# A Characterization of Strategic Complementarities

TL;DR: It is shown that generic 2 X 2 games either have no pure-strategy equilibria, or are GSC, a negative result because it implies that the predictive power of complementarities alone is very weak.

Abstract: I characterize games for which there is an order on strategies such that the game has strategic complementarities. I prove that, with some qualifications, games with a unique equilibrium have complementarities if and only if Cournot best-response dynamics has no cycles; and that all games with multiple equilibria have complementarities. As applications of my results, I show: 1. That generic 2X2 games either have no pure-strategy equilibria, or have complementarities. 2. That generic two-player finite ordinal potential games have complementarities.

## Summary (3 min read)

### 1 Introduction

- A game has strategic complementarities (Topkis 1979, Vives 1990) if, given an order on players’ strategies, an increase in one player’s strategy makes the other players want to increase their strategies.
- I present an example where no order that makes the game a GSC, avoids strictly dominated strategies in the interval prediction—this is further evidence that the interval prediction is problematic.
- In Section 2 I give some preliminary definitions and present the model that the results apply to.

### 2.2 The Model

- The reader should think of the results as dealing with the best-response function, or correspondence, of a game.
- The set of fixed points of φ—or, if best-responses are unique, of f—coincides with the set of pure-strategy Nash equilibria of Γ. I shall only deal with pure strategies and pure-strategy equilibria, but one can apply my results to the best-response correspondence of the mixed extension of a game.
- If players meet to play Γ over and over again, and if they, in each round of 4I focus on pure strategies because the GSC literature considers usually orders on pure strategies.
- Absence of cycles means that no Cournot best-response dynamics will get caught in a cycle.
- And global stability means that players that engage in Cournot best-response dynamics will eventually approach Nash equilibrium play.

### 3 Unique Equilibrium

- The same number of iterates away from e, which is slightly cumbersome but easy (see the proof of Theorem 3 for the details, and for the proof when X is infinite).
- Theorem 2 may have some mathematical interest, independently of the application to game theory that I emphasize.

### 3.1 Examples

- One implication of these results is that a game that satisfies the dominant diagonal condition in Gabay and Moulin (1980), and has thus a unique, globally stable, equilibrium, is a GSC.
- 6See section 3.3 for the difference between lattice and totally ordered strategy spaces.
- The intuitive idea that zero-sum games and GSC are different is therefore false.

### 3.2 Non-unique best-responses

- Theorem 2 for functions can be generalized to correspondences with finite values, but the characterization is not completely tight.
- Absence of cycles implies the existence of a total order that makes the correspondence weakly monotone, but a weakly monotone correspondence may have cycles, as weak monotonicity does not control all selections from the correspondence.
- I need the hypothesis of a correspondence with finite values to use non-standard analysis the way I do.
- If Γ is a game with unique best-responses, of course the bestresponses have finite values, and both weakly increasing and increasing in the strong set order coincide with monotone increasing.
- I have not been able to extend the characterization in Theorem 3 to arbitrary games with infinite strategy spaces and non-unique best-responses.

### 3.3 Totally ordered strategy space vs. a lattice strategy space,

- If a game’s best-response function does not have cycles, then the game is a GSC because there is a total order on strategies such that best responses are monotone increasing, and a totally ordered set is a lattice.
- In GSC where strategy spaces are non-complete lattices, best-responses may have cycles.
- In the usual definition of GSC—supermodular games—there is a link between the order and the topology on the strategy spaces that, among other things, ensures the existence of equilibria (by Tarski’s Theorem), and that if equilibrium is unique it must be globally stable.
- In infinite games, though, there may be GSC without equilibria, and GSC with a unique equilibrium that is not globally stable.
- I wish to emphasize that, despite this technical problem, for games with multiple equilibria, all of Vives’s, Topkis’s and Milgrom and Roberts’s results hold trivially with the order that I construct in section 4.

### 4 Multiple Equilibria

- I work out the details in Theorem 5, but besides answering the question in this paper, such a coarse order is uninteresting.
- Theorem 7 below says that, if a function f on a finite set has at least two different fixed points, then there is a lattice order such that f is monotone increasing, the interval prediction is the order interval between the fixed points, and any other lattice order that does the same work must have a larger interval prediction.
- The usual definitions of GSC ensure that best-responses are increasing in the strong set order, and that φ takes subcomplete- sublattice-values.
- The order that I construct in Theorem 5 does not guarantee that φ has these properties (unless, of course, φ is a function), but Topkis’s, Vives’s, Milgrom and Roberts’s, and Milgrom and Shannon’s results are true with the constructed order.
- Topkis, Vives and Zhou present examples where it is not a sublattice (see Echenique (2001), though).

### 5 Discussion of the interval prediction.

- I show that there are games for which the interval prediction is unavoidably vacuous, and that there are games for which the interval prediction must contain strictly dominated strategies.
- Theorem 7 delivers an order with the best possible interval prediction, but the construction of this order is in general quite cumbersome.
- To see this, suppose, for example, that (β, α) is the smallest strategy profile.
- All strategies in the coordination games are rationalizable.

### 6 Comparative Statics

- In a parameterized GSC, if a parameter t is complementary to players’ choices, there are selections of equilibria that are monotone increasing in t (Lippman, Mamer, and McCardle 1987, Sobel 1988, Villas-Boas 1997, Milgrom and Roberts 1990, Milgrom and Roberts 1994, Milgrom and Shannon 1994, Echenique 2002, Echenique and Sabarwal 2000).
- Using the framework in Milgrom and Roberts (1994), the question can be phrased as follows.
- I will rule out all possible cycles in best-response dynamics, except for the cycle involving (β, β).
- The authors have shown that all 2X2 games with a unique, strict Nash equilibrium are GSC.

### 8 Application 2: Ordinal Potential Games

- Potential games were studied in detail by Monderer and Shapley (1996) (see their paper for references to earlier work on potential games).
- My results shed some light on the relation between GSC and ordinal potential games.
- Let Γ be a finite two-player game with unique best responses.
- I do not discuss this here, the counterexamples are very simple.
- An ordinal potential game cannot have an infinite improvement path.

### 9.1 Proof of Theorem 3

- Steps 4 and 5 prove the first statement for arbitrary X by non-standard analysis methods.
- The idea is to embed X in a hyperfinite set, apply the result for finite sets to get an order that works in the hyperfinite set, and then restrict the order to X. Step 1.

### 9.2 Proof of Theorem 7

- In any case, x ∈ C∪A∪Be ∪Be, which establishes the claim.
- Be into Be, has exactly one fixed point, and no cycles.
- I now prove that (X,≤) is a complete lattice (steps 1 and 2), that f is monotone increasing (step 3), and that any other lattice order that makes f monotone increasing must have a larger interval prediction (step 4).
- I shall need the following two facts, the facts are immediate from the definition of ≤.
- So, f monotone increasing implies that f nM(x∗) fnM(z).

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### "A Characterization of Strategic Com..." refers background in this paper

...A detailed discussion of the concepts de ned in this subsection can be found in Topkis (1998)....

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...4 in Topkis (1998), there is a k ∈ N such that e = f (x)....

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...A detailed discussion of the concepts defined in this subsection can be found in Topkis (1998). A pair (X,≤), where X is a set and ≤ is a transitive, reflexive, antisymmetric binary relation, is a partially ordered set; (X,≤) is totally ordered if, for all x, y ∈ X, x ≤ y or y ≤ x (≤ is then a total order on X); (X,≤) is a lattice if whenever x, y ∈ X, both x ∧ y = inf {x, y} and x ∨ y = sup {x, y} exist in X....

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1,709 citations

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### "A Characterization of Strategic Com..." refers background in this paper

...A game has strategic complementarities (Topkis 1979, Vives 1990) if, given an order on players' strategies, an increase in one player's strategy makes the other players want to increase their strategies....

[...]

...Further, the set of all rationalizable strategies, and the set of limits of adaptive learning, is bounded below by the smallest equilibrium and above by the largest equilibrium (Vives 1990, Milgrom and Roberts 1990, Milgrom and Shannon 1994)....

[...]

...On the other hand, if is a GSC where each Si is a complete lattice, and best responses f are continuous, then a unique Nash equilibrium is globally stable (Vives 1990, Milgrom and Roberts 1990) so f has no cycles and there is a total order on strategies such that f is increasing....

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630 citations

### "A Characterization of Strategic Com..." refers background in this paper

...A game has strategic complementarities (Topkis 1979, Vives 1990) if, given an order on players' strategies, an increase in one player's strategy makes the other players want to increase their strategies....

[...]

...A detailed discussion of the concepts de ned in this subsection can be found in Topkis (1998). A pair (X; ), where X is a set and is a transitive, re exive, antisymmetric binary relation, is a partially ordered set; (X; ) is totally ordered if, for all x; y 2 X, x y or y x ( is then a total order on X); (X; ) is a lattice if whenever x; y 2 X, both x ^ y = inf fx; yg and x _ y = sup fx; yg exist in X....

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