Preferences + Utility

CHALLENGE: I have described rational choice. Remember, the purpose of a challenge is to provide people with an opportunity to network their ideas. So, what’s your favorite theorem? Creativity welcome!

Align left: The External [scientia]                   Align Right: The Internal [noumenon]

  1. There haven’t been any waves,
  2. Not even an undercurrent.
  3. The wind is quiet.
  4. Away are the waves,
  5. Of strength,
  6. Of this undercurrent.
  7. Ahhh, with the Political Pipeline at my back:
  8. What was that rational choice game?
  9. The teacher aloud in class,
  10. Today.

Preferences + Utility

1. Fundamental Assumptions [of rational choice theory]  A. Completeness—we suppose that there exists a set of states of the world (that we might have preferences over–go to Disney, Macedonia, stay home) / Alternatives / Bundles of goods, etc.  A1. says that people know their preferences among the set (of all possible choices).  So, if given a choice between alternatives X and Y, then only one of these statements can be true:   a. the actor prefers X to Y   b. the actor prefers Y to X   c. the actor is indifferent  A2.  Transitivity. Transitivity requires that the following be satisfied:   a. if a person prefers X to Y and Y to Z, then she prefers X to Z.   b. If a person prefers X to Y, and is indifferent to Y and Z, then she prefers X to Z.   c. if a person is indifferent between X and Y, and prefers Y to Z, then she prefers X to Z.   d. if a person is indifferent between X and Y, and indifferent Y to Z, then she indifferent  between Y and Z.

An intransitive preference ranking will—

cause incentives to be…

An easy target to see.  

i.e. X> Y, Y>Z, Z>X

A3. Revealed Preference—people always chose “the best” alternative for themselves  –the one you most favor given their constraints

SO, when we say “rational actor” then ALL we

Are saying is the former ASSUMPTIONS—

Nothing more, tweet.

2.  Formal model of preference:  A. notation.   Let X and Y be two alternatives.   Assume that there’s a group of actors number 1,2,3…   Symbolize the preferences of the “ith” person over X and Y, we write:    XRiY –> person i thinks X is at least as good as Y    XPiY–> person i strictly prefers X to Y   And  XIiY–> person i is indifferent between X and Y.

COMPLETENESS: for any pair of alternatives, XandY, either xRiY or YRiX.

TRANSITIVITY: For any three alternatives X1, Y1, and Z1, if XRiY and YRiZ, then XRiZ.

Definition: Acyclicity = the absence of cycles =   For any list of alternatives X1, X2….Xk if   X1PiX2 and X2PiX3 and…Xk-1PiXK then not XkPX1.

Let S represent actor i’s choice set. Let actor i’s set of best choices, denoted C(Ri,S), is defined as:  C(RiS)= {XisS | XRiY for all Y is S}  C = choice set-their actual highest ranked thing.

F(X,4)

B. Some basic properties of preferencesProposition: let S be a finite set of alternatives available to person i.  Suppose that i has complete and transitive preference order ordering Ri  Then, C(Ri,S) is non-empty.

Demonstrate that there is a best.

 PROOF

Choose 1 alternative, say X1,from S. If it is best, we are done  (we know its best once we check the alternatives).  If not, then there exists an alternative, say X2, for which X1RiX2 does not hold (that is to say that there is an X2 that is strictly preferred to X1).  By completeness, we know that some relationship exists.  So, by definition, X2 is strictly preferred to X1 (X2PiX1)  If X2 is best, we are done. If not, we can choose an X3 such that X3 is strictly preferred to X2 by the same argument above. We can then repeat the process over and over (we have a finite set) so this process can either terminate at a best choice (in which case we are done) or it can go on indefinitely. Since S has only a finite number of elements, for the process to go on indefinitely requires that it would eventually repeat. Therefore, there must be a violation or be a cycle of the form whereby X1PiX2 and X2PiX3…in order for there to be a cycle, you have to violate the transitivity assumption—you can’t violate the transitivity assumption. So you have to rule that you’re in a cycle violation. Remember, there is a best choice.

Which one of the options should become the law of the land (X, Y, or Z)?

Series of votes, X versus Y = X gets to Y 1   X versus Z   =  Z gets 2 votes and X gets 1   Y versus Z   =  Y gets 2 votes, Z gets 1.

There is not Condorcet here. It is a VOTING CYCLE!

Legislature needs to address this problem as SOCIAL CHOICE

Here’s that eight footer coming back at you! Get in the pipeline!

Amendment versus amendment:

The order in which things get framed IS choosing the winners and losers.[1]

“All was determined by the Amendment Procedure”

 Condorcet winner (defeats all the alternatives).

Still,

On the beach,

No more waves,

For me. 

In the distance,

Sunlight.

************

[1] See Arrow’s Theorem

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4 thoughts on “Preferences + Utility

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