Game Theory Lecture Notes

Transcription

1 Game Theory Lecture Notes Sérgio O. Parreiras Economics Department, UNC at Chapel Hill Spring, 2015

2 Outline Road Map Decision Problems Static Games Nash Equilibrium Pareto Efficiency Constrained Optimization Applications of Nash Equilibrium Other Solution Concepts Mixed Strategies Dynamic Games Bargaining Bayesian and Extensive Games of Incomplete Info Insurance Repeated Games Cooperative Game Theory

3 Game Theory: Binding Agreements Can players enter into bidding agreements?

4 Game Theory: Binding Agreements yes Cooperative GT. Can players enter into bidding agreements? no Non-Cooperative GT.

5 Non-Cooperative Games Uncertainty No Uncertainty Dynamic Static

6 Non-Cooperative Games Uncertainty No Uncertainty Dynamic Static 1. normal form games

7 Non-Cooperative Games Uncertainty No Uncertainty Dynamic 2. extensive games with perfect info. Static 1. normal form games

8 Non-Cooperative Games Uncertainty No Uncertainty Dynamic 2. extensive games with perfect info. Static 3. Bayesian games. 1. normal form games

9 Non-Cooperative Games Uncertainty No Uncertainty Dynamic 4. extensive games with imperfect info. 2. extensive games with perfect info. Static 3. Bayesian games. 1. normal form games

10 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

11 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

12 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

13 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

14 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

15 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

16 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

17 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

18 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

19 Strategic versus Decision Problems Decision Problems (studied by Operations Research) 1 Consumer only cares about prices not others actions 2 Competitive Firm can ignore the decision of other firms 3 Engineering Problems Strategic Problems (studied by Game Theory) 1 Agents care about the others decisions because... 2 their decisions affect the agents utility/profit/payoff. 3 put simply, there are externalities

20 Caveats The issue of weather a problem is strategic or non-strategic is a modeling choice Sometimes a seemingly non-strategic problem, like an engineering problem, may require strategic considerations. The Millenium Bridge wobblying. The Millenium Bridge (a possible explanation). Other times, a seemingly strategic problem, can be treated as a non-strategic one (no need tp over-analyze)

21 Caveats The issue of weather a problem is strategic or non-strategic is a modeling choice Sometimes a seemingly non-strategic problem, like an engineering problem, may require strategic considerations. The Millenium Bridge wobblying. The Millenium Bridge (a possible explanation). Other times, a seemingly strategic problem, can be treated as a non-strategic one (no need tp over-analyze)

22 Caveats The issue of weather a problem is strategic or non-strategic is a modeling choice Sometimes a seemingly non-strategic problem, like an engineering problem, may require strategic considerations. The Millenium Bridge wobblying. The Millenium Bridge (a possible explanation). Other times, a seemingly strategic problem, can be treated as a non-strategic one (no need tp over-analyze)

23 Caveats The issue of weather a problem is strategic or non-strategic is a modeling choice Sometimes a seemingly non-strategic problem, like an engineering problem, may require strategic considerations. The Millenium Bridge wobblying. The Millenium Bridge (a possible explanation). Other times, a seemingly strategic problem, can be treated as a non-strategic one (no need tp over-analyze)

24 Caveats The issue of weather a problem is strategic or non-strategic is a modeling choice Sometimes a seemingly non-strategic problem, like an engineering problem, may require strategic considerations. The Millenium Bridge wobblying. The Millenium Bridge (a possible explanation). Other times, a seemingly strategic problem, can be treated as a non-strategic one (no need tp over-analyze)

25 Looking back at Adam Smith 1 It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own interest. 2 People of the same trade seldom meet together, even for merriment and diversion, but the conversation ends in a conspiracy against the public, or in some contrivance to raise prices. It is impossible indeed to prevent such meetings, by any law which either could be executed, or would be consistent with liberty and justice. But though the law cannot hinder people of the same trade from sometimes assembling together, it ought to do nothing to facilitate such assemblies; much less to render them necessary.

26 Looking back at Adam Smith 1 It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own interest. 2 People of the same trade seldom meet together, even for merriment and diversion, but the conversation ends in a conspiracy against the public, or in some contrivance to raise prices. It is impossible indeed to prevent such meetings, by any law which either could be executed, or would be consistent with liberty and justice. But though the law cannot hinder people of the same trade from sometimes assembling together, it ought to do nothing to facilitate such assemblies; much less to render them necessary.

27 Decision Theory: Lotteries A lottery is a pair of outcomes and their respective probabilities: l = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )), where x k R and p k 0 for all k = 1,..., n and also p 1 + p p n = 1. x 1 x 2... x n p 1 p 2 p n l

28 The Certain Lottery, Expectation and Variance The lottery that gives outcome x with probability 1 (with certainty) is denoted: δ x = ((x), (1)). The expected value of the l 1 = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )) is: n E[l 1 ] = p 1 x 1 + p 2 x p n x n = p i x i ; and variance this lottery is Var[l 1 ] =p 1 (x 1 E[l 1 ]) 2 + p 2 (x 2 E[l 1 ]) p n (x n E[l 1 ]) 2 = n = p i (x i E[l 1 ]) 2. i=1 i=1

29 Composition of Lotteries Given two lotteries, l 1 = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )) and l 2 = (y 1, y 2,..., y m ), (q 1, q 2,..., q n )) and a number 0 < α < 1, one can create a compound lottery by choosing l 1 with probability α and l 2 with probability l 2.

30 Composition of Lotteries The compound lottery l plays l 1 with probability α and l 2 with probability l 2 : l = αl 1 (1 α)l 2 = = ((x 1, x 2, y 1, y 2 ), (αp, α(1 p), (1 α)q, (1 α)(1 q)). p x 1 l α 1 α l 1 l 2 1 p q x 2 y 1 1 q y 2

31 Composition of Lotteries The compound lottery l plays l 1 with probability α and l 2 with probability l 2 : l = αl 1 (1 α)l 2 = = ((x 1, x 2, y 1, y 2 ), (αp, α(1 p), (1 α)q, (1 α)(1 q)). x 1 l α p α (1 p) (1 α) q (1 α) (1 q) x 2 y 1 y 2

32 Preferences Over Lotteries Preferences or choices? Individual autonomy? Neurology?

33 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

34 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

35 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

36 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

37 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

38 Preferences Over Lotteries Given to lotteries l a and l b such that a decision maker (DM) chooses l a over l b, the following statements are equivalent: The DM judges l a no worst than l b (everyday language); The DM prefers l a to l b (economics language); l a l b (mathematics language). For simplicity we write: l a l b when l a l b but l b l a (strict preference) l a l b when l a l b and l b l a (indifference).

39 Preferences Over Lotteries A preference of the DM,, over the set of lotteries is just the DM s ranking of lotteries. We wish to have a numerical score that reflects the DM s ranking.

40 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

41 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

42 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

43 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

44 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3.

45 von Neuman & Morgenstern s Assumptions: Completeness For any two lotteries l 1 and l 2, l 1 l 2 and/or l 2 l 1. Transitivity For any lotteries l 1, l 2 and l 3, if l 1 l 2 and l 2 l 3 then l 1 l 3. Continuity If l 1 l 2 l 3 then exists p [0, 1] such that l 2 pl 1 (1 p)l 3. Independence If l 1 l 2 then for any l 3 and any 0 < p < 1, pl 1 (1 p)l 3 pl 2 (1 p)l 3. If satisfy all of of the above, there exists u : R R such that ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )) (y 1, y 2,..., y m ), (q 1, q 2,..., q n )) if and only if n u(x k ) p k > k=1 m u(y k ) q k. k=1

46 Expected Utility We write: U (l 1 ) = u(x 1 ) p u(x n ) p n and refer to U as the expected utility and to u as the: 1 utility for money 2 Bernoulli utility 3 von Neumann-Morgenstern utility (vn-m)

47 Expected Utility We write: U (l 1 ) = u(x 1 ) p u(x n ) p n, and refer to U as the expected utility and to u as the: 1 utility for money 2 Bernoulli utility 3 von Neumann-Morgenstern utility (vn-m)

48 Expected Utility We write: U (l 1 ) = u(x 1 ) p u(x n ) p n, and refer to U as the expected utility and to u as the: 1 utility for money 2 Bernoulli utility 3 von Neumann-Morgenstern utility (vn-m)

49 Extracting u from

50 Extracting u from

51 Extracting u from

52 Extracting u from

53 A Behavioral Look at Choice Anchoring Availability Representativeness Optimism and over confidence Gains and losses Status Quo Bias Framming

54 Risk Aversion Let s go back to expected utility theory, consider the two lotteries: l 1 = ((100, 200), ( 1 2, 1 2 ) and We have δ 150 = ((150), (1)) U (l 1 ) = u(150 50) u( ) 2 2 U (δ 150 ) = u(150) 1. and

55 Risk Aversion Let s go back to expected utility theory, consider the two lotteries: l 1 = ((100, 200), ( 1 2, 1 2 ) and We have δ 150 = ((150), (1)) U (l 1 ) = u(150 50) u( ) 2 2 U (δ 150 ) = u(150) 1. and

56 Risk Aversion Let s go back to expected utility theory, consider the two lotteries: l 1 = ((100, 200), ( 1 2, 1 2 ) and We have δ 150 = ((150), (1)) U (l 1 ) = u(150 50) u( ) 2 2 U (δ 150 ) = u(150) 1. and

57 Risk Aversion Let s go back to expected utility theory, consider the two lotteries: l 1 = ((100, 200), ( 1 2, 1 2 ) and We have δ 150 = ((150), (1)) U (l 1 ) = u(150 50) u( ) 2 2 U (δ 150 ) = u(150) 1. and

58 Risk Aversion [ u(200) u(150) U (l 1 ) U (δ 150 ) = 50 ] u(150) u(100) u u(200) u(100) x E[l 1 ]

59 Risk Aversion U (l 1 ) U (δ 150 ) = u u(200) u(150) 50 } {{ } Mu(150) u(150) u(100) } {{} 2 Mu(100) u(200) U (l 1 ) u(100) x E[l 1 ]

60 Risk Aversion U (l 1 ) U (δ 150 ) = u u(200) u(150) 50 } {{ } Mu(150) u(150) u(100) } {{} 2 Mu(100) u(200) U (l 1 ) u(100) x E[l 1 ]

61 Risk Aversion U (l 1 ) U (δ 150 ) = u u(200) u(150) 50 } {{ } Mu(150) u(150) u(100) } {{} 2 Mu(100) u(200) U (l 1 ) u(100) x E[l 1 ]

62 Risk Aversion U (l 1 ) U (δ 150 ) = u u(200) u(150) 50 } {{ } Mu(150) u(150) u(100) } {{} 2 Mu(100) u(200) U (l 1 ) u(100) x E[l 1 ]

63 Expected Utility Theory Attitudes Towards Risk 1 Diminishing marginal utility, u is concave, u < 0, the consumer is risk-averse. U (X) < u(e[x]) for all X 2 Increasing marginal utility, u is convex, u > 0, the consumer is risk-loving. U (X) > u(e[x]) for all X 3 Constant marginal utility, u is affine (linear plus a constant),u = 0, the consumer is risk-neutral, U (X) = u(e[x]) for all X

64 Expected Utility Theory Attitudes Towards Risk 1 Diminishing marginal utility, u is concave, u < 0, the consumer is risk-averse. U (X) < u(e[x]) for all X 2 Increasing marginal utility, u is convex, u > 0, the consumer is risk-loving. U (X) > u(e[x]) for all X 3 Constant marginal utility, u is affine (linear plus a constant),u = 0, the consumer is risk-neutral, U (X) = u(e[x]) for all X

65 Expected Utility Theory Attitudes Towards Risk 1 Diminishing marginal utility, u is concave, u < 0, the consumer is risk-averse. U (X) < u(e[x]) for all X 2 Increasing marginal utility, u is convex, u > 0, the consumer is risk-loving. U (X) > u(e[x]) for all X 3 Constant marginal utility, u is affine (linear plus a constant),u = 0, the consumer is risk-neutral, U (X) = u(e[x]) for all X

66 Expected Utility Theory Attitudes Towards Risk 1 Diminishing marginal utility, u is concave, u < 0, the consumer is risk-averse. U (X) < u(e[x]) for all X 2 Increasing marginal utility, u is convex, u > 0, the consumer is risk-loving. U (X) > u(e[x]) for all X 3 Constant marginal utility, u is affine (linear plus a constant),u = 0, the consumer is risk-neutral, U (X) = u(e[x]) for all X

67 Measuring the Degree of Risk-Aversion The Arrow-Pratt or Absolute Measure of Definition The Arrow-Pratt absolute measure of risk-aversion of an agent with VN-M utility u at wealth level w is: ρ u (w) = u (w) u (w). If for two individual with VN-M utilties u and ũ we have that ρ u (w) > ρũ(w) for all wealth levels w then we say that the agent with utility u is more risk-averse than the agent with utility ũ.

68 Measuring the Degree of Risk-Aversion The Arrow-Pratt or Absolute Measure of Definition The Arrow-Pratt absolute measure of risk-aversion of an agent with VN-M utility u at wealth level w is: ρ u (w) = u (w) u (w). If for two individual with VN-M utilties u and ũ we have that ρ u (w) > ρũ(w) for all wealth levels w then we say that the agent with utility u is more risk-averse than the agent with utility ũ.

69 Measuring the Degree of Risk-Aversion The Relative Measure of Risk-Aversion Definition The relative absolute measure of risk-aversion of an agent with VN-M utility u at wealth level w is: r u (w) = u (w) w u. (w)

70 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = S i is the set of strategy profiles or outcomes. i I u i is the payoff function of player i. How a player evaluates outcomes? i I

71 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = S i is the set of strategy profiles or outcomes. i I u i is the payoff function of player i. How a player evaluates outcomes? i I

72 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = S i is the set of strategy profiles or outcomes. i I u i is the payoff function of player i. How a player evaluates outcomes? i I

73 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = S i is the set of strategy profiles or outcomes. i I u i is the payoff function of player i. How a player evaluates outcomes? i I

74 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = S i is the set of strategy profiles or outcomes. i I u i is the payoff function of player i. How a player evaluates outcomes? i I

75 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = S i is the set of strategy profiles or outcomes. i I u i is the payoff function of player i. How a player evaluates outcomes? i I

76 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = i I i I S i is the set of strategy profiles or outcomes. What can happen in the game? u i is the payoff function of player i. How a player evaluates outcomes?

77 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = i I i I S i is the set of strategy profiles or outcomes. What can happen in the game? u i is the payoff function of player i. How a player evaluates outcomes?

78 Strategic Problems Static Games: The Normal Form Definition: A strategic game Γ (normal form game) is a triple: ( ( ) ) Γ = I, S i, i I u i : S i R i I I is the set of players. Who plays the game? e.g. I ={1, 2,..., n} or I =[0, 1] S i is the strategy set of player i. What a player can do? S = i I i I S i is the set of strategy profiles or outcomes. What can happen in the game? u i is the payoff function of player i. How a player evaluates outcomes?

79 Strategic Games Remarks A strategic game Γ = (I, S, u) is a noon-cooperative game. We assume players are not able to communicate or engage in bidding agreements unless the communication and/or agreements are modeled explicitly as strategies in S = S i. i I In the case of two players (#I = 2), the normal form representation of the game is referred as a bi-matrix game. When asked "model this as a strategic game", you are being asked to describe I, S i and u i for all i in I.

80 Profile Notation Say we have a game with three players, each player can choose between A, B or C. Consider the strategy profile s = (A, B, C) where: player 1 chooses A, s 1 = A; player 2 chooses B, s 2 = B; and player 3 chooses C, s 3 = C. Very often we may want to distinguish a player. For example, if we want to take the point of view of player 2 we write the previous strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)). In sum, s 2 = (s 1, s 3 ) is

81 Profile Notation Say we have a game with three players, each player can choose between A, B or C. Consider the strategy profile s = (A, B, C) where: player 1 chooses A, s 1 = A; player 2 chooses B, s 2 = B; and player 3 chooses C, s 3 = C. Very often we may want to distinguish a player. For example, if we want to take the point of view of player 2 we write the previous strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)). In sum, s 2 = (s 1, s 3 ) is

82 Profile Notation Say we have a game with three players, each player can choose between A, B or C. Consider the strategy profile s = (A, B, C) where: player 1 chooses A, s 1 = A; player 2 chooses B, s 2 = B; and player 3 chooses C, s 3 = C. Very often we may want to distinguish a player. For example, if we want to take the point of view of player 2 we write the previous strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)). In sum, s 2 = (s 1, s 3 ) is

83 Profile Notation Say we have a game with three players, each player can choose between A, B or C. Consider the strategy profile s = (A, B, C) where: player 1 chooses A, s 1 = A; player 2 chooses B, s 2 = B; and player 3 chooses C, s 3 = C. Very often we may want to distinguish a player. For example, if we want to take the point of view of player 2 we write the previous strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)). In sum, s 2 = (s 1, s 3 ) is

84 Profile Notation Say we have a game with three players, each player can choose between A, B or C. Consider the strategy profile s = (A, B, C) where: player 1 chooses A, s 1 = A; player 2 chooses B, s 2 = B; and player 3 chooses C, s 3 = C. Very often we may want to distinguish a player. For example, if we want to take the point of view of player 2 we write the previous strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)). In sum, s 2 = (s 1, s 3 ) is

85 Profile Notation Say we have a game with three players, each player can choose between A, B or C. Consider the strategy profile s = (A, B, C) where: player 1 chooses A, s 1 = A; player 2 chooses B, s 2 = B; and player 3 chooses C, s 3 = C. Very often we may want to distinguish a player. For example, if we want to take the point of view of player 2 we write the previous strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)). In sum, s 2 = (s 1, s 3 ) is

86 Profile Notation Say we have a game with three players, each player can choose between A, B or C. Consider the strategy profile s = (A, B, C) where: player 1 chooses A, s 1 = A; player 2 chooses B, s 2 = B; and player 3 chooses C, s 3 = C. Very often we may want to distinguish a player. For example, if we want to take the point of view of player 2 we write the previous strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)). In sum, s 2 = (s 1, s 3 ) is the strategy profile of players distinct from player 2

87 Examples of Strategic Games 1 The Battle of Sexes game. 2 Prisoners Dilemma 3 Cournot Duopoly 4 Bertrand Duopoly 5 The Stag-Hunt game (with 3 players)

88 The Battle of Sexes The strategic game (normal form) I = {1, 2} S 1 = S 2 = {B, S}, S = S 1 S 2 = {(B, B), (B, S), (S, B), (S, S)} u 1 (B, B) = u 2 (S, S) = 3, u 1 (S, S) = u 2 (B, B) = 2, u 1 (B, S) = u 1 (S, B) = 0, u 2 (B, S) = u 2 (S, B) = 0.

89 The Battle of Sexes The strategic game (normal form) I = {1, 2} S 1 = S 2 = {B, S}, S = S 1 S 2 = {(B, B), (B, S), (S, B), (S, S)} u 1 (B, B) = u 2 (S, S) = 3, u 1 (S, S) = u 2 (B, B) = 2, u 1 (B, S) = u 1 (S, B) = 0, u 2 (B, S) = u 2 (S, B) = 0.

90 The Battle of Sexes The strategic game (normal form) I = {1, 2} S 1 = S 2 = {B, S}, S = S 1 S 2 = {(B, B), (B, S), (S, B), (S, S)} u 1 (B, B) = u 2 (S, S) = 3, u 1 (S, S) = u 2 (B, B) = 2, u 1 (B, S) = u 1 (S, B) = 0, u 2 (B, S) = u 2 (S, B) = 0.

91 The Battle of Sexes The strategic game (normal form) Player 2 Player 1 B B S 0 I = {1, 2} S 1 = S 2 = {B, S}, S = S 1 S 2 = {(B, B), (B, S), (S, B), (S, S)} S u 1 (B, B) = u 2 (S, S) = 3, u 1 (S, S) = u 2 (B, B) = 2, u 1 (B, S) = u 1 (S, B) = 0, u 2 (B, S) = u 2 (S, B) = 0.

92 The Battle of Sexes The strategic game (normal form) Player 2 Player 1 B B S 0 I = {1, 2} S 1 = S 2 = {B, S}, S = S 1 S 2 = {(B, B), (B, S), (S, B), (S, S)} S u 1 (B, B) = u 2 (S, S) = 3, u 1 (S, S) = u 2 (B, B) = 2, u 1 (B, S) = u 1 (S, B) = 0, u 2 (B, S) = u 2 (S, B) = 0.

93 The Battle of Sexes The strategic game (normal form) Player 2 Player 1 B B 3 2 S S 0 3 I = {1, 2} S 1 = S 2 = {B, S}, S = S 1 S 2 = {(B, B), (B, S), (S, B), (S, S)} u 1 (B, B) = u 2 (S, S) = 3, u 1 (S, S) = u 2 (B, B) = 2, u 1 (B, S) = u 1 (S, B) = 0, u 2 (B, S) = u 2 (S, B) = 0.

94 The Prisoners Dilemma Player 2 Player 1 Effort Shirk Effort Shirk

95 Golden Balls Player 2 Player 1 SPLIT STEAL SPLIT Golden Balls YouTube video Radio Lab podcast STEAL

96 The Stag Hunt Player 2 Player 1 Stag Hare Stag Hare

97 Infinite Regress? Or, to change the metaphor slightly, professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees. Keynes, The General Theory of Employment, Interest, and Money.

98 Infinite Regress? Or, to change the metaphor slightly, professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees. Keynes, The General Theory of Employment, Interest, and Money.

99 Infinite Regress? Or, to change the metaphor slightly, professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees. Keynes, The General Theory of Employment, Interest, and Money.

100 Infinite Regress? Or, to change the metaphor slightly, professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees. Keynes, The General Theory of Employment, Interest, and Money.

101 Infinite Regress? Or, to change the metaphor slightly, professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees. Keynes, The General Theory of Employment, Interest, and Money.

102 NASH EQUILIBRIUM Definition: The strategy profile a = (a 1,..., a I ) S is a Nash Equilibrium, if and only if, u i (a ) u i (a i, a i), for all i in I and for all a i in S i.

103 NASH EQUILIBRIUM Definition: The strategy profile a = (a 1,..., a I ) S is a Nash Equilibrium, if and only if, u i (a ) u i (a i, a i), for all i in I and for all a i in S i.

104 NASH EQUILIBRIUM Definition: The strategy profile a = (a 1,..., a I ) S is a Nash Equilibrium, if and only if, u i (a ) u i (a i, a i), for all i in I and for all a i in S i.

105 NASH EQUILIBRIUM Definition: The strategy profile a = (a 1,..., a I ) S is a Nash Equilibrium, if and only if, u i (a ) u i (a i, a i), for all i in I and for all a i in S i.

106 Nash Equilibrium Learning by Doing Exercise A wrong definition (or suggestion) regarding what Nash equilibrium is (about). LBD Exercise 1. Model as a strategic game the matching problem described in the bar scene of the movie "A Beautiful Mind". 2. Compute all the Nash equilibrium of the game. 3. Explain why the behavior recommendation suggested by the character "nash" in the scene fails to be a Nash equilibrium.

107 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

108 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

109 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

110 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

111 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

112 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

113 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

114 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

115 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

116 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

117 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

118 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

119 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

120 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

121 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

122 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

123 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

124 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

125 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

126 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

127 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

128 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

129 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

130 More Examples Player 2 Player 1 t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

131 More Examples Player 2 Player 1 Nash equilibrium is (s2,t1) and not (5,5)! t 1 t 2 t 3 s 1 4,3 2,7 0,4 s 2 5,5 5,-1-4,-2

132 Cooperative Games: Pareto Efficiency Definition Definition: The strategy profile a = (a 1,..., a I ) S is a efficient, if and only if, u i (a ) < u i (a) for some player i and some alternative profile a then exists another player j I such that u j (a ) > u j (a). Remark: A Nash equilibrium may or may not be efficient. Most often, it will fail to be efficient because of the externalities in the strategic environment.

133 Cooperative Games: Pareto Efficiency Definition Definition: The strategy profile a = (a 1,..., a I ) S is a efficient, if and only if, u i (a ) < u i (a) for some player i and some alternative profile a then exists another player j I such that u j (a ) > u j (a). Remark: A Nash equilibrium may or may not be efficient. Most often, it will fail to be efficient because of the externalities in the strategic environment.

134 Cooperative Games: Pareto Efficiency Definition Definition: The strategy profile a = (a 1,..., a I ) S is a efficient, if and only if, u i (a ) < u i (a) for some player i and some alternative profile a then exists another player j I such that u j (a ) > u j (a). Remark: A Nash equilibrium may or may not be efficient. Most often, it will fail to be efficient because of the externalities in the strategic environment.

135 Welfare Functions Definition: A welfare function is an increasing map from the player s payoffs into the real line, W : R R (u 1, u 2,..., u I ) W (u 1, u 2,..., u I ) u i W 0 for every player i = 1, 2,..., I. An iso-welfare curve is a region in the utility space such that the welfare function is constant, W (u 1, u 2,..., u I ) =cte.

136 Welfare Functions Definition: A welfare function is an increasing map from the player s payoffs into the real line, W : R R (u 1, u 2,..., u I ) W (u 1, u 2,..., u I ) u i W 0 for every player i = 1, 2,..., I. An iso-welfare curve is a region in the utility space such that the welfare function is constant, W (u 1, u 2,..., u I ) =cte.

137 Pareto Efficiency and Welfare in payoff space Player 2 Player 1 u 2 Effort Shirk 10 9 Effort Shirk u 1

138 Pareto Efficiency and Welfare in payoff space W (u 1, u 2 ) = u 1 + u 2 u W = u 1

139 Pareto Efficiency and Welfare in payoff space W (u 1, u 2 ) = u 1 + u 2 u W = u 1

140 Pareto Efficiency and Welfare in payoff space W (u 1, u 2 ) = u 1 + u 2 u W = 12 u 1

141 Pareto Efficiency and Welfare in payoff space W (u 1, u 2 ) = u 1 + u 2 u W = u 1

142 Pareto Efficiency and Welfare in payoff space u u 1

143 Pareto Efficiency and Welfare in payoff space u makes all better-off u 1 not efficient!

144 Pareto Efficiency and Welfare in payoff space u u 1

145 Pareto Efficiency and Welfare in payoff space u no alternative makes all better-off u 1 efficient!

146 Pareto Efficiency and Welfare in payoff space u u 1

147 Pareto Efficiency and Welfare in payoff space u no alternative makes all better-off u 1 efficient!

148 Pareto Efficiency and Welfare in payoff space u u 1

149 Pareto Efficiency and Welfare in payoff space u no alternative makes all better-off u 1 efficient!

150 Pareto Efficiency & Welfare Functions If s = (s 1,..., s n ) S maximizes W (u(s)) then s is an efficient outcome and u(s) = (u 1 (s),..., u n (s)) is an efficient payoff. Conversely if the set of feasible payoffs U = {v R n : exists s such that u(s) = v} is convex then any efficient payoff maximizes a linear welfare function. A set A in R n is convex if the line joining any two elements of the set is inside the set, that is for any a and b in A, we must have that α a + (1 α) b is in A for all α [0, 1]. We call a weighted average of the vectors a and b, which is also a vector, α a + (1 α) b, a "convex combination of a and b". Example: a = (5, 7) and b = ( 1, 0) then α a +(1 α) b = (α 5+(1 α) ( 1), α 7+(1 α) 0)(6α 1, 7α).

151 Pareto Efficiency & Welfare Functions If s = (s 1,..., s n ) S maximizes W (u(s)) then s is an efficient outcome and u(s) = (u 1 (s),..., u n (s)) is an efficient payoff. Conversely if the set of feasible payoffs U = {v R n : exists s such that u(s) = v} is convex then any efficient payoff maximizes a linear welfare function. A set A in R n is convex if the line joining any two elements of the set is inside the set, that is for any a and b in A, we must have that α a + (1 α) b is in A for all α [0, 1]. We call a weighted average of the vectors a and b, which is also a vector, α a + (1 α) b, a "convex combination of a and b". Example: a = (5, 7) and b = ( 1, 0) then α a +(1 α) b = (α 5+(1 α) ( 1), α 7+(1 α) 0)(6α 1, 7α).

152 Pareto Efficiency & Welfare Functions If s = (s 1,..., s n ) S maximizes W (u(s)) then s is an efficient outcome and u(s) = (u 1 (s),..., u n (s)) is an efficient payoff. Conversely if the set of feasible payoffs U = {v R n : exists s such that u(s) = v} is convex then any efficient payoff maximizes a linear welfare function. A set A in R n is convex if the line joining any two elements of the set is inside the set, that is for any a and b in A, we must have that α a + (1 α) b is in A for all α [0, 1]. We call a weighted average of the vectors a and b, which is also a vector, α a + (1 α) b, a "convex combination of a and b". Example: a = (5, 7) and b = ( 1, 0) then α a +(1 α) b = (α 5+(1 α) ( 1), α 7+(1 α) 0)(6α 1, 7α).

153 Pareto Efficiency & Welfare Functions If s = (s 1,..., s n ) S maximizes W (u(s)) then s is an efficient outcome and u(s) = (u 1 (s),..., u n (s)) is an efficient payoff. Conversely if the set of feasible payoffs U = {v R n : exists s such that u(s) = v} is convex then any efficient payoff maximizes a linear welfare function. A set A in R n is convex if the line joining any two elements of the set is inside the set, that is for any a and b in A, we must have that α a + (1 α) b is in A for all α [0, 1]. We call a weighted average of the vectors a and b, which is also a vector, α a + (1 α) b, a "convex combination of a and b". Example: a = (5, 7) and b = ( 1, 0) then α a +(1 α) b = (α 5+(1 α) ( 1), α 7+(1 α) 0)(6α 1, 7α).

154 Finding the efficient frontier for a Cournot duopoly Profit functions are u 1 (q 1, q 2 ) = (1 q 1 q 2 c 1 ) q 1 and u 2 (q 1, q 2 ) = (1 q 1 q 2 c 2 ) q 2 Claim: Let (v 1, v 2 ) be such that [ ( ) ] 1 2 c1 v 1 0, 2 v 2 = then (v 1, v 2 ) is an efficient payoff. max u q 1,q 2 (q 1, q 2 ) 2 subject to u 1 (q 1,q 2 )=v 1

155 Finding the efficient frontier for a Cournot duopoly u 1 (q 1, q 2 ) = v 1 2q (1 c 1 q 2 )q 1 v 1 = 0 u 1 (q 1, q 2 ) = v 1 q 1 = (1 c 1 q 2 ) + (1 c 1 q 2 ) 2 4( 2)( v 1 ) 2 ( 2) q 1 = (1 c 1 q 2 ) (1 c 1 q 2 ) 2 8v 1 4

156 Nash Equilibrium Examples 1 Battle of Sexes 2 Cournot Duopoly 3 Stag-Hunt (with 3 players) 4 Prisioner s Dilemma 5 Coordination Games

157 Cournot Duoploy Two firms produce an homogeneous good at unit cost c. The market demand is P = α βq where Q = q 1 + q 2. Firms simultaneously chose their output quantities. Payoffs are: u 1 (q 1, q 2 ) = (α βq 1 βq 2 c) q 1 and u 2 (q 1, q 2 ) = (α βq 1 βq 2 c) q 2 Claim: ( α c (q 1, q 2 ) = 3β, α c ) is the unique Nash equilibrium. 3β

158 Cournot Duoploy Two firms produce an homogeneous good at unit cost c. The market demand is P = α βq where Q = q 1 + q 2. Firms simultaneously chose their output quantities. Payoffs are: u 1 (q 1, q 2 ) = (α βq 1 βq 2 c) q 1 and u 2 (q 1, q 2 ) = (α βq 1 βq 2 c) q 2 Claim: ( α c (q 1, q 2 ) = 3β, α c ) is the unique Nash equilibrium. 3β

159 Cournot Duoploy cont. ( α c Proving that (q 1, q 2 ) = 3β, α c ) is the unique Nash 3β equilibrium... u 1 (q 1, α c 3β ) = (α βq 1 β α c 3β c) q 1 = ( 2 3 (α c) βq 1) q 1

160 BEST RESPONSE FUNCTION Notation: For a set Z, we write 2 Z to denote set of all parts of the set Z. Definition: The best response of player i to a given action profile of the other players is denoted by BR i : S i 2 S i and defined by: BR i (a i ) = arg max u i (a i, a i ), a i S i In another words,

161 BEST RESPONSE FUNCTION Notation: For a set Z, we write 2 Z to denote set of all parts of the set Z. Definition: The best response of player i to a given action profile of the other players is denoted by BR i : S i 2 S i and defined by: BR i (a i ) = arg max u i (a i, a i ), a i S i In another words, a i BR i (a i ) if and only if for any â i S i u i (a i, a i ) u i (â i, a i ).

162 BEST RESPONSE FUNCTION Notation: For a set Z, we write 2 Z to denote set of all parts of the set Z. Definition: The best response of player i to a given action profile of the other players is denoted by BR i : S i 2 S i and defined by: BR i (a i ) = arg max u i (a i, a i ), a i S i In another words, a i BR i (a i ) if and only if for any â i S i u i (a i, a i ) u i (â i, a i ).

163 BEST RESPONSE FUNCTION It follows (almost immediately) from the definitions: The strategy profile a = (a 1,..., a I ) is a Nash Equilibrium, if and only if, a i BR i (a i) for all i I.

164 BEST RESPONSE FUNCTION It follows (almost immediately) from the definitions: The strategy profile a = (a 1,..., a I ) is a Nash Equilibrium, if and only if, a i BR i (a i) for all i I.

165 Finding Nash Equilibrium by Means of Best Responses For simplicity assume that for any player i and any action profile of the other players a i, i s best response is unique. In this case, the Nash equilibria are the solutions of the system of I equations : a i = BR i (a i) for all i I.

166 Finding Nash Equilibrium by Means of Best Responses For simplicity assume that for any player i and any action profile of the other players a i, i s best response is unique. In this case, the Nash equilibria are the solutions of the system of I equations : a i = BR i (a i) for all i I.

167 Best Responses Player 2 Player 1 Stag Hare Legend: BR 2 = BR 1 = Stag Hare S 2 H S S H S 1

168 Best Responses Player 2 Player 1 Stag Hare Legend: BR 2 = BR 1 = Stag Hare S 2 H S S H S 1

169 Best Responses Player 2 Player 1 Stag Hare Legend: BR 2 = BR 1 = Stag Hare S 2 H S S H S 1

170 Best Responses Player 2 Player 1 Stag Hare Legend: BR 2 = BR 1 = Stag Hare S 2 H S S H S 1

171 Best Responses Player 2 Player 1 Stag Hare Stag Legend: BR 2 = BR 1 = Hare S 2 H Nash eq. = all players are best responding, best response curves cross. S S 1

172 Best Responses Player 2 Player 1 Effort Shirk Legend: BR 2 = BR 1 = Effort Shirk S 2 S E E S S 1

173 Best Responses Player 2 Player 1 Effort Shirk Legend: BR 2 = BR 1 = Effort Shirk S 2 S E E S S 1

174 Best Responses Player 2 Player 1 Effort Shirk Legend: BR 2 = BR 1 = Effort Shirk S 2 S E E S S 1

175 Best Responses Player 2 Player 1 Effort Shirk Legend: BR 2 = BR 1 = Effort Shirk S 2 S E E S S 1

176 Best Responses Player 2 Player 1 Effort Shirk Effort Legend: BR 1 = BR 2 = Shirk S 2 S Nash eq. = all players are best responding, best response curves cross. E S 1

177 Computing best responses How to find BR i (s i ). 1 If S i is finite we just pick all the a i that deliver the highest value of u i (, s i ), where s i is given, and the set of such s i is the best response of player i to s i. 2 If S i is a infinite and more exactly a subset of R k we can use calculus provided u i is differentiable. For simplicity assume S i = [a, b] and u i differentiable in this case: 1 We call all x such that satisfy the first-order condition for an interior maximum (FOC), s i u i (x, s i ) = 0, the candidates for a maximum of u i (, s i ). 2 If we have, the corner condition s i u i (x, a) 0, we also say that a is a (corner) candidate for a maximum. 3 If we have, the corner condition s i u i (x, b) 0, we also say that b is a (corner) candidate for a maximum. 4 Amongst all the candidates for a maximum identified in a,b and c, we pick the one(s) that yield the highest value for u i (, s i ).