Basic Assumptions
To make the math easier, assume that the score of the winning player in every game is P. Obviously a point dumper will be unable to lose every game against a person with a constant score of P, and even if P is the average score of all the winners the numbers are a bit off. But this works well enough to first order.
The Equation
Let the score at the beginning of an N-game losing streak be S0. Sn is the score after N games. Assume that S0 <= 5*P so that the point loss will always be less than 100 points. Thus, every time you lose a game, you lose 20*(S/P) points. In other words,
S1 = S0 - 20*(S0/P) = S0 * (1 - 20/P)
S2 = S1 * (1 - 20/P) = S0 * (1 - 20/P)^2
Sn = S0 * (1 - 20/P)^N
Results
Now we can put in some numbers and reach some conclusions. Suppose a person has 1000 points and wants to get down to 1 point. Also suppose he manages to lose primarily to players with a score of very close to 1000. How many consecutive losses will it take?
(1/1000) = (1 - 20/1000)^N
ln(0.001) = N * ln(0.98)
N = 342 games.
In other words, a new recruit playing primarily against other new recruits would have to lose 342 straight games to get down to 1 point!
Same situation, but now he just wants to get down to 200 points so that when he plays against other people with scores of around 1000 he gets maximum points.
N = ln(0.2) / ln(0.98) = 80 games.
The following graph illustrates how many consecutive games one has to lose to drop one's score from 1000 to the given value (assuming his losses are to people with 1000 points):
As an additional curiosity, suppose that our illicit point dumper is trying to get down from 1000 points to 1 point once again. But somewhere around the 80 game mark (when he has 200 points), he accidentally wins one, bumping him back up to 300 points. How far back has he set himself?
To from from a score of 300 down to 1 takes 283 games. On top of the 81 games he's already played, that makes for a total of 363 losses plus the one accidental victory in the middle. That one victory set him back by 21 games.
Rounding Errors
There is one potential flaw in these calculations. At some point, the points lost in a game drops below 1. I'm really not sure how the algorithm on the board handles that situation. My calculations have assumed that your score is stored as a floating point number rather than an integer and that the points exchanged in a loss can be less than 1. Also, if you always lose an integer number of points, then your score will drop a lot faster if the equations say you should lose 1.55 points and you actually lose 2. What is the score threshold for this to introduce errors if the board algorithm is different from these assumptions?
20*(S/P) < 1
S < P/20
So if your score drops below 50 points (and you're still losing to players with 1000 points), the equations have you losing less than 1 point per game. If you always lose a minimum of 1 point with every loss, that means it will actually take only 49 losses to get from 50 down to 1 as opposed to the 194 losses the equations predict.
Conclusions
I know all of this is probably more math than anyone wants to see on a game site. Still, I've seen suggestions about introducing a different point minimum. I don't know that this math can help with developing a conclusion on that front. However, it does provide some reasonable levels for specific thresholds:
- It takes about 35 losses to get below 500 points. That type of record would be frustrating to a new player but is conceivable.
- After another 35 losses he'll drop below 250 points. I would suggest that players that reach a point level somewhere between 250 and 500 should receive some sort of offer of assistance, whether it be encouragement to check out the Society of Cooks or an offer to play team games with a higher ranking partner to help them figure out what they're doing wrong.
- On the other hand, getting your score down below 100 takes 114 straight losses (starting from a score of 1000), and I think this is a reasonable threshold to trigger an automatic investigation into point dumping. I'm not suggesting banning people that end up with scores that low, I just think they warrant some additional checking to make sure their low score is actually deserved.
<- img courtesy of Zoebear1
