Double counting
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Double counting in mathematics
Bijective proof
In combinatorics, double counting, also called two-way counting, is a proof technique that involves counting the size of a set in two ways in order to show that the two resulting expressions for the size of the set are equal. We describe a finite set X from two perspectives leading to two distinct expressions. Through the two perspectives, we demonstrate that each is to equal |X|.
Such a proof is sometimes called a bijective proof because the process necessarily provides a bijective mapping between two sets. Each of the two sets is closely related to its respective expression. This free bijective mapping may very well be non-trivial; in certain theorems, the bijective mapping is more relevant than the expressions' equivalence.
Example
For instance, consider the number of ways in which a committee can be formed from a total of n people, with from 0 through to n members:
Method 1: There are two possibilities for each person - they may or may not be on the committee. Therefore there are a total of 2 × 2 × ... × 2 (n times) = 2n possibilities.
Method 2: The size of the committee must be some number between 0 and n. The number of ways in which a committee of r people can be formed from a total of n people is nCr (this is a well known result; see binomial coefficient). Therefore the total number of ways is nCr summed over r = 0, 1, 2, ... n.
Equating the two expressions gives
- <math>\sum_{r=0}^n {}^nC_r = 2^n</math>
Handshaking lemma
An example of a theorem that is commonly proved with a double counting argument is the theorem that every graph contains an even number of vertices of odd degree. Let d(v) be the degree of vertex v. Every edge of the graph is incident to exactly two vertices, so by counting the number of edges incident to each vertex, we have counted each edge exactly twice. Therefore
- <math>\sum d(v) = 2e</math>
where e is the number of edges. The sum of the degrees of the vertices is therefore an even number, which could not happen if an odd number of the vertices had odd degree.
Another meaning
Double counting is also a fallacy in which, when counting events or occurrences in probability or in other areas, a solution counts events two or more times, resulting in an erroneous number of events or occurrences which is higher than the true result. For example, what is the probability of seeing a 5 when throwing a pair of dice? The erroneous argument goes as follows: The first die shows a 5 with probability 1/6; the second die shows a 5 with probability 1/6; therefore the probability of seeing a 5 is 1/6 + 1/6 = 1/3 = 12/36. However, the correct answer is 11/36, because the erroneous argument has double-counted the event where both dice show fives.
See also
Double counting in social accounting
Double-counting is used to refer not simply to a math problem in combinatorics, but also to a conceptual problem in social accounting practice, when the attempt is made to estimate the new value added by Gross Output, or the value of total investments.
What is the problem?
In the case of a small individual business, it is unlikely that an expenditure of funds, an input or output, or an income from production will be counted twice. If it happens, that's usually just bad accounting (a math error), or else a case of fraud.
But things are more complicated when we aggregate the accounts of many enterprises, households and government agencies ("institutional units" or transactors in social accounting language). Here, a conceptual problem arises.
The basic reason is that the income of one institutional unit is the expenditure of another, and the input of one institutional unit is the output of another.
If therefore we want to measure the total value-added by all institutional units, we need to devise a consistent system for grossing and netting the incomes and outlays of all units. Lacking such a system, we would end up double-counting incomes and expenditures of interacting units, exaggerating the quantity of value-added or investments.
Value theory
The system of gross and netting actually used, is ultimately based on a value theory, which specifies what may generally count as:
- comparable value (value equivalence)
- value used up
- conserved value
- transferred value
- newly created value
In other words, we cannot relate, group and aggregate prices in different ways without making some value-based assumptions that enable valid comparisons. Without those value assumptions, the aggregates themselves would be meaningless. Thus, when economists focus on market-prices, value assumptions are always in the back of their mind, even if they are not aware of that, and regard value theory as metaphysical.
Neo-classical economics rejects any value theory other than subjective marginal utility preferences, but social accountants who provide the empirical data for their economic science cannot regard value as simply subjective. Otherwise, anything can count as anything, according to subjective preference, and any old computation is permissible.
Counting units
Once the principles of the value theory are established, categories and counting units can be exactly and logically defined, as a basis for mathematical operations to aggregate the flows of incomes and expenditures. All flows can then be allocated to their appropriate category, without counting the same flow several times.
In fact, the value theory applied in national accounts is nowadays strongly influenced by the valuation principles of ordinary business accounts and the prevailing social relations governing economic exchange, often fixed by law. Thus, for example, it is argued that no new value can result from a unilateral transfer of funds, i.e. where funds are provided without anything being provided in return.
The implicit assumption made in national accounts, is that the account at the macro-level must be similar to that at the micro-level. Economic relations are regarded as broadly the same at the micro-level and the macro-level. An individual business buys and uses up inputs and produces outputs for sale; it has costs and revenues. Thus, in social accounting all transactors are treated in a similar way ("as if" they were a business). The accounts can be criticised for being eclectic in some ways, but that is not necessarily a problem; the aim of the exercise is to identify and categorise all flows, and the user can then reaggregate them in different ways.
Persistent double-counting problems
However, even if a consistent system of accounting rules is devised that conceptually eliminates double-counting, double-counting may technically still occur to some extent.
The first and most obvious reason is that, in actual accounting practice, boundary problems arise, because a flow of expenditures might be interpreted in different ways, from an accounting point of view. Sometimes, it will not be altogether clear which category a flow of expenditure belongs to exactly, it may not "fit" exactly into a category, or, it is technically impossible to separate out different flows in financial data, in such a way that is required by the social accounting system. This may mean that a flow is, in part or as a whole, inadvertently counted twice, because of difficulties with the data sources.
We might, for example, be easily able to identify an expenditure, yet this expenditure may not tally with the corresponding income that should exist, insofar as we can identify it (or vice versa). In that case, we have to make some assumptions or imputations based on what we do know, or can observe. Yet, some statistical discrepancies may remain.
Another reason has to do with the complexities of trade, in particular trade in services and international trade. Not only can it be difficult to correctly identify, survey and allocate particular financial incomes and expenditures, but also revaluations of assets occur, creating problems of how to value goods and services as such.
At the highest level, due to the expansion of foreign trade, a fraction of local value-added may consist of the local inflation of foreign-produced value-added, simply because imported foreign products are resold locally, at inflated prices, without any corresponding additional local production occurring. This may not necessarily create problems of double-counting locally, but if we want to estimate world GDP, we may face double-counting problems of some kind.
Does the World Bank double-count?
Interestingly, while world Gross Domestic Product (GDP) and world Gross National Income (GNI) are conceptually identical values for social accountants, the World Bank valuation of each of these aggregates differs by about one trillion US dollars, and the difference grows year by year.
The reason here however is not a double-counting error, but that different valuations of national currencies are used. Thus, the World Bank applies "ppp" (purchasing power parity) valuations to GDP, but an "Atlas Method" to estimate GNI. The argument is apparently that if world GDP is treated as an income, it will shrink (the World Bank cites 2004 GNI of $39.8 trillion and a GDP of $40.9 trillion, a discrepancy of $1.1 trillion).
One result of these different valuation methods, critics point out, is that it becomes impossible to compare GDP and GNI internationally with respect to the net international transfer of factor-income, which is excluded from national GDP, but included in national GNI.