The Pleb Economist #2: Politics is Provably Hard \ stacker news ~econ

The Pleb Economist #2: Politics is Provably Hard¹

Today I thought I'd share about a theorem that is close to my heart, because it's, more than anything else, the thing that made me fall in love with economics, warts and all. And that's Arrow's Impossibility Theorem.

Ken Arrow

Arrow's Impossibility Theorem is a result in the field of social choice theory. That's the study of how groups of individuals, each with their own distinct preferences, get together to make group decisions. Democratic elections are an obvious examples, but there are others as well: for example, how will the NBA decide on its MVP?

The basic setup goes like this: There are

individuals, each with their own personal ranking over

possible outcomes, but a single ranking over outcomes needs to be made for the whole group. The process which maps the individual rankings to the group ranking is known as a social choice function.

First preference plurality. These are systems in which every outcome is ranked according to how many first place votes it gets. Many countries use systems like this to determine their political leaders, including for the U.S. presidential elections.²

Borda count. These are systems in which outcomes are scored according to how many 1st, 2nd, 3rd, and so on, place votes they get. For example, out of 10 possible alternatives, each outcome scores 10 points for each 1st place vote, 9 points for each 2nd place vote, and so on. The candidates are then ranked according to their final score. The Borda Count isn't often used in political elections, but it's often used in sports. For example, the MLB uses a Borda Count system to decides its MVP, and the NCAA uses a Borda Count to rank college teams.

Both FPP and Borda Count are examples of social choice functions, but there are many more. The objective of social choice theory is to study social choice functions. Is there an optimal social choice function that can be used in every situation?

The revelation principle (or, the incentive to vote honestly)

Before moving on, let me address a question you might be wondering about. Does the social function require people to vote honestly? This is an important question for implementation, but from a theoretical point of view, it turns out not to matter.

It's been proven that any system where people can vote dishonestly is equivalent to another system where people are incentivized to vote honestly. Thus, in the analysis of social choice functions, we only need to restrict our attention to honest mechanisms.

This is known as the revelation principle, and social choice functions where people are properly incentivized to vote honestly are called incentive compatible social choice functions. Every non-incentive-compatible system has an equivalent incentive-compatible one.

Now let's ask whether we can design an optimal social choice function. Before doing so, we need to ask what makes a social choice function desirable? Here are a few properties that seem pretty non-controversially desirable.

Non-dictatorship. The social choice function reflects the rankings of more than one individual. In other words, the social choice function doesn't simply mimic the preferences of a single person in the group.

Pareto efficiency. This simply means that if everyone prefers candidate A to candidate B, then the social choice function will prefer candidate A to candidate B. Very non-controversial, right?

Independence of irrelevant alternatives (IIA). This means that the social choice function's relative ranking of A and B does not depend on where people place C in their rankings, as long as the relative positioning of A and B remain the same.

So for example, if 60% of people prefer A to B head-to-head, and the social ranking prefers A to B, then the introduction of a third option C can't flip the social ranking of A and B.
The American presidential election does not satisfy IIA. The introduction of a third party candidate can swing the election, even if the positional rankings of major party candidates stay the same in everyone's votes.

Arrow's Impossibility Theorem

And now we reach the climax of our article. The theorem. Arrow's Impossibility Theorem says that:

If there are at last three alternatives, then every Pareto Efficient, IIA social choice function is a dictatorship.

Somehow, we went from a very simple setup (

people collectively ranking

outcomes), and some very simple, non-controversial requirements, to an absolute result saying this is not possible.

The proof of the theorem is beyond the scope of this article, and honestly quite beyond the scope of my teaching abilities, but the result is quite surprising and mathematically proves the statement: politics is hard.

Applications

The main application is to put politics into perspective. Political fighting between interest groups is not going to end, and it's not going to be solved by better systems design. Systems design can perhaps improve things, but as Thomas Sowell likes to say: there are no solutions, only tradeoffs.
Thinking about Bitcoin, this also means that debates about the direction and design of Bitcoin probably can't be solved algorithmically. The human element will always be there, and if you want to see Bitcoin go in a certain direction you should be prepared to fight for it.
What about Bitcoin's network consensus mechanism? Isn't that a social choice function too? It's a consensus mechanism to rank competing chains. Curiously, this leads us to another result from social choice theory: the Vickrey-Clarke-Groves mechanism. The VCG mechanism says that we actually can have a socially optimal choice mechanism, as long as monetary transfers are allowed and the possibility of burning resources is available. The fascinating connection to Bitcoin comes from this burning of resources. One can think of Bitcoin's Proof-of-Work mechanism as a method of burning resources to establish consensus.³
Finally, a fanciful thought is that enlightened monarchy may be the most efficient political system; but only if the monarch happens to be a Pareto-Efficient monarch. But since that cannot be guaranteed, we can't practically implement such a thing here on earth.

Thank you for your attention to this lesson on Arrow's Impossibility Theorem. I hope you found the result as interesting and as mind-blowing as I did. Let me know what you think in the comments :)

Previous Pleb Economists

Pleb Economist #1: The Art of Economic Communication #839278

The Pleb Economist is a (hopefully) weekly column where I share my thoughts as a mainstream economist who is slightly jaded by mainstream economics. I also have a somewhat libertarian and rebellious bent, as well as a deep conviction in the utility of Bitcoin. ↩
Technically, it's a bit more complicated in the U.S. because of the primary system, but once the field has been winnowed down to the Democratic and Republican nominees and third party candidates, the final round of the election works like FPP. ↩
Can the VCG mechanism be applied to making decisions about Bitcoin's software? Perhaps, since the Bitcoin network allows for credible monetary commitments as well as credible burning of resources. I haven't really worked it out though, and such a system would put the power to decide Bitcoin's future in the hands of those who hold the most Bitcoin. ↩

113 sats \ 1 reply \ @Undisciplined 14 Jan

No way! Arrow's Impossibility Theorem is what convinced me to become an economist. I wrote about it for a project in a math capstone and was blown away.

32 sats \ 0 replies \ @SimpleStacker OP 14 Jan

Inspiring generations of economists, it seems

30 sats \ 1 reply \ @denlillaapan 21h

shiiiite, so happy to see this refresher.

It's been a while.

So, you mention at the end that, provided we can't guarantee enlightened monarch, what do you think that says relative to other systems of (political) rulership?

Saif has some witty phrase that monarchy is rule by random (genetics pretty randomly distributed), whereas democracy selects for less than that (i.e., the worst get on top, etc).

What do you think about that in regard to the impossibility theorem?

0 sats \ 0 replies \ @SimpleStacker OP 21h

Haha, Saif has a good point.

Saif is basically saying that only the worst people will seek out power (or have the capability of winning elections), and thus our democratic leaders are selected amongst the worst. There's two ways to relate this to Arrow's Theorem:

Arrow's Theorem doesn't say anything about the candidate pool, which is taken as given. So it can't really speak to Saif's quip.
But another way to look at it is that Arrow's Theorem generalizes to an exhaustive candidate pool (everyone in the country is a candidate), and the impossibility theorem still holds. So democracy is hard no matter the system or the process by which the candidate pool is generated.

30 sats \ 10 replies \ @Undisciplined 14 Jan

One of my main gripes with modern macro is their flagrant rejection of social choice theory. We have this beautiful theoretical work and then they just start by assuming it away with a representative preference function for the whole society.

30 sats \ 0 replies \ @denlillaapan 22h

yeeeeees

25 sats \ 8 replies \ @SimpleStacker OP 14 Jan

Yeah, i think modern macro, more than any other discipline, tends to reject anything that it can't conveniently render mathematically

Arrow's impossibility Theorem is an example of good use of math in economics, imo, since it's so general in its setup. I guess one thing it lacks is monetary transfers (hence VCG), and repeated interactions.

30 sats \ 7 replies \ @Undisciplined 14 Jan

I haven't read anything about integrating repeated interactions into the analysis. I suppose that allows for relaxing the dictator requirement to something like "the same person doesn't always get to decide".

Do you recall the minor scandal a few years ago surrounding people trying to sell their votes online? I'm guessing economists were the only people thinking "Yeah, why not?"

24 sats \ 6 replies \ @SimpleStacker OP 14 Jan

I've always wondered whether it would be cheaper to buy a vote directly than through the various lobbying and outreach efforts. I'm not sure how much I'd ask for my vote.

30 sats \ 5 replies \ @Undisciplined 14 Jan

Most importantly, the money would go directly to normal people, rather than clogging up every information medium with political ads.

One system I've thought about is one where voters pledge different amounts of money in the event that each candidate wins. Then, the winner is who has the most pledged from voters. What I love about that idea is that it sucks the political rents, right out of the system.

23 sats \ 4 replies \ @SimpleStacker OP 23h

I think the VCG works something like this. The winners pay the difference between their bids and the runner ups' bids. I think Bitcoin would make such systems more practical to implement, since the monetary commitments can be secured and verified by code.

30 sats \ 3 replies \ @Undisciplined 23h

Have you ever tried talking to a normal person about these ideas?

In my experience, they are not well-received (to say the least).

35 sats \ 2 replies \ @SimpleStacker OP 23h

I'm pretty selective on who I share my "econ" thoughts with. If they seem pretty open minded and enjoy thinking of unorthodox ideas, then I like talking about it. But otherwise, yeah, most people don't seem to have the patience for it.

view replies

30 sats \ 2 replies \ @Bell_curve 14 Jan

Rank choice voting?

I hate it for now. It’s the only reason the wrong candidate keeps winning in Alaska especially if no one gets 50 percent in the first round

38 sats \ 1 reply \ @SimpleStacker OP 14 Jan

Rank choice voting also fails IIA. It also tends to disfavor centrist candidates who are broadly acceptable but have few first place votes

43 sats \ 0 replies \ @Undisciplined 14 Jan

That depends on the scoring system. One of the interesting elements of social choice theory is that there is no "correct" scoring rubric.

I assume that means the dominant party pushes for the scoring metric that they assume will favor them most often.

Social Choice Theory: An Introduction

Examples of social choice functions

The revelation principle (or, the incentive to vote honestly)

Desirable properties of social choice functions

Arrow's Impossibility Theorem

Applications

Previous Pleb Economists

Footnotes