h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Very nice article. I haven't been through all the math yet, but I'll do so later.

Just one question, which I hope isn't too naive. At the top of the article, it says that "it undermines the basis of representative democracy and free market economics".

I can understand the implications for democracy itself, but I'm not sure about any implications for free market economics as such.

The basis of free market economics is an understanding of the laws of economics (assuming we've got them right, of course ). These 'laws' are not dependent upon the specific political system; they just *are*.

The implementation of a political system may effect the manner in which a free market system is put in place, of course, and to what extent a country's economy is allowed to follow a free-market path. Is this what you mean?

Anyway, nice article. I have mentioned Mr Arrow's theorem to people many times, and from now on I will refer them here.

(Steven Landsburg makes some mention of the theorem in his fascinating book _The Armchair Economist_.)

jd

Actually, my favourite voting impossibility theorem is the following: you cannot have a non-random strategy-free system.

Strategy free means that you should never desire to alter your vote based on knowledge of the votes of others. Non-random means that the same set of votes will always give the same result. And you can't have both.

I am surprised that the entry kinda skipped over the common example given of Arrow in practice: as follows. The votes are:

33% A>B>C
33% B>C>A
33% C>A>B
{1% abstain}

Suppose we elect A. Now, if we drop B from the election, we have a choice. We can change the winner to C, violating independance from irrelevant alternatives; or we can keep the winner as A.

This means that in a 33 A -> 66 C situation, we elect A. By symmetry, in a 33C -> 66 A situation we elect C. And that violates the second condition (which is technically known as 'monotonicity'), because by increasing support for A, we cause A to lose.

By symmetry, whatever you do will cause the identical problem. Hence, it is impossible to choose a candidate which satisfies all the criteria.

I'd be interested in the answer to Jim's question too - which is why I posted here originally. And then I got distracted...

The phrase "it undermines the basis of ... free market economics" isn't mine, and it's slightly misleading. It should say something like "it undermines the basis of welfare economics", which is turn the basis of all microeconomics and a lot of macroeconomics. The point is that in economics you have a "social welfare function", which represents the "happiness" of society as a whole, conceived of as (in some sense) the aggregate of the "happiness" of each individual. In the mathematical theory of economics, all of the assumptions of Arrow's theorem about the welfare function are made, and since there can be no such function satisfying all the conditions, the foundation of that portion of economics based on social welfare functions looks a little shaky. In fact, the theorem might actually be more relevant for economics than for democratic theory.

There are also some variants of the theory that I've heard about recently in the mathematical theory of Optimisation, but that's not really my field so I can't tell you much more about it.

Hmm.

First off - the variation of Arrow I learnt replaced the monotonic requirement with the majority requirement - that if there are two choices (X&Y), and a majority prefer X to Y, then you must select X. Of course, in economics this isn't the case - we are (supposedly) prepared to let 51% people suffer by a tiny amount, in order to make 49% of people incredibly happy.

According to that variation, there's no problem with economics, or with utilitarianism in general - but I'm unsure of this (probably more accurate) variation, so I may have to investigate further...

Yep - I definately can't see it. Actually, I'm not entirely clear I understand result two in the detailed mathslike bit, which doesn't help...

Anywho - if our social welfare function is just "add up the welfare function of each individual", where the individual's welfare function is just a mapping of policies to numbers (dollar values, perhaps), then what does it violate? I can't see that it violates any of them...

Adding up normalised social utilities, on the other hand, violates IIAC. For example, if a voter marks A=0%, B=90%, C=100% - and A drops out - then they'll change their marking to B=0%, C=100% - and that will possibly change the overall winner from B to C. But that's a seperate thing...

Lucinda, I'm not an enormous expert on economics. From the textbooks I've read, the social welfare function has to satisfy the conditions in the article (although in economics you'd be considering individuals' preferences between different allocations of resources rather than social policies). The monotonicity requirement is sometimes called the "pareto condition" in economics I think. It says that if you have an allocation of resources (call it X say), and you have another allocation of resources Y in which everyone is at least as well off as in X, and one person is strictly better off in Y than in X, then "society prefers" Y over X. This seems eminently sensible if you want to rule out wasteful allocations of resources.

I think you can probably get round some of these difficulties if you allow comparisons between numerical utility functions, although as far as I know this is considered illegal in economics. I think the problem is that even if I could say that I had 10 units of happiness and you said you had 15 units of happiness, our units of happiness can't be compared. In other words, you can't say that you're happier than I am just because you have more units of happiness (the scales might be different). Individual utility functions are only used for making comparisons between different allocations for each individual, not for making tradeoffs of one individuals interests against another's.

So, Arrow's theorem undermines utilitarianism and economics if you assume any of the following:

All individuals only have preference relations, no numerical utility functions.

All individuals have numerical utility functions but these cannot be compared between individuals (since a numerical utility function induces a preference relation and I think vice versa if you're disallowing comparisons between different individuals utilities).

However, if you assume that all individuals have comparable utility functions then you might be OK. If you think about it though, you're on pretty dodgy ground if you assert this. It's equivalent to saying that you can come up with a universal measure of happiness and compare two individuals and say definitely which is the happier (in all cases).

Ahh - gotcha!

Yeah - pareto condition is the name - they also talk about Pareto improvements - things you do which make life better for some people, and don't make life any worse for anyone.

I *thought* utilitarianism was the numbers - they're the people who try to go round putting a dollar value on love and suchlike - and the preferences are a seperate thing, with a seperate name - but I could be wrong.

Yeah - assuming there is a scale of happiness and comparisons can be made and all the rest IS hugely dodgy - especially when people assert that that scale is dollars! But people do do it - and as far as I'm aware, that's what economists mean when they talk about greatest happiness and suchlike. Myself, I've always thought of it as a simplifying assumption, rather than as truth. I wonder if this is a modern change to satisfy Arrow...