Introduction to Game Theory

1. Introduction to Game Theory G. Michael Youngblood, Ph.D.UT Arlington G. Michael Youngblood, Ph.D.

2. Objectives • Build Knowledge Breadth • Introduce concepts with key terms to facilitate further investigation • Seed research topics • Basics of how to apply scientific methods and rigor to game development and design G. Michael Youngblood, Ph.D.

3. What is a game? • Socratic white-board exercise G. Michael Youngblood, Ph.D.

4. Types of Games? • Socratic white-board exercise G. Michael Youngblood, Ph.D.

5. Types of Games • Board Games • Card Games • Video Games • Field Games G. Michael Youngblood, Ph.D.

6. Types of Computer Games • Adventure Games • Story-based games that usually rely on puzzle-solving to move action along • Action Games • Real-time games in which the player must react quickly to what is happening on the screen • Role-Playing Games (RPG) • Gamer generally directs a group of heros on a series of quests • Strategy Games • Require players to manage a limited set of resources to achieve a predetermined goal • Simulations • Games (?) that seek to emulate the real-world operating conditions of complicated machinery G. Michael Youngblood, Ph.D.

7. More Types of Computer Games • Sports Games • Games that let players vicariously participate in their favorite sport, either as a player or coach • Fighting Games • Two-person games in which each player controls a figure on the screen, using a combination of moves to attack his opponent and defend against his opponent’s attacks • Casual Games • Include adaptations of traditional games such as bridge, chess, hearts, and solitaire. They also include easy-to-play, short session games on the Web, such as Slingo, Poker, and Concentration • God Games (aka, software toys) • Games that have no real goal, other than to encourage the player to fool around with them just to see what happens G. Michael Youngblood, Ph.D.

8. Even More Types of Games • Educational Games • Those that teach while they entertain • Puzzle Games • Exist purely for the intellectual challenge of problem solving. • Online Games • Include any of the preceding genres, but their distinguishing feature is that tey’re played over the Internet • Serious Games? G. Michael Youngblood, Ph.D.

9. Game Features • Following Aristotle’s theory of categories there must be some commonality in all things called “games” • Common features • All games have rules • Strategy matters • There is an outcome • Outcome depends on the strategy chosen by the players (strategic interdependence) G. Michael Youngblood, Ph.D.

10. What is a game? • Any rule-governed situation with a well-defined outcome, characterized by strategic interdependence G. Michael Youngblood, Ph.D.

11. What is game theory? • Simple Socratic exercise G. Michael Youngblood, Ph.D.

12. Game Theory • A mathematical theory designed to model: • how rational individuals should behave • when individual outcomes are determined by collective behavior • strategic behavior • Rational usually means selfish --- but not always • Rich history, flourished during the Cold War • Traditionally viewed as a subject of economics • Subsequently applied by many fields • evolutionary biology, social psychology • Perhaps the branch of pure math most widely examined outside of the “hard” sciences G. Michael Youngblood, Ph.D.

13. Notation • One-person decisions with perfect information • Map • Perfect information • Extensive Form • Nodes (initial node) • Information set • Branches • Payoff (outcomes) • Backward induction • Normal Form G. Michael Youngblood, Ph.D.

14. Extensive Form G. Michael Youngblood, Ph.D.

15. Normal Form G. Michael Youngblood, Ph.D.

16. Games Like Chess G. Michael Youngblood, Ph.D.

17. GLC Theorem • Theorem on Games Like Chess: • In games like Chess, exactly one of the following is true: player 1 can guarantee a win, player 2 can guarantee a win, or each player can guarantee a draw. G. Michael Youngblood, Ph.D.

18. Proof • Case 1: Player 2 can reach a win in at least one of the payoff vectors u or v. Then player 2 so moves. In this case, if the play reaches player 2, he can guarantee a win. • Case 2: Player 2 cannot reach a win with either move, but he can reach a draw from at least one of the payoff vectors u or v. Then player 2 so moves. In this case, if play reaches player 2, he can guarantee a draw. • Case 3: Player 2 faces a loss no matter what move he makes: u = v = (w,l). In this case, player 1 can guarantee a win by moving left. The play reaches player 2, who is sure to lose. G. Michael Youngblood, Ph.D.

19. Games Nearly Like ChessExtensive Form G. Michael Youngblood, Ph.D.

20. Games Nearly Like ChessNormal Form G. Michael Youngblood, Ph.D.

21. Thoughts on game notation… • Ply • Static Board Evaluation • Nodes for an entire game • Stochastic games? G. Michael Youngblood, Ph.D.

22. Uncertainty • Described by means of a probability distribution • Expected value theory • Fair games have an expected value of zero to any player involved • Expected utility theory extends expected value theory to account for risk attitude G. Michael Youngblood, Ph.D.

23. Coalition-Function Form Payoff Triangle - Used to study gains of cooperation G. Michael Youngblood, Ph.D.

24. Game Type Concepts (1) • Zero-sum Games • Games in which the total of player’s utility in the game, for every combination of strategies (outcome), always adds to zero • Examples • Poker, Go, Chess • Constant-sum Game • Whatever the outcome, the player’s utilities add up to a constant sum. G. Michael Youngblood, Ph.D.

25. Game Type Concepts (2) • Symmetry • Games that look the same to all players • Each player has the same set of strategies • Every pair of players has the same utility function (i.e., when two players choose the same strategy, they get the same payoff) G. Michael Youngblood, Ph.D.

26. Game Type Concepts (3) • Simultaneous • Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them effectively simultaneous). • Sequential • Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. G. Michael Youngblood, Ph.D.

27. Game Type Concepts (4) • Perfect Information • A game is one of perfect information if all players know the moves previously made by all other players. Thus, only sequential games can be games of perfect information. • Not to be confused with complete information which requires that every player know the strategies and payoffs of the other players but not necessarily the actions. • Imperfect Information G. Michael Youngblood, Ph.D.

28. Game Type Concepts (5) • Infinitely-long • Games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed. • Focus is on a winning strategy G. Michael Youngblood, Ph.D.

29. Game Type Concepts (6) • Chance • Games that involve stochastic influence on the outcomes • Payoff becomes expected payoff • Probabilities attached to outcomes • More than one thing can happen G. Michael Youngblood, Ph.D.

30. Equilibria • A condition in which there are no forces (reasons) for change • Types • Nash • Correlated • Cooperative • Market • Bargaining G. Michael Youngblood, Ph.D.

31. Nash Equilibrium • John Nash (1950) • If there is a set of strategies with the property that no player can benefit by changing his strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium. G. Michael Youngblood, Ph.D.

32. Nash Equilibrium • Competition Game Example G. Michael Youngblood, Ph.D.

33. Correlated Equilibrium • Robert Aumann (1974) • The ideas is that a strategy set is chosen at random by some outside force. If neither player would want to deviate from the declared strategy, that strategy is correlated equilibrium. G. Michael Youngblood, Ph.D.

34. Strategy • A plan of action or approach for the desired result • Strategy profile is a set of strategies for each player which fully specifies all actions in a game • Many concepts • Dominance (strict and weak) G. Michael Youngblood, Ph.D.

35. Game Examples • Prisoner’s Dilemma • Principal V. Agent Game • Nash Bargaining Agent • Two-sided Market Game • Man, many others G. Michael Youngblood, Ph.D.

36. Prisoner’s Dilemma G. Michael Youngblood, Ph.D.

37. Research Opportunities • Apply classical game theory to computer games • Analyze strategy and equilibrium in existing games • … G. Michael Youngblood, Ph.D.

38. Other Opportunities • Imagine Cup • General Game Player (GGP) • Trading Agents Competition G. Michael Youngblood, Ph.D.

39. Prisoner’s Dilemma • Cooperate = deny the crime; defect = confess guilt of both • Claim that (defect, defect) is an equilibrium: • if I am definitely going to defect, you choose between -10 and -8 • so you will also defect • same logic applies to me • Note unilateral nature of equilibrium: • I fix a behavior or strategy for you, then choose my best response • Claim: no other pair of strategies is an equilibrium • But we would have been so much better off cooperating… • Looking ahead: what do people actually do? G. Michael Youngblood, Ph.D.

40. The World According to Nash • If > 2 actions, mixed strategy is a distribution on them • e.g. 1/3 rock, 1/3 paper, 1/3 scissors • Might also have > 2 players • A general mixed strategy is a vector P = (P[1], P[2],… P[n]): • P[i] is a distribution over the actions for player i • assume everyone knows all the distributions P[j] • but the “coin flips” used to select from P[i] known only to i • P is an equilibrium if: • for every i, P[i] is a best response to all the other P[j] • Nash 1950: every game has a mixed strategy equilibrium • no matter how many rows and columns there are • in fact, no matter how many players there are • Thus known as a Nash equilibrium • A major reason for Nash’s Nobel Prize in economics G. Michael Youngblood, Ph.D.

41. Facts about Nash Equilibria • While there is always at least one, there might be many • zero-sum games: all equilibria give the same payoffs to each player • non zero-sum: different equilibria may give different payoffs! • Equilibrium is a static notion • does not suggest how players might learn to play equilibrium • does not suggest how we might choose among multiple equilibria • Nash equilibrium is a strictly competitive notion • players cannot have “pre-play communication” • bargains, side payments, threats, collusions, etc. not allowed • Computing Nash equilibria for large games is difficult G. Michael Youngblood, Ph.D.