Introduction to  Game Theory Yale Braunstein Spring 2007
General approach <ul><li>Brief History of Game Theory  </li></ul><ul><li>Payoff Matrix </li></ul><ul><li>Types of Games  <...
Brief History of Game Theory <ul><li>1913 - E. Zermelo provides the first theorem of game theory; asserts that chess is st...
Rationality <ul><li>Assumptions:  </li></ul><ul><li>humans are rational beings </li></ul><ul><li>humans always seek the be...
Utility Theory <ul><li>Utility Theory based on: </li></ul><ul><li>rationality </li></ul><ul><li>maximization of utility </...
What is Game Theory?   <ul><li>Game theory is a study of how to mathematically determine the best strategy for given condi...
Game Theory <ul><li>Finding acceptable, if not optimal, strategies in conflict situations. </li></ul><ul><li>Abstraction o...
Why is game theory important? <ul><li>All intelligent beings make decisions all the time. </li></ul><ul><li>AI needs to pe...
Types of Games <ul><li>Sequential  vs.  Simultaneous moves </li></ul><ul><li>Single Play  vs.  Iterated  </li></ul><ul><li...
Zero-Sum Games <ul><li>The sum of the payoffs remains constant during the course of the game. </li></ul><ul><li>Two sides ...
Non-zero Sum Game <ul><li>The sum of payoffs is not constant during the course of game play. </li></ul><ul><li>Players may...
Games of Perfect Information <ul><li>The information concerning an opponent’s move is well known in advance. </li></ul><ul...
Imperfect Information <ul><li>Partial or no information concerning the opponent is given in advance to the player’s decisi...
Key Area of Interest <ul><li>chance </li></ul><ul><li>strategy </li></ul>Non-zero  Sum Imperfect  Information
Matrix Notation Notes: Player I's strategy A may be different from Player II's. P2 can be omitted if zero-sum game
Prisoner’s Dilemma &  Other famous games A sample of other games: Marriage Disarmament (my generals are  more irrational t...
Prisoner’s Dilemma 10 , 10 Blame Don't Blame Don't 20 , 0 0 , 20 1 , 1 Prisoner 1 Prisoner 2 Notes:  Higher payoffs (longe...
Games of Conflict <ul><li>Two sides competing against each other </li></ul><ul><li>Usually caused by complete lack of info...
Games of Co-operation <ul><li>Players may improve payoff through </li></ul><ul><li>communicating </li></ul><ul><li>forming...
Prisoner’s Dilemma with Iteration <ul><li>Infinite number of iterations </li></ul><ul><ul><li>Fear of retaliation </li></u...
Basic Strategies <ul><li>1. Plan ahead and look back  </li></ul><ul><li>2. Use a dominating strategy if possible </li></ul...
Plan ahead and look back Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent
If you have a dominating strategy,  use it Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Use s...
Eliminate any dominated strategy Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 ...
Look for any equilibrium <ul><li>Dominating Equilibrium </li></ul><ul><li>Minimax Equilibrium </li></ul><ul><li>Nash Equil...
Maximin & Minimax Equilibrium <ul><li>Minimax - to minimize the maximum loss (defensive) </li></ul><ul><li>Maximin - to ma...
Maximin & Minimax Equilibrium Strategies Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strateg...
Definition: Nash Equilibrium <ul><li>“ If there is a set of strategies with the property that no player can benefit by cha...
Is this a Nash Equilibrium? Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 M...
Cost to press button = 2 units When button is pressed, food given = 10 units Boxed Pigs Example
5 , 1 Press Wait Press Wait 9 , -1 4 , 4 0 , 0 Little Pig Big Pig Decisions, decisions...
Time for &quot;real-life&quot; decision making <ul><li>Holmes & Moriarity in &quot;The Final Problem&quot; </li></ul><ul><...
Mixed Strategy Safe 2 Safe 1 $ 0 $10,000 $100,000 Safe 1 Safe 2 $ 0
Mixed Strategy Solution
The Payoff Matrix  for Holmes & Moriarity Player #1 Player #2 Strategy #1 Strategy #2 Strategy #1 Strategy #2 Payoff (1,1)...
Evolutionary Game Theory <ul><li>Natural selection replaces rational behavior </li></ul><ul><li>Survival of the fittest </...
Hawk / Dove Game
Evolutionary Stable Strategy <ul><li>Introduced by Maynard Smith and Price (1973) </li></ul><ul><li>Strategy becomes stabl...
Hawk Dove Hawk Dove 2 2 10 0 10 0 -5 -5
 
Hawk Dove Hawk Dove 2 2 10 0 10 0 -5 -5
 
Where is game theory  currently used?  <ul><li>Ecology </li></ul><ul><li>Networks </li></ul><ul><li>Economics   </li></ul>
Limitations & Problems <ul><li>Assumes players always maximize their outcomes </li></ul><ul><li>Some outcomes are difficul...
Summary <ul><li>What is game theory? </li></ul><ul><ul><li>Abstraction modeling multi-person interactions </li></ul></ul><...
Sources <ul><li>Much more available on the web. </li></ul><ul><li>These slides (with changes and additions) adapted from: ...
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    1. 1. Introduction to Game Theory Yale Braunstein Spring 2007
    2. 2. General approach <ul><li>Brief History of Game Theory </li></ul><ul><li>Payoff Matrix </li></ul><ul><li>Types of Games </li></ul><ul><li>Basic Strategies </li></ul><ul><li>Evolutionary Concepts </li></ul><ul><li>Limitations and Problems </li></ul>
    3. 3. Brief History of Game Theory <ul><li>1913 - E. Zermelo provides the first theorem of game theory; asserts that chess is strictly determined </li></ul><ul><li>1928 - John von Neumann proves the minimax theorem </li></ul><ul><li>1944 - John von Neumann & Oskar Morgenstern write &quot;Theory of Games and Economic Behavior” </li></ul><ul><li>1950-1953 - John Nash describes Nash equilibrium </li></ul>
    4. 4. Rationality <ul><li>Assumptions: </li></ul><ul><li>humans are rational beings </li></ul><ul><li>humans always seek the best alternative in a set of possible choices </li></ul><ul><li>Why assume rationality? </li></ul><ul><li>narrow down the range of possibilities </li></ul><ul><li>predictability </li></ul>
    5. 5. Utility Theory <ul><li>Utility Theory based on: </li></ul><ul><li>rationality </li></ul><ul><li>maximization of utility </li></ul><ul><ul><li>may not be a linear function of income or wealth </li></ul></ul><ul><li>It is a quantification of a person's preferences with respect to certain objects. </li></ul>
    6. 6. What is Game Theory? <ul><li>Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome </li></ul>
    7. 7. Game Theory <ul><li>Finding acceptable, if not optimal, strategies in conflict situations. </li></ul><ul><li>Abstraction of real complex situation </li></ul><ul><li>Game theory is highly mathematical </li></ul><ul><li>Game theory assumes all human interactions can be understood and navigated by presumptions. </li></ul>
    8. 8. Why is game theory important? <ul><li>All intelligent beings make decisions all the time. </li></ul><ul><li>AI needs to perform these tasks as a result. </li></ul><ul><li>Helps us to analyze situations more rationally and formulate an acceptable alternative with respect to circumstance. </li></ul><ul><li>Useful in modeling strategic decision-making </li></ul><ul><ul><li>Games against opponents </li></ul></ul><ul><ul><li>Games against &quot;nature„ </li></ul></ul><ul><li>Provides structured insight into the value of information </li></ul>
    9. 9. Types of Games <ul><li>Sequential vs. Simultaneous moves </li></ul><ul><li>Single Play vs. Iterated </li></ul><ul><li>Zero vs. non-zero sum </li></ul><ul><li>Perfect vs. Imperfect information </li></ul><ul><li>Cooperative vs. conflict </li></ul>
    10. 10. Zero-Sum Games <ul><li>The sum of the payoffs remains constant during the course of the game. </li></ul><ul><li>Two sides in conflict </li></ul><ul><li>Being well informed always helps a player </li></ul>
    11. 11. Non-zero Sum Game <ul><li>The sum of payoffs is not constant during the course of game play. </li></ul><ul><li>Players may co-operate or compete </li></ul><ul><li>Being well informed may harm a player. </li></ul>
    12. 12. Games of Perfect Information <ul><li>The information concerning an opponent’s move is well known in advance. </li></ul><ul><li>All sequential move games are of this type. </li></ul>
    13. 13. Imperfect Information <ul><li>Partial or no information concerning the opponent is given in advance to the player’s decision. </li></ul><ul><li>Imperfect information may be diminished over time if the same game with the same opponent is to be repeated. </li></ul>
    14. 14. Key Area of Interest <ul><li>chance </li></ul><ul><li>strategy </li></ul>Non-zero Sum Imperfect Information
    15. 15. Matrix Notation Notes: Player I's strategy A may be different from Player II's. P2 can be omitted if zero-sum game
    16. 16. Prisoner’s Dilemma & Other famous games A sample of other games: Marriage Disarmament (my generals are more irrational than yours)
    17. 17. Prisoner’s Dilemma 10 , 10 Blame Don't Blame Don't 20 , 0 0 , 20 1 , 1 Prisoner 1 Prisoner 2 Notes: Higher payoffs (longer sentences) are bad. Non-cooperative equilibrium  Joint maximum Institutionalized “solutions” (a la criminal organizations, KSM) NCE Jt. max.
    18. 18. Games of Conflict <ul><li>Two sides competing against each other </li></ul><ul><li>Usually caused by complete lack of information about the opponent or the game </li></ul><ul><li>Characteristic of zero-sum games </li></ul>
    19. 19. Games of Co-operation <ul><li>Players may improve payoff through </li></ul><ul><li>communicating </li></ul><ul><li>forming binding coalitions & agreements </li></ul><ul><li>do not apply to zero-sum games </li></ul><ul><li>Prisoner’s Dilemma </li></ul><ul><li>with Cooperation </li></ul>
    20. 20. Prisoner’s Dilemma with Iteration <ul><li>Infinite number of iterations </li></ul><ul><ul><li>Fear of retaliation </li></ul></ul><ul><li>Fixed number of iteration </li></ul><ul><ul><li>Domino effect </li></ul></ul>
    21. 21. Basic Strategies <ul><li>1. Plan ahead and look back </li></ul><ul><li>2. Use a dominating strategy if possible </li></ul><ul><li>3. Eliminate any dominated strategies </li></ul><ul><li>4. Look for any equilibrium </li></ul><ul><li>5. Mix up the strategies </li></ul>
    22. 22. Plan ahead and look back Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent
    23. 23. If you have a dominating strategy, use it Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Use strategy 1
    24. 24. Eliminate any dominated strategy Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Eliminate strategy 2 as it’s dominated by strategy 1
    25. 25. Look for any equilibrium <ul><li>Dominating Equilibrium </li></ul><ul><li>Minimax Equilibrium </li></ul><ul><li>Nash Equilibrium </li></ul>
    26. 26. Maximin & Minimax Equilibrium <ul><li>Minimax - to minimize the maximum loss (defensive) </li></ul><ul><li>Maximin - to maximize the minimum gain (offensive) </li></ul><ul><li>Minimax = Maximin </li></ul>
    27. 27. Maximin & Minimax Equilibrium Strategies Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Min 1000 150 - 10 -15 160 Max
    28. 28. Definition: Nash Equilibrium <ul><li>“ If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium. “ </li></ul><ul><li>Source: http://www.lebow.drexel.edu/economics/mccain/game/game.html </li></ul>
    29. 29. Is this a Nash Equilibrium? Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Min 1000 150 - 10 -15 160 Max
    30. 30. Cost to press button = 2 units When button is pressed, food given = 10 units Boxed Pigs Example
    31. 31. 5 , 1 Press Wait Press Wait 9 , -1 4 , 4 0 , 0 Little Pig Big Pig Decisions, decisions...
    32. 32. Time for &quot;real-life&quot; decision making <ul><li>Holmes & Moriarity in &quot;The Final Problem&quot; </li></ul><ul><li>What would you do… </li></ul><ul><ul><li>If you were Holmes? </li></ul></ul><ul><ul><li>If you were Moriarity? </li></ul></ul><ul><li>Possibly interesting digressions? </li></ul><ul><ul><li>Why was Moriarity so evil? </li></ul></ul><ul><ul><li>What really happened? </li></ul></ul><ul><ul><ul><li>What do we mean by reality? </li></ul></ul></ul><ul><ul><ul><li>What changed the reality? </li></ul></ul></ul>
    33. 33. Mixed Strategy Safe 2 Safe 1 $ 0 $10,000 $100,000 Safe 1 Safe 2 $ 0
    34. 34. Mixed Strategy Solution
    35. 35. The Payoff Matrix for Holmes & Moriarity Player #1 Player #2 Strategy #1 Strategy #2 Strategy #1 Strategy #2 Payoff (1,1) Payoff (1,2) Payoff (2,1) Payoff (2,2) Canterbury Canterbury Dover Dover 0 100 50 0 Holmes Moriarty
    36. 36. Evolutionary Game Theory <ul><li>Natural selection replaces rational behavior </li></ul><ul><li>Survival of the fittest </li></ul><ul><li>Why use evolution to determine a strategy? </li></ul>
    37. 37. Hawk / Dove Game
    38. 38. Evolutionary Stable Strategy <ul><li>Introduced by Maynard Smith and Price (1973) </li></ul><ul><li>Strategy becomes stable throughout the population </li></ul><ul><li>Mutations becoming ineffective </li></ul>
    39. 39. Hawk Dove Hawk Dove 2 2 10 0 10 0 -5 -5
    40. 41. Hawk Dove Hawk Dove 2 2 10 0 10 0 -5 -5
    41. 43. Where is game theory currently used? <ul><li>Ecology </li></ul><ul><li>Networks </li></ul><ul><li>Economics </li></ul>
    42. 44. Limitations & Problems <ul><li>Assumes players always maximize their outcomes </li></ul><ul><li>Some outcomes are difficult to provide a utility for </li></ul><ul><li>Not all of the payoffs can be quantified </li></ul><ul><li>Not applicable to all problems </li></ul>
    43. 45. Summary <ul><li>What is game theory? </li></ul><ul><ul><li>Abstraction modeling multi-person interactions </li></ul></ul><ul><li>How is game theory applied? </li></ul><ul><ul><li>Payoff matrix contains each person’s utilities for various strategies </li></ul></ul><ul><li>Who uses game theory? </li></ul><ul><ul><li>Economists, Ecologists, Network people,... </li></ul></ul><ul><li>How is this related to AI? </li></ul><ul><ul><li>Provides a method to simulate a thinking agent </li></ul></ul>
    44. 46. Sources <ul><li>Much more available on the web. </li></ul><ul><li>These slides (with changes and additions) adapted from: http://pages.cpsc.ucalgary.ca/~jacob/Courses/Winter2000/CPSC533/Pages/index.html </li></ul><ul><li>Three interesting classics: </li></ul><ul><ul><li>John von Neumann & Oskar Morgenstern, Theory of Games & Economic Behavior (Princeton, 1944). </li></ul></ul><ul><ul><li>John McDonald, Strategy in Poker, Business & War (Norton, 1950) </li></ul></ul><ul><ul><li>Oskar Morgenstern, &quot;The Theory of Games,&quot; Scientific American , May 1949; translated as &quot;Theorie des Spiels,&quot; Die Amerikanische Rundschau , August 1949. </li></ul></ul>

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