Lecture 12. Game theory So far we discussed: roulette and blackjack Roulette: – Outcomes completely independent and random – Very little strategy (even.

Slides:

Advertisements

Similar presentations

Lecture 13. Poker Two basic concepts: Poker is a game of skill, not luck. If you want to win at poker, – make sure you are very skilled at the game, and.

Advertisements

Introduction to Game Theory

9.1 Strictly Determined Games Game theory is a relatively new branch of mathematics designed to help people who are in conflict situations determine the.

Name: Trương Hoài Anh Facebook: Quasar Hoaianh

Game Theory Assignment For all of these games, P1 chooses between the columns, and P2 chooses between the rows.

Module C1 Decision Models Uncertainty. What is a Decision Analysis Model? Decision Analysis Models is about making optimal decisions when the future is.

Game Theory S-1.

15 THEORY OF GAMES CHAPTER.

APPENDIX An Alternative View of the Payoff Matrix n Assume total maximum profits of all oligopolists is constant at 200 units. n Alternative policies.

Two-Player Zero-Sum Games

1 Chapter 4: Minimax Equilibrium in Zero Sum Game SCIT1003 Chapter 4: Minimax Equilibrium in Zero Sum Game Prof. Tsang.

A measurement of fairness game 1: A box contains 1red marble and 3 black marbles. Blindfolded, you select one marble. If you select the red marble, you.

MIT and James Orlin © Game Theory 2-person 0-sum (or constant sum) game theory 2-person game theory (e.g., prisoner’s dilemma)

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc

Game Theory, Part 1 Game theory applies to more than just games. Corporations use it to influence business decisions, and militaries use it to guide their.

© 2015 McGraw-Hill Education. All rights reserved. Chapter 15 Game Theory.

Game Theory: introduction and applications to computer networks Game Theory: introduction and applications to computer networks Zero-Sum Games (follow-up)

Part 3: The Minimax Theorem

Lecture 10. Simplified roulette European roulette has numbers 0,1,…36. xD0s

Minimax and Alpha-Beta Reduction Borrows from Spring 2006 CS 440 Lecture Slides.

Card Counting What is it, how does it work, and can I use it to pay for college? (and if it does work, do I even have to go to college?) Jeff O’Connell.

Lecture 1 - Introduction 1.  Introduction to Game Theory  Basic Game Theory Examples  Strategic Games  More Game Theory Examples  Equilibrium  Mixed.

Matrix Games Mahesh Arumugam Borzoo Bonakdarpour Ali Ebnenasir CSE 960: Selected Topics in Algorithms and Complexity Instructor: Dr. Torng.

An Introduction to Game Theory Part III: Strictly Competitive Games Bernhard Nebel.

Lectures in Microeconomics-Charles W. Upton Minimax Strategies.

UNR, MATH/STAT 352, Spring Head Tail Tossing a symmetric coin You are paying $1 How much should you get to make the game fair?

Game Theory Objectives:

Alpha-Beta Search. 2 Two-player games The object of a search is to find a path from the starting position to a goal position In a puzzle-type problem,

Some Definitions –Maxi-min (mini-max) strategy: The maxi-min strategy is the path with the highest possible (maximum) worst (minimum) outcome. –Dominant.

Minimax Strategies. Everyone who has studied a game like poker knows the importance of mixing strategies. –With a bad hand, you often fold –But you must.

Game Theory Statistics 802. Lecture Agenda Overview of games 2 player games representations 2 player zero-sum games Render/Stair/Hanna text CD QM for.

Honors Precalculus: Do Now 1.) You are betting with your friend. Every time you flip heads, your friend pays you a dollar. Every time you flip tails, you.

Game Theory Part 2: Zero Sum Games. Zero Sum Games The following matrix defines a zero-sum game. Notice the sum of the payoffs to each player, at every.

Strategic Decisions in Noncooperative Games Introduction to Game Theory.

Games. Adversaries Consider the process of reasoning when an adversary is trying to defeat our efforts In game playing situations one searches down the.

Game Theory, Part 2 Consider again the game that Sol and Tina were playing, but with a different payoff matrix: H T Tina H T Sol.

1 1 Slide © 2006 Thomson South-Western. All Rights Reserved. Slides prepared by JOHN LOUCKS St. Edward’s University.

When dealing with a model, we use the letter  for the mean. We write or, more often, replacing p by, Instead of , we can also write E(X ). ( Think of.

Part 3 Linear Programming

Decision Analysis Mary Whiteside. Decision Analysis Definitions Actions – alternative choices for a course of action Actions – alternative choices for.

1 What is Game Theory About? r Analysis of situations where conflict of interests is present r Goal is to prescribe how conflicts can be resolved 2 2 r.

Introduction to Game Theory Kamal Aboul-Hosn Cornell University Computers Playing Games.

WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6.

1 Optimizing Decisions over the Long-term in the Presence of Uncertain Response Edward Kambour.

Statistics Overview of games 2 player games representations 2 player zero-sum games Render/Stair/Hanna text CD QM for Windows software Modeling.

On Investor Behavior Objective Define and discuss the concept of rational behavior.

GAME THEORY Day 5. Minimax and Maximin Step 1. Write down the minimum entry in each row. Which one is the largest? Maximin Step 2. Write down the maximum.

9.2 Mixed Strategy Games In this section, we look at non-strictly determined games. For these type of games the payoff matrix has no saddle points.

Math Puzzles and Game Theory Final Exam Work through the slides, follow the directions, and fill in all required information.

By: Donté Howell Game Theory in Sports. What is Game Theory? It is a tool used to analyze strategic behavior and trying to maximize his/her payoff of.

Game Theory By Ben Cutting & Rohit Venkat.

Decisions under uncertainty and risk

Chapter 15: Game Theory: The Mathematics Lesson Plan of Competition

Game Theory Just last week:

Extensive-form games and how to solve them

Alpha-Beta Search.

Alpha-Beta Search.

Game Theory II Solutions 1

Alpha-Beta Search.

Alpha-Beta Search.

Chapter 15: Game Theory: The Mathematics Lesson Plan of Competition

Alpha-Beta Search.

Presentation transcript:

Lecture 12

Game theory So far we discussed: roulette and blackjack Roulette: – Outcomes completely independent and random – Very little strategy (even that is curbed by minimums and maximum bet rules) Blackjack – Dealers outcome completely random (no strategy) – There is some dependence between draws – Strategy can be useful (counting cards and adjusting play accordingly)

Game theory Poker – All the participant are following a strategy that uses the information based of other people What is a rational (best?) strategy? MAXIMIN – Select a strategy S – If the opponent new my strategy, what is the worst they can do to me? – Select the strategy that maximizes this worst case scenario

Understanding role of strategy Coin matching game: You and your friend each have a coin. You each secretly choose to show “heads” or “tails”. You simultaneously show your choice. The payoff: – If the coins are both “heads”, your friend pays you $1. – If the coins are both “tails”, your friend pays you $9. – If the coins do not match, you pay your friend $5. Is this a fair game?

Coin matching game you\friendHT H1-5 T 9 If both of us flip a coin: Expected gain = 1*0.5*0.5+(-5)*0.5*0.5+(-5)*0.5*0.5+9*0.5*0.5=0 (Looks fair but is it?) We do not need to flip a fair coin – we can choose any proportion of H/T we want! Consider our proportion of H p and friends proportion of H q. For now assume independence between us and friend Expected gain = 1*p*q+(-5)*p*(1-q)+(-5)*(1-p)*q+9*(1-p)*(1-q)=9-14p-14q+20pq How to select p?

Maximin Main idea: – We want select p to maximize expected win if positive we might also want to minimize variance – What is best depends on the friends strategy: (Depends on friends choices unknown to me) If I select p – if my friends selects all H gain =9-14p all T gain=-5+6p

Maximin Notice that smaller of the two lines is less than -0.8 The best option is p=0.7 – gain =-0.8 

Minimax (maximin for friend) Friends choices – What is best for the friend depends on my strategy (unknown to him) My friends select q – if I selects all H gain =9-14q all T gain=-5+6q Notice that larger (my gain his loss!) of the two less than -0.8 The best option is q=0.7 – expected gain =-0.8 (friend gains 0.8 on average) Nash equilibrium – if the other player is playing optimally there is nothing to gain from changing strategies