Optimism and Pessimism in Games

This article considers the impact of ambiguity in strategic situations. It extends the existing literature on games with ambiguity‐averse players by allowing for optimistic responses to ambiguity. We use the CEU model of ambiguity with a class of capacities introduced by Jaffrray and Philippe (Mathematics of Operations Research 22 (1997), 165–85), which allows us to distinguish ambiguity from ambiguity‐attitude, and propose a new solution concept, equilibrium under ambiguity (EUA), for players who may be characterized by ambiguity‐preference. Applying EUA, we study comparative statics of changes in ambiguity‐attitude in games with strategic complements. This extends work in Eichberger and Kelsey (Journal of Economic Theory 106 (2002), 436–66) on the effects of increasing ambiguity if players are ambiguity averse.


INTRODUCTION
Beginning with the seminal work of von Neumann and Morgenstern (1944), expected utility theory (EUT) has been closely related to the analysis of strategic decision-making. Equilibrium concepts for games combine consistency properties for the beliefs of players with assumptions about decision-making in the light of these beliefs. Many researchers (e.g., Allais, 1953;Ellsberg, 1961;Kahneman and Tversky, 1979) questioned the descriptive validity of EUT. Despite this, progress in applying the ensuing theoretical developments of Schmeidler (1989), Gilboa and Schmeidler (1989), and Sarin and Wakker (1992) to game theory has been slow.
There is a small literature about strategic behavior in games, reviewed below, which deals with the three main issues arising in games by departing from EUT: (1) How much consistency in beliefs does one want to impose in equilibrium?
(2) To what extent do beliefs about the opponents' behavior have to be independent? (3) How do attitudes toward ambiguity affect behavior in games?
In this article, our research will focus on the third question. The literature that followed Ellsberg (1961) has focused on ambiguity-aversion. As we will show, extending the analysis to include ambiguity-seeking requires us to reconsider the question of consistency in beliefs.
Considering optimistic attitudes toward ambiguity introduces nonconvexities in preferences. Thus, it is not possible to prove existence of equilibrium with standard techniques based on fixed-point theorems. Instead, we use lattice theory to demonstrate existence of equilibrium in games with increasing differences. 2 We view it as an advantage of this approach that it enables us to study existence and comparative statics in a common framework. Moreover, games with increasing differences have many economic applications, for instance, in industrial organization, macroeconomics, and public good provision.
1.1. Ambiguity and Ambiguity-Attitudes. Following Ellsberg's (1961) criticism of the subjective expected utility (SEU) approach of Savage (1954), both experimental and theoretical research have been directed toward a better understanding of ambiguity and modeling it in individual decision making. Ambiguity describes situations where individuals cannot or do not assign subjective probabilities to uncertain events. By now, there is also a substantial body of experimental evidence which shows that people behave differently in the presence of ambiguity (see, e.g., Camerer and Weber, 1992).
Most of the literature following Ellsberg (1961) assumed that individuals are ambiguity-averse, that is, they would pay some positive amount of money to avoid a situation where probabilities are poorly defined. Experimental evidence, however, does not uniformly confirm this view (see, for instance, Gonzalez and Wu, 1999;Kilka and Webe, 2001;Abdellaoui et al., 2005). Although many individuals behave cautiously when there is ambiguity, a significant group of individuals behave in the opposite way, a behavior which we call ambiguity-loving or ambiguity-seeking. Moreover, the same individual may express both ambiguity-seeking and ambiguity-aversion in different contexts. In an experimental study of ambiguity in games, Ivanov (2011) finds that more subjects are ambiguity-loving than ambiguity-averse.
In this article, we study the influence of ambiguity and ambiguity-attitude on behavior in games. Informally, ambiguity refers to how much uncertainty there is about the probabilities, and ambiguity-attitude measures how the individual reacts to unknown probabilities. We believe that ambiguity-attitude is a personal characteristic of an individual player and thus should be taken as exogenous.
We use Choquet expected utility (CEU), in which individuals have beliefs represented by capacities (nonadditive subjective probabilities). CEU represents preferences as maximizing the expected value of a utility function, where the expectation is represented as a Choquet integral (Choquet, 1953-54). These preferences have been axiomatized by Schmeidler (1989) and Sarin and Wakker (1992). Jaffray and Philippe (1997) propose a class of capacities (henceforth JP-capacities) that depend on a parameter which measures ambiguity-attitude. We use their approach to represent preferences of players by the Choquet integral with respect to a JP-capacity. 3 There is a small literature on strategic ambiguity. 4 Dow and Werlang (1994) use CEU preferences to represent players' ambiguity about their opponents' strategy choice. Best responses of a player depend on the capacity (i.e., their beliefs) and the degree of ambiguity embodied in it. In equilibrium, consistency requirements relating beliefs about opponents to their strategy choices have to be satisfied. Dow and Werlang (1994) represented the strategies an individual believes the opponents would play by the support of the capacity. 5 However, there is more than one way to define the support of a capacity. Equilibrium concepts in the literature differ mainly in the notion of support used to define an equilibrium. One of the surprising facts is that the support notions for capacities used in context with ambiguity-averse preferences are no longer suitable if players may be ambiguity-loving. Hence, we propose a new definition of support, which we believe is more appropriate, and use it as the basis of an equilibrium concept for games with ambiguity.
1.2. Ambiguity in Games with Positive Externalities. Many games in economics have prices or quantities as strategies that can be ordered in a natural way. Moreover, many of these applications are games of positive aggregate externalities with increasing differences. For such games, we show that an increase in ambiguity-seeking increases equilibrium strategies.
Intuitively, as a player becomes more ambiguity-loving, he will place more weight on outcomes which are perceived as good. If there are positive externalities, good outcomes are associated with opponents' playing high strategies. Increasing differences imply that raising the decisionweight on high strategies of the opponents will increase incentives to play a high strategy. Hence, the best response function of the player will shift up, which increases the equilibrium strategies of all players.
Strategic complementarity can lead to multiple equilibria. In this case, we show that for sufficient ambiguity, equilibrium will be unique. If players are sufficiently optimistic (respectively, pessimistic) the equilibrium strategies will be higher (respectively, lower) than in the highest (respectively, lowest) equilibrium without ambiguity. Ambiguity and ambiguity-attitude have distinct effects. Ambiguity causes the set of equilibria to collapse to a single equilibrium, whereas an increase (decrease) in optimism causes the set of equilibria to move up (down).
1.2.1. Organization of the article. In Section 2, we present our framework and definitions. Section 3 introduces the equilibrium concept for games with players of differing ambiguityattitudes. For the case of games with strategic complements, we prove the existence of equilibrium and derive the comparative statics results in Section 4. Related literature is discussed in Section 5. Section 6 contains our conclusions. Appendix Section A relates a number of alternative notions of the support of a capacity and contains the proofs for results on capacities. All other proofs are gathered in Section B. Our results are illustrated by an application to the centipede game, which can be found in the Appendix Subsection B.3.

MODELING OPTIMISM IN GAMES
We consider a game = N; (S i ), (u i ) : 1 i n with finite pure strategy sets S i for each player and payoff functions u i . The notation, s −i , indicates a strategy combination for all players except i. The space of all strategy profiles for i's opponents is denoted by S −i . The space of all strategy profiles is denoted by S. Player i has utility function u i : S → R, for i = 1, . . . , n. When convenient, we write u i (s) = u i (s i , s −i ).

Nonadditive Beliefs and Choquet Integrals.
In CEU, beliefs, ambiguity, and ambiguityattitude are represented as capacities, which assign nonadditive values to subsets of S −i . Formally, capacities are defined as follows.

DEFINITION 1. A capacity on S −i is a real-valued function ν on the subsets of S
The expected utility of the payoff obtained from a given act, with respect to a nonadditive belief, ν, can be found using the Choquet integral, defined below.
where the strategy profiles in S −i are numbered so that u i (s i , A simple though extreme example of a capacity is the complete uncertainty capacity. Intuitively it describes a situation where the decision maker knows which strategy profiles are possible but has no further information about their likelihood. EXAMPLE 1. The complete uncertainty capacity, ν 0 on S −i is defined by ν 0 (S −i ) = 1, ν 0 (A) = 0 for all A S −i .
Convex capacities can be associated in a natural way with a set of probability distributions called the core of the capacity. DEFINITION 4. Let ν be a capacity on S −i . The core, C(ν), is defined by Since a capacity and its dual represent upper and lower bounds for the probability distributions in the core, it is natural to define the degree of ambiguity of a player as follows.
DEFINITION 5. Let ν be a convex capacity on S −i . Define the maximal degree of ambiguity of These definitions are adapted from Dow and Werlang (1992). They provide upper and lower bounds on the amount of ambiguity which the individual perceives. Schmeidler (1989) shows that for a convex capacity, ν, the Choquet integral of a payoff function u i is equal to the minimum over the core of the expected value of u i , that is, , where E denotes the expected value of u i with respect to the probability distribution p on S −i . Indeed, Schmeidler (1989) argues that convex capacities represent ambiguity-aversion. More recently, Wakker (2001) has shown that convexity is implied by a generalized version of the Allais paradox. This provides another reason to take convex capacities as a representation of ambiguity and the CEU as the pessimistic evaluation of acts given this ambiguity.
2.2. Optimism, Pessimism, and JP-Capacities. This article is concerned with modeling both ambiguity-averse and ambiguity-seeking behavior. We achieve this by focusing on the class of JP-capacities introduced by Jaffray and Philippe (1997). DEFINITION 6. A capacity ν on S −i is a JP-capacity if there exists a convex capacity μ and α ∈ [0, 1], such that ν = αμ + (1 − α)μ. (Recall thatμ denotes the dual capacity of μ.) Ambiguity is represented by a convex capacity μ and its core, C(μ). A JP-capacity is a convex combination of the capacity μ and its dual. As the following proposition shows, the CEU of a JP-capacity is a weighted average of the minimum and the maximum expected utility over the set of probabilities in C(μ).
PROPOSITION 1. (JAFFRAY AND PHILIPPE, 1997). The CEU of a utility function u i with respect to a JP-capacity ν = αμ These preferences lie in the intersection of the CEU and α-Minimum Expected Utility (α-MEU) models. When beliefs are represented by JP-capacities, perceived ambiguity is represented by the capacity μ, whereas ambiguity-attitude is represented by α, with higher values of α corresponding to more ambiguity-aversion. If α = 1, then we obtain the Maximum Expected Utility (MEU) model axiomatized by Gilboa and Schmeidler (1989). For α = 0, we deal with a pure optimist, whereas in general, for α ∈ (0, 1), the player's preferences have both optimistic and pessimistic features. Hence, JP-capacities allow a distinction between ambiguity and ambiguity-attitude, which is formalized in the following definition. 6 DEFINITION 7. Let ν and ν be two capacities on S. We say that ν is more pessimistic than ν if for all A ⊆ S, ν(A) ν (A).
A neo-additive capacity describes a situation where the individual's "beliefs" are represented by a probability distribution π. However (s)he has some doubts about these beliefs. This ambiguity about the true probability distribution is reflected by the parameter δ. The highest possible level of ambiguity corresponds to δ = 1, whereas δ = 0 corresponds to no ambiguity. The Choquet expected value of a payoff function u i (s i , ·) with respect to the neo-additive-capacity ν is given by This expression is a weighted averaged of the highest payoff, the lowest payoff, and an average payoff. The response to ambiguity is partly optimistic represented by the weight given to the best outcome and partly pessimistic.

Modeling Players' Preferences.
To study the impact of ambiguity, especially ambiguityloving behavior, in games, it is necessary to make a clear distinction between ambiguity and ambiguity-attitude. There are, however, only a small number of models that allow one to do this.
(1) The α-MEU model, Marinacci (2002), which represents ambiguity by a set of probability distributions and ambiguity-attitude by the parameter α expressing the weight given to the worst possible expected utility. (2) CEU, Schmeidler (1989) and Sarin and Wakker (1992) in combination with JP-capacities.
Preferences are represented by a Choquet integral with respect to a nonadditive belief, ν = αμ + (1 − α)μ, where μ is a convex capacity. The parameter α can similarly be interpreted as a measure of ambiguity-attitude. (3) The smooth model or KMM-model, Klibanoff et al. (2005).
With the exception of the CEU model for the special case of a neo-additive capacity, none of these has an axiomatization in terms of preferences over Savage acts, which would allow one to distinguish ambiguity from the agent's ambiguity-attitudes. 8 In the case of CEU, preferences over acts determine the capacity ν uniquely but not necessarily the JP-form ν = αμ + (1 − α)μ, which would achieve such a separation. For the α-MEU model, there is no axiomatization so far in the standard single-time period Savage framework. 9 In α-MEU the set of priors is not well defined. Siniscalchi (2006) shows that there may be more than one set of priors and more than one α, which represent the same preferences. 10 In the axiomatization of the smooth model, the attitude toward ambiguity reflected in φ is determined by a second preference order over second-order acts, hence not derived from preferences over Savage acts alone. We consider the problem of the axiomatic separation of ambiguity-attitude and ambiguity as a challenge for future research.
Given the unresolved issues surrounding the question of how to distinguish ambiguity of beliefs from ambiguity-attitudes axiomatically, 11 we restrict attention to JP-capacities where this distinction between ambiguity-attitude, as reflected by the parameter α, and ambiguous beliefs, as represented by the convex part of the capacity μ, appears natural.

EQUILIBRIUM WITH OPTIMISM AND PESSIMISM
3.1. The Support of a Capacity. In the previous literature on games with ambiguity, the support of a player's beliefs represents the profile of strategies that he believes his opponents will play. An equilibrium is defined to occur when all profiles in the support consist only of best responses. In general, capacities reflect both ambiguity and ambiguity-attitudes. It is therefore necessary to separate ambiguity-attitudes from the ambiguous beliefs in order to find an appropriate support notion.
It is not possible for us to use existing definitions of the support unmodified since many of them have implicitly assumed ambiguity-aversion. Two definitions have been used for ambiguityaverse players with convex capacities: the Dow-Werlang (DW) support (Dow and Werlang, 1994), and the Marinacci (M) support (Marinacci, 2000). The DW-support of the capacity ν, supp DW ν is a set E ⊆ S −i , such that ν(S −i \E) = 0 and ν(F ) > 0, for all F such that S −i \E F . It always exists; however, it may not be unique. An example is the capacity of complete uncertainty in Example 1, where {s −i } is a support for any s −i ∈ S −i . Marinacci (2000) defines the support of a capacity ν to be the set of states with positive capacity, Provided it exists, supp M ν is always unique. However, this support can be empty.
Ryan (2002) studies notions of support for sets of probability distributions. Based on his work, we suggest the following definition of support for a convex capacity.
In Appendix Section A we show that, for convex capacities, the DW-support and the M-support coincide with the above notion if the DW-support is unique. Moreover, they also agree with the weak support for MEU defined by Ryan (2002). We believe that this consistency of the support notions for MEU and CEU is a strong argument for using this definition. In the context of games, states correspond to strategy profiles of the opponents. This support notion appears as 8 See Chateauneuf et al. (2007). 9 Ghirardato et al. (2004) have proposed a way to define a unique set of priors for the α-MEU model. However, as we argue in Eichberger et al. (2011), there are some problems with this approach when the state space is finite. Klibanoff et al. (2011) use an infinite time period model to axiomatize a version of α-MEU, which satisfies a symmetry assumption.
10 See especially the on-line appendix to Siniscalchi (2006). 11 For more discussion of this issue compare Epstein (1999) and Ghirardato and Marinacci (2002).
the natural choice because it does not require best-reply behavior against any strategy opponents may possibly play but only against those which are unquestionably played. A natural alternative definition is p ∈C(μ) supp p, which has been studied by Dow and Werlang (1991) and Lo (1996). They show that solution concepts for games based on this definition of support do not differ significantly from Nash equilibrium (henceforth NE). The alternative notion is therefore not compatible with modeling deviations from NE due to ambiguity. 12 It is not obvious how to extend the definition of support to nonconvex capacities. Consider the neo-additive capacity ν = δ(1 − α) + (1 − δ)π from Example 2. This capacity assigns positive values to all strategy profiles Hence, none of these concepts are suitable as a support notion for this capacity as they do not allow us to make a distinction between those strategies which a given player believes are possible for his opponents and others.
The problem is that, whenever there is even a small amount of optimism, neo-additive capacities assign positive capacity to all profiles. However, this does not mean the player "believes" in these profiles. Looking at profiles with positive capacity confounds belief and ambiguity-attitude, since optimism increases the capacity values assigned to all profiles. Sarin and Wakker (1998) argue that the beliefs of a decision-maker can be deduced from the weights used in the Choquet integral. Based on this criterion, one can show that a profile, s −i , always gets positive weight in the Choquet integral of a neo-additive capacity if and only if π(s −i ) > 0. For this reason, it seems desirable that supp ν = supp π in this case. This also makes sense in terms of our intuition. A neo-additive capacity describes a situation where the individual's beliefs, although expressed with some doubts, are represented by the probability distribution π. This intuition suggests that the set of profiles in which the player believes is given by supp π.
Thus, we argue that a necessary condition for a player to believe in a profile is that it should always get positive weight in the Choquet integral. This avoids the problem of confounding between ambiguity-attitude and belief described above. The set of all profiles with this property can be defined as follows.
DEFINITION 9. Define the set of decision-weight increasing profiles of a capacity v, A nonconvex capacity has a set of probabilities associated with it. These are the set of decision-weights used in evaluating the Choquet integral. Essentially B(ν) is the intersection of the supports of these weights. This is in the spirit of Sarin and Wakker (1998), who argue that an individual's beliefs can be deduced from these decision-weights. For a convex capacity the decision weights are the extreme points of the core. Thus B(ν) may be seen as a generalization of our definition of the support for a convex capacity.
One could use B(ν) as the definition of the support of a capacity. However, B(ν) depends on α. Since α is a measure of ambiguity-attitude, one would prefer the set of strategy profiles in which a player believes to be independent of it. These considerations lead us to propose a closely related set, instead. Hence, for JP-capacities, we propose a support notion which relates only to the convex part μ.
Our reason for defining the support of a JP-capacity in terms of its convex part is that a capacity and its dual are simply two ways of representing the same information. Since the JP-capacity ν is a weighted average of μ and its dual, μ does contain as much information about the player's beliefs as ν. The following result shows that all elements of supp JP ν always receive positive weight in the Choquet integral. Thus they meet the necessary condition described above.
If ν is a JP-capacity, B(ν) does not depend on the JP-representation. Thus, to a great extent, our support notion is independent of the parameters of the JP-representation. 14 We previously argued that for a neo-additive capacity, ν = δ(1 − α) + (1 − δ)π, our intuition required that the support of ν be supp π. The following result shows that our definition of support has this property.

Independent Beliefs and Equilibrium under Ambiguity.
In analogy to NE, we define equilibrium in beliefs under ambiguity (henceforth EUA) to be a situation where each player maximizes his (Choquet) expected utility given his ambiguous beliefs about the behavior of his opponents. These beliefs have to be reasonable in the sense that each player believes that his opponents play best responses. We interpret this as implying that the support of any given player's beliefs should be nonempty and consist only of best responses of his opponents. Let DEFINITION 11. An n-tuple of capacitiesν = ν 1 , . . . ,ν n is an equilibrium in beliefs under ambiguity if for all players, i ∈ I, If there is a strategy profileŝ = ŝ 1 , . . . .ŝ n such that for each playerŝ −i ∈ suppν i , we say thatŝ is an equilibrium strategy profile. Moreover, if for each player suppν i contains a single strategy profileŝ −i we say thatŝ is a singleton equilibrium.
In equilibrium, a player's evaluation of a particular strategy may, in part, depend on strategies of the opponents which do not lie in the support. We interpret these as events a player views as unlikely but which cannot be ruled out. This may reflect some doubts the player may have about the rationality of the opponents or whether he correctly understands the structure of the game.
In an EUA, players choose pure strategies and do not randomize. Nonsingleton equilibria cannot be interpreted as randomizations. In such an equilibrium, some player i will have two or more best responses. The support of other players' beliefs about i's play will contain some or all of them. Thus an equilibrium, where the support contains multiple strategy profiles, is an equilibrium in beliefs instead of in randomizations. If there are only two players and the beliefs are additive, then an EUA is a NE.
The model can accommodate observed behavior which is ruled out in NE. Players may be better off (in an expost sense) in an EUA than in the unique NE of a game. An example of this is the centipede game discussed in Appendix Subsection B.3.
For games with more than two players, however, an EUA with additive beliefs may not be a NE because players may not believe that their opponents act independently. In addition, it is possible for any two players to have different beliefs about the behavior of a third player. For NE, independence of beliefs follows immediately from the requirement that beliefs coincide with the (mixed) strategies actually played by the opponents. The independent choices of mixed strategies define a unique probability distribution on the product space of strategy sets. Both conditions fail for EUA beliefs.
It is well known (Denneberg, 2000, pp. 53-56) that there are several ways of extending the product of capacities from the Cartesian products of the strategy sets to general subsets of the product space. One possibility to obtain a notion of independent beliefs would be to apply the Möbius product (Ghirardato, 1997;Hendon et al., 1996) and use the JP-capacity of the Möbius product of μ as the relevant product capacity. 15

EXISTENCE AND COMPARATIVE STATICS
In this section, we prove the existence of equilibrium and study the comparative statics of changes in ambiguity and ambiguity-attitude on equilibrium.
4.1. Existence of Equilibrium. In many economic applications strategies are real numbers, such as prices or quantities, which have a natural order. Since strategy sets are finite, we identify strategy sets with an interval of the integers, 17 Increasing differences implies that a given player who perceives his opponents increase their strategy has an incentive to increase his own strategy as well. Bertrand oligopoly with linear demand and constant marginal cost provides an example of a game with increasing differences. Positive externalities and increasing differences will be a maintained hypothesis throughout the rest of the article. Negative externalities may be defined in an analogous way.
The following existence result is proved in Appendix Section B. Fix a vector of ambiguityattitude parameters α = (α 1 , . . . , α n ) ∈ [0, 1] n and maximal and minimal degrees of ambiguity (λ, γ) = ((λ 1 , γ 1 ), . . . , (λ n , γ n )), 0 γ i λ i 1; then there exists an EUA where players' beliefs are represented by JP-capacities with parameters α, satisfying λ i (μ i ) λ i and γ i (μ i ) γ i for all i ∈ I. THEOREM 1. Let be a game of positive externalities and increasing differences. Then for any exogenously given n-tuples of ambiguity-attitudes α, maximal degrees of ambiguity λ, and minimal degrees of ambiguity γ, (γ λ) the game has a singleton equilibrium in beliefs under ambiguity in JP-capacities ν = ν 1 , . . . , ν n , where ν i = α i μ i + (1 − α i )μ i for 1 i n. The convex capacity μ i has maximal degree of ambiguity at most λ i and minimal degree of ambiguity at least γ i .
In general, EUA will not be unique. In part, this arises because we have allowed a range for the degrees of ambiguity, that is, γ i < λ i . If this range were reduced, then EUA would have the same uniqueness properties as NE.

Comparative Statics.
Comparative statics exercises are difficult because the capacity represents three distinct concepts: the perceived ambiguity, the attitude to that ambiguity, and beliefs about the opponents' strategies. Moreover, these concepts are interrelated. For instance, if a player's ambiguity-attitude changes, this may cause him to play a different strategy. The opponents are likely to change their strategies in response, which would require the first player to revise his beliefs as well so as to maintain consistency.
To investigate the comparative statics of ambiguity-attitude, we need to vary ambiguityattitude while holding perceived ambiguity constant. We do this by placing exogenous bounds on the maximal and minimal degrees of ambiguity. The comparative static results do not depend on the values of these bounds despite the fact that they are exogenous. For the comparative static analysis we strengthen positive externalities to the following assumption.
This is a separability assumption. It says that a player only cares about a one-dimensional aggregate of his opponents' strategies. An example would be a situation of team production in which the utility of a given team member depends on his own labor input and the total input supplied by all other members of the team. Negative aggregate externalities may be defined in an analogous way.
The following comparative static result shows that an increase in pessimism will reduce the equilibrium strategies in games with positive aggregate externalities and increasing differences. If there are multiple equilibria, the strategies played in the highest and lowest equilibria will decrease. We assume that when the ambiguity-attitude of one player changes, the ambiguityattitudes of other players and the perceived ambiguity are held constant. THEOREM 2. Let be a game of positive aggregate externalities with increasing differences. Assume that beliefs are represented by JP-capacities and let α = (α i , α −i ) denote the vector of ambiguity-attitudes. Let s(α) (respectively,s(α)) denote the lowest (respectively, highest) equilibrium strategy profile when the minimal (respectively, maximal) degree of ambiguity is γ (respectively, λ). Then s(α) ands(α) are decreasing functions of α i .

Multiple Equilibria.
Strategic complementarity can give rise to multiple Nash equilibria. Under some assumptions, we can show if there are multiple equilibria without ambiguity and there is sufficient optimism (respectively, pessimism), equilibrium will be unique and will correspond to the highest (respectively, lowest) equilibrium without ambiguity. To prove this we need the following assumption.
This assumption is required for technical reasons. If the strategy space were continuous and utility were concave in the player's own strategy, it would be implied by our other assumptions. It says that the gaps in the discrete strategy space do not fall in the "wrong place." PROPOSITION 4. Consider a game, of positive externalities with increasing differences that satisfies Assumption 1. There existᾱ (respectively, α), 0 < α ᾱ < 1, andγ such that if the minimal degree of ambiguity is γ(μ i ) γ and α i ᾱ, (respectively, α) for 1 i n, then there is a unique singleton equilibrium with an equilibrium strategy profile that is smaller (respectively, larger) than the smallest (respectively, largest) equilibrium strategy profile without ambiguity.
Intuitively, as ambiguity increases a player will become less confident in the behavior of his opponents. Consequently he will respond less to perceived changes in their behavior. Thus, increasing ambiguity reduces the slope of the best response functions. When they are sufficiently flat it is only possible for them to intersect once. Hence, the equilibrium strategy profile is unique. Even when Assumption 1 is not satisfied, Lemma 13 shows that as ambiguity increases the Choquet expected payoffs tend to max Thus, the equilibrium payoffs will be unique even when the equilibrium strategies are not.
In a game with increasing differences and multiple Nash equilibria, increasing ambiguity causes the multiplicity of equilibria to disappear, whereas increasing ambiguity-aversion causes the equilibrium strategies to decrease. Hence, ambiguity and ambiguity-attitude have distinct effects.

LITERATURE REVIEW
In a broad view, the literature on games with ambiguity can be organized into two strands. One way to approach ambiguity in games is to interpret Nash equilibria as equilibria in beliefs. In this perspective, ambiguity concerns the behavior of the opponents. Players choose pure strategies but have ambiguous beliefs about the opponents' behavior. Equilibrium means that there is some degree of consistency between ambiguous beliefs and strategies played. The papers by Dow and Werlang (1994), Marinacci (2000), Eichberger and Kelsey (2000), and this article belong to this group. Haller (2000) noted the focus on a support notion for capacities which is a common feature of these equilibrium concepts.
The second way to approach equilibrium under ambiguity assumes that players choose mixed strategies. Hence, ambiguity concerns the mixed strategies of the opponents. Equilibrium, once again, requires some degree of consistency with the mixed strategies actually played. Klibanoff (1996), Lo (1996), and more recently Bade (2011), and Lehrer (2011) choose this approach. A characteristic feature of these papers is the "observational equivalence," as Bade (2011) calls it, between Nash equilibrium and these equilibrium notions for two-player games. Section 11 of Bade (2011) provides an excellent and extensive discussion of these equilibrium notions and illustrates their similarities and differences by examples. In particular, she highlights the fact that all these approaches focus on the case of ambiguity aversion. For the remainder of this review, we will concentrate on the more recent literature. Lehrer (2011) considers decision-making when individuals have incomplete information about the probabilities. In the simplest version, they may have a probability defined on a subalgebra of events instead of on all measurable events. In more complicated versions, the individuals may only know the expectations of certain random variables. The model is a special case of CEU with a convex capacity. It is less general than CEU since it is capable of modeling ambiguity-aversion but not ambiguity-seeking.
Lehrer proceeds to apply this model to games with a fixed partition of the strategy spaces. Players only know the probabilities of elements of the partition, but not those of individual strategies. The key difference from the present article is that players regard their own mixed strategies as being ambiguous. In contrast we assume that a given player views his opponents' behavior as potentially ambiguous but perceives no ambiguity about his own behavior.
Bade (2011) investigates whether a player can gain a strategic advantage by deliberately creating ambiguity. Given her assumptions, she finds the answer is no. Since her model uses general preferences over strategies, her results are applicable to a wide range of models. However, her results do not apply to our model because she makes two assumptions which our model does not satisfy. Firstly, her monotonicity assumption rules out the possibility that a state may have positive weight if associated with a bad outcome but zero weight if associated with a good outcome. CEU preferences do not, in general, satisfy this assumption. Secondly, like Lehrer, she assumes that players agree on the ambiguity of strategies. Hence, a player views his own action as being as ambiguous as the opponent does.
In a recent paper, Riedel and Sass (2011) study whether there is a strategic advantage to creating ambiguity. In their paper, players can commit to using an ambiguous randomizing device, like an urn with partial information over the characteristics of the balls in it. As equilibrium concept, they consider a Nash equilibrium in these "ambiguous strategies." Hence, players have common knowledge about "ambiguous strategies." Riedel and Sass (2011) show by example that "deliberate strategic ambiguity" may model behavior which can be observed in laboratory experiments.
In a dynamic repeated games context, Battigalli et al. (2012) use the smooth model of ambiguity. Individuals play the same game an infinite number of times in a strangers setting. Hence, repeated game effects do not arise. On the equilibrium path players choose best responses to the actual population-wide distribution of strategies played by opponents. Off the equilibrium path, however, non-Nash behavior can be sustained due to ambiguity about behavior there. Thus the set of equilibria is larger than the set of Nash equilibria. The dynamic equilibrium notions studied cannot be applied to games in strategic form directly. For the case of ambiguity aversion, Eichberger and Kelsey (2004) provide a first extension of the EUA concept to dynamic games which confirms the analysis of Battigalli et al. (2012) that ambiguous out-of-equilibrium beliefs may expand the set of Nash equilibria.

CONCLUSION
In this article, we have studied the impact of ambiguity in games with players who are not necessarily ambiguity-averse. We have extended previous work by proposing new definitions of support and equilibrium which allow for an ambiguity-loving (optimistic) attitude toward uncertainty. The new notion of equilibrium has already been successfully applied to the theoretical analysis of games in Eichberger et al. (2009) and to experimental studies in Eichberger et al. (2008) and Eichberger and Kelsey (2011).
Modeling optimistic responses to ambiguity introduces nonconvexities in preferences. Hence, existence proofs and comparative static results cannot be obtained with the same generality as in the convex case which has been treated in the literature. In this context, the article makes a number of innovations, for instance, developing techniques for analyzing the distinct effects of ambiguity and ambiguity-attitude. Compared to Eichberger and Kelsey (2002), we study a significantly broader class of games. In particular, we do not assume symmetry or concavity in a player's own strategy. Aggregate externalities are only assumed for the comparative statics section.
A couple of issues remain for future research. Extending the analysis to games with more than two players requires one to confront the problem of correlated beliefs. Lo (2009) argues for accepting correlations among beliefs about the opponents' behavior as the more adequate way of modeling a Nash equilibrium. Bade (2011) suggests notions of independence in beliefs which need to be related to and compared with notions of independence for capacities and multiple priors. This is a wide field for future research both theoretically and experimentally.
A second open question concerns the treatment of mixed strategies. Given preferences over strategies yielding ambiguous payoffs, mixed strategies can no longer be a straightforward extension of pure strategies. Whether mixed strategies should be treated as an ambiguity reducing device (Raiffa, 1961) or as a strategic element in its own right remains an unresolved issue. Earlier work on preferences for randomization under ambiguity by Eichberger and Kelsey (1996) and Klibanoff (2001) has found experimental support in Dominiak and Schnedler (2011) and a new theoretical perspective in Saito (2012). This work will lead to a reconsideration of the role of mixed strategies in games under ambiguity. APPENDIX A. Alternative Notions of Support. In Subsection 3.1, we introduced a support notion for convex capacities, supp μ, and argued that it is suitable because it coincides with all common definitions of support for convex capacities and a leading support notion for MEU. In this appendix, we will substantiate these claims with some formal results.
Ryan (2002) discusses several notions of a support for MEU preferences. 18 For a set P ⊆ (S −i ) of multiple priors, Ryan (2002, p. 56) defines a strong (respectively, weak) support of P as p ∈P supp p (respectively, p ∈P supp p), where supp p denotes the usual support of a probability distribution. The strong (respectively, weak) support comprises the strategy combinations of the opponents which have a positive probability under some (respectively, all) probability distributions from P. For convex capacities, preferences have both CEU and MEU representations. Hence, these support notions can be applied to a convex capacity, μ, in which case the weak support coincides with supp μ.
The following proposition shows that supp μ coincides with the M-support and the set of states that always receive positive weight in the Choquet integral, B(ν). Even the DW-support, which is in general nonunique, is closely related to the support of Definition 8 as Proposition 5 demonstrates.
PROOF OF PROPOSITION 2. We prove this by demonstrating that for all A If ν is a JP-capacity, B(ν) does not depend on the JP-representation. Thus, to a great extent, our support notion is independent of the parameters of the JP-representation.
We conclude this section with the proof of Proposition 3, which finds the support of a neoadditive capacity.
B. Games with Ambiguity. This section contains proofs of our existence and comparative statics results and some supplementary results. It uses techniques from Topkis (1998).
B.1. Existence. We start with a preliminary definition and lemma. DEFINITION 14. Suppose that B is a correspondence from a partially ordered set S to a lattice T such that for all s ∈ S, B(s) is a sublattice of T ; then we say that B is increasing if whenŝ >s, andt ∈ B(ŝ),t ∈ B(s), then min{t,t} ∈ B(s) and max{t,t} ∈ B(ŝ).

LEMMA 5. Let S be a lattice and let β : S → S be an increasing correspondence. Then
(1) β has a fixed point; (2) sup{s : β(s) s} is the greatest fixed point of β. 19 Recall C(μ) denotes the core of the capacity μ; see Definition 4. 20 Although C(μ) is an infinite set, the minimum must occur at one of the extremal points. The set of extremal points of a core is finite. Thus the minimum must be positive. 21 Recall ν 0 denotes the complete uncertainty capacity in Example 1.
PROOF. Let T = {s : β(s) s}. Note that T is nonempty since s ∈ T (where s = min S). Let s = sup T . By definition, if s > s , then Supposes ∈ T ; then β(s) s. Since β is increasing, β(s ) β(s) and β(s ) s s. Thus, β(β(s )) β(s ) s , which implies β(s ) ∈ T and hence s β(s ) s . 22 The last equation implies that s is a fixed point of β. Equation (A.1) implies that there is no greater fixed point.
PROOF OF THEOREM 1. Choose an n -tuple of parameters δ = δ 1 , . . . , δ n such that λ δ γ. Let νˆs −i i denote the neo-additive capacity on S −i defined by to be player i's (Choquet) expected utility from playing s i when his beliefs are represented by νˆs −i i , that is, is the best response of player i, if his beliefs are a neo-additive capacity which represents an ambiguous belief that his opponents will playŝ −i . Lemma 10 implies that V i (s i ,ŝ −i ) has increasing differences in s i ,ŝ −i ; hence β i is an increasing correspondence. (The proof is similar to that of Lemma 12.) Thus by Lemma 5, β has a fixed point s * . This implies ν s * = ν

B.2. Comparative statics proofs.
Correspondences on partially ordered sets. This section contains some results about increasing correspondences and selections from them. LEMMA 6. Suppose that B λ is an increasing correspondence from a partially ordered set S to a totally ordered set T for all λ in an index set ; thenB(s) = max λ∈ B λ (s) and B(s) = min λ∈ B λ (s) are increasing functions from S to T.

Finally Bˆλ(s) B(s), which establishes that B(ŝ) B(s).
The following lemma describes some properties of fixed points of functions on partially ordered sets. Constant contamination capacities. Below we define a special case of JP-capacities that arise naturally when considering pure equilibria in games.
DEFINITION 15 (Constant contamination). A capacity, ν˜s i (α i , δ i , ς i ), on S −i is said to display constant contamination (henceforth CC) if it may be written in the form denotes the probability distribution 24 on S −i , which assigns probability 1 tos −i and ς i is a convex capacity with supp ς i = ∅. To simplify notation we suppress the arguments (α i , δ i , ς i ) when it is convenient.
We interpret the capacity ν i (ς i , δ i , α i ) as describing a situation where player i "believes" that his opponents will play the pure strategy profiles −i but lacks confidence in this belief. The CC-capacity embodies a separation between beliefs represented by, π i , ambiguity-attitude represented by α i , and ambiguity represented by ς i and δ i . The parameter δ i determines the weight the individual gives to ambiguity. Higher values of δ i correspond to more ambiguity. The capacity ς i determines which strategy profiles the player regards as ambiguous. The following result finds the support of a CC capacity.
However, this is a contradiction, which establishes that supp μ i ⊆ {s −i }. The result follows.
The following lemma shows that any capacity which describes the equilibrium of a game is a CC-capacity. This provides a useful characterization of equilibrium beliefs. LEMMA 9. Let be a game with positive externalities and letν be a singleton equilibrium in JP-capacities of with equilibrium strategy profileŝ. Thenν is a profile of CC-capacities, that is, there exist convex capacities ζ i , 1 i n, with supp ζ i = ∅ and δ i , 1 i n, such that if we define PROOF. Sinceν is an equilibrium in JP-capacities, we may write the equilibrium beliefs of individual i in the formν i = α i μ i + (1 − α i )μ i for some convex capacity μ i . Define a capacity ζ i There are four cases to consider.
Ifŝ −i / ∈ A andŝ −i / ∈ B, then the claim follows from convexity of μ i , since ζ i = 1 1−δ i μ i for all four sets.
− δ i ) this proves convexity in this case. The remaining case can be established by similar reasoning. Since Increasing/decreasing differences. Recall that a game, = N;  (Choquet) expected payoff given that his beliefs are represented by the capacity which is increasing in s i since u i has increasing differences in s i , s −i .
We have used the fact that since there are positive aggregate externalities, all four integrands in the curly brackets are comonotonic. It is sufficient to show that the coefficient of α i is positive. This is equal to . Similarly the first k weights in the second sum in total are equal toς i (H M−k+1 ). Since ς i is convex, ; hence the weights in the first sum are first order stochastically dominated by those in the second. Thus, the first sum is smaller, which makes the overall expression negative. This establishes that and let m = min{y, z} and M = max{y, z}. Since It follows from Lemma 9 that the maximal (respectively, minimal) best response correspondence is the greatest (respectively, least) best response to all beliefs whose support is the pure strategy s −i with minimal (respectively, maximal) degree of ambiguity is at least γ (respectively, at most λ).
PROOF OF THEOREM 2. We prove the result for the highest equilibrium strategy. The lowest equilibrium strategy can be covered by a similar argument. Lemma 9 establishes that ifŝ is an equilibrium strategy profile when the minimal (respectively, maximal) degree of ambiguity is γ (respectively, λ), then there exist ζ i with λ i for 1 i n. Thus any given equilibrium, satisfying these constraints, is smaller than the largest fixed point of the maximal best response correspondenceB i (s −i , α i , γ i , λ i ).
Therefore sinces(α) is the profile of greatest equilibrium strategies it is the largest fixed point of the maximal best response function, that is,s(α) ∈B(s,ᾱ) and s(α) ∈ B(s, α). By Lemma 12, B i (s −i , α i , γ i , λ i ) is increasing in s −i and decreasing in α i . It follows from Lemma 7 thats(α) is decreasing in α.
Multiple equilibria. In this section, we show that equilibrium is unique if there is sufficient ambiguity.
LEMMA 13. Consider a game, , of positive externalities and increasing differences. There existsγ such that if the minimal degree of ambiguity is γ(μ i ) γ, then in any equilibrium ν = ν 1 , . . . , ν n , supp PROOF. Supposeŝ i ∈ A,s i / ∈ A. Number the strategy profiles of the opponents so that . Although in general σ r −i = s r −i , positive externalities imply that s 1 −i = σ 1 −i =s −i and s R −i = σ R −i = s −i . Suppose that the beliefs of individual i may be represented by a JP-capacity, ν i = α i μ i + (1 − α i )μ i . If i plays strategyŝ i , (s)he receives utility .
Similarly if i plays strategys i (s)he receives utility In the limit, asγ tends to 1, all the terms involving μ i tend to 0. Hence V i (ŝ i ) tends to α i u i (ŝ i , s THE CENTIPEDED GAME argmax s i ∈S i u i (s i ,s −i )). By Theorem 2, the resulting equilibrium is smaller (respectively, greater) than the lowest (respectively, highest) equilibrium without ambiguity.
B.3. The Centipede game. In this section, we illustrate our model by applying the EUA concept to a simplified version of the Ceentipede game. Two players, I = {1, 2}, with actions c ("continue") and e ("exit"), A := {c, a}, alternate in choosing their actions. We consider the case of three periods represented in Figure A1.
Denote by ce the strategy of Player 1 to continue at t = 1 and to exit at t = 3. Similarly, cc indicates the strategy to continue both at t = 1 and t = 3. In strategic form the game is represented by the following table: 25 Player 2 e c e 1, 0 1 , 0 Player 1 ce 0, 6 1 1 , 5 cc 0, 6 1 0 , 10 In the unique NE, Player 1 exits immediately and Player 2 chooses an arbitrary mixed strategy. Given the attractive payoffs after continuation, it is not surprising that most experimental studies of the Centipede game show that players continue for a sequence of moves, though not to the end of the game. 26 The following result provides conditions on the ambiguity-attitude parameters (α 1 , α 2 ) and players' beliefs represented by JP-capacities (μ 1 , μ 2 ) for an equilibrium in beliefs under ambiguity, where Player 1 continues for one stage, even though Player 2 will exit in stage 2.
Notice that the conditions of the claim require only a small degree of ambiguity for both players about the behavior of the opponent paired with a little optimism α 1 < 0 and α 2 < 1. Behavior in this equilibrium resembles the behavior observed in laboratory experiments of the Centipede game. Both players will be better off (ex post) given their behavior in this EUA than in the unique NE.