Going from vote share estimates to seat estimates

by Rajeeva Karandikar, Director, Chennai Mathematical Institute.

My previous blog posts (link, link), showed that opinion polls help us predict the vote shares of major parties (or alliances). This leads us to the next question: converting estimates of vote share into estimates of seats. This is much harder than meets the eye. Sampling is not done in all constituencies. In constituencies where we have respondents in our sample, the sample size is not large enough to predict the winner in these constituencies in isolation.

Hence, we have to build a mathematical model of voting behavior. It is widely believed that an individual's identity (caste, religion, economic status, gender, age) plays a role in his/her vote. Moreover, this behavior varies from state to state. So if one were to build a model incorporating these parameters, we will end up having a large number of free parameters, particularly as these correlations are likely to change from state to state. Data on these parameters at the constituency level is not available, as census data is compiled and reported at the district level. Thus, this approach is unlikely to yield a good result.

To get a tractable solution, we do not need to build a model for voting intention of an individual voter- it suffices to build a model for voting behavior of a constituency. One can assume that the socio-economic composition of a constituency does not undergo a major change in 5 years (this is true for most constituencies). We assume that the change in vote share for a given party in a constituency, from the previous election to the current one, is constant across a state, or a smaller geographic sub-region of a larger state. We call this the uniform swing assumption.

If the sample size at the state level, or a sub-region in a large state is adequate, we can estimate these vote shares via a methodologically proper poll.

Then using the uniform swing assumption described above, using actual data from the previous election and estimating vote shares of parties across a state or a sub-region of a state enables us to estimate vote shares of major parties in every constituency.

This is a crude model! In the historical experience, the reality has diverged from the uniform swing assumption. However, it turns out that with some further work, this model yields fairly good estimates of seats at the national level.

Consider a scenario where in one constituency, out of a sample of size 101, Candidate A gets 52 votes while candidate B gets 49 and in an adjacent constituency also on sample size 101, Candidate C gets 59 votes while candidate D gets 42. While we can be fairly confident that C will win, the same cannot be said of A. What is the best case scenario for B? The scenario is that A and B are almost neck-to-neck with B having a slight edge and yet a sample of size 101 shows B to be behind A by 3 votes. The probability of this happening is the same as the probability of seeing 49 or less heads in 101 tosses of a fair coin, which is 0.42. We assign B a winning probability of 0.42 and A a winning probability of 0.58. On the other hand, the probability of 42 or less heads in 101 tosses of a fair coin is 0.06. Thus, we assign a winning probability of 0.06 to candidate D while 0.94 to C.

This analogy can be extended to 3 or more significant candidates. We have been using this for the top three candidates: first the best case scenario for the third candidate, then the best case scenario for the second candidate and the remaining for the first candidate. This needs an assessment of the standard deviation of the vote estimates.

To summarise, based on an opinion poll (or our day-after poll), we obtain statewide vote shares and vote shares in sub-regions of a state and build vote estimates for all major parties in each constituency. Then we convert the vote share estimates in each constituency to predicted win probabilities for the top three candidates. Finally, we add up the probability of wins for a given party across all the 543 constituencies and this yields an estimate of the seats for the party.

This methodology, developed over 15 years ago, has yielded useful seat estimates. Of course, this element also has an errting to seat estimates go in opposite directions, which is lucky. Sometimes, the two errors conspire to go together and give bad results.

From October 2005 onwards, CNN-IBN, CSDS-Lokniti and I have done numerous poll projections. Most of these are based on post poll surveys, but occasionally these are also based on pre-election polls. Here is the listing of all such occasions: a comparison of what we said and what happened. I am giving vote share estimates and my seat estimate corresponding to that and the actual vote share and seats.

BIHAR(October, 2005)

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
JDU-BJP	36	37	127-137	147
RJD+	31	31	72-80	65
Others	33	32	29-39	31

ASSAM 2006

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
Congress	31	31	52-60	53
BJP	11	12	10-15	10
AGP	22	20	25-31	24
Others	36	37	26-35	39

TAMIL NADU 2006

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
AIADMK+	35	40	64-74	69
DMK+	45	45	157-167	163
DMDK	10	8	2-6	1
Others	10	7	-	1

KERALA 2006

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
LDF	51	49	107-117	98
UDF	41	43	25-31	42
Others	8	8	0-1	0

WEST BENGAL 2006

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
LF	53	50	230-240	235
INC	16	15	17-23	24
TMC+	27	29	32-40	31

PUNJAB 2007

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
SAD-BJP	41	45	50-60	68
Congress	41	41	50-60	44
Others	18	14	3-9	5

UTTARAKHAND 2007

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
Congress	31	30	21-27	21
BJP	34	32	33-39	35
Others	35	38	8-12	14

UTTAR PRADESH 2007

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
SP	25	25	99-111	97
BSP	29	30	152-168	206
BJP+	22	18	80-90	52
Congress	11	9	25-33	26
Others	13	18	21-27	26

GUJARAT 2007

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
BJP	47	49	92-100	117
Congress	42	39	77-85	62
Others	11	12	3-7	3

KARNATAKA 2008

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
BJP	30	34	79	110
Congress	35	35	86	80
JDS	21	19	45	28
Others	14	12	14	6

LOK SABHA 2009

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
UPA	36	36	210-225	262
NDA	28	24	180-195	159
Left	-	8	30-40	24
BSP	-	6	24-32	21
Others	-	26	-	77

BIHAR 2010

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
JDU-BJP	46	39	185-201	206
Congress	9	8	6-12	4
RJD-LJP	27	26	22-32	25
Others	18	27	9-19	8

ASSAM 2011

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
Congress	36	39	64-72	78
BJP	9	11	7-11	5
AGP	18	16	16-22	10
AIUDF	13	13	11-17	18
Others	24	21	12-20	15

KERALA 2011

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
LDF	36	45	69-77	68
UDF	45	46	63-71	72
Others	9	9	0	0

TAMIL NADU 2011

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
DMK+	44	39	102-114	31
AIDMK+	46	52	120-132	203
BJP Front	3	2	-	-
Others	7	7	-	-

WEST BENGAL 2011

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
Left	40	41	60-72	62
TMC+	50	48	222-234	227
Others	10	11	-	5

UTTARAKHAND 2012

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
Congress	39	34	31-41	32
BJP	32	33	22-32	31

PUNJAB 2012

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
SAD+BJP	41	42	51-63	68
Congress	40	40	48-60	46

MANIPUR 2012

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
Congress	30	42	24-32	42
TMC	14	17	7-13	7

UTTAR PRADESH 2012

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
SP	34	29	232-250	224
BSP	24	26	65-79	80
BJP+	14	15	36-44	47
Congress	12	12	28-38	28

GUJARAT 2012

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
BJP	48	48	129-141	116
Congress+	36	39	37-45	60
Others	16	13	4-10	6

HIMANCHAL PRADESH 2012

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
Congress	41	43	29-35	36
BJP	40	38	29-35	26

KARNATAKA 2013

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
BJP	23	20	39-49	40
Congress	37	37	117-129	122
JDS	20	20	34-44	40
Others	20	23	14-22	21

MADHYA PRADESH 2013

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
BJP	41	45	136-146	165
Congress	35	36	67-77	58
Others	24	19	13-21	7

RAJASTHAN 2013

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
BJP	43	45	126-136	162
Congress	33	33	49-57	21
Others	24	22	12-20	16

CHHATISGARH 2013

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
BJP	42	41	45-55	49
Congress+	38	40	32-40	39
Others	20	19	7-13	2

DELHI 2013

	Vote Estimate	Vote Actual	Seat Estimate	Seat Actual
BJP	33	34	32-42	31
Congress	23	24	9-17	8
AAP	27	30	13-21	28
Others	17	12	1-5	3