Lecture 1 - Game Theory

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This is the first of an 8 lecture series that I presented at University of Strathclyde in 2011/2012 as part of the final year AI course.

This lecture introduces the concept of a game, and the branch of mathematics known as Game Theory.

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Lecture 1 - Game Theory

  1. 1. 52.426 - 4th Year AI Game AI Luke Dicken Strathclyde AI and Games Group
  2. 2. Background • This is the 1st lecture in an 8 lecture series that constitutes the 2nd half of the course. • Target audience is a 4th year class that has had exposure to AI previously ‣ 3rd year - Agent-based systems ‣ 4th year (1st half) - Algorithms and Search, bin-packing • Although it is a Game AI module, the course itself is a general AI class, many non-games students.2
  3. 3. The Prisoners Dilemma • Imagine you and another person are arrested • Keep silent? Or betray the other person? • They have the same choice...3
  4. 4. Prisoners Dilemma Confess Silent P1 - 5yrs P1 - Free Confess P2 - 5yrs P2 - 20yrs P1 - 20yrs P1 - 1yr Silent P2 - Free P2 - 1yr4
  5. 5. Questions • Does it help to know the other person? • Is it better to be ignorant of your opponent than incorrectly predict their actions? • Do you want to minimise total time in jail, or your time in jail?5
  6. 6. The Odds/Evens Game • Player 1 picks up some number of marbles. • Player 2 guesses if amount is odd or even.6
  7. 7. The Odds/Evens Game Odd Even P1 - -1 P1 - +1 Odd P2 - +1 P2 - -1 P1 - +1 P1 - -1 Even P2 - -1 P2 - +17
  8. 8. Questions • Player 1 played odd last time, what should Player 2 guess this time? • Can Player 1 vary their strategy such that Player 2 can never guess it?8
  9. 9. Intro to Game Theory
  10. 10. Game Theory 101 • What weve just seen are examples "games" • Anytime we are talking about competing with other people for a reward, we can call it a "game" • Game Theory is a branch of mathematics that formally defines how best to play these games.10
  11. 11. 1 Player Games • Relatively trivial : A B C D 5 4 9 411
  12. 12. 2 Player Games • Things get more complicated when there’s a second player. • How can you predict what that person will do? • Can you ensure that you will do well regardless of the other player? • This is the essence of Game Theory.12
  13. 13. A Games "Sum" • Games can be "zero-sum" or "non-zero sum" • If a game is zero-sum then the two players are directly competing - for one to win X, the other must lose X • Contrast this a game where the two players are not completely opposed. ‣ E.g. Prisoners Dilemma • Zero-sum games allow us to make assumptions about how players will act but they are not the general case.13
  14. 14. 2 Player Zero-Sum Games • Although its a special case, this comes up very very often in the real world. ‣ Elections, gambling, corporate competition • Previously shown payoff for both players - in zero-sum this isn’t necessary ‣ The more Player 1 wins, the more Player 2 loses14
  15. 15. Equilibrium Points • A property of some games is that there is a single “solution” • If Player 1 changes strategy from their Equilibrium Strategy, they can only do worse (assuming Player 2 does not change) • Likewise Player 2 cannot change their strategy unilaterally and do any better either. • For both players, this is the best they can hope to achieve15
  16. 16. The “Value” of a Game • The “Value” of a game is “the rationally expected outcome” • For games that have equilibrium points, the Value is the reward of the equilibrium strategies. ‣ Player 1 can’t do worse than this value. ‣ Player 2 can prevent Player 1 from doing better.16
  17. 17. Political Example • Two candidates are deciding what position to take on an issue. • There are three options open to each of them ‣ Support X ‣ Support Y ‣ Duck the issue17
  18. 18. Political Example X Y Dodge X Y Dodge18
  19. 19. Political Example X Y Dodge X 45% 50% 40% Y 60% 55% 50% Dodge 45% 55% 40% Payoff Matrix wrt Player 1’s vote share19
  20. 20. Political Example • Whatever Player 1 does, Player 2 does best if they dodge the issue. • Whatever Player 2 does, Player 1 does best if they support Y.20
  21. 21. Dominant Strategies • Sometimes, a potential strategy choice is just bad. • Recall the 1-player game - one strategy was ALWAYS better. • This can happen in 2-player games too. • More formally, Strategy A dominates Strategy B iff for every move the opponent might choose, A always gives a better result. • Dominated strategies can safely be ignored then. ‣ A rational opponent would never play them, so you needn’t consider situations where they would.21
  22. 22. Domination i ii iii A 19 0 1 B 11 9 3 C 23 7 -322
  23. 23. Domination x ii iii A x 0 1 B x 9 3 C x 7 -3 iii dominates i (remember: from Player 2’s perspective, lower = better)23
  24. 24. Domination x ii iii x x x x B x 9 3 x x x x Now, B dominates both A and C Player 1 should choose B.24
  25. 25. Domination x x iii x x x x B x x 3 x x x x As Player 1 will choose B, Player 2 should choose iii Note that this is an equilibrium point25
  26. 26. Non-Zero Sum Games • Recall the Prisoner’s Dilemma problem. • In this game, the two players were not completely opposed ‣ Cooperation as well as competition • This means that a lot of the assumptions that we’ve made about what the players want to achieve don’t hold26
  27. 27. Prisoners Dilemma Confess Silent P1 - 5yrs P1 - Free Confess P2 - 5yrs P2 - 20yrs P1 - 20yrs P1 - 1yr Silent P2 - Free P2 - 1yr27
  28. 28. Some More Examples • Which would you prefer, a guaranteed £1 or an even chance at £3?28
  29. 29. Some More Examples • Suppose you lose concert tickets that cost you £40 to buy. Would you replace them for another £40 or do something else that night?29
  30. 30. Some More Examples • If 1% of people your age and health die in a given year, would you be prepared to pay £1,000 for £100,000 of life insurance?30
  31. 31. Some More Examples • You go to the store to buy a new video game costing £40. You find youve lost some money, also totalling £40, but you still have enough left to buy the game - do you?31
  32. 32. Some More Examples • Which would you prefer, a guaranteed £1,000,000 or an even chance at £3,000,000?32
  33. 33. Some More Examples • If 0.1% of people your age and health die in a given year, would you be prepared to pay £10 for £10,000 of life insurance?33
  34. 34. Something else is happening...
  35. 35. Utility Theory • "Utility" is an evaluation of how much use a particular result is. • It allows us to compare things "through the eyes of the player" rather than just mathematically. ‣ £1 and £3 are relatively interchangeable, and £1 is not significant. ‣ £1,000,000 is significant, and £3,000,000 is not three times as significant.35
  36. 36. Prisoners Dilemma Do we want an optimal solution for one player? Or for both? Confess Silent P1 - 5yrs P1 - Free Confess P2 - 5yrs P2 - 20yrs P1 - 20yrs P1 - 1yr Silent P2 - Free P2 - 1yr36
  37. 37. Irrational Actions • Utility functions for humans is beyond the scope of this session. • Behavioural Economics ‣ “Predictably Irrational” Dan Ariely • Be aware that players may not be rational. ‣ And we can exploit this to beat them even more :D37
  38. 38. Summary • Fundamentals of Game Theory • Rational play for 2 Player Zero Sum games • Difference of a Non-Zero Sum game • Introduction to irrational play38
  39. 39. Next Lecture • Fun With Probability! • How Spam Filters Work (Sort of) • Mixed Strategies in Games • ...And More39

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