This document summarizes the key findings of a research paper on the frequency of convergent games under best-response dynamics. The paper shows that:
1) The frequency of randomly generated games with a unique pure strategy Nash equilibrium goes to zero as the number of players or strategies increases.
2) Convergent games with fewer pure strategy Nash equilibria are more common than those with more equilibria.
3) For 2-player games with less than 10 strategies, games with a unique equilibrium are most common, but games with multiple equilibria are more likely for more than 10 strategies.
This document discusses applications of game theory in computer science, specifically in networking and algorithm analysis. It introduces fundamental game theory concepts like the Nash equilibrium. It then explores how game theory can be used to model network security as a stochastic game between a hacker and security team, allowing analysis of optimal strategies. It also explains Yao's minimax principle, which uses game theory to relate the complexities of deterministic and randomized algorithms by modeling them as players in a zero-sum game. By representing problems in game theoretic terms, complex issues can be analyzed to find solutions.
This document provides an introduction to game theory, including:
- Game theory mathematically determines optimal strategies given conditions to maximize outcomes.
- It has roots in ancient texts and was modernized in 1944. Famous examples include the Prisoner's Dilemma.
- Games involve players, strategies, and payoffs. Equilibria like Nash equilibria predict likely outcomes.
- Games can be simultaneous or sequential, affecting likely equilibria. Strategies can be pure or mixed.
Game theory seeks to analyze competing situations that arise from conflicts of interest. It examines scenarios of conflict to identify optimal strategies for decision makers. Game theory assumes importance from a managerial perspective, as businesses compete for market share. The theory can help determine rational behaviors in competitive situations where outcomes depend on interactions between decision makers and competitors. It provides insights to help businesses convert weaknesses and threats into opportunities and strengths to maximize profits.
Game Theory - Quantitative Analysis for Decision MakingIshita Bose
WHAT IS GAME THEORY?
HISTORY OF GAME THEORY
APPLICATIONS OF GAME THEORY
KEY ELEMENTS OF A GAME
TYPES OF GAME
NASH EQUILIBRIUM (NE)
PURE STRATEGIES AND MIXED STRATEGIES
2-PLAYERS ZERO-SUM GAMES
PRISONER’S DILEMMA
Game theory is the study of strategic decision makingManoj Ghorpade
Game theory is the study of strategic decision making between intelligent rational decision makers. It originated in economics and is used in fields like political science and psychology. Game theory analyzes interactions with both cooperative and non-cooperative games. Modern game theory began with John von Neumann's work on mixed strategy equilibria in two-person zero-sum games. Von Neumann and Oskar Morgenstern's 1944 book Theory of Games and Economic Behavior was influential in establishing game theory. Game theory has been widely applied, including in biology since the 1970s, and has helped explain behaviors in economics, politics and other fields.
This document provides an introduction to game theory. It discusses what game theory is, its essential features, and some key concepts in game theory including Nash equilibrium, backward induction, extensive form games, normal form games, mixed strategies, coordination games, zero-sum games, the prisoner's dilemma, chicken games, and repeated games. It also provides examples of applying game theory concepts to real-world situations such as the rivalry between Airbus and Boeing.
This document provides an overview of game theory concepts including its development, assumptions, classification of games, elements, significance, limitations, and methods for solving different types of games. Some key points:
- Game theory was developed in 1928 by John Von Neumann and Oscar Morgenstern to analyze decision-making involving two or more rational opponents.
- Games can be classified as two-person, n-person, zero-sum, non-zero-sum, pure-strategy, or mixed-strategy.
- Elements include the payoff matrix, dominance rules, optimal strategies, and the value of the game.
- Methods for solving games include using pure strategies if a saddle point exists, or mixed
This document provides an overview of game theory, including its founders John von Neumann and John Nash. Game theory is the study of strategic decision making among rational players where outcomes depend on the choices of all. It has applications in economics, politics, and biology. Key concepts discussed include Nash equilibrium, where no player benefits from changing strategies alone; the prisoner's dilemma game; and the tit-for-tat strategy of reciprocal cooperation and defection. The document outlines the assumptions, elements, and applications of game theory.
This document discusses applications of game theory in computer science, specifically in networking and algorithm analysis. It introduces fundamental game theory concepts like the Nash equilibrium. It then explores how game theory can be used to model network security as a stochastic game between a hacker and security team, allowing analysis of optimal strategies. It also explains Yao's minimax principle, which uses game theory to relate the complexities of deterministic and randomized algorithms by modeling them as players in a zero-sum game. By representing problems in game theoretic terms, complex issues can be analyzed to find solutions.
This document provides an introduction to game theory, including:
- Game theory mathematically determines optimal strategies given conditions to maximize outcomes.
- It has roots in ancient texts and was modernized in 1944. Famous examples include the Prisoner's Dilemma.
- Games involve players, strategies, and payoffs. Equilibria like Nash equilibria predict likely outcomes.
- Games can be simultaneous or sequential, affecting likely equilibria. Strategies can be pure or mixed.
Game theory seeks to analyze competing situations that arise from conflicts of interest. It examines scenarios of conflict to identify optimal strategies for decision makers. Game theory assumes importance from a managerial perspective, as businesses compete for market share. The theory can help determine rational behaviors in competitive situations where outcomes depend on interactions between decision makers and competitors. It provides insights to help businesses convert weaknesses and threats into opportunities and strengths to maximize profits.
Game Theory - Quantitative Analysis for Decision MakingIshita Bose
WHAT IS GAME THEORY?
HISTORY OF GAME THEORY
APPLICATIONS OF GAME THEORY
KEY ELEMENTS OF A GAME
TYPES OF GAME
NASH EQUILIBRIUM (NE)
PURE STRATEGIES AND MIXED STRATEGIES
2-PLAYERS ZERO-SUM GAMES
PRISONER’S DILEMMA
Game theory is the study of strategic decision makingManoj Ghorpade
Game theory is the study of strategic decision making between intelligent rational decision makers. It originated in economics and is used in fields like political science and psychology. Game theory analyzes interactions with both cooperative and non-cooperative games. Modern game theory began with John von Neumann's work on mixed strategy equilibria in two-person zero-sum games. Von Neumann and Oskar Morgenstern's 1944 book Theory of Games and Economic Behavior was influential in establishing game theory. Game theory has been widely applied, including in biology since the 1970s, and has helped explain behaviors in economics, politics and other fields.
This document provides an introduction to game theory. It discusses what game theory is, its essential features, and some key concepts in game theory including Nash equilibrium, backward induction, extensive form games, normal form games, mixed strategies, coordination games, zero-sum games, the prisoner's dilemma, chicken games, and repeated games. It also provides examples of applying game theory concepts to real-world situations such as the rivalry between Airbus and Boeing.
This document provides an overview of game theory concepts including its development, assumptions, classification of games, elements, significance, limitations, and methods for solving different types of games. Some key points:
- Game theory was developed in 1928 by John Von Neumann and Oscar Morgenstern to analyze decision-making involving two or more rational opponents.
- Games can be classified as two-person, n-person, zero-sum, non-zero-sum, pure-strategy, or mixed-strategy.
- Elements include the payoff matrix, dominance rules, optimal strategies, and the value of the game.
- Methods for solving games include using pure strategies if a saddle point exists, or mixed
This document provides an overview of game theory, including its founders John von Neumann and John Nash. Game theory is the study of strategic decision making among rational players where outcomes depend on the choices of all. It has applications in economics, politics, and biology. Key concepts discussed include Nash equilibrium, where no player benefits from changing strategies alone; the prisoner's dilemma game; and the tit-for-tat strategy of reciprocal cooperation and defection. The document outlines the assumptions, elements, and applications of game theory.
This document provides a brief introduction to game theory concepts, including normal form games, dominant strategies, and Nash equilibrium. It uses examples like the Prisoner's Dilemma and Cournot duopoly to illustrate these concepts. Normal form games represent strategic interactions through payoff matrices. Dominant strategies provide unambiguous best responses. Nash equilibrium is a prediction of strategies where no player benefits by deviating unilaterally. Multiple equilibria can exist in some games.
Game theory is a mathematical approach to modeling strategic interactions between rational decision-makers. It assumes humans seek the best outcomes and makes predictions based on payoff matrices showing players' rewards for different strategy combinations. Common applications include economics, politics, and analyzing conflict and cooperation situations like the Prisoner's Dilemma. Game theory also studies concepts like Nash equilibrium, mixed strategies, and evolutionary stable strategies.
Game theory is the study of strategic decision making where outcomes depend on the choices of multiple players. It originated in the 1920s and was popularized by John von Neumann. Game theory analyzes cooperative and non-cooperative games with various properties like the number of players, information available, and whether choices are simultaneous or sequential. Important concepts in game theory include Nash equilibrium, where no player can benefit by changing strategy alone, and prisoner's dilemma, where defecting dominates but collective cooperation yields higher payoffs. Game theory is now used widely in economics, politics, biology, and other fields involving interdependent actors.
Game theory is a branch of applied mathematics that analyzes strategic interactions between agents. It includes concepts like Nash equilibrium, mixed strategies, and coordination games. Game theory is used in economics, political science, biology, and other social sciences to model how individuals make decisions in strategic situations where outcomes depend on the decisions of others.
This presentation is an attempt to introduce Game Theory in one session. It's suitable for undergraduates. In practice, it's best used as a taster since only a portion of the material can be covered in an hour - topics can be chosen according to the interests of the class.
The main reference source used was 'Games, Theory and Applications' by L.C.Thomas. Further notes available at: http://bit.ly/nW6ULD
This document discusses a game theory example called the "grade game" to illustrate concepts like strategy, payoffs, and optimal choices. It presents a scenario where two students each privately choose to bid either A or B, and their grades depend on whether they match or differ from their paired partner's bid. The payoff matrix is shown, with outcomes like both getting a 7 if they match, or one getting a 10 and the other a 3 if they differ. It asks what students would choose, and discusses thinking about maximizing one's own outcome regardless of the partner, or trying to collude for higher joint grades. Overall it uses this game to introduce game theory ideas.
This document provides an introduction and overview of game theory. It describes key concepts in game theory including the elements of a game, complete and incomplete information, perfect and imperfect information, Nash equilibrium, simultaneous decisions, pure strategies and dominant strategies. It provides examples of classic games including the prisoner's dilemma, trade war, and battle of the sexes to illustrate these concepts. The prisoner's dilemma and trade war examples show how the games have dominant strategies that lead to a Nash equilibrium that is not optimal for either player.
This document provides an overview of game theory, including its history, basic concepts, types of strategies and equilibria, different types of games, and applications. It defines game theory as the mathematical analysis of conflict situations to determine optimal strategies. Key concepts explained include Nash equilibrium, mixed strategies, zero-sum games, repeated games, and sequential vs. simultaneous games. Applications of game theory discussed include economics, politics, biology, and artificial intelligence.
This document provides an overview of game theory and two-person zero-sum games. It defines key concepts such as players, strategies, payoffs, and classifications of games. It also describes the assumptions and solutions for pure strategy and mixed strategy games. Pure strategy games have a saddle point solution found using minimax and maximin rules. Mixed strategy games do not have a saddle point and require determining the optimal probabilities that players select each strategy.
Applications of game theory on event management Sameer Dhurat
Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science and biology.
In this presentation ,discussed regarding Application of game theory on Event Management with the help of Prisoner's Dilemma Game
Two-person zero-sum games involve two players where the total gains and losses sum to zero. Such games can be represented using a payoff matrix which shows the payoffs for each combination of strategies. The value of the game is the expected payoff when both players use optimal strategies. If a saddle point exists in the matrix, where the maximin and minimax values are equal, it represents the optimal strategies and value of the game.
Two-person zero-sum games involve two players where the total gains and losses sum to zero. Such games can be represented using a payoff matrix which shows the payoffs for each combination of strategies. The value of the game is the expected payoff when both players use optimal strategies. If a saddle point exists in the matrix, where the maximin and minimax values are equal, it represents the optimal strategies and value of the game.
Two-person zero-sum games involve two players where the total gains and losses sum to zero. Such games can be represented using a payoff matrix which shows the payoffs for each combination of strategies. The value of the game is the expected payoff when both players use optimal strategies. If a saddle point exists in the matrix, where the maximin and minimax values are equal, it represents the optimal strategies and value of the game.
Game theory is a mathematical tool used to describe strategic interactions between decision-makers. It involves players, strategies, and payoffs. A Nash equilibrium exists when no player can benefit by unilaterally changing their strategy given other players' strategies. The document provides examples of how game theory applies to wireless networks, including a forwarder's dilemma game about forwarding packets, a multiple access game about transmitting, and a joint packet forwarding game.
Game theory is a mathematical approach that analyzes strategic interactions between parties. It is used to understand situations where decision-makers are impacted by others' choices. A game has players, strategies, payoffs, and information. The Nash equilibrium predicts outcomes as the strategies where no player benefits by changing alone given others' choices. For example, in the Prisoner's Dilemma game about two suspects, confessing dominates remaining silent no matter what the other does, leading both to confess for a worse joint outcome than remaining silent.
This is a simple presentation on Game Theory in Network Security. I made it when I was searching for research points for my Master degree. Still searching for other research points. Any suggestions on research points in network security or network architecture? :)
Game theory is the study of strategic decision making. It involves analyzing interactions between players where the outcome for each player depends on the actions of all players. Key concepts in game theory include Nash equilibrium, where each player's strategy is the best response to the other players' strategies, and Prisoner's Dilemma, where the non-cooperative equilibrium results in a worse outcome for both players than if they had cooperated. Game theory is applied in economics, political science, biology, and many other fields to model strategic interactions.
This document provides an overview of game theory concepts taught in a university course. It defines game theory as the mathematics of human interactions and decision making. Key concepts discussed include Nash equilibrium, where each player adopts the optimal strategy given other players' strategies. Examples of applications are given in fields like economics, politics and biology. Different types of games and solutions concepts like mixed strategies are also introduced.
Game theory is the study of strategic decision making between two or more players under conditions of conflict or competition. A game involves players following a set of rules and receiving payoffs depending on the strategies chosen. Strategies include pure strategies that always select a particular action and mixed strategies that randomly select among pure strategies. The optimal strategies are those that maximize the minimum payoff for one player and minimize the maximum payoff for the other player. When the maximin and minimax values are equal, there is a saddle point representing the optimal strategies for both players.
This document provides an overview of mixed strategy Nash equilibrium and fixed point theorems. It defines mixed strategies as probability distributions over actions and mixed strategy Nash equilibrium as a profile of mixed strategies where no player can increase their expected payoff by deviating. Examples are given in 2x2 games. The document also reviews fixed point theorems like Kakutani and Brouwer that were used by Nash to prove the existence of mixed strategy Nash equilibria in finite games.
Game theory is used to analyze strategic decision-making situations involving multiple players under conditions of conflict or competition. It can help determine the best strategy for a firm given competitors' expected countermoves. Key concepts include pure and mixed strategies, optimal strategies, the value of the game, zero-sum and non-zero-sum games, and using payoff matrices to represent two-person zero-sum games and determine if a saddle point exists. When there is no saddle point, mixed strategies involving probabilities of different actions can determine the value of the game.
This document provides a brief introduction to game theory concepts, including normal form games, dominant strategies, and Nash equilibrium. It uses examples like the Prisoner's Dilemma and Cournot duopoly to illustrate these concepts. Normal form games represent strategic interactions through payoff matrices. Dominant strategies provide unambiguous best responses. Nash equilibrium is a prediction of strategies where no player benefits by deviating unilaterally. Multiple equilibria can exist in some games.
Game theory is a mathematical approach to modeling strategic interactions between rational decision-makers. It assumes humans seek the best outcomes and makes predictions based on payoff matrices showing players' rewards for different strategy combinations. Common applications include economics, politics, and analyzing conflict and cooperation situations like the Prisoner's Dilemma. Game theory also studies concepts like Nash equilibrium, mixed strategies, and evolutionary stable strategies.
Game theory is the study of strategic decision making where outcomes depend on the choices of multiple players. It originated in the 1920s and was popularized by John von Neumann. Game theory analyzes cooperative and non-cooperative games with various properties like the number of players, information available, and whether choices are simultaneous or sequential. Important concepts in game theory include Nash equilibrium, where no player can benefit by changing strategy alone, and prisoner's dilemma, where defecting dominates but collective cooperation yields higher payoffs. Game theory is now used widely in economics, politics, biology, and other fields involving interdependent actors.
Game theory is a branch of applied mathematics that analyzes strategic interactions between agents. It includes concepts like Nash equilibrium, mixed strategies, and coordination games. Game theory is used in economics, political science, biology, and other social sciences to model how individuals make decisions in strategic situations where outcomes depend on the decisions of others.
This presentation is an attempt to introduce Game Theory in one session. It's suitable for undergraduates. In practice, it's best used as a taster since only a portion of the material can be covered in an hour - topics can be chosen according to the interests of the class.
The main reference source used was 'Games, Theory and Applications' by L.C.Thomas. Further notes available at: http://bit.ly/nW6ULD
This document discusses a game theory example called the "grade game" to illustrate concepts like strategy, payoffs, and optimal choices. It presents a scenario where two students each privately choose to bid either A or B, and their grades depend on whether they match or differ from their paired partner's bid. The payoff matrix is shown, with outcomes like both getting a 7 if they match, or one getting a 10 and the other a 3 if they differ. It asks what students would choose, and discusses thinking about maximizing one's own outcome regardless of the partner, or trying to collude for higher joint grades. Overall it uses this game to introduce game theory ideas.
This document provides an introduction and overview of game theory. It describes key concepts in game theory including the elements of a game, complete and incomplete information, perfect and imperfect information, Nash equilibrium, simultaneous decisions, pure strategies and dominant strategies. It provides examples of classic games including the prisoner's dilemma, trade war, and battle of the sexes to illustrate these concepts. The prisoner's dilemma and trade war examples show how the games have dominant strategies that lead to a Nash equilibrium that is not optimal for either player.
This document provides an overview of game theory, including its history, basic concepts, types of strategies and equilibria, different types of games, and applications. It defines game theory as the mathematical analysis of conflict situations to determine optimal strategies. Key concepts explained include Nash equilibrium, mixed strategies, zero-sum games, repeated games, and sequential vs. simultaneous games. Applications of game theory discussed include economics, politics, biology, and artificial intelligence.
This document provides an overview of game theory and two-person zero-sum games. It defines key concepts such as players, strategies, payoffs, and classifications of games. It also describes the assumptions and solutions for pure strategy and mixed strategy games. Pure strategy games have a saddle point solution found using minimax and maximin rules. Mixed strategy games do not have a saddle point and require determining the optimal probabilities that players select each strategy.
Applications of game theory on event management Sameer Dhurat
Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science and biology.
In this presentation ,discussed regarding Application of game theory on Event Management with the help of Prisoner's Dilemma Game
Two-person zero-sum games involve two players where the total gains and losses sum to zero. Such games can be represented using a payoff matrix which shows the payoffs for each combination of strategies. The value of the game is the expected payoff when both players use optimal strategies. If a saddle point exists in the matrix, where the maximin and minimax values are equal, it represents the optimal strategies and value of the game.
Two-person zero-sum games involve two players where the total gains and losses sum to zero. Such games can be represented using a payoff matrix which shows the payoffs for each combination of strategies. The value of the game is the expected payoff when both players use optimal strategies. If a saddle point exists in the matrix, where the maximin and minimax values are equal, it represents the optimal strategies and value of the game.
Two-person zero-sum games involve two players where the total gains and losses sum to zero. Such games can be represented using a payoff matrix which shows the payoffs for each combination of strategies. The value of the game is the expected payoff when both players use optimal strategies. If a saddle point exists in the matrix, where the maximin and minimax values are equal, it represents the optimal strategies and value of the game.
Game theory is a mathematical tool used to describe strategic interactions between decision-makers. It involves players, strategies, and payoffs. A Nash equilibrium exists when no player can benefit by unilaterally changing their strategy given other players' strategies. The document provides examples of how game theory applies to wireless networks, including a forwarder's dilemma game about forwarding packets, a multiple access game about transmitting, and a joint packet forwarding game.
Game theory is a mathematical approach that analyzes strategic interactions between parties. It is used to understand situations where decision-makers are impacted by others' choices. A game has players, strategies, payoffs, and information. The Nash equilibrium predicts outcomes as the strategies where no player benefits by changing alone given others' choices. For example, in the Prisoner's Dilemma game about two suspects, confessing dominates remaining silent no matter what the other does, leading both to confess for a worse joint outcome than remaining silent.
This is a simple presentation on Game Theory in Network Security. I made it when I was searching for research points for my Master degree. Still searching for other research points. Any suggestions on research points in network security or network architecture? :)
Game theory is the study of strategic decision making. It involves analyzing interactions between players where the outcome for each player depends on the actions of all players. Key concepts in game theory include Nash equilibrium, where each player's strategy is the best response to the other players' strategies, and Prisoner's Dilemma, where the non-cooperative equilibrium results in a worse outcome for both players than if they had cooperated. Game theory is applied in economics, political science, biology, and many other fields to model strategic interactions.
This document provides an overview of game theory concepts taught in a university course. It defines game theory as the mathematics of human interactions and decision making. Key concepts discussed include Nash equilibrium, where each player adopts the optimal strategy given other players' strategies. Examples of applications are given in fields like economics, politics and biology. Different types of games and solutions concepts like mixed strategies are also introduced.
Game theory is the study of strategic decision making between two or more players under conditions of conflict or competition. A game involves players following a set of rules and receiving payoffs depending on the strategies chosen. Strategies include pure strategies that always select a particular action and mixed strategies that randomly select among pure strategies. The optimal strategies are those that maximize the minimum payoff for one player and minimize the maximum payoff for the other player. When the maximin and minimax values are equal, there is a saddle point representing the optimal strategies for both players.
This document provides an overview of mixed strategy Nash equilibrium and fixed point theorems. It defines mixed strategies as probability distributions over actions and mixed strategy Nash equilibrium as a profile of mixed strategies where no player can increase their expected payoff by deviating. Examples are given in 2x2 games. The document also reviews fixed point theorems like Kakutani and Brouwer that were used by Nash to prove the existence of mixed strategy Nash equilibria in finite games.
Game theory is used to analyze strategic decision-making situations involving multiple players under conditions of conflict or competition. It can help determine the best strategy for a firm given competitors' expected countermoves. Key concepts include pure and mixed strategies, optimal strategies, the value of the game, zero-sum and non-zero-sum games, and using payoff matrices to represent two-person zero-sum games and determine if a saddle point exists. When there is no saddle point, mixed strategies involving probabilities of different actions can determine the value of the game.
GAME THEORY NOTES FOR ECONOMICS HONOURS FOR ALL UNIVERSITIES BY SOURAV SIR'S ...SOURAV DAS
This document provides an overview of game theory concepts including definitions of key terms like mixed strategy, Nash equilibrium, payoff, perfect information, player, rationality, strategic form, strategy, zero-sum game, cooperative games, non-cooperative games, representation of games as normal form, pure vs mixed strategies, two-person zero-sum games, examples of zero-sum games, and using linear programming to determine optimal strategies in a game.
Optimization of Fuzzy Matrix Games of Order 4 X 3IJERA Editor
In this paper, we consider a solution for Fuzzy matrix game with fuzzy pay offs. The Solution of Fuzzy matrix games with pure strategies with maximin – minimax principle is discussed. A method takes advantage of the relationship between fuzzy sets and fuzzy matrix game theories can be offered for multicriteria decision making. Here, m x n pay off matrix is reduced to 4 x 3 pay off matrix.
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
The matrix game theory gives a mathematical background for dealing with competitive or antagonistic situations arise in many parts of real life. Matrix games have been extensively studied and successfully applied to many fields such as economics, business, management and e-commerce as well as advertising. This paper deals with two-person matrix games whose elements of pay-off matrix are fuzzy numbers. Then the corresponding matrix game has been converted into crisp game using defuzzification techniques. The value of the matrix game for each player is obtained by solving corresponding crisp game problems using the existing method. Finally, to illustrate the proposed methodology, a practical and realistic numerical example has been applied for different defuzzification methods and the obtained results have been compared
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
This document discusses solving fuzzy matrix games where the payoff elements are fuzzy numbers. It begins with definitions related to fuzzy sets and fuzzy numbers. A two-person zero-sum matrix game model is presented where the payoff matrix contains trapezoidal fuzzy numbers. The fuzzy game is converted to a crisp equivalent game using defuzzification techniques. Different defuzzification methods are applied to a numerical example and the results are compared. The key concepts of mixed strategies, maximin-minimax criteria and saddle points in fuzzy matrix games are also covered.
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
The matrix game theory gives a mathematical background for dealing with competitive or antagonistic situations arise in many parts of real life. Matrix games have been extensively studied and successfully applied to many fields such as economics, business, management and e-commerce as well as advertising. This paper deals with two-person matrix games whose elements of pay-off matrix are fuzzy numbers. Then the corresponding matrix game has been converted into crisp game using defuzzification techniques. The value of the matrix game for each player is obtained by solving corresponding crisp game problems using the existing method. Finally, to illustrate the proposed methodology, a practical and realistic numerical example has been applied for different defuzzification methods and the obtained results have been compared
A discussion of basic concepts from game theory, an incredibly useful lemma concerning auctions from mechanism design, and a discussion of TFNP, an interesting complexity class which captures search problems where an answer is guaranteed to exist, such as the problem of finding Nash equilibria in games
Abstract
In recent years there have been great strides in artificial intelligence (AI), with games often serving as challenge problems, benchmarks, and milestones for progress. Poker has served for decades as such a challenge problem. Past successes in such benchmarks, including poker, have been limited to two-player games. However, poker in particular is traditionally played with more than two players. Multiplayer games present fundamental additional issues beyond those in two-player games, and multiplayer poker is a recognized AI milestone. In this paper we present Pluribus, an AI that we show is stronger than top human professionals in six-player no-limit Texas hold’em poker, the most popular form of poker played by humans. - https://science.sciencemag.org/content/early/2019/07/10/science.aay2400
This document provides an overview of game theory concepts including:
- 2-player zero-sum games and minimax optimal strategies
- The minimax theorem which states that every 2-player zero-sum game has a value and optimal strategies for both players
- General-sum games and the concept of a Nash equilibrium as a stable pair of strategies where neither player benefits from deviating
- The proof of existence of Nash equilibria in general-sum games using Brouwer's fixed-point theorem
The document discusses Nash equilibrium, which is a solution concept in game theory where each player is making the best response given the other players' strategies. It provides an example of a simple game between two players, Tom and Sam, where choosing strategy A is the Nash equilibrium since neither player has an incentive to deviate. Mixed strategies are introduced where players randomize between different actions. An example game is used to illustrate finding the Nash equilibrium using mixed strategies. The document also discusses properties of Nash equilibria, including that every matrix game has at least one Nash equilibrium.
This document discusses game theory and provides examples of different types of games. It introduces the prisoner's dilemma game, which involves two players who must choose whether to cooperate with or betray each other. It also discusses finding Nash equilibria, including examples of mixed strategy equilibria in the Battle of the Sexes and Matching Pennies games. The document provides information on concepts such as dominant strategies, Pareto optimality, best responses, and expected utility in game theory.
This document summarizes research on extortion strategies in the Iterated Prisoner's Dilemma game. It introduces the Prisoner's Dilemma and describes how it can model situations like trench warfare. It explains how Press and Dyson showed that one player can unilaterally control the other player's payoff through a zero-determinant strategy. Using such a strategy, a player can extort their opponent by enforcing a relationship that gives themselves a higher payoff. The document demonstrates that an extorting player can always receive their maximum payoff, even against an adapting opponent who tries to change their strategy.
Beyond Nash Equilibrium - Correlated Equilibrium and Evolutionary Equilibrium Jie Bao
1. The document discusses several equilibrium concepts beyond Nash equilibrium including correlated equilibrium, evolutionary equilibrium, and Bayesian Nash equilibrium.
2. A correlated equilibrium is a generalization of Nash equilibrium where players' actions may be correlated based on a common signal, rather than being independent.
3. An evolutionary stable strategy is a strategy that, if adopted by a population, cannot be invaded by any alternative strategy. It is a refinement of Nash equilibrium from an evolutionary perspective.
This document provides an overview of game theory. It defines game theory as the study of how people interact and make decisions strategically, taking into account that each person's actions impact others. It discusses the history and key concepts of game theory, including players, strategies, payoffs, assumptions of rationality and perfect information. It provides examples of zero-sum and non-zero-sum games like the Prisoner's Dilemma. The document is intended to introduce game theory and its basic elements.
This document provides an overview of game theory. It defines game theory as the study of how people interact and make decisions in strategic situations, using mathematical models. It discusses the history and key concepts of game theory, including players, strategies, payoffs, assumptions of rationality and perfect information. It provides examples of zero-sum and non-zero-sum games like the Prisoner's Dilemma. The document also outlines the key elements of a game and different types of game theory, and discusses applications in economics, computer science, military strategy, biology and other fields.
1) Welfare economics and Pareto dominance are used to analyze outcomes at the aggregate level and understand social miscoordination.
2) Symmetric games have identical strategies for all players, making them easier to solve for Nash equilibria where everyone plays the same strategy.
3) Examples of symmetric games include Battle of the Sexes, Cournot duopoly, and Stag Hunt games with multiple players.
4) Level-k strategy models bounded rationality where players believe others make rational decisions up to a limited level of iterative reasoning.
This document outlines the key concepts in game theory. It introduces two-person zero-sum games and discusses optimal strategies, including the maximin and minimax principles. A game has a saddle point solution when the maximin value for one player equals the minimax value for the other player, indicating their optimal strategies and the game's value.
Similaire à Wiese heinrich2021 article-the_frequencyofconvergentgamesu (20)
Solution Manual For Financial Accounting, 8th Canadian Edition 2024, by Libby...Donc Test
Solution Manual For Financial Accounting, 8th Canadian Edition 2024, by Libby, Hodge, Verified Chapters 1 - 13, Complete Newest Version Solution Manual For Financial Accounting, 8th Canadian Edition by Libby, Hodge, Verified Chapters 1 - 13, Complete Newest Version Solution Manual For Financial Accounting 8th Canadian Edition Pdf Chapters Download Stuvia Solution Manual For Financial Accounting 8th Canadian Edition Ebook Download Stuvia Solution Manual For Financial Accounting 8th Canadian Edition Pdf Solution Manual For Financial Accounting 8th Canadian Edition Pdf Download Stuvia Financial Accounting 8th Canadian Edition Pdf Chapters Download Stuvia Financial Accounting 8th Canadian Edition Ebook Download Stuvia Financial Accounting 8th Canadian Edition Pdf Financial Accounting 8th Canadian Edition Pdf Download Stuvia
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
Vicinity Jobs’ data includes more than three million 2023 OJPs and thousands of skills. Most skills appear in less than 0.02% of job postings, so most postings rely on a small subset of commonly used terms, like teamwork.
Laura Adkins-Hackett, Economist, LMIC, and Sukriti Trehan, Data Scientist, LMIC, presented their research exploring trends in the skills listed in OJPs to develop a deeper understanding of in-demand skills. This research project uses pointwise mutual information and other methods to extract more information about common skills from the relationships between skills, occupations and regions.
Optimizing Net Interest Margin (NIM) in the Financial Sector (With Examples).pdfshruti1menon2
NIM is calculated as the difference between interest income earned and interest expenses paid, divided by interest-earning assets.
Importance: NIM serves as a critical measure of a financial institution's profitability and operational efficiency. It reflects how effectively the institution is utilizing its interest-earning assets to generate income while managing interest costs.
[4:55 p.m.] Bryan Oates
OJPs are becoming a critical resource for policy-makers and researchers who study the labour market. LMIC continues to work with Vicinity Jobs’ data on OJPs, which can be explored in our Canadian Job Trends Dashboard. Valuable insights have been gained through our analysis of OJP data, including LMIC research lead
Suzanne Spiteri’s recent report on improving the quality and accessibility of job postings to reduce employment barriers for neurodivergent people.
Decoding job postings: Improving accessibility for neurodivergent job seekers
Improving the quality and accessibility of job postings is one way to reduce employment barriers for neurodivergent people.
New Visa Rules for Tourists and Students in Thailand | Amit Kakkar Easy VisaAmit Kakkar
Discover essential details about Thailand's recent visa policy changes, tailored for tourists and students. Amit Kakkar Easy Visa provides a comprehensive overview of new requirements, application processes, and tips to ensure a smooth transition for all travelers.
5 Tips for Creating Standard Financial ReportsEasyReports
Well-crafted financial reports serve as vital tools for decision-making and transparency within an organization. By following the undermentioned tips, you can create standardized financial reports that effectively communicate your company's financial health and performance to stakeholders.
The Universal Account Number (UAN) by EPFO centralizes multiple PF accounts, simplifying management for Indian employees. It streamlines PF transfers, withdrawals, and KYC updates, providing transparency and reducing employer dependency. Despite challenges like digital literacy and internet access, UAN is vital for financial empowerment and efficient provident fund management in today's digital age.
Unlock Your Potential with NCVT MIS.pptxcosmo-soil
The NCVT MIS Certificate, issued by the National Council for Vocational Training (NCVT), is a crucial credential for skill development in India. Recognized nationwide, it verifies vocational training across diverse trades, enhancing employment prospects, standardizing training quality, and promoting self-employment. This certification is integral to India's growing labor force, fostering skill development and economic growth.
2. Dynamic Games and Applications
number of players and strategies has at least one MSNE (Nash [15,16]). This is not the case
for PSNEs.
Consider an n-player, m-strategy normal-form game and assume that players choose
their optimal strategy (facing previous optimal strategies of the opponents) in a clockwork
sequence—player 1 goes first, then player 2, etc. until its player 1’s turn again. We call a
game convergent, if starting from any initial strategy profile no player changes their strategy
under the described dynamic after a sufficiently large number of turns.
We describe such games by an n-partite graph with each node corresponding to a pure
strategy profile of the strategy choices of all but one player, and each edge corresponding to
the optimal strategy choice (best response). A PSNE corresponds to a shortest possible cycle
of length n.
In general, there are three types of games:
• Type A: Convergent games with a unique PSNE
• Type B: Convergent games with multiple PSNEs
• Type C: Non-convergent games
Type A games (for instance, the Prisoners’ Dilemma) are very easy to understand and
perfectly predictable. They converge to the PSNE. As we may re-arrange the strategies of the
players, Type B games are coordination games. An example of a Type C game is Matching
Pennies. Type B and Type C games have at least one MSNE.
We will investigate the likelihood of randomly created games that converge (Type A and
Type B) in the ensemble of games with a given number of players and a given number of
strategies available to each player. The frequencies can provide insights into predictability
and stability of equilibria in economic systems. For situations that are conveniently modelled
by low-dimensional (e.g. 2-player 2-strategy) games, predictability and stability properties
are often obvious. For more complicated biological interactions [8,9], bidding behaviour
[5], interactions on supply chains [3], trading behaviour in financial markets [4], or social
behaviour during a crisis (say the COVID-19 pandemic), this is different.
We will show that Type A and Type B games become less likely the more complex the
game is. Thus, modelling scenarios like climate change or financial market events with Type
A or Type B games would lead to misleading results. In spite of the involved difficulty, it
would be expedient to employ models that use Type C games.
1.1 Related Work
Several papers have considered aspects related to the number of PSNE in games with random
payoffs. We briefly consider the papers that dealt with random payoffs that are i.i.d. from a
continuous distribution.
Goldman [11] considered zero-sum 2-player games and showed that the probability of
having a PSNE goes to zero as the number of strategies grows. Goldberg et al. [10] considered
general 2-player games and showed that the probability of having at least one PSNE converges
to 1−exp(−1) as the number of strategies goes to infinity. Dresher [6] generalized this result
to the case of an arbitrary finite number of players. Powers [19] showed that, when the
number of strategies of at least two players goes to infinity, the distribution of the number of
PSNEs converges to Poisson(1). Stanford [20] derived an exact formula for the distribution of
the number of PSNEs in random games, and showed that for two-person symmetric games,
the number of symmetric and asymmetric PSNEs converges to a Poisson distribution [21].
McLennan [13] obtained a computationally implementable formula for the mean number of
Nash equilibria.
3. Dynamic Games and Applications
Alon et al. [1] studied the frequency of dominance-solvable games and obtained an exact
formula for the 2-player case. Dominance-solvable games are necessarily convergent, but
not vice versa, so we study a larger class of games (containing, for instance, coordination
games). The unique PSNE in Type A games are called Cournot stable; this class of games
was studied by Moulin [14].
Concerning the use of best-response structures as a tool to study convergence frequencies,
Pangallo et al. [17] and Pei and Takahaski [18] both obtained exact results for the frequency
of one or more PSNEs in the 2-player case. The authors in [12] use different methods to
bound the convergence frequency in multi-player games.
Finally, random games were studied in the context of theoretical biology, for instance, the
authors in [7] investigated the distribution of equilibria of an evolutionary dynamic.
1.2 Our Contribution
We introduce an n-partite graph describing the best responses of a game and use it to obtain
the frequency of randomly created games with a unique PSNE in the ensemble of n-player,
m-strategy games. These games are perfectly predictable. We then study games with more
than one PSNE, that are convergent under best-response dynamics, in which each player
successively chooses their optimal pure strategy. We show that convergent games with a
smaller number of PSNEs are more common than convergent games with a higher number of
PSNEs. We obtain an exact frequency for convergent 2-player games with any given number
of PSNEs. We finally highlight that for 2 players and less than 10 strategies, games with a
unique PSNE are more common than convergent games with multiple PSNEs, otherwise less
common.
2 Methods
2.1 Notation
A game with n ≥ 2 players and m ≥ 2 strategies available to each player is a tuple
(N, M, {ui }i∈N ) where N = {1, . . . , n} is the set of players, M = {1, . . . , m} the set
of strategies for each player, and ui : Mn → R a payoff function. A strategy profile
s = (s1, . . . , sn) ∈ Mn is a set of strategies for each player. An environment for player
i is a set s−i ∈ Mn−1 of strategies chosen by each player but i. A best response bi for player i
is a mapping from the set of environments of i to the set of non-empty subsets of i’s strategies
and is defined by
bi (s−i ) := arg maxsi ∈M ui (si , s−i ) .
A strategy profile s ∈ Mn is a pure strategy Nash equilibrium (PSNE) if for all i ∈ N and
all si ∈ M,
ui (s) ≥ ui (si , s−i ).
Equivalently, s ∈ Mn is a PSNE if for all i ∈ N and all si ∈ M, si ∈ bi (s−i ). A game
is non-degenerate, if for each player i and environment s−i , the best-response bi (s−i ) is a
singleton; we then write si = bi (s−i ). Similarly, a mixed strategy Nash equilibrium (MSNE)
is a strategy profile in mixed strategies.
4. Dynamic Games and Applications
Fig. 1 A 3-player, 2-strategy
game with one PSNE and the
corresponding 3-partite graph
representation. The best
responses corresponding to the
PSNE (I–IV–V) are highlighted
Pl. 3 V VI
I Pl. 2 III (0,0,0) (0,0,1)
Pl. 1 IV (1,1,1) (0,1,0)
II Pl. 2 III (1,0,1) (1,1,0)
IV (0,1,0) (1,0,1)
III-V III-VI IV-V IV-VI
I-V
I-VI
II-V
II-VI
I-III
I-IV
II-III
II-IV
2.2 Games as Graphs
The best-response structure of a game can be represented with a best-response digraph whose
vertex set is the set of strategy profiles Mn and whose edges are constructed as follows: for
each i ∈ N and each pair of distinct vertices s = (si , s−i ) and s = (s
i , s−i ), place a directed
edge from s to s if and only if s
i = bi (s−i ). There are edges only between strategy profiles
that differ in exactly one coordinate.
We now introduce an n-partite graph as an additional representation of the best responses
for a given fixed sequence of players. There is a total of nmn−1 nodes in n groups, each
group corresponding to a player and each node corresponding to an environment of a player.
At each node, a player chooses the best response; formally, the edges are constructed as
follows: for each pair (i, j) of players, where j moves directly after i, and each environment
s−i =
sj , s−i,− j
(where s−i,− j is s−i without the strategy choice of j), place a directed
edge from s−i to another environment s
− j = (s
i , s−i,− j ), if and only if
s
i = bi (s−i ). ()
As we can assume that games are non-degenerate, each node in a graph representing a game
has an out-degree of 1. A PSNE corresponds to a cycle of length n. Each player chooses among
m strategies at each node, thereby the total number of possible arrangements is mnmn−1
, each
equally likely.
We call the n-partite graph constructed as above but without the condition () the full
n-partite graph (see Fig. 5 (left)). Any n-partite graph corresponding to a given game is a
subgraph of the full n-partite graph. We will call a node free, if its out-degree is m, and fixed,
if its out-degree is 1.
Figure 1 shows a 3-player, 2-strategy game with one PSNE and the corresponding 3-partite
graph with playing sequence 1-2-3.
We could have relaxed the assumption of assuming only one specific sequence of play-
ers. However, PSNEs are stable under any playing sequence. Introducing random or other
playing sequences would make the digraph more complicated by adding more arcs without
changing the results. Dynamics where players choose their strategies simultaneously cannot
be represented by an n-partite graph.
5. Dynamic Games and Applications
3 Results
3.1 Type A: Convergent Games with a Unique PSNE
We generate n-player, m-strategy games at random by drawing mn tuples of payoffs from
a continuous distribution. This ensures that randomly created games are almost surely non-
degenerate. The exact type of distribution does not matter as long as payoffs are uncorrelated,
because the structure of the game only depends on which one in a set of values is the largest.
We leave games with correlated payoffs for future research, but results from the literature (
[17]) suggest that Type A games are less likely when payoffs are negatively correlated and
that Type B games (see below) become overwhelmingly likely with positive correlations.
Let pk
n,m denote the frequency of n-player, m-strategy convergent games with exactly k
PSNEs.
Theorem 1 The frequency of games with one unique PSNE in the ensemble is given by
p1
n,m = rn−1
+
m − 1
m − r
r
m
n−1
− 1
where r := m−1
mn + 1.
Note, that the frequency p1
n,m → 0 as the number of strategies or the number of players
goes to infinity, and that p1
n,m is decreasing in both n and m. For instance:
p1
2,m =
1
m
2 −
1
m
p1
3,m =
1
m2
3 −
3
m
+
3
m2
−
3
m3
+
1
m4
p1
4,m =
1
m3
4 −
4
m
+
6
m3
−
8
m4
+
2
m5
+
4
m6
−
6
m7
+
4
m8
−
1
m9
p1
5,m =
1
m4
5 −
5
m
+
10
m4
−
15
m5
+
5
m6
+
10
m8
−
20
m9
+
15
m10
−
5
m11
+
5
m12
−
10
m13
+
10
m14
−
5
m15
+
1
m16
.
Our result is different from the one in [20], where the frequency of games with one PSNE,
but possibly also mixed strategy Nash equilibria, converges to exp(−1) as the number of
strategies for at least two players goes to infinity.
Figure 2 shows the frequency of randomly created games with a unique PSNE. For com-
parison, we show frequencies obtained by numerically sampling over 500 randomly created
games with payoffs drawn from a normal distribution.
3.2 Type B: Convergent Games with Multiple PSNEs
We can bound the frequency of convergent games with more than one PSNE from above:
Theorem 2 For k1 k2, we have pk1
n,m pk2
n,m.
Theorems 1 and 2 imply that for every k, pk
n,m → 0 as the number of strategies or the
number of players goes to infinity. We computed for 3-player, 2-strategy games that p1
3,2 =
1984
4096 ≈ 48.43%, p2
3,2 = 828
4096 ≈ 20.21%, p3
3,2 = 56
4096 ≈ 1.37%, p4
3,2 = 2
4096 ≈ 0.049%.
6. Dynamic Games and Applications
Fig. 2 The frequency of randomly drawn games that have a unique PSNE
Fig. 3 The frequency of randomly drawn convergent 2-player games that have a given number of PSNEs
In two-player games, we can exactly state the frequency of games with k PSNEs.
Theorem 3 The frequency of 2-player, m-strategy convergent games with exactly k PSNEs
in the ensemble is given by
pk
2,m =
2m − k
m2k+2(k − 1)!
m!
(m − k)!
2
.
for k ≤ m, and is otherwise 0.
The frequency of drawing a 2-player convergent game (Type A or Type B) is then given by
m
k=1 pk
2,m, the frequency of Type B games only is
m
k=2 pk
2,m. Numerical evidence shows
that Type A games are more common than Type B games for m = 2, . . . , 9, and less common
for m ≥ 10.
Figures 3 and 4 show the frequency of randomly drawn convergent 2-player games that
have a given number of PSNEs.
7. Dynamic Games and Applications
Fig. 4 The frequency of randomly drawn convergent 2-player games that have a given number of PSNEs
where the frequency is log-scaled
4 Conclusion
We have investigated the frequency of games that are convergent under a best-response
dynamic, in which each player chooses their optimal pure strategy successively. Such games
may either be perfectly predictable, if they have a unique PSNE, or have multiple PSNEs.
We analytically computed the frequency of the first type by using a novel graph-theoretic
approach for describing games, and showed that if we let the number of players or the number
of strategies go to infinity, almost all games do not converge. We also showed that games
with a higher number of PSNEs are less common than games with a smaller number of
PSNEs. This calls the validity of simple models for complex scenarios into question. If a
simple scenario is to be modelled, a Type A game could be the right approach. However, for
complex scenarios, models based on Type A or Type B games can lead to misleading results.
Instead, techniques from agent-based modelling, network theory, or Bayesian statistics could
be employed.
For 2-player games, we gave an exact formula for the frequency of games with a given
number of PSNEs, and highlight that for less than 10 strategies, games with a unique PSNE
are more common than convergent games with multiple PSNEs, otherwise less common.
We believe that our graph-theoretic approach can generally be very useful to understand
complicated games. Extensions of this work would include finding the analytical frequency
of multi-player games with multiple pure Nash equilibria or with mixed Nash equilibria.
5 Proofs
Proof of Theorem 1 Consider the full n-partite graph for an n-player, m-strategy game. We
order the nodes in the following way: s−i s− j for different players i and j, if and only if
i j, and for the same player i, s−i s
−i under lexicographical ordering. Denote this full
n-partite graph by Gf = (V f, Ef), where V is the set of vertices and E is the set of edges.
The Laplacian matrix of a graph G = (V , E) without multiple edges and self-loops is
defined as the square matrix with side length |V | and
(L(G))i j =
⎧
⎪
⎨
⎪
⎩
δ+(i) if i = j
−1 if i = j, (i, j) ∈ E
0 if i = j, (i, j) /
∈ E
8. Dynamic Games and Applications
where δ+(i) is the out-degree of a node i. For Gf described above, the Laplacian matrix takes
the following form:
L Gf
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
D N1 0 0
0
0
0 Nn−1
S 0 0 D
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
where D, S, N1, . . . , Nn−1 are square matrices with side length mn−1 defined as follows:
– D = diag(m) is a diagonal matrix with m’s on the diagonal
– Nk = diag
K1, . . . , Kmk−1
is a blockmatrix with blockmatrices Kl on the diagonal,
where each Kl has side length mn−l and consists of m2 diagonal matrices diag(−1), each
with side length mn−l−1.
– S is more irregular,
(S)i j =
−1 if
i mod mn−2
=
j−1
m
0 otherwise.
For instance, in the case of 3-player, 2-strategy games, the Laplacian matrix corresponding
to Fig. 5 (left) is given by
L
Gf
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
2 0 0 0 −1 0 −1 0 0 0 0 0
0 2 0 0 0 −1 0 −1 0 0 0 0
0 0 2 0 −1 0 −1 0 0 0 0 0
0 0 0 2 0 −1 0 −1 0 0 0 0
0 0 0 0 2 0 0 0 −1 −1 0 0
0 0 0 0 0 2 0 0 −1 −1 0 0
0 0 0 0 0 0 2 0 0 0 −1 −1
0 0 0 0 0 0 0 2 0 0 −1 −1
−1 −1 0 0 0 0 0 0 2 0 0 0
0 0 −1 −1 0 0 0 0 0 2 0 0
−1 −1 0 0 0 0 0 0 0 0 2 0
0 0 −1 −1 0 0 0 0 0 0 0 2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
There are mn ways to choose the first PSNE, each fixing n nodes. Without loss of generality,
we choose the nodes where each player chooses their first strategy. We condense these n
nodes to a single node representing the PSNE, see Fig 5. The PSNE-node has an in-degree
of n(m − 1); we delete all outgoing edges. The resulting (n + 1)-partite graph consists of
nmn−1 −(n−1) nodes and will be denoted by Gc = (V c, Ec), where V c is the set of vertices
and Ec is the set of edges. All nodes except the PSNE-node are free.
We apply Kirchhoff’s theorem to Gc to get the number of spanning trees. This guaran-
tees that the game converges under clockwork best-response dynamics. Kirchhoff’s theorem
(applied to our problem) states that the number of spanning trees is the determinant of the
Laplacian matrix of Gc with the first row and column deleted, which corresponds to the
9. Dynamic Games and Applications
IV-VI
IV-V
III-VI
III-V
I-V
I-VI
II-V
II-VI
I-III
I-IV
II-III
II-IV
IV-VI
IV-V
III-VI
III-V
I-V
I-VI
II-V
II-VI
I-III
I-IV
II-III
II-IV
PSNE
Fig. 5 For 3-player, 2-strategy games the full graph Gf on the left and the condensed graph Gc on the right
PSNE-node. For a quadratic matrix A with side length n, we define
A to be the quadratic
matrix with side length (n − 1) obtained from A by deleting the first row and column.
For a general blockmatrix K =
A B
C D
, provided that A is invertible, we have
det K = det
D − C A−1
B
det A.
Applying this identity iteratively to
L (Gc) yields
det
L (Gc) = m
mn−1−1
(n−1)
· det
D −
1
mn−1
·
S ·
n−1
i=1
Ni
.
The matrix
S ·
i
Ni is given by
S ·
i
Ni
i j
= m − 1[1,mn−δ(i)−1]( j)
where
δ(i) := arg minp∈[1,n−1]
min
k∈[1,m p]
i − kmn−p−1
.
We simplify the matrix
D − 1
mn−1 ·
S ·
n−1
i=1
Ni by elementary row- and column-operations
to obtain a matrix A by the following algorithm:
10. Dynamic Games and Applications
Algorithm Simplifying
D − 1
mn−1 ·
S ·
n−1
i=1
Ni to obtain A
1. For p ∈ [1, . . . , n − 1]:
(a) For i ∈ [1, . . . , mn−1 − 1]:
i. If i = m p−1 or δ(i) = n − p, continue.
ii. Subtract the m p−1’s row from i.
2. For k ∈ [1, . . . , nm−1 − 1]:
(a) If for any p ∈ [0, . . . , n − 1], k|m p, continue.
(b) Add column k to column mn−δ(k)−1.
The determinant of the matrix A can be written as
det A = mmm−1−n
· det
A
for a matrix
A with side length (n − 1) and given by
A
i j
=
⎧
⎪
⎨
⎪
⎩
m + m−1
mn−1 − m−1
mi−1 i = j
m−1
mn−i+ j−1 − m−1
m j−1 i j
− m−1
m j−1 i j
Addingthei-thcolumnmultipliedby
− 1
m
tothe(i+1)-thcolumnfori = n−2, n−3, . . . , 1,
we get a matrix of the following form
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
D1 N 0 0
E2 D
0 0
N
En−1 0 0 D
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
where
D1 =
m − 1
mn−1
+ 1
Ei =
m − 1
mn−i
− (m − 1)
D = m
N = −
m − 1
mn
− 1.
To eliminate the N entries on the upper diagonal, we add the i-th row multiplied by
F := −
N
D
=
m−1
mn + 1
m
to the (i − 1)-th row for i = n − 1, . . . , 2. Then, the matrix is lower-triangular and the D1
entry is given by
D1 = D1 + F · E2 + · · · + Fn−2
· En−1
11. Dynamic Games and Applications
= D1 +
n−3
i=0
Fi+1
· Ei+2
= D1 +
n−3
i=0
Fi+1
m − 1
mn−i−2
−
n−3
i=0
Fi+1
(m − 1)
= D1 + F
m − 1
mn−2
n−3
i=0
(Fm)i
− F(m − 1)
n−3
i=0
Fi
= D1 + F
m − 1
mn−2
(Fm)n−2 − 1
Fm − 1
− F(m − 1)
Fn−2 − 1
F − 1
= m(r − 1) + m
rn−1
− r
−
r(m − 1)
r − m
r
m
n−2
− 1
+ 1
where r := m−1
mn + 1, and then
det
A = mn−2
·
D1.
Finally, the frequency of games with exactly one PSNE is given by
p1
n,m =
mn
mnmn−1
det
L (Gc) =
1
mmn−1−1
det A =
1
mn−1
det
A =
1
m
D1
where we have multiplied by the number of possible positions of the PSNE and divided by
the total number of possible arrangements. This completes the proof.
Proof of Theorem 2 Consider a full n-partite graph and assign k PSNEs, thereby fixing the
outgoing edges of kn nodes. We show that the number of possible realizations as a game
decreases, when adding another PSNE.
The number of ways we can add another PSNE (which is, in general, very complicated
to compute) is bounded from above by
mn−1 − k
m = mn − km, which is because there
are mn−1 − k free nodes for each player, each free node has an out-degree of m, and fixing
two nodes of an n-cycle fixes the remaining ones. However, adding a PSNE decreases the
number of possible realizations as a game by a factor of mn − 1, because the n nodes may
not form a cycle.
Induction over the number of added PSNEs completes the proof.
Proof of Theorem 3 It was shown in Austin [2] that the number of chromatic digraphs with
m nodes of each type, where each node has an out-degree one, and with a cycle of length 2k,
1 ≤ k ≤ m, is
(2m − k)
mm−k−1
2
m!
(m − k)!
2
.
Factoring out the number of ways to arrange k vertices on a cycle ((k + 1)!) and the total
number of possible arrangements (m2m), we get
pk
2,m =
2m − k
m2k+2(k − 1)!
m!
(m − k)!
2
.
This was given in Pangallo et al. [17] as a recursively defined formula.
12. Dynamic Games and Applications
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included in the
article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is
not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
References
1. Alon N, Rudov K, Yariv L (2020) Dominance solvability in random Games, preprint
2. Austin TL (1960) The enumeration of point labelled chromatic graphs and trees. Can J Math 12:535–545
3. Babu S, Mohan U (2018) An integrated approach to evaluating sustainability in supply chains using
evolutionary game theory. Comput Oper Res 89:269–283
4. Challet D, Marsili M, Zhang Y (2004) Minority games. Oxford University Press
5. Cheng L, Liu G, Huang H, Wang X, Chen Y, Zhang J, Meng A, Yang R, Yu T (2020) Equilibrium analysis
of general N-population multi-strategy games for generation-side long-term bidding: An evolutionary
game perspective, J Clean Prod, 276
6. Dresher M (1970) Probability of a pure equilibrium point in n-person games. J Comb Theory 8:134–145
7. Duong MH, Han TA (2016) Analysis of the expected density of internal equilibria in random evolutionary
multi-player multi-strategy games. J Math Biol 73(6–7):1727–1760
8. Duong MH, Han TA (2016) On the expected number of equilibria in a multi-player multi-strategy evolu-
tionary game. Dyn Games Appl 6:324–346
9. Gokhale CS, Traulsen A (2010) Evolutionary games in the multiverse. Proc Nat Acad Sci USA
107(12):5500–5504
10. Goldberg K, Goldman AJ, Newman M (1968) The probability of an equilibrium point. J Res Nat Bur
Standards Sect B 72B:93–101
11. Goldman AJ (1957) The probability of a saddlepoint. Am Math Monthly 64:729–730
12. Heinrich T et al. (2021) Best-response dynamics, playing sequences, and convergence to equilibrium in
random games. arXiv: 2101.04222
13. McLennan A (2005) The expected number of Nash equilibria of a normal form game. Econometrica
73(1):141–174
14. Moulin H (1984) Dominance solvability and Cournot stability. Math Soc Sci 7:83–102
15. Nash JF (1950) Equilibrium points in n-person games. Proc Nat Acad Sci USA 36:48–49
16. Nash JF (1951) Non-cooperative games. Ann Math 54(2):286–295
17. Pangallo M, Heinrich T, Farmer JD (2019) Best reply structure and equilibrium convergence in generic
games. Sci Adv 5(2):1–13
18. Pei T, Takahashi S (2019) Rationalizable strategies in random games. Games Econ Behav 118:110–125
19. Powers IY (1990) Limiting distributions of the number of pure strategy Nash equilibria in N-person
games. Internat J Game Theory 19(3):277–286
20. Stanford W (1995) A note on the probability of k pure Nash equilibria in matrix games. Games Econ
Behav 9(2):238–246
21. Stanford W (1996) The limit distribution of pure strategy Nash equilibria in symmetric bimatrix games.
Math Oper Res 21(3):726–733
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional affiliations.