[UPDATE: After posting this, Johan Gustafsson got in touch and it seems he and I have happened upon similar points via slightly different routes. His paper is here. He takes his axioms from Binmore's

*Rational Decisions*, who took them from Milnor's 'Games against Nature'. Hurwicz and Arrow also cite Milnor, but Hurwicz's original characterisation appeared before Milnor's paper, and he cites Chernoff's Cowles Commission Discussion Paper: Statistics No. 326A as the source of his axioms.]

In 1951, Leonid Hurwicz, a Polish-American economist who would go on to share the Nobel prize for his work on mechanism design, published a series of short notes as part of the Cowles Commission Discussion Paper series, where he introduced a new decision rule for choice in the face of massive uncertainty. The situations that interested him were those in which your evidence is so sparse that it does not allow you to assign probabilities to the different possible states of the world. These situations, he thought, fall outside the remit of Savage's expected utility theory.

The rule he proposed is called

*Hurwicz's Criterion of Realism*or just

*the Hurwicz Criterion*. He introduced it in the form in which it is usually stated in February 1951 in the Cowles Commission Discussion Paper: Statistics No. 356 -- the title was 'A Class of Criteria for Decision-Making under Ignorance'. The Hurwicz Criterion says that you should choose an option that maximises what I'll call its

*Hurwicz score*, which is a particular weighted average of its best-case utility and its worst-case utility. A little more formally: We follow Hurwicz and let an option be a function $a$ from a set $W$ of possible states of the world to the real numbers $\mathbb{R}$. Now, you begin by setting the weight $0 \leq \alpha \leq 1$ you wish to assign to the best-case utility of an option, and then you assign the remaining weight $1-\alpha$ to its worst-case. Then the Hurwicz score of option $a$ is just $$H^\alpha(a) := \alpha \max_{w \in W} a(w) + (1-\alpha) \min_{w \in W} a(w)$$

However, reading his other notes in the Cowles series that surround this brief three-page note, it's clear that Hurwicz's chief interest was not so much in this particular form of decision rule, but rather with any such rule that determines the optimal choices solely by looking at their best- and worst-case scenarios. The Hurwicz Criterion is one such rule, but there are others. You might, for instance, weight the best- and worst-cases not by fixed constant coefficients, but by coefficients that change with the minimum and maximum values, or change with the difference between them or with their ratio. One of the most interesting contributions of these papers that surround the one in which Hurwicz gives us his Criterion is a characterization of rules that depend only on best- and worst-case utilities. Hurwicz gave rather an inelegent initial version of that characterization in Cowles Commission Discussion Paper: Statistics No. 370, published at the end of 1951 -- the title was 'Optimality Criteria for Decision-Making under Ignorance'. Kenneth Arrow then seems to have helped clean it up, and they published the new version together in the Appendix of their edited volume, in which they contributed most of the chapters, often with co-authors,

*Studies in Resource Allocation*. The version with Arrow is still reasonably involved, but the idea is quite straightforward, and it is remarkable how strong a restriction Hurwicz obtains from seemingly weak and plausible axioms. This really seems to me a case where axioms that seem quite innocuous on their own can combine in interesting ways to make trouble. So I thought it might be interesting to give a simplified version that has all the central ideas.

Here's the framework:

*Possibilities and possible worlds.*Let $\Omega$ be the set of possibilities. A possible world is a set of possibilities--that is, a subset of $\Omega$. And a set $W$ of possible worlds is a partition of $\Omega$. That is, $W$ presents the possibilities at $\Omega$ at a certain level of grain. So if $\Omega = \{\omega_1, \omega_2, \omega_3\}$, then $\{\{\omega_1\}, \{\omega_2\}, \{\omega_3\}\}$ is the most fine-grained set of possible worlds, but there are coarser-grained sets as well, such as $\{\{\omega_1, \omega_2\}, \{\omega_3\}\}$ or $\{\{\omega_1\}, \{\omega_2, \omega_3\}\}$. (This is not quite how Hurwicz understands the relationship between different sets of possible states of the world -- he talks of deleting worlds rather than clumping them together, but I think this formalization better captures his idea.)

*Options*. For any set $W$ of possible worlds, an option defined on $W$ is simply a function from $W$ into the real numbers $\mathbb{R}$. So an option $a : W \rightarrow \mathbb{R}$ takes each world $w$ in $W$ and assigns a utility $a(w)$ to it. (Hurwicz refers to von Neumann and Morgenstern to motivate the assumption that utilities can be measured by real numbers.)

*Preferences*. For any set $W$ of possible worlds, there is a preference relation $\preceq_W$ over the options defined on $W$. (Hurwicz states his result in terms of optimal choices rather than preferences. But I think it's a bit easier to see what's going on if we state it in terms of preferences. There's then a further question as to which options are optimal given a particular preference ordering, but we needn't address that here.)

Hurwicz's goal was to lay down conditions on these preference relations such that the following would hold:

**Hurwicz's Rule**Suppose $a$ and $a'$ are options defined on $W$. Then

**(H1)**If

- $\min_w a(w) = \min_w a'(w)$
- $\max_w a(w) = \max_w a'(w)$

**(H2)**If

- $\min_w a(w) < \min_w a'(w)$
- $\max_w a(w) < \max_w a'(w)$

Here are the four conditions or axioms:

**(A1)**

**Structure**$\preceq_W$ is reflexive and transitive.

**(A2) Weak Dominance**

- If $a(w) \leq a'(w)$ for all $w$ in $W$, then $a \preceq_W a'$.
- If $a(w) < a'(w)$ for all $w$ in $W$, then $a \prec_W a'$.

**(A3) Permutation Invariance**For any set of worlds $W$ and any options $a, a'$ defined on $W$, if $\pi : W \cong W$ is a permutation of the worlds in $W$ and if $a'(w) = a(\pi(w))$ for all $w$ in $W$, then $a \sim_W a'$.

This just says that it doesn't matter to you which worlds receive which utilities -- all that matters are the utilities received.

**(A4) Coarse-Graining Invariance**Suppose $W = \{\ldots, w_1, w_2, \ldots\}$ is a set of possible worlds and suppose $a, a'$ are options on $W$ with $a(w_1) = a(w_2)$ and $a'(w_1) = a'(w_2)$. Then let $W' = \{\ldots, w_1 \cup w_2, \ldots\}$, so that $W'$ has the same worlds as $W$ except that, instead of $w_1$ and $w_2$, it has their union. And define options $b$ and $b'$ on $W'$ as follows: $b(w_1 \cup w_2) = a(w_1) = a(w_2)$ and $b'(w_1 \cup w_2) = a'(w_1) = a'(w_2)$, and $b(w) = a(w)$ and $b'(w) = a'(w)$ for all other worlds. Then $a \sim_W a'$ iff $b \sim_W b'$.

This says that if two options don't distinguish between two worlds, it shouldn't matter to you whether they are defined on a fine- or coarse-grained space of possible worlds.

Then we have the following theorem:

**Theorem**

**(Hurwicz)**(A1) + (A2) + (A3) + (A4) $\Rightarrow$ (H1) + (H2).

Here's the proof. Assume (A1) + (A2) + (A3) + (A4). First, we'll show that (H1) follows. We'll sketch the proof only for the case in which $W = \{w_1, w_2, w_3\}$, since that gives all the crucial moves. So denote an act on $W$ by a triple $(a(w_1), a(w_2), a(w_3))$. Now, suppose that $a$ and $a'$ are options defined on $W$ with the same minimum, $m$, and maximum, $M$. Let $n$ be the middle value of $a$ and $n'$ the middle value of $a'$.

Now, first note that

$$(m, m, M) \sim_W (m, M, M)$$ After all, $(m, m, M) \sim_W (M, m, m)$ by Permutation Invariance. And, by Coarse-Graining Invariance, $(m, M, M) \sim_W (M, m, m)$ iff $(m, M) \sim_{W'} (M, m)$, where $W' = \{w_1, w_2 \cup w_3\}$. And, by Permutation Invariance and the reflexivity of $\sim_{W'}$, $(m, M) \sim_{W'} (M, m)$. So $(m, M, M) \sim_W (M, m, m) \sim_W (m, m, M)$, as required. And now we have, by previous results, Permutation Invariance, and Weak Dominance:

$$a \sim_W (m, n, M) \preceq_W (m, M, M) \sim_W (m, m, M) \preceq_W (m, n', M) \sim_W a'$$

and

$$a' \sim_W (m, n', M) \preceq_W (m, M, M) \sim_W (m, m, M) \preceq_W (m, n, M) \sim_W a$$

And so, by transitivity, $a \sim_W a'$. That gives (H1).

For (H2), suppose $a$ has worst case $m$, middle case $n$, and best-case $M$, while $a'$ has worst case $m'$, middle case $n'$, and worst case $M'$. And suppose $m < m'$ and $M < M'$. Then$$a \sim_W (m, n, M) \preceq_W (m, M, M) \sim_W (m, m, M) \prec_W (m', n', M') \sim_W a'$$as required. $\Box$

In a follow-up blog post, I'd like to explore Hurwicz's conditions (A1-4) in more detail. I'm a fan of his approach, not least because I want to use something like his decision rule within the framework of accuracy-first epistemology to understand how we select our first credences -- our ur-priors or superbaby credences (see here). But I now think Hurwicz's focus on only the worst-case and best-case scenarios is too restrictive. So I have to grapple with the theorem I've just presented. That's what I hope to do in the next post. But here's a quick observation. (A1-4), while plausible at first sight, sail very close to inconsistency. For instance, (A1), (A3), and (A4) are inconsistent when combined with a slight strengthening of (A2). Suppose we add the following to (A2) to give (A2$^\star$):

3. If $a(w) \leq a'(w)$ for all $w$ in $W$ and $a(w) < a'(w)$ for some $w$ in $W$, then $a \prec_W a'$.

Then we have know from above that $(m, m, M) \sim_W (m, M, M)$, but (A2$^\star$) entails that $(m, m, M) \prec_W (m, M, M)$, which gives a contradiction.

What an interesting result. I suppose it would be easy to reframe it as an impossibility theorem? You could add another axiom along the lines of "it must be possible for the middle-cases to make a difference to the decision" and then the result would show that there is no decision rule that meets all the conditions.

ReplyDeleteSuch a result would seem intuitive. It would basically be saying: if you refuse to assign probabilities to scenarios, you will have no principled way of assigning weights to the middle-case scenarios, so you will have no adequate decision rule.

Thanks, Jonathan! Really nice point! In fact, after posting, Johan Gustafsson sent on a draft paper of his in which he gets the same result and does indeed reframe as an impossibility result. I've put a link in an update at the top of the post.

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