Cultural issues II: language

[Itzhak Gilboa]

In response to Joe's comment: I, for one, would be very interested in your thoughts on language and the state space, and I hope you'd post links when you have something written.

An aside on academic culture: a classical example of the dependence on language -- in particular, of simplicity judgments -- is Goodman's grue-bleen paradox.  For years I used to teach it as a "real" paradox, then I was convinced that it can be resolved and that "green" is really simpler than "grue" in a well-defined way.  (The argument is about language: the paradox shows that which theory is simpler depends on one's language.  But, as Solomonoff argued, using Kolmogorov's complexity, one can bound the extent to which two languages may differ in their "simpler than" ranking of theories.)

Then I found that some people thought that this resolution of Goodman's paradox was sort of obvious, and that others -- that it completely misses the point.  My sociological observation is that people who were brought up in statistics or computer science cultures, and who tend to think of data, or of digits on a tape, as primitives, would find that "green" is simpler than "grue".  By contrast, people who's mother tongue is logic, would typically think that the paradox is robust to these resolutions.  For the former, theories are generalizations, attempting to fit the primitive data.  For the latter, data are instantiations of idealized concepts, which are the primitives.  In a way, language strikes again: this time it's not the language modeled in Goodman's examples, but the language used by the theorists to discuss it.  The languages of logic and that of statistics/computer sciences are sufficiently different to change our notion of what is intuitive.

The difference between the theorists' languages, or cultures, is somewhat reminiscent of the difference between states of the world and logical propositions as the primitives of the model.  Here, again, one's academic upbringing might have far reaching implications on what one finds as intuitive.  For example, assumptions of measurability seem much more arbitrary if one thinks of a state space as the primitive, and much more natural if one starts with propositions for which the state space is a model.

Tzachi

On Fri, Dec 14, 2012 at 1:46 AM, halpern <hal...@cs.cornell.edu> wrote:

As someone who has worked both on the common prior assumption and on "logical" approaches to assigning probability (see, for example, "From statistics to beliefs" -- http://www.cs.cornell.edu/home/halpern/papers/bghk-aaai92.pdf -- and
"From statistical knowledge bases to degrees of beliefs" -- http://www.cs.cornell.edu/home/halpern/papers/statbel.pdf; the former is a conference paper that discusses three different "logical" approaches, while the latter is a journal paper that expands on one of them, which is Carnap's favored approach), I was somewhat surprised to see Carnap's approach suggested by Peter as a basis  for the common prior assumption.  Let me at least point out that it's far from obvious that Carnap's approach is objective.  Roughly, his approach says that we take all "descriptions of the world" to be equally likely.  But what counts as a description of the world depends very much on the language that you use to describe the world.  There's nothing objective about that.  One agent can talk about a scarf being colorful or drab; another might talk about the scarf being red or green. Different languages lead to different state spaces, and different priors on them.   In general, I don't think there's anything objective about the choice of language.  (As an aside, the role of language in game theory and decision theory is something I've been looking at closely for the past few years; I think there's much more to be said about that.)

There are also issues about the objects that we take to be equally likely; roughly speaking, the issue is what counts as a description of the world.   (This is made more precise in "From statistics to beliefs".)  Again, different choices lead to different priors.  Indeed, if I remember right, Carnap considered a continuum of possible priors indexed by a parameter \lambda. (We consider only three in "From statistics to beliefs", but two of them are different from those considered by Carnap.)

-- Joe

[Andrew Postlewaite]

David Dillenberger, Kareen Rozen and I have revised our paper "Optimism and Pessimism with Expected Utility". The abstract is below and the paper can be downloaded at http://www.ssc.upenn.edu/~apostlew/paper/pdf/Pessimism.pdf .  Any comments would be welcome.

Andy


Savage (1954) provides axioms on preferences over acts that are equivalent to the existence of a subjective expected utility representation. We show that there is a continuum of other "expected utility" representations in which for any act, the probability distribution over states depends on the corresponding outcomes and is first-order stochastically dominated by (dominates) the Savage distribution. We suggest that pessimism (optimism) can be captured by the stake-dependent probabilities in these alternate representations. We then extend the DM's preferences to be defined over both subjective acts and objective lotteries. Our result permits modeling ambiguity aversion in Ellsberg's two-urn experiment using pessimistic probability assessments, the same utility over prizes for lotteries and acts, and without relaxing Savage's axioms. An implication of our results is that the large body of existing research based on expected utility can, with a simple reinterpretation, be understood as modeling the behavior of optimistic or pessimistic decision makers.

--

Andrew Postlewaite   Department of Economics   University of Pennsylvania  Philadelphia, PA 19104       
Phone  215 898-7350   Fax  215 573-2057     http://www.ssc.upenn.edu/~apostlew/

 

[g charles-cadogan]

Prof. Postelwaite:

I made a cursory inspection of Dillenberger, at al (2012), and read Section 2 in its entirety. So my comments pertain to Section 2. Assuming without deciding that state dependent probability distributions are admissible, let p(x_1,x_2) be the probability distribution over states. Thus, we have the marginal distribution p(x_1,x_2|x_2) = p_1 and p(x_1,x_2|x_1)=p_2; and we should be able to recover x_1 and x_2 if we know p(.)--providing the simultaneous equations have a unique solution.  As a practical matter, one would need a parametric representation of p(.) based on x_1 and x_2 in order to accomplish that. However, I did not see such an example in the paper. Admittedly, Dillenberger, at al (2012) focus in on extension of Savage’s (1954, 1972) subjective expected utility (SEU).  Therefore, at the risk of comparing apples and oranges, I draw attention to the literature on prospect theory popularized by Kahneman and Tversky (1979), where an important paper by Tversky and Wakker (1995) shows how curvature properties of the inverted S-shaped probability weighting function identifies pessimistic and optimistic behavior. Moreover, empirical papers by Wilcox (2008) and Andersen, et al (2010) show how to estimate risk attitudes, like those in Dillenberger, at al (2012), in the context of a structural framework via maximum likelihood methods. In fact, Dillenberger, at al (2012) representation of P_1(x;p) is similar to Wilcox (2011) contextual utility. Additionally, a few weeks ago I posted a paper entitled “Group Representations for Decision Making under Risk and Uncertainty” which employs representation theory and or group theoretic methods to show how one can recover state dependent probability distributions, like that posited in Dillenberger, at al (2012), in the context of prospect theory. In particular, that paper plainly shows how curvature parameters and risk attitude by and through loss aversion is embedded in state dependent probabilities of the type posited in Dillenberger, at al (2012)  It would be interesting to see how Dillenberger, at al (2012) performs using a prospect theory paradigm.

References

Andersen, S.; Harrison, G. W.; Lau, M. I.; and Rustrom,  E. E. (2010), “Behavioral Econometrics for Psychologists,” Journal of Economic Psychology 31: 553-576

Cadogan, G. (2012), “Group Representations for Decision Making under Risk and Uncertainty”. Working Paper. Available at http://papers.ssrn.com/abstract=2189880

Dillenberger, D.;  Postlewaite, A.; and Rozen, K. (2012). Optimism and Pessimism with Expected Utility. Available at http://www.ssc.upenn.edu/~apostlew/paper/pdf/Pessimism.pdf

Kahneman, D. and A. Tversky (1979). Prospect theory: An analysis of decisions under risk. Econometrica 47(2), 263–291.

Savage, F. H. (1972). Foundations Of Statistics (2nd rev ed.). Mineola, NY:Dover Publications, Inc.

Tversky, A. and P. Wakker (1995, Nov.). Risk Attitudes and Decision Weights. Econometrica 63(6), 1255–1280.

Wilcox, N. (2008). Stochastic models for binary discrete choice under risk: A critical primer and econometric comparison. Research in experimental economics 12, 197–292. Special Issue: Risk Aversion in Experiments.

Wilcox, N. T. (2011). ‘Stochastically more risk averse’: A contextual theory of stochastic discrete choice under risk. Journal of Econometrics 162 (1), 89 – 104. Special Issue: The Economics and Econometrics of Risk.

--
You received this message because you are subscribed to the Google Groups "decision_theory_forum" group.
To post to this group, send email to decision_t...@googlegroups.com.
To unsubscribe from this group, send email to decision_theory_...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/decision_theory_forum?hl=en.