Graphical game theory

In game theory, the graphical form or graphical game is an alternate compact representation of strategic interactions that efficiently models situations where players' outcomes depend only on a subset of other players.[1] First formalized by Michael Kearns, Michael Littman, and Satinder Singh in 2001, this approach complements traditional representations such as the normal form and extensive form by leveraging concepts from graph theory to achieve more concise game descriptions.

In a graphical game representation, players are depicted as nodes in a graph, with edges connecting players whose decisions directly affect each other. Each player's utility function depends only on their own strategy and the strategies of their immediate neighbors in the graph, rather than on all players' actions. This framework is particularly valuable for modeling social network interactions, economic networks, and localized competitive scenarios where players primarily respond to those in their immediate vicinity.

The graphical approach offers significant advantages when representing large games with limited interaction patterns, as it can exponentially reduce the amount of information needed to fully describe the game. This compact representation facilitates more efficient computational analysis for complex multi-agent systems across fields such as artificial intelligence, economics, and network science.

## Contents

* 1 Formal definition
* 2 The size of the game's representation
* 3 An example
* 4 Nash equilibrium
* 5 References

## Formal definition

A graphical game is represented by a graph G, in which each player is represented by a node, and there is an edge between two nodes i and j iff their utility functions are dependent on the strategy which the other player will choose. Each node i in G has a function ui:{1…m}di+1->ℝ, where di is the degree of vertex i. ui specifies the utility of player i as a function of his strategy as well as those of his neighbors.

## The size of the game's representation

For a general n players game, in which each player has m possible strategies, the size of a normal form representation would be O(mn). The size of the graphical representation for this game is O(md) where d is the maximal node degree in the graph. If d≪n, then the graphical game representation is much smaller.

## An example

In case where each player's utility function depends only on one other player:

* The graphical form of the described game

The maximal degree of the graph is 1, and the game can be described as n functions (tables) of size m2. So, the total size of the input will be nm2.

## Nash equilibrium

Finding Nash equilibrium in a game takes exponential time in the size of the representation. If the graphical representation of the game is a tree, we can find the equilibrium in polynomial time. In the general case, where the maximal degree of a node is 3 or more, the problem is NP-complete.

## References

1. ↑ Kearns, Michael; Littman, Michael L.; Singh, Satinder (2 August 2001). "Graphical Models for Game Theory". Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence. pp. 253–260. https://dl.acm.org/doi/10.5555/2074022.2074054.

* Michael Kearns (2007) "Graphical Games". In Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory. Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0. http://www.cs.cmu.edu/~sandholm/cs15-892F13/algorithmic-game-theory.pdf.
* Michael Kearns, Michael L. Littman and Satinder Singh (2001) "Graphical Models for Game Theory".

* v
* t
* e

Topics in game theory

Definitions|

* Cooperative game
* Determinacy
* Escalation of commitment
* Extensive-form game
* First-player and second-player win
* Game complexity
* Graphical game
* Hierarchy of beliefs
* Information set
* Normal-form game
* Preference
* Sequential game
* Simultaneous game
* Simultaneous action selection
* Solved game
* Succinct game

Equilibrium
concepts|

* Nash equilibrium
* Subgame perfection
* Mertens-stable equilibrium
* Bayesian Nash equilibrium
* Perfect Bayesian equilibrium
* Trembling hand
* Proper equilibrium
* Epsilon-equilibrium
* Correlated equilibrium
* Sequential equilibrium
* Quasi-perfect equilibrium
* Evolutionarily stable strategy
* Risk dominance
* Core
* Shapley value
* Pareto efficiency
* Gibbs equilibrium
* Quantal response equilibrium
* Self-confirming equilibrium
* Strong Nash equilibrium
* Markov perfect equilibrium

Strategies|

* Dominant strategies
* Pure strategy
* Mixed strategy
* Strategy-stealing argument
* Tit for tat
* Grim trigger
* Collusion
* Backward induction
* Forward induction
* Markov strategy

Classes
of games|

* Symmetric game
* Perfect information
* Repeated game
* Signaling game
* Screening game
* Cheap talk
* Zero-sum game
* Mechanism design
* Bargaining problem
* Stochastic game
* n-player game
* Large Poisson game
* Nontransitive game
* Global game
* Strictly determined game
* Potential game

Games|

* Go
* Chess
* Infinite chess
* Checkers
* Tic-tac-toe
* Prisoner's dilemma
* Optional prisoner's dilemma
* Traveler's dilemma
* Coordination game
* Chicken
* Centipede game
* Volunteer's dilemma
* Dollar auction
* Battle of the sexes
* Stag hunt
* Matching pennies
* Ultimatum game
* Rock paper scissors
* Pirate game
* Dictator game
* Public goods game
* Blotto game
* War of attrition
* El Farol Bar problem
* Fair division
* Fair cake-cutting
* Cournot game
* Deadlock
* Diner's dilemma
* Guess 2/3 of the average
* Kuhn poker
* Nash bargaining game
* Prisoners and hats puzzle
* Trust game
* Princess and Monster game
* Rendezvous problem

Theorems|

* Arrow's impossibility theorem
* Aumann's agreement theorem
* Folk theorem
* Minimax theorem
* Nash's theorem
* Purification theorem
* Revelation principle
* Zermelo's theorem

Key
figures|

* Albert W. Tucker
* Amos Tversky
* Ariel Rubinstein
* Claude Shannon
* Daniel Kahneman
* David K. Levine
* David M. Kreps
* Donald B. Gillies
* Drew Fudenberg
* Eric Maskin
* Harold W. Kuhn
* Herbert Simon
* Hervé Moulin
* Jean Tirole
* Jean-François Mertens
* Jennifer Tour Chayes
* John Harsanyi
* John Maynard Smith
* Antoine Augustin Cournot
* John Nash
* John von Neumann
* Kenneth Arrow
* Kenneth Binmore
* Leonid Hurwicz
* Lloyd Shapley
* Melvin Dresher
* Merrill M. Flood
* Olga Bondareva
* Oskar Morgenstern
* Paul Milgrom
* Peyton Young
* Reinhard Selten
* Robert Axelrod
* Robert Aumann
* Robert B. Wilson
* Roger Myerson
* Samuel Bowles
* Suzanne Scotchmer
* Thomas Schelling
* William Vickrey

See also|

* All-pay auction
* Alpha–beta pruning
* Bertrand paradox
* Bounded rationality
* Combinatorial game theory
* Confrontation analysis
* Coopetition
* First-move advantage in chess
* Game mechanics
* Glossary of game theory
* List of game theorists
* List of games in game theory
* No-win situation
* Solving chess
* Topological game
* Tragedy of the commons
* Tyranny of small decisions

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Graphical game theory. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Graphical_game_theory&oldid=3985930"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template