Strategic complements

In economics and game theory, the decisions of two or more players are called strategic complements if they mutually reinforce one another, and they are called strategic substitutes if they mutually offset one another. These terms were originally coined by Bulow, Geanakoplos, and Klemperer (1985).

To clarify what is meant by 'reinforce' or 'offset', it is helpful to consider a situation in which the players all have similar choices to make, as in the paper of Bulow et al., where all the players are imperfectly competitive firms which are each choosing how much to produce. In that context, the production decisions are strategic complements if an increase in the production of one firm increases the marginal revenues of the others, because in that case the others will have an incentive to produce more too. This tends to be the case if there are sufficiently strong aggregate increasing returns to scale and/or the demand curves for the firms' products have a sufficiently low own-price elasticity. On the other hand, the production decisions are strategic substitutes if an increase in one firm's output decreases the marginal revenues of the others, giving them an incentive to produce less.

Calculus formulation
Mathematically, consider a symmetric game with two players that each have payoff function $$\Pi(x_i, x_j)$$, where $$x_i$$ represents the player's own decision, and $$x_j$$ represents the decision of the other player. Assume $$\Pi$$ is increasing and concave in the player's own strategy $$x_i$$. Under these assumptions, the two decisions are strategic complements if an increase in each player's own decision $$x_i$$ raises the marginal payoff $$\frac{d\Pi}{dx_j}$$ of the other player. In other words, the decisions are strategic complements if the second derivative $$\frac{d^2\Pi}{dx_idx_j}$$ is positive for $$i \neq j$$. Equivalently, this means that the function $$\Pi$$ is supermodular.

On the other hand, the decisions are strategic substitutes if $$\frac{d^2\Pi}{dx_idx_j}$$ is negative, that is, if $$\Pi$$ is submodular.