Supermodular

In mathematics, a function

is supermodular if

for all x, y ${\displaystyle \in }$ R^k, where x ${\displaystyle \uparrow }$ y denotes the componentwise maximum and x ${\displaystyle \downarrow }$ y the componentwise minimum of x and y.

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

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 ${\displaystyle \,f\,}$ defined over actions ${\displaystyle \,z_{i}\,}$ of two or more players ${\displaystyle i\in {1,2,\dots ,N}}$. Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: ${\displaystyle z_{i}\in [a,b]}$. In this context, supermodularity of ${\displaystyle \,f\,}$ implies that an increase in player ${\displaystyle \,i\,}$'s choice ${\displaystyle \,z_{i}\,}$ increases the marginal payoff ${\displaystyle df/dz_{j}}$ of action ${\displaystyle \,z_{j}\,}$ for all other players ${\displaystyle \,j\,}$. That is, if any player ${\displaystyle \,i\,}$ chooses a higher ${\displaystyle \,z_{i}\,}$, all other players ${\displaystyle \,j\,}$ have an incentive to raise their choices ${\displaystyle \,z_{j}\,}$ too. Following the terminology of Bulow, Geanakoplos, and Klemperer (1985), economists call this situation strategic complementarity, because players' strategies are complements to each other. This is the basic property underlying examples of multiple equilibria in coordination games.

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