rounding considered harmful - e-vote-id€¦ · dbms_random.normal random_nr from table...

Rounding Considered Harmful

Carsten Schurmann presented byPeter Rønne

IT University of CopenhagenDemTech

September 27, 2018

Carsten Schurmann (ITU, DemTech) Rounding Considered Harmful September 27, 2018 1 / 22

Article 76, Danish Election Law

(2) Each number of votes appearing as a result ofthe summation, shall be divided by 1 - 2 - 3 and soon until such number of divisions equivalent to themaximum number of seats expected to beallocated to the party or to the independentcandidate has been performed. The party or theindependent candidate having the highest resultingquotients shall be given the first seat in themulti-member constituency. The second highestquotient entails the second seat and so on and soforth, until all constituency seats in themulti-member constituency have been distributedamong the parties and the independent candidates.If two or more quotients are of equal size, lots shallbe drawn.


Inspecting the Implementation (in use)

select Obj_tillaegsmandater_landsdele

(Landsdel_id, Parti_id, Antal_stemmer,

Antal_kredsmandater, Kvotient_nr, Kvotient, Random_nr)

bulk collect into Temptab

from (select Landsdel_id,

Parti_id,

Antal_stemmer,

Antal_kredsmandater,

V_divisor Kvotient_nr,

round (Antal_stemmer / V_divisor) Kvotient,

dbms_random.Normal Random_nr

from table (P_col_tillaegsman_landsdele));

[Courtesy of Denmark Statistics]


Compensatory Seat Allocation, Parliament Election 2007


Observations

I The implementation of the Danish Seat Allocation algorithm rounds

I It looks like the error incurred by rounding is “neglibile”

But

I The quotients can get rather small (divisors > 30)

I The algorithm may erroneously draw lots

I Consequence: A wrong election result?


Summary of the talk

Do not round!


Overview of this Talk

1 Electoral Systems based on D’Hondt

2 Two Errors Due to Rounding

3 A Close Call

4 A Monte-Carlo Simulation

5 Conclusion


Electoral Systems based on D’Hondt


History

I Voting rule for Party-List Proportional Representation

I Seat Allocation Algorithm used in more than 50 countries

I Introduced by Thomas Jefferson for United States House of Representatives1791

I Introduced in Europe by Victor D’Hondt 1878

I Table computation using divisors 1, 2, 3, 4 . . .

I Estimates how many votes are necessary to win a seat

I Related Voting Rule: Sainte-Lague (different divisors 1,3,5,7 . . . )


D’Hondt and Drawing Lots

Without Rounding

A B C

Tallies 999 500 1501

Divisor 1 999 500 1501Divisor 2 499.5 250 750.5Divisor 3 333 166.6 500.3

Seat allocation (in order)

1. C,A,C,C

With Rounding

A B C

Tallies 999 500 1501

Divisor 1 999 500 1501Divisor 2 500 250 750Divisor 3 333 167 500

Seat allocation (in order)

1. C,A,C,A

2. C,A,C,B

3. C,A,C,C


Genuine and False Ties

Genuine Tie

A tie between two quotients is genuine if it is not caused by numeric effects, such asrounding.

False Tie

A tie between two quotients is false if it is caused by numeric effects, such asrounding.


Two Errors Due to Rounding


Error 1: Incorrect Allocation Order

Situation: More than k seats are to be allocated.For a k-way false tie the probability of allocating seats in the wrong order is

p =1

k!

Error 2: Incorrect Seat Allocation

Situation: Fewer than k seats are to be allocated.For a k-way false tie the probability of allocating seats correctly is

p =1(nk

)


A Close Call


Error 1 Situation Observed, Parliament Election 2007


Hypothetical Error 2 Situation for Seats and 41, 42 (not in table)


A Monte-Carlo Simulation


Research Question

How likely is it that a Rounding Seat Allocation System commits to Error 1 and 2?


A Monto-Carlo Simulation for Error 1

Simulation parameters

I 8 parties

I Vote totals uniformly drawn from 20, 000− 400, 000

I 1, 000, 000 runs

Error 1: 9 out of 10 election results have 2-way false tie

n = 1 2 3 4

E (false n-way tie) 173.17545 0.904989 0.0048 0.000022


A Monto-Carlo Simulation for Error 2

Simulation parameters

I 8 parties

I Vote totals uniformly drawn from 20, 000− 400, 000

I 1, 000, 000 runs

Error 2: The probability of a false 2-way tie is roughly 1.5

n = 1 2 3 4

# observations 984, 089 15, 766 144 1


Conclusion


Conclusion

I Do not round!

I Recall that if b, d 6= 0 the following holds

a

b<

c

dif and only if a · d < c · b

Homework

Check if your country has implemented seat allocation correctly! Fix it, if not.Note: This homework only applies to delegates from d’Hondt and Sainte-Laguecountries


rounding considered harmful - e-vote-id€¦ · dbms_random.normal random_nr from table...

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