Game Theory Lecture Notes

Transcription

1 Game Theory Lecture Notes Sérgio O. Parreiras Economics Department, UNC at Chapel Hill Fall, 2017

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

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 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 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 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 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 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 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 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 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 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 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 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

28 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

29 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

30 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

31 Choice Under Uncertainty States of the World Definition: State of the World A state of the world ω is a complete description of reality. The set of all possible sets of the world is denote by Ω. The above definition is too ambitious to be of any use. In practice, we assume Ω is small (finite). Because we are forced to simplify reality: You should think of a state of the world ω as describing all relevant information to some decision making problem.

32 States of the World Example 1 Decision Problem In a sunny morning, choose whether to carry an umbrella, a parasol or nothing to work. Possible States of The World Ω = {ω 1, ω 3, ω 3 }. ω 1 The afternoon is also sunny. ω 2 The afternoon is cloudy but it does not rain. ω 3 The afternoon is rainy.

33 States of the World Example 1 Decision Problem Buy, sell or take no action regarding Alibaba shares in the NYSE. Possible States of The World Ω = {ω 1, ω 3, ω 3, ω 4 }. ω 1 % BABA > 20%. ω 2 20% > % BABA > 5%. ω 3 5%> % BABA > 0%. ω 4 0%> % BABA > 10%. where above, % BABA is the rate of change of Alibaba s stock price, p 1 p 0 p 0, where p 1 is the future (six months ahead) price and p 0 is the current price.

34 Knowledge Is there any point to which you would wish to draw my attention? To the curious incident of the dog in the night-time. The dog did nothing in the night-time. That was the curious incident, remarked Sherlock Holmes. Source: Doyle, A. C. (1894). The Memoirs of Sherlock Holmes. London, England: George Newnes.

35 Knowledge Reports that say that something hasn t happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns? the ones we don t know we don t know. And if one looks throughout the history of our country and other free countries, it is the latter category that tend to be the difficult ones. Donald Rumsfeld, February 12, 2002.

36 Information Partitions Individual i s knowledge about the states of the world is represented by a partition P of the set Ω. Definition A partition P of Ω is a collection of subsets of Ω such that: 1. If A P and B P and A B then A B =. 2. A P A = P. If two states are in the same element of our knowledge partition, it means we cannot distinguish between these states. But if two states are if different elements of the partition, we can distinguish them.

37 Information Partitions Individual i s knowledge about the states of the world is represented by a partition P of the set Ω. Definition A partition P of Ω is a collection of subsets of Ω such that: 1. If A P and B P and A B then A B =. 2. A P A = P. If two states are in the same element of our knowledge partition, it means we cannot distinguish between these states. But if two states are if different elements of the partition, we can distinguish them.

38 Information partition Weather in the afternoon example Possible States of The World Ω = {ω 1, ω 2, ω 3 }. ω 1 The afternoon is also sunny. ω 2 The afternoon is cloudy but it does not rain. ω 3 The afternoon is rainy. a) P a = {{ω 1 }, {ω 2, ω 3 }}, we know if the afternoon is sunny or not, but if it is not sunny, we cannot tell if it is rainy or cloudy. b) P b = {{ω 1, ω 2, ω 3 }}, we do not know anything about the weather in the afternoon. c) P c = {{ω 1, ω 2 }, {ω 3 }}, we know if the afternoon is rainy or not, but if it is not rainy, we cannot tell if it is sunny or cloudy.

39 Information Partitions Cheryl s birthday Albert and Bernard just met Cheryl. "When s your birthday?" Albert asked Cheryl. Cheryl thought a second and said, "I m not going to tell you, but I ll give you some clues." She wrote down a list of 10 dates: May 15, May 16, May 19 June 17, June 18 July 14, July 16 August 14, August 15, August 17 "My birthday is one of these," she said. Then Cheryl whispered in Albert s ear the month and only the month of her birthday. To Bernard, she whispered the day, and only the day. "Can you figure it out now?" she asked Albert. Albert: I don t know when your birthday is, but I know Bernard doesn t know, either. Bernard: I didn t know originally, but now I do. Albert: Well, now I know, too! When is Cheryl s birthday?

40 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17

41 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 "whispered in Albert s ear the month"

42 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 "To Bernard, she whispered the day"

43 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 "whispered in Albert s ear the month" "To Bernard, she whispered the day"

44 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 "whispered in Albert s ear the month" "To Bernard, she whispered the day" Albert: I don t know when your birthday is, but I know Bernard doesn t know, either.

45 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 "whispered in Albert s ear the month" "To Bernard, she whispered the day" Albert: I don t know when your birthday is, but I know Bernard doesn t know, either. Bernard: I didn t know originally, but now I do.

46 MAY15 MAY16 MAY19 JUN 17 JUN18 JUL14 JUL16 AUG14 AUG15 AUG17 "whispered in Albert s ear the month" "To Bernard, she whispered the day" Albert: I don t know when your birthday is, but I know Bernard doesn t know, either. Bernard: I didn t know originally, but now I do. Albert: Well, now I know, too!

47 Lotteries Assume we have n states of the world. Definition: A lottery is a list of prizes and probabilities, l = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )), where x k R m is the prize the lottery gives when state k occurs, and p k 0 is the probability that state k occurs.

48 Lotteries Assume we have n states of the world. Definition: A lottery is a list of prizes and probabilities, l = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )), where x k R m is the prize the lottery gives when state k occurs, and p k 0 is the probability that state k occurs.

49 Lotteries In practice, it is very useful to depict lotteries as a graph. x 1 x 2... x n p 1 p 2 p n l

50 Lotteries In practice, it is very useful to depict lotteries as a graph. x 1 x 2... x n p 1 p 2 p n l

51 The Certain Lottery The lottery that gives prize x with probability one (with certainty) is denoted by: δ x = ((x), (1)). 1 x δ x

52 Expectation and Variance The expected value of a lottery l 1 = ((x 1, x 2,..., x n ), (p 1, p 2,..., p n )) is: E[l 1 ] = p 1 x 1 + p 2 x p n x n = and the variance of a lottery l is: n p i x i ; 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

53 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 α in the interval (0, 1), we can create a compound lottery l that plays l 1 with probability α and l 2 with probability 1 α. 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 x 2 y 1 y 2

54 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

55 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

56 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 DM 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).

57 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 DM 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).

58 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 DM 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).

59 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 DM 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).

60 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 DM 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).

61 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 DM 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).

62 Preferences Over Lotteries A preference of the DM, DM, over the set of lotteries is just the DM s ranking of lotteries. We wish (for convenience) a numerical score that reflects the DM s ranking.

63 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.

64 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.

65 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.

66 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.

67 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.

68 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 m u(x k ) p k > u(y k ) q k. k=1 k=1

69 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 (vnm)

70 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 (vnm)

71 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 (vnm)

72 How to use expected utility 1. List the states of the world: ω 1, ω 2,..., ω n. 2. List states probabilities: p 1, p 2,...,p n. 3. For each possible action: a) List its possible outcomes (prizea): x 1, x 2,..., x n. b) Obtain outcomes utilities: u(x 1 ), u(x 2 ),..., u(x n ). c) Calculate the expectation of the utility: p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n ). 4. Choose action that delivers the highest expected utility.

73 How to use expected utility 1. List the states of the world: ω 1, ω 2,..., ω n. 2. List states probabilities: p 1, p 2,...,p n. 3. For each possible action: a) List its possible outcomes (prizea): x 1, x 2,..., x n. b) Obtain outcomes utilities: u(x 1 ), u(x 2 ),..., u(x n ). c) Calculate the expectation of the utility: p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n ). 4. Choose action that delivers the highest expected utility.

74 How to use expected utility 1. List the states of the world: ω 1, ω 2,..., ω n. 2. List states probabilities: p 1, p 2,...,p n. 3. For each possible action: a) List its possible outcomes (prizea): x 1, x 2,..., x n. b) Obtain outcomes utilities: u(x 1 ), u(x 2 ),..., u(x n ). c) Calculate the expectation of the utility: p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n ). 4. Choose action that delivers the highest expected utility.

75 How to use expected utility 1. List the states of the world: ω 1, ω 2,..., ω n. 2. List states probabilities: p 1, p 2,...,p n. 3. For each possible action: a) List its possible outcomes (prizea): x 1, x 2,..., x n. b) Obtain outcomes utilities: u(x 1 ), u(x 2 ),..., u(x n ). c) Calculate the expectation of the utility: p 1 u(x 1 ) + p 2 u(x 2 ) p n u(x n ). 4. Choose action that delivers the highest expected utility.

76 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 2000x x 2. She has to choose between insurance or no insurance. 1. What are the states? 2. What are the states probabilities? 3. If she does not insure, what are her prizes in each state? 4. If she does not insure, what is her utility in each state? 5. What is the expected utility of not insuring? 6. What is the expected utility of insuring?

77 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 2000x x 2. She has to choose between insurance or no insurance. 1. What are the states? 2. What are the states probabilities? 3. If she does not insure, what are her prizes in each state? 4. If she does not insure, what is her utility in each state? 5. What is the expected utility of not insuring? 6. What is the expected utility of insuring?

78 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 2000x x 2. She has to choose between insurance or no insurance. 1. What are the states? 2. What are the states probabilities? 3. If she does not insure, what are her prizes in each state? 4. If she does not insure, what is her utility in each state? 5. What is the expected utility of not insuring? 6. What is the expected utility of insuring?

79 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 2000x x 2. She has to choose between insurance or no insurance. 1. What are the states? 2. What are the states probabilities? 3. If she does not insure, what are her prizes in each state? 4. If she does not insure, what is her utility in each state? 5. What is the expected utility of not insuring? 6. What is the expected utility of insuring?

80 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 2000x x 2. She has to choose between insurance or no insurance. 1. What are the states? 2. What are the states probabilities? 3. If she does not insure, what are her prizes in each state? 4. If she does not insure, what is her utility in each state? 5. What is the expected utility of not insuring? 6. What is the expected utility of insuring?

81 How to use expected utility An Example Suzan has to decide whether to insure (or not) her laptop which is worth $ 1,500. The insurance premium is $ 20 and the deductible is $ 200. The probability of theft or accident is 1/25 and the probability of nothing happening is 24/25. Her utility for money is u(x) = 2000x x 2. She has to choose between insurance or no insurance. 1. What are the states? 2. What are the states probabilities? 3. If she does not insure, what are her prizes in each state? 4. If she does not insure, what is her utility in each state? 5. What is the expected utility of not insuring? 6. What is the expected utility of insuring?

82 Extracting u from

83 Extracting u from

84 Extracting u from

85 Extracting u from

86 Risk Aversion Let s go back to expected utility theory, consider the two lotteries: l 1 = ((100, 102), ( 1 2, 1 2 ) and We have δ 101 = ((101), (1)) U (l 1 ) = u(100) u(102) 1 2 U (δ 101 ) = u(101) 1. and

87 Risk Aversion Let s go back to expected utility theory, consider the two lotteries: l 1 = ((100, 102), ( 1 2, 1 2 ) and We have δ 101 = ((101), (1)) U (l 1 ) = u(100) u(102) 1 2 U (δ 101 ) = u(101) 1. and

88 Risk Aversion U (l 1 ) U (δ 101 ) = [u(102) u(101) u(101) u(100)] 1 2 U (l 1 ) = u(100) u(102) 1 2 and U (δ 101 ) = u(101).

89 Risk Aversion U (l 1 ) U (δ 101 ) = u(102) u(101) u(101) u(100) 1 }{{}}{{} 2 Mu(101) Mu(100) U (l 1 ) = u(100) u(102) 1 2 and U (δ 101 ) = u(101).

90 Risk Aversion U (l 1 ) U (δ 101 ) = u(102) u(101) u(101) u(100) 1 }{{}}{{} 2 Mu(101) Mu(100) U (l 1 ) > U (δ 101 ) Mu(101) > Mu(100). U (l 1 ) = u(100) u(102) 1 2 and U (δ 101 ) = u(101).

91 Risk Aversion U (l 1 ) U (δ 101 ) = u(102) u(101) u(101) u(100) 1 }{{}}{{} 2 Mu(101) Mu(100) U (l 1 ) > U (δ 101 ) Mu(101) > Mu(100). U (l 1 ) = U (δ 101 ) Mu(101) = Mu(100). U (l 1 ) < U (δ 101 ) Mu(101) < Mu(100). U (l 1 ) = u(100) u(102) 1 2 and U (δ 101 ) = u(101).

92 Risk Aversion u u(102) u(100) x U (l 1 ) E[l 1 ] U (δ 101 ) = u(101)

93 Risk Aversion u u(102) 1 2 u(100) u(102) u(100) x U (l 1 ) E[l 1 ] U (δ 101 ) = u(101)

94 Risk Aversion u u(102) 1 2 u(100) u(102) u(100) x U (l 1 ) E[l 1 ] U (δ 101 ) = u(101)

95 Risk Aversion u u(102) 1 2 u(100) u(102) u(100) x U (l 1 ) E[l 1 ] U (δ 101 ) = u(101)

96 Risk Aversion u u(102) 1 2 u(100) u(102) u(100) x U (l 1 ) E[l 1 ] U (δ 101 ) = u(101)

97 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

98 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

99 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

100 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

101 Measuring the Degree of Risk-Aversion The Arrow-Pratt or Absolute Measure of Risk Aversion 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 utilities 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.

102 Measuring the Degree of Risk-Aversion The Arrow-Pratt or Absolute Measure of Risk Aversion 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 utilities 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.

103 Measuring the Degree of Risk-Aversion The Relative Measure of Risk-Aversion We are not covering this material, please skip this slide... 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)

104 Defining 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? typically 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

105 Defining 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? typically 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

106 Defining 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? typically 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

107 Defining 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? typically 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

108 Defining 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? typically 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

109 Defining 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? typically 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

110 Defining 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? typically 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?

111 Defining 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? typically 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?

112 Defining 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? typically 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?

113 Profile Notation We borrow set theory language to describe a game but the profile notation is particular to game theory. 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. Sometimes we may want to distinguish a player. We may want to take the point of view of player 2 In this case we write the strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)), In sum, s 2 = (s 1, s 3 ) is

114 Profile Notation We borrow set theory language to describe a game but the profile notation is particular to game theory. 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. Sometimes we may want to distinguish a player. We may want to take the point of view of player 2 In this case we write the strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)), In sum, s 2 = (s 1, s 3 ) is

115 Profile Notation We borrow set theory language to describe a game but the profile notation is particular to game theory. 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. Sometimes we may want to distinguish a player. We may want to take the point of view of player 2 In this case we write the strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)), In sum, s 2 = (s 1, s 3 ) is

116 Profile Notation We borrow set theory language to describe a game but the profile notation is particular to game theory. 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. Sometimes we may want to distinguish a player. We may want to take the point of view of player 2 In this case we write the strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)), In sum, s 2 = (s 1, s 3 ) is

117 Profile Notation We borrow set theory language to describe a game but the profile notation is particular to game theory. 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. Sometimes we may want to distinguish a player. We may want to take the point of view of player 2 In this case we write the strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)), In sum, s 2 = (s 1, s 3 ) is

118 Profile Notation We borrow set theory language to describe a game but the profile notation is particular to game theory. 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. Sometimes we may want to distinguish a player. We may want to take the point of view of player 2 In this case we write the strategy profile s = (A, B, C) as s = (B, s 2 ) = (B, (A, C)), In sum, s 2 = (s 1, s 3 ) is

119 Profile Notation We borrow set theory language to describe a game but the profile notation is particular to game theory. 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. Sometimes we may want to distinguish a player. We may want to take the point of view of player 2 In this case we write the 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

120 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 playera)

121 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.

122 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.

123 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.

124 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.

125 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.

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

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

128 Golden Balls Player 2 Player 1 SPLIT STEAL SPLIT STEAL

129 The Stag Hunt Player 2 Player 1 Stag Hare Stag Hare

130 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.

131 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.

132 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.

133 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.

134 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.

135 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.

136 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.

137 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.

138 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.

139 Nash Equilibrium A wrong definition (or suggestion) regarding what Nash equilibrium is (about).

140 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

141 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