Von Neumann–Morgenstern independence axiom vs. Savage independence theorm

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?

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.

This answer corrects my previous answer (almost 10 years old) which misleadingly compared Vnm's model with a state-dependent model, as opposed to Savage's model which the OP asks about.

The difference between the two axioms is they belong to different models of decision under uncertainty, each aimed at answering different kinds of questions.

In VnM's model (Mas-Collel, section 6.B), the primitives are:

• A set of $n$ outcomes $C$,
• Lotteries $L = (p_1, \dots, p_n)$ representing the probability that outcome $n$ occurs, and
• A preference relation $\succeq$ (binary relation) over the collection $\mathcal{L}$ of lotteries.

Informally, VnM's model is meant to represent a situation where a decision-maker chooses between different "gambles" (represented by lotteries in $\mathcal{L}$) based on $\succeq$. By "based on", we mean that when faced with two lotteries $L$ and $L'$, the decision-maker picks $L$ iff $L \succ L'$.

Importantly, in VnM's model, there's no question of where these gambles "come from", and an implicit understanding that the decision-maker knows the probabilities associated with a lottery $L$ when making choices. In practice, this may represent situations similar to playing a casino gamble the decision-maker knows the odds of.

The question VnM ask (and answer with their famous representation theorem) is what set of "reasonable" restrictions can be put on $\succeq$ to guarantee that it can be represented by a utility function of the expected utility form. That is, what set of restrictions on $\succeq$ guarantee the existence of numbers $(u^\succeq_1, \dots, u^\succeq_n)$ such that

$$L \succeq L'$$

if and only if

$$U(L) = u^\succeq_1 p_1 + \dots + u^\succeq_n p_n \geq u^\succeq_1 p'_1 + \dots + u^\succeq_n p'_n = U(L').$$

To this end, one of the "reasonable" restrictions they consider is the independence axiom mentioned by the OP, namely:

For all $L,L',L''$ and all $\alpha \in (0,1)$, we have $L \succ L'$ if and only if $\alpha L + (1-\alpha) L'' \succ L' + (1-\alpha) L''$.

In Savages's model (here I present a version that is a mix of Mas-Collel, section 6.F and https://plato.stanford.edu/entries/decision-theory/#VNMRepThe, to allow for non-monetary outcomes and better comparison with VnM), the primitives are:

• A set of $n$ outcomes $C$,
• A set of $m$ states of the world $S$,
• Collections of random variables over outcomes $X = (X_1,\dots, X_m)$ known as "acts", and
• A preference relation (binary relation) $\succeq$ over the collection $\mathcal{X}$ of acts.

Informally, Savage's model represent a situation where a decision-maker chooses between different "plans of actions" (represented by the acts in $\mathcal{X}$) based on $\succeq$ (again, by "based on", we mean that when faced with the choice between acts $X$ and $X'$, the decision-maker implements $X$ iff $X \succ X'$). That is, Savage's framework is meant to model a situation in which a decision-maker chooses between "actions" rather than "gambles", with an "act" $X$ being thought of as a contingent plan for action in every possible state of the world (with each action leading to a certain distribution of outcomes in each state).

Importantly, in Savage's model, the probabilities of different states materializing are not part of the primitives of the model. Decision-makers simply choose between contingent plans of actions $X$ based on some preference $\succeq$.

The question Savage asks is precisely what "consistency" restrictions on $\succeq$ make $\succeq$ representable by a utility function that suggests behavior consistent with (subjective) probabilistic beliefs about the states of the world.

The idea is if there exists:

• numbers $(\pi^\succeq_1, \dots, \pi^\succeq_m)$ with $\sum_{i=1}^m \pi^\succeq_i = 1$, and
• numbers $(u_1^\succeq, \dots, u_n^\succeq)$,

such that

$$X \succeq X'$$

if and only if

$$U(X) = \sum_c \sum_s u_c^\succeq \pi^\succeq_s X_s(c) \geq \sum_c \sum_s u^\succeq_c \pi^\succeq_s X'_s(c) = U(X'),$$

then the behavior induced by $\succeq$ is indistinguishable from the behavior of a decision-maker who

• holds (subjective) probabilistic beliefs $(\pi^\succeq_1, \dots, \pi^\succeq_m)$ over the realization of the states,
• assigns utilities $(u^\succeq_1, \dots, u^\succeq_n)$ to the different outcomes in $C$, and
• chooses between contingent plans of actions based on the (subjective) expected utility characterized by $(\pi_1, \dots, \pi_m)$ and $(u_1, \dots, u_n)$.

In other words, decisions based on $\succeq$ are "as if" they were based on some presumed probability distribution over states of the world. In this sense, probabilities would be derived from (or "revealed by") actions, which some view as offering an action-based subjective foundation for probabilities themselves.

One of the restrictions Savage impose to this end (i.e., determining the conditions on $\succeq$ that guarantee the kind of representability described above) is the independence axiom mentioned by the OP, namely:

For all $X,X',X''$ and all $\alpha \in (0,1)$, we have $X \succ X'$ if and only if $\alpha X + (1-\alpha) X'' \succ X' + (1-\alpha) X''$ (where acts like $\alpha X + (1-\alpha) X''$ are defined as $[\alpha X_1 + (1-\alpha) X_1, \dots, \alpha X_m + (1-\alpha) X_m'']$).