Strategic Complements and Strategic Substitutes

In economics and game theory, the decisions of two or more players are called strategic complements if they reinforce one another and strategic substitutes if they offset one another. When decisions are strategic complements, a player who sees a rival do more of something wants to do more of it too. When they are strategic substitutes, a player who sees a rival do more wants to do less.

The terms were coined by Jeremy Bulow, John Geanakoplos and Paul Klemperer in their 1985 paper "Multimarket Oligopoly: Strategic Substitutes and Complements," published in the Journal of Political Economy. The idea itself is older and the mathematics behind it has a separate history in operations research, but the vocabulary they introduced spread quickly and is now standard throughout economics.

The distinction sounds simple, but it is one of the most useful classifications in applied game theory. Whether actions are complements or substitutes determines whether a firm benefits from committing to be aggressive or to be soft, whether a government should subsidize or tax its exporters, whether an economy can get stuck in a bad equilibrium, and whether simple learning processes converge. It also connects to a well-developed body of mathematics, the theory of supermodular games, which guarantees that games with strategic complements have equilibria with especially tidy properties.

This article explains the definitions, the history, the core mathematics in plain notation, several fully worked examples, the main applications, and common misunderstandings.

The basic idea

Imagine two coffee shops on the same street that each choose a price. If the shop across the street raises its price, some of its customers drift over to you. You now face stronger demand, and the best response is usually to raise your own price a little too. A higher price by your rival makes a higher price more attractive for you. Prices here are strategic complements.

Now imagine two farms choosing how much wheat to grow for the same market. If the other farm grows more, the market price falls, and each extra bushel you grow is worth less. The best response is to grow a bit less. More output by your rival makes more output less attractive for you. Quantities here are strategic substitutes.

The test is always the same: how does a rival's action change the marginal benefit of my own action? If a rival's increase raises my marginal benefit, the actions are complements. If it lowers my marginal benefit, they are substitutes.

A useful shortcut is the slope of the best-response function, the rule that tells a player the best action given what the rival does. Under strategic complements, best responses slope upward: when the rival does more, so do I. Under strategic substitutes, best responses slope downward.

History

Before the vocabulary

Economists studied oligopoly long before the terms existed. Augustin Cournot's model of quantity competition (1838) and Joseph Bertrand's critique emphasizing price competition (1883) are the two classic cases, and they turn out to be the textbook examples of strategic substitutes and complements respectively. What was missing was a general way to say why strategic moves that help a firm in one model hurt it in the other.

Fudenberg and Tirole, 1984

Drew Fudenberg and Jean Tirole's short 1984 paper "The Fat-Cat Effect, the Puppy-Dog Ploy, and the Lean and Hungry Look," in the American Economic Review Papers and Proceedings (volume 74, number 2, pages 361 to 366), analyzed how a firm should invest before competing with a rival. Their colorful labels described different strategic investment patterns, and which one is optimal depends on whether the later competition involves actions that are complements or substitutes. Their taxonomy is described in detail later in this article.

Bulow, Geanakoplos and Klemperer, 1985

Bulow, Geanakoplos and Klemperer's paper (Journal of Political Economy, volume 93, number 3, pages 488 to 511) studied firms that operate in more than one market. Its starting observation, as described in the working-paper abstract circulated through the Cowles Foundation, is that actions a firm takes in one market can affect its profitability in another market even without any joint economies or diseconomies of production. The reason is that a move in one market can change the firm's marginal costs, and that changes the behavior of competitors in the second market.

Whether such cross-market effects help or hurt the firm depends on whether more "aggressive" play by one firm in a market, such as a lower price or higher quantity, lowers or raises competing firms' marginal profitability in that market. The paper named these two cases strategic substitutes and strategic complements, and noted that both are compatible with either price competition or quantity competition. That last point matters: the terms are about the sign of the interaction, not about whether firms choose prices or quantities. The paper applied the ideas to market entry, rational retaliation as an entry barrier, international trade and learning curves, and observed that with strategic complements firms may strategically underinvest in fixed costs.

Cooper and John, 1988

Russell Cooper and Andrew John's "Coordinating Coordination Failures in Keynesian Models" (Quarterly Journal of Economics, volume 103, number 3, pages 441 to 463) brought the concept into macroeconomics. They argued that strategic complementarity is the basic property behind multiple equilibria in coordination games, and that this unifies a range of Keynesian-flavored models in which an economy can settle into a low-activity equilibrium even though a higher-activity one exists.

Supermodular games: Topkis, Vives, Milgrom and Roberts

The mathematical foundation came from a different direction. Donald Topkis, working in operations research, studied optimization over lattices and in 1979 published "Equilibrium Points in Nonzero-Sum n-Person Submodular Games" in the SIAM Journal on Control and Optimization (volume 17, number 6, pages 773 to 787). Xavier Vives developed the economic theory in "Nash equilibrium with strategic complementarities" (Journal of Mathematical Economics, 1990, volume 19, number 3, pages 305 to 321). Paul Milgrom and John Roberts's "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities" (Econometrica, 1990, volume 58, number 6, pages 1255 to 1277) showed that in these games the sets of pure-strategy Nash equilibria, correlated equilibria and rationalizable strategies all share the same upper and lower bounds, and that a broad class of adaptive learning processes, including best-response dynamics and Bayesian learning, eventually settles within those bounds. Their abstract lists applications to oligopoly, macroeconomic coordination failures, arms races, bank runs, technology adoption and diffusion, R&D competition and pretrial bargaining.

Later contributions include Paul Milgrom and Chris Shannon's work on monotone comparative statics (Econometrica, 1994), which generalized the conditions needed, and Federico Echenique's 2007 algorithm for finding all equilibria of games with strategic complements (Journal of Economic Theory, volume 135, number 1). Vives's 1999 book Oligopoly Pricing: Old Ideas and New Tools and his 2005 survey "Complementarities and Games: New Developments" in the Journal of Economic Literature are standard overviews.

Definitions in plain notation

The calculus version

Take a two-player game where player i chooses a number x_i (a price, a quantity, an effort level) and gets payoff Π_i(x_i, x_j), which depends on both choices. Assume the payoff is concave in the player's own choice, so the best response is found by setting the derivative with respect to x_i equal to zero.

The decisions are:

• Strategic complements if the cross-partial derivative ∂²Π_i / ∂x_i ∂x_j is positive.
• Strategic substitutes if ∂²Π_i / ∂x_i ∂x_j is negative.

The cross-partial measures how the marginal payoff of my action (∂Π_i/∂x_i) changes when the rival's action x_j goes up. Positive means the rival's action raises my marginal return.

Link to best-response slopes

Write BR_i(x_j) for player i's best response. It satisfies the first-order condition ∂Π_i/∂x_i = 0. Differentiating that condition with respect to x_j gives the slope of the best response:

slope of BR_i = − (∂²Π_i / ∂x_i ∂x_j) / (∂²Π_i / ∂x_i²)

Concavity means the denominator ∂²Π_i/∂x_i² is negative. So the slope has the same sign as the cross-partial. Upward-sloping best responses mean complements; downward-sloping means substitutes.

The general version: increasing differences

Calculus is convenient but not necessary. A function Π_i(x_i, x_j) has increasing differences if, whenever x_i' > x_i and x_j' > x_j,

Π_i(x_i', x_j') − Π_i(x_i, x_j') ≥ Π_i(x_i', x_j) − Π_i(x_i, x_j)

In words: the gain from switching to the higher action is at least as large when the rival also takes the higher action. This version works for discrete choices (adopt or not, enter or not) and does not require differentiability. When payoffs are smooth, increasing differences is equivalent to a non-negative cross-partial.

A closely related condition is supermodularity, which applies when a player's own choice has several dimensions (for example a price and an advertising level). Supermodularity requires that the different components of the player's own choice are complements with each other. With one-dimensional choices, it holds automatically, and only increasing differences across players is needed.

Supermodular games

A supermodular game (also called a game with strategic complementarities) is one where, roughly:

1. Each player's set of strategies is a complete lattice, a technical condition satisfied for example by closed intervals of numbers or products of them.
2. Each player's payoff is supermodular in that player's own strategy.
3. Each player's payoff has increasing differences between that player's own strategy and the other players' strategies.

Topkis's work and its successors established several results for these games:

• Existence: A pure-strategy Nash equilibrium exists. No need for mixed strategies or for the convexity assumptions of the usual existence theorems.
• Extremal equilibria: There is a largest and a smallest equilibrium. Every other equilibrium lies between them.
• Convergence: Starting from the lowest possible strategy profile and repeatedly playing best responses produces an increasing sequence that converges to the smallest equilibrium. Starting from the highest profile produces a decreasing sequence that converges to the largest.
• Rationalizability bounds (Milgrom and Roberts): Every strategy that survives iterated elimination of strictly dominated strategies lies between the smallest and largest equilibria. If the two coincide, so there is a unique equilibrium, then the game is dominance solvable.
• Monotone comparative statics: If an outside parameter t raises every player's marginal payoff (payoffs have increasing differences in own action and t), then the largest and smallest equilibria both increase with t.

The existence result rests on Alfred Tarski's fixed-point theorem for lattices (published in the Pacific Journal of Mathematics in 1955): an increasing function from a complete lattice to itself has a fixed point, and the fixed points themselves form a complete lattice. Under strategic complements, the joint best-response map is increasing, so Tarski's theorem applies.

Games with strategic substitutes do not enjoy the same general guarantees. Two-player games with strategic substitutes can be turned into games of complements by reversing the order on one player's strategy (counting one player's quantity "downward"), so some results carry over in that case, but with three or more players this trick no longer works in general.

Worked examples

All the numbers in this section are hypothetical and chosen for clean arithmetic.

Example 1: Cournot quantity competition (substitutes)

Two firms produce an identical good. Market price is P = 10 − q₁ − q₂, and each firm has a constant marginal cost of 2.

Firm 1's profit is Π₁ = (10 − q₁ − q₂ − 2) q₁ = (8 − q₁ − q₂) q₁.

• Marginal profit: ∂Π₁/∂q₁ = 8 − 2q₁ − q₂.
• Cross-partial: ∂²Π₁/∂q₁∂q₂ = −1, which is negative. Quantities are strategic substitutes.
• Best response: setting marginal profit to zero gives q₁ = (8 − q₂)/2. The slope is −1/2.

By symmetry, the equilibrium has q₁ = q₂ = q with q = (8 − q)/2, so q = 8/3 ≈ 2.67. The price is 10 − 16/3 ≈ 4.67, and each firm earns (4.67 − 2) × 2.67 = 64/9 ≈ 7.11.

Example 2: Differentiated Bertrand price competition (complements)

Two firms sell similar but not identical products. Demand for firm i is q_i = 10 − p_i + 0.5 p_j, and marginal cost is 2.

Firm i's profit is Π_i = (p_i − 2)(10 − p_i + 0.5 p_j).

• Marginal profit: ∂Π_i/∂p_i = 10 − p_i + 0.5 p_j − (p_i − 2) = 12 − 2p_i + 0.5 p_j.
• Cross-partial: ∂²Π_i/∂p_i∂p_j = 0.5, which is positive. Prices are strategic complements.
• Best response: p_i = 6 + 0.25 p_j. The slope is +0.25.

The symmetric equilibrium solves p = 6 + 0.25p, so p = 8. Each firm sells 10 − 8 + 4 = 6 units and earns (8 − 2) × 6 = 36.

For comparison, if the two firms could coordinate prices to maximize joint profit, they would choose a common price p to maximize 2(p − 2)(10 − 0.5p). Setting the derivative 2[(10 − 0.5p) − 0.5(p − 2)] = 2(11 − p) to zero gives p = 11, with each firm selling 4.5 units and earning 40.5. The competitive equilibrium price (8) is below the joint-profit price (11), as usual.

Example 3: Best-response iteration converges from both sides

Because prices in Example 2 are strategic complements, the convergence result for supermodular games can be seen directly.

Start both firms at the lowest sensible price, marginal cost 2, and let each repeatedly play a best response to the other's previous price:

Starting from below, prices rise every round. Starting from above, they fall every round. Both sequences close in on the equilibrium of 8. Here the smallest and largest equilibria coincide, so the equilibrium is unique and, by the Milgrom–Roberts result, the game is dominance solvable within this range of prices.

Example 4: Commitment helps differently in the two cases

Suppose one firm can commit to its action first, before the other responds. This is a Stackelberg game.

Cournot (substitutes). If firm 1 commits to a quantity first, it anticipates firm 2's response q₂ = (8 − q₁)/2. Firm 1 maximizes (8 − q₁ − (8 − q₁)/2) q₁ = ((8 − q₁)/2) q₁, which gives q₁ = 4. Then q₂ = 2, the price is 10 − 6 = 4, firm 1 earns (4 − 2) × 4 = 8 and firm 2 earns (4 − 2) × 2 = 4. Compared with simultaneous play (7.11 each), the leader gains and the follower loses. Committing to be aggressive (high output) pays, because the rival responds by backing off.

Bertrand (complements). If firm 1 commits to a price first, it anticipates p₂ = 6 + 0.25 p₁. Its demand becomes 10 − p₁ + 0.5(6 + 0.25 p₁) = 13 − 0.875 p₁, so it maximizes (p₁ − 2)(13 − 0.875 p₁). The derivative is 13 − 0.875 p₁ − 0.875(p₁ − 2) = 14.75 − 1.75 p₁, which is zero at p₁ = 59/7 ≈ 8.43. Then p₂ ≈ 8.11.

The leader raises its price above the simultaneous-move level, the follower raises its price in response, and both are better off, the follower more so. Under strategic complements, committing to be soft (a high price) pays, because the rival responds by softening too.

This contrast, aggressive commitment under substitutes and soft commitment under complements, is the central strategic lesson of the concept.

Example 4b: How a cost shock spreads to the rival

The classification also predicts how a shock to one firm spreads to the other. Suppose firm 1's marginal cost rises from 2 to 3 while firm 2's stays at 2.

Bertrand (complements). Firm 1's marginal profit becomes 10 − p₁ + 0.5 p₂ − (p₁ − 3) = 13 − 2p₁ + 0.5 p₂, so its best response becomes p₁ = 6.5 + 0.25 p₂. Firm 2's best response is unchanged at p₂ = 6 + 0.25 p₁. Substituting gives p₁ = 6.5 + 0.25(6 + 0.25 p₁) = 8 + 0.0625 p₁, so p₁ = 8 / 0.9375 ≈ 8.53, and then p₂ ≈ 6 + 2.13 = 8.13.

Cournot (substitutes). Firm 1's best response becomes q₁ = (7 − q₂)/2, while firm 2's stays q₂ = (8 − q₁)/2. Substituting gives q₁ = (7 − 4 + q₁/2)/2 = 1.5 + q₁/4, so q₁ = 2, and then q₂ = 3.

Under complements, the shocked firm's move is echoed by its rival. Under substitutes, the rival partly offsets it. This is the monotone comparative statics result in a small, concrete form: when a parameter raises one player's incentive to take a higher action, and actions are complements, every player's equilibrium action rises.

Example 5: Technology adoption and multiple equilibria

Two firms decide whether to adopt a new technology standard. Adoption costs something, but it pays off only if the other firm adopts too. Payoffs (firm 1, firm 2):

Firm 1's gain from adopting is 2 − 0 = 2 if firm 2 adopts, and −1 − 0 = −1 if firm 2 does not. The gain is higher when the rival adopts, so the game has increasing differences: adoption decisions are strategic complements.

There are two pure-strategy equilibria: both adopt (2, 2) and neither adopts (0, 0). Both are self-consistent, but one is better for everyone. This is a coordination failure in the sense Cooper and John described. There is also a mixed-strategy equilibrium. If firm 2 adopts with probability q, firm 1 is indifferent when 2q − (1 − q) = 0, which gives q = 1/3. In that equilibrium each firm adopts with probability one-third.

Example 6: The social multiplier

Strategic complements amplify shocks. Suppose each of two players chooses an effort level e_i with payoff

u_i = θ e_i − e_i²/2 + β e_i e_j

where θ is an individual taste for effort and β measures how much a partner's effort raises the return to one's own. The cross-partial is β, so efforts are complements when β is positive.

The best response is e_i = θ + β e_j. In a symmetric equilibrium e = θ + βe, so e = θ / (1 − β), provided β is less than 1.

With θ = 1 and β = 0.5, equilibrium effort is 2. If θ rises to 1.1 for both players, equilibrium effort rises to 2.2. The direct effect of the taste change on each player, holding the partner fixed, is only 0.1; the full change is 0.2. The extra 0.1 comes from each player responding to the other's higher effort. The factor 1/(1 − β) = 2 is a "social multiplier." Under strategic substitutes (β negative), the same calculation gives a factor below 1: responses partly cancel the shock.

Applications

Strategic investment: the Fudenberg–Tirole taxonomy

Before competing, a firm can often make an investment that changes its own later behavior, for example building capacity, cutting costs, advertising or acquiring customers. Such an investment has a direct effect on profit and a strategic effect through the rival's response. Whether the strategic effect favors overinvesting or underinvesting depends on two things: whether the investment makes the firm "tough" (more aggressive toward the rival later) or "soft," and whether later actions are strategic complements or substitutes.

When the incumbent wants to accommodate the rival rather than drive it out, the standard taxonomy, presented in Jean Tirole's textbook The Theory of Industrial Organization (1988) and following Fudenberg and Tirole (1984), is:

When the incumbent wants to deter entry, it wants to look tough regardless, so it overinvests if investment makes it tough ("top dog") and underinvests if investment makes it soft ("lean and hungry").

Example 4 above illustrates the logic: in price competition (complements), looking soft encourages the rival to be soft too, which helps you.

Strategic trade policy

The distinction played a prominent role in the 1980s debate over strategic trade policy. James Brander and Barbara Spencer (1985, Journal of International Economics) showed that, in a Cournot model where a domestic and a foreign firm compete in a third market, a government can raise national welfare by subsidizing its exporter, because the subsidy commits the domestic firm to higher output and the foreign rival backs off. Jonathan Eaton and Gene Grossman (1986, Quarterly Journal of Economics) showed that if the firms compete in prices instead (strategic complements), the welfare-improving policy flips: an export tax, which softens the domestic firm and induces the rival to raise its price as well. The policy recommendation depends on the sign of the strategic interaction, which is hard to know in practice. This sensitivity is one reason many economists became skeptical of strategic trade policy as a practical guide.

Macroeconomic coordination

Many macroeconomic models feature strategic complementarity. If other firms hire and produce more, demand for my product rises and I want to produce more. Peter Diamond's 1982 search-equilibrium model (Journal of Political Economy), in which the payoff to producing depends on how easy it is to find trading partners, is a well-known example with multiple equilibria. Cooper and John (1988) showed that in models of this type, when there are positive spillovers and strategic complementarity, the possible equilibria can be ranked: everyone prefers the high-activity equilibrium, but the economy can get stuck at a lower one. This gives one formal account of "animal spirits" and self-fulfilling pessimism.

Bank runs and currency attacks

In Douglas Diamond and Philip Dybvig's 1983 model of bank runs (Journal of Political Economy), depositors' withdrawal decisions are strategic complements: if many others withdraw, the bank is more likely to fail, and withdrawing early becomes more attractive. There are two equilibria, one with a run and one without. Self-fulfilling currency attacks have a similar structure.

The theory of global games, developed by Hans Carlsson and Eric van Damme (Econometrica, 1993) and applied by Stephen Morris and Hyun Song Shin to currency attacks (American Economic Review, 1998), shows that adding a small amount of private uncertainty about economic fundamentals to such games can select a unique equilibrium. The strategic complementarity is essential to how these arguments work: it allows equilibrium strategies to be pinned down by iterated dominance, much as in the Milgrom–Roberts results.

Network effects and technology adoption

Products with network effects, such as communication platforms, payment systems and software standards, are more valuable when more people use them. Adoption decisions are strategic complements, which is why such markets often tip toward one standard and why early adoption can snowball. Example 5 above is the simplest version.

Arms races and conflict

If a rival country builds more weapons, the marginal security value of one's own weapons may rise, making arms levels strategic complements. Milgrom and Roberts listed arms races among their applications. Not every conflict model has this property; in some, a stronger rival makes one's own investment less worthwhile, which makes them substitutes.

Public goods and free riding

Contributions to a public good are often strategic substitutes: if others contribute more, the marginal value of my own contribution falls and I may contribute less. This is the basic free-rider logic. It contrasts with peer-effects settings like Example 6, where a partner's effort raises the return to one's own.

Peer effects

In education, crime and labor markets, researchers study whether a person's behavior responds positively to that of peers. If it does, choices are strategic complements and the social multiplier applies. Measuring this is notoriously difficult, as discussed below.

When are Cournot quantities complements, and Bertrand prices substitutes?

The textbook association, Cournot with substitutes and Bertrand with complements, holds for standard linear demand and constant costs, but it is not a law.

Under Cournot competition, firm 1's marginal revenue is P(Q) + q₁ P′(Q), where Q is total output. The effect of q₂ on this marginal revenue is P′(Q) + q₁ P″(Q). The first term is negative because demand slopes down. The second term depends on the curvature of demand. If demand is sufficiently convex (P″ large and positive), the sum can be positive, making quantities strategic complements. The Wikipedia article on the topic notes that firms' outputs tend to be complements when there are strong aggregate increasing returns to scale and/or low own-price elasticity of demand.

Under price competition, prices are usually strategic complements, but unusual demand or cost structures can reverse this. And when the goods are complementary products (like hardware and software made by different firms), a rival's price cut raises demand for your product, which changes the analysis again.

The general lesson from Bulow, Geanakoplos and Klemperer is that the classification depends on the sign of the cross effect, which must be checked in each model, not assumed from whether firms choose prices or quantities.

Many players, dynamics and learning

Most of the examples above have two players, but the concept extends directly to many. With n players, player i's actions are strategic complements to the others' if raising any rival's action raises player i's marginal payoff. Many economic models are aggregative: each player cares only about their own action and some total or average of everyone else's, such as industry output or the average price. In such models the question reduces to the sign of the cross effect between a player's action and the aggregate.

The two-player trick of flipping one player's order to turn substitutes into complements does not generalize. If firm 1 and firm 2 are substitutes and firm 2 and firm 3 are substitutes, flipping firm 2 makes both pairs complements, but then firms 1 and 3, which are also substitutes, cannot be fixed at the same time. This is why Cournot oligopoly with three or more firms does not in general inherit the lattice results that hold for duopoly, and why existence and stability results for many-firm Cournot models rely on other tools.

Dynamics are another area where the distinction shows up. Under strategic complements, simple adjustment processes such as best-response dynamics or fictitious play move in a monotone way, as Example 3 illustrates, and Milgrom and Roberts showed that a broad class of adaptive learning rules eventually settle between the smallest and largest equilibria. Under strategic substitutes, the same processes can overshoot: if every player expects others to produce a lot, each produces little, which then leads each to expect low output and produce a lot, and so on. With two players and linear best responses of slope −1/2, as in Example 1, the oscillations die out, but with steeper responses or more players they need not.

Empirical identification

Knowing whether actions are complements or substitutes in a real market is often what matters for policy, but it is hard to measure.

The reflection problem. Charles Manski's 1993 paper "Identification of Endogenous Social Effects: The Reflection Problem" (Review of Economic Studies) showed that when people's outcomes move together, it can be impossible to tell whether this reflects people responding to each other (a genuine strategic interaction), people sharing common characteristics, or people facing common shocks. A correlation between my choice and my neighbor's does not by itself show that our choices are strategic complements.

Approaches researchers use. Common strategies include natural experiments that shift one player's incentives but not another's, structural estimation of demand and cost to compute best-response slopes, and laboratory experiments in which payoff functions are controlled by design. Each has its own assumptions.

Local versus global. The cross-partial can change sign at different points. A market can display complements in one range of prices and substitutes in another. Empirical studies typically estimate the sign near observed outcomes, which may not hold under large policy changes.

Common misconceptions

"Strategic complements are complementary goods." No. Complementary goods (like coffee and sugar) are a statement about consumer demand. Strategic complements are a statement about how one player's action affects another player's marginal payoff. Two coffee shops selling substitute products can have prices that are strategic complements, as in Example 2.

"Price competition always means complements and quantity competition always means substitutes." These are the standard cases under linear demand, but Bulow, Geanakoplos and Klemperer emphasized that either type of competition is compatible with either kind of interaction.

"Strategic complements means players benefit from each other's actions." That is a different property, called a positive spillover or positive externality: whether my payoff level rises when you do more. Strategic complementarity is about my marginal payoff. The two can go together (as in Example 5) or not. In Example 2, a rival's higher price raises my profit (positive spillover) and raises my marginal return to a higher price (complements). In a pollution game, your extra pollution may lower my payoff while raising my incentive to pollute, so there could be complements with a negative spillover.

"Games with strategic complements always have multiple equilibria." They can, but they need not. Example 2 has a unique equilibrium. What complements guarantee is that equilibria, however many there are, are ordered, with a largest and a smallest.

"Games with substitutes have no pure equilibrium." Many do; Example 1 has a unique one. They simply lack the general lattice guarantees that come with complementarity.

"Supermodular games require differentiability." The general theory uses increasing differences on lattices and works for discrete choices such as adopt or not adopt.

Comparison table

Frequently asked questions

Who coined the terms? Jeremy Bulow, John Geanakoplos and Paul Klemperer, in their 1985 Journal of Political Economy paper "Multimarket Oligopoly: Strategic Substitutes and Complements."

What is the quickest test for which case I'm in? Ask whether your best response goes up or down when your rival does more. Up means complements, down means substitutes. Mathematically, check the sign of the cross-partial derivative of your payoff with respect to your action and the rival's action.

What is a supermodular game? A game in which strategy sets are lattices and payoffs satisfy supermodularity in a player's own strategy and increasing differences between a player's strategy and others' strategies. These are the formal "games with strategic complements." Donald Topkis's 1979 paper first formalized them.

Why do supermodular games always have a pure-strategy equilibrium? Because the joint best-response map is increasing, and Tarski's fixed-point theorem guarantees that an increasing map on a complete lattice has fixed points, which are Nash equilibria.

Can the same game have complements for one player and substitutes for another? Yes. In some entry or leadership models, one player's best response slopes up while the other's slopes down. The classification can apply player by player.

Why does it matter for policy? Because many policy prescriptions reverse depending on the sign. The Brander–Spencer and Eaton–Grossman results on export subsidies versus export taxes are the classic case.

Is strategic complementarity the same as increasing returns? No. Increasing returns is about how a single producer's costs or output change with scale. Strategic complementarity is about how players' incentives interact. Aggregate increasing returns can be one source of strategic complementarity, as noted in the Wikipedia summary, but the concepts are distinct.

How does it relate to the multiplier in macroeconomics? Under strategic complements, a shock that nudges everyone's incentive in the same direction is amplified, because each player responds to others' responses. Example 6 shows the arithmetic. Cooper and John used this logic to connect game theory to Keynesian ideas.

Are there computational tools for these games? Yes. Besides best-response iteration, which finds the smallest and largest equilibria, Federico Echenique (2007) gave algorithms for finding all equilibria in games of strategic complements. Chuangyin Dang, Qi Qi and Yinyu Ye (2020) have studied the computational complexity of finding Tarski fixed points, which is relevant to how hard supermodular games are to solve.

Key terms

Summary

Strategic complements and substitutes classify how players' choices interact: whether one player's action makes another's action more or less attractive at the margin. The distinction, named by Bulow, Geanakoplos and Klemperer in 1985, explains why firms in price competition benefit from appearing soft while firms in quantity competition benefit from appearing tough, why the best trade policy can flip from subsidy to tax, and why economies can fall into coordination failures. Games of strategic complements have a powerful mathematical structure, worked out by Topkis, Vives, Milgrom and Roberts and others: pure equilibria exist, they are ordered, best-response dynamics converge to the extremes, and comparative statics are monotone. The main practical difficulty is empirical: knowing which case applies in a real market requires evidence on how incentives respond, and that evidence is hard to obtain.

Sources

• Strategic complements (Wikipedia)
• Bulow, Geanakoplos and Klemperer, "Multimarket Oligopoly: Strategic Substitutes and Complements" (Cowles Foundation working paper, EconPapers)
• Milgrom and Roberts (1990), "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica (Econometric Society)
• Fudenberg and Tirole (1984), "The Fat-Cat Effect, the Puppy-Dog Ploy, and the Lean and Hungry Look" (EconBiz record)
• Rasmusen, "The Fudenberg-Tirole Taxonomy" lecture notes
• Topkis (1979), "Equilibrium Points in Nonzero-Sum n-Person Submodular Games," SIAM Journal on Control and Optimization
• Vives (1990), "Nash equilibrium with strategic complementarities," Journal of Mathematical Economics90005-T)