Personal Finance Decisions with Untruthful Advisors: an Agent-Based Model
aa r X i v : . [ q -f i n . C P ] S e p P ER SONAL F INANCE D EC ISIONS WITH U NTRUTHFUL A DVISORS : AN A GENT -B ASED M ODEL
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Loretta Mastroeni Dept. of Economics
Roma Tre UniversityVia Silvio D’Amico 77, 00145 Rome, Italy [email protected]
Maurizio Naldi
Dept. of Law, Economics, Politics and Modern languagesLUMSA UniversityVia Marcantonio Colonna 19, 00192 Rome, Italy [email protected]
Pierluigi Vellucci
Dept. of EconomicsRoma Tre UniversityVia Silvio D’Amico 77, 00145 Rome, Italy [email protected]
September 17, 2019 A BSTRACT
Investors usually resort to financial advisors to improve their investment process until the point ofcomplete delegation on investment decisions. Surely, financial advice is potentially a correctingfactor in investment decisions but, in the past, the media and regulators blamed biased advisors formanipulating the expectations of naive investors. In order to give an analytic formulation of theproblem, we present an Agent-Based Model formed by individual investors and a financial advisor.We parametrize the games by considering a compromise for the financial advisor (between a suffi-cient reward by bank and to keep his/her reputation), and a compromise for the customers (betweenthe desired return and the proposed return by advisor). Then we obtain the Nash equilibria and thebest response functions of the resulting game. We also describe the parameter regions in which thesepoints result acceptable equilibria and the greediness/naivety of the customers emerge naturally fromthe model. Finally, we focus on the efficiency of the best Nash equilibrium. K eywords Opinion dynamics · Agents-based model · Personal finance · Price of Stability
Decisions concerning personal finance are taken by individuals on the basis of a variety of factors. For example,the investment decision process appears to incorporate a broad range of variables that may influence the individualinvestor’s behaviour, such as the perceived ethics of a firm, and recommendations from individual stock brokers orfriends/coworkers [1]. Actually, investors usually resort to financial advisors to improve their investment process untilthe point of complete delegation on investment decisions. Surely, financial advice is potentially a correcting factor ininvestment decisions [2].Nevertheless, in the aftermath of the recent financial bubbles, the media and the regulators usually placed much ofthe blame on biased advisors for manipulating the expectations of naive investors [3]. According to this view, ananalyst may receive incentives to generate biased, optimistic forecasts while naive individual investors are unable torecognize that these biased recommendations are motivated by incentives to sell financial products. This means that,when asked for a professional advice (i.e. an opinion), advisors may not straightforwardly state what they truly think,but rather be tempted to misrepresent their opinion to conform to the bank they are paid by. In any case, the role of
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17, 2019financial advisors, as well as that of other influencers, is to be properly accounted for in an analysis of personal financedecisions.However, though the decisions taken in personal finance have been a subject of interest in a number of papers (see, e.g.,[4, 5, 6, 7, 8]), so far the adopted framework has considered an individual acting without interaction with influencersof any kind.In this paper, we wish instead to consider such interactions, with the aim of understanding how the opinions ofan individual investor may change under the influence of his/her advisors, considering the aims of the stakeholdersinvolved. For this purpose we formulate an Agent-Based Model (ABM) that includes three classes of agents: a bank,a financial advisor, a set of investors or customers. This model mimics the environment an individual investor findswhen it manageshis/her investment through the local branch of a bank, where a financial advisor oversees a group ofthe bank’s customers. The different aims of the stakeholders are recognized by resorting to a game-theoretic model,where each personal investment/advise corresponds to selecting a strategy, and the agents’ payoff depends on thestrategies chosen by him/herself and other players. We refer to this game as the personal finance game and providethe following contributions: • We introduce an ABM to address the personal finance game (Section 3). The major advantage that we expectfrom the adoption of an analytic framework like ABM is that analytic derivations of the properties of themodel can be equally used as descriptive and as prescriptive tools, as widely noticed in the literature (e.g. [9],and also [10, 11, 12, 13, 14, 15]). • We obtain the best response functions and the Nash equilibria of the resulting game (Section 4.1). We alsodescribe the parameter regions in which these points represent acceptable equilibria (Section 4.1.1). Surpris-ingly, without constraints on the returns (except for the fact that the proposed returns by advisor differ fromthe one desired by customers), the equilibrium is reached when the customer expects a return bigger than theone the advisor proposes instead. • We provide a mathematical description of the boundary of the utility functions domain D (Section 4.2). • We introduce the social welfare in this context and obtain an analytic formulation of the Price of Stability (thegeneralization of the Price of Anarchy when more equilibria are present) for our ABM (Section 5) thanks tothe analytic formulation of D (given in the previous point). Our paper is related to the literature framework of opinion formation games that relax the assumption of truthfulness (a.k.a as honesty) in the process of opinion formation, allowing game players to express some opinions which neednot coincide with their true opinions.The players whose opinions we wish to model are represented as nodes of a social network (i.e., vertices on a graph),where the links between the nodes represent the direct influence between players in forming their opinions.We therefore introduce a connected undirected graph G = ( V , E ) be with |V| = n and for each edge e = ( i, j ) ∈ E let w ij ≥ be its weight. Let W = [ w ij ] ij be the matrix of weights. Every vertex of the graph (i.e., each player or agent)is characterized by an internal opinion s i and a stated opinion z i . The set of neighbors of agent i in the social networkrepresented by the graph G is denoted by N ( i ) .This game can be expressed as an instance ( G, W, s , z ) that combines a weighted graph ( G, W ) and the vectorsof opinions s = ( s , . . . , s n ) and z = ( z , . . . , z n ) , which are attributes of the nodes. The internal opinion s i isunchanged and not affected by opinion updates, while each player’s strategy is represented by his/her stated opinion z i , which may be different from his/her s i and gets updated [16, 17, 18, 19, 20, 21, 22, 23, 24]. The paper by Buechelet al. [25] differs from all these papers by mainly as it considers true and stated opinions evolving over time accordingto different laws.In [25] the utility of agent i depends on the distance of true opinion s i to stated opinion z i as well as on the distanceof stated opinion z i to group opinion q i . Bindel et al. [16] study the price of anarchy — the ratio between the costof the Nash equilibrium and the cost of the optimal solution — in a game of opinion formation. They assume thatperson i has an internal opinion s i , which remains unchanged from external influences, and a stated opinion z i whichis updated as a weighted sum of his/her’ neiughbours’ stated opinions z i = s i + P j ∈ N ( i ) w ij z j P j ∈ N ( i ) w ij (1)2 PREPRINT - S
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17, 2019where w i,j ≥ . Both opinions are assumed to be real numbers. Updating z i as in (1) allows to minimize the costfunction ( z i − s i ) + X j ∈ N ( i ) w ij ( z j − z i ) . (2)Both papers, [16] and [25], are inspired by classical models due to DeGroot [26] and Friedkin-Johnsen [27]. Also in[25] both opinions z i and s i are assumed to be real numbers.Gionis et al. [17] follow the framework of Bindel et al. [16], by considering equations (1) and (2) as update ruleand personal cost function. The internal and external opinions have been modeled as real values in the interval [0 , .Gionis et al. study the CAMPAIGN problem, whose goal is to identify a set of target nodes T , whose positive opinionabout an information item will maximize the overall positive opinion for the item in the social network. The objectivefunction to maximize is therefore g ( z ) = P ni =1 z i .Bhawalkar et al. [18] analyze the equilibrium outcomes of symmetric co-evolutionary game and the K-nearest neigh-bor (K-NN) game, distinguishing between internal and stated opinions with the usual symbols s i and z i (which arereal numbers). In the K-NN game, each agent has exactly K friends, so the interaction is of the nearest neighbors typeand the size of N ( i ) is exactly K for each agent i .Ferraioli et al. [19] continue the study of Bindel et al. by simplifying their model to the case of binary opinion z i ,which can be found in the individual’s voting intention in a referendum, while s i ∈ [0 , . They study best-responsedynamics and show upper and lower bounds on the convergence to Nash equilibria.The cost function considered by Chierichetti et al. [20] (where update rules for z i and s i are not present) replacesthe quadratic terms in [16] by distances in a discrete metric space while s i belongs to a discrete set (binary in somespecial cases) and z i ∈ R . The authors adopt a strategy z minimizing the social cost function as an optimal solutionand establish bounds on the price of stability Auletta et al. [21] consider a personal cost that is defined through a monotone non-decreasing function of z , assumingbinary z i and s i (without update rules for them). The authors called that class generalized discrete preference games .In particular, they show that every game with two strategies per agent that admits a generalized ordinal potential isstructurally equivalent (in particular, better-response equivalent) to a generalized discrete preference game. In anotherwork [22], the same authors consider the game in which agents are utility maximizers, z i , s i ∈ { , } and address thequestions of price of stability/price of anarchy of a game in terms of the social welfare: SW ( z ) := P i u i ( z ) .Bilò et al. [23] focus on the case in which, for each player i , the innate opinion s i ∈ [0 , , while the expressed opinion z i ∈ { , } . They define a cost-minimization n -player game. Bilò et al. show that any game in this class always admitsan ordinal potential that implies the existence of pure Nash equilibria and convergence of better-response dynamicsstarting from any arbitrary strategy profile. The social optimum is obtained with respect to the problem of minimizingthe sum of the players’ costs. They also focus on the efficiency losses due to selfish behavior and give upper and lowerbounds on the price of anarchy and lower bounds on the price of stability.In [24], Chen et al. bound the price of anarchy for a game in which both s i and z i are real numbers. In this Section we introduce an ABM to study the personal finance game . In our ABM there are three classes ofagents: • a bank ( B ); • a financial advisor ( A ); • a set of n customers or customers ( CL i , i = 1 , . . . , n ).Our model falls in the literature framework of opinion formation games where game players can express some opinionsand may change them according to the interactions with the other agents. For some of them the opinions they expressneed not coincide with their true opinions. The opinions concern investment decisions. The authors refer to this class of games as discrete preference games . The price of stability is a measure of the game efficiency that is commonly adopted instead of the price of anarchy whenmultiple Nash equilibria are present, and is defined as on the ratio between the social cost of the best Nash equilibrium and theoptimal solution. We return to the subject in Section 5. PREPRINT - S
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17, 2019The aim of bank B is to steer the customers towards a particular investment decision, represented by an opinion w ∈ R + . For example, w could concern the decision to buy a security S rather than a different security S or otherfinancial instruments.The financial advisor A expresses an opinion s ∈ R + which need not coincide with his/her true opinion x ∈ R + ,respectively referred to in the following as the stated opinion and the internal opinion . The financial advisor maytherefore be untruthful . A is paid by B and he/she gives advice (by way of s ) to customers when invited to do so,but the stated opinion s might not perfectly correspond to the one recommended by bank, w (to preserve his/her goodreputation, for instance).customers have their opinions c i , i = 1 , . . . , n , which fall within the range [ d i , s ] where s is the stated opinion and d i ≤ s is a positive lower bound, which represents the opinion that the customer i would assume if there wasn’t anyinteraction with A. The opinions of all customers are collected in c := ( c , . . . , c n ) ∈ R n .Opinions c i , i = 1 , . . . , n and s change over time, i.e. c i = c i ( t ) i = 1 , . . . , n and s = s ( t ) , while w , x and d arefixed over time. However, we assume that all the opinions lie within the range [0 , .In the spirit of the models [28, 25], we consider a utility function for A that depends on the incentive to be truthful (theintrinsic part) and the incentive to steer the customers towards w (the remunerative part). The incentive to be truthfulcould be related to the Advisor’s conscience or to the desire of the Advisor to keep his/her reputation. Additionally,we assume that the utility function for A also depends on the desire to influence the customers. The resulting utilityfunction is supposed to be a quadratic form in the opinions and to be additive.Thus, the utility of the financial advisor depends on the distance of his/her true opinion x to his/her stated opinion s aswell as on the distance of the bank’s desired investment decision w to customers opinions c and the distance between s and c : u A ( c , s, w, x ) = − α ( s − x ) − β n X i =1 ( w − c i ) − γ n X i =1 ( s − c i ) , (3)where α, β, γ > ; β is the remuneration coefficient for A and is paid by bank B . The more customers eventually buythe security pushed forward by the bank B , the more the advisor A is remunerated.The advisor’s strategic leverage is the stated opinion s , and his/her aim is to maximize his/her utility: max s u A ( c , s, w, x ) . (4)To define the utility of customers CL i , i = 1 , . . . , n , we introduce the following returns on their investments: • r s , which is the return proposed by A to all the customers; • r d i , which is the return that each customer considers that he/she can achieve through a “good” investmentdecision.The customer i would like to get r d i but he/she does not completely trust herself (the customer is not assumed to bea financial expert)) and moves towards r s . In general, two possible situations may occur: r s ≤ r d i and r s > r d i . Inthe first, A proposes to the customer i a return that is less (or equal) than expectations of CL i while in the second wehave the opposite. The rationales for the two cases are respectively that the financial advisor is able to find a betterinvestment than the customers due to his/her superior financial expertise or that customers have unrealistic expectationsdue to their poor knowledge of financial markets.We assume that the utility of each customer i depends on his/her lack of agreement with the advisor A . This cognitivedissonance [16] provides customers with an incentive to modify their behavior to reduce the “cost” of this lack ofconsensus. Remember that customers have their opinions c i , which fall within the range [ d i , s ] where advisor’s statedopinion s is a positive upper bound for them, and that parameter d i represents the opinion that the customer i wouldassume if there weren’t any interaction with A. Thus, the utility of CL i assumes value r s if c i = s but depends on thedistance of his/her opinion c i to the advisor’s stated opinion s in all the other cases: u CL i ( c i , d i , s ) = r d i + c i − d i s − d i ( r s − r d i ) − ζ ( s − c i ) , (5)where ζ > ( ∀ i = 1 , . . . , n ) represents the sensitivity of customer to cognitive dissonance. Let us observe that thefollowing consequences hold:(a) ≤ c i − d i s − d i ≤ since d i ≤ c i ≤ s ; 4 PREPRINT - S
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17, 2019(b) r d i + c i − d i s − d i ( r s − r d i ) ≥ for c i ∈ [ d i , s ] because of consequence (a) and r s , r d i > ;(c) h r d i + c i − d i s − d i ( r s − r d i ) i c i = d i = r d i and h r d i + c i − d i s − d i ( r s − r d i ) i c i = s = r s → u CL i ( s, d i , s ) = r s and u CL i ( d i , d i , s ) = r d i − ζ ( d i − c i ) (in the latter case the cost of lack of consensus is maximum). The model described in Section 3 describes the opinion dynamics of a financial advisor and his/her customers, wherethe opinions are influenced by each other’s choice. This interaction can therefore be considered as a strategic game,where the players are the financial advisor and his/her customers (the bank’s role is just to set the fixed aim w and theincentive β ) and their strategic leverages are respectively the stated opinion s and the opinions c ′ i s .In this Section we solve the personal finance game by deriving the Nash equilibria. We also investigate their admissi-bility, i.e., their compatibility with the constraints embedded in the model. We now find the Nash equilibrium of the n + 1 -player game ( n customers plus one financial advisor) using their bestresponse functions. The best response functions aim at maximizing the players’ utilities: max s u A ( c , s, w, x )max c i u CL i ( c i , d i , s ) , i = 1 , . . . , n . (6)For any c i , we obtain the optimal s by zeroing the derivative of the utility ∂u A ( c , s, w, x ) ∂s = − α ( s − x ) − γ n X i =1 ( s − c i )= − αs + 2 αx − γns + 2 γ n X i =1 c i = 0 . (7)Turning to customers utility and fixing s , we obtain ∂u CL i ( c i , d i , s ) ∂c i = r s − r d i s − d i + 2 ζ ( s − c i ) . (8)Then the system (6) that expresses best response functions becomes: ( − αs + 2 αx − γns + 2 γ P ni =1 c i = 0 r s − r di s − d i + 2 ζ ( s − c i ) = 0 . (9)The solution of the system of linear equations is the advisor’s best response function s = 1 α + γn αx + γ n X i =1 c i ! , (10)which is a linear function of the customers’ opinions. Similarly, the customers’ best response function is given by c i = r s − r d i ζ ( s − d i ) + s . (11)For the sake of simplicity, consider the special case where all the customers have the same initial opinion and expecta-tions, i.e., r d i = r d and d i = d ∀ i = 1 , . . . , n , so that all the customers take the same investment decision, i.e., c i = c .Sometimes we will denote this case as the case of homogeneous investors .Then, the best response functions become simply s = αx + γncα + γn (12)and c = r s − r d ζ ( s − d ) + s . (13)5 PREPRINT - S
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17, 2019 . . . . . . . . Customer’s opinion A dv i s o r’ ss t a t e dop i n i on financial advisorcustomerFigure 1: Best response functions of financial advisor (blue) and customers (red). Parameters: r s − r d = − . , ζ = 10 , d = 0 . , α = 0 . , x = 0 . , γ = 0 . , n = 1 .We plot the best response functions of both players in Fig. 1 for a sample case. We see that the best response functionof the financial advisor in Equation (12) is a linear function of the customers’ opinion, with a slope γnα + γn < , whilethe best response function of the customers in Equation (13) is the sum of an angle bisector and a homographic functionwith a vertical asymptote at s = d . The Nash equilibria are represented by the intersection of the two curves. Note thepresence of two Nash equilibria.In order to compute the the equilibria, we solve the system (9) for d i = d . Here, by substitution we obtain − αs + 2 αx + γζ n r s − r d s − d = 0 , (14)from which αs − α ( d + x ) s + 2 αxd − γnζ ( r s − r d ) = 0 , (15)for s = d . Accordingly, the two Nash equilibria are: P ∗ = ( s ∗ , c ∗ , . . . , c ∗ n ) = (cid:18) a, ζ r s − r d a − d + a, . . . , ζ r s − r d a − d + a (cid:19) P † = ( s † , c † , . . . , c † n ) = (cid:18) b, ζ r s − r d b − d + b, . . . , ζ r s − r d b − d + b (cid:19) , (16)where a = d + x r ( d − x ) + 2 γnαζ ( r s − r d ) b = d + x − r ( d − x ) + 2 γnαζ ( r s − r d ) (17)are the roots of quadratic equation (15).We can now examine the dependence of the Nash equilibria on the model parameters, recalling that γ measuresthe importance of the advisor’s influence on customers, α measures the importance of truthfulness, β measures theimportance of remuneration for the advisor’s choice, and ζ measures the importance of belief in the advisor’s statedopinion (i.e., the cognitive dissonance). The curves are shown in Figs. 2-4, where the parameters are held fixedexcepting that of interest. The curves shows how the two equilibria move, with an arrow indicating the direction ofgrowth of the parameter of interest. A triangular region is shown as bounded by the two straight lines: equilibriafalling outside that region are not acceptable since they violate the constraint on customers’ opinion ( d ≤ c ≤ s ). Impact of cognitive dissonance
In response to changes in ζ , Nash equilibria are placed along the dash-dotted line in Fig. 2. When ζ increases, both6 PREPRINT - S
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25 0 . . . . . acceptable regionAdvisor’s stated opinion C u s t o m e r’ s op i n i on P ∗ P † Figure 2: Nash equilibria’ dependence on increasing values of ζ . Fixed parameters: r s − r d = − . , d = 0 . , α = 0 . , x = 0 . , γ = 0 . , n = 1 . For ζ = 5 we obtain no real Nash equilibria. . .
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35 0 . − . − − . . acceptable regionAdvisor’s stated opinion C u s t o m e r’ s op i n i on P ∗ P † Figure 3: Nash equilibria’ dependence on increasing values of α . Fixed parameters: r s − r d = − . , d = 0 . , ζ = 10 , x = 0 . , γ = 0 . , n = 1 . 7 PREPRINT - S
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35 0 . − . . . acceptable regionAdvisor’s stated opinion C u s t o m e r’ s op i n i on P ∗ P † Figure 4: Nash equilibria’ dependence on decreasing values of γ . Fixed parameters: r s − r d = − . , d = 0 . , ζ = 10 , x = 0 . , α = 0 . , n = 1 . For γ = 0 . we obtain no real Nash equilibria.equilibria tend to pull away and to accumulate in different areas. This is because, if we fix n = 1 and assume e.g. d < x , as ζ tends to ∞ the equilibria P ∗ and P † in (16) become lim ζ → + ∞ (cid:18) a, ζ r s − r d a − d + a (cid:19) = ( x, x )lim ζ → + ∞ (cid:18) b, ζ r s − r d b − d + b (cid:19) = (cid:18) d, d − α ( x − d ) γ (cid:19) . (18)As ζ tends to ∞ , the second of (18) represents an unreachable limit because there exists a number ¯ ζ > such that, foreach ζ > ¯ ζ , the support of the curve (cid:18) b, ζ r s − r d b − d + b (cid:19) where b = d + x − r ( d − x ) + 2 γnαζ ( r s − r d ) (19)goes out the triangular acceptance region (see Fig. 2, the squared branch). If we assume that, for ζ = ¯ ζ , the curve (cid:16) b, ζ r s − r d b − d + b (cid:17) falls right onto the horizontal side of the triangle, we define (cid:18) b, ζ r s − r d b − d + b (cid:19) where b = d + x − s ( d − x ) + 2 γnα ¯ ζ ( r s − r d ) (20)the last useful equilibrium . The critical value ¯ ζ can be found by imposing ζ r s − r d b − d + b = d where b = d + x − s ( d − x ) + 2 γnα ¯ ζ ( r s − r d ) (21)because c = d represents the horizontal side of the triangular acceptance region (let us remember that d ≤ c ≤ s ).Then, from (21) it is easily to prove that ¯ ζ = − (cid:18) α + γα (cid:19) r s − r d ( x − d ) . (22)The value in (22) is positive if r s < r d . Calculated in (22), the last useful equilibrium is (cid:18) d + x − | ( d − x )( α − γ ) | α + γ , d (cid:19) . (23)8 PREPRINT - S
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Remark . For the sake of simplicity, Eq. (18) have been obtained for the particular case d < x . Anyway, it ispossible to prove that lim ζ → + ∞ (cid:18) a, ζ r s − r d a − d + a (cid:19) = (cid:18) d + x | d − x | ,d + x | d − x | − α x − d − | x − d | γ (cid:19) lim ζ → + ∞ (cid:18) b, ζ r s − r d b − d + b (cid:19) = (cid:18) d + x − | d − x | ,d + x − | d − x | − α x − d + | x − d | γ (cid:19) . (24) Impact of advisor’s truthfulness
A similar result also applies to the case α → + ∞ . Actually, if we assume d < x , as α tends to ∞ the equilibria P ∗ in(16) is lim α → + ∞ (cid:18) a, ζ r s − r d a − d + a (cid:19) = ( x, x ) (25)while P † is excluded because this time lim α → + ∞ ζ r s − r d b − d + b = ∞ . (26)This case corresponds to an advisor that is very sensitive to the difference between his/her stated and true opinion, i.e.to the difference between what he/she says and what he/she really thinks. See Fig. 3. Impact of trust in the advisor
Fig. 4 shows that, as γ approaches to zero, only one equilibria survives. Actually, for n = 1 and d < x , as γ → theequilibria P ∗ in (16) is lim γ → (cid:18) a, ζ r s − r d a − d + a (cid:19) = (cid:18) x, r s − r d ζ ( x − d ) + x (cid:19) (27)while P † is excluded because lim γ → ζ r s − r d b − d + b = ∞ . (28)Since γ tells us how wide is the advisor’s desire to influence the customers, when γ = 0 (and all the other parametersare fixed and = 0 ) his/her equilibrium is represented by his/her internal opinion. (There is no desire to influence thecustomers, then there is no reason to tell a lie.) Observe also that r s − r d ζ ( x − d ) + x < x if r s < r d ; in this situation the Nashequilibrium is acceptable and the equilibrium solution for customer is different from the advisor’s internal opinion. Remark . In the above bullet list we have considered α , ζ → + ∞ and γ → . The same considerations remaintrue, in approximation, if we substitute α , ζ → + ∞ and γ → with finite values such that γnαζ r s − r d ( d − x ) ≪ . E.g. in(17): s γnαζ r s − r d ( d − x ) ≈ γnαζ r s − r d ( d − x ) + . . . (29)whenever γnαζ | r s − r d | ( d − x ) < .Let us conclude this section by considering the dependence of Nash equilibria on the increasing measure of importanceof the influence on customers, γ , which is depicted in Fig. 5. Observe that, if γ gets too big, both equilibria becomenot real. In this Section we derive the parameter region in which Nash equilibria, obtained in Section 4.1, stay coherent withopinion variable definition. For example, since c , . . . , c n , s shall fall within the range [0 , , the coordinates of Nashequilibria are constrained between and , and hence we obtain the conditions on parameters to ensure that. We willfocus on the special case , i.e. r d i = r d and d i = d ∀ i = 1 , . . . , n .In our model we distinguish between strategic variables and parameters. The first of these are c i , i = 1 , , . . . , n , and s , while the latter are d , x , w , n , α , β , γ , ζ , r d , r s . The set of all numeric values that they can assume are called,respectively, the domain D and the admissible parameter region R .9 PREPRINT - S
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65 0 . . . . . acceptable region s c P ∗ P † Figure 5: Nash equilibria’ dependence on γ , like in Fig. 4 but for increasing values of the parameter. Fixed parameters: r s − r d = − . , d = 0 . , ζ = 100 , x = 0 . , α = 8 , n = 1 .From the assumptions on opinion variables and parameters, we have that D := (cid:8) ( c , . . . , c n , s ) (cid:12)(cid:12) d ≤ c i ≤ s ∀ i = 1 , . . . , n, s ∈ [ d, (cid:9) , (30) R := (cid:8) ( d, x, w, n, β, γ, ζ, α, r s , r d ) (cid:12)(cid:12) d, x, w ∈ [0 , ,n ∈ N , β, γ, ζ, α > , r d , r s ∈ [0 , } . (31)Let us denote with Y n = Y × Y × · · · × Y | {z } n = { ( y , . . . , y n ) | y i ∈ Y for every i ∈ { , . . . , n }} (32)the n -ary Cartesian power of a set Y . Hence, the domain can be rewritten as D = [ d, s ] n × [ d, , (33)Let us derive conditions on the parameters (in other words, subsets of R ) which ensure the existence of P ∗ and P † , bydistinguishing between two cases, (A) r s = r d and (B) r s = r d . Case (A) , r s = r d . In view of this, Nash equilibria becomes: P ∗ = ( s ∗ , c ∗ , . . . , c ∗ n ) = (cid:18) d + x | d − x | , . . . , d + x | d − x | (cid:19) P † = ( s † , c † , . . . , c † n ) = (cid:18) d + x − | d − x | , . . . , d + x − | d − x | (cid:19) . (34)We have the following result: Proposition 4.3.
Let r s = r d . The Personal Finance Game admits the following Nash equilibria: P ∗ = ( x, . . . , x ) , P † = ( d, . . . , d ) for d < xP ∗ = P † for d = xP ∗ = ( d, . . . , d ) for d > x (35) Proof.
By virtue of constraints (30), the admissible parameter regions in which P ∗ and P † are acceptable Nashequilibria are described by: R ∗ = n ≤ d ≤ d + x + | d − x | ≤ o , R † = n ≤ d ≤ d + x − | d − x | ≤ o (36)andwhere we denoted by R ∗ , R † ⊆ R these regions ( R ∗ for P ∗ and R † for P † ). (cid:3) Case (B) , r s = r d . Let us denoted by R ∗ , R † ⊆ R the admissible parameter regions in which, respectively, P ∗ and P † are acceptable Nash equilibria. Then 10 PREPRINT - S
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Proposition 4.4.
Let r s = r d , then R ∗ =[0 , x ) × [0 , × N × (0 , + ∞ ) ×× h(cid:16) (0 , γn ) × [ r d − r (1) d , r d ) × [0 , (cid:17) ∪ (cid:16) [ γn, + ∞ ) × [ r d − r (2) d , r d ) (cid:17) × [0 , i (37) and R † = [0 , x ) × [0 , × N × (0 , + ∞ ) × (0 , γn ) × [ r d − r (1) d , r d − r (2) d ] × [0 , (38) where with r (1) d , r (2) d we denoted, respectively, αζ γn ( x − d ) and ζα (cid:16) x − dα + γn (cid:17) .Proof. The completed proof is given in Appendix. (cid:3)
Remark . The intersection R † ∩ R ∗ is not empty. Then, for parameters values in R † ∩ R ∗ , the two Nash equilibria P ∗ and P † are both acceptable (see e.g. Fig. 1). Remark . Focusing on r s range in the equation of admissible parameter region R ∗ ∪ R † , the equations (37) and(38), we notice that r s < r d . The customer i , that in our paper is a small investor, could be naive about incentivesand expects a return bigger than the one advisor A proposes instead to her. However, A, who can choose to expressan opinion which need not coincide with his/her true opinion, is supposed to be unbiased : they do not manipulate theexpectations of naive investors, which can be translated in the condition r s = r d . See, e.g., [3] for the role of advisorsand their communication process with investors in generating divergence of opinion, [29] for evidence on investorreaction to recommendations and [30] for evidence on analyst incentives. D sc s c c Figure 6: Graphical representation of ∂ D for two ( n = 1 , on the left) and three ( n = 2 , on the right) dimensionalspaces. The boundaries are highlighted in different colors.The following result characterizes mathematically the boundary of the set D described in (30) and denoted by ∂ D . Agraphical representation of ∂ D for two and three dimensional spaces is depicted in Fig. 6, where the boundaries arehighlighted in different colors. Mathematically, since D := (cid:8) ( c , s ) (cid:12)(cid:12) d ≤ c ≤ s ≤ (cid:9) , n = 1 (39)and D := (cid:8) ( c , c , s ) (cid:12)(cid:12) d ≤ c ≤ s ≤ , d ≤ c ≤ s ≤ (cid:9) , n = 2 (40)they are described respectively by ∂ D := { c = d , s ∈ [ d, } ∪ { s = 1 , c ∈ [ d, } ∪ { s = c , c ∈ [ d, } (41)11 PREPRINT - S
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17, 2019for n = 1 , and ∂ D := { c = d , s ∈ [ d, , d ≤ c ≤ s } ∪ { d ≤ c ≤ s , c = d , s ∈ [ d, } (42) ∪ { s = 1 , c , c ∈ [ d, } ∪ { s = c , c ∈ [ d, , d ≤ c ≤ c } (43) ∪ { s = c , c ∈ [ d, , d ≤ c ≤ c } (44)for n = 2 . Another example (four dimensional space) has equation D = (cid:8) ( c , c , c , s ) (cid:12)(cid:12) d ≤ c ≤ s ≤ , d ≤ c ≤ s ≤ , d ≤ c ≤ s ≤ (cid:9) , (45)for n = 3 and then ∂ D := { c = d , s ∈ [ d, , d ≤ c ≤ s , d ≤ c ≤ s } ∪ (46) ∪ { c = d , s ∈ [ d, , d ≤ c ≤ s , d ≤ c ≤ s } ∪ (47) ∪ { c = d , s ∈ [ d, , d ≤ c ≤ s , d ≤ c ≤ s } ∪ { s = 1 , c , c , c ∈ [ d, } ∪ (48) ∪ { s = c , c ∈ [ d, , d ≤ c ≤ c , d ≤ c ≤ c } (49) ∪ { s = c , c ∈ [ d, , d ≤ c ≤ c , d ≤ c ≤ c } (50) ∪ { s = c , c ∈ [ d, , d ≤ c ≤ c , d ≤ c ≤ c } . (51) Proposition 4.7.
The boundary of domain D , described by (30), is ∂ D = n ( c , . . . , c n , s ) ∈ [ d, n +1 :max i =1 ,...,n c i ≤ s ∧ " n Y i =1 ( c i − d ) ( s − c i ) ( s −
1) = 0 o . (52) Proof.
Let us consider the set (30), where d ≤ c i ≤ s ≤ ∀ i = 1 , . . . , n . Fix j ∈ { , , . . . , n } and assume max i =1 ,...,n c i = c j .Fix, for example, c = d then we still have d ≤ c i ≤ s ≤ , i.e. d ≤ c i ≤ s ∧ d ≤ s ≤ , ∀ i = 2 , . . . , n . Relation d ≤ s ≤ and d ≤ c i ≤ are verified by definition while the truthfulness of c i ≤ s is ensured by c j ≤ s , because c i ≤ c j ∀ i = 1 , . . . , n by definition. The same can be concluded for every c i = d .Finally, we note that if s = 1 then max i =1 ,...,n c i ≤ which corresponds to require ( c , . . . , c n ) ∈ [ d, n . (cid:3) In Section 4,we have seen that our game may have at most two Nash equilibria. Those equilibria represent the outcomeof the strategic interaction of the players, i.e. the advisor and the customers (the individual investors), to maximizetheir own utilities. However, their decisions may differ from what could be achieved if the overall maximum utilitywould be sought. Therefore, the utility achieved under a Nash equilibrium could be globally not efficient. In the caseswith a single Nash equilibrium, this loss of efficiency can be computed through the Price of Anarchy. In the case ofmore Nash equilibria, like ours, that concept has been generalized into the Price of Stability (PoS) [31]. In this section,we compute the Price of Stability for our game.For the price of stability we adopt the definition of [31]:PoS = value of best equilibriumvalue of optimal solution . (53)12 PREPRINT - S
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17, 2019Let us denote u A ( c , s, w, x ) := u A ( c , s ) and u CL i ( c i , d i , s ) := u CL i ( c i ) . We now calculate the utility functionsoutcomes in Nash equilibria P ∗ and P † . Then: u A ( P ∗ ) = − α ( a − x ) − βn (cid:18) w − ζ r s − r d a − d − a (cid:19) − γn ζ (cid:18) r s − r d a − d (cid:19) u CL i ( P ∗ ) = r s + 14 ζ (cid:18) r s − r d a − d (cid:19) (54)and u A ( P † ) = − α ( b − x ) − βn (cid:18) w − ζ r s − r d b − d − b (cid:19) − γn ζ (cid:18) r s − r d b − d (cid:19) u CL i ( P † ) = r s + 14 ζ (cid:18) r s − r d b − d (cid:19) . (55)The social welfare, i.e. the total utility of the agents, is: SW ( c , s ) = u A ( c , s ) + n X i =1 u CL i ( c i , s )= − α ( s − x ) − β n X i =1 ( w − c i ) − ( γ + ζ ) n X i =1 ( s − c i ) + r d n + r s − r d s − d " − dn + n X i =1 c i . (56)Because of the mixed terms in c i and s , the optimal solution of i -th customer depends on the choices done by financialadvisor.Whether the maxima of SW belong to D or ∂ D , is a question that is addressed and fully solved by Proposition 5.1below.In the following, for any complex number z = x + iy where x and y are real numbers, the absolute value or modulusof z is denoted | z | and is defined by | z | = p x + y .The following preliminary result concerns the roots of a quartic equation. A general method for solving quarticequations is found in Cardano’s Ars Magna, but it is attributed to Cardano’s assistant Ludovico Ferrari (1522-1565)[32]. Proposition 5.1.
Let us consider ω z + ω z + ω z + ω = 0 , (57) where ω , ω > , ω , ω ∈ R ( ω , ω both negative or both positive). Let also ∆ = 256 ω ω − ω ω ω ω − ω ω − ω ω ω ω − ω ω − ω ω D = 64 ω ω − ω ω ω − ω , P = − ω , R = ω + 8 ω ω . (58) The following are proved:(i) All the roots of (57) are not-real if and only if ∆ > and D > .(ii) There exists at least one root of (57) which has positive real part.(iii) Let Ω = n ω , ω ,ω , ω : ω − | ω | − | ω | + ω > , ω − | ω | − | ω | < , (∆ ≤ or D ≤ o (59) be a subset of the admissible parameter region R . Then, ∀ ω i ∈ Ω all the roots of (57) have modulus > .Proof. The completed proof is given in Appendix. (cid:3)
We now turn our attention to finding the maximum of the function ( c , s ) → SW ( c , s ) in the set D described in (30).13 PREPRINT - S
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Theorem 5.2.
Let ∆ , D, P, R and Ω like in Proposition 5.1. Let also SW be the social welfare function as in (56) andlet ω = n ( r d − r s ) ω = 2 βn ( r d − r s )( d − w ) ω = 0 ω = 4 [ βn ( d − w )( γ + ζ ) + α ( d − x )( β + γ + ζ )] ω = 4 [ βn ( γ + ζ ) + α ( β + γ + ζ )] . (60) Then the following claims hold.i) If ∆ > and D > , or ω i ∈ Ω ∀ i = 0 , . . . , , the function SW attains its maximum in a point belonging to ∂ D .ii) Let y = s − d ∈ [0 , . In all the other cases in which SW results concave, the function attains its maximumin a point ( c , . . . , c n , s ) ∈ D such that ω y + ω y + ω y + ω y + ω = 0 (61) and c = · · · = c n = 2 βw + 2( γ + ζ ) s + r s − r d s − d β + 2( γ + ζ ) . (62) Proof.
Let us first consider the maximum points of SW that are internal to D , namely in int ( D ) := (cid:8) ( c , . . . , c n , s ) (cid:12)(cid:12) d < c i < s ∀ i = 1 , . . . , n, s ∈ [ d, (cid:9) . (63)Being SW of class C ∞ , the maximum points in int ( D ) can be found amongst the stationary points, in other wordsamongst the points ( c , . . . , c n , s ) ∈ int ( D ) such that ∇ SW ( c , . . . , c n , s ) = (0 , . . . , . We have that ∂SW ( c ,...,c n ,s ) ∂c = 0 . . . ∂SW ( c ,...,c n ,s ) ∂c n = 0 ∂SW ( c ,...,c n ,s ) ∂s = 0 , (64)thus c = · · · = c n = 2 βw + 2( γ + ζ ) s + r s − r d s − d β + 2( γ + ζ ) (65)and − α ( s − x ) − γ + ζ ) n s − βw + 2( γ + ζ ) s + r s − r d s − d β + 2( γ + ζ ) ! + − r s − r d ( s − d ) " β ( w − d ) + 2( γ + ζ )( s − d ) + r s − r d s − d β + 2( γ + ζ ) n = 0 (66)Rearrange the terms of the latter equation: α ( s − x ) (2 β + 2( γ + ζ ))( s − d ) ++ 2( γ + ζ ) n (cid:2) β ( s − w )( s − d ) − ( r s − r d )( s − d ) (cid:3) ++ ( r s − r d ) (cid:2) β ( w − d )( s − d ) + 2( γ + ζ )( s − d ) + ( r s − r d ) (cid:3) n = 0 (67)which, substituting the new variable y = s − d ∈ [0 , , yields the polynomial equation (61), where ω , ..., ω aredescribed in (60). This proves claim ii).However, as we can see, ω and ω are ≥ . Accordingly, by assumption of claim i) and from Proposition 5.1 thesocial welfare function does not assume (admissible) maxima in D and so we have to focus on ∂ D . And this provesclaim i). (cid:3) PREPRINT - S
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17, 2019According to Theorem 5.2, let us denote the maximum values assumed by function SW with SW M and assume that itis global. Then, from the definition of PoS described by Eq. (53), we havePoS = max (cid:8) u A ( P ∗ ) + P ni =1 u CL i ( P ∗ ) , u A ( P † ) + P ni =1 u CL i ( P † ) (cid:9) SW M (68)where the utility functions estimated in P ∗ and P † are shown in (54) and (55). One can look at Eq. (68) as how muchthe central authority can earn if he/she can intervene in the game, helping the players (advisor, customers) converge toa good Nash equilibrium. Investors usually resort to financial advisors (paid by a bank) to improve their investment process until the point ofcomplete delegation on investment decisions. Surely, financial advice is potentially a correcting factor in investmentdecisions but, in the past, the media and regulators blamed biased advisors for manipulating the expectations of naiveinvestors. Then we wondered whether that was indeed the case and we built an ABM for the communication processbetween bank, advisors and investors.We defined a compromise for the financial advisor (between a sufficient reward by bank and to keep his/her reputation),and a compromise for the customers (between the desired return and the proposed return by advisor). In this way, thenotion of PoS — which we also analytically formulated — naturally arisen in our model.Moreover, we obtained two Nash equilibria and the best response functions of the resulting game. Anyway, one ofthese equilibria is not always acceptable: • The presence of a very truthful advisor translates into the presence of only one Nash equilibria (represented byhis/her internal opinion). This case corresponds to an advisor that is very sensitive to the difference betweenhis/her stated and true opinion, i.e. to the difference between what he/she says and what he/she really thinks. • If the advisor’s desire to influence the customers is irrelevant, only one equilibria survives and his/her equi-librium is represented by his/her internal opinion. • The same equilibria associated to advisor’s internal opinion survives when the sensitivity of customer tocognitive dissonance becomes strong. Cognitive dissonance provides customers with an incentive to modifytheir behavior to reduce the “cost” of the lack of agreement between advisor and customers. Then this casecorresponds to customers that are very sensitive to the difference between his/her opinion and advisor’s statedopinion.Then, by describing the parameter regions in which both equilibria result acceptable, we shown that they exist whethercustomers expect a return bigger than what advisor proposes instead to them: we can say that greediness/naivety ofthe customers emerge naturally from the model.The results of the paper concern the special case of homogeneous investors. It would be interesting to extend theresults of the paper to the more general case of non-homogeneous investors . Moreover, although a number of analyticresults obtained here for this special case, a closed-form expression of PoS seems to be unmanageable. We feel that awidespread use of simulation tools may help the understanding these open questions in future works.
A Appendix
A.1 Proof of Proposition 5.1
According to the theory of quartic equations [33], all the roots of (57) are non-real only in the following cases • ∆ > and D > • ∆ > and P > • ∆ = 0 and D = 0 and P > and R = 0 .Since P = − ω < , we see that all the roots of (57) are non-real if and only if ∆ > and D > .15 PREPRINT - S
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17, 2019For the proof of (ii) let f ( z ) be LHS of (57). Then, f ′ ( z ) = 4 ω z + 3 ω z + ω . We have that f ′ ( z ) = 0 has onlyone real root because the discriminant of the cubic equation f ′ ( z ) = 0 , ∆ (see [33]), is negative ∆ = − ω ω − ω ω < . (69)It follows that the number of the real roots of the quartic equation (57) is at most two.Our proof proceeds by reductio ad absurdum. Let us assume that all the roots of (57) have negative real part. The rootsmay be a , b , c + di , c − di where a , b , c , d ∈ R + with a < , b < , c < , d > or a + bi , a − bi , c + di , c − di where a , b , c , d ∈ R + with a < , b > , c < , d > .Since the coefficient of z is , we get, by Vieta’s formulas (see “Newton’s Identities” in [34]), ω = ab + 2 ac + 2 bc + c + d (70)for the first case and ω = a + b + 4 ac + c + d (71)for the second. In both cases, the LHS equals while the RHS is positive. This is impossible.For the proof of (iii) we already know, from (ii), that f ′ ( z ) = 0 has only one real root. Besides, we have f ′′ ( z ) = 0 ⇐⇒ z = − ω ω and z = 0 .Now, depending on the sign of ω and ω we have two different cases. Case 1 . Let ω > and ω > . Since f ′ (0) = ω > , we see that f ′ ( z ) = 0 has only one real root z = α where α < . It is necessary that f ( −
1) = ω − ω − ω + ω > and that f ( z ) = 0 has at least one real root, i.e. ∆ ≤ or D ≤ from (i). Since we have − ω ω < and f ′ (0) = ω > considering graphs in Fig. 7, we see that it is necessarythat f ′ ( −
1) = − ω + 3 ω + ω > . − − − α zy − − − xy Figure 7: On the left: a graph of f ′ ( z ) . On the right: a graph of f ( z ) . As we can see, f ′ ( α ) = 0 , f ′ ( − > , f ′ (0) > and f ( − > .On the other hand, if f ( − > , f ′ ( − > and (∆ ≤ or D ≤ then, we see that all the real roots of (57) havemodulus greater than 1. Case 2 . Let ω < and ω < . Since f ′ (0) = ω < , we see that f ′ ( z ) = 0 has only one real root z = β where β > . It is necessary that f (1) = ω + ω + ω + ω > and that f ( z ) = 0 has at least one real root, i.e. ∆ ≤ or D ≤ from (i). Since we have < − ω ω and f ′ (0) = ω < considering graphs in Fig. 8, we see that it is necessarythat f ′ (1) = 4 ω + 3 ω + ω < .On the other hand, if f (1) > , f ′ (1) < and ( ∆ ≤ or D ≤ ) then, we see that all the real roots of (57) havemodulus greater than .From the two cases, all the real roots of (57) have modulus > whenever ω , . . . , ω belong to the subset Ω . (cid:3) A.2 Proof of Proposition 4.4
Let’s start to prove (37) and focus on (30). Then the admissible parameter region in which P ∗ is an acceptable Nashequilibrium derives from the following inequalities: ≤ d ≤ ζ r s − r d a − d + a ≤ a ≤ (72)16 PREPRINT - S
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17, 2019 − − − − β zy xy Figure 8: On the left: a graph of f ′ ( z ) . On the right: a graph of f ( z ) . As we can see, f ′ ( β ) = 0 , f ′ (0) < , f ′ (1) < and f (1) > .from which a ≤ a ≥ d ≤ ad ≤ ζ r s − r d a − d + a ζ r s − r d a − d + a ≤ a ⇒ a ≤ a ≥ d ≤ ad − a ≤ ζ r s − r d a − d ζ r s − r d a − d ≤ (73)By third equation of (73) — d ≤ a — from a ≥ and, from a ’s definition in (17), the system (73) becomes d + x r ( d − x ) + 2 γnαζ ( r s − r d ) ≤ (74) d ≤ d + x r ( d − x ) + 2 γnαζ ( r s − r d ) (75) r d − ζ (cid:18) x − d r ( d − x ) + 2 γnαζ ( r s − r d ) (cid:19) ≤ r s < r d (76)Let us consider inequality (74): r ( d − x ) + 2 γnαζ ( r s − r d ) ≤ − d − x . (77)It has solution: ( d − x ) + γnαζ ( r s − r d ) ≥ d + x ≤ which is checked because d, x ∈ [0 , r s ≤ r d + αζγn (1 − d − x + dx ) (78)which can be simplified in r d − αζ γn ( x − d ) ≤ r s ≤ r d + 2 αζγn (1 − d )(1 − x ) . (79)Let us consider inequality (75), r ( d − x ) + 2 γnαζ ( r s − r d ) ≥ d − x , (80)whose solution is ( d − x ) + γnαζ ( r s − r d ) ≥ d − x ≥ d − x ) + γnαζ ( r s − r d ) ≥ ( d − x ) ∪ ( ( d − x ) + γnαζ ( r s − r d ) ≥ d − x < (81)i.e. ( d − x ) + γnαζ ( r s − r d ) ≥ d − x ≥ γnαζ ( r s − r d ) ≥ ∪ ( ( d − x ) + γnαζ ( r s − r d ) ≥ d − x < (82)17 PREPRINT - S
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17, 2019But the third inequality of the first system of (82) is at odds with (76) and, hence, the solution of (75) comes down to ( r s ≥ r d − αζ γn ( d − x ) d < x (83)Let us consider inequality (76), which can be rewritten as follows r d − ζ " d − x ) + 2 γnαζ ( r s − r d )++ 2( x − d ) r ( d − x ) + 2 γnαζ ( r s − r d ) ≤ r s (84)and, since d < x — by (83) —, it follows that r ( d − x ) + 2 γnαζ ( r s − r d ) ≥ − (cid:18) ζ + γnζα (cid:19) r s − r d x − d − ( x − d ) (85)from which ( d − x ) + γnαζ ( r s − r d ) ≥ − (cid:16) ζ + γnζα (cid:17) r s − r d x − d − ( x − d ) ≥ d − x ) + γnαζ ( r s − r d ) ≥ (cid:16)(cid:16) ζ + γnζα (cid:17) r s − r d x − d + ( x − d ) (cid:17) (86) ∪ ( ( d − x ) + γnαζ ( r s − r d ) ≥ − (cid:16) ζ + γnζα (cid:17) r s − r d x − d − ( x − d ) < (87)It should be noted that d = x . Actually, if we substituted d = x in Eq. (84), we would get r d − r s + γnα ( r d − r s ) ≤ which does not have solutions for r s < r d .We can rewrite system (86) in the following way r s ≥ r d − αζ γn ( x − d ) r s ≤ r d − ζαα + γn ( x − d ) ( d − x ) + γnαζ ( r s − r d ) ≥ (cid:16)(cid:16) ζ + γnζα (cid:17) r s − r d x − d + ( x − d ) (cid:17) (88)The solutions of the first two inequalities intersect if α > γn while for the last one we have γnαζ ( r s − r d ) ≥ (cid:18) ζ + γnζα (cid:19) (cid:18) r s − r d x − d (cid:19) + 2 (cid:18) ζ + γnζα (cid:19) ( r s − r d ) (89)which becomes ≥ (cid:18) ζ + γnζα (cid:19) (cid:18) r s − r d x − d (cid:19) + 2 ζ ( r s − r d ) , (90)whose solution is r d − ζα (cid:18) x − dα + γn (cid:19) ≤ r s < r d (91)Then system (86) has an empty solution if α < γn and becomes r d − αζ γn ( x − d ) ≤ r s ≤ r d − ζαα + γn ( x − d ) r d − ζα (cid:16) x − dα + γn (cid:17) ≤ r s < r d (92)for α > γn . The solution of (86) is empty for α < γn but, for α > γn it is r d − ζα (cid:18) x − dα + γn (cid:19) ≤ r s ≤ r d − ζαα + γn ( x − d ) (93)18 PREPRINT - S
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17, 2019since we have that r d − αζ γn ( x − d ) ≤ r d − ζα (cid:16) x − dα + γn (cid:17) for each parameters’ values. We now focus on system(87): ( r s ≥ r d − αζ γn ( x − d ) (cid:16) ζ + γnζα (cid:17) r s − r d x − d + ( x − d ) > ⇒ ( r s ≥ r d − αζ γn ( x − d ) r s > r d − ζαα + γn ( x − d ) (94)whose solution is (cid:26) r s ≥ r d − αζ γn ( x − d ) , α < γn (cid:27) ∪ (cid:26) r s > r d − ζαα + γn ( x − d ) , α > γn (cid:27) . (95)Accordingly, by merging (93) — for α > γn — and (95) we obtain the solution of (76): (cid:26) r s ≥ r d − αζ γn ( x − d ) , α < γn (cid:27) ∪ (cid:26) r s > r d − ζαα + γn ( x − d ) ∪ r d − ζα (cid:18) x − dα + γn (cid:19) ≤ r s ≤ r d − ζαα + γn ( x − d ) , α > γn ) . (96)Hence, the solution of (76) is (cid:26) r s ≥ r d − αζ γn ( x − d ) , α < γn (cid:27) ∪ ( r s ≥ r d − ζα (cid:18) x − dα + γn (cid:19) , α > γn ) (97)Let us observe that, by (76), the system (74)-(76) admits solution only if − ( x − d ) ≤ − d − x + dx ) , which isactually equivalent to ( x + d − ≥ . Moreover, r d ≥ r d − αζ γn ( x − d ) . By substituting (79), (83) and (97) in thesystem (74)-(76) we obtain r d − αζ γn ( x − d ) ≤ r s ≤ r d + αζγn (1 − d )(1 − x ) d < xα < γnr d − αζ γn ( x − d ) ≤ r s < r d (98)and r d − αζ γn ( x − d ) ≤ r s ≤ r d + αζγn (1 − d )(1 − x ) d < xα > γnr d − ζα (cid:16) x − dα + γn (cid:17) ≤ r s < r d (99)Since d, x ∈ [0 , we have that, for both systems, r d + αζγn (1 − d )(1 − x ) ≥ r d . Moreover, to (98) and (99) shouldbe added the conditions r d , r s ∈ [0 , . Hence we obtained thesis (37).We now come to prove (38), by focusing on (30). Then the admissible parameter region in which P † is an acceptableNash equilibrium derives from the following inequalities: ≤ d ≤ ζ r s − r d b − d + b ≤ b ≤ (100)from which b ≤ b ≥ d ≤ bd ≤ ζ r s − r d b − d + b ζ r s − r d b − d + b ≤ b ⇒ b ≤ b ≥ d ≤ bd − b ≤ ζ r s − r d b − d ζ r s − r d b − d ≤ (101)By third formula of system (101) — d ≤ b — we can rearrange its fourth and fifth equations as, respectively, − ( b − d ) ≤ ζ ( r s − r d ) and r s < r d . Moreover, b ≥ if d + x ≥ r ( d − x ) + 2 γnαζ ( r s − r d ) ⇒ d + x ≥ d − x ) + γnαζ ( r s − r d ) ≥ d + x ) ≥ ( d − x ) + γnαζ ( r s − r d ) (102)19 PREPRINT - S
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17, 2019but since d, x ≥ and r s < r d , the latter becomes r d − αζ γn ( x − d ) ≤ r s . Then the system (101) can be rewritten as b ≤ d ≤ b ζ ( r s − r d ) + ( b − d ) ≥ r d − αζ γn ( x − d ) ≤ r s < r d (103)or, from b ’s definition in (17) d + x − r ( d − x ) + 2 γnαζ ( r s − r d ) ≤ (104) d ≤ d + x − r ( d − x ) + 2 γnαζ ( r s − r d ) (105) ζ ( r s − r d ) + (cid:18) x − d − r ( d − x ) + 2 γnαζ ( r s − r d ) (cid:19) ≥ (106) r d − αζ γn ( x − d ) ≤ r s < r d (107)Let us consider inequality (104): r ( d − x ) + 2 γnαζ ( r s − r d ) ≥ d + x − . (108)We have ( d − x ) + γnαζ ( r s − r d ) ≥ d + x − ≥ d − x ) + γnαζ ( r s − r d ) ≥ ( d + x − ∪ ( ( d − x ) + γnαζ ( r s − r d ) ≥ d + x − ≤ (109)The first system does not admit solution because d, x ∈ [0 , while the solution of the second one comes down to ( d − x ) + 2 γnαζ ( r s − r d ) ≥ (110)which hence is the solution of inequality (104).Let us consider inequality (105), r ( d − x ) + 2 γnαζ ( r s − r d ) ≤ x − d , (111)whose solution is x − d ≥ d − x ) + γnαζ ( r s − r d ) ≥ d − x ) + γnαζ ( r s − r d ) ≤ ( x − d ) ⇒ ( x ≥ dr d − αζ γn ( x − d ) ≤ r s < r d (112)Let us consider inequality (106): ζ ( r s − r d ) + (cid:18) x − d − r ( d − x ) + 2 γnαζ ( r s − r d ) (cid:19) ≥ . (113)It can be rewritten as follows r s − r d ζ + ( x − d ) (cid:20) ( d − x ) + 2 γnαζ ( r s − r d ) (cid:21) + − x − d r ( d − x ) + 2 γnαζ ( r s − r d ) ≥ (114)and, since d < x — by (112) —, we obtain r ( d − x ) + 2 γnαζ ( r s − r d ) ≤ x − d + (cid:18) ζ + γnζα (cid:19) r s − r d x − d (115)20 PREPRINT - S
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17, 2019from which ( d − x ) + γnαζ ( r s − r d ) ≥ x − d + (cid:16) ζ + γnζα (cid:17) r s − r d x − d ≥ d − x ) + γnαζ ( r s − r d ) ≤ (cid:16)(cid:16) ζ + γnζα (cid:17) r s − r d x − d + ( x − d ) (cid:17) (116)It should be noted that d = x because, if we substituted d = x in Eq. (114), we would get ( r s − r d ) (cid:2) γnα (cid:3) ≥ which does not have solutions for r s < r d .We can rewrite system (116) in the following way r s ≥ r d − αζ γn ( x − d ) r s ≥ r d − ζαα + γn ( x − d ) ( d − x ) + γnαζ ( r s − r d ) ≤ (cid:16)(cid:16) ζ + γnζα (cid:17) r s − r d x − d + ( x − d ) (cid:17) (117)For the last inequality we have γnαζ ( r s − r d ) ≤ (cid:18) ζ + γnζα (cid:19) (cid:18) r s − r d x − d (cid:19) + 2 (cid:18) ζ + γnζα (cid:19) ( r s − r d ) (118)which becomes ≤ (cid:18) ζ + γnζα (cid:19) (cid:18) r s − r d x − d (cid:19) + 2 ζ ( r s − r d ) , (119)whose (acceptable) solution is r s ≤ r d − ζα (cid:18) x − dα + γn (cid:19) . (120)Hence we rewrite system (117) as r s ≥ r d − αζ γn ( x − d ) r s ≥ r d − ζαα + γn ( x − d ) r s ≤ r d − ζα (cid:16) x − dα + γn (cid:17) (121)Since r d − ζαα + γn ( x − d ) ≤ r d − αζ γn ( x − d ) ≤ r d − ζα (cid:18) x − dα + γn (cid:19) ≤ r d , (122)for α < γn , and r d − αζ γn ( x − d ) ≤ r d − ζα (cid:18) x − dα + γn (cid:19) ≤ r d − ζαα + γn ( x − d ) ≤ r d , (123)for α > γn , the solution of system (121) and then of the inequality (106), is empty if α > γn but it equals r d − αζ γn ( x − d ) ≤ r s ≤ r d − ζα (cid:18) x − dα + γn (cid:19) if α < γn (124)Accordingly, by substituting (110), (112) and (124) into the system (104)-(107) we obtain d < xα < γnr d − αζ γn ( x − d ) ≤ r s ≤ r d − ζα (cid:16) x − dα + γn (cid:17) r d − αζ γn ( x − d ) ≤ r s < r d (125)which turns into d < xα < γnr d − αζ γn ( x − d ) ≤ r s ≤ r d − ζα (cid:16) x − dα + γn (cid:17) (126)Hence we obtained thesis (38). (cid:3) PREPRINT - S
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