To make things weird, although Frisch was fully aware of the importance of random distribution in economics relations, he never mention the randomness in binary preference relations!

How to define preference? I learned that $x\succ y$ means the decision maker is more likely to choose $x$, but I don't know the sources of this.

I found the following definition on Wikipedia:

Suppose a person is offered apples (x) and oranges (y), and is asked to choose one of the two.

Under several repetitions of this experiment (and assuming laboratory conditions controlling outside factors), if the scientist observes that apples are chosen 51% of the time it would mean that $x\succ y$. If half of the time oranges are chosen, then $x\sim y$. Finally, if 51% of the time she chooses oranges it means that $y\succ x$. Preference is here being identified with a greater frequency of choice.

I see that Block, Marschak, Tversky, and Luce use this definition, but their first occupation is not economist I guess. My question is, did Frisch (or other old famous economists except Debreu) used this definition?

Debreu wrote: "The relation $p(a, b) > 1/2$ is read "$a$ is preferred to $b$"

For example, see "Topological Methods in Cardinal Utility Theory" and " STOCHASTIC CHOICE AND CARDINAL UTILITY (Econometrica)"

Tversky use the same thing, for example, in his 1969 "Intransitivity of preferences".

To clarify, the "observable definition", emphasized in the title, means an equivalent condition of "$x$ is preferred $y$" that is economically observable, economically intuitive, experimentally or empirically testable. Of course, everyone who exposed to any microeconomics courses understands the mathematical definition of preference. In contrast, this question asks what is the meaning of "preference" in economic experiments and real world.

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    $\begingroup$ I learned that $x\sim y$ means the decision maker is more likely to choose $x$--I don't think this is correct. The most standard/common approach is to take preferences as primitive, then assume that the decision maker (usually) chooses based on her preferences. That is, choices follow from preferences, rather than preferences being defined based on choices. $\endgroup$
    – user18
    Oct 10 '20 at 3:13
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    $\begingroup$ The interpretation given in the "51% example" is something made up by some random Wikipedia contributor. I do not think most economists would endorse that particular interpretation. There are any number of other possible interpretations of the observed results (e.g. the subject's tastes have changed, conditions have changed). $\endgroup$
    – user18
    Oct 10 '20 at 3:16
  • $\begingroup$ @KennyLJ Debreu did use this definition. But he is more like a mathematician than economist. $\endgroup$
    – High GPA
    Oct 10 '20 at 4:10
  • $\begingroup$ @KennyLJ For your first comment, do you mean that: if $x\succsim y$, then the agent is more likely to choose $x$? $\endgroup$
    – High GPA
    Oct 11 '20 at 13:17
  • $\begingroup$ The definition of $x\succ y$ used by Debreu is that $x\succeq y$ holds but not $y\succeq x$. $\endgroup$ Nov 10 '20 at 19:10

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