1 Introduction to Game Theory (Deterministic Model) Non-cooperative Game Theory with Complete Information Prof. Dr. Jinxing Xie Department of Mathematical Sciences Tsinghua University, Beijing , China math.tsinghua.edu.cn Voice: (86-10) Fax: (86-10) Office: Rm. 1202, New Science Building

2 What is Game Theory?  “ No man is an island ”  Study of rational behavior in interactive or interdependent situations  Bad news: Knowing game theory does not guarantee winning  Good news: Framework for thinking about strategic interaction

3 Games We Play  Group projectsfree-riding, reputation  Flat tire coordination  GPA trap prisoner ’ s dilemma  Tennis / Baseball mixed strategies  Mean professors commitment  Traffic congestion  Dating information manipulation

4 Games Businesses Play  Patent races game of chicken  Drug testing mixed strategies  FCC spectrum auctions  Market entrycommitment  OPEC output choice collusion & enforcement  Stock options compensation schemes  Internet pricingmarket design

5 Why Study Game Theory? Because the press tells us to … “ As for the firms that want to get their hands on a sliver of the airwaves, their best bet is to go out first and hire themselves a good game theorist. ” The Economist, July 23,1994 p. 70 “ Game Theory, long an intellectual pastime, came into its own as a business tool. ” Forbes, July 3, 1995, p. 62. “ Game theory is hot. ” The Wall Street Journal, 13 February 1995, p. A14

6 Why Study Game Theory?  Because we can formulate effective strategy …  Because we can predict the outcome of strategic situations …  Because we can select or design the best game for us to be playing …

7 Why Study Game Theory? McKinsey:  John Stuckey & David White - Sydney  “ To help predict competitor behavior and determine optimal strategy, our consulting teams use techniques such as pay-off matrices and competitive games. ”  Tom Copeland - Director of Corporate Finance  “ Game theory can explain why oligopolies tend to be unprofitable, the cycle of over capacity and overbuilding, and the tendency to execute real options earlier than optimal. ”

8 Outline -- Concepts  Recognizing the game  Rules of the game  Simultaneous games Anticipating rival ’ s moves  Sequential games Looking forward – reasoning back  Mixed strategies Sensibility of being unpredictable  Repeated games Cooperation and agreeing to agree

9 Outline -- Applications  Winning the game  Commitment Credibility, threats, and promises  Information Strategic use of information  Bargaining Gaining the upper hand in negotiation  Auctions Design and Participation

10 Interactive Decision Theory  Decision theory  You are self-interested and selfish  Game theory  So is everyone else “ If it ’ s true that we are here to help others, then what exactly are the others here for? ” - George Carlin

11 The Golden Rule COMMANDMENT Never assume that your opponents’ behavior is fixed. Predict their reaction to your behavior.

12 The Matrix of Game Theory Non-cooperative Dynamic Games Non-cooperative Static Games

13 The Matrix of Non-cooperation Game Complete (Full) Information Incomplete Information Simultaneous Move (Static) Nash Equilibrium Bayesian Equilibrium Sequential Move (Dynamic) Subgame Perfect (Nash) Equilibrium Subgame Perfect Bayesian Equilibrium

14 Definition of a Game  Must consider the strategic (Normal) environment  Who are the PLAYERS? (Decision makers)  What STRATEGIES are available? (Feasible actions)  What are the PAYOFFS? (Objectives)  Rules of the game  What is the time-frame for decisions?  What is the nature of the conflict?  What is the nature of interaction?  What information is available?

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16 The Assumptions  Rationality  Players aim to maximize their payoffs  Players are perfect calculators  Common knowledge  Each player knows the rules of the game  Each player knows that each player knows the rules  Each player knows that each player knows that each player knows the rules  Each player knows that each player knows that each player knows that each player knows the rules  Each player knows that each player knows that each player knows that each player knows that each player knows the rules  Etc. etc. etc.

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34 Cournot’s Model of Oligopoly (Cournot, 1838)  Single good produced by n firms  Cost to firm i of producing q i units: C i (q i ), where C i is nonnegative and increasing  If firms’ total output is Q then market price is P(Q), where P is nonincreasing  Profit of firm i, as a function of all the firms’ outputs:

35 Cournot’s Model of Oligopoly Strategic (normal) form game:  players: firms  each firm’s set of actions: set of all possible outputs  each firm’s preferences are represented by its profit

36 Example: Duopoly  two firms  Inverse demand: constant unit cost: C i (q i ) = cq i, where c < 

37 Example: Duopoly

38 Example: Duopoly

39 Example: Duopoly  Best response function is: Same for firm 2: b 2 (q) = b 1 (q) for all q.

40 Example: Duopoly

41 Example: Duopoly Nash equilibrium: Pair (q* 1, q* 2 ) of outputs such that each firm’s action is a best response to the other firm’s action or q* 1 = b 1 (q* 2 ) and q* 2 = b 2 (q* 1 ) Solution: q 1 = (  − c − q 2 )/2 and q 2 = (  − c − q 1 )/2 q* 1 = q* 2 = (  − c)/3

42 Example: Duopoly

43 Example: Duopoly Conclusion:  Game has unique Nash equilibrium: (q* 1, q* 2 ) = ((  − c)/3, (  − c)/3)  At equilibrium, P* = (  + 2c)/3, each firm’s profit is  =  (  − c) 2 )/9  Total output 2(  − c)/3 lies between monopoly output (  − c)/2 and competitive output  − c.

44 Cournot’s Model of Oligopoly: Notes  Dependence of Nash equilibrium on number of firms  Comparison of Nash equilibrium with collusive outcomes (monopoly): If there are only one firm in the market: Max (  -q-c)q  q* = (  -c)/2 < 2(  -c)/3  If P(Q) =  -bQ: Cournot’s model gives (q* 1, q* 2 ) = ((  − c)/3b, (  − c)/3b) P* = (  + 2c)/3 (unchanged)  =  (  − c) 2 )/9b

45 Bertrand’s Model of Oligopoly (Bertrand, 1883)  Strategic variable price rather than output.  Single good produced by n firms  Cost to firm i of producing q i units: C i (q i ), where C i is nonnegative and increasing  If price is p, demand is D(p)  Consumers buy from firm with lowest price  Firms produce what is demanded

46 Bertrand’s Model of Oligopoly (Bertrand, 1883) Strategic game:  players: firms  each firm’s set of actions: set of all possible prices  each firm’s preferences are represented by its profit

47 Example: Duopoly  2 firms  C i (q i ) = c q i for i = 1, 2  D(p) =  − p  p =  - D  P in [0, ∞], or actually in [0,  ]?

48 Example: Duopoly Nash Equilibrium: (p 1, p 2 ) = (c, c) total quantity produced =  − c (?) If each firm charges a price of c then the other firm can do no better than charge a price of c also (if it raises its price it sells no output, while if it lowers its price it makes a loss), so (c, c) is a Nash equilibrium.

49 Example: Duopoly No other pair (p 1, p 2 ) is a Nash equilibrium since  If p i < c then the firm whose price is lowest (or either firm, if the prices are the same) can increase its profit (to zero) by raising its price to c  If p i = c and p j > c then firm i is better off increasing its price slightly  if p i ≥ p j > c then firm i can increase its profit by lowering p i to some price between c and p j (e.g. to slightly below p j if D(p j ) > 0 or to p monop if p j > p monop ).

50 Bertrand’s Model: Notes If D(p) has the form: p =  − bD : Nash Equilibrium unchanged: (p 1, p 2 ) = (c, c) total quantity produced = (  − c)/b (0?)  If the products produced by two firms are non-identical: D i (p i )=  − p i + bp j (i=2-j)  p i * =(  + c + bp j * )  p 1 * =p 2 * =(  + c ) / (2-b)

51 Hotelling’s Model of Electoral Competition  Several candidates run for political office  Each candidate chooses a policy position  Each citizen, who has preferences over policy positions, votes for one of the candidates  Candidate who obtains the most votes wins.

52 Hotelling’s Model of Electoral Competition Strategic game:  Players: candidates  Set of actions of each candidate: set of possible positions  Each candidate gets the votes of all citizens who prefer her position to the other candidates’ positions; each candidate prefers to win than to tie than to lose. Note: Citizens are not players in this game.

53 Example  Two candidates  Set of possible positions is a (one-dimensional) interval.  Each voter has a single favorite position, on each side of which her distaste for other positions increases equally.  Unique median favorite position m among the voters: the favorite positions of half of the voters are at most m, and the favorite positions of the other half of the voters are at least m.

54 Example Direct argument for Nash equilibrium (m, m) is an equilibrium: if either candidate chooses a different position she loses. No other pair of positions is a Nash equilibrium:  If one candidate loses then she can do better by moving to m (where she either wins or ties for first place)  If the candidates tie (because their positions are either the same or symmetric about m), then either candidate can do better by moving to m, where she wins.

55 Sequential Game “ Life must be understood backward, but … it must be lived forward. ” - Soren Kierkegaard

56 Games of Chicken  A monopolist faces a potential entrant  Monopolist can accommodate or fight  Potential entrant can enter or stay out Monopolist AccommodateFight In 50, 50-50, -50 Out 0, 100 Potential Entrant

57 Equilibrium  Use best reply method to find equilibria Monopolist AccommodateFight In , 50-50, -50 Out 100 0, , 100 Potential Entrant

58 Importance of Order  Two equilibria exist  ( In, Accommodate )  ( Out, Fight )  Only one makes temporal sense  Fight is a threat, but not credible  Not sequentially rational  Simultaneous outcomes may not make sense for sequential games.

59 Sequential Games E out in M fight acc 0, , , 50 The Extensive Form

60 Looking Forward …  Entrant makes the first move:  Must consider how monopolist will respond  If enter:  Monopolist accommodates M fight acc -50, , 50

61  Now consider entrant ’ s move  Only ( In, Accommodate ) is sequentially rational … And Reasoning Back E out in M 0, , 50 acc

62 Sequential Rationality COMMANDMENT Look forward and reason back. Anticipate what your rivals will do tomorrow in response to your actions today

63 Solving Sequential Games: Backward Induction  Start with the last move in the game  Determine what that player will do  Trim the tree  Eliminate the dominated strategies  This results in a simpler game  Repeat the procedure

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73 Equilibrium  What is likely to happen when rational players interact in a game?  Type of equilibrium depends on the game  Simultaneous or sequential  Perfect or limited information  Concept always the same:  Each player is playing the best response to other players ’ actions  No unilateral motive to change  Self-enforcing

74 Summary  Recognizing that you are in a game  Identifying players, strategies, payoffs  Understanding the rules  Manipulating the rules  Nash Equilibrium  Subgame Perfect Equilibrium  Best response strategy (function)  Backward Induction  Searching for possible outcomes