An algorithm for dynamical games with fractal-like

Contemporary Mathematics

An algorithm for dynamical games with fractal-like trajectories David Carfì and Angela Ricciardello

Abstract. In this paper, we propose an algorithm to represent the payoff trajectory of two-player discrete-time dynamical games. Specifically, we consider discrete dynamical games which can be modeled as sequences of normal-form games (the states of the dynamical game) with payoff functions of class C 1 . In this context, the payoff evolution of such type of dynamical games is the sequence of the payoff spaces of their game-states and the payoff trajectory of such games is the union of the members of the evolution. The formulation of the algorithm is motivated - especially in several applicative contexts such as Economics, Finance, Politics, Management Sciences, Medicine and so on ... by the need of a complete knowledge of the payoff evolution (problem which is still open in the most part of the cases), when the real problem requires a Complete Analysis of the interactions, beyond the study of just the Nash equilibria. We consider, to prove the efficiency and strength of our method, the development (by the algorithm itself) of some non-linear dynamical games taken from applications to Microeconomics and Finance. The dynamical games that we shall examine are already deeply studied and represented, at least at their initial state - by the application of the topological method presented by Carfì in [7] - in several applicative papers by Carfì, Musolino and Perrone (see [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]) by a long, quite indirect and step by step implementations of other standard computational softwares (such as AutoCad, Derive, Grapher, Graph and Maxima) or following a pure mathematical way (see for example [8]): on the contrary, our algorithm provides the direct and one shot graphical representation of the entire evolution of those games (by movies) and consequently of the entire trajectory. Moreover, the applicative games we consider in the paper (inspired and suggested by Economics and Finance) have a natural dynamics having fractal-like trajectories.

Keywords: Discrete dynamical games; payoff evolution; payoff trajectories; economic games; financial games; Pareto boundary; Complete study of a game; fractallike geometry. 1. Introduction 1.1. Brief history of the related past researches. In 2009, D. Carfì and A. Ricciardello (see [25]) presented a new computational procedure to determine the payoff spaces of non-parametric differentiable normal form games. Then, the authors applied a new procedure (see [1]) to numerically determine an original type c

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of 3-dimensional representation of the payoff space of a normal-form C 1 parametric game, with two players. Moreover, the method in [25] has been pointed out in [26] and assumed with the aim of realizing a numerical procedure providing the geometrical representation of the payoff scenarios of C 1 -families of normal-form games, with two players. 1.2. Aims of the paper. Our study pertains discrete families of normalform C 1 -games with 2 players, whose payoff functions are defined on intervals of the real Euclidean 2-space. This study includes also games whose payoff functions present a parameter varying in a discrete set. In [23, 24, 27, 28] David Carfì et al analyze also parametric games where the parameter set is interpreted as a coopetitive strategy space. Our analysis of discrete parametric games allows us, also, to pass from the payoff representation of standard normal-form games (see, for this classic games, [3, 2, 49, 50]) to some types of coopetitive extensions. 1.3. Structure of the paper. To ease the reader, in the first section of the paper we bring to mind terminology and some definitions, while in the second part, the method proposed in [7] and applied in the development of our algorithm, is presented. The application of our algorithm to several examples concludes the paper. 1.4. Motivation of the paper: the complete study of a game. Game theory has proved a powerful tool to suggest strategies that must be employed by rational individuals in competitive and cooperative environments. Nevertheless, in the great part of current applied literature about the subject, the methodologies used are essentially taken from the finite Game Theory and devoted to the study of Nash equilibria; this precludes several more deep applications, studies and developments. On the contrary, we want to concentrate our attention on infinite differentiable games, which are models more complex and much more adherent to the real human, economic and financial interactions: this is the final task of the Complete Analysis of a Differentiable Game. Its first goal is the precise knowledge of the Pareto boundaries (maximal and minimal) of the payoff space, this knowledge will allow us to evaluate the quality of the different Nash equilibria (by the distances from the Nash equilibria themselves to Pareto boundaries, with respect to appropriate metrics), in order to determine some focal equilibrium points (in the sense of Meyerson) collectively more satisfactory than each other. Moreover, the complete knowledge of the payoff-space will allow to develop explicitly the cooperative phase of the game and the various bargaining problems rising from the strategic interaction of the tourist firms (Nash bargaining problem, Kalai-Smorodinski bargaining problem and so on). The complete study of an infinite differentiable game f , introduced in [2] and [6] by D. Carfì , consists of the following points of investigation: 0. Structure analysis of the game 0.1) classify the game (linearity, symmetries, invertibility, ...); 0.2) find the critical zone of the game and its image by f ; 0.3) determine the biloss space im(f ); 0.4) determine inf and sup of the game f and see if they are shadow optima; 1. Pareto analysis of the game 1.1) determine the Pareto boundaries ∂ ∗ f and ∂∗ f of f ; 1.2) determine the inverse images by f of the Pareto boundaries;

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1.3) specify the control of each player upon the boundaries; 1.4) specify the noncooperative reachability of the Pareto boundaries; 1.5) find possible Pareto solutions and crosses; 2. Nash (Selfish) analysis 2.1) 2.2) 2.3) 2.4) 2.5)

find best reply correspondences and Nash equilibria; study the existence of Nash equilibria (Brouwer and Kakutany); find Nash equilibria, if any; evaluate non-cooperative reachability of Nash equilibria; evaluate the position of Nash equilibria with respect to ∂ ∗ f and ∂∗ f ;

3. Devotion analysis 3.1) find devotion correspondences and devotion equilibria; 3.2) specify the efficiency and noncooperative reachability of devotion equilibria; 3.3) confrontation of the devotion equilibrium with the Nash equilibrium; 4. Dominant analysis 4.1) find dominant strategies, if any; 4.2) find strict and dominant Nash equilibria; 4.3) reduce the game by elimination of dominated strategies; 5. Conservative analysis 5.1) 5.2) 5.3) 5.4) 5.5)

find conservative values and worst loss functions of the players; find conservative strategies and crosses; find all the conservative parts of the game (in bistrategy and biloss spaces); find core of the game and conservative knots; evaluate Nash equilibria by the core and the conservative bivalue;

6. Offensive analysis 6.1) 6.2) 6.3) 6.4) 6.5) 6.6)

find worst offensive correspondences and offensive equilibria; evaluate non-cooperative reachability of offensive equilibria; evaluate the position of offensive equilibria with respect to ∂ ∗ f and ∂∗ f ; find worst offensive strategies of any player against the other player; find possible dominant offensive strategies; confront Nash equilibria with offensive equilibria;

7. Cooperative analysis 7.1) 7.2) 7.3) 7.4) 7.5) 7.6)

find the best compromises (Kalai-Smorodinsky solutions) and their bilosses; find the elementary core best compromise and corresponding biloss; find the Nash bargaining solutions and corresponding bilosses; find the solutions with closest bilosses to the shadow minimum; find the maximum collective utility solutions; study the transferable utility case.

8. Solution analysis 8.1) confront the possible non-cooperative solutions among them; 8.2) confront the possible cooperative solutions among them; 8.3) confront noncooperative and cooperative solution. 1.5. Confrontation with other papers in Game Theory literature. For what concerns the confrontation with other papers in the Game Theory literature we observe that: • the dynamical games that we shall examine are already deeply studied and represented, at least at their initial state - by the application of the topological method presented by Carfì in [7] - in several applicative papers by Carfì, Musolino and Perrone (see [10], [11], [12], [13], [14], [15] [16], [20], [21], [22]) by

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a long, quite indirect and step by step implementations of other standard computational softwares (such as AutoCad, Derive, Grapher, Graph and Maxima) or following a pure mathematical way (see for example [8]): on the contrary, our algorithm provides the direct and one shot graphical representation of the entire evolution of those games (by movies) and consequently of the entire trajectory. • the standard literature on game theory does not present algorithms devoted to the graphical representation and computation of the payoff spaces, but essentially devoted to the determination of Nash equilibria, their stabilities and their approximations, see for example [4, 5, 6], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48] and [51, 52, 53, 54, 55, 56, 57, 58]. 2. Preliminaries and notations In order to help the reader and increase the level of readability of the paper, we recall some notations and definitions about n-player games in normal-form, presented yet in [7, 1]. Although the below definition seems, at a first sight, different from the standard one (presented, for example, in [49]), we desire to note that it is substantially the same; on the other hand, the definition in this new form underlines that a normal-form game is nothing but a vector-valued function and that any possible exam or solution of a normalform games attains, indeed, to this functional nature. After the new definition, we shall comment the equivalence of the two forms of the definition. Definition 1 (of game in normal-form). Let E = (Ei )n i=1 be an ordered family of non-empty sets. We call n-person game in normal-form, upon the support E, each function f : × E → Rn , where × E denotes the Cartesian product ×n i=1 Ei of the family E. The set Ei is called the strategy set of player i, for every index i of the family E, and the product × E is called the strategy profile space, or the n-strategy space, of the game. Remark. First of all we recall a standard form definition of normal-form game: Definition. A strategic game consists of a system (N, E, f ), where: 1. a finite set N (the set of players) of cardinality n is canonically identified with the set of the first n positive integers; 2. E is an ordered family of nonempty sets, E = (Ei )i∈N , where, for each player i in N, the nonempty set Ei is the set of actions available to player i; 3. f is an ordered family of real functions f = (fi )i∈N , where, for each player i in N , the function fi : × E → R is the utility function of player i (inducing a preference relation on the Cartesian product × E := ×j∈N Ej (the preference relation of player i on the whole strategy space).  Well, it is quite clear that the above system (N, E, f ) is nothing but a redundant form of the family f itself, which we prefer to consider in its vector-valued functional nature f : ×j∈N Ej → Rn : x 7→ (fi (x))i∈N . Terminology. Together with the previous definition of game in normal form, we have to introduce some terminologies: • the set {i}n i=1 of the first n positive integers is said the set of players of the game; • each element of the Cartesian product × E is said a strategy profile, or n-strategy, of the game; • the image of the function f , i.e., the set f (× E) of all real n-vectors of type f (x), with x in the strategy profile space × E, is called the n-payoff space, or simply the payoff space, of the game f . Moreover, we recall the definition of Pareto boundary whose main properties have been presented in [9]. By the way, the maximal boundary of a subset T of the Euclidean space Rn is the set of those s ∈ T which are not strictly less than any other element of T .

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Definition 2 (of Pareto boundary). The Pareto maximal boundary of a game f is the subset of the n-strategy space of those n-strategies x such that the corresponding payoff f (x) is maximal in the n-payoff space, with respect to the usual order of the euclidean n-space Rn . If S denotes the strategy space × E, we shall denote the maximal boundary of the n-payoff space by ∂f (S) and the maximal boundary of the game by ∂ f (S) or by ∂(f ) . In other terms, the maximal boundary ∂ f (S) of the game is the reciprocal image (by the function f ) of the maximal boundary of the payoff space f (S). We shall use analogous terminologies and notations for the minimal Pareto boundary. Remark (on the definition of Pareto boundary). Also in the case of this definition, essentially the definition of maximal (Pareto) boundary is the standard one, unless perhaps the name Pareto: it is nothing more that the set of maximal elements in the standard pre-order set sense, that is the set of all elements that are not strictly less than other elements of the set itself. The only circumstance to point out is that the natural pre-order of the strategy set × E is that induced by the standard point-wise order of the image f (S) by means of the function f , that is the reciprocal image of the point-wise order on f (S) via f .

3. The method for C 1 games In this paper, we deal with normal-form game f defined on the product of n compact and non-degenerate intervals of the real line, and such that f is the restriction to the n-strategy space of a C 1 function defined on an open set of Rn containing the n-strategy space S (which, in this case, is a compact infinite part of the n-space Rn ). Details are in [7, 25], but in the following we recall some basic notions. 3.1. Topological boundary. For easy of the not-specialized reader, we recall that the topological boundary of a subset S of a topological space (X, τ ) is the set of those points x of the space X such that every neighborhood of x contains at least one point of S and at least one point in the complement of S. Observe that the topological boundary of the support X of the topological space (X, τ ) is empty (in the topological space itself). The key theorem of our method is the following one, we invite the reader to pay much attention to the topologies used below. Theorem 1. Let f be a C 1 function defined upon an open set O of the euclidean space Rn and with values in Rn . Then, for every part S of the open set O, the topological boundary of the image of S by the function f , in the topological space f (O) (i.e. with respect to the relativization of the Euclidean topology to f (O)) is contained in the union f (∂O S) ∪ f (C), that is ∂f (O) f (S) ⊆ f (∂O S) ∪ f (C), where: (1) C is the critical set of the function f in S (that is the set of all points x of S such that the Jacobian matrix Jf (x) is not invertible); (2) ∂O S is the topological boundary of S in O (with respect to the relative topology of O). Note. Observe for example the following trivial case. Let O be the unit open ball B(02 , 1) of the plane and let f be the canonical set-immersion (injection) of O into the plane R2 (that is the function f : O → R2 : x 7→ x). If S := O, then f (S) = O; the boundary of f (S) in f (O) is empty (since f (O) = O), the boundary of S in O is empty too, and the theorem gives the trivial inclusion ∅ ⊆ ∅. Note. We note, however, that when S is a compact subset of the open set O (it doesn’t matter in what topology...), then the boundaries of S and f (S) in O and f (O) coincides with the boundaries of S and f (S) in Rn .

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4. Two players parametric games In this section we shall introduce the definitions of parametric games, as it is employed in the following. Definition 3. Let E = (Et )t∈T and F = (Ft )t∈T be two families of non empty sets and let f = (ft )t∈T be a family of functions, where ft : Et × Ft → R2 , for each t ∈ T . We define parametric gain game over the strategy pair (E, F ) and with family of payoff functions f the pair G = (f, >), where the symbol > stands for the usual strict upper order of the Euclidean plane R2 . We define the payoff space of the parametric game G as the union of all the payoff spaces of the game family g = ((ft , >))t∈T , that is, as the union of the payoff family P = (ft (Et × Ft ))t∈T . Dynamics. We will refer to the above family g as to the dynamical path of the game G, since we can see it as a curve of games: g : T → g(T ) : t 7→ (ft , >). 4.1. Payoff set-dynamics. We note also that the family P can be identified with the multi-valued path in R2 p : T → R2 : t 7→ ft (Et × Ft ), (multivalued means that to each value t ∈ T the mappings p associates a subset of the plane, and not one unique single point of it) and that the graph of this path p is a subset of the Cartesian product T × R2 , on the other hand, the trace of the curve p, is a subset of the plane and it is the union of all the values of the multi-valued path p. 5. The algorithm 5.1. The game framework of the algorithm. In particular we are concentrated on the following specific kind of parametric game: • parametric games in which the families E and F consist of only one set, respectively. In the latter case, we can identify a parametric game with a pair (f, >), where f is a function from a Cartesian product T × E × F into the plane R2 , where T , E and F are three non-empty sets. Definition 4. When the triple (T, E, F ) is a triple of subsets of normed spaces, we define the parametric game (f, >) of class C 1 if the function f is of class C 1 . 5.2. Structure of the algorithm. The algorithm for the representation of the payoff trajectory of dynamical game generalizes the procedure presented in [1, 25], for discrete dynamical games of the type (fn )n∈N . In particular, it has been extended to sequences (of payoff functions) recursively defined. We define, for all (x, y) in the strategy space S := E × F := [x1 , x3 ] × [y1 , y3 ] ⊂ R2 , ( f0 (x, y) = (f0 1 (x, y), f0 2 (x, y)) fn (x, y) = (fn1 (x, y), fn2 (x, y)), for all integers n ≥ 1. Note that if, for each n, the function fn is defined by means of the function fn−1 , then it has to be evaluated. Our aim is to represent the payoff family scenario, varying the parameter n ∈ N, performing the iteration for n varying from 1 to a fixed natural number N , with N ∈ N fixed a priori sufficiently great.

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To this order, all the points in the topological boundary T of the strategy space S and the critical zone Cn have to be transformed by using each payoff function fn . Thus, let be Tn0 = {fn (x1 , y)}y∈F ∪ {fn (x, y1 )}x∈E ∪ {fn (x3 , y)}y∈F ∪ {fn (x, y3 )}x∈E , be the transformation of the topological boundary. Moreover, let us denote Cn ={(x, y) ∈ S : ∂1 fn1 (x, y) · ∂2 fn2 (x, y) − ∂2 fn1 (x, y) · ∂1 fn2 (x, y) = 0} Cn0 ={(fn1 (x, y), fn2 (x, y)) : (x, y) ∈ Cn } the critical zone and its transformation, respectively. It has been proved that the topologicalSboundary of the payoff scenario of the whole dynamical game is contained in the 0 0 union N n=0 (Tn ∪ Cn ). Taking into account the introduced notation, our algorithm can be summarized in few steps as follows. INPUT: E = [x1 , x3 ], F = [y1 , y3 ], N (Maximum

number

of

Iteration)

f0 (x, y) and fn (x, y) ∀n ≥ 1(Payoff function) PROCESSING: FOR n=0 to N - Evaluation of function fn , for every n, if necessary. fn (x, y) = (fn1 (x, y), fn2 (x, y)) - Transformation of the topological boundary T by fn : representation (plot) of Tn0 , as defined above; - Evaluation of the critical zone Cn : representation (plot) of the inverse image (det ◦Jfn )← (0); - Transformation of the critical zone: plot of the image Cn0 := fn (Cn ), described above; - Payoff Space Pn of the game fn : Pn is the fill in of the union Tn0 ∪ Cn0 . END Payoff space of the dynamical game G = (fn ): let f : E × F × N → R2 defined by f (x, y, n) := fn (x, y), for every (x, y, n), we plot S of the image f (E × F × N) as the union N n=0 Pn . OUTPUT: plots of the payoff boundary scenario family scenario and of payoff scenario family f (E × F × N). 5.3. Principal aims of the algorithm. Our algorithm gives us: • the dynamical evolution of the payoff family P , in the sense of the dynamical evolution (in real time) when we consider the parameter set T as the real time straight-line (this by movies); • the trace of this dynamical path, i.e., the very payoff space of the parametric game G. 6. Examples In the following subsections we shall consider the following examples: 1. the parametric game G = (fa )a∈T , defined by fa (x, y) = ||(1, a)||−2 (x(1 − 2x + y), y(1 − 4y + x)) + φ(a)(1, 1), for all x, y ∈ [0, 1] and a ∈ [0, +∞[, where φ(a) := a(1 + a)−1 . 2. the parametric game G = (fa )a∈T , defined by fa (x, y) = ||(1, a)||−2 (x(1 − 2x + y), y(1 − 4y + x)) + ie−iφ(a) , for all x, y in [0, 1] and a ∈ [0, +∞], where φ(a) := a(1 + a)−1 .

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3. the parametric game G = (fa )a∈T , defined by fa (x, y) = 4(1 + a3 )−1 (1 − x − y)(x, y) + g(a), with g(a) = (2φ(a), −(1/6)a−2 φ(a)2 ((a − 3)2 + 9)), for all x, y in [0, 1] and a in [0, +∞], where φ(a) := a(1 + a)−1 . 4. the parametric game G = (fn )n∈N , defined by fn (x, y) = (1/3)n f0 (x, y) + (3/2)(1 − (1/3)n )w where f0 (x, y) = (−(1/2)(1 − x)y, xy) and w = (1/3, 2/3). 5. the parametric game G = (fn )n∈N , defined by fn+1 (x, y) = an+1 f0 (x, y) + an w where f0 (x, y) = (−(1/2)(1 − x)y, xy), with a−1 = 0, a0 = 1, an+1 = 1 + (1/3)an and w = (1/3, 2/3). 6.1. First game. Here, we present a parametric game of Bertrand type (already represented, at the initial state, in [21], by a long procedure, using Maxima), whose strategy sets are E = F = [0, 1], the parameter set is T = R≥ and the a-biloss (disutility) function is defined by fa (x, y) = ||(1, a)||−2 (x(1 − 2x + y), y(1 − 4y + x)) + φ(a)(1, 1), for all x, y in [0, 1] and a in [0, +∞[, where φ(a) := a(1 + a)−1 . The payoff scenario path of the discrete subfamily-game G0 := (fn )n∈N is depicted in Figure 1 (up to n = 10, for sake of simplicity). This payoff scenario path is the union of the payoff scenario evolution family (fn (E × F ))n∈N , which can be seen as a multi-valued discrete dynamical path γ : N → R2 : n 7→ fn (E × F ).

Figure 1. Game 1. Payoff trajectory of the subgame G0 .

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6.2. Second game. In this subsection, we present another Bertrand-type parametric game, where strategy sets are E = F = [0, 1], the parameter set is T = R≥ and the a-biloss (disutility) function is defined by fa (x, y) = ||(1, a)||−2 (x(1 − 2x + y), y(1 − 4y + x)) + ie−iφ(a) , for all x, y in [0, 1] and a in [0, +∞], where φ(a) := a(1 + a)−1 . The payoff scenario path of the discrete subfamily-game G0 := (fn )n∈N is depicted in Figure 2 (up to n = 10, for sake of simplicity).

Figure 2. Game 2. Payoff trajectory of the subgame G0 .

6.3. Third game. In this subsection, we present a Cournot-type parametric game, where strategy sets are E = F = [0, 1], the parameter set is T = [2, +∞] and the a-payoff function is defined by fa (x, y) = c(x, y, a) + (2a/(1 + a), −((1/6)(a − 3)2 + 3/2)/(1 + a)2 ), where c(x, y, a) = 4(1 − x − y)(1 + a3 )−1 (x, y), for all x, y ∈ [0, 1] and a ∈ T . We shall consider, this time, two subsequences of the above game. The payoff scenario path of the discrete subfamily-game G0 := (fn )n∈N is depicted in Figure 3 and 4 (up to n = 20, for sake of simplicity).

The payoff scenario path of the discrete subfamily-game G00 := (f2n )n∈N is depicted in Figure 5 and in Figure 6 (up to n = 10, for sake of simplicity).

6.4. Fourth game. We present a discrete parametric game (proposed in financial literature by D. Carfì and F. Musolino), with strategy sets E = [0, 1], F = [−1, 1] and the n-payoff function is defined by fn (x, y) = (1/3)n f0 (x, y) + (3/2)(1 − (1/3)n w)

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Figure 3. Game 3. Payoff boundary trajectory of sub-game G0 .

Figure 4. Game 3. Payoff trajectory of the sub-game G0 .

where f0 (x, y) = (−(1/2)y(1 − x), xy) and w = (1/3, 2/3). The payoff scenario path of the discrete family-game G := (fn )n∈N is depicted in Figure 7. Otherwise, setting E = [−1, 1] and F = [−1, 1], the resulting payoff trajectory changes, as shown in Figure 8

6.5. Fifth game. Let us consider another Carfì-Musolino financial game, that defined by G = (fn )n∈N with fn : E × F → R2 , with fn+1 (x, y) = an+1 f0 (x, y) + an w,

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Figure 5. Game 3. Payoff boundary trajectory of the sub-game G00 .

Figure 6. Game 3. Payoff trajectory of the sub-game G00 .

for every (x, y) in E × F , where: f0 (x, y) = (−(1/2)y(1 − x), xy); with a−1 = 0, a0 = 1, an+1 = 1 + (1/3)an and w = (1/3, 2/3). If E = [−1, 1] and F = [−1, 1], the payoff trajectory is illustrated in Figure 9. Figure 10 refers to case E = [0, 1] and F = [−1, 1], with fn : [0, 1] × [−1, 1] → R2 . Setting E = [0, 1], F = [−1, 1] and an+1 = 1 + 0.9an , we obtain the trajectory represented in Figure 11. At the end, if we assume E = [−1, 1] and F = [−1, 1], with an+1 = 1 + n + 0.33an , we get the payoff trajectory illustrated in Figure 12

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Figure 7. Game 4. Payoff trajectory of the fourth game.

Figure 8. Game 4. Payoff trajectory of the fourth game extended to E = [−1, 1] and F = [−1, 1].

AN ALGORITHM FOR DYNAMICAL GAMES WITH FRACTAL-LIKE TRAJECTORIES

Figure 9. Game 5. Payoff trajectory of the fifth game (E = F = [−1, 1]).

Figure 10. Game 5. Payoff trajectory of the fifth game restricted to E = [0, 1] and F = [−1, 1].

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Figure 11. Game 5. Payoff trajectory of the fifth game with E = [0, 1] F = [−1, 1] and an+1 = 1 + 0.9an .

Figure 12. Game 5. Payoff Space of the fifth game for E = [0, 1] F = [−1, 1] and an+1 = 1 + n + 0.3an ..

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7. Final Remarks The Bertrand and Cournot games proposed above (in the example 1,2 and 3) belong to the classic economic games presented in the literature by Carfì and Perrone in [20, 21, 22], the model proposed there is quite general and the specific examples we propose here are not particularly distinguished Bertrand/Cournot games, we fixed the constants only for sake of representability. Any other Bertrand or Cournot type game is of the same nature and the algorithm is straightforwardly good for any choice of the constants. By the way, we have chosen those particular games to compare the results of our algorithm to the studies performed by the software Maxima (already published in those Carfì-Perrone papers) with very long and intricate procedures. Note that Carfì and Perrone study only the initial state of the economic dynamical game, but from this initial graphical representation is possible to deduce all the sequence by contractions and translations: our algorithm here gives in one shot all the dynamics without further considerations and operations. The Financial games studied in the last games derive from a complex and wide tentative of Carfì and Musolino (see [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]) to give a robust stability to the financial markets under speculative attacks. In this cases Carfì and Musolino study (by softwares Graph and Grapher) all the evolution of the financial games, so that the confrontation with the representations of our paper is total, not only partial. 8. Resume In [25], a new procedure to determine the payoff scenarios of non-parametric differentiable games has been presented; then this new procedure has been applied in [1] to numerically determine a new 3-dimensional representation of the payoff spaces of continuous families of normal-form C 1 -games, with two players, families indexed by a compact interval of the real line. Moreover, the method in [7] has been pointed out in [26] with the aim of realizing a numerical procedure providing, finally, the real geometrical representation of the payoff scenarios of C 1 -families of normal-form C 1 -games, with two players, families indexed by a compact interval of the real line. In this present work, the method in [7] is applied to realize an algorithm for the representation of the payoff space trajectories of discrete families of normal-form C 1 games, that is a numerical procedure providing the real geometrical representation of the payoff scenarios of sequences of normal-form C 1 -games. 9. Conclusions In this paper, we have proposed a new algorithm able to represent in one shot the payoff trajectory of two-player discrete-time dynamical games. Specifically: • we consider discrete dynamical games which can be modeled as sequences of normal-form games (the states of the dynamical game) with payoff functions of class C 1 . • In this context, the payoff trajectory of such type of dynamical games is the sequence of the payoff spaces of their game-states. • The formulation of the algorithm is motivated - especially in several applicative contexts such as Economics, Finance, Politics, Management Sciences, Medicine and so on ... - by the need of a complete knowledge of the payoff evolution (problem which is still open in the most part of the cases), especially when the real problem requires a Complete Analysis beyond the study of just the Nash equilibria. • We consider, to prove the efficiency and strength of our method, the development (by the algorithm itself) of some non-linear parametric games taken from applications to Micro-Economics and Finance.

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• The dynamical games that we shall examine are already deeply studied and represented, at their initial state, by the application of the topological method presented by Carfì in [7], in several applicative papers by Carfì, Musolino and Perrone (see [10], [11], [12], [13], [14], [16], [20]) by a long, indirect and step by step implementations of other standard computational softwares (such as Autocad, Derive, Grapher, Graph and Maxima) or from a pure mathematical way (see for example [8]); • contrary to the classic softwares present in the literature, our algorithm provides the direct and one shot graphical representation of the entire evolution of those games. • Finally, the applicative games we consider in the paper (inspired by Economics and Finance) have a natural dynamics having fractal-like trajectories. Acknowledgement. The authors wish to thank an anonymous referee that helped very much to deeply improve the paper. References [1] S. Agreste, D. Carfì, A. Ricciardello, An algorithm for payoff space in C 1 parametric games, Applied Sciences (APPS), vol. 13 (2011). [2] J. P. Aubin, Mathematical Methods of Game and Economic Theory, North-Holland. [3] J. P. Aubin, Optima and Equilibria, Springer Verlag. [4] N. Bhat, K. Leyton-Brown, Computing Nash equilibria of action-graph games, Proceedings of the 20th conference on Uncertainty in artificial intelligence. AUAI Press, 2004. [5] T. Basar, et al, Dynamic noncooperative game theory, Vol. 200. California, CA: Academic press, 1995. [6] M. Bloem, T. Alpcan, T. Basar, A Stackelberg game for power control and channel allocation in cognitive radio networks, Proceedings of the 2nd international conference on Performance evaluation methodologies and tools. ICST (Institute for Computer Sciences, SocialInformatics and Telecommunications Engineering), 2007. [7] D. Carfì, Payoff space of C 1 Games, Applied Sciences 11, 35-47 (2009). [8] D. Carfì, Differentiable game complete analysis for tourism firm decisions, in Proceedings of the 2009 International Conference on Tourism and Workshop on Sustainable Tourism within High Risk Areas of Environmental Crisis, Messina, Italy, 2009. [9] D. Carfì, Optimal boundaries for decisions, AAPP, Vol. LXXXVI issue 1, 2008, pp. 1-12. [10] D. Carfì, F. Musolino (2011), Fair redistribution in financial markets: a game theory complete analysis, Journal of Advanced Studies in Finance (JASF) 2, 74. [11] D. Carfì, F. Musolino (2012), A coopetitive approach to financial markets stabilization and risk management, in Advances in Computational Intelligence, pp. 578-592 (Springer-Verlag, Berlin, 2012); book serie: Communications in Computer and Information Science. [12] D. Carfì, F. Musolino (2012), Game theory and speculation on government bonds, Economic Modelling (Elsevier), 29 (6), pp. 2417-2426. [13] D. Carfì, F. Musolino (2013), Dynamical Stabilization of Currency Market with Fractal-like Trajectories, pre-print. [14] D. Carfì, F. Musolino (2013), Game Theory Application of Monti’s Proposal for European Government Bonds Stabilization,Applied Sciences, vol.15. [15] D. Carfì, F. Musolino (2012), Credit crunch in the Euro area: a coopetitive multi-agent solution, in Multicriteria and Multiagent Decision Making with Applications to Economic and Social Sciences, vol. 310, book serie:Studies in Fuzziness and Soft Computing. ISBN: 978-3-642-35634-6. [16] D. Carfì, F. Musolino (2013), A Coopetitive-dynamical game model for currency markets stabilization, joint proceedings of Fractal Geometry in Pure and Applied Mathematics: in Memory of Benoit Mandelbrot (Boston, January 2012), Analysis, Fractal Geometry, Dynamical Systems, and Economics (Messina, November 2011), Geometry and Analysis on Fractal Spaces (Hawaii, March 2012), submitted. [17] D. Carfì, F. Musolino (2013), Model of Possible Cooperation in Financial Markets in Presence of Tax on Speculative Transactions, AAPP|Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., vol 91, issue 1 (2013).

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