Family critical of my Come Dine with Me scoring system. My family and I have been arguing for years about a fun 'Come Dine with Me' competition we held between the six of us.
If you're not familiar with the show, a group of people all host a dinner party, and the remaining group members give the night a score out of 10. After everybody has hosted their dinner party, the scores are added up and a winner (and order of losers) is determined.
Before we started, I told everybody that, no matter how good the first dinner party was, I would be scoring it a 5. My argument for this was that it gave me the maximum amount of room (thereabouts) to base the subsequent scores relative to this first score.
If the first party was great, I didn't want to give it a 10, because of the next one was better then I'd have nowhere to go.
Since the game was completely self-contained (e.g. we weren't using the winner's score to compete against another family) then my argument is that the winner and order of loses would remain exactly the same.
My family didn't accept this and said I wasn't being fair and that this scoring system wasn't in the spirit of the game. We ended up cancelling the entire game after the first night and we've argued about it ever since. This happened around seven years ago.
My family and I are far from being mathematics scholars, so I wondered if people could confirm, in layman's terms, whether I was right. And perhaps someone could explain why I'm right better than I can :)
 A: This is fair only if
a) everyone else ranks the first party as a $5$ too, or
b) you don't take part in the competion (that is, you never host a party - you act merely as a judge for everyone else's parties, in which case it's is fair for all of them).
Here's the reasoning for why your system  is unfair:
Say all the parties were equally good, but "objectively" better than a $5/10$, say $8/10$.
Then, you would receive $8 \times 5 = 40$ overall points. However, everyone else would receive $8 \times 4 + 5 = 37$ points ($5$ from you, because you're comparing to the first party, and $8$ from everyone else). So, you would unfairly win in spite of the fact that everyone hosted equally good parties.
a) is fair for everyone, because once the first party gets $5 \times 5 = 25$ points, everyone rates all following parties using $5$ as a guideline, so all points for all the remaining parties are equally offset. That is, comparing to $25$ becomes fair.
b) is fair for everyone (assuming you don't take part), because you're not in the system, so you just act like an 'external judge' that differentiates between everyone else in your family.
I think the way to go forward for next time is for everyone to give their points at the end of all parties, giving what they think is the best party a 10. This solves both issues - everyone will have 'room to judge to other parties' and because everyone ranks the best party at 10, there won't be any unfair offset.
A: The main problem is that the scale has a maximum of 10. This implies that the numbers 1 to 10 are evenly divided from bad to perfect, making a 5 out of 10 very poor. By using such a scale you already have two reference points, 1 and 10, with which to compare the quality, and adding a third reference point (5 for the first meal) would make the scale non-linear.
Here are three ways to deal with the problem.


*

*Use a scale without a maximum. The first meal is given a default score, of say 100, and all other meals are judged relative to that (and to the zero score). Unfortunately this does not work well with multiple judges, as each will essentially have their own scale (based on how good they felt the first meal was). You would have to normalise each scale (e.g. divide by the maximum score) before averaging the judges' scores.

*Use a finer grained scale. If the first meal was very good, give it for example an 8.5 which still gives you enough slots above it to place every meal to come.

*Accept that you cannot differentiate between every meal, and that you will have to give some meals the same scores. It is not your job to rank them, it is your job to score them. The ranking only comes after the scores of all the judges have been averaged, resulting in a more precise score.
