Kalai-Smorodinsky Balances for N-Tuples of Interfering Elements

Abstract

The study proposed here builds up a game model (with associated algorithms) in a specific non-linear interfering scenario, with n possible interacting elements. Our examination provides optimal Kalai-Smorodinsky compromise solution n-tuples, for the game, whose components indicate active principle quantity percentages. We solve the problem by using the Carfì’s payoff analysis method for differentiable payoff functions. Moreover we implement Matlab algorithms for the construction and representation of the payoff spaces and for the finding of Kalai-Smorodinsky solutions. The software for the determination of graphs are adopted, but not presented here explicitly. The core section of the paper, completely studies the game in the n-dimensional case, by finding the critical zone of the game in its Cartesian form. At this aim, we need to prove a theorem and a lemma about the Jacobian determinant of the n-game. In the same section, we write down the intersection of the critical zone and the Kalai-Smorodinsky straight-line. In the Appendix 1 we solve the problem in closed form for the 2 dimensional case and numerically for $n > 2$. Our methods works also for games with non-convex payoff space. Finally, in a particular highly symmetrical case, we solve analytically the Kalai-Smorodinsky compromise problem in all cases. We provide some applications of the obtained results, particularly to economic problems.

Appendices

Appendix 1: The Algorithm

Here the Matlab command list follows.

1.1 Kalai-Smorodinsky Solution in 2 Dimension

which provides:

1.2 Numerical Sample in the Case $n=10$

which provides:

Appendix 2: Differentiable Games and Pareto Boundary

In this paper we use a general method introduced by one of the authors in [7] and referred in literature as Complete Study of a Differentiable Game. Indeed, in the great part of the current Game Theory literature, the study of a game in normal form consists essentially in the determination of the Nash equilibria (in mixed strategies) and in the analysis of their stability properties (see for instance [2, 3, 29, 30]). This approach cannot provide a complete and global view of the game and of all its possible feasible solutions. Indeed, to figure out how the game determines the interaction among the players, a deeper knowledge of at least the payoff space of the game appears not only useful but mandatory. For instance, it appears of the greatest interest to know the positions of the payoff profiles corresponding to the Nash equilibria in the whole of the payoff space of the game and the knowledge of these relative locations requires the knowledge of the entire payoff space. The need of better knowing the general shape of the payoff profile space becomes inevitable, when the problem to solve in the game reveals a bargaining one, at least at the level of the maximal boundary of the payoff space: in fact, the exact determination of bargaining solutions (compromise solutions) needs the analytical determination of the maximal Pareto boundary. In the paper [7], Carfì presented a general method to find an explicit expression of the topological boundary of the payoff space of the game. In that paper Carfì followed the way shown in [6], in order to construct the theoretical bases for Decisions in Economics and Finance by means of algebraic, topological and differentiable structures.

1.1 Preliminaries and Notations

Here we reconsider, for convenience of the reader, the study conducted in the paper [23]. 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 [1, 7]. Although the below definition seems, at a first sight, different from the standard one (presented, for example, in [29]), 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 normal-form games attains, indeed, to this functional nature.

Definition 1

(definition of a game in normal form). Let $E = (E_i)_{i=1}^n$ be an ordered family of non-empty sets. We call n-person game in normal-form, upon the support E, each function $f:\ ^{\times }E \rightarrow \mathbb {R}^n$, where $^{\times }E$ denotes the Cartesian product $\times _{i=1}^n E_i$ of the family E. The set $E_i$ is called the strategy set of player i, for every index i of the family E, and the product $^{\times }E$ is called the strategy profile space, or the n-strategy space, of the game.

Remark 2

First of all we recall a standard form definition of normal-form game.

Definition 2

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, $E = (E_i)_{i\in N}$, where, for each player i in N, the nonempty set $E_i$ is the set of actions available to player i;

• 3. f is an ordered family of real functions $f = (f_i)_{i\in N}$, where, for each player i in N, the function $f_i : \ ^\times E \rightarrow \mathbb {R}$ is the utility function of player i (inducing a preference relation on the Cartesian product $^\times E := \times _{j \in N} E_j$ (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

$$ f : \times _{j \in N} E_j \rightarrow \mathbb {R}^n : x \mapsto (f_i(x))_{i \in N}.$$

Terminology. Together with the previous definition of game in normal form, we need to introduce some terminologies:

• the set $\{ i\} _{i = 1}^{n}$ of the first n positive integers is said the set of players of the game;

• each element of the Cartesian product $^{\times }E$ is said a strategy profile, or n-strategy, of the game;

• the image of the function f, i.e., the set $f( ^{\times }E)$ of all real n-vectors of type f(x), with x in the strategy profile space $^{\times }E$, 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 [6]. By the way, the maximal boundary of a subset T of the Euclidean space $\mathbb {R}^n$ is the set of those $s\in T$ which are not strictly less than any other element of T.

Definition 3

(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 $\mathbb {R}^n$. If S denotes the strategy space $^{\times }E$, we shall denote the maximal boundary of the n-payoff space by $\overline{\partial }f(S)$ and the maximal boundary of the game by $\overline{\partial }_{f}(S)$ or by $\overline{\partial }(f)$. In other terms, the maximal boundary $\overline{\partial }_{f}(S)$ of the game is the reciprocal (inverse) 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 3

(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 $^{\times }E$ 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 (Bourbaki’s term for inverse image) of the point-wise order on f(S) via f.

The Method for $\varvec{C^{1}}$ 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 $\mathbb {R}^{n}$ containing the n-strategy space S (which, in this case, is a compact infinite part of the n-space $\mathbb {R}^{n}$). Details can be found in [7, 20, 21] but in the following we recall some basic notions.

Topological Boundary. The key theorem of our method is the following one, we invite the reader to pay much attention to the topologies used below.

Theorem 2

Let f be a $C^{1}$ function defined upon an open set O of the euclidean space $\mathbb {R}^{n}$ and with values in $\mathbb {R}^n$. 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

$$f(\partial _O S)\cup f(C),$$

that is

$$\partial _{f(O)} f(S) \subseteq f(\partial _O S)\cup f(C),$$

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 $J_{f}(x)$ is not invertible);

• 2. $\partial _O S$ is the topological boundary of S in O (with respect to the relative topology of O).

We strongly invite the reader to see the definitions and remarks about and around Theorem 2 in [7, 20, 21, 23].