Paradox of compromise illustrated by neighbours painting houses

Question

If I remember correctly, there was a problem illustrating the paradox of compromise (and to some extent democracy?) where there are two neighbours that want to paint their houses.

They both want to paint their house their favourite colour (e.g. red and green),

but that would be an eyesore to the other. With no option,

they decide to paint it a colour they both dislike, but don't hate as much as the other person's favourite colour (e.g. yellow).

It showed that sometimes compromise can lead to a position where both parties are worse off, compared to had no compromise been made in first place.

I did digging but I don't remember how this is called or find an article related to it, I just get house painting ads.

Answer 1

The paint color metaphor was used a while ago during Brexit debates, when "sources" suggested that indicative votes "will leave us with an Auf Wiedersehen, Pet Brexit'". Auf Wiedersehen, Pet is an old TV show about seven British construction workers. In one of the episodes they pick a color to paint their shed by making lists of preferences. The winner is yellow, which no one wanted, but some marked as a second choice. The story ran in major UK newspapers with headlines like Daily Mail's May Warned She Could End up with an 'Auf Wiedersehen, Pet' Brexit No-one Wants:

"They told the paper: 'There is a scene in Auf Wiedersehen, Pet where the builders pick a colour to paint their shed and end up with yellow. 'They are all baffled because no-one voted for yellow, but it turns out that two people put it down as their second choice. 'So there is an issue with a ranking system, as it comes with the inherent danger that you end up with a result that no-one wanted.'"

The "indicative votes" were supposed to produce an outcome by a sequence of votes to choose among, roughly, hard Brexit, soft Brexit and no Brexit. So this is what is being illustrated. But is it paradoxical? I think not. An outcome that nobody likes but can live with is the definition of compromise, not something unexpected. And in the OP example the builders are "worse off" compared to what? Painting both houses their color? But that was never in the cards.

What really happened during Brexit is closer to the Condorcet (a.k.a. thwarted majorities) paradox of preferential voting, where roughly a third of population supported each option, so that there was always a majority to reject every option. And, depending on sequencing the "indicative votes", any option could have been arranged to win. Fishburn and Brams discuss other paradoxes of preferential voting, but Auf Wiedersehen, Pet does not really display any of its paradoxical features in particular.

Nor does it have a catch that makes social cooperation games with rational but inferior outcomes interesting. The most famous is the prisoner's dilemma, but arguably closer to Auf Wiedersehen, Pet is the stag hunt. There each player has to choose to hunt a hare, which they can do on their own, or a stag, for which cooperation of the other player is required. As in preferential voting, the players do not know the other's choice. The "safe" option leads to the inferior outcome with two hares and no stag. Except in Auf Wiedersehen, Pet, we end up with the stag, if the hares mean getting their favorite color for their house, and the stag is an analog of yellow. Whether it is or not depends on how much they dislike their house being yellow vs their neighbor's house being red or green. But either way there is no clearly superior alternative to create a paradox.

Answer 2

I don't recall this particular example as being notable in any way — i.e., it's not frequently used in and of itself, or the preferred case of some particular theoretician that I know of — but it is typical of the kind of case often used in rational actor theory and game theory. A cursory google search turned up nothing directly relevant. Such is life...

I will point out that this example shares the common weaknesses of RAT and GT, in the presumption that individual preferences are:

• Unambiguous and conscious
• Well-ordered and enumerable
• Immutable in both intensive and extensive properties
• Independent or other preferences and considerations

All of which are questionable restrictions. For instance, this model precludes the possibility that the neighbors might reach a compromise color and decide after the fact that that they both prefer the compromise color to the originals, and precludes the possibility that the neighbors might get more value from reaching a compromise with their neighbor (with all the future benefits that entails) than is lost because they did not get their preferred color choice. The framing of the question explicitly casts compromise as a loss-loss, which is far from realistic.

Answer 3

Your interpretation is wrong. Living entities group in order to increase the probabilities of survival, at the price of losing some advantages available if living alone.

So, both persons on your example are not worse. They have more probabilities of survival, at the cost of painting their houses in a color they dislike, or not being able to build their gardens on the middle of the street. Nothing extraordinary.

Answer 4

The example is a poor one as the yellow house compromises IS the best collective outcome. However, the poor example, does not obviate the paradox which was intended.

I had a professor who pointed out the paradox, and it shows up with preference profiles which are non-linear. His example was voting for a policy, where one could have the govt implement 0 of something, 1/2 unit of something, or 1 unit of something. If the vote distribution is 4, 2, 4, and the policy ends up at the compromise middle, IE 1/2 of the action, in most cases, where preferences are basically linear, then the half-way median WOULD be a least-bad compromise.

But some policies have radically extreme outcomes, and in those cases, a compromise is disaster. The example the professor offered, was going to war. NOT going to war -- saves lives and one's nation's economy, but MIGHT sacrifice one's ally nations, and/or one's principles. Going to war could put one's sovereignty, population, and economy at risk of destruction, but might be worth that risk. Going to war HALFWAY -- brings all the risks of war, but by starving the war effort of resources, drastically reduces the hope of victory. A halfway war -- if that is the outcome of the 4/2/4 vote -- is a lose/lose.

I don't know the game theory reference where you will find this, but I hope this description helps you find it.

Answer 5

If they had not compromised then each neighbour's house would be an 'eyesore' or together they would make an eyesore. (The example seems indeterminate as to which applies.) If they do compromise then each neighbour's house is less than, or less of, an eyesore. So they appear to benefit from the compromise: the result is less unfavourable than if they had not compromised. Non-compromise where an eyesore results is what makes both parties worse off.