A paper on no-betting Pareto
Itzhak Gilboa
David Schmeidler
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marc henry
g charles-cadogan
Polak, Benjamin
When I see nails everywhere, I can never find a hammer.
Teddy Seidenfeld
On Tzachi's advice, but with hesitation for pointing to what is a null-event by most of your academic lights, here are two papers from the Journal of Philosophy of an ongoing debate, about Pareto, between Isaac Levi on the one side, and Jay Kadane, Mark Schervish, and me on the other.
On the Shared Preferences of Two Bayesian Decision Makers; Seidenfeld, Kadane and Schervish, Journal of Philosophy, Vol. 86, No. 5 (May, 1989), pp. 225-244. http://www.jstor.org/stable/2027108
Pareto Unanimity and Consensus; Levi, Journal of Philosophy, Vol. 87, No. 9 (Sep., 1990), pp. 481-492 http://www.jstor.org/stable/2026970
Holiday wishes,
Teddy
Hammond, Peter
Dear Almost-all,
On Tzachi's advice, but with hesitation for pointing to what is a null-event by most of your academic lights, here are two papers from the Journal of Philosophy of an ongoing debate, about Pareto, between Isaac Levi on the one side, and Jay Kadane, Mark Schervish, and me on the other.
On the Shared Preferences of Two Bayesian Decision Makers; Seidenfeld, Kadane and Schervish, Journal of Philosophy, Vol. 86, No. 5 (May, 1989), pp. 225-244. http://www.jstor.org/stable/2027108
Pareto Unanimity and Consensus; Levi, Journal of Philosophy, Vol. 87, No. 9 (Sep., 1990), pp. 481-492http://www.jstor.org/stable/2026970
Holiday wishes,
Teddy
On 12/9/12 8:06 AM, Itzhak Gilboa wrote:
Dear all,
Larry Samuelson, David Schmeidler, and I completed a paper on Pareto domination under different subjective beliefs, trying to distinguish between risk sharing and pure betting. (The paper started from a note "A difficulty with Pareto domination", but it contains quite a bit more than did the note.)
The link is
and the abstract is below. Comments are most welcome!
Best,
Tzachi
________
No-Betting Pareto Dominanceby Itzhak Gilboa, Larry Samuelson, and David SchmeidlerAbstractWe argue that, in the presence of uncertainty, the notion of Pareto dominance is not as compelling as under certainty. In particular, voluntary trade that is based on di erences in tastes is commonly acceptedas favorable, because no agent involved in it can be wrong about her tastes. By contrast, voluntary trade that is based on incompatible beliefs may indicate that at least one agent is wrong abouther beliefs. We propose a weaker, No-Betting, notion of Pareto domination, which requires, on top of unanimity of preference, the existence of shared beliefs that can rationalize such preference for each agent.
Charles Manski
It may be relevant to mention that I reporting some findings on ex post Pareto dominance with heterogeneous beliefs and decision rules in
C. Manski (2010), "When consensus choice dominates individualism: Jensen's inequality and collective decisions under uncertainty," Quantitative Economics, 1, 187 - 202. http://onlinelibrary.wiley.com/doi/10.3982/QE5/pdf
Here is the abstract: "Research on collective provision of private goods has focused on distributional considerations. This paper studies a class of problems of decision under uncertainty in which an efficiency argument for collective choice emerges from the mathematics of aggregating individual payoffs. Consider decision making when each member of a population has the same objective function, which depends on an unknown state of nature. If agents knew the state of nature, they would make the same decision. However, they may have different beliefs or may use different decision criteria to cope with their incomplete knowledge. Hence, they may choose different actions even though they share the same objective. Let the set of feasible actions be convex and the objective function be concave in actions, for all states of nature. Then Jensen's inequality implies that consensus choice of the mean privately chosen action yields a larger mean payoff than does individualistic decision making, in all states of nature. If payoffs are transferable, the mean payoff from consensus choice may be allocated to Pareto dominate individualistic decision making. I develop these ideas. I also use Jensen's inequality to show that a planner with the power to assign actions to the members of the population should not diversify. Finally, I give a version of the collective-choice result that holds with consensus choice of the median rather than mean action."
Section 3 of the article gives the Pareto dominance finding, summarized as follows: "Section 3 shows that if payoffs are transferable, the mean payoff realized by collective choice of the consensus action may be allocated across the population so that collective choice Pareto dominates individualistic decision making in all states of nature. A Pareto dominating collective-choice mechanism is implementable if agents truthfully reveal the actions they would choose individualistically. I give conditions under which truthful revelation is incentive compatible."
Chuck
Professor Charles F. Manski
Department of Economics, Northwestern University
2001 Sheridan Road, Evanston, IL 60208 USA
phone 1-847-491-8223, fax 1-847-491-7001
email: cfma...@northwestern.edu
Sent: Sunday, December 09, 2012 10:52 AM
Peter Wakker
Dear all,
The common prior assumption is close to what is called the logical view of probability. Carnap wrote several books on it. The idea is that probability assignments can have the same status as logic. They are then objective, and the same for any two individuals who learned the same things during their lives.
Best regards, Peter
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hykel hosni
Two groups led by Jeff Paris and Alena Vencovska' (Manchester) and
Jon Williamson (Kent) are very actively working on
revamping/fixing/pushing forward Carnap's probability logic. Albeit
from a logical point of view, they defend a subjective interpretation
of probability. So the "common prior assumption" is best seen as an
intersubjective (rather than objective) feature of probability
assignments in those framework.
Details of recent/ongoing work can be found on
http://www.maths.manchester.ac.uk/~jeff/
and
http://www.kent.ac.uk/secl/philosophy/jw/2012/fobetil/
Best wishes,
hykel
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Hykel Hosni
Scuola Normale Superiore, Pisa and
CPNSS - London School of Economics
http://homepage.sns.it/hosni
g charles-cadogan
Fry, Robert L.
Edwin Jaynes at Wash U. was a friend of mine and let me try to
add to this discussion.
I believe that it was in a book review by Jaynes of the book The
Algebra of Probable Inference by Richard Cox that Jaynes
attributed to Cox the distinction of resolving the proper
interpretation of probability theory as being a generalization
of logic and Boolean Algebra. In this book (1961) in actually
in a much earlier paper Probability, Frequency, and Reasonable Expectation
which in 1946 derives probability theory using functional analysis
as a unique consequence of Boolean Algebra. This is all quite
profound and basic but pales in light of Cox’s last paper.
Cox was a physics professor at Johns Hopkins University for over
50 years (where I which and how I became aware of his work).
In his last paper titled “Of Inference and Inquiry” presented at
the first Maximum Entropy at MIT in 1978. In this paper, he
develops a Boolean algebra of questions – a logic complementary
to and consistent with standard Boolean Algebra as we know it.
Cox’s joint algebra of “assertions” and “questions” is quite powerful
and well-suited to understanding decision making with uncertain
(or certain) information. I have spent some 20 years extending
this into a general theory of computation and intelligent systems.
It works something like this.
Information flow is a dynamic process of what when a system poses a
question to its environment. A question consists of a suite of
answers pre-defined within the system, e.g., L={is the cat alive,
is the cat dead} = {a,d}. Use upper-case to denote questions.
Likewise, questions answered by the system represent decisions
like A={take the up escalator, take the down escalator}.
One can then consider Boolean combination of information and
decision-making questions, e.g., in the Shell Game:
S=”Under which of three shells lies the pea?” = {s1, s2,s3}
A = “Which shell should I select?” = {a1, a2, a3}
Then in Cox’s lingo, A v S is the disjunction of these questions
and is the actionable [common] information available for making this decision
which is based on the acuity of the player in watching the other player
interleave the shells in attempt to confuse the first player. Information
and decisions [control] are both considered and manipulated within
the local subjective frame of the system. In its extension,
one can exploit information theory but in a reverse sense as developed
by Shannon.
In conventional information theory, the source answers a question
of what to transmit (decision) and the receiver answers the question of what is
received (information).
In a generalized decision viz information theory, the environment
answers questions posed by a system and in turn uses this information
to make decisions on what to do (answering a question). These
two paradigms are exactly complementary to one another. One can
consider the probability of deciding a upon observing information b,
i.e., p(a|b).
Anyway, I have found this a very powerful and useful way of understanding
and designing systems that make decisions with uncertain information
which ostensibly is always the case in practice. I have many published
papers and seminars on this approach to decision making within
intelligent systems. Basically, information theory and decision [control]
theory are two incarnations of the same thing.
Bob Fry @JHU
From: decision_t...@googlegroups.com [mailto:decision_t...@googlegroups.com] On Behalf Of g charles-cadogan
Sent: Wednesday, December 12, 2012 7:33 PM
To: decision_t...@googlegroups.com
g charles-cadogan
The incipient thread began with Peter Wakker’s assertion that the “common prior assumption is close to what is called the logical view of probability. Carnap wrote several books on it.” There is also a recent survey paper by Sandy Zabel entitled “Carnap And The Logic Of InductiveInference”, Handbook of the History of Logic. 2011;10:265-309
As I read the prior comment by Robert U. Fry, I was reminded of [the attached] Jaynes (l973), "The well posed problem," Foundations of Physics, 3:477-493 solution to Bertrand’s paradox via the use of the maximum entropy principle. That solution provides a very poignant example of what can happen when a minimalist view of information is taken. Essentially, Bertrand’s (l888) paradox involves an equilateral triangle inscribed in a circle and the question ‘what is the probability that a randomly selected chord is longer than the length of a side of the triangle?’. It turns out that there is no unique answer to the question depending on the [prior] information that an analyst provides regarding the orientation of the triangle. Jaynes showed that an objective probability can be derived by using only the information provided by the question without any “common prior assumption”. It would be interesting to see what if anything would happen if the maximum entropy principle was applied to Gilboa and Schmielder (l989), “Maxmin expected utility with non-unique prior”—assuming that it has not already been done.
Itzhak Gilboa
Generally speaking, I think that Bayesian reasoning, like logic, is a fantastic tool for sorting out our intuition, for reasoning coherently, etc. Indeed, I can think of no better tool for dealing with philosophical paradoxes. But the Bayesian approach has a weakness that's not shared by logic: it can't represent ignorance. It has no way to say "I don't have the foggiest idea".
Tzachi“By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty ... The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence ... About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know.”
Fry, Robert L.
Dealing with ignorance is quite straightforward.
Probability theory deals with knowledge and what is subjectively known.
It is the degree to which one assertion say a (perhaps an observation)
implies another assertion b (perhaps a decision).
Entropy (in the info theory realm) deals with subjective ignorance and
what is unknown relative to what could be.
Suppose one is in a large department store and you are shopping for
perfume for your wife form Xmas. You are on the 2nd floor and
you know that the perfume is either on the first or 3rd floor. Being
a guy, you don’t ask where. You have no prior knowledge which
of the 2 floors the perfume is on and your question is F={h,l} it
is on the higher floor or lower floor.
Your decision is D={u,d}, i.e., take the “up” escalator” or take
the “down” escalator.
Your uncertainty is H(F) = 1 bit. Your actionable information is
the mutual information I(F;D)=0. In Cox’s jargon,
I(F;D) = b(FvD) = the “bearing of the information F on the
decision D.”
Think of a drinking glass partially full of water. The water and its
level represent knowledge. The empty void ignorance or
uncertainty. If empty (as in the example above) then the
entropy is maximum and all knowledge or information states
are equally probable. If full, then a particular knowledge
or decision is certain.
Both information and decisions are defined in the local frame
and therein related to one another in the subjective
formation of decisions.
Furthermore, probability theory and information theory are fully
symmetric and complementary, e.g.,
p( a v b)=p(a) + p(b) – p(a^b)
b(A^B) = b(A) + b(B) – b(AvB) which in information theory reads as H(A,B) = H(A) +H(B) – I(A;B)
From: decision_t...@googlegroups.com [mailto:decision_t...@googlegroups.com] On Behalf Of Itzhak Gilboa
Sent: Thursday, December 13, 2012 7:49 AM
To: decision_t...@googlegroups.com
Subject: Re: [DT_Forum] Common prior and Carnap's logical view of probability
This is very interesting. Yet, the simple question of ignorance remains: suppose that I'm asked whether all Arbodytes are Cylcophines, and that I've never heard these terms before. How would I form a Bayesian prior for this proposition?
Clearly, the 50%-50% is tempting, but then I'd have to deal with the question of all Cyclophines being Arbodytes, all super-Arbodytes being pseudo-Cyclophines etc. We can't assign 50% probability to every proposition about which we know nothing. (Just as we can't assume that every random variable about which we know nothing has a uniform distribution.)
Clearly, the problem is only aggravated when we go back in time/information. Maybe one has some guesses about "Arbodytes" and "Cyclophines", or some empirical frequencies on inclusion relations between words picked up at random from a dictionary. But how would an embryo form such a prior? Indeed, how would it form a prior over the type it is going to be born into?
Generally speaking, I think that Bayesian reasoning, like logic, is a fantastic tool for sorting out our intuition, for reasoning coherently, etc. Indeed, I can think of no better tool for dealing with philosophical paradoxes. But the Bayesian approach has a weakness that's not shared by logic: it can't represent ignorance. It has no way to say "I don't have the foggiest idea".
This is why I share the view that it is sometimes less rational to entertain (and act upon) Bayesian beliefs than to admit that one knows nothing. Relatedly, this is why I find the CPA (Common Prior Assumption) precisely this: an assumption. For the same reason that, in some situations, one can't convince another that a particular prior is the "correct" one, one may not be able to convince oneself thatthis is the prior one should indeed have, and one may be more rational in admitting that one simply does not know.
Importantly, many economic questions are of this nature. I guess the most convincing appeal to authority here would be quoting Keynes (1937):
“By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty ... The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence ... About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know.”
Tzachi
--
Itzhak Gilboa
Itzhak Gilboa
This question is addressed in the following (old) paper of mine: http://faculty.fuqua.duke.edu/~rnau/incoherence_of_agreeing_to_disagree.pdf. I have a rather different view on this issue from the one presented in the classic papers you mentioned (as well as many others). They typically start from Harsanyi's common prior assumption and then proceed to analyze what would happen if beliefs were to diverge through the receipt of private information from sources with agreed-upon likelihood functions. To me, the common prior assumption is fundamentally incompatible with Bayesianism, despite the fact that it is the foundation of most "Bayesian" models in game theory and information economics and even financial economics. One element of Bayesianism is the use of probabilities to represent beliefs and the requirement that those probabilistic beliefs should be updated by Bayes' rule in situations where it applies. But there is another element, namely that the decision maker's prior probability distribution is in general a personal probability distribution that is at least partly derived from information and experiences and mental calculations known only to him or her. From the viewpoint of others, an agent's prior probability distribution is subjective and idiosyncratic and primitive--no amount of communication can fully explain where the numbers came from let alone produce agreement on them. Correct me if I am wrong, but I think this is what de Finetti and Savage would have said. About the only situations in which it makes sense to assume that agents will have identical subjective prior probabilities are ones in which the probabilities are in fact objective (e.g., games of chance) or in which they are based on the mutual acceptance of probabilities provided by others with much better information (e.g., computerized weather predictions) or in which they are only vaguely quantified (e.g., we both feel it is "around 50-50" or "one in ten"). More generally, if we try to infer each other's beliefs from observations of behavior (e.g., betting or investing or insuring), and the events are of personal importance to us, then our revealed probabilities will be distorted by state-dependent marginal utilities for money (in the case of de Finetti's model) or by interpersonal differences in the definition of "consequences" (in the case of Savage's model) . What we will observe under such conditions are each other's "risk-neutral probabilities", which are products of personal subjective probabilities and relative marginal utilities for money, rather than true personal probabilities. This is exactly what goes on in the stock market. The common prior assumption DOES make sense when applied to risk neutral probabilities, because it is the dual of the condition of no-arbitrage. This is the appropriate way to interpret de Finetti's fundamental theorem of probability and the fundamental theorem of asset pricing in the case of risk averse agents with subjective beliefs. So, from my perspective, what ought to be assumed in many of these information models (particularly the ones that try to explain what is going on in markets) is that agents who have had the opportunity to communicate in the language of betting or trading will have common prior risk neutral probabilities. Now suppose they obtain additional private information from a source whose reliability is described by a likelihood function that is agreed upon (e.g., an experiment or survey) AND (very importantly) in which they have no prior stakes, i.e., they don't care about the result of the experiment per se, given the state of nature. Then it follows that their posterior risk-neutral probabilities should be mutually consistent, because risk-neutral probabilities are updated by Bayes' rule when the agent's likelihood function is not itself contaminated by state-dependent marginal utilities for money. This is the generalization of Aumann's agreeing-to-disagree theorem that is discussed in my paper. So, from my perspective, the various Bayesian no-trade theorems and solution concepts and asset pricing models make sense when the "prior" probabilities in them are risk-neutral probabilities but not when they are subjective probabilities that are measures of pure belief, except in the special cases that I mentioned above, which are comparatively uninteresting. Another issue that is potentially relevant here is that the agents may have incomplete preferences (which in general they will) and/or they have limited opportunities to communicate (e.g., they may observe some bets that others are willing to take but not all bets). In such cases everyone's revealed probabilities are imprecise to some extent, and there the requirement of no-arbitrage is (only) that the intervals of risk-neutral probabilities of different agents should have a non-empty intersection.
Amarante Massimiliano
suppose that I'm asked whether all Arbodytes are Cylcophines, and that I've
never heard these terms before. How would I form a Bayesian prior for this
proposition?
In fact, the 50-50 thing, or more generally the uniform prior as a representation of ignorance is major BS. But, just like any major BS, it is hard to die.
‘’Learning’’ (for real) means to discover new properties. These enlarge the class of your mesurable sets (possibly with the aid of logic) and, one day, this class might include Arbodytes and Cylcophines. Only then, you will be able to assign probabilities.
max
-----Original Message-----
From: decision_t...@googlegroups.com on behalf of Itzhak Gilboa
Sent: Thu 13/12/2012 7.48
To: decision_t...@googlegroups.com
Subject: Re: [DT_Forum] Common prior and Carnap's logical view of probability
Tzachi
Fry, Robert L.
You have to know what an Arbodyte and Cyclophines are before we can reason with them.
halpern
"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
> One cannot formulate logical or probabilistic or info theoretic statements about assertions like u mention that are undefined within the local frame of a system. These are meaningless. To do this these must be synthesized within the system thru learning - the internal formation of the ways we distinguish our environment in terms of what we can subjectively ask or answer (decide). Either type of question is defined by the set of possible answers to it.
>
> You have to know what an Arbodyte and Cyclophines are before we can reason with them.
>
> ----- Original Message -----
> From: Amarante Massimiliano [mailto:massimilia...@umontreal.ca]
> Sent: Thursday, December 13, 2012 08:58 AM
> To: decision_t...@googlegroups.com <decision_t...@googlegroups.com>
> Subject: RE: [DT_Forum] Common prior and Carnap's logical view of probability
>
>> This is very interesting. Yet, the simple question of ignorance remains:
> suppose that I'm asked whether all Arbodytes are Cylcophines, and that I've
> never heard these terms before. How would I form a Bayesian prior for this
> proposition?
>
>
Fry, Robert L.
Thank you for sharing this Tzachi. I am giving a partial response to this
below. Very interesting to hear about Carnap’s work and now
understanding it.
First, I have to say that I agree with this 100% in regards to the assessment of Carnap. Suppose if you will that Carnap did understand things at a more fundamental and physical/logical level but that he could not quantify this nor explain how to do so. Until your posting I knew nothing of Carnap’s work although I had heard of him.
This quantification is what I believe Cox accomplished; that in his view logic, probability, and entropy provide a holistic and coherent way of quantifying the relativistic nature of how system make decisions given uncertain information.. Cox gained much perspective from a much older source. Felix Cohen, a well-known lawyer in Indian [American] Law within the US State Department. Google him. His father is quite famous too – a famous logician. Anyway, in 1929 he published a paper titled “What is a Question?” within the philosophical magazine “The Monist.” In this paper Cohen develops the idea that we are at a tremendous loss for not having a logic of Questions that lay on par with that of conventional logic and Boolean Algebra.
Cohen said that one can only ask a question if one knew in advance all its possible answers. Cox accomplished this goal in his last paper. Around 1978 Cox told a good friend of mine (Dr. Robert Evans) that he thought it would likely take 50 or more years for people understand what he was trying to say. Another good friend, Dr. Myron Tribus who used Jaynes’ work to prove that all of thermodynamics can be derived from information theory (see http://en.wikipedia.org/wiki/Myron_Tribus and “Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications.” I guess I am showing my age.
Let me give you two living exemplars. Only one now since an already too protracted production.
Consider a protozoan-like creature swimming in water. It has cilia that can propel it forwards (longitudinal motion only along its current 3-D body inertial orientation). So random current are constantly re-orienting the body alignment of this single-celled organism – agent if you will although I hate that term. It cilia are activated by it’s local and subjective decision to do so. Cilia are either activated; or not. Let us define this subjective decision by the question C={a,@a} where a is the decision “Activate my Cilia!” and @a is the opposite decision “Do not Activate my Cilia!” The decision a and the alternate decision @a have the following logical properties.
If one decision is made then the other is not. If one decision is not made, then the other one is. These together comprise the properties of being both logically exhaustive AND logically exclusive. Together these logical and physical properties make a and @a complementary assertions or decisions. As shown by George Spencer-Brown, the British logician in his brilliant “Laws of Form” that these two properties together give rise to the structure and form of Boolean Algebra. This is what Cox built on as well although he did not seem aware of Spencer-Brown’s work.
Now back to the Protozoan-like creature.
It has a sensitivity to ambient light such that this sensing capability with some internalized threshold says that either light is or is not present. This sensing capability provides an information conduit or portal through which the creature can extract information from its environment. Like it decision space C, it has a locally defined (threshold and all) capability of having its environment answer the question L={l, @l} where l is the assertion l=”I see light!” and @l=”I don’t see light!” and where by the way L = “Is there light?” = {l, @l}. Like Cohen said, the protozoan system has to know the possible answers to the question it poses and the question it answers.
Let us consider the possible behaviors of the creature. Assume that algae-like food consumable by the creature is only readily available if it is growing and has plenty of light. The possible behaviors can be delineated:
1. Never do anything! You’ll die eventually.
@l à @a
l à @a
2. Hey, might have something here
@là@a
làa
3. You are going to die and do it quick
@làa
là @a
4. You will be wasting a lot of energy and probably die in the outcome
@iàa
làa
Note that everything has to do with information, decisions, energy, and survival. Only behavior 2. Can lead to a system perpetuating itself and successfully exploiting its environment. I use the notation à meaning “logically implies.” Probability the is relative degree to which the creature makes decisions, That is, p(a|l) is a locally determined probabilistic measure of the frequency with which the system (much better than agent) makes the decision a given it observes l.
The evolution of p(a|l), p(a|@l), p(@a|@l) and p(@l|l) will describe how a system evolves so as to continue its existence within its local environment. That is, it learning. Some will die and some will live. These that tend to learn to navigate towards the life will most probabilistically live.
Decisions are ultimately made in response to available information. How that information is best acquired relevant to the decisions to be made can be seen to be a process. In particular, a thermodynamic process and probably best cast as a computational process. Anyway, this is the view that I have and can expand upon if interest exists. Cortical neurons provide an especially useful and powerful exemplar I can show in further detail and have published. Neurons are physically decoupled from one another and therefore represent another form of a physical agent (please, just a physical system). How they distinguish their environment (other neurons) and affect its environment (other neurons) is very useful I believe. How they dynamically define the question(s) they ask and the questions they answer (decide) and how they match the rate it acquires information to the rate it can make decisions.
Bob
From: decision_t...@googlegroups.com [mailto:decision_t...@googlegroups.com] On Behalf Of Itzhak Gilboa
Sent: Thursday, December 13, 2012 8:54 AM
To: decision_t...@googlegroups.com
Subject: Re: [DT_Forum] Common prior and Carnap's logical view of probability
Hello again,
--
Bob Nau
--Bob
Konrad Grabiszewski
This multi-track discussion is fascinating, especially, for someone who is not very familiar with deep philosophical issues of science(s). I truly enjoy following it but would appreciate a help with the following two issues:
To Robert Fry:
When you say “You have to know what an Arbodyte and Cyclophines are before we can reason with them,” what do you mean by “knowing what an Arbodyte is”? Is such a statement somehow formally defined in (modal) logic? Do I begin with a set of primitive properties/propositions (e.g., p = has six legs, q = has a tail, r = eats roses) with which I am able to uniquely define an Arbodyte as A = {p,q,r}? Then, I say that knowing what an Arbodyte is means that KA if and only if K{p,q,r}? But, how do I define “knowing what “has six legs”” is? Maybe I don’t? This reminds me of a lecture from high school on the 18th century Polish encyclopedia in which a “horse” was defined more-or-less as “what a horse is everyone knows.” So, as in this encyclopedia, do I presume that “has six legs” is an obvious statement that everyone understands?
To Tzachi:
Going back to Tzchai’s intial issue: “Suppose that I'm asked whether all Arbodytes are Cylcophines, and that I've never heard these terms before. How would I form a Bayesian prior for this proposition?” If one was to subscribe to the “as if” methodology of economics, one could say that I need to re-phrase your question as a decision problem. I.e., prepare bets on “Arbodytes \subset Cylcophines”, “super-Arbodytes \subset pseudo-Cyclophines” and so on. Then, I force you to rank the bets and, from that, I derive your prior. I give that prior to you and say “you behave as if having such a prior.” Maybe I am not fully understanding your question, but it seems to me that your question has more of a normative appeal: How would/should I form a prior? If this is the case, then I would say that your question implies that there is the correct way to form a prior. This bothers me a little bit as it leads to comparison “Is Ann’s prior better than Bob’s?” which I don’t really understand. Same as in: “Should I make my decision relying on Savage or Gilboa-Schmeidler? Which is better?” I don’t understand how one could rank models.
Thank you,
Konrad
tzachi...@gmail.com
Konrad Grabiszewski
Dear Tzachi,
But if you are after the normative side of construction of prior, then why not to look at something simpler than Arbodyte/Cyclophines? In the example you gave, there seem to be two issues:
(1) how should an agent construct a prior if he knows what Arbodyte/Cyclophines are?
(2) how should an agent construct a prior if he does not know what Arbodyte/Cyclophines are?
(I am referring to Robert Fry’s comment about knowing what Arbodyte/Cyclophines are.)
It would seem that (1) is a simpler task than (2). Suppose that I replace your Arbodyte/Cyclophines by the following: “Suppose that I'm asked whether or not there will be a revolution in Mexico in 2013. How would I form a Bayesian prior for this proposition?”
(Presume that “revolution in Mexico” is a well-specified/described event so we avoid the issue of “knowing what revolution in Mexico is” raised by Robert Fry.)
How would you approach forming a prior in this problem? You can look at past Mexican and non-Mexican revolutions and try to find the factors which determine a probability of revolution. This is not perfect (in fact, might be pretty useless) since the reasons of past revolutions might have nothing to do with current social situation in Mexico. Next, is there the way to form a prior in this problem? Could I say that your prior is better than mine?
Best,
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Fry, Robert L.
Hi Konrad,
To know is a very basic notion from my perspective as follows.
Imagine I am showing you and you are looking at a black image;
maybe like pure black screen. I ask you what you see and you say
“Nothing.”
Suppose that I had inserted the near-black colored large-font word
“Something!” Your relative level of visual acuity does not allow
you to detect and read this word however. You subjective
distinguish nothing.
Now, split the screen vertically with the left side still black but the
right side white. You can now easily distinguish the two sides of this
image. If you indicate one side you will have not indicated the other.
If you do not indicate one side, then you will have indicated the
other. These two properties (mutual exclusion and exhaustion) give
the global property of the logical complementarity. That is, physical
and logical are one and the same thing operationally within any
physical system that poses a binary question or answers a binary
decision (also a question). Thus, physicalàlogical.
But this physicality is within the subjective frame of a physical system
that can acquire information from its environment (that which is
not the physical system) and the has the ability to made decisions
that conduct actions allowing the same physical system to control the
future.
To know is to subjectively be able to distinguish. In this case ha has the
question you need to be able to answer is “Would I know a Arbodyte
or Cyclophines if you saw one?” ha ha. Do you know what the heck these
darn things are?
Learning is the formation of new ways of collecting and processing
information from your environment and new ways of doing things
that make efficient use of collected information. That is, the physical
system evolves it ability distinguish its subjective environment both in
terms of what it can know about it and what it can do to control it.
I know this is a protracted answer but the answer to the question
of what it means to know is at the same time simple yet conceptually difficult.
It is no different than the Yin-Yang in Taoism in that one can only know high
if one also knows low.
In my view, logica, its rules, and theorems are dynamic rules of subjective computation
within and by physical systems.
Bob
From: decision_t...@googlegroups.com [mailto:decision_t...@googlegroups.com] On Behalf Of Konrad Grabiszewski
Sent: Friday, December 14, 2012 12:22 PM
To: decision_t...@googlegroups.com
Subject: [DT_Forum] Re: Subjective probability and the common prior assumption
This multi-track discussion is fascinating, especially, for someone who is not very familiar with deep philosophical issues of science(s). I truly enjoy following it but would appreciate a help with the following two issues:
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Konrad Grabiszewski
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Fry, Robert L.
The answer to your question is simple.
By assumption, you know that there are horses, mules, and donkeys.
You admit to the possible existence of all three kinds of animals
and that the animals you are looking at (consisting only of these
3 types for now). One of these animals is presented to you and
you want to determine which it is – an inquiry?
W={h,m,d} defines your internally defined states that represent
possible answers to your inquiry.
As you gaze at the animal using your eyes I guess for acquiring
the information visually that you will use to make this determination.
In the case of a horse or of a donkey, your visual acuity is such to allow
you to determine these with confidence. If a mule, then you
cannot say whether it is any of the animals. Your eyes give you
no information bearing on resolving this question. This is a different
issue from knowing horses, mules, and donkeys exist. You did
not have these distinctions at birth or perhaps for some time. I am
not sure I do now.
From: decision_t...@googlegroups.com [mailto:decision_t...@googlegroups.com] On Behalf Of Konrad Grabiszewski
Sent: Friday, December 14, 2012 7:42 PM
To: decision_t...@googlegroups.com
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Ken Binmore
original gut feelings into a consistent system of beliefs
by the use of the intellect. The same reasoning can be
employed to explain {\em subjective equilibria,} provided
that we insist that players massage the beliefs they
attribute to other players along with their own. The
result will be that all the beliefs they attribute to the
players will be derivable from a {\em common prior.}
However, the argument doesn't imply that it will be common
knowledge that all players have the {\em same\/} common
prior, which is a standard assumption in some contexts.
--
Ambrus, Attila
The above just implies my belief (which you should probably ignore, as there are much more educated people than me expressing opinion) that based on purely logical grounds it is impossible to give an answer what a *rational* decision-maker should do when his/her information is very sparse/limited. I do think it is an important empirical question though what people do in those situations, as I think they do encounter them a lot of the times. Maybe people's behavior in such situations can be derived by some kind of logical reasoning. But I think it is an empirical question whether people use purely logical reasoning how to behave in situations like this.
Best, Attila
Konrad Grabiszewski
Dear Robert,
Thank you again for a detailed answer. Additional doubts/questions emerge.
1. First, regarding “exists”, I believe that in the original exercise that Tzachi posted
Suppose that I'm asked whether all Arbodytes are Cylcophines, and that I've never heard these terms before. How would I form a Bayesian prior for this proposition?
it is implicitly assumed that Arbodytes exist. If they do not exist, then the exercise is simple: assign prior 1 to the proposition “all Arbodytes are Cylcophines” as the set of Arbodytes is the empty set. At least, in my understanding, “something exists” means that there is a non-empty set containing that something.
[In fact, since people believe that “if something is not on the Internet, then it does not exist,” we can help Tzachi to form a prior by googling “Arbodytes.” There are two hits: this forum and… Tzachi’s presentation. That is, we can deduce that “Arbodytes” is a made-up word and they do not exist; as such, the set of “Arbodytes” is the empty set. Prior found…]
2. Second, when you insist on using eyes to distinguish among objects I presume that “vision” is just an abbreviation of “all instruments which help to distinguish objects.” There are plenty of examples where I have two distinct things, I know what each one of them is, but I can’t separate them by using eyes (e.g., jazz & rock; cup of tea without sugar & cup of tea with four spoons of sugar; a male from North Dakota & a male from South Dakota).
3. Third, what bothers me is your statement “You have to know what an Arbodyte and Cyclophines are before we can reason with them.” (It’s probably my fault of misinterpreting your words.) I can reason about problems involving Arbodytes and Cyclophines without knowing what they are:
Suppose that I'm told that four out of every ten Arbodytes are Cylcophines. What is the probability that the next Arbodyte I meet is a Cylcophine? How would I form a Bayesian prior for this proposition?
Still, we do not know what Arbodytes and Cyclophines are but there seems to be no problem reasoning about this puzzle. I would say that 0.4 is a nice way to form a prior given the data you have.
I presume that Tzachi’s puzzle is about having no information. I.e., there is a state space W = {x,y} and a possibility correspondence (i.e., Aumann model) such that P(x) = P(y) = W. So it’s not about not knowing what “x” and “y” are. Nor is it about {x} and {y} being non-measurable (they are). It’s about a rule/method/algorithm of forming a prior on such W. It’s possible I don’t understand Tzachi’s question, but if I do, then I would say that any prior is a “good” prior.
Regards,
Konrad
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Fry, Robert L.
1. First, regarding “exists”, I believe that in the original exercise that Tzachi posted
Suppose that I'm asked whether all Arbodytes are Cylcophines, and that I've never heard these terms before. How would I form a Bayesian prior for this proposition?
it is implicitly assumed that Arbodytes exist. If they do not exist, then the exercise is simple: assign prior 1 to the proposition “all Arbodytes are Cylcophines” as the set of Arbodytes is the empty set. At least, in my understanding, “something exists” means that there is a non-empty set containing that something.
>>Yes, exists within the subjective frame of a system
[In fact, since people believe that “if something is not on the Internet, then it does not exist,” we can help Tzachi to form a prior by googling “Arbodytes.” There are two hits: this forum and… Tzachi’s presentation. That is, we can deduce that “Arbodytes” is a made-up word and they do not exist; as such, the set of “Arbodytes” is the empty set. Prior found…]
2. Second, when you insist on using eyes to distinguish among objects I presume that “vision” is just an abbreviation of “all instruments which help to distinguish objects.”
>> No, just eyes. Additional information from ancillary sensors or testings, e.g., DNA
can be fused with visual information, again within the local frame of the same system
to make the determination regarding what kind of animal is boing looked at.
There are plenty of examples where I have two distinct things, I know what each one of them is, but I can’t separate them by using eyes (e.g., jazz & rock; cup of tea without sugar & cup of tea with four spoons of sugar; a male from North Dakota & a male from South Dakota).
3. Third, what bothers me is your statement “You have to know what an Arbodyte and Cyclophines are before we can reason with them.” (It’s probably my fault of misinterpreting your words.) I can reason about problems involving Arbodytes and Cyclophines without knowing what they are:
Please be explicit. If you know what these items are then forge ahead and
rationalize regarding them. To me, they are fictitious quantities affording no margin
for rationalization. That is the point. Are these defined within your local frame?
Priors are typically established as the new a posterior given new likelihood
information in the form of a measurements combined with a previous prior
through Bayes’ Th. This iterative executed means that whatever the
original prior was is means in that it has evolved to a new state of
subjective knowledge given information extracted by the system from
its environment as guided by Bayes; Theorem. I’m just saying the
issue of what is the objective prior is principally academic and experience
takes care of this problem.
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Konrad Grabiszewski
Dear Robert,
1. Re “just eyes”: If you insist that “vision” is the way to distinguish objects then, for example, I don’t know how eyes could separate jazz from rock. I also do believe that blind people are able to distinguish among objects but your approach would suggest they aren’t. Sounds rather strange to me.
2. Re “Please be explicit”: In the Tzachi’s puzzle you don’t reason about Arbodytes or Cyclophines but about a proposition/sentence containing words “Arbodytes” and “Cyclophines.” These are not the same issues. In order to solve Tzachi’s puzzle, you don’t need to know what Arbodytes or Cyclophines are. Example: Let “p” be some primitive proposition. I can reason about sentence q = ((p) and (not p)), and conclude that q is false, without knowing what p means.
Best,
Konrad
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Fry, Robert L.
I actually agree with points 1 and 2. Regarding 1., this whole argument depends
on that existence assumption. The ability to distinguish does not mean
you have the sensory information to do so or do so in all instances. I tried to
say this in my one post.
Point 3 also depends on what you assume you know. You are divinely
give information that you being a rationally person can process regardless
whether its Arbodytes, Cyclophines, or anything. You still
cannot reason about their existence or possible relationships to one
another.
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