# How to make a fair model for competition please?

1. Feb 11, 2014

### Viktoria

Hi,
English is not my native language.

I manage first time a competition on social media, where:
1. people bring a photo and gather like on it. The top 3 likes will go to final,
2. in the final, they play with a small robot as a vacuum cleaner to clean an area of 1mX1.5m which has 1 pin in every 5cm square of it. Each player will have 5 min to use the robot.

I must model so that it will be fair: more chance of making mistake (leaving uncleaned squares) the highest the likes, but how to relate these numbers to squares so it is fair?

Currently the top 3 are:
a. 84
b. 60
c. 51

I'm sure this is an easy math problem only I can't solve it on the Earth! I don't know even how to google to find similar cases to read about.

2. Feb 11, 2014

### Simon Bridge

You want to handicap the person with the most likes so they have to work harder at the next challenge?
(I don't see how that's fair!)

That would depend on the details of the robot etc - why not just give them a bigger area to clean?

3. Feb 11, 2014

### Viktoria

By fair, I meant that the one who gathered more like, has a better chance on the final.

Using like on photos is only this first time as we didn't have enough time to make a filtering device for the first stage of the competition. Next time we replace the first stage with an online game.

So, the problem is somehow relate the like quantity they gathered, to the quantity of left items they fail to collect with the robot, the higher the like, the less penalty.

How to come to a formula is what I don't know :(

Edit:
Why handicap? If person has more likes, game ignores more mistake!

4. Feb 11, 2014

### Simon Bridge

Oh probably just English use: you said -
"more chance of making mistakes" suggests the puzzle is harder
"the highest likes" suggests people who do well on the first stage
put it together and you had said that you want people who do well in the first part to have a harder puzzle.

But you want to handicap the people with the least likes right?

Why is it fair that people who do well on the first task have an easier time on the second one?

To favor the people with more likes, then, give them a smaller area to clean - or just fewer obstacles (pegs) or give them a faster cleaning robot or one that is more manoverable or something? The exact formula will depend on the exact strategy you pick for favoring the people with the most likes.

Anyway - you should have a situation like so:

You have N players numbered 1...N.
In the first task, the nth player gets kn likes.
... then you can rank the players according to likes: the nth player gets rank Rn - if they came first their rank is 1 etc.

The minimum number of likes is kmin
The maximum is kmax

Example: say their are 4 players, then: N=4
They get some likes so that: k1=54, k2=105, k3=23, k4=72 or k=[54,105,23,72]
Ordering the players from highest to lowest gives: [2, 4, 1, 3]
Ranking them gives: R1=3, R2=1, R3=4, R4=2 or R=[3, 1, 4, 2]
kmax=105
kmin=23
... got it so far?

Lets say you choose a strategy where the person with the most likes cleans the least area.

The maximum area to clean is Amax and the minimum is Amin

Then you want players with $k_n=k_{max}$ to have to clean area $A_{min}$ and players with $k_n=k_{min}$ to have to clean area $A_{min}$ and players with something in between to have to clean an area in between. There are lots of equations that do that - the simplest one is a straight line.

You know how to find the equation of a straight line right?

You could choose a scoring method ... if the nth player cleans An area in a fixed time T, then the bigger area gets a bigger score. So if each square cm is worth 1 point, then the initial score is just the area cleaned. Then you multiply that score by a factor determined by the players rank to get the final score Sn for the player. i.e. $S_n=(N-R_n)A_n$ ... would be a very crude approach that does what you want.

Scoring methods are usually the simplest and most flexible but you need a clear idea of how big a reward you want to give people for the different tasks and how that influences the gameplay - especially if you display a running score during play.

Note: if you have 1000 players and/or they get loads of likes then it is probably going to be difficult for anyone to catch up if they fall behind in the first part. You have to decide how important that is for your competition and adjust the handicapping method accordingly.

Last edited: Feb 11, 2014
5. Feb 12, 2014

### Viktoria

Well I don't know how to best appreciate your complete, clear, easy to learn, polished...reply! Thank you! It was what I came to this forum for!

I didn't think of this! Why the puzzle gets harder?
1. I give them 5min each to clean the area
2. the area is the same for every one
3. after 5min, the top player left 10 items on the ground, second player 5 and third one also 5.
4. I conclude that the sorting order remains (top player won, second is still on second place too).

as you said later:
This is just like I gave the top player 10 items "just fewer obstacles", isn't it?!

you wrote crystal clear! Could be understood even without having a brain at all!

Sure! (may even understand a little more of math :-))

Now as final notes:
I think I can't get the right interpretation of "handicap" you use (not fluent in English). Could you give a new definition please:
1. when you write "handicapping the player"
2. "handicapping method"

P.S.
You know, physically implementing the "more obstacles" strategy is easier: for "less area" I must "paint" the ground in different ways, where for "more obstacles" I just put more items on the area!

Thanks really!

6. Feb 12, 2014

### Simon Bridge

Its the way the English language works.
If there is more chance to make a mistake, then it is harder to avoid a mistake, so you make more mistakes, so it is harder to do well in the task.

It's just not what you meant to say that's all.

That's right - if each player has to clear an area of some objects, then the player with the most objects has the biggest chance of making a mistake (missing an object) so the task is hardest for him.

A handicap is a penalty given to a player to make the game harder.

In real life, a game if considered "fair" if, at the start of the game, there is an equal chance for each player to win. The game is considered "just" (as in "justice") if the player who most deserves to win has the best chance of winning.

So, to make a game "fair" the better players are given some sort of penalty.

The word has come to refer to any "built in" penalty that makes things hard - so having a broken leg would be a handicap to walking.

So when you handicap a player, you make that players task harder.
The "handicap method" is the exact way that the player's game is made harder.

One way you could use this in your game is to score players by the number of seconds they take to clean their areas - using the same number of objects and area for each... then you subtract the number of likes they got in the previous challenge. Lowest overall score wins.

If T is the number of seconds to clean the area, and L is the number of likes, then the score is S=T-L
But there is a problem if it is easy to get likes but hard to clean the areas or something.
In which case you may want to "weight" the different parts so S=aT-bL adjusting a and b to suit. i.e. if you want the number of likes to have a big impact on the score you make b > a.

7. Feb 12, 2014

### Viktoria

Brilliant!

I started to remember back that 1 chapter on game theory that I read in the probability and statistics by John Frond, years ago in school, well, use it or lose it. Perhaps must get a book on game theory or go to corsera and get a course on it (I think I've seen something there).

Thank you for the solution:
I've now clear imagination which way to go to try to provide them fair and just game!

:-)

8. Feb 12, 2014

### Simon Bridge

No worries. Have fun.