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.
Game theory is the study of strategic decision making between interdependent parties. It analyzes situations where players make decisions that will impact outcomes for themselves and others. The document provides examples of classic game theory scenarios like the prisoner's dilemma and discusses concepts like dominant strategies, Nash equilibriums, and mixed strategies. It also presents a two-player "two-finger Morra game" to illustrate game theory principles.
Being an Iranian Woman Today イラン人女性として現代に生きるということParisa Mehran
In this presentation, I will talk about the challenges and opportunities of being an Iranian woman today. I will challenge existing stereotypes and misunderstandings about Iranian women by sharing personal narratives and by recounting stories of Iranian women in power and leadership.
この発表では、イラン人女性として現代を生きる上での困難と機会についてお話します。イラン人女性に対しては様々な偏見や誤解がありますが、今日は私の個人的な経験や、リーダーシップを持って力強く生きているイラン人女性たちのことをお話することで、ステレオタイプを打ち破っていきたいと思います。
The document discusses concepts in social network analysis including measuring networks through embedding measures and positions/roles of nodes. It covers network measures such as reciprocity, transitivity, clustering, density, and the E-I index. It also discusses positions like structural equivalence and regular equivalence and how to compute positional similarity through adjacency matrices.
This document discusses various interpolation methods:
- Newton's divided differences method uses finite differences to determine polynomial coefficients that fit scattered data points. Lagrange polynomials provide an alternative formulation.
- Spline interpolation smooths transitions by fitting different lower order polynomials to intervals between data points, maintaining continuity of derivatives at knots.
- Inverse interpolation finds the independent variable value corresponding to a given dependent variable value, using normal interpolation and root finding.
This document defines and provides examples of different types of graphs and digraphs. It defines a graph as an ordered pair of vertices and edges, and defines a digraph as a graph where edges have a direction from one vertex to another. It provides examples of simple, symmetric, asymmetric, complete, and balanced digraphs. It also defines in-degree, out-degree, and total degree of vertices in digraphs.
Game theory is a strategic decision-making process that models interactions between two or more players. It is commonly used in economics to analyze industries and competition between firms. Key concepts in game theory include games, players, strategies, payoffs, information sets, equilibriums, assumptions of rationality and payoff maximization. Common solution techniques include backwards induction, Nash equilibriums, mixed strategies, and minimax strategies. Examples discussed include prisoner's dilemmas, zero-sum games, and dominance.
Interactive Recommender Systems with Netflix and SpotifyChris Johnson
Interactive recommender systems enable the user to steer the received recommendations in the desired direction through explicit interaction with the system. In the larger ecosystem of recommender systems used on a website, it is positioned between a lean-back recommendation experience and an active search for a specific piece of content. Besides this aspect, we will discuss several parts that are especially important for interactive recommender systems, including the following: design of the user interface and its tight integration with the algorithm in the back-end; computational efficiency of the recommender algorithm; as well as choosing the right balance between exploiting the feedback from the user as to provide relevant recommendations, and enabling the user to explore the catalog and steer the recommendations in the desired direction.
In particular, we will explore the field of interactive video and music recommendations and their application at Netflix and Spotify. We outline some of the user-experiences built, and discuss the approaches followed to tackle the various aspects of interactive recommendations. We present our insights from user studies and A/B tests.
The tutorial targets researchers and practitioners in the field of recommender systems, and will give the participants a unique opportunity to learn about the various aspects of interactive recommender systems in the video and music domain. The tutorial assumes familiarity with the common methods of recommender systems.
Game theory is the study of strategic decision making between interdependent parties. It analyzes situations where players make decisions that will impact outcomes for themselves and others. The document provides examples of classic game theory scenarios like the prisoner's dilemma and discusses concepts like dominant strategies, Nash equilibriums, and mixed strategies. It also presents a two-player "two-finger Morra game" to illustrate game theory principles.
Being an Iranian Woman Today イラン人女性として現代に生きるということParisa Mehran
In this presentation, I will talk about the challenges and opportunities of being an Iranian woman today. I will challenge existing stereotypes and misunderstandings about Iranian women by sharing personal narratives and by recounting stories of Iranian women in power and leadership.
この発表では、イラン人女性として現代を生きる上での困難と機会についてお話します。イラン人女性に対しては様々な偏見や誤解がありますが、今日は私の個人的な経験や、リーダーシップを持って力強く生きているイラン人女性たちのことをお話することで、ステレオタイプを打ち破っていきたいと思います。
The document discusses concepts in social network analysis including measuring networks through embedding measures and positions/roles of nodes. It covers network measures such as reciprocity, transitivity, clustering, density, and the E-I index. It also discusses positions like structural equivalence and regular equivalence and how to compute positional similarity through adjacency matrices.
This document discusses various interpolation methods:
- Newton's divided differences method uses finite differences to determine polynomial coefficients that fit scattered data points. Lagrange polynomials provide an alternative formulation.
- Spline interpolation smooths transitions by fitting different lower order polynomials to intervals between data points, maintaining continuity of derivatives at knots.
- Inverse interpolation finds the independent variable value corresponding to a given dependent variable value, using normal interpolation and root finding.
This document defines and provides examples of different types of graphs and digraphs. It defines a graph as an ordered pair of vertices and edges, and defines a digraph as a graph where edges have a direction from one vertex to another. It provides examples of simple, symmetric, asymmetric, complete, and balanced digraphs. It also defines in-degree, out-degree, and total degree of vertices in digraphs.
Game theory is a strategic decision-making process that models interactions between two or more players. It is commonly used in economics to analyze industries and competition between firms. Key concepts in game theory include games, players, strategies, payoffs, information sets, equilibriums, assumptions of rationality and payoff maximization. Common solution techniques include backwards induction, Nash equilibriums, mixed strategies, and minimax strategies. Examples discussed include prisoner's dilemmas, zero-sum games, and dominance.
Interactive Recommender Systems with Netflix and SpotifyChris Johnson
Interactive recommender systems enable the user to steer the received recommendations in the desired direction through explicit interaction with the system. In the larger ecosystem of recommender systems used on a website, it is positioned between a lean-back recommendation experience and an active search for a specific piece of content. Besides this aspect, we will discuss several parts that are especially important for interactive recommender systems, including the following: design of the user interface and its tight integration with the algorithm in the back-end; computational efficiency of the recommender algorithm; as well as choosing the right balance between exploiting the feedback from the user as to provide relevant recommendations, and enabling the user to explore the catalog and steer the recommendations in the desired direction.
In particular, we will explore the field of interactive video and music recommendations and their application at Netflix and Spotify. We outline some of the user-experiences built, and discuss the approaches followed to tackle the various aspects of interactive recommendations. We present our insights from user studies and A/B tests.
The tutorial targets researchers and practitioners in the field of recommender systems, and will give the participants a unique opportunity to learn about the various aspects of interactive recommender systems in the video and music domain. The tutorial assumes familiarity with the common methods of recommender systems.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
This document provides an introduction to logistic regression. It begins by explaining how linear regression is not suitable for classification problems and introduces the logistic function which maps linear regression output between 0 and 1. This probability value can then be used for classification by setting a threshold of 0.5. The logistic function models the odds ratio as a linear function, allowing logistic regression to be used for binary classification. It can also be extended to multiclass classification problems. The document demonstrates how logistic regression finds a decision boundary to separate classes and how its syntax works in scikit-learn using common error metrics to evaluate performance.
1. The document defines different types of graphs including simple graphs, multigraphs, pseudographs, directed graphs, and mixed graphs. It provides examples of how graphs can model real-world networks and relationships.
2. Graph terminology is defined, including adjacent vertices, neighborhoods, degrees of vertices, and the handshaking theorem. Examples show how to calculate degrees and neighborhoods in sample graphs.
3. Directed graphs are discussed, defining in-degrees and out-degrees of vertices in directed graphs.
National power is a key component of international politics and refers to a nation's ability to secure its national interests and objectives. It involves the capacity to use force or the threat of force over other nations. National power is a combination of military power, economic power, and psychological power. It is derived from various elements including a nation's geography, natural resources, population, economic development, technology, military capabilities, and other factors. Nations can exercise national power through methods like persuasion, reward, punishment, manipulation, and force in their interactions with other countries.
PCA is a dimensionality reduction technique that uses linear transformations to project high-dimensional data onto a lower-dimensional space while retaining as much information as possible. It works by identifying patterns in data and expressing the data in such a way as to highlight their similarities and differences. Specifically, PCA uses linear combinations of the original variables to extract the most important patterns from the data in the form of principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible.
Binary Class and Multi Class Strategies for Machine LearningPaxcel Technologies
This presentation discusses the following -
Possible strategies to follow when working on a new machine learning problem.
The common problems with classifiers (how to detect them and eliminate them).
Popular approaches on how to use binary classifiers to problems with multi class classification.
This document discusses different types of matchings in graphs. A matching is a set of edges without common vertices. A maximum matching contains the largest possible number of edges. A maximal matching is one where no additional edges can be added without violating the matching property. A perfect matching is where every vertex is incident to exactly one edge, making it both maximum and maximal. The document provides definitions and properties of different matchings in graphs.
1. The ultimatum game involves two players dividing a fixed amount, with the first player proposing a division and the second deciding to accept or reject.
2. Early experiments found people offered more, around 40-50% of the amount, inconsistent with the rational prediction of minimal offers being accepted.
3. Further experiments explored factors like fairness, rejection fears, and information in bargaining, finding both explanations valid and information promoting fairness.
I made a PPT about a popular blockchain game, evaluated its business model, game mechanism, target customer, and identified the moats of the blockchain game, and researched how its underlying technology, Unreal Engine 5, enhanced the overall gaming experience.
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices and edges connecting pairs of vertices. There are many types of graphs including trees, which are connected acyclic graphs. Spanning trees are subgraphs of a graph that connect all vertices using the minimum number of edges. Key concepts in graph theory include paths, connectedness, cycles, and isomorphism between graphs.
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
It includes:
Introduction to Graphs
Applications
Graph representation
Graph terminology
Graph operations
Adding vertex and edge in Adjacency matrix representation using C++ program
Adjacency List implementation in C++
Homework Problems
References
El documento presenta información sobre la resolución de ecuaciones diofánticas. Explica que se requieren 4 pasos: 1) calcular el MCD de los coeficientes, 2) comprobar si la ecuación tiene soluciones enteras, 3) hallar la solución particular, y 4) determinar la solución general. Luego, resuelve 2 ejemplos de ecuaciones diofánticas siguiendo estos pasos.
Soft power: A conceptual appraisal of the power of attractionFidel525104
This lecture covers discussion surrounding the concept of power, the place of soft power within broader discussions about power, and how soft power is operationalized.
This document provides information about solving systems of linear equations through various methods such as graphing, substitution, and elimination. It defines what a linear system is and explains the concepts of consistent and inconsistent systems. Graphing is discussed as a way to find the point where two lines intersect. The substitution and elimination methods are described step-by-step with examples shown of using each method to solve sample systems of equations. Additional topics covered include slope, matrix notation, and an example of using a matrix to perform a Hill cipher encryption on a short plaintext message.
This document provides an overview of the Naive Bayes classifier machine learning algorithm. It begins by introducing Bayesian methods and Bayes' theorem. It then explains the basic probability formulas used in Naive Bayes. The document demonstrates how to calculate the probability that a patient has cancer using Bayes' theorem. It describes how Naive Bayes finds the maximum a posteriori hypothesis and deals with issues like parameter estimation and conditional independence. Finally, it works through an example predicting whether to play tennis using a sample training dataset.
This document provides an overview of linear equations and how to solve them. It defines key characteristics of linear equations, such as variables only having an exponent of 1 and no terms being multiplied together. Various examples of linear and non-linear equations are shown. The document then explains that solving linear equations involves finding the value of the variable that makes both sides equal. It provides step-by-step instructions for solving one-step equations and works through examples of solving multi-step equations. Students are given a checkpoint to practice solving equations on their own before moving to an independent practice section.
This document discusses generative adversarial networks (GANs) and the LAPGAN model. It explains that GANs use two neural networks, a generator and discriminator, that compete against each other. The generator learns to generate fake images to fool the discriminator, while the discriminator learns to distinguish real from fake images. LAPGAN improves upon GANs by using a Laplacian pyramid to decompose images into multiple scales, with separate generator and discriminator networks for each scale. This allows LAPGAN to generate sharper images by focusing on edges and conditional information at each scale.
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 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.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
This document provides an introduction to logistic regression. It begins by explaining how linear regression is not suitable for classification problems and introduces the logistic function which maps linear regression output between 0 and 1. This probability value can then be used for classification by setting a threshold of 0.5. The logistic function models the odds ratio as a linear function, allowing logistic regression to be used for binary classification. It can also be extended to multiclass classification problems. The document demonstrates how logistic regression finds a decision boundary to separate classes and how its syntax works in scikit-learn using common error metrics to evaluate performance.
1. The document defines different types of graphs including simple graphs, multigraphs, pseudographs, directed graphs, and mixed graphs. It provides examples of how graphs can model real-world networks and relationships.
2. Graph terminology is defined, including adjacent vertices, neighborhoods, degrees of vertices, and the handshaking theorem. Examples show how to calculate degrees and neighborhoods in sample graphs.
3. Directed graphs are discussed, defining in-degrees and out-degrees of vertices in directed graphs.
National power is a key component of international politics and refers to a nation's ability to secure its national interests and objectives. It involves the capacity to use force or the threat of force over other nations. National power is a combination of military power, economic power, and psychological power. It is derived from various elements including a nation's geography, natural resources, population, economic development, technology, military capabilities, and other factors. Nations can exercise national power through methods like persuasion, reward, punishment, manipulation, and force in their interactions with other countries.
PCA is a dimensionality reduction technique that uses linear transformations to project high-dimensional data onto a lower-dimensional space while retaining as much information as possible. It works by identifying patterns in data and expressing the data in such a way as to highlight their similarities and differences. Specifically, PCA uses linear combinations of the original variables to extract the most important patterns from the data in the form of principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible.
Binary Class and Multi Class Strategies for Machine LearningPaxcel Technologies
This presentation discusses the following -
Possible strategies to follow when working on a new machine learning problem.
The common problems with classifiers (how to detect them and eliminate them).
Popular approaches on how to use binary classifiers to problems with multi class classification.
This document discusses different types of matchings in graphs. A matching is a set of edges without common vertices. A maximum matching contains the largest possible number of edges. A maximal matching is one where no additional edges can be added without violating the matching property. A perfect matching is where every vertex is incident to exactly one edge, making it both maximum and maximal. The document provides definitions and properties of different matchings in graphs.
1. The ultimatum game involves two players dividing a fixed amount, with the first player proposing a division and the second deciding to accept or reject.
2. Early experiments found people offered more, around 40-50% of the amount, inconsistent with the rational prediction of minimal offers being accepted.
3. Further experiments explored factors like fairness, rejection fears, and information in bargaining, finding both explanations valid and information promoting fairness.
I made a PPT about a popular blockchain game, evaluated its business model, game mechanism, target customer, and identified the moats of the blockchain game, and researched how its underlying technology, Unreal Engine 5, enhanced the overall gaming experience.
Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices and edges connecting pairs of vertices. There are many types of graphs including trees, which are connected acyclic graphs. Spanning trees are subgraphs of a graph that connect all vertices using the minimum number of edges. Key concepts in graph theory include paths, connectedness, cycles, and isomorphism between graphs.
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
It includes:
Introduction to Graphs
Applications
Graph representation
Graph terminology
Graph operations
Adding vertex and edge in Adjacency matrix representation using C++ program
Adjacency List implementation in C++
Homework Problems
References
El documento presenta información sobre la resolución de ecuaciones diofánticas. Explica que se requieren 4 pasos: 1) calcular el MCD de los coeficientes, 2) comprobar si la ecuación tiene soluciones enteras, 3) hallar la solución particular, y 4) determinar la solución general. Luego, resuelve 2 ejemplos de ecuaciones diofánticas siguiendo estos pasos.
Soft power: A conceptual appraisal of the power of attractionFidel525104
This lecture covers discussion surrounding the concept of power, the place of soft power within broader discussions about power, and how soft power is operationalized.
This document provides information about solving systems of linear equations through various methods such as graphing, substitution, and elimination. It defines what a linear system is and explains the concepts of consistent and inconsistent systems. Graphing is discussed as a way to find the point where two lines intersect. The substitution and elimination methods are described step-by-step with examples shown of using each method to solve sample systems of equations. Additional topics covered include slope, matrix notation, and an example of using a matrix to perform a Hill cipher encryption on a short plaintext message.
This document provides an overview of the Naive Bayes classifier machine learning algorithm. It begins by introducing Bayesian methods and Bayes' theorem. It then explains the basic probability formulas used in Naive Bayes. The document demonstrates how to calculate the probability that a patient has cancer using Bayes' theorem. It describes how Naive Bayes finds the maximum a posteriori hypothesis and deals with issues like parameter estimation and conditional independence. Finally, it works through an example predicting whether to play tennis using a sample training dataset.
This document provides an overview of linear equations and how to solve them. It defines key characteristics of linear equations, such as variables only having an exponent of 1 and no terms being multiplied together. Various examples of linear and non-linear equations are shown. The document then explains that solving linear equations involves finding the value of the variable that makes both sides equal. It provides step-by-step instructions for solving one-step equations and works through examples of solving multi-step equations. Students are given a checkpoint to practice solving equations on their own before moving to an independent practice section.
This document discusses generative adversarial networks (GANs) and the LAPGAN model. It explains that GANs use two neural networks, a generator and discriminator, that compete against each other. The generator learns to generate fake images to fool the discriminator, while the discriminator learns to distinguish real from fake images. LAPGAN improves upon GANs by using a Laplacian pyramid to decompose images into multiple scales, with separate generator and discriminator networks for each scale. This allows LAPGAN to generate sharper images by focusing on edges and conditional information at each scale.
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 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.
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.
The document discusses games with incomplete information and how they are modeled. It notes:
- Games in the real world often have incomplete information where players do not know each other's payoffs or strategies.
- Harsanyi's approach models incomplete information using random variables for each player's preferences that are privately observed but have a commonly known probability distribution. This transforms incomplete information into imperfect information.
- A Bayesian game formally represents this setting, where a player's utility depends on their type drawn from a probability distribution. A Bayesian Nash equilibrium requires optimal strategies given beliefs over other players' types.
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.
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
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
Game theory is the mathematical modeling of strategic decision making between rational actors. Standard game theory assumes all players are rational and have complete information. However, criticisms note real world games are complex and player rationality is not guaranteed. The Prisoner's Dilemma shows the Nash equilibrium is not necessarily Pareto optimal. In the Battle of the Sexes, there are multiple Nash equilibria but the outcome cannot be predicted with certainty as it is a one-shot, non-sequential game. Iterated elimination of strictly dominated strategies is one method to solve games by removing irrational strategies.
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.
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.
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 provides an overview of key concepts in game theory, including:
1. It defines dominance, best response, Nash equilibrium, and Pareto dominance.
2. It gives an example game to illustrate dominance and best responses, showing that strategy C dominates R for player 2.
3. Nash equilibrium is defined as a strategy profile where each player's choice is a best response to the other players' choices. The battle of the sexes game is used to show multiple Nash equilibria can exist.
This document provides an overview of game theory concepts including strategic form games, extensive form games, dominated and dominant strategies, Nash equilibrium, prisoner's dilemma, Bayesian games, Bayesian-Nash equilibrium, and Schelling points. It discusses how these concepts are relevant for analyzing behaviors and equilibria on blockchain platforms, noting that payoffs must represent all gains to participants rather than just tokens.
Game theory is a strategic decision-making process that models interactions between two or more players. It is commonly used in economics to analyze industries and competition between firms. Key concepts in game theory include games, players, strategies, payoffs, information sets, equilibriums, assumptions of rationality and payoff maximization. Common applications discussed are the prisoner's dilemma, zero-sum games, dominant and dominated strategies, Nash equilibriums, mixed strategies, and minimax strategies.
OR PPT 280322 maximin final - nikhil tiwari.pptxVivekSaurabh7
Maximin and Minimax strategies are approaches used in game theory to determine the optimal strategy for players. Maximin selects the strategy that maximizes the minimum possible payoff, while Minimax selects the strategy that minimizes the maximum possible loss. These strategies are applied in zero-sum games under the assumptions of rational opponents. The strategies can be modeled using decision trees and algorithms like Minimax to analyze games like Tic Tac Toe and determine the best move.
This document provides an overview of game theory concepts including its assumptions, classifications, elements, significance, and limitations. It also describes methods for solving different types of games such as the prisoner's dilemma, 2-person zero-sum games, and pure strategy games. Game theory analyzes strategic decision making among interdependent parties and can provide insights into situations involving conflict or competition between rational opponents.
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.
The document provides an introduction to game theory and its applications in computer networks. It discusses key concepts in game theory such as games, strategies, outcomes, preferences and equilibria. It also covers solution concepts for different types of games including dominance, saddle points, mixed strategies and Nash equilibrium. Examples of applications discussed include routing games, congestion control games and wireless games.
Nash equilibrium and Pareto optimality are discussed and compared using examples from game theory. In the Battle of Sexes game, the Nash equilibria coincide with the Pareto efficient outcomes. However, in other games like the Prisoner's Dilemma and the Tragedy of the Commons, the Nash equilibria are not Pareto optimal and there is a possibility of Pareto improvement by playing different strategies. Braess's Paradox is also discussed where adding more resources to a network can paradoxically decrease overall performance at equilibrium due to drivers taking suboptimal routes.
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2. Nau: Game Theory 2
Introduction
In Chapter 6 we looked at 2-player perfect-information zero-sum games
We’ll now look at games that might have one or more of the following:
> 2 players
imperfect information
nonzero-sum outcomes
3. Prisoner’s Dilemma
Nau: Game Theory 3
The Prisoner’s Dilemma
Scenario: the police have arrested two suspects for a crime.
They tell each prisoner they’ll reduce his/her prison sentence if he/she
betrays the other prisoner.
Each prisoner must choose between two actions:
cooperate with the other prisoner, i.e., don’t betray him/her
defect (betray the other prisoner).
Payoff = – (years in prison):
Each player has only two strategies,
each of which is a single action
Non-zero-sum
Imperfect information: neither player knows the other’s move until after
both players have moved
Agent 2
Agent 1
C D
C –2, –2 –5, 0
D 0, –5 –4, –4
4. Nau: Game Theory 4
The Prisoner’s Dilemma
Add 5 to each payoff, so that the numbers are all ≥ 0
These payoffs encode the same preferences
Prisoner’s Dilemma:
Agent 2
Agent 1
Note: the book represents payoff matrices in a non-standard way
It puts Agent 1 where I have Agent 2, and vice versa
C D
C 3, 3 0, 5
D 5, 0 1, 1
Prisoner’s Dilemma:
Agent 2
Agent 1
C D
C –2, –2 –5, 0
D 0, –5 –4, –4
5. Nau: Game Theory 5
How to reason about games?
In single-agent decision theory, look at an optimal strategy
Maximize the agent’s expected payoff in its environment
With multiple agents, the best strategy depends on others’ choices
Deal with this by identifying certain subsets of outcomes called solution
concepts
Some solution concepts:
Dominant strategy equilibrium
Pareto optimality
Nash equilibrium
6. Nau: Game Theory 6
Strategies
Suppose the agents agent 1, agent 2, …, agent n
For each i, let Si = {all possible strategies for agent i}
si will always refer to a strategy in Si
A strategy profile is an n-tuple S = (s1, …, sn), one strategy for each agent
Utility Ui(S) = payoff for agent i if the strategy profile is S
si strongly dominates si' if agent i always does better with si than si'
si weakly dominates si' if agent i never does worse with si than si', and
there is at least one case where agent i does better with si than si',
7. Nau: Game Theory 7
Dominant Strategy Equilibrium
si is a (strongly, weakly) dominant strategy if it (strongly, weakly)
dominates every si' ∈ Si
Dominant strategy equilibrium:
A set of strategies (s1, …, sn) such that each si is dominant for agent i
Thus agent i will do best by using si rather than a different strategy,
regardless of what strategies the other players use
In the prisoner’s dilemma, there is one dominant strategy equilibrium:
both players defect
Prisoner’s Dilemma:
Agent 2
Agent 1
C D
C 3, 3 0, 5
D 5, 0 1, 1
8. Nau: Game Theory 8
Pareto Optimality
Strategy profile S Pareto dominates a strategy profile Sʹ′ if
no agent gets a worse payoff with S than with Sʹ′,
i.e., Ui(S) ≥ Ui(Sʹ′) for all i ,
at least one agent gets a better payoff with S than with Sʹ′,
i.e., Ui(S) > Ui(Sʹ′) for at least one i
Strategy profile s is Pareto optimal, or strictly Pareto efficient, if there’s
no strategy s' that Pareto dominates s
Every game has at least one Pareto optimal profile
Always at least one Pareto optimal profile in which the strategies are
pure
9. Prisoner’s Dilemma
Nau: Game Theory 9
Example
The Prisoner’s Dilemma
(C,C) is Pareto optimal
No profile gives both players
a higher payoff
(D,C) is Pareto optimal
No profile gives player 1 a higher payoff
(D,C) is Pareto optimal - same argument
(D,D) is Pareto dominated by (C,C)
Agent 2
Agent 1
But ironically, (D,D) is the dominant strategy equilibrium
C D
C 3, 3 0, 5
D 5, 0 1, 1
10. Nau: Game Theory 10
Pure and Mixed Strategies
Pure strategy: select a single action and play it
Each row or column of a payoff matrix represents both an action and a
pure strategy
Mixed strategy: randomize over the set of available actions according to
some probability distribution
Let Ai = {all possible actions for agent i}, and ai be any action in Ai
si (aj ) = probability that action aj will be played under mixed strategy si
The support of si is
support(si) = {actions in Ai that have probability > 0 under si}
A pure strategy is a special case of a mixed strategy
support consists of a single action
Fully mixed strategy: every action has probability > 0
i.e., support(si) = Ai
11. Nau: Game Theory 11
Expected Utility
A payoff matrix only gives payoffs for pure-strategy profiles
Generalization to mixed strategies uses expected utility
Let S = (s1, …, sn) be a profile of mixed strategies
For every action profile (a1, a2, …, an), multiply its probability and its
utility
• Ui (a1, …, an) s1(a1) s2(a2) … sn(an)
The expected utility for agent i is
€
Σ s1 a1 ( ) s2 a2 ( ) … sn an ( )
Ui s1,…, sn ( ) = Ui a1,…,an ( )
(a1 ,…,an )∈A
12. Nau: Game Theory 12
Best Response
Some notation:
If S = (s1, …, sn) is a strategy profile, then S−i = (s1, …, si−1, si+1, …, sn),
• i.e., S–i is strategy profile S without agent i’s strategy
If si' is any strategy for agent i, then
• (si' , S−i ) = (s1, …, si−1, si', si+1, …, sn)
Hence (si , S−i ) = S
si is a best response to S−i if
Ui (si , S−i ) ≥ Ui (si', S−i ) for every strategy si' available to agent i
si is a unique best response to S−i if
Ui (si , S−i ) > Ui (si', S−i ) for every si' ≠ si
13. Nash Equilibrium
A strategy profile s = (s1, …, sn) is a Nash equilibrium if for every i,
Nau: Game Theory 13
si is a best response to S−i , i.e., no agent can do
better by unilaterally changing his/her strategy
Theorem (Nash, 1951): Every game with a finite number of agents and
action profiles has at least one Nash equilibrium
In the Prisoner’s Dilemma, (D,D)
is a Nash equilibrium
If either agent unilaterally switches
to a different strategy, his/her
expected utility goes below 1
A dominant strategy equilibrium is
always a Nash equilibrium
Prisoner’s Dilemma
Agent 2
Agent 1
C D
C 3, 3 0, 5
D 5, 0 1, 1
14. Nau: Game Theory 14
Battle of the Sexes
Example
Two agents need to
coordinate their actions, but
they have different preferences
Original scenario:
• husband prefers football
• wife prefers opera
Another scenario:
Husband
Wife
• Two nations must act together to deal with an international crisis
• They prefer different solutions
This game has two pure-strategy Nash equilibria (circled above)
and one mixed-strategy Nash equilibrium
How to find the mixed-strategy Nash equilibrium?
Opera Football
Opera 2, 1 0, 0
Football 0, 0 1, 2
Nash equilibria
15. Finding Mixed-Strategy Equilibria
Nau: Game Theory 15
Generally it’s tricky to compute mixed-strategy equilibria
But easy if we can identify the support of the equilibrium strategies
Suppose a best response to S–i is a mixed strategy s whose support
includes ≥ 2 actions
Then every action a in support(s) must have the same expected utility
Ui(a,S–i)
• If some action a* in support(s) had a higher expected utility than
the others, then it would be a better response than s
Thus any mixture of the actions in support(s) is a best response
16. Battle of the Sexes
Suppose both agents randomize, and the husband’s mixed strategy sh is
Opera Football
Opera 2, 1 0, 0
Football 0, 0 1, 2
Nau: Game Theory 16
sh(Opera) = p; sh(Football) = 1 – p
Expected utilities of the wife’s actions:
Uw(Football, sh) = 0p + 1(1 − p)
Uw(Opera, sh) = 2p
Husband
Wife
If the wife mixes between her two actions, they must have the same
expected utility
If one of the actions had a better expected utility, she’d do better with a
pure strategy that always used that action
Thus 0p + 1(1 – p) = 2p, so p = 1/3
So the husband’s mixed strategy is sh(Opera) = 1/3; sh(Football) = 2/3
17. Nau: Game Theory 17
Battle of the Sexes
A similar calculation shows that the wife’s mixed strategy sw is
sw(Opera) = 2/3, sw(Football) = 1/3
In this equilibrium,
P(wife gets 2, husband gets 1)
= (2/3) (1/3) = 2/9
P(wife gets 1, husband gets 2)
= (1/3) (2/3) = 2/9
P(both get 0) = (1/3)(1/3) + (2/3)(2/3) = 5/9
Husband
Wife
Thus the expected utility for each agent is 2/3
Pareto-dominated by both of the pure-strategy equilibria
In each of them, one agent gets 1 and the other gets 2
Opera Football
Opera 2, 1 0, 0
Football 0, 0 1, 2
18. Nau: Game Theory 18
Finding Nash Equilibria
Matching Pennies
Each agent has a penny
Each agent independently chooses to display
his/her penny heads up or tails up
Easy to see that in this game, no pure strategy
could be part of a Nash equilibrium
For each combination of pure strategies, one of the agents can do better
by changing his/her strategy
• for (Heads,Heads), agent 2 can do better by switching to Tails
• for (Heads,Tails), agent 1 can do better by switching to Tails
• for (Tails,Tails), agent 2 can do better by switching to Heads
• for (Tails,Heads), agent 1 can do better by switching to Heads
But there’s a mixed-strategy equilibrium:
(s,s), where s(Heads) = s(Tails) = ½
Agent 2
Agent 1
Heads Tails
Heads 1, –1 –1, 1
Tails –1, 1 1, –1
19. Nau: Game Theory 19
A Real-World Example
Penalty kicks in soccer
A kicker and a goalie in a penalty kick
Kicker can kick left or right
Goalie can jump to left or right
Kicker scores iff he/she kicks to one
side and goalie jumps to the other
Analogy to Matching Pennies
• If you use a pure strategy and the other agent uses his/her best
response, the other agent will win
• If you kick or jump in either direction with equal probability,
the opponent can’t exploit your strategy
20. Another Interpretation of Mixed Strategies
Nau: Game Theory 20
Another interpretation of mixed strategies is that
Each agent’s strategy is deterministic
But each agent has uncertainty regarding the other’s strategy
Agent i’s mixed strategy is everyone else’s assessment of how likely i is to
play each pure strategy
Example:
In a series of soccer penalty kicks, the kicker could kick left or right in
a deterministic pattern that the goalie thinks is random
21. Nau: Game Theory 21
Two-Finger Morra
There are several versions of this game
Here’s the one the book uses:
Each agent holds up 1 or 2 fingers
If the total number of fingers is odd
• Agent 1 gets that many points
If the total number of fingers is even
• Agent 2 gets that many points
Agent 1 has no dominant strategy
Agent 2 plays 1 => agent 1’s best response is 2
Agent 2 plays 2 => agent 1’s best response is 1
Similarly, agent 2 has no dominant strategy
Thus there’s no pure-strategy Nash equilibrium
Look for a mixed-strategy equilibrium
Agent 2
Agent 1
1 finger 2 fingers
1 finger –2, 2 3, –3
2 fingers 3, –3 –4, 4
22. 1 finger 2 fingers
1 finger –2, 2 3, –3
2 fingers 3, –3 –4, 4
Nau: Game Theory 22
Two-Finger Morra
Let p1 = P(agent 1 plays 1 finger)
and p2 = P(agent 2 plays 1 finger)
Suppose 0 < p1 < 1 and 0 < p2 < 1
If this is a mixed-strategy equilibrium, then
Agent 2
Agent 1
1 finger and 2 fingers must have the same expected utility for agent 1
• Agent 1 plays 1 finger => expected utility is –2p2 + 3(1−p2) = 3 – 5p2
• Agent 1 plays 2 fingers => expected utility is 3p2 – 4(1−p2) = 7p2 – 4
• Thus 3 – 5p2 = 7p2 – 4, so p2 = 7/12
• Agent 1’s expected utility is 3–5(7/12) = 1/12
1 finger and 2 fingers must also have the same expected utility for agent 2
• Agent 2 plays 1 finger => expected utility is 2p1 – 3(1−p1) = 5p1 – 3
• Agent 2 plays 2 fingers => expected utility is –3p1 + 4(1−p1) = 4 – 7p1
• Thus 5p1 – 3 = 4 – 7p1 , so p1 = 7/12
• Agent 2’s expected utility is 5(7/12) – 3 = –1/12
23. Nau: Game Theory 23
Another Real-World Example
Road Networks
Suppose that 1,000 drivers wish to
travel from S (start) to D (destination)
Two possible paths:
• S→A→D and S→B→D
S
The roads S→A and B→D are very long and very wide
• t = 50 minutes for each, no matter how many drivers
The roads S→B and A→D are very short and very narrow
• Time for each = (number of cars)/25
Nash equilibrium:
• 500 cars go through A, 500 cars through B
• Everyone’s time is 50 + 500/25 = 70 minutes
• If a single driver changes to the other route
› There now are 501 cars on that route, so his/her time goes up
D
t = cars/25
t = cars/25
t = 50
t = 50
B
A
24. Nau: Game Theory 24
Braess’s Paradox
Suppose we add a new road from B to A
It’s so wide and so short that it takes 0 minutes
New Nash equilibrium:
All 1000 cars go S→B→A→D
Time is 1000/25 + 1000/25 = 80 minutes
To see that this is an equilibrium:
If driver goes S→A→D, his/her cost is 50 + 40 = 90 minutes
If driver goes S→B→D, his/her cost is 40 + 50 = 90 minutes
Both are dominated by S→B→A→D
To see that it’s the only Nash equilibrium:
For every traffic pattern, compute the times a driver would get on all
three routes
In every case, S→B→A→D dominates S→A→D and S→B→D
Carelessly adding capacity can actually be hurtful!
S
D
t = cars/25
t = cars/25
t = 50
t = 50
B
A
t = 0
25. Nau: Game Theory 25
Braess’s Paradox in practice
From an article about Seoul, South Korea:
“The idea was sown in 1999,” Hwang says. “We had experienced a
strange thing. We had three tunnels in the city and one needed to be
shut down. Bizarrely, we found that that car volumes dropped. I
thought this was odd. We discovered it was a case of ‘Braess paradox’,
which says that by taking away space in an urban area you can actually
increase the flow of traffic, and, by implication, by adding extra
capacity to a road network you can reduce overall performance.”
John Vidal, “Heart and soul of the city”, The Guardian, Nov. 1, 2006
http://www.guardian.co.uk/environment/2006/nov/01/society.travelsenvironmentalimpact
26. Nau: Game Theory 26
The p-Beauty Contest
Consider the following game:
Each player chooses a number in the range from 0 to 100
The winner(s) are whoever chose a number that’s closest to 2/3 of the
average
This game is famous among economists and game theorists
It’s called the p-beauty contest
I used p = 2/3
27. Elimination of Dominated Strategies
A strategy si is (strongly, weakly) dominated for an agent i if some other
Nau: Game Theory 27
strategy sʹ′i strictly dominates si
A strictly dominated strategy can’t be a best response to any move
So we can eliminate it (remove it from the payoff matrix)
Once a pure strategy is eliminated, another strategy may become dominated
This elimination can be repeated
28. Iterated Elimination of Dominated Strategies
Iteratively eliminate strategies that can never be a best response if the other agents
Nau: Game Theory 28
play rationally
All numbers ≤ 100 => 2/3(average) < 67
=> Any rational agent will choose a number < 67
All rational choices ≤ 67 => 2/3(average) < 45
=> Any rational agent will choose a number < 45
All rational choices ≤ 45 => 2/3(average) < 30
=> Any rational agent will choose a number < 30
. . .
Nash equilibrium: everyone chooses 0
29. Nau: Game Theory 29
p-Beauty Contest Results
(2/3)(average) = 21
winner = Giovanni
30. Another Example of
p-Beauty Contest Results
Average = 32.93
2/3 of the average = 21.95
Winner: anonymous xx
Nau: Game Theory 30
31. Nau: Game Theory 31
We aren’t rational
We aren’t game-theoretically rational agents
Huge literature on behavioral economics going back to about 1979
Many cases where humans (or aggregations of humans) tend to make
different decisions than the game-theoretically optimal ones
Daniel Kahneman received the 2002 Nobel Prize in Economics for his
work on that topic
32. Nau: Game Theory 32
Choosing “Irrational” Strategies
Why choose a non-equilibrium strategy?
Limitations in reasoning ability
• Didn’t calculate the Nash equilibrium correctly
• Don’t know how to calculate it
• Don’t even know the concept
Hidden payoffs
• Other things may be more important than winning
› Want to be helpful
› Want to see what happens
› Want to create mischief
Agent modeling (next slide)
33. Nau: Game Theory 33
Agent Modeling
A Nash equilibrium strategy is best for you
if the other agents also use their Nash equilibrium strategies
In many cases, the other agents won’t use Nash equilibrium
strategies
If you can forecast their actions accurately, you may be
able to do much better than the Nash equilibrium strategy
Example: repeated games
34. Repeated Games
Used by game theorists, economists, social and behavioral scientists
as highly simplified models of various real-world situations
Repeated Stag Hunt Nau: Game Theory 34
Roshambo
Iterated Chicken Game
Repeated
Matching Pennies
Iterated Prisoner’s Dilemma Repeated
Ultimatum Game
Iterated Battle
of the Sexes
35. Prisoner’s Dilemma:
Agent 2
Agent 1
C D
C 3, 3 0, 5
D 5, 0 1, 1
Iterated Prisoner’s Dilemma,
with 2 iterations:
Nau: Game Theory 35
Repeated Games
In repeated games, some game G is played
multiple times by the same set of agents
G is called the stage game
Each occurrence of G is called
an iteration or a round
Usually each agent knows what all
the agents did in the previous iterations,
but not what they’re doing in the
current iteration
Usually each agent’s
payoff function is additive Agent 1: Agent 2:
C
D C
Round 1: C
Round 2:
Total payoff: 3+5 = 8 3+0 = 3
36. Roshambo (Rock, Paper, Scissors)
Nau: Game Theory 36
A1
A2
Rock Paper Scissors
Rock 0, 0 –1, 1 1, –1
Paper 1, –1 0, 0 –1, 1
Scissors –1, 1 1, –1 0, 0
Nash equilibrium for the stage game:
choose randomly, P=1/3 for each move
Nash equilibrium for the repeated game:
always choose randomly, P=1/3 for each move
Expected payoff = 0
Let’s see how that works out in practice …
37. Roshambo (Rock, Paper, Scissors)
Nau: Game Theory 37
A1
A2
Rock Paper Scissors
Rock 0, 0 –1, 1 1, –1
Paper 1, –1 0, 0 –1, 1
Scissors –1, 1 1, –1 0, 0
1999 international roshambo programming competition
www.cs.ualberta.ca/~darse/rsbpc1.html
Round-robin tournament:
• 55 programs, 1000 iterations
for each pair of programs
• Lowest possible score = –55000,
highest possible score = 55000
Average over 25 tournaments:
• Highest score (Iocaine Powder): 13038
• Lowest score (Cheesebot): –36006
Very different from the game-theoretic prediction
38. Nau: Game Theory 38
Opponent Modeling
A Nash equilibrium strategy is best for you
if the other agents also use their Nash equilibrium strategies
In many cases, the other agents won’t use Nash equilibrium strategies
If you can forecast their actions accurately, you may be able to do
much better than the Nash equilibrium strategy
One reason why the other agents might not use their Nash equilibrium
strategies:
Because they may be trying to forecast your actions too
39. Iterated Prisoner’s Dilemma
Prisoner’s Dilemma:
Agent 2
Nash equilibrium
Nau: Game Theory 39
Agent 1
C D
C 3, 3 0, 5
D 5, 0 1, 1
Multiple iterations of the Prisoner’s Dilemma
Widely used to study the emergence of
cooperative behavior among agents
e.g., Axelrod (1984), The Evolution of Cooperation
Axelrod ran a famous set of tournaments
People contributed strategies
encoded as computer programs
Axelrod played them against each other
If I defect now, he might punish
me by defecting next time
40. » It could establish and maintain cooperations with many other agents
» It could prevent malicious agents from taking advantage of it
TFT Tester
C
C
C
C
C
Nau: Game Theory 40
TFT with Other Agents
TFT TFT
C
C
C
C
C
C
C
C C
…
…
C
C
C
C
C
In Axelrod’s tournaments, TFT usually did best
TFT Grim
C
C
C
C
C
C
C
C C
…
…
C
C
C
C
C
C
C
C C
…
…
D
C
D
C
C
TFT AllD
C
D
D
D
D
D
D
D D
…
…
D
D
D
D
D
TFT AllC
C
C
C
C
C
C
C
C C
...
...
C
C
C
C
C
41. Nau: Game Theory 41
Example:
A real-world example of the IPD, described in Axelrod’s book:
World War I trench warfare
Incentive to cooperate:
If I attack the other side, then they’ll retaliate and I’ll get hurt
If I don’t attack, maybe they won’t either
Result: evolution of cooperation
Although the two infantries were supposed to be enemies, they
avoided attacking each other
42. Nau: Game Theory 42
IPD with Noise
In noisy environments,
There’s a nonzero probability (e.g., 10%)
that a “noise gremlin” will change some
of the actions
• Cooperate (C) becomes Defect (D),
and vice versa
Can use this to model accidents
Compute the score using the changed
action
Can also model misinterpretations
Compute the score using the original
action
C C
C C
C D
…
C
…
Noise
Did he really
intend to do that?
43. Nau: Game Theory 43
Example of Noise
Story from a British army officer in World War I:
I was having tea with A Company when we heard a lot of shouting and went
out to investigate. We found our men and the Germans standing on their
respective parapets. Suddenly a salvo arrived but did no damage.
Naturally both sides got down and our men started swearing at the Germans,
when all at once a brave German got onto his parapet and shouted out:
“We are very sorry about that; we hope no one was hurt. It is not our
fault. It is that damned Prussian artillery.”
The salvo wasn’t the German infantry’s intention
They didn’t expect it nor desire it
44. Nau: Game Theory 44
Consider two agents
who both use TFT
One accident or
misinterpretation
can cause a long
string of retaliations
C C
C
C
C
D C
C
C
C
C D
. . .
. . .
C
D
C
D
D
Noise"
Retaliation
Retaliation
Retaliation
Retaliation
Noise Makes it Difficult
to Maintain Cooperation
45. Some Strategies for the Noisy IPD
Nau: Game Theory 45
Principle: be more forgiving in the face of defections
Tit-For-Two-Tats (TFTT)
» Retaliate only if the other agent defects twice in a row
• Can tolerate isolated instances of defections, but susceptible to exploitation
of its generosity
• Beaten by the TESTER strategy I described earlier
Generous Tit-For-Tat (GTFT)
» Forgive randomly: small probability of cooperation if the other agent defects
» Better than TFTT at avoiding exploitation, but worse at maintaining cooperation
Pavlov
» Win-Stay, Lose-Shift
• Repeat previous move if I earn 3 or 5 points in the previous iteration
• Reverse previous move if I earn 0 or 1 points in the previous iteration
» Thus if the other agent defects continuously, Pavlov will alternatively cooperate
and defect
46. Nau: Game Theory 46
Discussion
The British army officer’s story:
a German shouted, ``We are very sorry about that; we hope no one was
hurt. It is not our fault. It is that damned Prussian artillery.”
The apology avoided a conflict
It was convincing because it was consistent with the German infantry’s
past behavior
The British had ample evidence that the German infantry wanted to
keep the peace
If you can tell which actions are affected by noise, you can avoid reacting
to the noise
IPD agents often behave deterministically
For others to cooperate with you it helps if you’re predictable
This makes it feasible to build a model from observed behavior
47. Au & Nau. Accident or intention:
That is the question (in the iterated
prisoner’s dilemma). AAMAS, 2006.
Au & Nau. Is it accidental or
intentional? A symbolic approach to the
noisy iterated prisoner’s dilemma. In G.
Kendall (ed.), The Iterated Prisoners
Dilemma: 20 Years On. World
Scientific, 2007.
Nau: Game Theory 47
The DBS Agent
Work by my recent PhD graduate, Tsz-Chiu Au
Now a postdoc at University of Texas
From the other agent’s recent behavior,
build a model π of the other agent’s strategy
A set of rules giving the probability
of each action in various situations
Use the model to filter noise
Observed move contradicts the model => assume the observed move
is noise
Detect changes in the other agent’s strategy
Observed move contradicts the model too many times => assume
they’ve changed their strategy; recompute the model
Use the model to help plan our next move
Game-tree search, using π to predict the other agent’s moves
48. Nau: Game Theory 48
20th Anniversary IPD Competition
http://www.prisoners-dilemma.com
Category 2: IPD with noise
165 programs participated
DBS dominated the
top 10 places
Two agents scored
higher than DBS
They both used
master-and-slaves strategies
49. Nau: Game Theory 49
Master & Slaves Strategy
Each participant could submit up to 20 programs
Some submitted programs that could recognize each other
(by communicating pre-arranged sequences of Cs and Ds)
The 20 programs worked as a team
• 1 master, 19 slaves
When a slave plays with its master
• Slave cooperates, master defects
=> maximizes the master’s payoff
When a slave plays with
an agent not in its team
• It defects
=> minimizes the other
agent’s payoff
My goons give
me
all their
money …
… and they
beat up
everyone else
50. Nau: Game Theory 50
Comparison
Analysis
Each master-slaves team’s average score was much lower than DBS’s
If BWIN and IMM01 had each been restricted to ≤ 10 slaves,
DBS would have placed 1st
Without any slaves, BWIN and IMM01 would have done badly
In contrast, DBS had no slaves
DBS established cooperation
with many other agents
DBS did this despite the noise,
because it filtered out the noise
51. Nau: Game Theory 51
Summary
Dominant strategies and dominant strategy equilibria
Prisoner’s dilemma
Pareto optimality
Best responses and Nash equilibria
Battle of the Sexes, Matching Pennies, Two-Finger Morra
Real-world examples
Soccer penalty kicks, road networks (Braess’s Paradox)
Repeated games and opponent modeling
roshambo (rock-paper-scissors
iterated prisoner’s dilemma with noise
• opponent models based on observed behavior
• detection and removal of noise, game-tree search
20th anniversary IPD competition