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博弈论 （ GAME THEORY ） LECTURE 1

Tom H. Luan （栾浩） tom.luan@xidian.edu.cn CES, Xidian University. 博弈论 （ GAME THEORY ） LECTURE 1. Resource. Lecturer : Tom H. Luan ( 栾浩 ) Office: 行政辅楼 318 Web: web.xidian.edu.cn/luanhao Email: tom.luan@ xidian.edu.cn Text Robert Gibbons, 1992, A Primer in Game Theory , Prentice Hall.

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博弈论 （ GAME THEORY ） LECTURE 1

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  1. Tom H. Luan （栾浩） tom.luan@xidian.edu.cn CES, Xidian University 博弈论 （GAME THEORY）LECTURE 1

  2. Resource • Lecturer : Tom H. Luan (栾浩) • Office: 行政辅楼 318 • Web: web.xidian.edu.cn/luanhao • Email: tom.luan@xidian.edu.cn • Text • Robert Gibbons, 1992, A Primer in Game Theory, Prentice Hall

  3. Grading • 40%: Reading (阅读)：阅读一篇IEEE Transactions论文，完成以下工作： • 小报告（30%）： • 总结论文内容，涵盖题目、文章出处、问题介绍、博弈论建模、算法设计、结论，以及最具代表性仿真2附图，完成一个4页纸中文报告 • 演示（ 10% ）：10页幻灯片（包括首页与感谢页），描述所调研内容 • 60%：Project （项目）：完成一个报告 • 选择一：调研。针对一个领域课题，如云计算服务器访问控制，调研使用博弈论方法对此课题的研究，完成一个15页纸调研报告。 • 选择二：设计。结合自己的科研课题，使用博弈论方法进行建模，并给出初步解决思路，完成一个7页纸报告。 • 报告格式： IEEE Journal格式，单栏，1.5倍行距， 12pt字体

  4. Outline of Static Games of Complete Information • Introduction to games • Normal-form (or strategic-form) representation • Iterated elimination of strictly dominated strategies • Nash equilibrium • Applications of Nash equilibrium • Mixed strategy Nash equilibrium

  5. 什么是博弈论？ • 什么是博弈？任何需要顾及到个体利益的决策过程，都是博弈 • 博弈的特点： • 分布式：一般没有中央控制单元，博弈者各自为政 • 多成员：至少包含两名或以上博弈者 • 相互联系：一名博弈者的决定，可能会影响其他博弈者 • 对完整博弈过程的一种逻辑的分析 • 博弈过程：如大国博弈（贸易出口），或个人博弈（股票投资） • 一种逻辑：没个博弈者（Game Player）都是理性的获取其最大利益 • 分析： • 如何制定博弈规则，从而形成最佳的系统（如联合国机制）？ • 每个博弈者，应当如何制定对其最优的策略，从而最大化其收益？

  6. 稳定是关键！ • 什么是博弈？任何需要顾及到个体利益的决策过程，都是博弈 • 博弈的特点： • 分布式：一般没有中央控制单元，博弈者各自为政 • 多成员：至少包含两名或以上博弈者 • 相互联系：一名博弈者的决定，可能会影响其他博弈者 • 基础是稳定（纳什均衡（Nash Equilibrium）） • 目标是利益最大化（ Pareto Optimal ）

  7. What is game theory? • We focus on games where: • There are at least two rational players • Each player has more than one choices • The outcome depends on the strategies chosen by all players; there is strategic interaction • Strategic externality • Example: Six people go to a restaurant. • Each person pays his/her own meal – a simple decision problem • Before the meal, every person agrees to split the bill evenly among them – a game

  8. A Beautiful Mind • 约翰·纳什，生于1928年6月13日。著名经济学家、博弈论创始人，因对博弈论和经济学产生了重大影响，而获得1994年诺贝尔经济学奖。2015年5月23日，于美国新泽西州逝世 • 1950年于其仅27页的博士论文中提出重要发现，这就是后来被称为“纳什均衡”的博弈理论

  9. What is game theory? • Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) • Game theory has applications • Economics/Politics/Sociology/Law/Biology • The “double helix” and unifying tool for social scientists

  10. Prisoner 2 Mum Confess Mum Prisoner 1 Confess Classic Example: Prisoners’ Dilemma • Two suspects held in separate cells are charged with a major crime. However, there is not enough evidence. • Both suspects are told the following policy: • If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. • If both confess then both will be sentenced to jail for six months. • If one confesses but the other does not, then the confessor will be released but the other will be sentenced to jail for nine months.

  11. Pat Opera Prize Fight Opera Chris Prize Fight Example: The battle of the sexes • At the separate workplaces, Chris and Pat must choose to attend either an opera or a prize fight in the evening. • Both Chris and Pat know the following: • Both would like to spend the evening together. • But Chris prefers the opera. • Pat prefers the prize fight. • Non-zero-sum game

  12. Player 2 Head Tail Head Player 1 Tail Example: Matching pennies • Each of the two players has a penny. • Two players must simultaneously choose whether to show the Head or the Tail. • Both players know the following rules: • If two pennies match (both heads or both tails) then player 2 wins player 1’s penny. • Otherwise, player 1 wins player 2’s penny. • Zero-sum game: no way for collaboration

  13. A set of players (at least two players) For each player, a set of strategies/actions Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies {Player 1, Player 2, ... Player n} S1 S2 ... Sn ui(s1, s2, ...sn), for alls1S1, s2S2, ... snSn. Static (or simultaneous-move) games of complete information A static (or simultaneous-move) game consists of:

  14. Static (or simultaneous-move) games of complete information • Simultaneous-move • Each player chooses his/her strategy without knowledge of others’ choices. • Complete information (on game’s structure) • Each player’s strategies and payoff function are common knowledge among all the players. • Assumptions on the players • Rationality • Players aim to maximize their payoffs • Players are perfect calculators • Each player knows that other players are rational

  15. Static (or simultaneous-move) games of complete information • The players cooperate? • No. Only non-cooperative games • Methodological individualism • The timing • Each player i chooses his/her strategy siwithout knowledge of others’ choices. • Then each player i receives his/her payoff ui(s1, s2, ..., sn). • The game ends.

  16. Definition: normal-form or strategic-form representation • The normal-form (or strategic-form) representation of a game G specifies: • A finite set of players {1, 2, ..., n}, • players’ strategy spaces S1 S2 ... Sn and • their payoff functions u1 u2 ... unwhere ui : S1 × S2 × ...× Sn→R.

  17. Normal-form representation: 2-player game • Bi-matrix representation • 2 players: Player 1 and Player 2 • Each player has a finite number of strategies • Example:S1={s11, s12, s13} S2={s21, s22} Bi-matrix: a matrix with two elements per cell

  18. Prisoner 2 Players Strategies Mum Confess Mum Prisoner 1 Confess Classic example: Prisoners’ Dilemma:normal-form representation • Set of players: {Prisoner 1, Prisoner 2} • Sets of strategies:S1= S2= {Mum, Confess} • Payoff functions:u1(M, M)=-1, u1(M, C)=-9, u1(C, M)=0, u1(C, C)=-6;u2(M, M)=-1, u2(M, C)=0, u2(C, M)=-9, u2(C, C)=-6 Payoffs

  19. Pat Opera Prize Fight Opera Chris Prize Fight Example: The battle of the sexes • Normal (or strategic) form representation: • Set of players: { Chris, Pat } (={Player 1, Player 2}) • Sets of strategies: S1= S2 = { Opera, Prize Fight} • Payoff functions:u1(O, O)=2, u1(O, F)=0, u1(F, O)=0, u1(F, O)=1;u2(O, O)=1, u2(O, F)=0, u2(F, O)=0, u2(F, F)=2

  20. Player 2 Head Tail Head Player 1 Tail Example: Matching pennies • Normal (or strategic) form representation: • Set of players: {Player 1, Player 2} • Sets of strategies: S1= S2 = { Head, Tail } • Payoff functions:u1(H, H)=-1, u1(H, T)=1, u1(T, H)=1, u1(H, T)=-1;u2(H, H)=1, u2(H, T)=-1, u2(T, H)=-1, u2(T, T)=1

  21. Example: Tourists & Natives • Only two bars (bar 1, bar 2) in a city • Can charge price of $2, $4, or $5 • 6000 tourists pick a bar randomly • 4000 natives select the lowest price bar • Example 1: Both charge $2 • each gets 5,000 customers and $10,000 • Example 2: Bar 1 charges $4, Bar 2 charges $5 • Bar 1 gets 3000+4000=7,000 customers and $28,000 • Bar 2 gets 3000 customers and $15,000

  22. Example: Cournot model of duopoly • A product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. Each firm chooses the quantity without knowing the other firm has chosen. • The market price is P(Q)=a-Q, where Q=q1+q2. • The cost to firm i of producing quantity qi is Ci(qi)=cqi. The normal-form representation: • Set of players: { Firm 1, Firm 2} • Sets of strategies: S1=[0, +∞), S2=[0, +∞) • Payoff functions: u1(q1, q2)=q1(a-(q1+q2)-c), u2(q1, q2)=q2(a-(q1+q2)-c)

  23. One More Example • Each of n players selects a number between 0 and 100 simultaneously. Let xidenote the number selected by player i. • Let y denote the average of these numbers • Player i’s payoff = xi – 3y/5 • The normal-form representation:

  24. Prisoner 2 Players Strategies Mum Confess Mum Prisoner 1 Confess Solving Prisoners’ Dilemma • Confess always does better whatever the other player chooses • Dominated strategy • There exists another strategy which always does better regardless of other players’ choices Payoffs

  25. si” is strictly better than si’ Prisoner 2 Mum Confess regardless of other players’ choices Mum Prisoner 1 Confess Definition: strictly dominated strategy

  26. Example • Two firms, Reynolds and Philip, share some market • Each firm earns $60 million from its customers if neither do advertising • Advertising costs a firm $20 million • Advertising captures $30 million from competitor

  27. 2-player game with finite strategies • S1={s11, s12, s13} S2={s21, s22} • s11is strictly dominated bys12if u1(s11,s21)<u1(s12,s21) andu1(s11,s22)<u1(s12,s22). • s21 is strictly dominated bys22ifu2(s1i,s21) < u2(s1i,s22), for i = 1, 2, 3

  28. si” is at least as good as si’ Player 2 L R regardless of other players’ choices U Player 1 B Definition: weakly dominated strategy

  29. Strictly and weakly dominated strategy • A rational player never chooses a strictly dominated strategy. Hence, any strictly dominated strategy can be eliminated. • A rational player may choose a weakly dominated strategy. • The order of elimination does not matter for strict dominance elimination (pin down the same equilibrium), but does for weak one.

  30. Iterated elimination of strictly dominated strategies • If a strategy is strictly dominated, eliminate it • The size and complexity of the game is reduced • Eliminate any strictly dominated strategies from the reduced game • Continue doing so successively • Rationalizable equilibrium

  31. Player 2 Left Middle Up Player 1 Down Iterated elimination of strictly dominated strategies: an example Player 2 Left Middle Right Up Player 1 Down

  32. Example: Tourists & Natives • Only two bars (bar 1, bar 2) in a city • Can charge price of $2, $4, or $5 • 6000 tourists pick a bar randomly • 4000 natives select the lowest price bar • Example 1: Both charge $2 • each gets 5,000 customers and $10,000 • Example 2: Bar 1 charges $4, Bar 2 charges $5 • Bar 1 gets 3000+4000=7,000 customers and $28,000 • Bar 2 gets 3000 customers and $15,000

  33. Example: Tourists & Natives Payoffs are in thousands of dollars

  34. Example: Tourists & Natives • Inefficiency again as in Prisoners’ dilemma • Some games have no rationalizable equilbrium (co-ordination games) • Strategic ambiguity

  35. One More Example • Each of n players selects a number between 0 and 100 simultaneously. Let xidenote the number selected by player i. • Let y denote the average of these numbers • Player i’s payoff = xi – 3y/5

  36. One More Example • The normal-form representation: • Players: {player 1, player 2, ..., player n} • Strategies: Si=[0, 100], for i = 1, 2, ..., n. • Payoff functions: ui(x1, x2, ..., xn) = xi – 3y/5 • Is there any dominated strategy? • What numbers should be selected? • What if ui(x1, x2, ..., xn) = xi – 6y/5

  37. New solution concept: Nash equilibrium The combination of strategies (B, R) has the following property: • Player 1 CANNOT do better by choosing a strategy different from B, given that player 2 chooses R. • Player 2 CANNOT do better by choosing a strategy different from R, given that player 1 chooses B.

  38. New solution concept: Nash equilibrium The combination of strategies (B’, R’) has the following property: • Player 1 CANNOT do better by choosing a strategy different from B’, given that player 2 chooses R’. • Player 2 CANNOT do better by choosing a strategy different from R’, given that player 1 chooses B’.

  39. Nash Equilibrium: idea • Nash equilibrium • A set of strategies, one for each player, such that each player’s strategy is best for her, given that all other players are playing their equilibrium strategies • Two interpretations of Nash Equilibrium • Probability based one • Population based one • Symmetric games • Asymmetric games

  40. Given others’ choices, player i cannot be better-off if she deviates from si* (cf: dominated strategy) Definition: Nash Equilibrium

  41. 2-player game with finite strategies • S1={s11, s12, s13} S2={s21, s22} • (s11,s21)is a Nash equilibrium if u1(s11,s21) u1(s12,s21), u1(s11,s21) u1(s13,s21) andu2(s11,s21) u2(s11,s22).

  42. Player 2 Left Middle Up Player 1 Down Finding a Nash equilibrium: cell-by-cell inspection Player 2 Left Middle Right Up Player 1 Down

  43. Example: Tourists & Natives Payoffs are in thousands of dollars

  44. One More Example • The normal-form representation: • Players: {player 1, player 2, ..., player n} • Strategies: Si=[0, 100], for i = 1, 2, ..., n. • Payoff functions: ui(x1, x2, ..., xn) = xi – 3y/5 • What is the Nash equilibrium?

  45. Best response function: example • If Player 2 chooses L’ then Player 1’s best strategy is M’ • If Player 2 chooses C’ then Player 1’s best strategy is T’ • If Player 2 chooses R’ then Player 1’s best strategy is B’ • If Player 1 chooses B’ then Player 2’s best strategy is R’ • Best response: the best strategy one player can play, given the strategies chosen by all other players

  46. Example: Tourists & Natives • what is Bar 1’s best response to Bar 2’s strategy of $2, $4 or $5? • what is Bar 2’s best response to Bar 1’s strategy of $2, $4 or $5? Payoffs are in thousands of dollars

  47. 2-player game with finite strategies • S1={s11, s12, s13} S2={s21, s22} • Player 1’s strategy s11is her best response to Player 2’s strategys21if u1(s11,s21) u1(s12,s21) andu1(s11,s21) u1(s13,s21).

  48. Prisoner 2 Mum Confess Mum Prisoner 1 Confess Using best response function to find Nash equilibrium • In a 2-player game, ( s1, s2 ) is a Nash equilibrium if and only if player 1’s strategy s1 is her best response to player 2’s strategy s2, and player 2’s strategy s2 is her best response to player 1’s strategy s1.

  49. Using best response function to find Nash equilibrium: example • M’ is Player 1’s best response to Player 2’s strategy L’ • T’ is Player 1’s best response to Player 2’s strategy C’ • B’ is Player 1’s best response to Player 2’s strategy R’ • L’ is Player 2’s best response to Player 1’s strategy T’ • C’ is Player 2’s best response to Player 1’s strategy M’ • R’ is Player 2’s best response to Player 1’s strategy B’

  50. Example: Tourists & Natives Payoffs are in thousands of dollars Use best response function to find the Nash equilibrium.

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