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Author Topic: Is non transitive preference - irrational?
seagull
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If I say that I:

1. I Prefer A to B
2. I prefer B to C
3. I prefer C to A

Does that mean that I am being irrational?

Are there circumstances or examples in which it would be clearly rational to have those preferences?

Are there circumstances or examples in which it would be clearly irrational to have those preferences?

In those examples: is there a way top take advantage of the non transitive preference?

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Aris Katsaris
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> Does that mean that I am being irrational?

Yes.

> Are there circumstances or examples in which it would be clearly rational to have those preferences?

No.

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LetterRip
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Nope, economists and philosophers have examined non transitive preference and have shown cases where it is rational.

See this blog post,

http://michele-journal.blogspot.com/2007/05/non-transitivity-with-no-irrationality.html

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Aris Katsaris
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> In those examples: is there a way top take advantage of the non transitive preference?

If you mean to be taken advantage, yes:
Let's say that:
You prefer A over B so that you'd be willing to give X dollars to be given A instead of B.
You prefer B over C so that you'd be willing to give Y dollars to be given B instead of C.
You prefer C over A so that you'd be willing to give Z dollars to be given C instead of A.

I begin by offering you C for free. Then I get you to give me Y dollars, to take back C and offer you B instead. Then I get you to give me X dollars so that I give you A instead of B. Then I get you to give me Z dollars so that I give you C instead of A.

So now you're back to having C (my initial offer), except that it's no longer free, you've given me X+Y+Z dollars, all according to your preferences. I repeat the process until I'm rich and you're penniless.

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Aris Katsaris
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The example you provided LetterRip explicitly specifies "weak preference" (an >= relation) -- contrasts it to the "strong preference" that seagull is speaking about.
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seagull
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Aris, there are examples in experimental game theory where people actually paid good money for such a cycle. And the strange thing is that even AFTER they realized that its a sure way to lose money they only decreased the sum they were willing to pay but did not stop.

I read it in a book, but can't find an online reference right now.

I tend to agree with you that those people were irrational but they didn't think they were. Which makes me wonder how many irrational things I am doing that seem rational to me even though they might not be.

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seagull
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1. I Prefer Rock to Scissors
2. I prefer Scissors to paper
3. I prefer Paper to Rock

And it's a strong - not a weak preference.
In the context is of a rock-paper-scissors game where the winner gets money, I'd even be willing to pay for a winning outcome and I think that not paying for it would be irrational.

Aris' argument about "taking advantage" is valid but not relevant in this case because it can not be used in the context I described.

But I am still not sure that my own position is rational. That is why I started this thread.

[ December 16, 2012, 10:03 AM: Message edited by: seagull ]

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AI Wessex
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I think Seagull's original problem is irrational if A, B and C represent complete components of a closed system. Aris' example introduces a value assignment for each of the elements. Taken as a series of transactions the step-wise preference choices are not unreasonable, but the result is non-optimal. The failure was not planning ahead to see what the implications of making short-term choices would be. The result is due to nondeterministic selection rather than irrationality, since the individual choices were themselves reasonable. The complete process only looks irrational to an outside observer who has complete knowledge.

In artificial intelligence tactics this would be succumbing to a hill-climbing fallacy that only takes local factors into consideration and only considers a single pathway through the options. If your goal was to get the most desired element at the least cost you should have picked C and stuck with it (highest value, least cost and shortest path).

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seagull
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quote:
if A, B and C represent complete components of a closed system
Al, can you please clarify what you mean by that phrase.

There are precise mathematical definitions for what a complete relation and I think that completeness in LetterRip's link refers to the mathematical definition of completeness. But completeness of the preference relation has nothing to do with "complete components".

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AI Wessex
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Right, I'm not being precise enough and would have to strain really hard to get there [Smile] . I mean that there is complete information about each component, no extenuating circumstances or preference values. If a >= b >= c, then it is not possible for c >= a. If you therefore have all of the information and still end up choosing a you have made an irrational choice.

But I think comparing a mathematical system that is closed and (in my usage) complete doesn't overlap the real world as much as we would like. AI acknowledges that by using search to find value in "messy" systems. In the real world we use trial and error and "levels of confidence" to resolve a great many problems. That is the point I was trying to make.

[Not trying to derail] Another thread I would like to see is whether people believe that humans are rational or even of a single mind. Personally, I think the dominance of reason in human action is a fairly late breaking development. [/Not trying to derail]

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seagull
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quote:
If a >= b >= c, then it is not possible for c >= a.
This statement is simply false.
There are two true statements:

If ">" is a strong transitive relation then a>b>c, then it is not possible for c>a.

If a>=b>=c, then it is possible that c >= a, if ">=" is a weak transitive then a=b=c is the only way to satisfy all three relationships.

[ December 16, 2012, 11:09 AM: Message edited by: seagull ]

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Aris Katsaris
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quote:
1. I Prefer Rock to Scissors
2. I prefer Scissors to paper
3. I prefer Paper to Rock

And it's a strong - not a weak preference.

You're misusing the word "prefer" - what you actually mean is:

"1. I prefer to have Rock, when the other guy has Scissors"
"2. I prefer to have Scissors, when the other guy has Paper"
"3. I prefer to have Paper, when the other guy has Rock".

You aren't actually comparing different scenarios A, B, C which you prefer to each other -- you're just stating a preference for A when the other guy has B, a preference for B when the other guy has C, and a preference for C when the other guy has A.

That's a different issue altogether, unrelated to your initial question.

[ December 16, 2012, 02:45 PM: Message edited by: Aris Katsaris ]

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Aris Katsaris
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quote:
If a >= b >= c, then it is not possible for c >= a.
Let:
a = 5
b = 5
c = 5

In that case:
a >= b >= c
and
c >= a

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seagull
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Aris,
I agree with the validity of your interpretation in the context that you describe.

What I am having trouble with is the following statements:
quote:
You're misusing the word "prefer" - what you actually mean is:
...
That's a different issue altogether, unrelated to your initial question.

I prefer not to tell you "what you actually mean".
I would rather ask you to clarify:

Was your statement intended to imply that you know what I mean better than I do?

My "initial question" said nothing about "scenarios" but it did expressly leave the question open as to what "prefer" means in different circumstances or examples.

Why do you think that I am misusing the word "prefer"? Wouldn't it be just as valid for me to say that unnecessarily restricting the meaning of the word "prefer" to fit your interpretation would be misusing it?

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seagull
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In the context of a democratic vote between two options, would it be valid to say that the voters (or public or electorate) "prefer" the option that got the majority of the votes?

[ December 16, 2012, 04:21 PM: Message edited by: seagull ]

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KidTokyo
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If A, B, and C are of variable value in different contexts, then it's not irrational. Preference is functionally always context-based, i.e., preferred for a particular purpose. I don't think "objective preferences" actually exist, and only that would make the statement irrational, as though preferences were actually assigning fixed quantitative values that are context-independent.
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Aris Katsaris
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quote:
Was your statement intended to imply that you know what I mean better than I do?
I think that you are more likely to know better what you mean, but *I* am more likely to know better how to express what you mean.

quote:
Wouldn't it be just as valid for me to say that unnecessarily restricting the meaning of the word "prefer" to fit your interpretation would be misusing it?
No, it wouldn't be. You don't prefer Rock over Scissors when the other guy has *Paper* -- in that particular circumstance you very clearly prefer Scissors.

You example usage was very clearly *bad usage of the words*, and I won't pretend that all usages of words are equally valid just to be nice.

quote:
In the context of a democratic vote between two options, would it be valid to say that the voters (or public or electorate) "prefer" the option that got the majority of the votes?
No, even for a single individual (let alone fuzzy groups) a vote indicates preference but doesn't prove it. A person might have checked the wrong ballot, a person might have felt peer-pressured or threatened to vote a certain way, a person might have voted tactically, a person might have voted randomly, a person might have a preference that wasn't available in the ballot, etc, etc, etc.

[ December 16, 2012, 04:59 PM: Message edited by: Aris Katsaris ]

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AI Wessex
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The problem I'm having in this discussion is that people flip back and forth between mathematical relationships and "preferences" and between open and closed systems. I'll sit back and watch the rest of the show to see if you all can get on the same page with each other so I can learn something from it.
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seagull
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Aris, you seem to be missing the point.
Given the choice between any two of Rock, Paper and Scissors where I get one and my opponent gets the other, the preference is clear and my statements are not a "misuse" of the words.

In order to make your argument valid you need to force it into a context in which you know what the other person has and use words like "when" that indicate sequence.

In a game like Rock paper scissors players would prefer to wait until they know what the other person has. Preferring to wait is a valid and rational strategy which transcends the difficulty of non-transitive preference and trivializes my initial question. There are circumstances where the option to wait is not available - if you wait until you know what the other person has it is considered cheating.

Given any pair of Rock, Paper or Scissors I can state a "strong preference" for winning which transcends the timing/foreknowledge/cheating issue at the cost of giving up transitivity of the preference relation. In doing so I am using an alternate definition of the word "prefer" from the one you have in mind. My alternate definition is not a choice for a solipsistic personal vocabulary. It is not only valid and meaningful but also common enough in economic and mathematical literature that I do not understand why you consider it to be "bad usage of words".

I will be the first to agree with you that the two definitions are incompatible and that it is important to know the context before we try to deduce the rationality of such preferences. This was exactly the point of my initial post.

I am interested in other circumstances and examples that demonstrate the difficulty of applying rational thinking to the notion of preference.

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seagull
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Al, flipping back and forth between different meanings for the same symbol is the whole point of symbolic math (and possibly of symbolic language as well).

Symbolic language is one of the foundations of AI (and underlying computation models) so I am interested in understanding how we interpret symbols like "prefer" or A,B,C and especially in the alternate meaningful interpretations.

In the context of math there are clear and distinct definitions for the terms complete/incomplete and open/closed. A complete relationship between two objects in a set is one that is defined for every pair of objects. If the relationship is not well defined for some pairs it is said to be incomplete.

The words open/closed on the other hand have a specific meaning in set theory which is tangential to this discussion (as far as I can tell). They also have a very different meaning in chemistry/thermodynamics which is irrelevant but may be closer to what you meant. Set theory language is not the first alternate definition of the words open/closed that you will find in an English dictionary, but in the context of rational study of preference relationships it is more meaningful than the thermodynamic (or open/closed door) meanings of the words.

There are sets that are both open and closed but that is a topic for a different discussion.

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Aris Katsaris
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This is absurd. You have no understanding of how variables such as A, B, C are to be used. They must be replaced precisely.

"I prefer Rock to Scissors" does NOT bloody well mean "I prefer to choose Rock if my opponent has chosen Scissors". You can tell the difference because the latter sentence has lots and lots of different words to describe the different statement.

I am done with this discussion.

[ December 16, 2012, 09:05 PM: Message edited by: Aris Katsaris ]

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