You’re stuck in a motel because of a business meeting. It’s 6 o’ the clock post meridian, and you’re hungry. The town in which you must linger is small and the dining choices stunted. It’s either a hamburger at Wendy’s, or a hamburger at McDonald’s. Wendy’s is better, you think, because their soft-serve frozen drink is called a “Frosty” and McDonald’s isn’t.
The phone book says Burger King (hamburger) is an option, and they have hats; plus the creepy guy they had in their ads in oddly intriguing. Burger King wins over Wendy’s. Yet your dear old mother was a McDonald’s lady from way back. So, if you had to choose—and you do—you would pick McDonald’s over Burger King.
Thus far, our situation in the head-to-head competitions is this:
W > MC,
BK > W,
MC > BK.
If we were to string these out into one line, we’d get this:
BK > W > MC > BK.
This is so because, from the first line, W > MC, and from the third line M > BK; but from the second, you decided BK > W. There is no joy to be found. Your decision is intransitive, irreparably so: no matter what you decide, you’ll be violating one of your preferences. You’ll be accepting another, of course: you’ll be happy and sad simultaneously.
You may think that they solution lies in looking outside the (such as it is) food quality of these dining establishments—perhaps one is twice as far as the other, and you’re low on gas—but this is not so. Because even factoring them in, it is still possible that the ordering remains the same. (You can always, of course, change your order.)
It’s not the system that broken, it’s that the utopia of deciding upon the optimal vote is impossible. Not always, mind; but sometimes, and more often than you would like. Intransitive systems are not that rare. Think of the childhood game of rock, paper, scissors (to which there is devoted at least one society). There is no best, or ideal solution that guarantees a winner in this game. Neither expected value, game theory, nor any other branch of mathematics can help here. But there may be help in multitudes.
Turns out that you have two comrades with you, similarly exiled: Bob and Randolph. And you, knowing the forlorn solution, have decided to reorder your preferences to BK > W > MC (you figure your mother will never know). You call up Bob and Randy and ask them theirs. Here’s how they respond:
You: BK > W > MC,
Bob: W > MC > BK,
Randy: MC > BK > W.
For first place, each restaurant has one vote. Again, no joy. But how about if we count the votes with what’s called the Borda system, after the chevalier, incongruously named the sailor? This assigns an arbitrary number of points for first place, fewer points for second, and so on, to produce a weighted count.
It’s easy to see—if not, look harder—that no matter how many points you pick for first, second, and third place, each dining establishment will have an equal weight for first place. Thus, the only way to decide is by somebody taking on the role of dictating where you’ll eat. And when he does—it may even be you that does the dictating—somebody is going to be unhappy: somebody is going to get their last place choice.
Similar dispiriting results can be found in any election in which there are more than two choices. Kenneth Arrow quantified one such result in a well known impossibility theorem, which said that if a voting system had three or more choices, and that it met certain eminently sensible desiderata, there always existed a “dictator”, by which he meant a person whose ordered preferences are substituted for all other voters’. This dictator might be an accidental and not a manipulative or domineering one.
The complexities of Arrow’s theorem are interesting, but in an important sense, they are beside the point. For we can prove that there will always be an unhappy voter or voters in nearly any election. We can show authoritatively, that is, that the human state is imperfectible, as measured by satisfaction.
Take any vote in which the choices are two or more and there are two or more voters. Then, it frequently occurs that unanimity is absent. And that’s it. The people on the losing end of the vote go away unhappy necessarily. This result is so banal that we scarcely notice it, but its consequences are deep and fundamental.
The unhappiness of the losers may be temporary, and may even be assuaged by “enlightening” them such that they change their vote (if only in their mind). But the unhappiness is there and it is real, and it may be enduring.
It doesn’t change your result but you left out: Go Hungry > MC
We can’t tell the difference, in this scheme, between those who would rather go hungry from those who simply aren’t hungry.
Similarly, when I didn’t vote at the recent election it was active non-voting not passive. But, under the present scheme, I am indistinguishable from those who just don’t give a damn.
There’s also the question of “none of the options are my preferred option”. In the burger example, perhaps what I really want is to go to A&W, but there’s none in the entire state. We are assuming here that one of the available options is exactly my preferred option, either as a diner or as a voter, but that is rarely the case in fast food or in elections.
I have a good friend Professor of physics in a famous Paris University .
He is doing research precisely in this domain :
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“Abstract.
The dynamics of spreading of the minority opinion in public debates (a reform proposal, a behavior change, a military retaliation) is studied using a di usion reaction model. People move by discrete step on a landscape of random geometry shaped by social life (oces, houses, bars, and restaurants). A perfect world is considered with no advantage to the minority. A one person-one argument principle is applied to determine locally individual mind changes. In case of equality, a collective doubt is evoked which in turn favors the Status Quo. Starting from a large in favor of the proposal initial majority, repeated random size local discussions are found to drive the majority reversal along the minority hostile
view. Total opinion refusal is completed within few days. Recent national collective issues are revisited.
The model may apply to rumor and fear propagation.”
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Due to his deep knowledge of such issues , he is obviously a climate skeptic .
A few weeks ago he sent me for comments a draft where he studies the propagation of climatic fearmongering and comes with an interesting concept separating people in 2 categories : those who will change their mind (open) and those who won’t (closed) .
He demonstrates that a propagation of an opinion say “CAGW” will be significantly accelerated if the initial CAGW population has a greater proportion of “closed” members than the non CAGW initial population .
This is interesting because a very minoritary opinion whose supporters are majoritarily “closed” will quickly overwhelm a majority opinion whose supporters are “open” .
The abstract above considered that the whole population was open , e.g opinions were formed in rationnal 1 to 1 and 1 to N discussions succeeding in time .
The “closed” category he added now is such as it keeps its opinion regardless the number of discussions and regardless the opinions of other people .
His model shows that CAGW opinions will quickly propagate even starting from very low initial positions . However as soon as appears a “closed” population opposing CAGW , CAGW disappears as fast as it came because it corresponds to a minority opinion in the “open” part of the population .
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And yes , all this is not very far from the problem whether I will eat in Tour d’Argent , in McDonald or chez Troisgros .
TomVonk: Not to be a wise guy but I’m curious. Physics? I would’ve guessed a couple of other disciplines, but not the study of matter, et al.
No, gcb, I’m pretty sure that was a point in Briggs’ example. That OF THE AVAILABLE CHOICES, what is your preference. You might prefer Red Robin or Steak & Shake burgers but if there’s none within a hundred miles, you’re going to be out of luck. Briggs’ point was that: you have # choices: 1, 2, 3. What’s your preference of those choices?
Now Rich has a point about the fourth option. But that also becomes mute when you have other people involved. Say you abstain from voting, then Bob & Randy will drag you off to Mc (which wins the weighted voting).
I gave up on economics because of this theorem and focused on decision-making from a broader viewpoint.
The bottom line is that we can ask reasonable questions for which there might be no reasonable answers.
This is a three player version ‘the battle of the sexes’ game. In the two player game, you and your wife want to do different things, but it is more important to both of you, that whatever you do, you want to do it together. The recommended strategy for that game is to speak first and have a reputation for being stubborn.
If we were dogs, the ‘alpha’ of the group would make the decision. He will chose what he desires for himself, and life will be better if we eat than if we starve. As long as the alpha provides decent leadership, we will willingly follow. Would we be happier if someone just made the choice for us?
We would also be happier if we had fewer choices? If the one of the burger establishments didn’t exist in this town, we could clearly make the best decision, and the group could maximize utility. Modern living with unbounded choices creates angst. Yet in every situation, wouldn’t you rather have more choices?
The most difficult decisions are frequently the most trivial. When the expected outcomes are nearly relatively equal, we suffer from analysis paralysis. We are sitting in this stinky motel room discussing the relative advantages of MickeyD vs. Le Roi de Burger when we could be eating our fried / broiled deliciousness.
This is a serious problem. Consider the problem of democratic elections. Somehow the dictator usually exists and the result is often that the people get the opposite of what they want.
The dictator can prevent someone else from getting a majority unless the dictator’s pet legislation is passed. The dictator is, by definition, not the choice of the majority and his policies are not the ones the majority would support or desire.
The result is often perverse, corrupt and sometimes bordering on evil. I leave it to you, as an exercise, to supply your own examples.
If you can mathematically design an election/legislative system that solves this problem, you deserve the Nobel Peace Prize.
The unhappiness of the losers may be temporary, and may even be assuaged by “enlightening†them such that they change their vote (if only in their mind). But the unhappiness is there and it is real, and it may be enduring.
In the case of the hamburgers, suppose a decision was reached by arbitrarily picking the picker (perhaps a game of rock paper scissors). Randy won and chose McDonald’s, thus Bob and I agreed to have dinner at the Golden Arches amid differing levels of dissatisfaction. After returning to the hotel, however, Bob decided that McDonald’s hamburgers (at least at this particular franchise) were better than he remembered. With the benefit of experience, our ordering was revised, and thus a real winner emerged:
Me: BK > W > MC
Bob: MC > W > BK
Randy: MC > BK > W
The happiness of people is fickle and subject to so many variables (a point touched on in the blog on the value of controls vs. randomization) that I don’t see the point of using it to evaluate voting systems.
49erDweet
TomVonk: Not to be a wise guy but I’m curious. Physics? I would’ve guessed a couple of other disciplines, but not the study of matter, et al.
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Well yes . An opinion is a state . The state changes by interactions . Hence changes of opinion correspond to dynamics in the state space .
He came to apply methods of diffusion reaction (hardcore physics) to the dynamics in the state space .
It’s a way to introduce some science and mathematics in a domain that is dominated by medias , polls and witch doctors .
Of course there are simplifying assumptions and approximations but it gives rather good results .
Obviously witch doctors and environmentalists don’t like him much .
Hello from Australia where we all just lost in the last election. Try as we did we just couldn’t end up with no politicians.
Craig, you have my sympathies.
Randy won and chose McDonald’s, thus Bob and I agreed to have dinner at the Golden Arches amid differing levels of dissatisfaction.
And they measured dissatisfaction how?
The measure of dissatisfaction, like happiness, was self-reported and entirely subjective.