Strategic complements
Encyclopedia
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 curve
s 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 game
s.
with two players that each have payoff function
, where 
represents the player's own decision, and 
represents the decision of the other player. Assume 
is increasing and concave
in the player's own strategy
. Under these assumptions, the two decisions are strategic complements if an increase in each player's own decision 
raises the marginal payoff 
of the other player. In other words, the decisions are strategic complements if the second derivative 
is positive for 
. Equivalently, this means that the function 
is supermodular
.
On the other hand, the decisions are strategic substitutes if
is negative, that is, if 
is submodular
.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
and game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, 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
Returns to scale
In economics, returns to scale and economies of scale are related terms that describe what happens as the scale of production increases in the long run, when all input levels including physical capital usage are variable...
and/or the demand curve
Demand curve
In economics, the demand curve is the graph depicting the relationship between the price of a certain commodity, and the amount of it that consumers are willing and able to purchase at that given price. It is a graphic representation of a demand schedule...
s for the firms' products have a sufficiently low own-price elasticity
Price elasticity of demand
Price elasticity of demand is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price...
. 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
Russell W. Cooper (economist)
Russell W. Cooper is an American macroeconomist who is the Fred Hofheinz Professor of Economics at the University of Texas, Austin. He has previously held academic positions at Yale University, the University of Iowa, and Boston University. He is best known for his work on coordination games...
and Andrew John, strategic complementarity is the basic property underlying examples of multiple equilibria in coordination game
Coordination game
In game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies...
s.
Calculus formulation
Mathematically, consider a symmetric gameSymmetric 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 the strategies, then a game is symmetric. ...
with two players that each have payoff function
, where represents the player's own decision, and represents the decision of the other player. Assume is increasing and concaveConcave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
in the player's own strategy
. Under these assumptions, the two decisions are strategic complements if an increase in each player's own decision raises the marginal payoff of the other player. In other words, the decisions are strategic complements if the second derivative is positive for . Equivalently, this means that the function is supermodularSupermodular
In mathematics, a functionf\colon R^k \to Ris supermodular iff + f \geq f + ffor all x, y \isin Rk, where x \vee y denotes the componentwise maximum and x \wedge 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...
.
On the other hand, the decisions are strategic substitutes if
is negative, that is, if is submodularSupermodular
In mathematics, a functionf\colon R^k \to Ris supermodular iff + f \geq f + ffor all x, y \isin Rk, where x \vee y denotes the componentwise maximum and x \wedge 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...
.
See also
- SupermodularSupermodularIn mathematics, a functionf\colon R^k \to Ris supermodular iff + f \geq f + ffor all x, y \isin Rk, where x \vee y denotes the componentwise maximum and x \wedge 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...
- Coordination gameCoordination gameIn game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies...
- Coordination failure (economics)Coordination failure (economics)In economics coordination failure is a concept that can explain recessions through the failure of firms and other price setters to coordinate. In an economic system with multiple equilibria, coordination failure occurs when a group of firms could achieve a more desirable equilibrium but fail to...
- Uniqueness or multiplicity of equilibrium
- Multiplier (economics)Multiplier (economics)In economics, the fiscal multiplier is the ratio of a change in national income to the change in government spending that causes it. More generally, the exogenous spending multiplier is the ratio of a change in national income to any autonomous change in spending In economics, the fiscal...