MATH 1020 Chapter 1: Introduction to Game theory

1. MATH 1020Chapter 1: Introduction to Game theory • Dr. Tsang

2. Why do we like games? amusement, thrill and the hope to win uncertainty – course and result of a game

3. Reasons for uncertainty randomness combinatorial multiplicity imperfect information

4. Three types of games bridge

6. Game Theory 博弈论 • Game theory is the study of how people interact and make decisions. • This broad definition applies to most of the social sciences, but game theory applies mathematical models to this interaction under the assumption that each person's behavior impacts the well-being of all other participants in the game. These models are often quite simplified abstractions of real-world interactions.

7. A cultural comment • The Chinese translation “博弈论” may be a little bit misleading. • Games are serious stuffs in western culture. • The Great Game: the strategic rivalry and conflict between the British Empire and the Russian Empire for supremacy in Central Asia (1813-1907). • Wargaming: informal name for military simulations, in which theories/tactics of warfare can be tested and refined without the need for actual hostilities.

8. The Great Game: Political cartoon depicting the Afghan Emir Sher Ali with his "friends" the Russian Bear and British Lion (1878)

9. What is Game Theory? Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome “how rationalindividuals make decisions when they are aware that their actions affect each other and when each individual takes this into account”

11. Brief History of Game Theory • Game theoretic notions go back thousands of years (Sun Tzu‘s writings孙子兵法) • 1913 - E. Zermelo provides the first theorem of game theory; asserts that chess is strictly determined • 1928 - John von Neumann proves the minimax theorem • 1944 - John von Neumann & Oskar Morgenstern write "Theory of Games and Economic Behavior” • 1950-1953 - John Nash describes Nash equilibrium (Nobel price 1994)

13. Rationality Assumptions: • humans are rational beings • humans always seek the best alternative in a set of possible choices Why assume rationality? • narrow down the range of possibilities • predictability

14. Utility Theory Utility Theory based on: • rationality • maximization of utility • may not be a linear function of income or wealth Utility is a quantification of a person's preferences with respect to certain behavior as oppose to other possible ones.

15. Game Theory in the Real World • Economists • innovated antitrust policy • auctions of radio spectrum licenses for cell phone • program that matches medical residents to hospitals. • Computer scientists • new software algorithms and routing protocols • Game AI • Military strategists • nuclear policy and notions of strategic deterrence. • Sports coaching staffs • run versus pass or pitch fast balls versus sliders. • Biologists • what species have the greatest likelihood of extinction.

16. What are the Games in Game Theory? • For Game Theory, our focus is on games where: • There are 2 or more players. • There is some choice of action where strategy matters. • The game has one or more outcomes, e.g. someone wins, someone loses. • The outcome depends on the strategies chosen by all players; there is strategic interaction. • What does this rule out? • Games of pure chance, e.g. lotteries, slot machines. (Strategies don't matter). • Games without strategic interaction between players, e.g. Solitaire.

17. Game Theory • Finding acceptable, if not optimal, strategies in conflict situations. • An abstraction of real complex situation • Assumes all human interactions can be understood and navigated by presumptions • players are interdependent • uncertainty: opponent’s actions are not entirely predictable • players take actions to maximize their gain/utilities

18. Types of games • zero-sum or non-zero-sum [if the total payoff of the players is always 0] • cooperative or non-cooperative [if players can communicate with each other] • complete or incomplete information [if all the players know the same information] • two-person or n-person • Sequential vs. Simultaneous moves • Single Play vs. Iterated

19. Essential Elements of a Game • The players • how many players are there? • does nature/chance play a role? • A complete description of what the players can do – the set of all possible actions. • The information that players have available when choosing their actions • A description of the payoff consequences for each player for every possible combination of actions chosen by all players playing the game. • A description of all players’ preferences over payoffs.

20. Normal Form Representation of Games A common way of representing games, especially simultaneous games, is the normal form representation, which uses a table structure called a payoff matrixto represent the available strategies (or actions) and the payoffs.

21. A payoff matrix: “to Ad or not to Ad” PLAYERS STRATEGIES PAYOFFS

22. The Prisoners' Dilemma囚徒困境 • Two players, prisoners 1, 2. • Each prisoner has two possible actions. • Prisoner 1: Don't Confess, Confess • Prisoner 2: Don't Confess, Confess • Players choose actions simultaneously without knowing the action chosen by the other. • Payoff consequences quantified in prison years. • If neither confesses, each gets 3 year • If both confess, each gets 5 years • If 1 confesses, he goes free and other gets 10 years • Prisoner 1 payoff first, followed by prisoner 2 payoff • Payoffs are negative, it is the years of loss of freedom

23. Prisoners’ Dilemma: payoff matrix 2 1

25. Prisoner’s Dilemma : Example of Non-Zero Sum Game • A zero-sum game is one in which the players' interests are in direct conflict, e.g. in football, one team wins and the other loses; payoffs sum to zero. • A game is non-zero-sum, if players interests are not always in direct conflict, so that there are opportunities for both to gain. • For example, when both players choose Don't Confess in the Prisoners' Dilemma

26. Zero-Sum Games • The sum of the payoffs remains constant during the course of the game. • Two sides in conflict • Being well informed always helps a player

27. Non-zero Sum Game • The sum of payoffs is not constant during the course of game play. • Some nonzero-sum games are positive sum and some are negative sum • Players may co-operate or compete.

28. Information • Players have perfect information if they know exactly what has happened every time a decision needs to be made, e.g. in Chess. • Otherwise, the game is one of imperfect information.

29. Imperfect Information • Partial or no information concerning the opponent is given in advance to the player’s decision, e.g. Prisoner’s Dilemma. • Imperfect information may be diminished over time if the same game with the same opponent is to be repeated.

30. Games of Perfect Information • The information concerning an opponent’s move is well known in advance, e.g. chess. • All sequential move games are of this type.

31. Games of Co-operation Players may improve payoff through • communicating • forming binding coalitions & agreements • do not apply to zero-sum games Prisoner’s Dilemma with Cooperation

32. Games of Conflict • Two sides competing against each other • Usually caused by complete lack of information about the opponent or the game • Characteristic of zero-sum games

33. Example of zero-sum game Matching Pennies Mis-matcher matcher

34. Rock-Paper-Scissors

35. Zero-sum game matrices are sometimes expressed with only one number in each box, in which case each entry is interpreted as a gain for row-player and a loss for column-player.

36. Strategies • A strategy is a “complete plan of action” that fully determines the player's behavior, a decision rule or set of instructions about which actions a player should take following all possible histories up to that stage. • The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). • A strategy on the other hand is a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game.

37. Dominant or dominated strategy A strategy S for a player A is dominantif it is always the best strategies for player A no matter what strategies other players will take. A strategy S for a player A is dominatedif it is always one of the worst strategies for player A no matter what strategies other players will take.

38. Use strategy 1 If you have a dominantstrategy, use it! Opponent Strategy 1 Strategy 2 Strategy 1 150 1000 You Strategy 2 25 - 10

39. Dominance Solvable • If each player has a dominant strategy, the game is dominance solvable COMMANDMENT If you have a dominant strategy, use it. Expect your opponent to use his/her dominant strategy if he/she has one.

40. Only one player has a Dominant Strategy • For The Economist: • G dominant, S dominated • Dominated Strategy: • There exists another strategy which always does better regardless of opponents’ actions

41. How to recognize a Dominant Strategy • To determine if the row player has any dominant strategy • Underline the maximum payoff in each column • If the underlined numbers all appear in a row, then it is the dominant strategy for the row player No dominant strategy for the row player in this example.

42. How to recognize a Dominant Strategy • To determine if the column player has any dominant strategy • Underline the maximum payoff in each row • If the underlined numbers all appear in a column, then it is the dominant strategy for the column player There is a dominant strategy for the column player in this example.

43. If there is no dominant strategy • Does any player have a dominant strategy? • If there is none, ask “Does any player have a dominated strategy?” • If yes, then • Eliminate the dominated strategies • Reduce the normal-form game • Iterate the above procedure

44. Eliminate strategy 2 as it’s dominated by strategy 1 Eliminate any dominated strategy Opponent Strategy 1 Strategy 2 Strategy 1 150 1000 You Strategy 2 25 - 10 160 -15 Strategy 3

45. Successive Elimination of Dominated Strategies • If a strategy is dominated, eliminate it • The size and complexity of the game is reduced • Eliminate any dominant strategies from the reduced game • Continue doing so successively

46. Example: Two competing Bars • Two bars (bar 1, bar 2) compete • Can charge price of $2, $4, or $5 for a drink • 6000 tourists pick a bar randomly • 4000 natives select the lowest price bar No dominant strategy for the both players. Bar 2

47. Bar 2 Successive Elimination of Dominated Strategies Bar 2 $2 $4 $5 $2 10 , , 10 14 , , 12 14 , , 15 Bar 1 Bar 1 $4 20 , , 20 28 , , 15 12 , , 14 $5 15 , , 28 25 , , 25 15 , , 14

48. An example for Successive Elimination of strictly dominated strategies, or the process of iterated dominance

49. Equilibrium • The interaction of all players' strategies results in an outcome that we call "equilibrium." • Traditional applications of game theory attempt to find equilibria in games. • In an equilibrium, each player is playing the strategy that is a "best response" to the strategies of the other players. No one is likely to change his strategy given the strategic choices of the others. • Equilibrium is not: • The best possible outcome. Equilibrium in the one-shot prisoners' dilemma is for both players to confess. • A situation where players always choose the same action. Sometimes equilibrium will involve changing action choices (known as a mixed strategy equilibrium).

50. Definition: Nash Equilibrium “If there is a set of strategies with the property that no player can benefit by changing his/her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.” Source: http://www.lebow.drexel.edu/economics/mccain/game/game.html