Illegal Subletting and Deterrent Penalties

Illegal subletting of HDB flats should be punished. Severely. It is a violation of the spirit of public housing. The question is, what is the appropriate penalty in monetary terms?

This is the expected number of months of rent that an illegal subletter expects to get:

(1-p^12N)/(1-p)

where p is the probability of not getting caught in a given month and N is the rental life of the unit.

This expression may not quite look right, but I assure you it can be obtained by summing

Sum_{[k=0 to 12N-1]} (k+1) p^k(1-p)

which considers cases where an illegal subletter is caught and stops and

Sum_{[k=12N to ∞]} 12N p^k(1-p)

which corresponds to the case where the illegal subletter is not caught and obtains rentals for every month of the rental life of the unit. [As for why they are equal, I don't have any deep intuition beyond just working it out. I was so skeptical, I did the sum explicitly in Excel and did a comparison. It does check out.]

This, multiplied by the monthly rental, is the minimum fine that should be levied to make illegal subletters indifferent between obeying the law and trying their luck. Raising it a bit more would tip the scales in favour of the law.

Since the rental life of flats tend to be long, for any small probability of getting caught in any given month, the expected number of months of rental accrued is very high. For instance, if the probability of not getting caught in a given month is 0.995, then for a flat with a rental life of 20 years, the probability of not getting caught at all is about 0.3 (and the expected number of months of rent accrued is 139.94).

At a glance:

Expected Months of Rental Collected		Rental Life N (years)
		5	10	15	20
p	0.9975	55.7815	103.784	145.0924	180.6402
	0.995	51.9478	90.4027	118.8694	139.9422
	0.9925	48.4603	79.3076	98.9434	111.4425
	0.99	45.2843	70.062	83.6192	91.0371
	0.9875	42.3889	62.3175	71.6868	76.0916
	0.985	39.7462	55.796	62.277	64.8941
	0.9825	37.3314	50.2742	54.7615	56.3172
	0.98	35.1223	45.5731	48.6828	49.6081
	0.9775	33.0989	41.5482	43.7051	44.2557
	0.975	31.2434	38.083	39.5803	39.9081
	0.9725	29.5396	35.083	36.1233	36.3185
	0.97	27.9731	32.4714	33.1947	33.311
	0.9675	26.5311	30.1855	30.6888	30.7582
	0.965	25.2019	28.174	28.5246	28.5659
	0.9625	23.9751	26.395	26.6392	26.6639
	0.96	22.8412	24.8136	24.9839	24.9986
	0.9575	21.7919	23.4011	23.5199	23.5287
	0.955	20.8194	22.1337	22.2166	22.2219
	0.9525	19.9171	20.9914	21.0493	21.0525
	0.95	19.0786	19.9576	19.998	19.9999

Probability of Non-detection		Rental Life N (years)
		5	10	15	20
p	0.9975	0.8605	0.7405	0.6373	0.5484
	0.995	0.7403	0.548	0.4057	0.3003
	0.9925	0.6365	0.4052	0.2579	0.1642
	0.99	0.5472	0.2994	0.1638	0.0896
	0.9875	0.4701	0.221	0.1039	0.0489
	0.985	0.4038	0.1631	0.0658	0.0266
	0.9825	0.3467	0.1202	0.0417	0.0144
	0.98	0.2976	0.0885	0.0263	7.84E-3
	0.9775	0.2553	0.0652	0.0166	4.25E-3
	0.975	0.2189	0.0479	0.0105	2.30E-3
	0.9725	0.1877	0.0352	6.61E-3	1.24E-3
	0.97	0.1608	0.0259	4.16E-3	6.69E-4
	0.9675	0.1377	0.019	2.61E-3	3.60E-4
	0.965	0.1179	0.0139	1.64E-3	1.93E-4
	0.9625	0.1009	0.0102	1.03E-3	1.04E-4
	0.96	0.0864	7.46E-3	6.44E-4	5.56E-5
	0.9575	0.0738	5.45E-3	4.03E-4	2.97E-5
	0.955	0.0631	3.98E-3	2.52E-4	1.59E-5
	0.9525	0.0539	2.91E-3	1.57E-4	8.46E-6
	0.95	0.0461	2.12E-3	9.78E-5	4.50E-6

To use these tables, one might first make a ballpark estimate of the probability of an illegal subletter never getting caught given he/she illegally rents out part of his/her flat for some number of years. Using the second table, a value for p can be estimated. Then the first table can be used to identify the right ballpark of a deterrent fine.

In any event, all this tells us that since it is very unlikely to ever get caught, the rental that an illegal subletter expects to accrue is extremely high. Therefore, unless there are stiff penalties such as large enough fines or repossession of the flat by HDB, illegal subletters will continue to ply their trade.