Supermodular function explained
In mathematics, a supermodular function is a function on a lattice that, informally, has the property of being characterized by "increasing differences." Seen from the point of set functions, this can also be viewed as a relationship of "increasing returns", where adding more elements to a subset increases its valuation. In economics, supermodular functions are often used as a formal expression of complementarity in preferences among goods. Supermodular functions are studied and have applications in game theory, economics, lattice theory, combinatorial optimization, and machine learning.
Definition
Let
be a
lattice. A real-valued function
is called
supermodular if
f(x\veey)+f(x\wedgey)\geqf(x)+f(y)
for all
.
[1] If the inequality is strict, then
is
strictly supermodular on
. If
is (strictly) supermodular then
f is called (
strictly) submodular. A function that is both submodular and supermodular is called
modular. This corresponds to the inequality being changed to an equality.
We can also define supermodular functions where the underlying lattice is the vector space
. Then the function
is
supermodular if
f(x\uparrowy)+f(x\downarrowy)\geqf(x)+f(y)
for all
,
, where
denotes the componentwise maximum and
the componentwise minimum of
and
.
If f is twice continuously differentiable, then supermodularity is equivalent to the condition[2]
| \partial2f |
| \partialzi\partialzj |
\geq0foralli ≠ j.
Supermodularity in economics and game theory
The concept of supermodularity is used in the social sciences to analyze how one agent's decision affects the incentives of others.
Consider a symmetric game with a smooth payoff function
defined over actions
of two or more players
. Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval:
. In this context, supermodularity of
implies that an increase in player
's choice
increases the marginal payoff
of action
for all other players
. That is, if any player
chooses a higher
, all other players
have an incentive to raise their choices
too. Following the terminology of Bulow,
Geanakoplos, and
Klemperer (1985), economists call this situation
strategic complementarity, because players' strategies are complements to each other.
[3] This is the basic property underlying examples of multiple equilibria in
coordination games.
[4] The opposite case of supermodularity of
, called submodularity, corresponds to the situation of
strategic substitutability. An increase in
lowers the marginal payoff to all other player's choices
, so strategies are substitutes. That is, if
chooses a higher
, other players have an incentive to pick a
lower
.
For example, Bulow et al. consider the interactions of many imperfectly competitive firms. When an increase in output by one firm raises the marginal revenues of the other firms, production decisions are strategic complements. When an increase in output by one firm lowers the marginal revenues of the other firms, production decisions are strategic substitutes.
A supermodular utility function is often related to complementary goods. However, this view is disputed.[5]
Supermodular set functions
Supermodularity can also be defined for set functions, which are functions defined over subsets of a larger set. Many properties of submodular set functions can be rephrased to apply to supermodular set functions.
Intuitively, a supermodular function over a set of subsets demonstrates "increasing returns". This means that if each subset is assigned a real number that corresponds to its value, the value of a subset will always be less than the value of a larger subset which contains it. Alternatively, this means that as we add elements to a set, we increase its value.
Definition
Let
be a finite set. A set function
is
supermodular if it satifies the following (equivalent) conditions:
f(A)+f(B)\leqf(A\capB)+f(A\cupB)
for all
.
f(A\cup\{v\})-f(A)\leqf(B\cup\{v\})-f(B)
for all
, where
.
A set function
is submodular if
is supermodular, and modular if it is both supermodular and submodular.
Additional Facts
is modular and
is submodular, then
is a supermodular function.
- A non-negative supermodular function is also a superadditive function.
Optimization Techniques
There are specialized techniques for optimizing submodular functions. Theory and enumeration algorithms for finding local and global maxima (minima) of submodular (supermodular) functions can be found in "Maximization of submodular functions: Theory and enumeration algorithms", B. Goldengorin.[6]
See also
Notes and References
- Book: Supermodularity and complementarity . 1998 . Princeton University Press . 978-0-691-03244-3 . Topkis . Donald M. . Frontiers of economic research . Princeton, N.J.
- The equivalence between the definition of supermodularity and its calculus formulation is sometimes called Topkis' characterization theorem. See Paul . Milgrom . John . Roberts . 1990 . Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities . . 58 . 6 . 1255–1277 [p. 1261] . 2938316 . 10.2307/2938316 .
- Jeremy I. . Bulow . John D. . Geanakoplos . Paul D. . Klemperer . 1985 . Multimarket Oligopoly: Strategic Substitutes and Complements . . 93 . 3 . 488–511 . 10.1086/261312 . 10.1.1.541.2368 . 154872708 .
- Russell . Cooper . Andrew . John . 1988 . Coordinating coordination failures in Keynesian models . . 103 . 3 . 441–463 . 10.2307/1885539 . 1885539 .
- 10.1016/j.jet.2008.06.004 . Supermodularity and preferences . . 144 . 3 . 1004 . 2009 . Chambers . Christopher P. . Echenique . Federico . 10.1.1.122.6861 .
- Goldengorin . Boris . 2009-10-01 . Maximization of submodular functions: Theory and enumeration algorithms . European Journal of Operational Research . en . 198 . 1 . 102–112 . 10.1016/j.ejor.2008.08.022 . 0377-2217. subscription .
