🚀 go-pugleaf

	With all the talk in both ca.environment and sci.econ about 
markets/governments ability to solve public policy problems, I decided
it might be useful to post a result from mathematical game theory.

	What follows is a presentation of Arrow's social choice theorem.
We wish to ask the question: how should resources be allocated in a 
society? And see if there is in fact any reasonable answer.


Define(Resource): Undefined. A primitive concept, like "point" in set theory.

Define(Allocation): An allocation is a desciption of the use or disposition
of all resources. Resources included everything: private property, public
property, shared goods (air, light, water, etc...).

Define(Preference ordering): A preference ordering is an order on the set
of all possible allocations. We will say A P B to mean A is prefferred to B.
Preference orderings are transitive, i.e. if A P B and B P C then A P C.
Preference orderings are complete, i.e. for any two possible allocations
A,B either A P B or B P A. Although semantically this sounds strange, for 
the logic to work out A P A (i.e. A is "prefferred" to itself) for all A.

Note: nothing has been said about who or what has preferences...

Define(agent): An agent
			(1) has a preference ordering, defined above.
			(2) may choose an action from some set of actions
				which may be different for each agent.
			(3) allways chooses the action which produces
				the most preferred outcome by that agent.

Notes:
	An agent need not necessarily mean a "human."
	In fact, very few humans are perfect agents as descirbed above.
	(3) does not necessarily imply "selfish" behavior. I may be altruistic
and have preferences where I prefer an allocation that gives me less and others
more of certain resources. 

Define(society): A set of agents.

Define(pareto-superior, abbreviated hereafter PS):
	For any two allocations A,B, A is Pareto-superior to B (A PS B)
if and only if all agents in society prefer A to B. PS is NOT COMPLETE.
i.e. there are plenty of allocations A,B for which neither A PS B or B PS A.

Define(social preference function):
	A social preference function is a function of the preference 
orderings of all agents in society and produces a new preference ordering
(which is transitive, etc...) which we will call the social preference 
function. 

Note: we shall show that social preference functions with certain nice
properties do NOT exist.

Theorem: (Arrow, 1963, Social Choice and Individual Values).

If there are 3 or more possible allocations, then NO social preference function
satisfies the following 4 conditions simultaneously:

(1): Completeness. The social preference is allways defined, no matter how
individual preferences vary (note that the preference can change with changes
in the individuals preference). It is defined over all possible allocations.
i.e. if you knew the social choice preference function, and I reported
everyones' preferences and the possible allocations, you could order the 
possible allocations from best to worst, from "societys" point of view.

(2): Pareto-efficiency. (A better name might be unanimous consent...)
If A P B for all agents in society, then for the social preference A P B.

(3): Independence from irrelevant alternatives. If the social preference
function gives A P B with a certain set of individual preferences, then
the social preference function gives B P A only if some individuals 
change from A P B to B P A or vice versa. If individuals change their 
preferences on C and D (not equal to A,B) then it doesn't affect the social
preference of A vs. B.

(4): Absence of a dictator: There is not an individual agent such that A P B
for the agent implies A P B for society, no matter what others' preferences
are. 

Proof:

Assume 1,2,3. Arrow proves mathematically that a dictator exists.
See Arrow's work, or 

Kreps, A Course in Microeconomic Theory, pp.178-180.

or 

a good game theory textbook that deals with social choice.

I considered posting the actual proof, but its long, tedious, and some of 
the symbols would be needed to translated to phrases...

There is also a book by Ordeshook that address this.

What does this do for your thoughts about using political methods, of
any variety, to determine public policy about anything (the environment,
transit, education, etc...)??

Paul Brewer.