NAME
Math::Round::Fair - distribute rounding errors fairly
SYNOPSIS
use Math::Round::Fair 'round_fair', 'round_adjacent';
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
my @rounded = round_adjacent(0.95, 0.65, 0.41, 0.99);
# @rounded will be one of the following:
# 59% of the time: 1, 1, 0, 1
# 35% of the time: 1, 0, 1, 1
# 5% of the time: 0, 1, 1, 1
# 1% of the time: 1, 1, 1, 0
DESCRIPTION
This module provides two exportable functions, "round_fair", which
allocates an integer value, fairly distributing rounding errors, and
"round_adjacent", which takes a list of real numbers and rounds them up,
or down, to an adjacent integer, fairly. Both functions return a list of
fairly rounded integer values.
"round_fair" and "round_adjacent" round up, or down, randomly, where the
probability of rounding up is equal to the fraction to round. For
example, 0.5 will round to 1.0 with a probability of 0.5. 0.3 will round
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.
The algorithm employed by this module uses 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.
round_adjacent(@input_values)
Returns a list of integer values, each of which is numerically
adjacent to the corresponding element of @input_values, and whose
total is numerically adjacent to the total of @input_values.
The expected value of each output value is equal to the
corresponding element of @input_values (within a small error margin
due to the limited machine precision).
CAVEATS
* A number of in-situ integrity checks are enabled by setting
$ENV{MATH_ROUND_FAIR_DEBUG} before loading "Math::Round::Fair". These
integrity checks increase runtime by approximately one-third. Set
$ENV{MATH_ROUND_FAIR_DEBUG} to 1 to enable integrity checks, 2 for
some extra debug output, 0, or unset to disable the checks. By
default, the integrity checks are disabled.
* The algorithm that satisfies these constraints is not necessarily
unique, and the implementation may change over time.
* Randomness is obtained via calls to rand(). You might want to call
srand() first. The number of invocations to rand() per call may change
in subsequent versions.
* The rounding of each element in the list is *not* independent of the
rounding of the other elements. This is the price that you pay for
guaranteeing that the total is also fair and accurate.
AUTHORS
Marc Mims , Anders Johnson
LICENSE
Copyright (c) 2009-2010 Marc Mims
This is free software. You may use it, distributed it, and modify it
under the same terms as Perl itself.