NAME
Math::Round::Fair - distribute rounding errors fairly
SYNOPSIS
use Math::Round::Fair 'round_fair';
my $cents = 7;
my @weights = (1, 2, 3, 2, 1);
my @allocation = round_fair($cents, @weights);
print "@allocation\n";
# output will be one of the following:
# 0 1 3 2 1
# 0 2 2 2 1
# 0 2 3 1 1
# 0 2 3 2 0
# 1 1 2 2 1
# 1 1 3 1 1
# 1 1 3 2 0
# 1 2 2 1 1
# 1 2 2 2 0
my @total;
for ( 1..900 ) {
@allocation = round_fair($cents, @weights);
@total[$_] += @allocation[$_] for 0..$#allocation;
}
print "@total\n";
# output will be *near* 700 1400 2100 1400 700, e.g.:
# 698 1411 2096 1418 677
DESCRIPTION
This module provides a single, exportable function, "round_fair", which
allocates an integer value, fairly distributing rounding errors.
"round_fair" rounds up, or down, randomly, where the probability of
rounding up is equal to the fraction to round. For example, "round_fair"
will round 0.5 to 1.0 with a probability of 0.5. It will round 0.3 to
1.0 3 out of 10 times and to zero 7 out of 10 times, on average.
Consider the problem of distributing one indivisible item, for example a
penny, across three evenly weighted accounts, A, B, and C.
Using a naive approach, none of the accounts will receive an allocation
since the allocated portion to each is 1/3 and 1/3 rounds to zero. We
are left with 1 unallocated item.
Another approach is to adjust the basis at each step. We start with 1
item to allocate to 3 accounts. 1/3 rounds to 0, so account A receives
no allocation, and we drop it from consideration. Now, we have 2
accounts and one item to allocate. 1/2 rounds to 1, so we allocate 1
item to account B. Account C gets no allocation since there is nothing
left to allocate.
But what happens if we allocate one item to the same three accounts
10,000 times? Ideally, two accounts should end up with 3,333 items and
one should end up with 3,334 items.
Using the naive approach, all three accounts receive no allocation since
at each round the allocation is 1/3 which rounds to zero. Using the
second method, account A and account C will receive no allocation, and
account B will receive a total allocation of 10,000 items. Account B
always receives the benefit of the rounding errors using the second
method.
"round_fair" uses an algorithm with randomness to ensure a fair
distribution of rounding errors. In our example problem, we start with 1
item to allocate. We calculate account A's share, 1/3. Since it is less
than one item, we give it a 1/3 chance of rounding up (and, therefore, a
2/3 chance of rounding down). It wins the allocation 1/3 of the time.
2/3 of the time we continue to B. We calculate B's allocation as 1/2
(since there are only 2 accounts remaining and one item to allocate). B
rounds up 1/2 of 2/3 (or 1/3) of the time and down 1/2 of 2/3 (or 1/3)
of the time. If neither A nor B rounds up (which occurs 2/3 * 1/2, or
1/3 of the time), C's allocation is calculated as 1/1 since we have one
item to allocate and only one account to allocate it to. So, 1/3 of the
time C receives the benefit of the rounding error. We never end up with
any unallocated items.
This algorithm works for any number of weighted allocations.
round_fair($value, @weights)
Returns a list of integer values that sum to $value where each
return value is a portion of $value allocated by the respective
weights in @weights. The number of return values is equal to the
number of elements in @weights
$value must be an integer.
AUTHOR
Marc Mims
LICENSE
Copyright (c) 2009 Marc Mims
This is free software. You may use it, distributed it, and modify it
under the same terms as Perl itself.