On Hurwicz preferences in psychological games

De Marco, Guiseppe; Romaniello, Maria; Roviello, Alba Article On Hurwicz preferences in psychological games Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: De Marco, Guiseppe; Romaniello, Maria; Roviello, Alba (2024) : On Hurwicz preferences in psychological games, Games, ISSN 2073-4336, MDPI, Basel, Vol. 15, Iss. 4, pp. 1-26, https://doi.org/10.3390/g15040027 This Version is available at: https://hdl.handle.net/10419/330096 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Citation: De Marco, G.; Romaniello, M.; Roviello, A. On Hurwicz Preferences in Psychological Games. Games 2024,15, 27. https://doi.org/ 10.3390/g15040027 Academic Editor: Ulrich Berger Received: 24 June 2024 Revised: 22 July 2024 Accepted: 26 July 2024 Published: 30 July 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). games Article On Hurwicz Preferences in Psychological Games Giuseppe De Marco 1,2,*, Maria Romaniello 3and Alba Roviello 4 1Department of Management and Quantitative Sciences, University of Napoli Parthenope, Via Generale Parisi 13, 80132 Napoli, Italy 2Center for Studies in Economics and Finance, University of Napoli Federico II, Via Cupa Cinthia, 80126 Napoli, Italy 3 Department of Economics, University Campania Vanvitelli, Corso Gran Priorato di Malta, 81043 Capua, Italy; [email protected] 4Department of Economics and Statistical Sciences, University of Napoli Federico II, Via Cupa Cinthia, 80126 Napoli, Italy; [email protected] *Correspondence: giuseppe.demar[email protected] Abstract: The literature on strategic ambiguity in classical games provides generalized notions of equilibrium in which each player best responds to ambiguous or imprecise beliefs about his opponents’ strategic choices. In a recent paper, strategic ambiguity has been extended to psychological games, by taking into account ambiguous hierarchies of beliefs and max–min preferences. Given that this kind of preference seems too restrictive as a general method to evaluate decisions, in this paper we extend the analysis by taking into account α -max–min preferences in which decisions are evaluated by a convex combination of the worst-case (with weight α ) and the best-case (with weight 1 −α ) scenarios. We define the α -max–min psychological Nash equilibrium; an illustrative example shows that the set of equilibria is affected by the parameter α and the larger the ambiguity, the greater the effect. We also provide a result of stability of the equilibria with respect to perturbations that involve the attitudes toward ambiguity, the structure of ambiguity, and the payoff functions: converging sequences of equilibria of perturbed games converge to equilibria of the unperturbed game as the perturbation vanishes. Surprisingly, a final example shows that the existence of equilibria is not guaranteed for every value of α. Keywords: psychological games; ambiguous beliefs; α-MEU; equilibrium existence 1. Introduction It is well-known that the Nash equilibrium concept for strategic games prescribes the following: (i) each player chooses his best strategy in response to the beliefs he has about his opponents’ strategic choices; (ii) each player’s beliefs are correct; that is, each player believes with probability 1 that opponents will follow their equilibrium strategies. The evidence arising from decision theory tells us that beliefs cannot always be assumed to be correct. The literature that focuses on the issue of strategic ambiguity in classical strategic form games provides generalized notions of equilibrium in which each player best responds to ambiguous or imprecise beliefs about his opponents’ strategic choices, i.e., beliefs may take the form of a capacity or a set of probability distributions (see [ 1 – 6 ] and references therein). There might be many sources of strategic ambiguity in a game: for example, Lehrer [3] focuses on the case in which players do not have precise knowledge of the mixed strategy chosen by each of the other players but rather know only the probability of some subsets of pure strategies, not being aware of the precise subdivision of probabilities within those subsets. In [ 7 ], the study of strategic ambiguity has been extended to psychological games by looking at ambiguous or imprecise hierarchies of beliefs. Psychological games provide a generalization of classical games, aiming to explicitly take into account the emotions, opinions, and intentions of the decision-makers in the strategic interaction 1 . This class of games is characterized by the assumption that each player’s payoff depends on his hierarchy of beliefs, i.e., it depends Games 2024,15, 27. https://doi.org/10.3390/g15040027 https://www.mdpi.com/journal/games Games 2024,15, 27 2 of 26 not only on what every player does but also on what he thinks every player believes, on what he thinks every player believes the others believe, and so on. The main solution for psychological games is presented in Geanakoplos et al. [14] and is based on the idea that the entire hierarchy of beliefs of each player must be correct in equilibrium. Since beliefs about opponents’ strategic choices can be regarded as first-order beliefs, the literature on strategic ambiguity substantially looks at games in which first-order beliefs are ambiguous. Ref. [ 7 ], instead, looks at ambiguity regarding the entire hierarchy of beliefs as, for instance, partial knowledge may appear directly in the second-order (or higherorder) beliefs or strategic ambiguity produces ambiguous higher-order beliefs as a natural consequence. Therefore, the function that maps strategic profiles to the correct hierarchies of beliefs, which is used in the classical definition of psychological Nash equilibria, is therein replaced by a set-valued map (called ambiguous belief correspondence), which maps strategic profiles to the subsets of those hierarchies of beliefs that players perceive to be consistent with the corresponding strategy profile. In the corresponding equilibrium notion presented in [ 7 ], players are assumed to be completely pessimistic as they are endowed with max–min preferences (also called MEU preferences; see [ 15 ]); each player maximizes (with respect to his own strategy) the minimum expected utility computed along the graph of the ambiguous belief correspondence whose values, in turn, depend on the entire strategy profile. The max–min approach turns out to be analytically convenient; furthermore, it has a clear axiomatic foundation. Nevertheless, it seems to be too restrictive as a general approach because only the “worst-case scenario” is relevant for the evaluation of a decision so the analysis is limited to an extreme form of pessimism. The restrictiveness of the MEU model can be naturally overcome by considering the so-called α -max–min preferences (also called α -MEU or Hurwicz preferences), first introduced in [ 16 ]. In this model, decisions are evaluated by a convex combination of the worst-case (with weight α ) and the best-case (with weight 1 −α ) scenarios. This type of preference was widely analyzed and applied in several different settings in order to include a larger spectrum of ambiguous attitudes (see [17–20] to quote a few). The literature shows that the generality of the model is affected once preferences are restricted only to the max–min approach; in fact, the behavior of agents with an intermediate attitude toward ambiguity cannot be fully explained by max–min preferences 2 . Hurwicz preferences, instead, are much more general and exploitable when dealing with model uncertainty, as they allow for analysis of the decisions of agents with a larger variety of personal characteristics. This perspective is key when looking at several economic and financial problems, such as establishing the design of optimal insurance contracts or investigating duopolistic competitions (see, for example, [ 22 , 23 ]). In fact, these papers show that the characterization of equilibria explicitly depends on the degree of pessimism/optimism α . Finally, other theoretical papers take into account the Hurwicz preferences, for instance [ 24 , 25 ], where the problems of information processing and awareness, respectively, have been addressed. In this paper, we extend the analysis of psychological games under ambiguity to α -max– min preferences and provide the notion of α -MEU psychological Nash equilibrium ( α -PNE) for situations in which players have Hurwicz preferences. The weights α that characterize the attitudes of the players toward ambiguity turn out to be key to understanding how equilibria change according to the players’ degree of pessimism/optimism. We present an illustrative example, showing not only that the set of equilibria depends on the parameter α but also that differences are emphasized by the amount of ambiguity in the game: the larger the ambiguity, the greater the differences. The example highlights another relevant feature: equilibria corresponding to a given value of α cannot always be approached by sequences of equilibria of games in which the parameter α is slightly perturbed, meaning that equilibria are unstable with respect to perturbations on the degree of pessimism/optimism. From the mathematical point of view, this implies a lack of lower semi-continuity of psychological Nash equilibria under the Hurwicz preferences. The failure of this property is not surprising since Games 2024,15, 27 3 of 26 the lack of lower semi-continuity of the equilibrium correspondence is a common feature in most of the game models. We show, instead, that the α -PNE correspondence satisfies upper semi-continuity-like stability: converging sequences of equilibria of perturbed games converge to equilibria of the unperturbed game as the perturbation vanishes. The issue of the upper semi-continuity properties of equilibria has been largely investigated in the literature for classical games (see for instance [ 26 – 28 ] and references therein) and is a key property to build refinements of equilibria based on stability with respect to trembles. In this paper, we obtain the stability of equilibria under general perturbations that involve, simultaneously, the attitudes toward ambiguity (that is, the parameters α ), the structure of ambiguity, and the payoff functions. In particular, this result allows for selection criteria (for PNE under ambiguity) based on stability properties with respect to perturbations on the weights α. The most surprising feature of α -PNE is, however, a negative result. Although for psychological Nash equilibria and psychological Nash equilibria under max–min preferences an existence result was obtained under standard assumptions, in this paper we provide a counterexample in which a game has no α -PNE. This negative result comes from the fact that the best reply correspondence of the summary utility function (used to obtain equilibrium existence) does not have convex images and therefore fixed points, in general. The paper is organized as follows: Section 2defines the game and the equilibrium concept. Section 3presents the illustrative example while Section 4is dedicated to the upper-semi-continuity property of equilibria. In Section 5, the issue of the lack of existence of equilibria is studied. All the proofs are relegated to Appendix A. Appendix B, instead, is devoted to complementary results concerning the problem of the (non)-existence of equilibria in games with Hurwicz preferences and non-psychological payoffs. 2. Model and Equilibria We consider a finite set of players I={ 1, . . . n} , and, for each player i , we denote with Ai={a1 i , . . . , ak(i) i} the (finite) pure strategy set of player i . As usual, the set of strategic profiles A is the Cartesian product of the strategy sets of each player, which is A=A1× · · · × An=∏i∈IAi , and A−i=A1× · · · × Ai−1×Ai+1× · · · × An=∏j=iAj . Let Σi be the set of mixed strategies of player i , where each mixed strategy σi∈Σi is a nonnegative vector σi= (σi(ai))ai∈Ai∈Rk(i) +such that ∑ai∈Aiσi(ai) = 1. Denote also with Σ=∏i∈IΣi and with Σ−i=∏j=iΣj . We use (σi , σ−i) with σi∈Σi and σ−i∈Σ−i to represent σ∈Σ. 2.1. Hierarchies of Beliefs The belief structure is constructed following [ 14 ]. Recall that, for any topological space S , ∆(S) denotes the set of Borel probability measures on S . For every player i and for every k∈N,k>1, the k-th order beliefs set is defined recursively as follows: B1 i=∆(Σ−i), B2 i=∆(Σ−i×B1 −i), . . . Bk i:=∆(Σ−i×B1 −i×B2 −i× · · · × Bk−1 −i), . . . where Bk −i:=∏j=iBk j . The set of all hierarchies of beliefs of player i is Bi=∏∞ k=1Bk i . Note that for every k , Bk i is compact and can be metrized as a separable metric space. Consequently, since Bi is a countable product of separable and compact metric spaces, it is also a separable and compact metric space3. Games 2024,15, 27 4 of 26 We will restrict the attention to the subset of collectively coherent beliefs Bi⊂Bi , i.e., the compact set of beliefs of player i in which he is sure that it is common knowledge that beliefs are coherent. Precisely, a belief bi= (b1 i , b2 i , . . . )∈Bi is said to be coherent if, for every k∈N , the marginal probability of bk+1 i on Σ−i×B1 −i×B2 −i× · · · × Bk−1 −i coincides with bk i, which is marg(bk+1 i,Σ−i×B1 −i×B2 −i× · · · × Bk−1 −i) = bk i. You can find the construction of the set of collectively coherent beliefs in [ 14 ] and the proof of its compactness in [ 7 ]. Throughout the remainder of the paper, with an abuse of notation, we will denote with Bi the set of collectively coherent beliefs or any of its compact subsets. As in [ 7 ], we allow for ambiguity in the beliefs; therefore, beliefs are compact subsets Ki⊆Bi . We denote with Ki the set of all compact subsets of Bi . This choice allows consideration of the ambiguity players encounter during the game due to uncertainty about other players’ actions and beliefs: the agent does not have a precise belief bi but knows that the belief can be any bi∈Ki . If Ki is a singleton, then the belief is not ambiguous, leading the theory back to the standard case. 2.2. Game and Equilibria Following the model in [ 14 ], each agent i is endowed with a utility function of the form ui:Bi×Σ→R, (1) depending not only on the mixed strategy profile but also on the agent’s beliefs: ui(bi , σ) represents the payoff to player i if he believed bi and the strategy profile σ is actually played. Indeed, fixing bi , ui(bi , ·) can be (but not necessarily) the classical expected utility function as it is assumed in [ 14 ]. As agents face set-valued beliefs Ki∈Ki , they have a set-valued payoff {ui(bi,σ)}bi∈Ki for every given ambiguous belief Ki∈Ki and strategy profile σ∈Σ . There are several ways in which the agents’ ambiguity might be solved depending on the agents’ attitudes toward ambiguity. In [ 7 ], the case was considered where players are ambiguity-averse, modeling the utility functions as max–min preferences. To encompass a broad spectrum of ambiguity attitudes, this paper focuses on the so-called α -max–min preferences, which allow us to range from an ambiguity-seeking attitude (as α= 0) to an ambiguity aversion attitude (as α= 1). In this framework, each agent i has a utility function of the following form: Uα i:Ki×Σ→Rdefined, for αi∈[0, 1], by Uα i(Ki,σ)=αiinf bi∈Ki ui(bi,σ)+ (1−αi)"sup bi∈Ki ui(bi,σ)#∀(Ki,σ)∈Ki×Σ, (2) where α denotes the vector α= (α1 , . . . , αn)∈[ 0, 1 ]n . Now, it is possible to define the game. Definition 1. An α-MEU normal form psychological game is defined by Gα={A1,· · · ,An,Uα 1,· · · ,Uα n} where the utility functions Uα iare defined as in Formula (2) for every i ∈N. In the models of strategic ambiguity where players have partial knowledge of the strategies played by their opponents, players’ beliefs depend on the actual strategy and take the form of set-valued maps (correspondences) from the set of strategic profiles to the set of probability distributions over opponents’ strategies (see [ 3 , 5 , 29 ]). In [ 7 ], this approach was generalized to hierarchies of beliefs: agent i is endowed with a set-valued Games 2024,15, 27 5 of 26 map γi:Σ⇝Bi (called ambiguous belief correspondence of player i ), where each image γi(σ) is a non-empty and compact set, i.e., ∅=γi(σ)∈Ki∀σ∈Σ. Each subset γi(σ)⊆Bi provides the set of hierarchies of beliefs that player i perceives to be consistent given the strategy profile σ . The set-valued maps γi are exogenous and have different structures depending on the specific problem; therefore, they can be considered as parameters of the game. Remark 1. It is well-known that uncertainty and partial ignorance can be modeled with several different tools beyond sets of probability measures, such as the Choquet capacities and belief functions (see [ 2 , 17 , 30 – 35 ]). Due to its generality and the exogeneity of the set-valued maps γi , our model can embrace several specific cases. For example, one can consider γi to be defined as the core of a belief function associated with player i , which is a non-empty and compact set. In the special case of precise probability, the core, and, consequently, the image set of γi , reduces to a single probability measure, giving back the non-ambiguous case. In [ 7 ], a link with partially specified probabilities [ 3 ] was also constructed. In this paper, we follow the approach in [7]: Definition 2. An α -MEU psychological Nash equilibrium (henceforth, α -PNE) of the game Gα with belief correspondences γ= (γ1 , . . . , γn) is a pair (K∗ , σ∗) , where K∗= (K∗ 1 , . . . , K∗ n) with K∗ i⊆Biand σ∗∈Σsuch that, for every player i: (i) K∗ i=γi(σ∗); (ii) Uα i(K∗ i,σ∗)⩾Uα i(K∗ i,(σi,σ∗ −i)) for every σi∈Σi. In this case, we can also say that (γ(σ∗),σ∗)is an α-MEU psychological Nash equilibrium. Remark 2. The definition of α -PNE captures, in a natural way, the main features of the classical equilibrium notions since condition (ii) requires that the equilibrium strategy of each player is optimal given his beliefs and condition (i) requires that beliefs must satisfy a consistency condition with the equilibrium strategy profile that is characterized by the belief correspondences γi . However, the nature of this latter consistency condition differentiates it from the concept of psychological Nash equilibrium, which, in turn, inherits from the classical Nash equilibrium the requirement that beliefs must be correct in equilibrium. The α -PNE concept is based on a different perspective that is similar to the one in the definition of self-confirming equilibrium (SCE) in4[5]. As clearly explained by the authors 5 : “...in a SCE, agents best respond to confirmed probabilistic beliefs. Confirmed means that their beliefs are consistent with the evidence they can collect, given the strategies they adopt.... The key difference between SCE and Nash equilibrium is that, in an SCE, agents may have incorrect beliefs because many possible underlying distributions are consistent with the empirical frequencies they observe.” From the mathematical point of view, in a SCE, beliefs are parametrized by feedback functions that give rise to information about opponents’ strategic profiles, and by a prior belief in these strategic profiles. Such feedback functions and prior beliefs together provide the beliefs that the players perceive to be consistent with the strategy profile, for every profile. In practice, beliefs are represented by a set-valued map (that depends on the strategy profile) that is called identification correspondence. This correspondence can be regarded as a particular case of the belief correspondence used in the definition of α -PNE when we look only at first-order beliefs. Moreover, a version of SCE under uncertainty, namely the MSCE concept (in mixed strategies), can be regarded as a α -PNE (for α= 1) in the case in which the opponents’ strategies are replaced by first-order (ambiguous) beliefs in the expected utility of each player. Naturally, there are differences between α -PNE and SCE. First, in the former, higher-order beliefs enter explicitly the utility function in an arbitrary way, while, in the latter, just the firstorder beliefs enter the classical expected utility function in place of opponents’ strategies. Most importantly, the SCE involves a richer structure of beliefs with respect to α -PNE as, in an SCE, Games 2024,15, 27 6 of 26 it fully captures two different scenarios: the first one is a repeated game in which there are no intertemporal strategic links between the plays, while the second is the (so-called) large population scenario in which there is a large society of individuals who recurrently play a given game. In the α -PNE, beliefs correspondences represent a generic mathematical tool that can generalize different models like the Identification Correspondence quoted above or like some imprecise perturbations of correct beliefs, as illustrated in the example in Section 3below, in order to run a robustness analysis on the unperturbed equilibria. Finally, the nature of the definition of equilibrium differs between α -PNE and SCE as the latter always has the classical Nash equilibrium concept as a refinement while the relation between α -PNE and the classical psychological Nash equilibrium depends on the specific model taken into account. 2.3. Summary Utility Functions Similar to [ 14 ], α -PNE has a characterization as Nash equilibria. Let wα i:Σ×Σ→R be the summary utility function defined by the following: wα i(σ,τ) = Uα i(γi(σ),τ) = αiinf bi∈γi(σ)ui(bi,τ)+ (1−αi)"sup bi∈γi(σ) ui(bi,τ)#∀(σ,τ)∈Σ×Σ. (3) Then, the following immediately follows from the definition: Lemma 1. The profile (γ(σ∗) , σ∗) is an α -MEU psychological Nash equilibrium if and only if, for every player i, wα i(σ∗,(σ∗ i,σ∗ −i)) ⩾wα i(σ∗,(yi,σ∗ −i)) ∀yi∈Σi. (4) Remark 3. In [ 14 ], the equilibrium beliefs of each agent i are described by the correct beliefs function βi:Σ→Bi which, for every σ∈Σ , specifies the unique hierarchy of beliefs of player i that is correct, given σ . Now, if we replace γi with βi , in Definition 2, we retrieve the definition of classical psychological Nash equilibria. On the other hand, if we replace γi with βi in (3), we obtain the original summary utility function defined in [14]. 3. An Illustrative Example In this section, we present an example of a psychological game under ambiguity in which players have Hurwicz preferences. The goal is twofold: on the one hand, we aim to put definitions to work and show how to find psychological Nash equilibria under ambiguity in simple models. On the other hand, the example highlights in which way the equilibria may be sensitive to variations in the amount of the structure of ambiguity in the game and the attitudes of the players toward ambiguity. More precisely, we consider a specific form of ambiguity: players’ beliefs are provided by a perturbation of the correct belief function that takes the form of a ball of radius ε around the correct belief. This approach resembles the contamination model approach and allows us to analyze the sensitivity of α -PNE with respect to the unique parameter ε . Moreover, as the attitude toward ambiguity of each player i is parametrized by the corresponding value of αi , we study the sensitivity of equilibria with respect to αi. The game considered in the example is the bravery game that was first analyzed in the framework of standard psychological games by [ 14 ]. In [ 7 ], it was shown that allowing for ambiguous hierarchies of beliefs may significantly affect the set of equilibria when players are endowed with max–min preferences. In this work, we study the game with respect to the double parametrization εand αi. Example 1. The game is described as follows: Player 1 (John) has to publicly make a decision, and he is concerned about what Player 2 (Anne) will think about him. He can either be bold, exposing himself to the possibility of danger, or he can opt for a timid decision; therefore, John’s pure strategy set is A1={Bold,Timid} . Anne is inactive during the whole interaction but her beliefs about John have an impact on John’s behavior; indeed, his payoff depends not only on what he does but also Games 2024,15, 27 7 of 26 on what he believes Anne thinks he will do. Suppose that John chooses Bold with probability p and Timid with probability 1 −p . We consider the case in which John cares only about the expectation ˜ q of his belief about the expectation q of Anne’s first-order belief. Moreover, John would rather be timid, unless he thinks Anne is expecting him to be bold, in which case he prefers not to disappoint her. Anne prefers to think of her friend as bold, and it is better for her if he opts for the bold decision. The game and payoffs are described below: John 3(1−˜ q), 1 −q Timid 1−p 2−˜ q, 2(1+q) Bold p Equilibria without Ambiguity Since Anne is a non-active player, the mixed strategy profile is given only by John’s mixed strategy p . With the abuse of notation, the correct belief functions are defined as follows: β2(p) = p tells that the expectation of Anne’s first-order correct beliefs about John’s strategy ( p ) must be equal to p ; β1(p) = p shows that the expectation of John’s correct second-order beliefs about Anne’s expectation of the correct first-order belief about John’s strategy (p) must be equal to p as well. The expected utility of John takes the following form: u1(˜ q,p) = p(2−˜ q) + 3(1−p)(1−˜ q) = p(2˜ q−1) + 3(1−˜ q). In the case of non-ambiguous beliefs, the game has three psychological equilibria, as shown in [14]: - p=1=˜ q=q: John chooses to be Bold; - p=0=˜ q=q: John chooses to be Timid; - p=1/2 =˜ q=q: John randomizes with probability p =1/2. The Game in Case of Ambiguity Now, we assume that John has ambiguous beliefs; in particular, John’s belief is represented by the map γε 1(p) = [p−ε , p+ε]∩[ 0, 1 ] with 0 <ε⩽ 1. We look at the equilibria of the game in case players (John, in this case) have α -MEU preferences. In particular, we show in which way the different attitudes toward ambiguity affect the equilibrium behavior. In order to compute John’s summary utility function, we first compute, for every pair of John’s mixed strategies (p , y) , the following: arg min ˜ q∈γε 1(p) u1(˜ q,y) = (˜ q′∈[0, 1]|u1(˜ q′,y) = min ˜ q∈γε 1(p)u1(˜ q,y)), arg max ˜ q∈γε 1(p) u1(˜ q,y) = (˜ q′∈[0, 1]|u1(˜ q′,y) = max ˜ q∈γε 1(p)u1(˜ q,y)). We have the following: arg min ˜ q∈γε 1(p) u1(˜ q,y) = arg min ˜ q∈[p−ε,p+ε]∩[0,1] [˜ q(2y−3) + 3−y] = min{p+ε, 1},∀y∈[0, 1]. Games 2024,15, 27 8 of 26 Similarly, arg max ˜ q∈γε 1(p) u1(˜ q,y) = arg max ˜ q∈[p−ε,p+ε]∩[0,1] [˜ q(2y−3) + 3−y] = max{p−ε, 0},∀y∈[0, 1]. If p+:=min{p+ε, 1} and p−:=max{p−ε, 0} , for every pair of John’s mixed strategies (p,y)and for every α∈[0, 1], we have the following: wα 1(p,y) = α"min ˜ q∈γε 1(p)˜ q(2y−3) + 3−y#+ (1−α)"max ˜ q∈γε 1(p)˜ q(2y−3) + 3−y#= α[p+(2y−3) + 3−y]+(1−α)[p−(2y−3) + 3−y] = y[2α(p+−p−) + 2p−−1]−3α(p+−p−) + 3(1−p−). α-PNE Recall that p gives α-PNE if and only if wα 1(p,p)⩾wα 1(p,y)∀y∈[0, 1],α∈[0, 1]. It is clear that equilibria depend on α and ε . Below, we provide a full characterization of all the (α-PNE) equilibria. First, denote with ˜ p=1 2+ε(1−2α),p∗=1 2α−ε,ˆ p=1−2α 2−2α+ε. It follows that Lemma 2. Let 0<ε⩽1 2. (i) If ε<1/4, then, for every α∈[0, 1], the α-PNE are: p =0, p=1, p=˜ p; (ii) If ε⩾1/4, then: –for α∈h0, 1 −1 4εh, the α-PNE are: p =0, p=1, p=ˆ p; –for α∈h1−1 4ε,1 4εi, the α-PNE are: p =0, p=1, p=˜ p; –for α∈i1 4ε, 1i, the α-PNE are: p =0, p=1, p=p∗; where ˆ p=˜ p if α=1−1 4εand ˜ p=p∗if α=1 4ε. Proof. See Appendix A. Lemma 3. Let 1 2<ε⩽1. Then, - for α∈h0, 1 −1 2εh, the unique α-PNE is: p =0; - for α∈h1−1 2ε,1 2h, the α-PNE are: p =0, p=1, p=ˆ p; - for α=1 2, the α-PNE are: p =0, p=1and every p ∈[1−ε,ε]; - for α∈i1 2,1 2εi, the α-PNE are p =0, p=1, p=p∗; - for α∈i1 2ε, 1i, the unique α-PNE is: p =1; where ˆ p=1if α=1−1 2εand p∗=0if α=1 2ε. Proof. See Appendix A. We can summarize the results in the following table, which is filled with the values of the parameter α, ensuring the existence of the corresponding equilibrium, as follows (Table 1): Games 2024,15, 27 15 of 26 (i) For α∈h0, 1 −1 4εi, wα ε(p, 0)>wα ε(p,y)∀y∈]0, 1], if p<ˆ p; wα ε(p, 1)>wα ε(p,y)∀y∈[0, 1[, if p>ˆ p; wα ε(p,y) = 3/2 ∀y∈[0, 1], if p=ˆ p. Therefore, we obtain the following equilibria: p=0, p=1, and p=ˆ p. (ii) For α∈i1−1 4ε,1 4εh, wα ε(p, 0)>wα ε(p,y)∀y∈]0, 1], if p<˜ p; wα ε(p, 1)>wα ε(p,y)∀y∈[0, 1[, if p>˜ p; wα ε(p,y) = 3/2 ∀y∈[0, 1], if p=˜ p. Therefore, we obtain the following equilibria: p=0, p=1, and p=˜ p. (iii) For α∈h1 4ε, 1i, wα ε(p, 0)>wα ε(p,y)∀y∈]0, 1], if p<p∗; wα ε(p, 1)>wα ε(p,y)∀y∈[0, 1[, if p>p∗; wα ε(p,y) = 3/2 ∀y∈[0, 1], if p=p∗. Therefore, we obtain the following equilibria: p=0, p=1, and p=p∗. Appendix A.2. Proof of Lemma 3 Suppose 1 2<ε⩽ 1. In this case, the summary utility function takes the following form: wα ε(p,y) =      y[2α(p+ε)−1]−3α(p+ε) + 3)if 0 ⩽p⩽1−ε y[2α−1]−3α+3 if 1 −ε<p<ε. y[(p−ε)(2−2α) + 2α−1]+ (3−3α)(1−p+ε)if ε⩽p⩽1, Note that 0 <1−1 2ε⩽1 2⩽1 2ε<1; consider again p∗=1 2α−εand ˆ p=1−2α 2−2α+ε. (a) Denote with h1(y):=y[2α(p+ε)−1]− 3 α(p+ε) + 3) . If α= 0, the function h1(y) decreases throughout the entire interval [ 0, 1 ] . If α> 0, h1(y) decreases on [ 0, 1 ] if p<p∗ , is constant on [ 0, 1 ] if p=p∗ , and increases on [ 0, 1 ] if p>p∗ . Moreover, p∗⩾0 if and only if α∈i0, 1 2εi, while p∗⩽1−εif and only if α∈h1 2, 1i. (b) Denote with h2(y):=y[ 2 α− 1 ]− 3 α+ 3; the function h2(y) decreases throughout the entire interval [ 0, 1 ] if α<1 2 , is constant on the interval [ 0, 1 ] if α=1 2 , and increases throughout the entire interval [0, 1]if α>1 2. (c) Denote with h3(y):=y[(p−ε)(2−2α) + 2α−1]+ ( 3 − 3 α)( 1 −p+ε) . If α= 1, the function h3(y) increases throughout the entire interval [ 0, 1 ] . If α< 1, the function h3(y) decreases on [ 0, 1 ] for p<ˆ p , is constant on [ 0, 1 ] for p=ˆ p , and increases throughout the entire interval [ 0, 1 ] if p>ˆ p . Moreover, ˆ p⩽ 1 if and only if α∈ h1−1 2ε, 1h, while ˆ p⩾εif and only if α∈h0, 1 2i. It follows that - If α∈h0, 1 −1 2εh, the function y→wα ε(p,y)decreases on [0, 1]for every p∈[0, 1]. - If α= 1 −1 2ε , then the function y→wα ε(p , y) decreases on [ 0, 1 ] for p<ˆ p= 1, and is constant in [0, 1]for p=ˆ p=1. - If α∈i1−1 2ε,1 2h , the function y→wα ε(p , y) decreases on [ 0, 1 ] for p<ˆ p , is constant on [0, 1]for p=ˆ p, and increases on [0, 1]for p>ˆ p. Games 2024,15, 27 16 of 26 - If α=1 2 , the function y→wα ε(p , y) decreases on [ 0, 1 ] for p< 1 −ε , is constant in [ 0, 1 ] for 1 −ε⩽p⩽ε, and increases on [0, 1]for p>ε. - If α∈i1 2,1 2εh , the function y→wα ε(p , y) decreases on [ 0, 1 ] for p<p∗ , is constant on [0, 1]for p=p∗, increases on [0, 1]for p>p∗. - If α=1 2ε , the function y→wα ε(p , y) increases on [ 0, 1 ] for p>p∗= 0, and is constant for p=p∗=0. - If α∈i1 2ε, 1i, the function y→wα ε(p,y)increases on [0, 1]for every p∈[0, 1]. The equilibria are computed as follows: (i) For α∈h0, 1 −1 2εh, wα ε(p, 0)>wα ε(p,y)∀y∈]0, 1], for all p∈[0, 1]. Therefore, we only have the following equilibrium: p=0. (ii) For α∈h1−1 2ε,1 2h, wα ε(p, 0)>wα ε(p,y)∀y∈]0, 1], if p<ˆ p; wα ε(p, 1)>wα ε(p,y)∀y∈[0, 1[, if p>ˆ p; wα ε(p,y) = 3/2 ∀y∈[0, 1], if p=ˆ p. Therefore, we have the following equilibria: p= 0, p= 1, and p=ˆ p . Note that for α=1−1 2εwe obtain the following: ˆ p=1. (iii) For α=1 2, wα ε(p, 0)>wα ε(p,y)∀y∈]0, 1], if p<1−ε; wα ε(p, 1)>wα ε(p,y)∀y∈[0, 1[, if p>ε; wα ε(p,y) = 3/2 ∀y∈[0, 1], if 1 −ε⩽p⩽ε. In this case, we have an infinite number of equilibria: p= 0, p= 1, and every p∈[1−ε,ε]. (iv) For α∈i1 2,1 2εi, wα ε(p, 0)>wα ε(p,y)∀y∈]0, 1], if p<p∗; wα ε(p, 1)>wα ε(p,y)∀y∈[0, 1[, if p>p∗; wα ε(p,y) = 3/2 ∀y∈[0, 1], if p=p∗. Therefore, we have three equilibria: p= 0, p= 1, and p=p∗ . Note that for α=1 2ε , p∗=0. v) For α∈i1 2ε, 1i, wα ε(p, 1)>wα ε(p,y)∀y∈[0, 1[, for all p∈[0, 1]. Therefore, we have a unique equilibrium: p=1. Appendix A.3. Proof of Theorem 1 For every player i and every ν∈N , let wαν i,ν be the summary utility function of the game Gαν ν, i.e., wαν i,ν(σ,τ):=αi,ν"inf bi∈γi,ν(σ)ui,ν(bi,τ)#+ (1−αi,ν)"sup bi∈γi,ν(σ) ui,ν(bi,τ)#∀(σ,τ)∈Σ×Σ, Games 2024,15, 27 17 of 26 and wα ibe the summary utility function of the game Gα, i.e., wα i(σ,τ):=αiinf bi∈γi(σ)ui(bi,τ)+ (1−αi)"sup bi∈γi(σ) ui(bi,τ)#∀(σ,τ)∈Σ×Σ. The continuous convergence of the sequence of functions {wαν i,ν}ν∈N to the function wα i , for every i∈I , guarantees the result. In fact, if {σ∗ ν}ν∈N⊂Σ is a sequence converging to σ∗∈Σ such that, for every ν∈N , (γν(σ∗ ν) , σ∗ ν) is an α -MEU psychological Nash equilibrium of Gαν ν, then it follows that, for every player i, wαν i,ν(σ∗ ν,σ∗ ν)⩾wαν i,ν(σ∗ ν,(yi,σ∗ −i,ν)) ∀yi∈Σi. Applying the continuous convergence of {wαν i,ν}ν∈Nto wα i, we obtain the following: wα i(σ∗,σ∗) = lim ν→∞wαν i,ν(σ∗ ν,σ∗ ν)⩾lim ν→∞wαν i,ν(σ∗ ν,(yi,σ∗ −i,ν)) = wα i(σ∗,(yi,σ∗ −i)) ∀yi∈Σi. This latter inequality implies that (γ(σ∗) , σ∗) is an α -MEU psychological Nash equilibrium of Gα . Therefore, the proof reduces when verifying the continuous convergence of {wαν i,ν}ν∈N to wα i . That is, we need to check that for every (σ , τ)∈Σ×Σ and for every sequence {(σν,τν)}ν∈Nconverging to (σ,τ)we obtain the following inequalities: lim sup ν→∞ wαν i,ν(σν,τν)⩽wα i(σ,τ)⩽lim inf ν→∞wαν i,ν(σν,τν). (A1) Denote with wm i,ν(σ,τ) = inf bi∈γi,ν(σ)ui,ν(bi,τ),wM i,ν(σ,τ) = sup bi∈γi,ν(σ) ui,ν(bi,τ), and wm i(σ,τ) = inf bi∈γi(σ)ui(bi,τ),wM i(σ,τ) = sup bi∈γi(σ) ui(bi,τ). Consider (σ , τ)∈Σ×Σ and take a sequence {(σν , τν)}ν∈N converging to (σ , τ) . Now we prove the following: lim sup ν→∞ wm i,ν(σν,τν)⩽wm i(σ,τ)⩽lim inf ν→∞wm i,ν(σν,τν), and lim sup ν→∞ wM i,ν(σν,τν)⩽wM i(σ,τ)⩽lim inf ν→∞wM i,ν(σν,τν). First, we show that wm i(σ,τ)⩽lim inf ν→∞wm i,ν(σν,τν)resp. wM i(σ,τ)⩾lim sup ν→∞ wm i,ν(σν,τν). Suppose by contradiction that we have the following: wm i(σ,τ)>lim inf ν→∞wm i,ν(σν,τν)resp. wM i(σ,τ)<lim sup ν→∞ wm i,ν(σν,τν). (A2) This means that along a subsequence {(σνk,τνk)}k∈N, we have the following: lim k→∞wm i,νk(σνk,τνk)<wm i(σ,τ)resp. lim k→∞wM i,νk(σνk,τνk)>wM i(σ,τ). (A3) Games 2024,15, 27 18 of 26 Additionally, the continuity of ui and ui,ν for every ν and the compactness of the images of γi and γi,ν , for every ν , guarantee the following: exist bm i∈γi(σ) and bm i,ν∈γi,ν(σν) , (resp. bM i∈γi(σ)and bM i,ν∈γi,ν(σν)), for every ν, such that we have the following: wm i(σ,τ) = ui(bm i,τ) = inf bi∈γi(σ)ui(bi,τ), resp. wM i(σ,τ) = ui(bM i,τ) = sup bi∈γi(σ) ui(bi,τ)! and wm i,ν(σν,τν) = ui,ν(bm i,ν,τν) = inf bi,ν∈γi,ν(σν)ui,ν(bi,ν,τν), resp. wM i,ν(σν,τν) = ui,ν(bM i,ν,τν) = sup bi,ν∈γi,ν(σν) ui,ν(bi,ν,τν)!. Consider the sequence of beliefs {bm i,νk}k∈N , (resp. {bM i,νk}k∈N ), obtained along the subsequence {(σνk , τνk)}k∈N , as in (A3) . The sequence {bm i,νk}k∈N , (resp. {bM i,νk}k∈N ), has a subsequence {bm i,νh}h∈N , (resp. {bM i,νh}h∈N ), which converges to a point ˆ bm i∈Bi , (resp. ˆ bM i∈Bi ), since Bi is compact. The point ˆ bm i , (resp. ˆ bM i ), actually belongs to γi(σ) . In fact, by definition, the upper limit Lim sup ν→∞ γi,ν(σν) contains the limits of every converging subsequence of {bm i,νk}k∈N, (resp. {bM i,νk}k∈N); that is, ˆ bm i,ˆ bM i∈Lim sup ν→∞ γi,ν(σν). Moreover, {γi,ν}ν∈N is sequentially upper-convergent to γi , meaning that Lim sup ν→∞ γi,ν(σν)⊆ γi(σ) ; therefore, ˆ bm i , ˆ bM i∈γi(σ) . By construction ui(bm i , τ)⩽ui(ˆ bm i , τ) (resp. ui(bM i , τ)⩾ ui(ˆ bM i , τ) ). The sequence {ui,ν}ν∈N sequentially converges to ui ; since (bm i,νh , τνh)→(ˆ bm i , τ) , (resp. (bM i,νh,τνh)→(ˆ bM i,τ)), we obtain the following: ui(ˆ bm i,τ) = lim h→∞ui,νh(bm i,νh,τνh),resp. ui(ˆ bM i,τ) = lim h→∞ui,νh(bM i,νh,τνh). Hence, wm i(σ,τ) = ui(bm i,τ)⩽ui(ˆ bm i,τ) = lim h→∞ui,νh(bm i,νh,τνh) = lim h→∞wm i,νh(σνh,τνh), resp. wM i(σ,τ) = ui(bM i,τ)⩾ui(ˆ bM i,τ) = lim h→∞ui,νh(bM i,νh,τνh) = lim h→∞wM i,νh(σνh,τνh). Then, inequality (A3) implies that wm i(σ,τ)⩽lim h→∞wm i,νh(σνh,τνh)<wm i(σ,τ), resp. wM i(σ,τ)⩾lim h→∞wM i,νh(σνh,τνh)>wM i(σ,τ), which results in a contradiction. So, wm i(σ,τ)⩽lim inf ν→∞wm i,ν(σν,τν),resp. wM i(σ,τ)⩾lim sup ν→∞ wM i,ν(σν,τν). Now we show the following: wm i(σ,τ)⩾lim sup ν→∞ wm i,ν(σν,τν),resp. wM i(σ,τ)⩽lim inf ν→∞wM i,ν(σν,τν). Games 2024,15, 27 19 of 26 Let bm i∈γi(σ)(resp. bM i∈γi(σ)) be such that ui(bm i,τ) = inf bi∈γi(σ)ui(bi,τ) = wm i(σ,τ) resp. ui(bM i,τ) = sup bi∈γi(σ) ui(bi,τ) = wM i(σ,τ)!. The points bm i and bM i exist because of the continuity of ui and the compactness of γi(σ) for every σ∈Σ. Since the sequence {γi,ν}ν∈Nis sequentially convergent to γi, i.e., γi(σ)⊆Lim inf ν→∞γi,ν(σν), then, by definition, there exists a sequence {ˆ bm i,ν}ν∈N converging to bm i , (resp. {ˆ bM i,ν}ν∈N converging to bM i), such that, for every ν,ˆ bm i,ν∈γi,ν(σν)(resp. ˆ bM i,ν∈γi,ν(σν)). The sequence {ui,ν}ν∈Nsequentially converges to ui; it follows that lim sup ν→∞ ui,ν(ˆ bm i,ν,τν)⩽ui(bm i,τ),resp. lim inf ν→∞ui,ν(ˆ bM i,ν,τν)⩾ui(bM i,τ). By construction, for every ν∈N, we have the following: wm i,ν(σν,τν)⩽ui,ν(ˆ bm i,ν,τν),resp. wM i,ν(σν,τν)⩾ui,ν(ˆ bM i,ν,τν). This finally implies the following: lim sup ν→∞ wm i,ν(σν,τν)⩽lim sup ν→∞ ui,ν(ˆ bm i,ν,τν)⩽ui(bm i,τ) = wm i(σ,τ), resp. lim inf ν→∞wM i,ν(σν,τν)⩾lim sup ν→∞ ui,ν(ˆ bM i,ν,τν)⩾ui(bM i,τ) = wM i(σ,τ). So, we obtain the following: lim sup ν→∞ wm i,ν(σν,τν)⩽wm i(σ,τ)⩽lim inf ν→∞wm i,ν(σν,τν) and lim sup ν→∞ wM i,ν(σν,τν)⩽wM i(σ,τ)⩽lim inf ν→∞wM i,ν(σν,τν). Hence, from the properties of the upper and lower limits, we obtain the following: lim sup ν→∞ wαν i,ν(σν,τν) = lim sup ν→∞hαi,νwm i,ν(σν,τν) + (1−αi,ν)wM i,ν(σν,τν)i⩽ lim sup ν→∞ αi,νwm i,ν(σν,τν) + lim sup ν→∞ (1−αi,ν)wM i,ν(σν,τν)⩽ lim sup ν→∞ αi,νlim sup ν→∞ wm i,ν(σν,τν)+lim sup ν→∞ (1−αi,ν)lim sup ν→∞ wM i,ν(σν,τν)⩽ αiwm i(σ,τ) + (1−αi)wM i(σ,τ), and αiwm i(σ,τ) + (1−αi)wM i(σ,τ)⩽ lim inf ν→∞αi,νlim inf ν→∞wm i,ν(σν,τν)+lim inf ν→∞(1−αi,ν)lim inf ν→∞wM i,ν(σν,τν)⩽ lim inf ν→∞αi,νwm i,ν(σν,τν) + lim inf ν→∞(1−αi,ν)wM i,ν(σν,τν)⩽ Games 2024,15, 27 20 of 26 lim inf ν→∞hαi,νwm i,ν(σν,τν) + (1−αi,ν)wM i,ν(σν,τν)i=lim inf ν→∞wαν i,ν(σν,τν). Condition (A1) is satisfied and {wαν i,ν}ν∈Ncontinuously converges to wα i. Appendix A.4. Proof of Lemma 4 For every pair of strategic profiles (p,r)and (x,y), we have the following: Uα 1(γ1(x,y),(p,r)) = αmin ˜ q∈γ1(x,y)u1(˜ q,(p,r))+ (1−α)"max ˜ q∈γ1(x,y)u1(˜ q,(p,r))#. Recalling the form of γ1, we obtain the following: arg min ˜ q∈γ1(x,y) u1(˜ q,(p,r)) = arg min ˜ q∈[0,1] u1(˜ q,(p,r)) = arg min ˜ q∈[0,1] [2˜ q(1−p−r)−8pr +5p+5r−3] =      0 if p<1−r, [0, 1]if p=1−r, 1 if p>1−r, and arg max ˜ q∈γ1(x,y) u1(˜ q,(p,r)) = arg max ˜ q∈[0,1] u1(˜ q,(p,r)) = arg max ˜ q∈[0,1] [2˜ q(1−p−r)−8pr +5p+5r−3] =      1 if p<1−r, [0, 1]if p=1−r, 0 if p>1−r. Therefore, given the two strategic profiles (x,y)and (p,r), min ˜ q∈γ1(x,y)u1(˜ q,(p,r)) = (−8pr +5p+5r−3 if p⩽1−r, −8pr +3p+3r−1 if p>1−r, and max ˜ q∈γ1(x,y)u1(˜ q,(p,r)) = (−8pr +5p+5r−3 if p⩾1−r, −8pr +3p+3r−1 if p<1−r. Hence, the summary utility function has the following form: wα 1((x,y),(p,r)) = Uα 1(γ1(x,y),(p,r)) =      p(5−8r−2α) + 5r−3−2αr+2αif p>1−r, p(3−8r+2α) + 3r−1+2αr−2αif p<1−r, p(5−8r) + 5r−3 if p=1−r. Recall that BRα 1(r) = {p∈Σ1|wα 1((p,r),(p,r)) ⩾wα 1((p,r),(x,r)),∀x∈Σ1}. In order to construct BRα 1(r), note the following: • If p> 1 −r , the function wα 1((x , y) , (p , r)) increases on p for r<5−2α 8 , is constant for r=5−2α 8, decreasing for r>5−2α 8; • If p< 1 −r , the function wα 1((x , y) , (p , r)) increases on p for r<3+2α 8 , is constant for r=3+2α 8, decreasing for r>3+2α 8; • If p= 1 −r , the function wα 1((x , y) , (p , r)) is constant on p since wα 1((x , y) , (p , r)) = 8r2−8r+2. Games 2024,15, 27 21 of 26 We need to distinguish three cases: • Suppose α<1/2; in this case, we have that 3+2α 8<5−2α 8. Therefore, for every (x,y), we have the following: - If r<3+2α 8 , then wα 1((x , y) , (· , r)) increases on [ 0, 1 ] and attains the maximum for p=1. - If r=3+2α 8 , then wα 1((x , y) , (· , r)) is constant on [ 0, 1 −r] , increases on [ 1 −r , 1 ] , and attains the maximum for p=1. - If 3+2α 8<r<1 2 , then wα 1((x , y) , (· , r)) decreases on [ 0, 1 −r] , increases on [ 1 −r , 1 ] , and attains the maximum for p=1. - If r=1 2 , then wα 1((x , y) , (· , r)) decreases on [ 0, 1 −r] , increases on [ 1 −r , 1 ] , and attains the maximum for p=1 and p=0. - If 1 / 2 <r<5−2α 8 , then wα 1((x , y) , (· , r)) decreases on [ 0, 1 −r] , increases on [1−r, 1], and attains the maximum for p=0. - If r=5−2α 8 , then wα 1((x , y) , (· , r)) decreases on [ 0, 1 −r] , is constant on [ 1 −r , 1 ] , and attains the maximum for p=0. - If r>5−2α 8 , then wα 1((x , y) , (p , r)) decreases on [ 0, 1 ] and attains the maximum for p=0. It follows that BRα 1(r) =      1 if r∈[0, 1/2[, {0, 1}if r=1/2, 0 if r∈]1/2, 1]. • Suppose α=1/2; in this case, we have that 3+2α 8=5−2α 8=1 2. Therefore, for every (x,y), we have the following: - If r<1 2 , then wα 1((x , y) , (· , r)) , increases on [ 0, 1 ] , and attains the maximum for p=1. - If r=1 2 , then wα 1((x , y) , (· , r)) is constant on [ 0, 1 ] ; therefore, every p∈[ 0, 1 ] is a maximum point. - If r>1 2 , then wα 1((x , y) , (· , r)) decreases on [ 0, 1 ] and attains the maximum for p=0. In this case, BRα 1(r) =      1 if r∈[0, 1/2[, [0, 1]if r=1/2, 0 if r∈]1/2, 1]. • Suppose α>1/2; in this case, we have that 3+2α 8>5−2α 8. Therefore, for every (x,y), we have the following: - If r<5−2α 8 , then wα 1((x , y) , (· , r)) increases on [ 0, 1 ] and attains the maximum for p=1. - If r=5−2α 8 , then wα 1((x , y) , (· , r)) increases on [ 0, 1 −r] and is constant on [ 1 −r , 1 ] ; therefore, every p∈[1−r, 1]is a maximum point. - If 5−2α 8<r<3+2α 8 , then wα 1((x , y) , (· , r)) increases on [ 0, 1 −r] , decreases in [1−r, 1], and attains the maximum for p=1−r. Games 2024,15, 27 22 of 26 - If r=3+2α 8 , then wα 1((x , y) , (· , r)) is constant on [ 0, 1 −r] and decreases on [ 1 −r , 1 ] ; therefore, every p∈[0, 1 −r]is a maximum point. - If r>3+2α 8 , then wα 1((x , y) , (· , r)) decreases on [ 0, 1 ] and attains the maximum for p=0. In this case, BRα 1(r) =                1 if r∈0, 5−2α 8, 3+2α 8, 1if r=5−2α 8, 1−rif r∈5−2α 8,3+2α 8, 0, 5−2α 8if r=3+2α 8, 0 if r∈3+2α 8, 1. The equilibria computation follows immediately from the fixed points of the best reply correspondences. Appendix B The counterexample in Section 5highlights that the lack of existence of equilibria depends on the lack of convexity of the best reply correspondence, which, in turn, depends on the lack of quasi-concavity of the player’s utility with respect to their own strategy. Now, it is clear that the lack of quasi-concavity is due to the presence (in the utility function) of the max operator, with respect to beliefs. In the quoted example, only second-order beliefs play a role in a player’s utility. Now, since second-order beliefs directly depend on the player’s strategy, the natural question is whether the presence of the max operator with respect only to first-order beliefs still affects the quasi-concavity of the utility functions in the classical models under strategic ambiguity (i.e., without psychological utilities). Below, we analyze this issue. In order to focus on the mathematical problem, we consider the following simple form of strategic ambiguity: for every mixed strategy profile (σi , σ−i) , agent i does not observe precisely σ−i but considers as consistent any belief ( q ) in a given subset γi(σi , σ−i)⊆Σi . For each belief q∈γi(σi,σ−i), his expected utility would be11 ui(σi,q) = ∑ ai∈Ai σi(ai)ui(ai,q) = ∑ ai∈Ai σi(ai)"∑ a−i∈A−i q(a−i)ˆ ui(ai,a−i)# so that an optimistic player. player i (i.e. αi= 0 ) , has the following utility function Ui:Σ→Rdefined by the following: Ui(σi,σ−i) = max q∈γi(σi,σ−i)ui(σi,q)∀(σi,σ−i)∈Σ, where γi:Σ⇝Σ−iis the belief correspondence. Now, it follows immediately that the function Ui is concave with respect to σi if the set-valued map M:Σ⇝Σ−i, defined by Mi(σi,σ−i) = arg max q∈γi(σi,σ−i) ui(σi,q)∀(σi,σ−i)∈Σ depends only on σ−i, i.e., Mi(σ′ i,σ−i) = Mi(σ′′ i,σ−i)∀σ′ i,σ′′ i∈Σi. (A4) In fact, in this case, we immediately have that—for any arbitrary σ−i —there exists ˆ q , such that, for every t∈[0, 1], ˆ q∈Mi(tσ′ i+ (1−t)σ′′ i,σ−i); Games 2024,15, 27 23 of 26 so Ui(σ′ i,σ−i) = ui(σ′ i,ˆ q),Ui(σ′′ i,σ−i) = ui(σ′′ i,ˆ q),Ui(tσ′ i+ (1−t)σ′′ i,σ−i) = ui(tσ′ i+ (1−t)σ′′ i,ˆ q) and Ui(tσ′ i+ (1−t)σ′′ i,σ−i) = tUi(σ′ i,σ−i) + (1−t)Ui(σ′′ i,σ−i), implying that Uiis concave with respect to σi. When condition (A4) does not hold, utilities are not always quasi-concave. So, the best reply correspondences do not have convex values and equilibria may not exist, as shown in the example below. Example A1. We consider a two-player game: the pure strategy set of Player 1 (Anne) is A1= {Accept , Reject} and the pure strategy set of Player 2 (John) is A2={Accept , Reject} . The game is represented as follows (Table A1): Table A1. Game form for Example A1. John Anne Accept Reject Accept 1, 0 0, 2 Reject 0, 1 2, 0 As usual, p is the mixed strategy of Player 1, where, with an abuse of notation, p is the probability of Accept and 1 −p is the probability of Reject . Similarly, r is the mixed strategy of Player 2, where, with an abuse of notation, r is the probability of Accept and 1 −r is the probability of Reject . John’s utility is assumed to be the classical expected utility without ambiguity. Anne, on the other hand, faces ambiguity: she does not observe r but, if the chosen strategy profile is (p , r) , she perceives as consistent all the beliefs q (where q is the probability of John’s Accept and 1 −r is the probability of John’s Reject ) that belong to γ1(p,r), where γ1is Anne’s belief correspondence defined by the following12: γ1(p,r) = (r if p ∈[0, 1/2[, [0, 1]if p ∈[1/2, 1]. For every strategy (p) and belief (q), the expected utility of Anne is as follows: u1(p,q) = 3pq −2p−2q+2. Now, let α1=0be Anne’s ambiguity attitude parameter. We have the following: arg max q∈γ1(p,r) u1(q,(p,r)) = arg max q∈γ1(p,r) [3pq −2p−2q+2] =            r if p ∈[0, 1/2[, 0if p ∈[1/2, 2/3[, [0, 1]if p =2/3, 1if p ∈]2/3, 1]. Therefore, U1(p,r) = max q∈γ1(p,r)u1(p,q) =      3pr −2p−2r+2if p ∈[0, 1/2[, −2p+2if p ∈[1/2, 2/3], p if p ∈]2/3, 1] Games 2024,15, 27 24 of 26 Now, we compute Anne’s best reply: BR1(r) = {p∈[0, 1]|U1(p,r)⩾U1(x,r),∀x∈[0, 1]}. Note that we have the following: - CASE p∈[ 1 / 2, 1 ] , the function U1 attains the maximum value 1 in the two maximum points p=1/2, p=1, for every r ∈[0, 1], i.e., max p∈[1/2,1]U1(p,r) = U1(1/2, r) = U1(1, r) = 1, ∀r∈[0, 1]. - CASE p ∈[0, 1/2[the function (a) strictly decreases on p for r ∈[0, 2/3[and attains the maximum for p =0, so max p∈[0,1/2[U1(p,r) = 2−2r. Note that 2−2r>1⇐⇒ r∈[0, 1/2[ (b) is constant on p for r =2/3: U1(p, 2/3) = 2/3 ∀p∈[0, 1] (c) strictly increases on p for r∈] 2 / 3, 1 ] and attains no maximum points because the domain is not closed, but, in this case, we have the following: U1(p,r) = 3pr −2p−2r+2<3 2r−1−2r+2=1−1 2r<1. We have the following: BR1(r) =      {0}if r ∈[0, 1/2[, {0, 1/2, 1}if r =1/2, {1/2, 1}if r ∈]1/2, 1]. The best reply of Player 2 can be easily computed, as there are no psychological effects: BR2(p) =      1if p ∈[0, 1/3[, [0, 1]if p =1/3, 0if p ∈]1/3, 1]. It immediately follows that there are no equilibria. Notes 1 The literature on psychological games has increased considerably in the past decades; we recall [ 8 ] for further theoretical findings, refs. [9–11] for some applications, and [12,13] for surveys on psychological games and references. 2Optimistic and intermediate attitudes actually have strong empirical support (see for example [21]). 3See [7] for additional details on the topological and metric structure of the beliefs space. 4 The self-confirming (or conjectural) equilibrium was first studied in [ 36 , 37 ]. In [ 5 ], the definition was extended by taking into account different attitudes toward ambiguity or model uncertainty. 5See the Introduction section in [5]. 6 Note that, lower and upper semi-continuous set-valued maps (or correspondences) are also often denoted in the literature, respectively, as lower and upper hemi-continuous set-valued maps. However, we follow the notation in the book [38]. 7 The game considered in Example 3.4 in [ 7 ] is different from the one presented in the present paper; however, ambiguous hierarchies of beliefs have the same structure. 8For technical reasons, we consider the case where functions take values in R=R∪ {−∞,+∞}.