CHARACTERISING EQUILIBRIUM SELECTION IN GLOBAL GAMES WITH STRATEGIC COMPLEMENTARITIES. 1. Introduction

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1 CHARACTERISING EQUILIBRIUM SELECTION IN GLOBAL GAMES WITH STRATEGIC COMPLEMENTARITIES CHRISTIAN BASTECK* TIJMEN R. DANIËLS FRANK HEINEMANN TECHNISCHE UNIVERSITÄT BERLIN Abstract. Global games are widely used to predict behaviour in games with strategic complementarities and multiple equilibria. We establish two results on the global game selection. First, we show that, for any supermodular complete information game, it is independent of the payoff functions chosen for the game s global game embedding, though (as is well-known) it may depend on the noise distribution. Second, we give a simple sufficient criterion to derive the selection and establish noise independence in many action games by decomposing them into games with smaller action spaces, to which we may often apply simple criteria. We also report in which small games noise independence may be established by counting the number of players or actions. Keywords: global games, equilibrium selection, strategic complementarities. JEL codes: C72, D Introduction Games of strategic complementarities, for instance models of financial crises or network externalities, often have multiple equilibria. An important issue, from a theoretical as well as a policy perspective, is how to predict the equilibrium that will be played. One widely used approach to predict behaviour in such games is to turn them into a global game. Extend a complete information game to an incomplete information game with a one-dimensional state space, such that the original game can be viewed as one particular realisation of the random state variable. This state is usually interpreted as an economic fundamental. Each player receives a noisy private signal about the true state. Then, under certain supermodularity and monotonicity conditions, 1 Frankel, Morris and Pauzner [10] ( FMP ) prove limit uniqueness: as the noise in private signals vanishes, for almost all realisations of the state parameter players coordinate on a Nash equilibrium of the complete information game given by the payoffs at the true state. This global game selection ( GGS ) may be used as a prediction and to derive comparative statics results. Applications include models of speculative attacks (Morris and Shin [17], Cukierman, Date: First version: December This revised version: March *Corresponding author: christian.basteck@tu-berlin.com. We are grateful to an anonymous referee and associate editor whose suggestions have enabled us to improve the content and presentation of our paper substantially. We further thank Satoru Takahashi, participants at the Econometric Society world congress 2010 and the meeting of the European Economic Association 2010 for helpful comments. We acknowledge support from DFG through SFB649 and from a joint DFG-ANR grant. 1 FMP assume that there exists an ordering on actions such that players incentive to switch to a greater action is monotonic in both the state and the actions of others; and that at sufficiently low (high) states, each player s least (greatest) action is strictly dominant. See Section 2 for precise definitions. 1

2 Goldstein and Spiegel [9], Guimarães and Morris [12], Corsetti, Dasgupta, Morris and Shin [7], Corsetti, Guimarães and Roubini [8]), banking crises (Goldstein [11], Rochet and Vives [22]), and investment problems (Steiner and Sákovics [23]). Experiments by Heinemann, Nagel and Ockenfels [13, 14] show the GGS is useful to predict subjects behaviour. Unfortunately, there is no generic way to extend a complete information game to a global game, and which equilibrium is selected may depend on the details of the chosen extension. It is well-known that the GGS may depend on the distribution of the noisy private signals. If multiple equilibria are replaced by multiple global game selections, this reduces the value of the GGS as a selection criterion. FMP provide some conditions under which the GGS is noise independent, that is, independent of this distribution. Furthermore, even though FMP prove limit uniqueness for a broad class of games, most applications are to binary-action games. In symmetric binary-action games, the GGS can be easily determined as a player s best reply to the belief that the fraction of other players choosing either action is uniformly distributed (see FMP or Morris and Shin [18]); a generalisation for a class of asymmetric binary-action games is given by Steiner and Sákovics [23]. In this paper, we extend results on global games in two directions. First, we show that for any supermodular complete information game, the GGS is independent of the extended payoff function, as long as it satisfies the usual supermodularity assumptions. Hence, the GGS does not depend, for example, on the choice of economic fundamental used as a state parameter in the global game. In fact, it does not even matter whether the chosen extension is continuous. This is important, because many applications are regime change models in which the state affects payoffs discontinuously, while the results by FMP require continuity. Furthermore, our result implies that dependence on the distribution of private signals is the only source of multiplicity for the GGS. Second, we provide a new and simple method to determine the GGS and check its noise independence in games with many actions. The GGS is noise independent if the game can be suitably decomposed into smaller noise independent games. For example, we may split up a n-action game into many binary-action games and apply simple known criteria for deriving the GGS. If the smaller games are noise independent and their selections point in direction of the same action profile, then this action profile is the noise independent selection of the larger game. Our second result gives a new heuristic for the global game equilibrium prediction that lends itself to economic applications. For instance, introducing a third option in a binary-action game may change the selection between the two original equilibria. We give an example of a refinancing model where introducing collateralised loans as a third option besides withdrawing and extending unsecured loans changes the GGS from the inefficient withdrawal equilibrium to that of unsecured refinancing. Other models of interest may have many actions and potentially a large number of equilibria. As an example, we analyse a generalised version of Bryant s [3] minimum effort game, 2

3 to which none of the known easy heuristics may be applied. By decomposing it into binary-action games, we may derive the GGS and establish its noise independence straightforwardly. The decomposition of large games is particularly useful, because noise independence is typically easier to establish for smaller games, often simply by counting the number of players and actions. In their seminal paper introducing global games, Carlsson and Van Damme [6] proved that all 2 2 games with multiple equilibria are noise independent. We prove the same of supermodular 2 n action games. Conversely, we give a counterexample to noise independence in a symmetric three-player, three-action game. Together with known results by Carlsson [4], FMP and Basteck and Daniëls [1], this completes the characterisation of supermodular games which noise independence can be established by counting. Our paper proceeds as follows. Section 2 contains preliminaries. The rest of the paper is organised around a characterisation of the GGS given in Section 3. Instead of analysing a sequence of global games with vanishing noise, we show that we may determine the GGS from a single incomplete information game with simple payoffs. This game is independent of the state-dependent payoff function that is chosen as an extension of the original game. We use the characterisation to prove generic uniqueness of the GGS and extend it to discontinuous global games. In Section 4 we discuss the decomposition method and establish our results on noise independence. Section 5 contains applications. Section 6 concludes. Proofs are in the appendix Setting and Definitions In this paper we consider games with a finite set of players I, who have finite action sets A i = {0, 1,..., m i }, i I, which we endow with the natural ordering. We define the joint action space A as i I A i, and write A i for j i A j. For a = (a i ) i I A and a = (a i ) i I we write a a if and only if a i a i for all i I. The least and greatest action profiles in A are denoted by m and 0. A complete information game g is specified by its real-valued payoff functions g i (a i, a i ), i I, where a i denotes i s action and a i A i denotes the opposing action profile. A game g is supermodular if for all i, a i a i, and a i a i, (1) g i (a i, a i ) g i (a i, a i ) g i (a i, a i) g i (a i, a i), implying a player s best reply is weakly monotonic in the actions of her opponents (see Topkis [24]; note that property (1) is what FMP refer to as strategic complementarities). We define a global game G(v) as in FMP [10]: consider payoff functions u i (a i, a i, θ) that depend on an additional state parameter θ R. For each θ, let the game given by the u i (, θ) be a supermodular game (Assumption A1). In addition, we assume there are dominance regions: there exist thresholds θ < θ such that the smallest and greatest actions in A i are strictly dominant

4 when payoffs are given by, respectively, u i (, θ) and u i (, θ) (Assumption A2). Furthermore, each u i satisfies state monotonicity in the sense that greater states make greater actions more appealing (Assumption A3): there exists K > 0 such that for all a i a i and θ θ θ θ we have 0 K(a i a i )(θ θ) ( u i (a i, a i, θ ) u i (a i, a i, θ ) ) ( u i (a i, a i, θ) u i (a i, a i, θ) ). Finally, we assume each u i is continuous with respect to θ (Assumption A4). In the global game G(v), the state θ is realised according to a continuous density φ with connected support that includes the values θ and θ in its interior. Players observe it with some noise, and then act simultaneously. Formally, let f = ( f i ) i I denote a tuple of probability densities, whose supports are subsets of [ 1, 1]. Each player i receives a private signal x 2 2 i = θ + vη i, where each η i is drawn independently according to the density f i, and v (0, 1] is a scaling parameter. Thus, a global game G(v) is specified by the payoff functions u i, a prior distribution of states φ, a noise structure f, and a scaling parameter v, each of which is common knowledge among players. A strategy s i for G(v) is a function that maps a player s signal to an action. 2 Joint strategy profiles are denoted by s = (s i ) i I and s i = (s j ) j I {i}. Slightly abusing notation, denote by s(x) the joint action profile obtained when each player receives the same signal x i = x. We write s s if and only if s(x) s (x) for all x. A strategy profile s is increasing, if each s i is increasing in x i ; it is a (Bayes-Nash) equilibrium if each s i (x i ) is a best reply against s i, given u i and using Bayes s rule to derive the conditional densities of θ and x i, given x i. Following FMP, we also define the simplified global game G (v), which differs from G(v) in that players payoffs are given by u i (a i, a i, x i ) (thus players payoffs depend directly on their signal), and that the prior φ is uniform on a large closed interval U containing [θ, θ]. FMP show that the equilibrium strategy profile in G (v) is unique up to its points of discontinuity (Lemma A1 in FMP) but G(v) may have multiple equilibrium strategy profiles. However, the key result on global games says that as v 0, the equilibrium strategy profiles of the games G(v) and G (v) all converge pointwise to the same limiting strategy profile, which is increasing and unique up to its finitely many discontinuities (Theorem 1 in FMP). 4 Theorem (Limit Uniqueness). The global games G(v) and G (v) have an essentially unique, common limit equilibrium strategy profile as v 0. More precisely, there exists an increasing pure strategy profile s such that, for each v > 0, if s v is an equilibrium strategy profile of G(v), and s v is the unique equilibrium strategy profile of G (v), then lim v 0 s v (x) = lim v 0 s v(x) = s(x) for all x except possibly at the finitely many discontinuities of s. 2 Focusing on pure strategies is without loss of generality and simplifies notation. Supermodularity implies that greatest and least equilibrium strategy profiles exist in pure strategies and are bounds on any mixed strategy profile. Since these equilibrium profiles converge to a common limit (see the theorem below), no mixed strategy profile survives as v 0.

5 Since the limit strategy profile s is well-defined up to its points of continuity, we use s and s to denote the right and left continuous versions of s, respectively. Finally, we will make use of the following well known fact about supermodular incomplete information games. For a given game, let β(s) denote the joint greatest best reply to s, i.e. the profile in which each player uses her greatest best reply to s i. Then β is monotonic in s. Moreover, if s β(s), then the upper best reply iteration s, β(s), β(β(s)),... converges monotonically to an equilibrium strategy profile. This applies in particular to G(v) and G (v) Equilibrium Selection via Global Games Consider a supermodular complete information game g. Often g has multiple Nash equilibria, in which case a global game gives a method to resolve equilibrium indeterminacy. Let G(v) be a global game with payoff functions u i such that g i (a i, a i ) = u i (a i, a i, θ ) for some θ (and satisfying the usual assumptions, outlined in Section 2). We say that g is embedded in G(v) at θ. We may view the global game embedding G(v) as a tool for equilibrium refinement. If as the noise in private signals vanishes the global game s limit strategy profile s is continuous at θ, s(θ ) determines a unique Nash equilibrium of the complete information game g (see Theorem 2 in FMP), which we refer to as the global game selection (GGS, following Heinemann et al. [14]). If s is not continuous at θ then its left and right continuous versions determine two distinct Nash equilibria, s(θ ) and s(θ ). We refer to s(θ ) and s(θ ) as the least and greatest GGS, respectively. 3 In this section, we give a simple characterisation of the selection, which implies that it depends only on the payoff functions of g and the noise distribution of private signals f of the global game, while the additional modelling choices of the global game, the prior φ and payoff functions u i (, θ), θ θ, do not affect it. We also show that (for fixed f ) the least GGS is identical to the greatest GGS for almost all supermodular games. Thus, the GGS is generically unique. Let us first note that this approach can by applied to any supermodular game. Lemma 1. For any supermodular game g, there exists a global game G(v) such that g is embedded in it. To see this, consider u i given by u i (a i, a i, θ) := g i (a i, a i ) + θa i, for i I, a i A i and a i A i. One may straightforwardly verify that u i satisfies assumptions (A1) (A4). Choose some prior φ with sufficiently wide support and some noise distribution f, and our claim is satisfied at θ = A Characterisation of the Global Game Selection Without loss of generality, we may assume that g is embedded in G(v) at θ = 0. We will show how the GGS given by s(0) may be determined without analysing a sequence of incomplete 3 Continuity of the payoff functions of G(v) implies that s(θ ) and s(θ ) are Nash equilibria of g.

6 6 a A a i g i (a i, a i ) s v 0 0 A Figure 1. The incomplete information game e v x information games with increasingly precise noise structures. Instead, we introduce a new incomplete information game that allows us to establish the GGS more directly. Following [10], consider an alternative incomplete information game e v with payoff functions ũ i as follows (Figure 1): a i if x i < 0, ũ i (a i, a i, x i ) := g i (a i, a i ) if x i 0. As in the simplified global game G (v), assume θ is uniformly distributed over a large interval 4 U R and each player i receives a private signal x i = θ + vη i, where each η i is drawn according to the density f i. Now, for any x i, the best reply under the payoff function u i (, x i ) is weakly greater than under ũ i (, x i ) for negative signals because the payoff ( a i ) implies that the least action strictly dominates under ũ i, for x i = 0 because u i is identical to ũ i, and for positive signals since a player s incentive to increase her action increases in x i under u i but is constant under ũ i. For this reason, we can use e v to find a lower bound for the GGS. Let us denote by a the greatest action profile such that, for some v > 0 and x > 0, there exists an equilibrium strategy profile s v in e v such that s v (vx) = a. Since v is only a scale parameter, for each v v there also exists an equilibrium strategy profile s v in e v such that s v (vx) = a. By our comparison of ũ i and u i above, and standard results on supermodular games, a best reply iteration in G (v) starting from s v gives a monotonic sequence that (necessarily) converges to its unique equilibrium strategy profile s v. Note that s v(vx) a. Thus, letting v 0, we find that for s (the right continuous version of the limiting strategy profile of G (v) as v 0), we must have s(0) a. Hence the greatest GGS must be weakly greater than a. This type of argument to establish a lower bound on the GGS is used in the proof of theorem 4 in FMP. We will establish a converse result: the action profile a also determines an upper bound for the greatest GGS. We sketch our argument as follows. Consider the simplified global game G (v) and assume that the limit strategy profile s is continuous at 0. For sufficiently small values of v, the equilibrium profile s v of G (v) will be constant and equal to some Nash equilibrium a of g in a neighbourhood of 0. We may truncate G (v) and s v by restricting them to signals x i 0, 4 We assume that [ 1, A ] U for reasons that will become apparent immediately below. Note that in the simplified global game G (v) we can always extend U without any essential impact on the game. 1

7 and the truncated s v will still be an equilibrium profile in the truncated version of G (v) at low signals, beliefs are unaffected, and for signals close to 0 players will play a, as those with nearby signals do the same. Both e v and the truncated G (v) contain a lower dominance region and agree on the payoff functions g i = u i (, 0) at their highest signal. However, at in-between signals, payoffs in e v give an incentive to play greater actions compared with the payoffs in G (v). Therefore, the greatest equilibrium profile of e v determines an upper bound on s v and, hence, also on the GGS. In our proof, we formalise this argument and extend it to the points of discontinuity of s. Taken together, this allows us to pin down the greatest GGS s(0) completely by analysing the much simpler game e v. As the scale factor plays no essential role in determining a, we may as well fix v = 1 and drop the index. Note that in e, the greatest equilibrium strategy profile s is always constant for signals above A, i.e. for signals greater than the cardinality of the joint action set. This is because by standard results on supermodular games, s must consist of monotonic step functions, where at each step at least one player switches to a higher action. As a player with signal x i knows that all her opponents receive signals within [x i 1, x i + 1], the distance between the steps must be less than 1. Thus, if one wants to determine a, and hence the greatest GGS, one may evaluate s at x = A. Dually, we define the game e in which the payoffs ũ i are given by g i if x i 0 and by a i if x i > 0. We then consider its least equilibrium strategy profile and denote by a the least action profile prescribed by it. This pins down the least GGS. Combining everything, we have: Theorem 2. Suppose g is embedded in a global game G(v) at θ. If s is the essentially unique 5 limit equilibrium strategy profile of G(v), then s(θ ) = a and s(θ ) = a. Thus, by implication, for any value of the state parameter θ, the least and greatest GGS at θ depend only on the noise structure f and on the complete information game given by the payoff functions u i (, θ). Theorem 2 will be our main workhorse in what follows. In our notation below, the incomplete information games e and e and action profiles a and a always correspond to an associated complete information game g and a noise structure f that should be clear from the context. If we need to parametrise games or vary noise structures, we will spell this out explicitly. Equilibrium selection in a global game G(v) is often described as an infection process, starting from its dominance regions. The game e inherits the payoffs of g for most signals but, fitting with this description, adds a seed of pessimism, as players know that action 0 will be played for low signals. Conversely, the game e tests stability against a seed of optimism. In order for a Nash equilibrium a of g to be a GGS, it must be immune to both tests. Remarkably, there always exists such an equilibrium, and (as we show below) generically, it is unique. 5 Recall that s is unique up to its points of discontinuity; s is its right continuous and s its left continuous version. 7

8 The irrelevance of the prior distribution φ for establishing the GGS was already shown by FMP. It may be surprising that the extended payoff function u i of G(v) is also irrelevant. If one thinks of global game selection in terms of an infection process, one might imagine that if g is embedded in G(v) at some θ close to the lower dominance threshold θ, this may influence the GGS in such a way that it selects a lower equilibrium compared to a global game in which g is embedded close to θ. However, Theorem 2 tells us this is not the case. A practical way to think about Theorem 2 is the following. In economic applications, the state parameter θ is typically interpreted as an economic fundamental affecting the decision problem of players. But there may be several economic variables that are candidates for the parameter θ. Theorem 2 tells us that the choice of the fundamental used to perturb the decision problem is irrelevant: the GGS will be the same The Global Game Selection is Generically Unique FMP define a global game as a family of supermodular games ordered along a one-dimensional state space, and find that the GGS is unique at almost all states. Theorem 2 says that for any given supermodular game, the least and greatest GGS depend only on the choice of the noise distribution f and can be determined without reference to any particular global game embedding. So, we will complement FMP s observation by showing that (given f ) the GGS is generically unique for all supermodular games. In particular, a supermodular game picked at random has a unique GGS with probability 1. More precisely, consider the set of games with player set I and joint action set A, which is naturally identified with the Euclidean space R I A. Let S R I A be the subset of supermodular games. The set S is closed and forms a proper convex cone 6 in R I A. For any fixed f, we may write S f := {g S the GGS is unique}, and denote by S f its complement in S. We prove the set S f is small relative to S f, both in measure and in a topological sense. Proposition 3. For any noise structure f, S f is open and dense in S, while S f is closed and nowhere dense in S. Moreover, S f is of infinite Lebesgue measure, while S f is a null set. Recall that S f is dense in S if each g S f can be arbitrarily approximated by games in S f. As jumps in the limit strategy profile of a global game are isolated, we can approximate any g S f if we embed it in a global game G(v) with noise structure f and choose a sequence of games in S f along the one-dimensional state space of G(v). In the proof, we also establish S f is open in S, and thus S f is closed and nowhere dense. We then show that S f has Lebesgue measure zero, by applying a result that connects its topological properties to its measure. 6 S is closed under addition and non-negative scalar multiplication of payoff functions, and has non-empty interior.

9 Global Game Selection in Discontinuous Global Games Proposition 3 indicates that generic uniqueness holds in any global game, provided that changes in the parameter θ sufficiently perturb the game whenever the GGS is not unique, even if assumptions (A3) and (A4) are not fully satisfied. While these assumptions are mathematically convenient, from an applied point of view they are not always desirable. Therefore, we will investigate how they can be weakened. For intuition, consider the following n-player binary-action game: player i α > ξ α ξ where α is the proportion of players who choose 0. It is an exemplary regime change game, where players payoffs depend on whether they reach the critical mass ξ. Intuitively, there are at least two ways to perturb the game: one can perturb the payoffs as in Lemma 1, or one can perturb the critical threshold by setting ξ = θ. In the latter case, which is often considered in the applied literature (e.g. [7, 12, 17, 23]), the embedding payoff function u(a, ) is not continuous in θ (A4) and violates state monotonicity (A3); for a given action profile a, payoffs are unchanged for most changes in θ and jump at the point where a change in θ leads to a regime change. However, even in this case, the following assumption is satisfied: (A3 ) Weak state monotonicity. Greater states make greater actions weakly more appealing: for all θ < θ, i, a i, and a i < a i, u i (a i, a i, θ) u i (a i, a i, θ) u i (a i, a i, θ ) u i (a i, a i, θ ). Let G(v) be a generalised global game, with payoffs u i differing from an ordinary global game in that they satisfy (A3 ) but not necessarily (A3) and (A4). By standard results on supermodular games, for each v > 0, the least and greatest equilibrium strategy profiles š v and ŝ v nevertheless exist. We define their pointwise limits ŝ = lim sup v 0 ŝ v and š = lim inf v 0 š v. The selection in the generalised global game G(v) may be determined using e as in Theorem 2. To be more precise, for each θ, denote the complete information game with payoffs u i (, θ) by g θ. For each g θ, we may determine the action profiles a θ and a θ from the associated incomplete information games e θ and e θ. Note that, regarded as functions of θ, (A3 ) guarantees that a θ and a θ are increasing and thus have only finitely many discontinuities. Proposition 4. Consider a generalised global game. For θ such that a θ = a θ = a, we have ŝ(θ) = š(θ) = a, except possibly at the finitely many discontinuities of a θ and a θ. To prove this result, we sandwich the payoff functions of G(v) between that of two regular global games that approximate it. Then, we may use Theorem 2 to show that their limit strategy profiles must agree on the GGS at θ. This pins down the limit strategy profile of G(v) as well.

10 10 Symmetric Games Asymmetric Games actions: 2 each 3 each 4 each actions: 2 each 2 n 3 each 2 players a b c 2 players a * b 3 players c * 3 players d n/a n players c n players e n/a Always noise independent Counterexample to noise independence exists. Noise independent Counterexample to noise independence exists. * For This empty paper, cells Section noise 4. dependence a Carlsson follows and Van from Damme. a counterexample b Oyama in and a smaller Takahashi. game. c Frankel, This paper. Morris a Carlsson and Pauzner. and Van Damme d Carlsson. [6]. b Basteck e Corsetti andetdaniëls al. [1]. c Frankel, Morris and Pauzner [10]. d Carlsson [4]. e Corsetti et al. [7] Table 1. Noise (In)dependence Table 1. Noise (In)dependence in Supermodular Games 4. Noise Independence A game g embedded in a global game G(v) at θ is called noise independent if the global game s limit strategy profile s takes on the same values at θ regardless of the choice of the noise distribution f. Theorem 2 implies that noise independence is well-defined as a property of g: a supermodular game is noise independent in one global game embedding if and only if it is noise independent in every other. Many small games, with few players or few actions, are noise independent. Typically, for such games there are also easy heuristics to find the GGS. For instance, a well-known elementary condition to judge whether a game is noise independent is the p-dominance criterion. Definition. Given a tuple p = (p i ) i I, an action profile a is said to be p-dominant if each player i who expects her opponents to play a i with probability p i would choose a i as a best reply. One can show that if g has an action profile a that is p-dominant with i I p i < 1, then this is the unique, noise independent GGS. For some games, an even simpler way to determine noise independence is to count the number of players and actions. Table 1 summarises when this is possible; for supermodular games indicated with a, noise independence always holds. Table 1 also indicates that noise independence may fail quickly as we enlarge the action sets of players. In this section, we will show that a game may nevertheless be noise independent if we can suitably decompose it into smaller games. We will also use our characterisation to prove two further results for small games, completing the table by the entries indicated in bold A Decomposition Approach to Noise Independence We will start by making a more basic observation: in certain games with large action spaces, the GGS may be determined by looking at smaller games. Let us introduce some terminology. 1 Definition. Consider a supermodular game g with joint action set A. For action profiles a a, we define [a, a ] := {ã A a ã a }. The restricted games g [a, a ] and e [a, a ] are given by

11 11 a A s a restricted game 0 x 0 A Figure 2. Exploiting the GGS of restricted games restricting the joint action set of g and of its associated incomplete information game e to [a, a ]. Figure 2 now illustrates the idea. If certain action profiles a and a are played in equilibrium strategy profiles of the restricted incomplete information games e [0, a] and e [a, a ], we can patch these profiles together to obtain a strategy profile in the larger game e that attains the action profile a. As in Section 3, we may use this to establish the GGS. We may also do this iteratively: Lemma 5. Fix a supermodular game g and noise structure f. An action profile a n is the unique global game selection, if there is a sequence 0 = a 0 a 1 a n a n+1 a m = m such that (i) a j is the unique global game selection in g [a j 1, a j ] for all j n, and (ii) a j 1 is the unique global game selection in g [a j 1, a j ] for all j > n. A noise independence result follows as an immediate corollary. If the restricted games in the hypothesis of Lemma 5 are noise independent, we can use the same decomposition of g regardless of f, and g must be noise independent. Theorem 6. Fix a supermodular game g. An action profile a n is the unique noise independent global game selection, if there is a sequence 0 = a 0 a 1 a n a n+1 a m = m such that (i) a j is the unique noise independent global game selection in g [a j 1, a j ] for all j n, and (ii) a j 1 is the unique noise independent global game selection in g [a j 1, a j ] for all j > n. In Section 5 we apply these results to derive the noise independent GGS in two examples. The concept of p-dominance reveals an interesting connection between noise independence and the literature on robustness to incomplete information, introduced by Kajii and Morris [15]. Proposition 2.7 and 3.8 in Oyama and Tercieux [21] together imply that if a supermodular game g can be decomposed as above into restricted games, each of which has a strict p- dominant equilibrium with sufficiently small p, uniform across these games, then a n is the unique equilibrium of g that is robust to incomplete information. The formal link to our theorem runs via the observation that if an equilibrium is robust to incomplete information, then it is also the noise independent GGS 7 so that the conclusion of Theorem 6 follows. However, Theorem 6 allows application of a range of known criteria for noise independence 7 See Oury and Tercieux [19] or our working paper version [2]; Morris and Shin [18] provide a heuristic argument. 1

12 besides p-dominance; for instance, the fact that all symmetric 3 3 games, symmetric n-player binary-action games, and (as we show shortly) 2 n games are noise independent, none of which are equivalent to the p-dominance criterion. Indeed, we may apply a different criterion to each restricted game. Thus, the conditions under which our theorem applies are strictly more general. The conclusion of Theorem 6 follows independently of whether a n is robust to incomplete information, which is a sufficient, but not a necessary condition for noise independence (see the combined results of Basteck and Daniëls [1] and Oyama and Takahashi [20]). Our theorem establishes a more direct result about noise independence in global games Small Games Obviously, in order to fruitfully apply Theorem 6, we need as many basic conditions as possible that guarantee noise independent selection for restricted games. We will use our characterisation to prove two new results for small games: one positive result, and one negative. 2 n games. We now show that supermodular 2 n games are always noise independent. Let g be a supermodular game with I = {1, 2}, A 1 = {0, 1}, A 2 = {0, 1,..., m 2 }. For an arbitrary noise distribution f, consider the associated incomplete information game e and its greatest equilibrium strategy profile s, in which players jointly play a = (a 1, a 2 ) at sufficiently high signals. If a i = 0 for some i I, player i plays 0 for all signals. Thus a j, j i must be j s greatest best reply to 0. Then, the joint strategy given by s i (x) = 0 for all x, and s j (x) = 0 if x < 0 and s j (x) = a j if x 0 is an equilibrium under any other noise structure f. Alternatively, suppose a 1 = 1 and a 2 = k > 0. Then s may be identified with a threshold z 1 1 and a tuple of thresholds z 1 2, z2 2,..., zk 2, at which players 1 and 2 switch to greater actions, respectively. The opposing action distribution faced by player 1 at x 1 = z 1 1 is determined by the probabilities (2) Pr(x 2 < z j 2 x 1 = z 1 1 ) =: P j, j {1,..., k}. Now, observe the opposing action distribution that player 2 faces at each of her thresholds z j 2 is also given by these probabilities, since for all j {1,..., k} we find Pr(x 1 > z 1 1 x 2 = z j 2 ) = Pr(x 1 x 2 > z 1 1 z j 2 x 2 = z j 2 ) = Pr(x 1 x 2 > z 1 1 z j 2 x 1 = z 1 1 ) = Pr(x 2 < z j 2 x 1 = z 1 1 ) = P j, where the second equality follows from the uniform prior distribution of θ. Now, if we consider any other noise structure f and associated game e, we can always construct an increasing strategy profile s in which players jointly play (1, a 2 ) for sufficiently high signals, by putting z 1 1 = 1 and arranging the remaining k thresholds {z1 2, z2 2,..., zk 2 } such that the k independent equations in (2) hold under the new noise structure.

13 In this way, the action distributions that players face at their thresholds remain unchanged. Since they were willing to switch to greater actions given these beliefs, a best reply iteration in e starting at s is monotonic and leads to an equilibrium strategy profile s s of e. By Theorem 2 the greatest GGS under f, a, is weakly greater than a, the greatest GGS under f. Since f and f were arbitrary, we find that a = a. Dually, a = a. So we have proved: Proposition 7. Any supermodular 2 n game is noise independent. Symmetric games. Noise independence may fail in symmetric three-player, threeaction games. For example, in the following game g, the GGS depends on the noise structure: player i players I {i} (0, 0) (0, 1) (1, 1) (0, 2) (1, 2) (2, 2) If players noise terms η i all have identical distributions, then (0, 0, 0) will be the least GGS. To see this, consider the game e, in which payoffs are given by g for negative signals and 2 is dominant for positive signals. If all players switch between actions 0 and 2 at x i = 0, they hold the Laplacian belief at x i, i.e. they expect to face each of the opposing action profiles (0, 0), (0, 2), (2, 2) with probability 1. As expected payoff to any action is zero, they are indifferent at 3 their threshold x i = 0. Thus, (0, 0, 0) is the (necessarily) least equilibrium strategy profile in e. If, instead, we consider the noise structure in which η 1 and η 2 are uniformly distributed on [ 1 2, 1 2 ], while player 3 receives a very precise signal x 3 θ, then the unique GGS is (2, 2, 2). To see this, consider the game e endowed with this noise structure and suppose players 1 and 2 switch from 0 to 2 at the signal x i = 0, and player 3 switches to action 1 at x 3 = 1 and to action at x 3 = 1. Let 0,1 10 i (x i ) denote player i s payoff difference of playing 1 rather than 0 given the signal x i, and define 1,2 i (x i ) and 0,2 i (x i ) analogously. Let us first examine the degenerate case x 3 = θ, where player 3 is informed about the true state. We find that for i {1, 2}, 0,2 i (0) = , 1,2 i (0) = 2197, and 0, ( ) = 1 50, 1,2 3 ( 1 10 ) = 2 25, so that all players strictly prefer greater actions at their thresholds. Thus (using an upper-best reply iteration) a = (2, 2, 2) under this noise structure. Since players strictly prefer greater actions, this must remain the case if we draw x 3 from [θ ε, θ + ε] and choose ε sufficiently small this only slightly perturbs expected payoffs. And even if we very slightly perturb the payoffs of g, the expected payoff differences at the thresholds remain positive. Thus (2, 2, 2) is the greatest GGS for all games in some neighbourhood of the game g. This implies that in any global game embedding of g with this noise distribution, (2, 2, 2) is the unique GGS. 13

14 14 5. Applications It remains to show that our results may be applied in economically interesting settings. We report two examples. (In this section, n N also abbreviates the action profile where all players use n.) Refinancing Game. Consider a game g in which 3 lenders decide whether or not to refinance a firm invested in a long term project. Each lender can lend one unit of cash. The firm promises a repayment of R h at maturity. However, if the firm cannot fully refinance the project, it may not be able to repay in full. It may repay R l < R h, or default outright and not repay at all. The firm has the option to issue collateralised debt that gives lenders an additional payoff of c in both the partial and the complete default scenarios, but yields a lower repayment R m < R h when the project succeeds, as the provision of collateral is costly for the firm. The following matrix summarises the possible outcomes, where we identify the decision to give an unsecured loan with action 2, a secured loan with action 1, and the decision not to lend at all with action 0. lender i lenders I {i} (0, 0) (0, 1), (0, 2) or (1, 1) (1, 2) or (2, 2) c R l + c R m 2 0 R l R h For R l = 2 3 and R h = 2, and in the absence of collateralised debt, the GGS implies that lenders will not finance the project recall that in symmetric binary-action games, the GGS is the best reply to the belief assigning equal probability to opponents profiles (0, 0), (0, 2) and (2, 2). Introducing collateralised loans may change this result. For example, if c = 2 3 and R m = 3 2, we find that in the restricted game g [0, 1] (where lenders choose between 0 and 1), the GGS is 1, as it is the best reply to the belief assigning equal probability to (0, 0), (0, 1) and (1, 1). Similarly, the GGS in g [1, 2] is 2. Thus, by Lemma 5, the GGS in g is 2: lenders will be willing to give unsecured loans once we include option 1. Note that this result holds regardless of the noise structure, as noise independence is inherited from the two symmetric binary-action games by Theorem 6, even though symmetric games are not in general noise independent. Asymmetric Minimum Effort Game. Each player i I produces an intermediate good that is a necessary input for a final good. Players choose their production level a i {0,..., m} and production of the final good good is a min := min{a i i I}. Players individual payoff functions are given by u i (a i, a i ) = b i (a min ) c i (a i ), where b i and c i are increasing benefit and cost functions. This game typically has many equilibria. It generalises a model of Bryant [3] studied by Van Huyck et al. [25] and Carlsson and Ganslandt [5]. 8 8 Carlsson and Ganslandt [5] consider a form of trembling hand perfection in a symmetric version of this model; Van Huyck et al. [25] experimentally test a variant with linear and symmetric payoffs.

15 To solve the game for the GGS, we decompose it into m restricted binary-action games with joint action sets {k 1, k} I, 0 < k m. We will determine the GGS in these restricted games. By Theorem 2, the GGS equals k if and only if in the associated game e [k 1, k] we can construct an increasing strategy profile such that players choose action k for sufficiently high signals, and from which a best reply iteration leads upwards. 9 Our solution of this model is based on the following lemma (which actually makes a more general point about binary-action games). Lemma 8. Consider a strategy profile for e [k 1, k], given by thresholds z = (z i ) i I such that each player i switches from action k 1 to action k at z i [0, A ]. Let P i (z) be the probability that player i attaches to all her opponents playing action k given her opponents thresholds, when she receives the signal z i. Then the beliefs at the thresholds z i satisfy i I P i (z) = 1. The greatest GGS equals k if and only if we can adjust the thresholds z i such that each player prefers to play k at her threshold; it is unique if they can be made to strictly prefer this, i.e. (3) p i := c i(k) c i (k 1) b i (k) b i (k 1) < P i(z), From Lemma 8, we deduce that summing (3) over all players gives c i (k) c i (k 1) (4) b i (k) b i (k 1) < 1. i I Thus, a necessary condition for k to be the unique GGS is that the aggregate increase in players cost-benefit ratios due to everyone switching from k 1 to k is strictly smaller than 1. This is also sufficient, since k is p-dominant with p i < 1 when (4) holds, implying noise independent selection. Conversely, if (4) holds with the inequality reversed, k 1 must (necessarily) be the unique GGS though in this case the restricted game is generally not p-dominant solvable. Provided the left hand side of (4) crosses 1 at most once and from below for increasing k = 1, 2,..., m, this decomposition yields a generically unique, noise independent, GGS. (A sufficient, but not necessary, condition is that the individual c i are convex and b i are concave.) Conclusion We have shown how, for any supermodular complete information game, we may deduce its global game selection directly using solely the complete information game s payoffs and the global game s noise distribution (Theorem 2). For almost all games this gives a unique global game selection (Proposition 3). Our results may be used to establish selection under weakened assumptions on the global game (Proposition 4), which is useful from an applied perspective. From a practical point of view, our most powerful result is Theorem 6. It implies that the 9 Recall e [k 1, k] denotes the incomplete information game with with action sets restricted to [k 1, k], in which θ is drawn uniformly from U [ 1, A ] and k 1 is dominant for low signals. See Section 3.

16 global game selection may be derived by decomposing a many-action game into smaller games, for which existing heuristics and noise independence results can be applied. As we showed in Section 5, simplified conditions for the global game selection and a manageable heuristic to derive it in many action games make it easier to apply the theory global games. That should facilitate research on topics where strategic complementarities are crucial. 16 Appendix Proof of Theorem 2. The lower bound argument used in FMP and discussed in Section 3.1 yields s(θ ) a and, dually, s(θ ) a. We now prove the converse: s(θ ) a and, dually, s(θ ) a. We prove the first inequality; the second follows by duality. Without loss of generality assume θ = 0. Consider the simplified global game G (v) and its right continuous, increasing equilibrium strategy profile s v that converges to the limit strategy profile s as v 0. If s is continuous at θ, then there exist v and δ > 0 such that for all v < v and x [θ δ, θ + δ] we have s v(x) = s(θ ) =: a. Next, fix some v < min{δ, v} and consider e v, where players receive signals x i = θ + vη i and θ is distributed uniformly on the interval U, as in the simplified global game G (v). Assume that in the game e v players use the (truncated) strategy profile s given by: s v(x) if x θ, s (x) = s v(θ ) if x > θ. For any player i, and any signal x i < 0, action 0 is dominant both in e v and in G (v). So, in e v, for x < 0, the greatest best reply β(s )(x) = 0 = s v(x) = s (x). For x i [0, θ ], i s opponents receive signals smaller than θ + δ and follow s v i in ev. Since the distributions of players signals are identical in G (v) and e v, but i s payoff function is given by u i (, θ ) in e v and by u i (, x i ) in G (v), supermodularity implies that in e v, for 0 x θ, β(s )(x) = β(s v)(x) s v(x) = s (x). For x i > θ, player i s opponents receive signals greater than θ δ and hence play a i in the game ev. As a is a Nash equilibrium of g, in the game e v, for x > θ, β(s )(x) a = s (x). In sum, in the game e v, β(s ) s. Hence, an upper-best reply iteration starting at s yields a monotonically increasing sequence of strategy profiles that converges to an equilibrium profile s s. It follows that s (θ ) s (θ ) = s v(θ ) = a = s(θ ). As we define a as the greatest action profile played in any equilibrium strategy profile in (any) e v, we have a s(θ ) = s(θ ). If s is not continuous at θ, chose a sequence {θ n } n N converging to θ from above such that s is continuous at each θ n. By the first part of this proof, for each g n embedded at θ n, the greatest equilibrium strategy profile s n of the associated e n satisfies s n ( A ) s(θ ) (recall that we may fix v = 1 and evaluate s n ( A ) to find the greatest action profile played). As s n is an equilibrium in e n, ( i I, a i A i, x i 0, ui (s n i (x i), s n i (x i), θ n ) u i (a i, s n i (x i), θ n ) ) π i (x i x i ) dx i 0, R I 1

17 where π i ( x i ) is the conditional density over opponents signals. By the state monotonicity assumption (A3), the sequence of profiles s n converges to s = inf{s n n N} in monotonically decreasing fashion. By the dominated convergence theorem we find: ( i I, a i A i, x i 0, ui (s i (x i ), s i(x i ), θ ) u i (a i, s i(x i ), θ ) ) π i (x i x i ) dx i 0. R I 1 So s is an equilibrium strategy profile in the game e associated with g. As s prescribes an action profile weakly greater than s(θ ) at the signal A, we conclude that a s(θ ). Proof of Proposition 3. Fix f. In this proof, for any game g S, let us denote its associated incomplete information games by e(g), e (g) and the least and greatest GGS by a(g) and a(g). Also, for any game g S we define a global game embedding as in Lemma 1 by setting u g i (a i, a i, θ) := g i (a i, a i ) + θa i. Given this embedding, let g θ denote the complete information game embedded at θ. For r > 0, let B r (g) denote the open ball in R I A with radius r around g. To prove that S f is dense in S we may show that if g S f, there is a game arbitrarily close to g for which the GGS is unique. But this is always true, as the limit strategy profile of the embedding given by the payoffs u g i has only finitely many discontinuities. To prove that S f is open in S, note that if this limit strategy profile is continuous at 0, it must be constant over some interval [ 2ε, 2ε], as A is finite. Then, by the following result, S f is open: Claim 9. If g S f, and a = a(g θ ) = a(g θ ) for all θ [ 2ε, 2ε] for some ε > 0, then a = a(g ) = a(g ) for all supermodular games g in an ε-neighbourhood of g. Proof. Let g B ε (g) S. For all i, a i, and a i a i we find u g i (a i, a i, 2ε) u g i (a i, a i, 2ε) = g i (a i, a i ) g i (a i, a i ) 2ε(a i a i) g i (a i, a i ) g i (a i, a i ) 2ε g i(a i, a i ) g i(a i, a i ). For any opposing action distribution, this implies that the greatest best reply under g is weakly greater than under g 2ε. Hence, the greatest equilibrium profile in e(g ), is weakly greater than the greatest equilibrium profile in e(g 2ε ). So a(g ) a(g 2ε ) = a. Using a symmetric argument, we establish that a(g ) a. Dually, we may show that a(g ) = a, proving the claim. As S f is open and dense in S, its complement S f is closed and nowhere dense in S. Now, we may also establish measure theoretic genericity. A subset P of R I A is called porous if there are λ (0, 1) and k > 0 such that for any g P and ε (0, k), there exists some g R I A such that B λε (g ) B ε (g) P. Any porous subset of R I A is a Lebesgue null set ([16], p ). Let: 17 S f k := {g S f g θ S f, θ [ k, 0) (0, k]}. We will prove that S f k is porous. Assume g S f k and choose ε (0, k). Setting g := g ε 2, we

18 know that the GGS will be unique and identical to that of g for all games {g θ S θ ( ε 2, ε 2 )}. By Claim 9, we know that the GGS is unique for all supermodular games in an ε 4 -neighbourhood, thus B ε 4 (g ) S f k =. Setting λ = 1, we have for ε (0, k) that B 4 λε(g ) B ε (g) S f k, i.e., is a countable union of Lebesgue null sets and hence S f k is porous. Thus S f = {k Q k>0} S f k a null set itself. To see that, by contrast, S is of infinite Lebesgue measure, note its interior is non-empty. In addition, for each ball B contained in S, we find another ball of greater measure, disjoint from B, by multiplying the payoffs of games in B with a sufficiently large constant. Proof of Proposition 4. Consider a generalised game with noise structure f and let, for some fixed θ, a θ = a θ = a, with both a θ and a θ continuous at θ. Note that ŝ š, so it suffices to show that š(θ ) a ŝ(θ ). We will prove the first inequality; the second follows by duality. To do so, we will compare the payoff functions u i satisfying (A1) (A3 ) to payoff functions u i satisfying (A1) (A4) so that we may apply Theorem 2. For two complete information games, we write g 1 g 2 if for all i, a i < a i and a i their corresponding payoff functions satisfy g 1 i (a i, a i ) g 1 i (a i, a i ) < g 2 i (a i, a i ) g 2 i (a i, a i ). Note that, given any opposing action distribution, the smallest best reply under g 1 will be weakly smaller than the smallest best reply under g 2. Now, consider the game g θ ε, with ε chosen such that a = a θ ε = a θ ε (using continuity of a θ and a θ at θ ). Next, consider the supermodular game g given by g i (a i, a i ) = u i (a i, a i, θ ε) ka i, k > 0. By Claim 9, we can choose k such that the GGS in g is unique and equals a. Furthermore, without loss of generality, assume that there exist extreme values ˇθ < θ and θ < ˆθ such that we have a chain of four games satisfying gˇθ g g θ gˆθ. Using this chain, we construct u : u i (a i, a i, ˇθ) if θ < θ ε, u i(a i, a i, θ) = θ θ u ε i(a i, a i, ˇθ) + θ (θ ε) g ε i (a i, a i ) if θ ε θ < θ, ˆθ θ ˆθ θ g i (a i, a i ) + θ θ ˆθ θ u i (a i, a i, θ ) if θ < θ < ˆθ, (ˆθ + 1 θ)u i (a i, a i, θ ) + (θ ˆθ)u i (a i, a i, ˆθ) if ˆθ θ < ˆθ + 1, u i (a i, a i, ˆθ) if ˆθ + 1 θ, Comparing u and u, we see that the dominance regions have been shifted to the right, gˇθ is now embedded at θ ε, g at θ, g θ at ˆθ, gˆθ at ˆθ + 1 and the remaining games are linear interpolations. Thus, for any θ, the smallest best reply under u is greater than under u. Also, as payoffs are linearly interpolated between gˇθ g g θ gˆθ, payoff differences are piecewise linear in θ, thus satisfy (A3). Clearly, (A4) is satisfied as well. Finally, consider the global game G (v) with the newly constructed payoff function u, and the same noise structure f and prior φ as G(v). For any v > 0, G(v) has a smallest equilibrium strategy 18