I’m negotiating an agreement with someone and need to device a formula for reducing the person’s compensation if he fails to meet certain deadlines. I’m looking for a formula that never reaches zero, and the best I’ve been able to devise is X-(X(Z/(Z+X)))) where X is the value of compensation, Y is the number of days allotted and Z is the number of extra days required. However, that calculation imposes a much higher penalty on the first missed day than the fifth. Ideally, the the penalty should increase over time without the possibility of it equalling or exceeding the total amount of compensation. Is that even possible?

I can understand the logic of the Birthday Paradox, but the mathematics are beyond my grasp.

You need to decide whether this relationship is linear or something else. Do you want the penalty to be equally severe, more severe, or less severe the later that the work becomes? Do you want a dollar amount cap on the penalty (i.e., have a minimum guaranteed compensation no matter what)?

I’ve set up an Excel workbook with three different formulas and charts that show how it plays out.

I think you *do * want compensation to approach zero as Z approaches infinity, or some large, unacceptably-late number. However, I have provided formulas that allow a limit of the penalty to some percentage of X. In these examples, the maximum penalty is reached when Z = Y, but you can tune the parameters to get it how you want it.

You can download it by right-clicking on this link.

The point is that the formula can be tweaked to give you something that will decay the compensation from 100% to 0 in a finite time, and that gets more severe as the project becomes later (up to a point, but after that they’ve lost enough that big declines aren’t possible anyway).

Say that you wanted to give a maximum of 60 days after the deadline. If the original compensation is C, the compensation n days after the deadline is C*(344/345)[sup]n(n + 1)/2[/sup] if n > 2 (and just C when n = 1). That gives a breakdown like this:

```
Days Compensation
Late Multiplier
1 1.00
2 1.00
3 0.99
4 0.98
5 0.97
6 0.96
7 0.94
8 0.92
9 0.90
10 0.88
11 0.85
12 0.83
13 0.80
14 0.77
15 0.74
16 0.71
17 0.67
18 0.64
19 0.61
20 0.58
21 0.54
22 0.51
23 0.48
24 0.45
25 0.42
26 0.39
27 0.36
28 0.33
29 0.31
30 0.28
31 0.26
32 0.24
33 0.22
34 0.20
35 0.18
36 0.16
37 0.14
38 0.13
39 0.12
40 0.10
41 0.09
42 0.08
43 0.07
44 0.06
45 0.06
46 0.05
47 0.04
48 0.04
49 0.03
50 0.03
51 0.02
52 0.02
53 0.02
54 0.02
55 0.01
56 0.01
57 0.01
58 0.01
59 0.01
60 0.01
```

Basing the penalty on actual costs or damages incurred by you due to the late performance will probably be fair. The wikipepdia entry on ‘Liquidated Damages’ is probably a decent place to start.

Thanks, **CookingWithGas** and **ultrafilter**. Your work is enormously helpful.

Ultrafilter, your formula makes sense, except I’m unclear where the numbers (344/345) come from.

You have not said anything about the nature of this contract. I suggest that you investigate further about liquidated damages and what courts consider reasonable. Face it, if you are looking at actually implementing such a contract clause, the relationship at that point is probably adversarial. It might end up in court, so be certain it will hold up in court.

It looks like that is just a constant he chose. If you choose a smaller one, such as 1/2, his compensation will decrease more rapidly with lateness. If you choose a bigger one, such as 999/1000, the decrease will happen slower.

This is the reason. 344/345 was the largest value which caused the compensation on day 60 to round to .01 and on day 61 to 0.