In
mathematics, a function
:
is supermodular if
:
for all
,
, where
denotes the componentwise maximum and
the componentwise minimum of
and
.
If −''f'' is supermodular then ''f'' is called submodular, and if the inequality is changed to an equality the function is modular.
If ''f'' is twice continuously differentiable, then supermodularity is equivalent to the condition
:
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
In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If one can change the identities of the players without changing the payoff to ...
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: