Editing Subadditivity (section)

==Examples in various domains==

===Entropy===

[[Entropy]] plays a fundamental role in [[information theory]] and [[statistical physics]], as well as in [[quantum mechanics]] in a generalized formulation due to [[von Neumann entropy|von Neumann]].
Entropy appears always as a subadditive quantity in all of its formulations, meaning the entropy of a supersystem or a set union of random variables is always less or equal than the sum of the entropies of its individual components.
Additionally, entropy in physics satisfies several more strict inequalities such as the Strong Subadditivity of Entropy in classical statistical mechanics and its [[Strong subadditivity of quantum entropy|quantum analog]].

===Economics===

Subadditivity is an essential property of some particular [[Cost curve|cost function]]s. It is, generally, a [[necessary and sufficient condition]] for the verification of a [[natural monopoly]]. It implies that production from only one firm is socially less expensive (in terms of average costs) than production of a fraction of the original quantity by an equal number of firms.

[[Economies of scale]] are represented by subadditive [[average cost]] functions.

Except in the case of complementary goods, the price of goods (as a function of quantity) must be subadditive. Otherwise, if the sum of the cost of two items is cheaper than the cost of the bundle of two of them together, then nobody would ever buy the bundle, effectively causing the price of the bundle to "become" the sum of the prices of the two separate items.  Thus proving that it is not a sufficient condition for a natural monopoly; since the unit of exchange may not be the actual cost of an item.  This situation is familiar to everyone in the political arena where some minority asserts that the loss of some particular freedom at some particular level of government means that many governments are better; whereas the majority assert that there is some other correct unit of cost.{{citation needed|date=October 2015}}

===Finance===
Subadditivity is one of the desirable properties of [[coherent risk measure]]s in [[risk management]].<ref name="Rau-Bredow">{{Cite journal | doi = 10.3390/risks7030091| title = Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures| year = 2019| last1 = Rau-Bredow | first1 = H. | journal = Risks| volume = 7| issue = 3|pages = 91| doi-access = free| hdl = 10419/257929| hdl-access = free}}</ref> The economic intuition behind risk measure subadditivity is that a portfolio risk exposure should, at worst, simply equal the sum of the risk exposures of the individual positions that compose the portfolio. The lack of subadditivity is one of the main critiques of [[Value at risk|VaR]] models which do not rely on the assumption of [[Normal distribution|normality]] of risk factors. The Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio <math> V </math> at the confidence level <math> 1-p </math> is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss,
<math display="block"> \text{VaR}_p \equiv z_{p}\sigma_{\Delta V} =  z_{p}\sqrt{\sigma_x^2+\sigma_y^2+2\rho_{xy}\sigma_x \sigma_y} </math>
where <math> z_p </math> is the inverse of the normal [[cumulative distribution function]] at probability level <math> p </math>, <math> \sigma_x^2,\sigma_y^2 </math> are the individual positions returns variances and <math> \rho_{xy} </math> is the [[Pearson correlation coefficient|linear correlation measure]] between the two individual positions returns. Since [[variance]] is always positive,
<math display="block"> \sqrt{\sigma_x^2+\sigma_y^2+2\rho_{xy}\sigma_x \sigma_y} \leq \sigma_x + \sigma_y  </math>
Thus the Gaussian VaR is subadditive for any value of <math> \rho_{xy} \in [-1,1] </math> and, in particular, it equals the sum of the individual risk exposures when <math> \rho_{xy}=1 </math> which is the case of no diversification effects on portfolio risk.

===Thermodynamics===
Subadditivity occurs in the thermodynamic properties of non-[[ideal solution]]s and mixtures like the excess [[molar volume]] and [[heat of mixing]] or excess enthalpy.

===Combinatorics on words===
A factorial [[Formal language|language]] <math>L</math> is one where if a [[String (computer science)|word]] is in <math>L</math>, then all [[Substring|factors]] of that word are also in <math>L</math>.   In [[combinatorics on words]], a common problem is to determine the number <math>A(n)</math> of length-<math>n</math> words in a factorial language.  Clearly <math>A(m+n) \leq A(m)A(n)</math>, so <math>\log A(n)</math> is subadditive, and hence Fekete's lemma can be used to estimate the growth of <math>A(n)</math>.<ref name=shur>{{cite journal|last=Shur|first=Arseny|title=Growth properties of power-free languages|journal=Computer Science Review|date=2012|volume=6|issue=5–6|pages=187–208|doi=10.1016/j.cosrev.2012.09.001}}</ref>

For every <math>k \geq 1</math>, sample two strings of length <math>n</math> uniformly at random on the alphabet <math>1, 2, ..., k</math>. The expected length of the [[longest common subsequence]] is a ''super''-additive function of <math>n</math>, and thus there exists a number <math>\gamma_k \geq 0</math>, such that the expected length grows as <math>\sim \gamma_k n</math>. By checking the case with <math>n=1</math>, we easily have <math>\frac 1k < \gamma_k \leq 1</math>. The exact value of even <math>\gamma_2</math>, however, is only known to be between 0.788 and 0.827.<ref>{{Cite journal |last=Lueker |first=George S. |date=May 2009 |title=Improved bounds on the average length of longest common subsequences |url=https://dl.acm.org/doi/10.1145/1516512.1516519 |journal=Journal of the ACM |language=en |volume=56 |issue=3 |pages=1–38 |doi=10.1145/1516512.1516519 |s2cid=7232681 |issn=0004-5411|url-access=subscription }}</ref>