sometimes economists are aware that a relationship is not linear, but they estimate a linear model anyway because the resulting approximation is easier to interpret ("If you increase $x_1$ by 1, then your $y$ decreases by 0.3 on average!").
Best Answer
For the dual problem:
If $L\succ M$ and $\alpha\in (0,1)$ then by Ax3, $$L\,=\,\alpha L+(1-\alpha)L\,\succ\, \alpha L+(1-\alpha)M\,\succ\,\alpha M+(1-\alpha)M\,=\,M \,.$$
If also $\beta\in (0,\alpha)$ is given, then applying the same once again to $N:=\alpha L+(1-\alpha)M\,\succ M$ and $\gamma:=\frac\beta\alpha\in (0,1)$ we get $$N\,\succ\,\gamma N+(1-\gamma)M\,=\,\beta L+(1-\beta)M\,.$$ Swapping the role of $\alpha,\beta$ shows that if $\alpha<\beta$ then $\alpha L+(1-\alpha)M \prec \beta L+(1-\beta)M$, so it ensures $\alpha>\beta$ in Ax4.