Supermodular function

In mathematics, a function In mathematics, a function is supermodular if for all x {displaystyle x} , y ∈ R k {displaystyle yin mathbb {R} ^{k}} , where x ↑ y {displaystyle xuparrow y} denotes the componentwise maximum and x ↓ y {displaystyle xdownarrow y} the componentwise minimum of x {displaystyle x} and y {displaystyle 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 f {displaystyle ,f} defined over actions z i {displaystyle ,z_{i}} of two or more players i ∈ 1 , 2 , … , N {displaystyle iin {1,2,dots ,N}} . Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: z i ∈ [ a , b ] {displaystyle z_{i}in } . In this context, supermodularity of f {displaystyle ,f} implies that an increase in player i {displaystyle ,i} 's choice z i {displaystyle ,z_{i}} increases the marginal payoff d f / d z j {displaystyle df/dz_{j}} of action z j {displaystyle ,z_{j}} for all other players j {displaystyle ,j} . That is, if any player i {displaystyle ,i} chooses a higher z i {displaystyle ,z_{i}} , all other players j {displaystyle ,j} have an incentive to raise their choices z j {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. The opposite case of submodularity of f {displaystyle ,f} corresponds to the situation of strategic substitutability. An increase in z i {displaystyle ,z_{i}} lowers the marginal payoff to all other player's choices z j {displaystyle ,z_{j}} , so strategies are substitutes. That is, if i {displaystyle ,i} chooses a higher z i {displaystyle ,z_{i}} , other players have an incentive to pick a lower z j {displaystyle ,z_{j}} . 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.