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= Temporized Equilibria =









\begin{abstract}
This article introduces a paradigm for the calculation of an alternative solution to the Nash equilibrium applicable to the class of games <math>G</math> and develops mathematical methods for the analysis of these games.

The concepts of deterrence, equilibrium and temporized solution will be introduced by means of mathematical definitions.

The main result is the demonstration of the existence of an equilibrium point for each <math>G</math> game, while other results concern with the geometrical structure of the solution and with the restriction to symmetrical games in order to create a rigorous characterization of these situations.

The topics to be treated have a significance that goes beyond the mere mathematical discussion to attain a level of universal understanding that aims to describe some of the most profound aspects of human society, religion, politics and in some ways, of consciousness itself.
\end{abstract}




==Introduction==
The theory of correlated equilibria developed by Aumann and Schelling, allows the calculation of the solution of a game, using a stratagem, which makes asymmetric and incomplete information provided separately to the players self-enforced, in other words stable, according to a certain definition of equilibrium.

On the contrary our theory is based on the concept of information deterrent that, as we shall see in the next section, is incomplete but symmetrical and provided jointly to all the players.

The basic idea is to give all players an extra information at the beginning of the game, which is used by them to recalculate the expected payoff.

The concepts of deterrent and temporized equilibrium, are the basis of our discussion. For both we will provide a general and rigorous definition. In particular, the concept of equilibrium will be seen as a characterization of the equilibrium for a finite, non-cooperative and non-zero-sum game.

In the next section we will mathematically prove the existence of the equilibrium point and provide methodological tools for the calculation of the temporized solution; we will discuss the properties of this point with regard to the concepts of Pareto-optimality and superrationality.

As an example of application of our theory, we include the solution to the prisoner's dilemma using the concept of temporized equilibrium.

Finally, the interpretations and applications of concepts exposed, will be treated in the concluding section of this article.

==Definitions and terminology==
In this section we define the fundamental ideas and we point out the terminology that will be used in the treatment. We will make reference to the concepts of finite, non-zero-sum game in strategic form, mixed strategy, payoff function and Nash equilibrium, according to the definitions furnished in \cite {Nash00}.

To relieve the notation, we will use the appellative \textit {"game <math> G </math>"} referring to finite, non-zero-sum games in strategic form, in which for every player, the coefficients of the non linear terms of the utility function after rearranging some terms, are all positive or all negative or all zero.

As we will see subsequently, this last property, that may appear restrictive, will allow us to describe in appropriated way a huge class of games.

Further more we will use indifferently the notations <math>f_j</math> and <math>f_j \left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^j \ldots x_{n_j-1}^j \right)</math> to indicate the utility function for the player <math>j</math>.

We now define the concept of ''deterrent'' , trying to make it as much general as possible to provide a definition able to nimbly embrace concepts relative to different fields.

\begin{definition}[Deterrent]
Piece of information, indication, tip or public consciousness, simmetric and incomplete, that is known to all the players of a <math>G</math> game at the beginning of the game, that induce the players to reconsider their utility functions in order to downsize the target set of such functions and then calculate the solutions of the game.
\end{definition}

It is evident that the presence of the ''deterrent'' affects the players perception of the game. This concept is fundamental in our theory, in fact it represents the element for which the players reconsider their strategic choices for a ''temporized equilibrium'' . In the following we define the idea of ''temporized equilibrium'' .

\begin{definition}[Temporized Equilibrium]
A temporized equilibrium is a Nash equilibrium, for which the equilibrium strategies identify a point on the hyperplane <math>H</math> defined by the following system:
\begin{displaymath}
H : \left\{
\begin{array}{l}
k_1 \frac{f_1\left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^r \ldots x_{n_r-1}^r\right)}{\max \left\lbrace f_1\left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^r \ldots x_{n_r-1}^r\right) \right\rbrace } = t\\
\cdots\\
k_r \frac{f_r\left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^r \ldots x_{n_r-1}^r\right)}{\max \left\lbrace f_r\left(x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^r \ldots x_{n_r-1}^r\right) \right\rbrace } = t
\end{array} \right.
\end{displaymath}
where <math>f_j \left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^j \ldots x_{n_j-1}^j \right)</math> represents the utility function for the player <math>j</math>, <math>r \in \mathbb{N}</math> indicates the number of players, <math>k_j \in \mathbb{R}^+</math> are the proportionality coefficients and <math>n_j</math> identify the available actions for every player <math>j</math>. The <math>x_s^j</math> are the porbability distributions assigned by the player <math>j</math> to the action <math>s</math>, for which the following properties hold: <math>x_s^j \in [0,1] \ \forall s,j</math> e <math>\sum_s x_s^j=1 \ \forall j</math>.
\end{definition}

From 2.2 we understand that the temporized equilibrium inherits the stability features from the definition of Nash equilibrium and adds to it more specific connotates.

\begin{definition}[Variables associated to the <math>j</math>-th player]
We define variables associated to the <math>j</math>-th player, the probability distributions <math>x_s^j</math> and we define functions that take as argument the variables associated to the <math>j</math>-th player the functions <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> in which the arguments are expressed exclusively in terms of variables associated to the player <math>j</math>, that follow the relations <math>h_j</math> resulting from the calculus of the components of the hyperplane <math>H</math>. Obviously the subsequent holds:
<center><math>
f_j \left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^j \ldots x_{n_j-1}^j \right) = f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right),\ \forall j
</math></center>
\end{definition}

In the folowing sections, we will justify the definitions given and we will build our theory mathematically.

==Existence of the temporized equilibrium==
We now lay the foundations on which out theory is based. First we enunciate adn demonstrate a theorem of general validity, that is necessary to our demosntration of the existence of the temporized equilibrium. We will build an hyperplane that will be used to give a charaterisation and geometrical accuracy to the solution.
\begin{theorem}
In every game <math>G</math> with <math>r</math> players, in which every player <math>j \in [1,r] \subset \mathbb{N}</math> can make <math>n_j</math> actions, exist <math>r</math> coefficients <math>k_j \in \mathbb{R}^+</math> that makes the components of the hyperplane <math>H</math>, identified by the following system, continous:
\begin{displaymath}
H : \left\{
\begin{array}{l}
k_1 \frac{f_1\left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^r \ldots x_{n_r-1}^r\right)}{\max \left\lbrace f_1\left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^r \ldots x_{n_r-1}^r\right) \right\rbrace } = t\\
\cdots\\
k_r \frac{f_r\left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^r \ldots x_{n_r-1}^r\right)}{\max \left\lbrace f_r\left(x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^r \ldots x_{n_r-1}^r\right) \right\rbrace } = t
\end{array} \right.
\end{displaymath}

\end{theorem}
\begin{proof}
Let us demonstrate the theorem in case of two player, <math>r=2</math>, each one having <math>n_1=n_2=2</math> actions. The utility functions are: <math>f_1(x_1^1,x_2^1)</math> e <math>f_2(x_1^2,x_2^2)</math>, where <math>x_2^1=1-x_1^1</math> e <math>x_2^2=1-x_1^2</math>; the bimatrix associated to the game is:
\begin{displaymath}
U =
\left( \begin{array}{cc}
a_{11}, b_{11} & a_{12}, b_{12} \\
a_{21}, b_{21} & a_{22}, b_{22}
\end{array} \right)
\end{displaymath}
Let's write the hyperplane <math>H</math> with the utility functions of the game:
\begin{displaymath}
H : \left\{
\begin{array}{l}
k_1 \frac{a_{11} x_1^1 x_1^2 + a_{12} x_1^1 \left( 1-x_1^2 \right) + a_{21} \left( 1-x_1^1 \right) x_1^2 + a_{22} \left( 1-x_1^1 \right) \left( 1-x_1^2 \right)}{\max\left\lbrace a_{11}, a_{12}, a_{21}, a_{22}\right\rbrace } = t\\
k_2 \frac{b_{11} x_1^1 x_1^2 + b_{12} x_1^1 \left( 1-x_1^2 \right) + b_{21} \left( 1-x_1^1 \right) x_1^2 + b_{22} \left( 1-x_1^1 \right) \left( 1-x_1^2 \right)}{\max\left\lbrace b_{11}, b_{12}, b_{21}, b_{22}\right\rbrace} = t
\end{array} \right.
\end{displaymath}
Let us suppose that <math>k_1=1</math> and call <math>A=\max\left\lbrace a_{11}, a_{12}, a_{21}, a_{22}\right\rbrace </math> and <math>B=\max\left\lbrace b_{11}, b_{12}, b_{21}, b_{22}\right\rbrace</math>, solve the system, obtaining, after a little calculation, the equation of the hyperplane <math>H</math>:
\begin{multline}
x_1^1x_1^2\left(\frac{a_{11}-a_{12}-a_{21}+a_{22}}{A}-\frac{k_2 \left(b_{11}-b_{12}-b_{21}+b_{22}\right)}{B}\right)+
\\
+ x_1^1\left( \frac{a_{12}-a_{22}}{A} - \frac{k_2 \left(b_{12}-b_{22}\right)}{B} \right) + x_1^2\left( \frac{a_{21}-a_{22}}{A} - \frac{k_2\left( b_{21} - b_{22} \right)}{B}\right) +
\\
+ a_{22}/A - k_2 b_{22}/{B} = 0
\label{hyper}
\end{multline}
In order to guarantee that every component of the equation (\ref{hyper}) is continuous, it is sufficient to set to zero the coefficient of the non-linear term. To do this it is necessary to solve the following equation:
<center><math>
\frac{a_{11}-a_{12}-a_{21}+a_{22}}{A} - \frac{k_2 \left(b_{11}-b_{12}-b_{21}+b_{22}\right)}{B}=0
</math></center>
In case <math>A</math> or <math>B</math> are zero, it is possible to sum a quantity <math>u \neq 0</math> to the coefficient of <math>U</math> before the calculus and then subtract them at the end. If the identity holds:
<center><math>
\frac{a_{11}-a_{12}-a_{21}+a_{22}}{A} = \frac{b_{11}-b_{12}-b_{21}+b_{22}}{B}
</math></center>
then by definition <math>k_2=1</math>.
\end{proof}
The extension of this demonstration to generic <math>G</math> games is trivial, in fact it is sufficient to find the values of the coefficient <math>k_j</math> that make the non-linear terms of the hyperplan components equations null. If all the components of <math>H</math> are continuos <math>\forall k_j \in \mathbb{R}^+</math>, then by definition <math>k_j=1 \ \forall j \in [1,r]</math>.

The number of players and of the possibile actions for each player affect the number of veriable in the system that ramain unsolved, that represent the number of dimensions of the hyperplane that we have already discussed. For <math>H</math> holds the following:
\begin{proposition}
The hyperplane <math>H</math> has <math>d(H)=\sum_{i=1}^r (n_i -1)+1-r</math> dimension.
\end{proposition}
\begin{proof}
Every player has an utility function, so the system <math>H</math> has <math>r</math> equations. To every action is associated a probability function, the number of variables in the function <math>f_j</math> is <math>n_i-1</math> for every player <math>i</math>. At the total we have to sum the variable <math>t</math> and then the variables that remain parametrically expressed are <math>d(H)=\sum_{i=1}^r (n_i -1)+1-r</math>. This value represents the number of dimensions of the hyperplane.
\end{proof}

Now that we have terminated with the description of the hyperplane <math>H</math>, by expressing it as a a function of its components and derving information on the number of dimensions, we observe how this structure properties influence the player's utility functions characteristics.
\begin{proposition}
The continuity of the hyperplane components <math>H</math> is a sufficient condition for the continuity of the player's utility functions <math>k_jf_j</math> calculted on <math>H</math>.
\end{proposition}
\begin{proof}
The utility functions <math>f_j</math> are <math>j</math> grade polynoms and so they are obviously continous. So the functions <math>f_j</math> continuity calculated on <math>H</math> that take as arguments the variables <math>x_s^j</math> associated to the <math>j</math>-th player, depends exclusively on the continuity of the hyperplane components.
\end{proof}

By contruction of the hyperplane <math>H</math>, we know that it is that geometrical place in which the utility functions (moltiplied by the coefficient <math>k_j</math>) are the same. Let us use the definition 2.3S to enunciate the following:
\begin{theorem}
Stationary points of the utility functions <math>f_j</math> calculated on the hyperplane <math>H</math>, that take as arguments the variables <math>x_s^j</math> associated to the <math>j</math>-th player, are coincident <math>\forall j \in [1,r] \subset \mathbb{N}</math>.
\end{theorem}
\begin{proof}
By theorem 3.1, the hyperplane <math>H</math> is the place of points in which the functions <math>k_jf_j</math> are equivalent. By definition 2.3, we can express the <math>f_j \left( x_1^1 \ldots x_{n_1-1}^1 \ldots x_1^j \ldots x_{n_j-1}^j \right)</math> as exclusive functions of the variables related to the player <math>j</math>, calculated on the hyperplane, <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math>, without changing the properties, (<math>h_j</math> is a continous function <math>\forall j \in [1,r]</math>). We now can conclude that <math>H</math> is the place of points in wich the functions <math>k_jf_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> are equivalent, for this reason aven the stationary points of such functions coincide. By the following derivation rule: <math>D\left[ k f(x) \right] = k D\left[ f(x) \right]</math>, we can write:
<center><math>
D\left[k_jf_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)\right]=k_j D\left[f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)\right]
</math></center>
Now, by the definition of stationary point, we write from the previous:
<center><math>
k_j D\left[f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)\right]=0
</math></center>
Since the coefficients <math>k_j > 0</math>, the equation has the same solutions of the following: <math>D\left[f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)\right]=0</math>, and that holds for every <math>j</math>. We have demonstrated that the stationary points of the functions <math>k_jf_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> and <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> coincide <math>\forall j</math> and so, by the transitive property, the stationary point of the <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> are identical <math>\forall j</math>.
\end{proof}
We know that the components of the utility functions are continous and defined on a compact set (Def. 2.2), so the following holds:
\begin{corollary}
The maxima of the utility functions <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math>, calculated on <math>H</math>, define univocally a point on such hyperplane.
\end{corollary}
\begin{proof}
By contruction the functions <math>k_jf_j</math> calculated on the hyperplane <math>H</math>, are continous and defined on a compact set, since <math>x_s^j \in [0,1] \ \forall s,j</math>, by the Weierstrass' theorem they admit an absolute maximum <math>M \in H</math>. By the theorem 3.1 the <math>k_jf_j(M)</math> have the same value <math>\forall j</math>. By the previous theorem, we know that the <math>k_jf_j</math> have the same stationary points, so <math>M</math> is a stationary point for all the <math>k_jf_j</math>. Let us use the following property of the funtion <math>argmax</math>:
<center><math>
argmax(kf(x))=argmax(f(x)),\ with\ k>0
</math></center>
that allow us to say that, maximize the <math>k_jf_j</math> is equivalent to maximize the <math>f_j</math>. As we did in the demonstration of the theorem 3.4 we can express the <math>f_j</math> as function of the variables associated to the player <math>j</math>, without alter the properties of such functions and so we can conclude that maximize the <math>f_j</math> is equivalent to maximize the <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math>. We apply the function <math>argmax</math> a to these and so we obtain the coordinates of the maximum point, expressed in cartesian coordinates, since <math>H</math> is defined as a compact subspace of the Euclidean space <math>\mathbb{E}^r</math>:
<center><math>
M \left(argmax\left( f_1 \left( h_1\left( x_1^1 \ldots x_{n_1 - 1}^1 \right) \right) \right), \ldots, argmax\left( f_r \left( h_r\left( x_1^r \ldots x_{n_r - 1}^r \right) \right) \right) \right) \in H
</math></center>
that is the geometrical point whose coordinates are the maximum point of the utility functions <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> that take as arguments the variables <math>x_s^j</math> associated to the player <math>j</math>, <math>\forall j</math>.
\end{proof}

We have built the mathematical structures that will allow us dto demonstrate the existence of the temporized equilibrium. Let us enunciate the cardinal theorem for our theory and, give a demonstration that uses that instruments just exposed.
\begin{theorem}
For every <math>G</math> game exsits a temporized equilibrium.
\end{theorem}
\begin{proof}
This theorem is a direct consequence of the previous theorems. For every <math>G</math> game hold the theorem 3.1, that states the continuity of the hyperplane <math>H</math> components, that, by the proposition 3.3 is a sufficient condition for the continuity of the <math>f_j</math> calculated on <math>H</math>; by theorem 3.4 we know that the players' utility functions, <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math>, calculated on the hyperplane have the same stationary points and more on by the corollary 3.5 we know that the maximum of the <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> individute a point <math>M</math> on <math>H</math>. We need now to verify the stability of the point <math>M</math>, for concluding that it is an equilibrium point. We call <math>f_j\left( argmax\left( f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right) \right) \right)</math> the value of <math>f_j</math> in the point <math>M</math>. It is evident that the relation holds:
<center><math>
f_j\left( argmax\left( f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right) \right) \right) \geq f_j\left( l\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)\ \forall l,j
</math></center>
with <math>l</math> that represents a generic function on the variables <math>\left( x_1^j \ldots x_{n_j - 1}^j \right)</math>; this means that a temporized equilibrium is a Nash equilibrium, in which the utility that the player <math>J</math> can obtain by playing <math>M</math> is grather or equal than the utility that he could obtain by playing any other strategy, and this holds <math>\forall j</math>. So the point <math>M</math> is stable and we can say that the coordinates of <math>M</math> are the temporized equilibrium profiles of every player, for a <math>G</math> game.
\end{proof}
We have shown that in every <math>G</math> game exists a temporized equilibrium; starting from the theorem 3.6 demonstration, we can get some properties of the equilibrium point.

We will treat in this section the Pareto-optimality, while we leave to the next section the considerations about superrationality.
===Pareto-optimality===
A fundamental characteristic of the temporized equilibrium is ''Pareto-optimality'' . We have a paretian optimus (alternatively said ''allocative efficiency'' ) when it is not possible any reorganization of the payoff distribution that can increase everone conditions. In such situations, the utility of a player can be increased only decreasing the utility of some other players; this means that no player can increase his condition if none decrease his own one.

The Pareto-optimality property depends directle on the theorem 3.5; since the point <math>M</math> is the maximum point for the utility function of every player and, the strategies profile by which every player choose the equilibrium is possible, it is evident that the temporized equilibrium is the best choice that every player can make in the game, according with the definition of Pareto-optimality.


==Superrationality==
Superrationality is an alternative way of thinking. The first assumption is that the answer to a symmetric problem will be the same for all the superrational player. Further more this equality will be taken in consideration before knowing what the strategy will be. The strategy will be found by maximizing the payoff for every player, assuming that every player will use the same strategy. Since the superrational player knows that the other superrational players will do the same thing, whatever it would be, there will be only two choices (in the case of a <math>2 \times 2</math> game). Both will cooperate or both will defect respect to the value of the superrational answer.

In the calculus of the temporized equilibria every player assume that even the other players will use the deterrent information to calculate the impact that it has on the game e only after that he makes the choice, without any agreements with the other players or further hypothesis about their rationality. The stability of the temporized equilibrium is given by the fact that it is a Nash equilibrium, Pareto-optimal. So, even if the superrationality is different from the classical concepts of game theory, the research of temporized equilibrium in symmetric games brings to the same results that one can obtain using the definition of superrationality.

Let us see two theorems applicable to symmetric games that will allow us to conclude that the superrational behaviour Vcould be obtained by rational players, according to the definition given by games theory, using the temporized equilibria theory; but before that let us start with a consideration that holds for non cooperative games:
\begin{remark}
Every finite, non zero sum and symmetric game, is a <math>G</math> game.
\end{remark}

Thank to this remark, we can expose the following:
\begin{corollary}
In every symmetric <math>G</math> game, for which the hypothesis of the theorem 3.1 hold, the coefficients <math>k_j</math> are all equal to 1.
\end{corollary}
\begin{proof}
The bimatrix associated to a symmetric game is anti-symmetric. Using the dispositions of theorem 3.1 it results evident that by equalling the utility functions of the players and reordering the terms, the coefficients of the non linear terms go to zero, since the coefficients of the bimatrix are the same for every player for every value <math>k_j</math>. It follows that, by definition, when the coefficients of the non linear terms go to zero, the <math>k_j</math> have value 1.
\end{proof}

As direct consequence of the corrolary just exposed, we express the following theorem that will be fundamental for the considerations that we will make at the end of this section, in relation with the concept of superrationality.
\begin{theorem}
In every symmetric <math>G</math> game, the temporized equilibrium strategy is the same for every player.
\end{theorem}
\begin{proof}
By the corollary 4.2 we know that the coefficients <math>k_j</math> in a symmetric <math>G</math> game are equal to 1, so <math>k_jf_j = f_j</math>. On the hyperplane <math>H</math> the functions <math>k_jf_j</math> are equivalent <math>\forall j</math>, so in our case the functions <math>f_j</math> are all equals <math>\forall j</math>. By the theorem 3.4 the stationary points of the <math>f_j</math> coincide and more on the values of the <math>f_j</math> in such points are the same <math>\forall j</math>. Since the <math>f_j</math> are continous, one of the common stationary point is an absolute maximum <math>M^*</math>. Once again it is possible to express the <math>f_j</math> as exclusive function of the variables associated to the player <math>j</math>. After this operation, it holds the condition for which the <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> are identical. At this point by applying the <math>argmax</math> function to every <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math>, we obtain that the <math>argmax\left( f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right) \right)</math> are identical <math>\forall j</math> and so by the theorem 3.5 the players have an equilibrium strategy that uniquely individuate a point <math>M^*</math> on <math>H</math>, that we have demostrate that it is the same for all.
\end{proof}
In the characteristics of the temporized equilibrium for symmetric <math>G</math> game, stands the demonstration of the statements made at the beginning of this section. In fact, since the equilibrium strategy is the same for all te players and the value of the game at the temporized equilibrium is Pareto-optimal, the aswer of the rational players is the one that would be given by superrational players to the same game.

In the next section we will show, in a classical game theory example, a practical case of temporized equilibrium calculus, that should clarify the aspects related to superrationality.


==Temporized solution to the prisoner's dilemma==
The prisoner's dilemma is a <math>G</math> game example very famous in literature, in which two criminals are accused of robbery. The detectives arrest both, close them in two different cells and don't allow them to communicate. They give them two choices: confess or not confess. Every strategy profile provide different payoff, according to the following bimatrix:
\begin{displaymath}
\left( \begin{array}{cc}
5, 5 & 0, 10 \\
10, 0 & 1, 1
\end{array} \right)
\end{displaymath}
where the first/second player chosing the first row/column choose to confess while chosing the second he decide to do not confess. The solution for this game, using the notion of Nash equilibrium, is (not confess, not confess) whith expected payoff 1 for both.

Since every player has the possibility to gain 10, the solution just proposed seems to be not very convenient, even if stable. Let us see what will happen, if the deterrent information would be furnished to the two players.

Let us suppose the a detective in a ''very bad temper'' , say to both criminals, after having exposed the classical game, that their possibility to obtain the minimum number of year in prison will decrease according to the same proportional law and, without the possiblity for them to make any agreements, he makes them go in separated cells.

At this point the two prisoners, in possession of the deterrent information (the detective menace), before making any choice, rationally decide to understand the entity of the information reagarding to the game payoff. (they complete the information)

Firstly the calculate the hyperplane <math>H</math> equation:

\begin{displaymath}
H : \left\{
\begin{array}{l}
k_1 \frac{5 x_1^1 x_1^2 + 0 x_1^1 \left( 1-x_1^2 \right) + 10 \left( 1-x_1^1 \right) x_1^2 + 1 \left( 1-x_1^1 \right) \left( 1-x_1^2 \right)}{10} = t\\
k_2 \frac{5 x_1^1 x_1^2 + 10 x_1^1 \left( 1-x_1^2 \right) + 0 \left( 1-x_1^1 \right) x_1^2 + 1 \left( 1-x_1^1 \right) \left( 1-x_1^2 \right)}{10} = t
\end{array} \right.
\end{displaymath}
The deterrent furnishes the fundamental information about the coefficients <math>k_j</math>; since the possibilities decrease for both according to the same proportions, it follows that <math>k_j=1 \ j=1,2</math>. So solving the system we obtain: <math>x_1=x_2</math> that is clearly linear. At this point the two players calculate separated their <math>f_j \left( h_j\left( x_1^j \ldots x_{n_j - 1}^j \right) \right)</math> obtaining:
<center><math>\begin{matrix}
f_1(x_1)=-4x_1^2+8x_1+1 \nonumber\\
f_2(x_2)=-4x_2^2+8x_2+1 \nonumber
\end{matrix}</math></center>
Now the two players will try the strategy that maximize their own utility function. In figure it is shown the behaviour of such functions:
\begin{figure}[h]
\includegraphics[width=0.60\textwidth]{prigio.jpg}
\caption{<math>f(x_1)=-4x_1^2+8x_1+1</math> e <math>f(x_2)=-4x_2^2+8x_2+1</math>}
\end{figure}

It is evident that maximizing on <math>x_1</math> for the first and <math>x_2</math> for the second, both player obtain that the optimal strategy is <math>x_1=x_2=1</math>, since it brings a payoff equal to 5 for both. By the theorems exposed in the previous sections, we know that the point <math>M(1,1) \in H</math> is a Nash equilibrium of the temporized game, so it is stable and Pareto-optimal. The temporized solution suggest the use of the strategy <math>x_1=x_2=1</math> with an expected payoff equals to 5 for both, while the Nash solution suggest the strategy <math>x_1=x_2=0</math> with an expected payoff equals to 1 for both.

In relation to the superrational concept, let us say something more.

The answer of two superrational player to the prisoner's dilemma, in the classical formulation, is the one to play the strategy <math>x_1=x_2=1</math>, after making some consideration about the razional ''faculty'' of the opponent. In the temporized paradigm, the player's choices do not need any hypothesis regarding the rationality characteristics of the other, but it uses the classical concept of rationality that, by adding the deterrent information, makes the players doing the operations shown in this section to arrive to the conclusions that are identical to the ones obtainable by superrational players, using elements provided by games theory.


==Interpretations and applications==
The arguments exposed, leave place for many considerations and, interpretations that are free from the specific topic in which they have been formulated. In this section we will discuss a summary exposition of such implications.

Firstly, the temporized solution, unify the best result for a player, with the best result for every player. This property is a consequence of the Pareto-optimality of the solution; in fact the equilibrium point, represents the strategies profile for which every player obtain the maximum from the game, for himself and for the others. It exists a sort of social significance in the concept of temporized equilibrium, so it is social optimal and in the meanwhile individually optimal. In other words, by chosing the temporized equilibrium, the players can obtain, from the game, the best result for all.

A second consideration regards the importance of the incompleteness of the deterrent information, in fact in the incompleteness of the deterrent stays the needing of the player to build the structure previously exposed, that brings to the calculus of the equilibrium. We can conclude that it not necessary to give an asymmetric and incomplete information (as in the correlated equilbria theory), but it is sufficient to give the players an incomplete information for using the rationality principle in order to obtain alternative solution concepts.

In the previous sections we have demonstrated which are the mathematical mechanisms that induce the players to choose the temporized solution; in this section we give a more ''philosofical'' significance to the deterrent information. It gives to the players a sort of group ''consciousness'' , that is observable in the characteristics of hte equilibrium point; this consciousness, even if we are treating non-cooperative games, oblige the players to play strategy profile profoundly related. As we have affirmed at the beginning of this section, in the temporized solution, the individual maximum coincides with the group maximum and in this property stays the consciousness that we cited previously. The rationality it is not sufficient to obtain the personal and global optimum, but it is necessary to have ''something more'' . In the model proposed by our theory, this ''something more'' is induced by the deterrent. There exist a lot of real example in shich such situation is evident, from social relationships to commercial ones, from religion to politics and so on.

In all the situations in which some entities are in contact to obtain a common solution and in which it is present a trusted supervisor that gives the deterrent information, we are in temporized situation; so back to the previous examples, in social relationships the deterrent may be umanity, in commercial ones, the deterrent may be the control structures, for example, to respect the ''antitrust'' laws, in religion it may be god and in politics the deterrent may be represented by the state.

As we have seen the interpretations and the real cases in which we encounter the concept of temporized equilibrium are very common and are part of big a various fields. The applicative possibilities of the theory of non-cooperative games, are covered by our theory, but differently from the classical theory, our one build the concept of consciousness/deterrent and so it describes better some situations that are tipical in the social behaviour. The study of <math>G</math> game in temporized context, finds natural applications in the study of solutions concept altertive t the Nash equilibrium.

Some applications, maybe science fictional, could be the study of the choice of punishment systems as an answer to criminal behaviour or the creation of a super entity that should manage juridical relastions between individuals and furnish the deterrent information in order to obtain the best result for the society (that as we have said conicides with the personal best).

Beyond the category of non-cooperative games, our theory cna find applications in the study of cooperative games; if we assign the meaning of ''"partecipation to the coalition"'' to the coefficient <math>k_j</math> we should characterize the optimal strategies of every player as a part of a coalition.

\begin{thebibliography}{000}
\bibitem{Nash00} Non-cooperative games, ''John Nash'' , May 1950, ''A dissertation presented to the faculty of Princeton University for the Degree of Doctor in Philosophy''
\label{Nash1950}
\end{thebibliography}

\end{document}

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