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 see what is meant by 'reinforce' or 'offset', consider a situation in which the players all have similar choices to make, as in the paper of Bulow et al., where the players are all imperfectly competitive firms that must each decide how much to produce. Then the production decisions are strategic complements if an increase in the production of one firm increases the marginal revenues of the others, because that gives the others 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.

According to Russell Cooper and Andrew John, strategic complementarity is the basic property underlying examples of multiple equilibria in coordination games.

Calculus formulation

Mathematically, consider a symmetric game is increasing and concave is supermodular.

is submodular.

Example

is given by

in market 2 is given by

, we must have

Using vector calculus, geometric algebra, or differential geometry can be calculated in terms of second partial derivatives of the payoff functions:

,

This, as price is increased in market 1, Firm x sells more in market 1 and less in market 2, while firm y sells more in market 2. If the Cournot equilibrium of this model is calculated explicitly, we find

Supermodular games

A game with strategic complements is also called a supermodular game. This was first formalized by Topkis, and studied by Vives. There are efficient algorithms for finding pure-strategy Nash equilibria in such games.

Supermodular games are a particularly important and well-behaved class of games because the principle of strategic complementarity ensures that players' best response functions are monotonic (non-decreasing).

See also

• Supermodular

• Coordination game

• Coordination failure (economics)

• Uniqueness or multiplicity of equilibrium

• Multiplier (economics)

• Aggregative game

References