Wolfgang Schwarz

Values and consequences in economics and quantum mechanics

One of the novelties in Richard Jeffrey's "Logic of Decision" (1965) was to unify the space over which probabilities and values are defined: both probability and desirability are distributed over the space of possible worlds, of ways things might be. By contrast, in earlier theories like that of Savage, probabilities were defined over states (or events) and utilities over consequences, which were taken to be distinct kinds of things. Technically, this difference between Savage and Jeffrey isn't terribly important as long as anything an agent may care about can be found in the set of 'consequences'. However, the distinction and the labeling in Savage's treatment carries a danger to overlook the complexity of human values. This has, I believe, led to a number of serious mistakes.

Here are some things one might value, or desire: getting 10 Euros, eating ice-cream, rolling down a hill, being healthy, being poor, owning a donkey.

Here are some other things: having learnt Latin as a child, having discovered the incompleteness theorems, not descending from apes, getting 10 Euros next month, being payed as much as one deserves, there being trees in two hundred years, the world being created by an intelligent being.

Savage-style theories, which still dominate in economics, tend to ignore desires of the second type. Thus if I prefer 10 Euros over 10 apples today, but 10 apples over 10 Euros tomorrow, then many economic models would say that my preferences have changed. However, let A- be the proposition that I receive 10 apples now while not having received 10 Euros yesterday; let A+ be the proposition that I receive 10 apples now while having received 10 Euros yesterday. Suppose my preference order both today and tomorrow is A+ > 10 Euros > A-. Then I will take the 10 Euros today and the 10 apples tomorrow, and my preferences won't have changed. Similarly, as Robert Aumann pointed out to Savage, it can make a big difference to the desirability of receiving $100 whether one's wife survives a dangerous operation. In general, the 'consequences' to which real people assign values aren't just simple, temporary events ('receiving $100'), but entire histories that may involve the distant past and future and the well-being of other people. They are entire ways things might be. The alleged distinction between states and consequences therefore becomes insubstantial (as Aumann also noticed).

The misconception that values are attached only to simple, temporary events shows up (for example) in the Discounted Utility model, the currently dominant framework for policy evaluation. Suppose values are only assigned to things like 'getting 10 Euros', irrespective of what happens earlier or later. How then should we make decisions whose consequences extend far into the future (say, whether to act on climate change or not)? A natural thought is to aggregate the costs and benefits for our successors and choose the option with the highest (expected) aggregated utility. If we care more about closer than about distant successors, we might add a 'discount rate', giving less weight to the costs and benefits for more distant successors. This is the Discounted Utility model, and it is supposed to reflect our actual judgements regarding long-term decisions.

Since in reality our values are not restricted to 'getting n Euros', it is unsurprising that the Discounted Utility model systematically fails in empirical tests (see Frederick et al 2002 for a survey). The things we value include entire histories, and there is no reason why the value we attach to a history should be proportional to the aggregate of goods we receive over this history. Many things I value about histories (like an end of the Darfur war) have nothing to do with goods I receive; some even concern times long after I'm dead. As for the distribution of goods in my life, my preferences are not at all determined by (discounted) aggregation: some goods I'd rather have now, others later, and often I'd prefer an even distribution over getting everything tomorrow.

Like other expected utility theories, the Discounted Utility model can be derived from 'qualitative' axioms about preferences, in this case preferences over histories. Such axioms were found by Tjalling Koopmans (1960). Economists tend to find them "intuitively compelling" (e.g. Rick and Loewenstein 2008, p.144), though they clearly reveal the misconception about values. For instance, Koopmans's axiom of stationarity holds that agents prefer history h1 over h2 iff they prefer the extended history a,h1 (extended by prefixing a) over the extended history a,h2. But couldn't I prefer having a glass of wine before going to bed over having water, but have the reverse preference if both alternatives are prefixed by drinking a bottle of vodka? (Koopmans, it should be mentioned, was well aware of these limitations, and had little faith in the Discounted Utility model.)

Next, the interpretation of quantum mechanics. On the Everett ('many worlds') account, every physically possible result of a quantum measurement actually occurs, but each on its own 'branch' of the universe. What is traditionally understood as the chance of a particular outcome is now represented as the weight of the relevant branch. The problem then is to explain why these weights can play the role of probabilities in theory confirmation and rational action. David Deutsch (1999) and David Wallace (in several papers) claim to have found this explanation, by proving that rational agents who know that their world is about to fission into several distinct successors s, each with a particular quantum weight w(s), will choose actions that maximize the average utility on each branch, weighted by the branch weights. Branch weights therefore play exactly the role of subjective probability in guiding rational action. (See Greaves 2006 for an introduction to the Deutsch-Wallace program).

We don't need to follow the proof very closely to see where it rests on Savage's misconception about human values. The basic idea is to start with axioms concerning qualitative preferences among measurements and their consequences -- much like Koopmans's axioms concerning preferences among histories -- and show that these axioms are uniquely represented by a distribution of probabilities and values in which the probabilities match the branch weights. (I find the proof in Wallace 2003 particularly clear; see Wallace 2002, Wallace 2005, and Greaves 2004 for variations.) Somewhat more precisely, let a game be a triple of a physical state, an operator (on the relevant Hilbert space -- think of something like momentum or position), and a function returning monetary values depending on the operator's value for the state. A physical process realises a game <s,o,p> iff it consists in performing a measurement of o on s and handing out money in accordance with p and the measurement result (on each branch). Suppose now that rational agents have a preference order over games. Under apparently harmless assumptions, it can be shown that this order is uniquely represented by a utility function that identifies the utility of a game with the expectation of its payoff relative to the branch weights, i.e. with \sum_c w(c)c, where w(c) is the sum of the weights of branches with payoff c. This means that in choosing between games, rational agents will act as if they were uncertain about which outcome would occur, and as if they distribute their credence in accordance with the branch weights.

There are two problems with this, both due to the Savage-style division between states and consequences (payoffs). To see the first, note that agents don't actually face choices between 'games' (triples of a state, an operator and a function), but only between actions -- physical processes that at most realise a game. Deutsch (implicitly) and Wallace (explicitly) assume that the utility of such an action equals the utility of any game it realises. The specification of the game must therefore contain everything the agent may care about in a given realisation. Why should that be so? Because it is assumed that all the agent cares about is the payoff, which is specified in the game.

As long as all we care about are things like 'receiving 10 Euros', this is fine. But what if I care about what kind of process made me receive 10 Euros -- say, how much carbon dioxide it produced, or whether it killed my entire family? Then the desirability of an action is not determined by what games it realises. When Wallace (2003:23) claims that "there is no rational justification" for preferences among games with the same payoff, he overlooks history-dependent values. (What if we replace the 'payoffs' with everything an agent might care about? A game is then, in effect, a centered specification of an entire branching universe, and multiple realisation becomes impossible. Unfortunately, the 'apparently harmless' axioms required to derive the probability rule then cease to make sense.)

The other problem is to assume that the present value of a bet equals the expected payoff we get from it later. If this were true then, in order to determine the value of a quantum bet, we'd first have to figure out what we should think about the payoff we can expect to receive in cases where we're about to undergo branching. Proponents of the Deutsch-Wallace program therefore spend a lot of ink debating issues about personal identity and expectations in the face of fission. However, in a Jeffrey-style theory, the value of a bet is not the expectation of its later payoff. The value of a bet -- just like the value of any other action -- is the expectation of its current utility. This works because utilities are defined for things like 'receiving 10 Euros tomorrow'.

So let's look at the present utilities I might assign to hypotheses involving branching. I might give high marks to scenarios where I receive money on branches with high weight. But I might just as well prefer receiving money on branches with low weight. As mentioned above, I might also care about what sort of event has caused the branching. And I might care about what happens on branches on which I don't even exist. There is no reason to assume that rational preferences are determined by whether or not I will exist on a branch, and what payoffs I will receive there. To think that today's value of 'receiving 10 Euros tomorrow' is simply a matter of tomorrow's value of 'receiving 10 Euros' is to repeat the mistake of the Discounted Utility model.