Blossom (functional): Difference between revisions

Revision as of 12:07, 9 March 2013

In mathematics, and in particular the study of game theory, a function is graph continuous if it exhibits the following properties. The concept was originally defined by Partha Dasgupta and Eric Maskin in 1986 and is a version of continuity that finds application in the study of continuous games.

Notation and preliminaries

Consider a game with $N$ agents with agent $i$ having strategy $A_{i} \subseteq ℝ$ ; write $𝐚$ for an N-tuple of actions (i.e. $𝐚 \in \prod_{j = 1}^{N} A_{j}$ ) and $𝐚_{- i} = (a_{1}, a_{2}, \dots, a_{i - 1}, a_{i + 1}, \dots, a_{N})$ as the vector of all agents' actions apart from agent $i$ .

Let $U_{i} : A_{i} ⟶ ℝ$ be the payoff function for agent $i$ .

A game is defined as $[(A_{i}, U_{i}); i = 1, \dots, N]$ . If a graph is continuous you should connect it if it's not then don't connect it.

Definition

Function $U_{i} : A ⟶ ℝ$ is graph continuous if for all $𝐚 \in A$ there exists a function $F_{i} : A_{- i} ⟶ A_{i}$ such that $U_{i} (F_{i} (𝐚_{- i}), 𝐚_{- i})$ is continuous at $𝐚_{- i}$ .

Dasgupta and Maskin named this property "graph continuity" because, if one plots a graph of a player's payoff as a function of his own strategy (keeping the other players' strategies fixed), then a graph-continuous payoff function will result in this graph changing continuously as one varies the strategies of the other players.

The property is interesting in view of the following theorem.

If, for $1 \leq i \leq N$ , $A_{i} \subseteq ℝ^{m}$ is non-empty, convex, and compact; and if $U_{i} : A ⟶ ℝ$ is quasi-concave in $a_{i}$ , upper semi-continuous in $𝐚$ , and graph continuous, then the game $[(A_{i}, U_{i}); i = 1, \dots, N]$ possesses a pure strategy Nash equilibrium.

References

Partha Dasgupta and Eric Maskin 1986. The existence of equilibrium in discontinuous economic games, I: theory. The Review of Economic Studies, 53(1):1-26

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+{{orphan|date=September 2009}}
+In [[mathematics]], and in particular the study of [[game theory]], a [[function (mathematics)|function]] is '''graph continuous''' if it exhibits the following properties.  The concept was originally defined by [[Partha Dasgupta]] and [[Eric Maskin]] in 1986 and is a version of [[continuous function|continuity]] that finds application in the study of [[continuous game]]s.
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+==Notation and preliminaries==
+Consider a [[game]] with <math>N</math> agents with agent <math>i</math> having strategy <math>A_i\subseteq\Bbb{R}</math>; write <math>\mathbf{a}</math> for an N-tuple of actions (i.e. <math>\mathbf{a}\in\prod_{j=1}^NA_j</math>) and <math>\mathbf{a}_{-i}=(a_1,a_2,\ldots,a_{i-1},a_{i+1},\ldots,a_N)</math> as the vector of all agents' actions apart from agent <math>i</math>.
+Let <math>U_i:A_i\longrightarrow\Bbb{R}</math> be the payoff function for agent <math>i</math>.
+A '''game''' is defined as <math>[(A_i,U_i); i=1,\ldots,N]</math>.
+If a graph is continuous you should connect it if it's not then don't connect it.
+==Definition==
+Function <math>U_i:A\longrightarrow\Bbb{R}</math> is '''graph continuous''' if for all <math>\mathbf{a}\in A</math> there exists a function <math>F_i:A_{-i}\longrightarrow A_i</math> such that <math>U_i(F_i(\mathbf{a}_{-i}),\mathbf{a}_{-i})</math> is continuous at <math>\mathbf{a}_{-i}</math>.
+Dasgupta and Maskin named this property "graph continuity" because, if one plots a graph of a player's payoff as a function of his own strategy (keeping the other players' strategies fixed), then a graph-continuous payoff function will result in this graph changing continuously as one varies the strategies of the other players.
+The property is interesting in view of the following theorem.
+If, for <math>1\leq i\leq N</math>, <math>A_i\subseteq\Bbb{R}^m</math> is non-empty, [[Convex function|convex]], and [[compact set|compact]]; and if <math>U_i:A\longrightarrow\Bbb{R}</math> is [[quasi-concave function|quasi-concave]] in <math>a_i</math>, [[upper semi-continuous]] in <math>\mathbf{a}</math>, and graph continuous, then the game <math>[(A_i,U_i); i=1,\ldots,N]</math> possesses a [[pure strategy]] [[Nash equilibrium]].
+==References==
+* [[Partha Dasgupta]] and [[Eric Maskin]] 1986.  ''The existence of equilibrium in discontinuous economic games, I: theory''.  The Review of Economic Studies, 53(1):1-26
+{{DEFAULTSORT:Graph Continuous Function}}
+[[Category:Game theory]]

Blossom (functional): Difference between revisions

en>Meredyth

Revision as of 12:07, 9 March 2013

Notation and preliminaries

Definition

References

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Blossom (functional): Difference between revisions

en>Meredyth

Revision as of 12:07, 9 March 2013

Notation and preliminaries

Definition

References

Navigation menu

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