An unusual voting problem?

“Four Weddings” is a reality based TV show that appears in America on the cable channel TLC. Yes a TV show: not a researcher, not someone who has recently solved a long-standing open problem. Just a TV show.

Today I want to discuss a curious math puzzle that underlines this show.

The show raises an interesting puzzle about voting schemes:

How can we have a fair mechanism when all the voters have a direct stake in the outcome?

So let’s take a look at the show, since I assume not all of you are familiar with it. I do admit to watching it regularly—it’s fun. Besides the American version there are many others including a Finnish version known as “Neljät Häät” (Four Weddings), a German version called “4 Hochzeiten und eine Traumreise” (4 Weddings and one dream journey), and a French version called “4 Mariages pour 1 Lune de Miel” (4 Weddings for 1 Honeymoon). The last two remind me most of the 1994 British movie “Four Weddngs and a Funeral” but there is no real connection.

There is keen interest worldwide, it seems, in weddings as they are a major life event. And of course, they are filled with lots of beautifully dressed people, lots of great displays of food and music, and lots of fun.

## The Show

Like many shows, “Four Weddings” is based on a British show—do all good shows originate in the UK? Four brides, initially strangers, meet and then attend each others’ weddings. Each then scores the others weddings on various aspects: bridal gown, venue, food, and so on. Then the bride with the highest score wins a dream honeymoon. Of course there is the small unstated issue that the honeymoon, no matter how exotic, happens well after the actual wedding. Oh well.

The scoring method varies from season to season and also from country to country. But higher scores are better, and the brides get a chance on camera to explain why they scored how they did. A typical comment might be: I loved the venue, the food, but the music was terrible.

You get to see four different weddings, which is the main attraction in watching the show. Usually each wedding is a bit out there: you see weddings with unusual themes, with unusual venues, and other unusual features. If you are not ready to have an interesting wedding, to spend some extra time in making it special, then you have little chance of winning.

## The Puzzle

The puzzle to me is really simple: why would the brides rate each other fairly? They all want to win the honeymoon, the prize, so why ever give high ratings? Indeed.

There have been some discussions on the web on what makes the scoring work. Some have noticed that the most expensive weddings usually win.

Clearly the game-theoretic optimal move seems to be to give all the other brides low scores and hope the others act fairly. The trouble with this method is that you look bad—and who wants to look bad on a TV show that millions might see? Can we make a model that accounts for this? It does not have to embrace possible psychological factors at all—it just has to do well at predicting the observed ratings on the show.

## A Solution Idea

I have thought about this somewhat and have a small idea. Perhaps some of you who are better at mechanism design could work out a scoring method that actually works. My idea is to penalize a score that is much lower than the others. A simple version could be something like this: Suppose the brides are Alice, Betty, Carol, and Dawn. If Alice’s wedding gets scores like this:

Betty: 7
Carol: 6
Dawn: 3,

then perhaps we deduct a point from Dawn’s total. She clearly is too low on Alice. Can we make some system like this really work?

## A Related Game?

I have been discussing with Ken and his student Tamal Biswas some further applications of their work on chess to decision making. Their latest paper opens with a discussion of “level-${k}$ thinking” and the “11-20 money game” introduced in a recent paper by Ayala Arad and Ariel Rubinstein.

In the game each player independently selects a number ${n}$ between 11 and 20 and instantly receives ${\n}$. In addition, if one player chose a number exactly \$1 below the other’s number, that person receives a bonus of \$20 more. Thus if one player chooses the naively maximizing value \$20, the other can profit by choosing \$19 instead. The first player however could sniff out that strategy by secretly choosing \$18 instead of \$20. If the second player thinks and suspects that, the \$19 can be revised down to \$17. And so on in what sometimes becomes a race for the bottom, although the Nash equilibrium assigns non-zero probability only to the values \$15 through \$20.

In this game the level ${k}$ of thinking what the opponent might do is simply represented by the value ${k = 20 - n}$. There is now a rich literature of studies of how real human players deviate from the Nash equilibrium, though they come closer to it under conditions of severe time pressure. The connection sought by Ken and Tamal relates ${k}$ to search depth in chess—that is, to how many moves a player looks ahead.

Ken does not know whether anyone has intensively treated the extension to ${m > 2}$ players. The following seems to be the most relevant way to define this:

1. If the lowest player is \$1 below the second-lowest, then the lowest player gets the \$20 bonus, else nobody does.
2. If the lowest is lower by 2 or more then that person gets nothing—not even the original ${\n}$.

It would be interesting to study this with ${m=4}$ and compare the results to the observed behavior in the show “Four Weddings.” Could something like this be going on? Or are the brides simply being true to their own standards and gushing with admiration where merited? It could be interesting either way, whether they match or deviate from the projections of a simple game-theoretic model with scoring like this.

## Open Problems

What is the right scoring method here? Is it possible to find one?

August 2, 2015 11:14 am

Why do you assume the low score is wrong? Looking at one wedding gives you nothing to make that assumption. Maybe if your total over all the other weddings is low then you get adjusted, or better just make the values into rank scores – 1st, second, third so that the actual value of the scoring doesn’t matter. That would seem fairer to me, although you likely get more ties.

August 2, 2015 11:36 am

Then maybe dividing 10 points among the other contestants; this ensures you’re not being cheap, while allowing for a finer notation.

• August 2, 2015 2:34 pm

I think the problem with this is that (supposing the others are being truthful) you should give the best wedding the lowest score – this increases your chances of getting first place. Btw, I personally seriously question the depth of the strategies employed by the participants of the show. I wonder how many of them (and the spectators) realize that giving a high score lowers their chances of winning.

August 2, 2015 7:57 pm

The American show (at least) uses rank as part of their scoring to solve just that. I don’t know if its enough to outweigh the nominal scores they also give.

2. August 2, 2015 2:42 pm

This reminds me a bit of the various correction for guessing formulae for grading student exams. Questions have been raised over whether such a procedure could ever be fair.

• August 10, 2015 11:16 am

Correction: Get rid of multiple-choice exams.

3. August 2, 2015 8:48 pm

One amendment (likely to have been conceived before), is to have the initial best-wedding scoring be open (everyone sees everyone else’s scoring), then have a second scoring to select, not the best wedding, but the two best judges (and the “best judge” scoring should be blinded).

The prize awarded to the two “best judges” should be worth significantly less than a full honeymoon, yet worth significantly more than no prize at all … moreover the two “best judges” should receive an invitation to appear in the next round of selection.

This pay-it-forward reward systems fosters traits that many cultures value, such as institutional continuity, aesthetic traditions, and democratic leavening.

4. August 2, 2015 11:37 pm

A modified Borda Count (with punishment for against the trend voting) was proposed for telescope time allocation (http://arxiv.org/pdf/0906.1943.pdf).

Not sure it would work when there are only 4 voters, though.

5. August 3, 2015 10:27 am

It seems that the optimal strategy depends on what one perceives the cost of humiliation on national television is. Suppose we assume that all players have perfect judgement and can judge a wedding objectively (in other words can predict how others will judge a wedding). They would assign each wedding an “objective” score. In other words, the “objective” score is the average rating of the other two players. Then, they would decrease this objective score according to how much they value public opinion (i.e. assuming that deviation from the mean score is what causes public disdain). Of course, given rational players who are following similar strategies, then, a person could guess how much someone else would decrease their score and decrease their score accordingly. This means that even in this scheme, the equilibrium score would be 0 provided the audience dislikes outliers who score significantly below the others. (Although, in reality, the audience may still view everyone poorly even if everyone gave a score of 0.) If all players are perfectly rational, then they should all give scores of 0 and there would be no “Four Weddings” TV show. As mentioned before, a more optimal scoring scheme would reward non-zero rankings. What if we give the top ranker of each wedding and the winner with the wedding with the greatest rankings equal probability of winning the honeymoon? Or the top rankers some probability lower than the probability of the player with the wedding with the greatest ranking?

6. August 3, 2015 10:38 am

Reblogged this on Sublime Illusions and commented:
Interesting post about fair scoring with stakes involved. As it is, given rational players, I don’t think the show should exist at all. At least, I don’t see a rational incentive for players to rate other weddings any score other than 0. But perhaps, there’s a way to amend the voting scheme so that there is an incentive for non-zero voting.