2.1 Chicken Revisited: Consider the game of chicken in Section 12.2.1 with the = parameters R...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
2.1 Chicken Revisited: Consider the game of chicken in Section 12.2.1 with the = parameters R = 8, H = 16, and L 0 as described there. A preacher, who knows some game theory, decides to use this model to claim that moving to a society in which all parents are lenient will have detrimental effects on the behavior of teenagers. Does equilibrium analysis support this claim? What if R = 8, H = 0, and L= 16? Com 122 252 Chapter 12 Bayesian Games 12.2 12.2.1 Just as in (12.2) we take the expectations of player i over the realizations of type 8, when player i knows his own type (hence the conditional expectation takes the form E-181, which is conditional on the realization of 8). This notation is more general in the sense that it is not restricted to a finite type space and instead can apply to a continuum of types for each player. For example, each player may have an infinite number of possible types drawn from an interval e, -10.0₁ with cumulative distribution F(8) and density function (0) F(0). In this case the expected payoff of player i will be written as an integral (more precisely, -1 integrals) that forms the expectations over the realizations of the other players types, and accordingly their actions as derived from their strategies. Examples This section offers two examples as illustrations of Bayesian games of incomplete information. In the first example each player has a finite number of actions and types, while in the second each player has a continuum of types. Sections 12.3-12.5 present some important applications of Bayesian games. Teenagers and the Game of Chicken It is often the case that a player who finds himself in some game of conflict will hold out and suffer in order to appear strong, instead of letting his rival get the better of him. Be it firms in the marketplace, politicians in government, countries at war, or even kids on the playground, the optimal behavior will depend on some combination of each player's tendency to be aggressive and his belief about his opponent's tendency to be aggressive. To illustrate this idea, consider a simple game of aggression that is known to many teenagers: the game of "chicken." The 1955 film Rebel Without a Cause features James Dean as a juvenile delinquent and introduced to the silver screen one rather dangerous variant of the game. In the movie two teenagers simultaneously drive their cars toward the edge of a cliff. The first one to jump out is considered chicken and loses the contest. (In the movie one dies, and coincidentally Dean himself died in a car accident just before the film was released.) Many other films have featured variations on this game of chicken, a well-known example being Grease (1978), starring John Travolta. The following example is yet another variant of the game. Two teenagers, players 1 and 2. have borrowed their parents' cars and decided to play the game of chicken as follows: They drive toward each other in the middle of a street, and just before impact they must simultaneously choose whether to be chicken and swerve to the right, or continue driving head on. If both are chicken then both gain no respect from their friends but suffer no losses; thus both get a payoff of 0. If i continues to drive while ji plays chicken then i gains all the respect, which is a payoff of R. and j gets no respect, which is worth 0. In this case both players suffer no additional losses. Finally if both continue to drive head on then they split the respect (because respect is considered to be relative), but an accident is bound to happen and they will at least be reprimanded by their parents, if not seriously injured. An accident imposes a personal loss of k on each player, so the payoff to each one is 4-k. 0 D d OR OSR-L RO 0.5R-L d C) LL C C 0 0 LH Nature D 2 d C c/ 0 0 d C HL d 2 HH D C d 0 R 0.5R-H R 0 0.58-L 6 C D d C/ d 0R 0.5R-L 00058- 0R0058- RO 0.5R-H FIGURE 12.3 The game of chicken with incomplete information. There is, however, a potential difference between these two youngsters: The punishment, k, depends on the type of parents they have. For each kid, parents can be either harsh (H) or lenient (L) with equal probability, and the draws from Nature on the types of parents are independently distributed. This means that the likelihood that player i's parents are harsh is equal to and is independent of the type of parents that player j has. If player i's parents are harsh then they will beat the living daylights out of their child, which imposes a high cost of an accident, denoted by k=H. If instead the parents are lenient then they will give their child a long lecture on why his behavior is unacceptable, which imposes a lower cost of an accident, denoted by k=L<H. Each kid knows the type of his parents but does not know the type of his opponent's parents. The distribution of types is common knowledge (this is the common prior assumption). The extensive form of this game is depicted in Figure 12.3. There are four states of Nature, 0,0₂ € (LL, LH, HL, HH) that denote the types of player 1 and player 2, respectively, and each of these four states occurs with equal probability because of the probability distributions described earlier. The structure of the information sets follows from the knowledge of the players when they make their moves. Player 1. for example, cannot distinguish between the states of Nature LL and LH (his type is L), nor can he distinguish between HL and H H (his type is H). However, he knows what his own type is, and hence he can distinguish between these two pairs of states. A similar logic explains the information sets of player 2. The action sets of each player are A₁ = (C. D) and A₂ = (c, d), where C (or c) stands for "chicken" and D (or d) stands for "drive." From the extensive form in Figure 12.3 we can derive the matrix form as follows. A strategy for player 1 is denoted by xy = S₁ = (CC, CD. DC. DD), where x is what he does if he is an L type and y is what he does if he is an H type. Similarly S₂=(cc, cd, de, dd). The payoffs are calculated using the probabilities over the states of Nature together with the strategies of the players. For example, if playe 1 chooses CD and player 2 chooses dd then the expected payoffs for playet Ev,(CD, dd)= x vy(C. d; L) +x (C, d; L) + x (D. d; H)+D. +4× 254 Chapter 12 Bayesian Games = Player 1 and the expected payoffs for player 2 are = x₂(C. d; L) + x v₂(C. d; H) + 1 × 1₂ (D. d; L) + x 03/0.48) - **+ ** + 1 × (-¹) + 1 × ( -#) - ² -4 -4 3R Eu, (CD, dd) = In a similar way we can calculate the payoffs from all the other combinations of pur strategies for each player (this would be useful practice for you if things still feel a bit shaky mathematically). This results in the following matrix-form Bayesian game cc ed 0.4 CC CD 4.0-4.*-* DC 4.0 *-.*-# DD 0,0 Player 1 Player 2 de се 0. 第一些,婆一季 *-.*-4 R.O-4-4-4-4-4-4-4 To solve for the Bayesian Nash equilibria we need to have more information about the parameters R, H, and L. Assume that R= 8, H=16, and L=0, which results in the following matrix-form game: Player 2 de ed 0,4 0,4 CC 0,0 CD 4,0 -1.-1 DC 4,0 3.-1 DD 8,0 2.-6 dd 0, R 4-4.*-4-4 4-4.*-1-1 4-4-4-4-4-4 -1.3 3,3 2,2 dd 0,8 -6,2 2,2 -4,14 A quick analysis reveals that the game has a unique pure-strategy Bayesian Nash equilibrium: (DC, de). That is, the children of lenient parents will continue driving head on, while those of harsh parents will swerve to avoid the costly consequences. If the payoffs we assumed are indeed representative of this situation then the view that harsh upbringing yields better outcomes can be supported. Perhaps, however, children of lenient parents learn somehow to respect their parents' property (the car), while children of harsh parents, despite anticipating the pain of a harsh punishment, do not respect their parents' property. In light of this different description, we might assume 12.2.2 12.2 Examples 255 that L=16 and H=0, which would result in the opposite prediction. Hence, as in any game, we need to get the payoffs right in order to feel comfortable offering policy recommendations. 2.1 Chicken Revisited: Consider the game of chicken in Section 12.2.1 with the = parameters R = 8, H = 16, and L 0 as described there. A preacher, who knows some game theory, decides to use this model to claim that moving to a society in which all parents are lenient will have detrimental effects on the behavior of teenagers. Does equilibrium analysis support this claim? What if R = 8, H = 0, and L= 16? Com 122 252 Chapter 12 Bayesian Games 12.2 12.2.1 Just as in (12.2) we take the expectations of player i over the realizations of type 8, when player i knows his own type (hence the conditional expectation takes the form E-181, which is conditional on the realization of 8). This notation is more general in the sense that it is not restricted to a finite type space and instead can apply to a continuum of types for each player. For example, each player may have an infinite number of possible types drawn from an interval e, -10.0₁ with cumulative distribution F(8) and density function (0) F(0). In this case the expected payoff of player i will be written as an integral (more precisely, -1 integrals) that forms the expectations over the realizations of the other players types, and accordingly their actions as derived from their strategies. Examples This section offers two examples as illustrations of Bayesian games of incomplete information. In the first example each player has a finite number of actions and types, while in the second each player has a continuum of types. Sections 12.3-12.5 present some important applications of Bayesian games. Teenagers and the Game of Chicken It is often the case that a player who finds himself in some game of conflict will hold out and suffer in order to appear strong, instead of letting his rival get the better of him. Be it firms in the marketplace, politicians in government, countries at war, or even kids on the playground, the optimal behavior will depend on some combination of each player's tendency to be aggressive and his belief about his opponent's tendency to be aggressive. To illustrate this idea, consider a simple game of aggression that is known to many teenagers: the game of "chicken." The 1955 film Rebel Without a Cause features James Dean as a juvenile delinquent and introduced to the silver screen one rather dangerous variant of the game. In the movie two teenagers simultaneously drive their cars toward the edge of a cliff. The first one to jump out is considered chicken and loses the contest. (In the movie one dies, and coincidentally Dean himself died in a car accident just before the film was released.) Many other films have featured variations on this game of chicken, a well-known example being Grease (1978), starring John Travolta. The following example is yet another variant of the game. Two teenagers, players 1 and 2. have borrowed their parents' cars and decided to play the game of chicken as follows: They drive toward each other in the middle of a street, and just before impact they must simultaneously choose whether to be chicken and swerve to the right, or continue driving head on. If both are chicken then both gain no respect from their friends but suffer no losses; thus both get a payoff of 0. If i continues to drive while ji plays chicken then i gains all the respect, which is a payoff of R. and j gets no respect, which is worth 0. In this case both players suffer no additional losses. Finally if both continue to drive head on then they split the respect (because respect is considered to be relative), but an accident is bound to happen and they will at least be reprimanded by their parents, if not seriously injured. An accident imposes a personal loss of k on each player, so the payoff to each one is 4-k. 0 D d OR OSR-L RO 0.5R-L d C) LL C C 0 0 LH Nature D 2 d C c/ 0 0 d C HL d 2 HH D C d 0 R 0.5R-H R 0 0.58-L 6 C D d C/ d 0R 0.5R-L 00058- 0R0058- RO 0.5R-H FIGURE 12.3 The game of chicken with incomplete information. There is, however, a potential difference between these two youngsters: The punishment, k, depends on the type of parents they have. For each kid, parents can be either harsh (H) or lenient (L) with equal probability, and the draws from Nature on the types of parents are independently distributed. This means that the likelihood that player i's parents are harsh is equal to and is independent of the type of parents that player j has. If player i's parents are harsh then they will beat the living daylights out of their child, which imposes a high cost of an accident, denoted by k=H. If instead the parents are lenient then they will give their child a long lecture on why his behavior is unacceptable, which imposes a lower cost of an accident, denoted by k=L<H. Each kid knows the type of his parents but does not know the type of his opponent's parents. The distribution of types is common knowledge (this is the common prior assumption). The extensive form of this game is depicted in Figure 12.3. There are four states of Nature, 0,0₂ € (LL, LH, HL, HH) that denote the types of player 1 and player 2, respectively, and each of these four states occurs with equal probability because of the probability distributions described earlier. The structure of the information sets follows from the knowledge of the players when they make their moves. Player 1. for example, cannot distinguish between the states of Nature LL and LH (his type is L), nor can he distinguish between HL and H H (his type is H). However, he knows what his own type is, and hence he can distinguish between these two pairs of states. A similar logic explains the information sets of player 2. The action sets of each player are A₁ = (C. D) and A₂ = (c, d), where C (or c) stands for "chicken" and D (or d) stands for "drive." From the extensive form in Figure 12.3 we can derive the matrix form as follows. A strategy for player 1 is denoted by xy = S₁ = (CC, CD. DC. DD), where x is what he does if he is an L type and y is what he does if he is an H type. Similarly S₂=(cc, cd, de, dd). The payoffs are calculated using the probabilities over the states of Nature together with the strategies of the players. For example, if playe 1 chooses CD and player 2 chooses dd then the expected payoffs for playet Ev,(CD, dd)= x vy(C. d; L) +x (C, d; L) + x (D. d; H)+D. +4× 254 Chapter 12 Bayesian Games = Player 1 and the expected payoffs for player 2 are = x₂(C. d; L) + x v₂(C. d; H) + 1 × 1₂ (D. d; L) + x 03/0.48) - **+ ** + 1 × (-¹) + 1 × ( -#) - ² -4 -4 3R Eu, (CD, dd) = In a similar way we can calculate the payoffs from all the other combinations of pur strategies for each player (this would be useful practice for you if things still feel a bit shaky mathematically). This results in the following matrix-form Bayesian game cc ed 0.4 CC CD 4.0-4.*-* DC 4.0 *-.*-# DD 0,0 Player 1 Player 2 de се 0. 第一些,婆一季 *-.*-4 R.O-4-4-4-4-4-4-4 To solve for the Bayesian Nash equilibria we need to have more information about the parameters R, H, and L. Assume that R= 8, H=16, and L=0, which results in the following matrix-form game: Player 2 de ed 0,4 0,4 CC 0,0 CD 4,0 -1.-1 DC 4,0 3.-1 DD 8,0 2.-6 dd 0, R 4-4.*-4-4 4-4.*-1-1 4-4-4-4-4-4 -1.3 3,3 2,2 dd 0,8 -6,2 2,2 -4,14 A quick analysis reveals that the game has a unique pure-strategy Bayesian Nash equilibrium: (DC, de). That is, the children of lenient parents will continue driving head on, while those of harsh parents will swerve to avoid the costly consequences. If the payoffs we assumed are indeed representative of this situation then the view that harsh upbringing yields better outcomes can be supported. Perhaps, however, children of lenient parents learn somehow to respect their parents' property (the car), while children of harsh parents, despite anticipating the pain of a harsh punishment, do not respect their parents' property. In light of this different description, we might assume 12.2.2 12.2 Examples 255 that L=16 and H=0, which would result in the opposite prediction. Hence, as in any game, we need to get the payoffs right in order to feel comfortable offering policy recommendations.
Expert Answer:
Answer rating: 100% (QA)
Introduction The provided text introduces the game of chicken a scenario often observed in conflicts where players choose to endure potential harm rather than appear weak The specific example involves ... View the full answer
Related Book For
Discrete and Combinatorial Mathematics An Applied Introduction
ISBN: 978-0201726343
5th edition
Authors: Ralph P. Grimaldi
Posted Date:
Students also viewed these accounting questions
-
Consider the game of Roulette as described in Example 3.44. (a) If the game is played once, what is the probability the outcome is (i) High or odd. (ii) Low or black? (b) If the game is played twice,...
-
Consider the game of tennis when deuce is reached. If a player wins the next point, he has advantage. On the following point, he either wins the game or the game returns to deuce. Assume that for any...
-
At the Racho San Joaquin Golf Course adjacent to the campus, the average number of strokes for a golfer to finish the course (i.e. complete one game of golf) is normally distributed with mean 105 and...
-
Carey Company is borrowing $200,000 for one year at 12 percent from Second Intrastate Bank. The bank requires a 20 percent compensating balance. What is the effective rate of interest? What would the...
-
Hanes Sport Wear is facing increased price competition, and its market price has fallen by $3. What approach besides target costing can Hanes use to reduce manufacturing cost?
-
Saturated R-134a vapor enters a compressor at 6°F. At compressor exit, the specific entropy is the same as that at the inlet, and the pressure is 80 psia. Determine the R-134a exit temperature...
-
Describe or define the following key performance indicators (KPIs) from the statement of cash flow: 1. Cash flow from operating activities 2. Cash flow from investing activities 3. Cash flow from...
-
Lonyear Corporation has collected the following information related to its December 31, 2014, balance sheet. Prepare the assets section of Lonyear Corporations balancesheet. Accounts receivable...
-
4. The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 1313 varieties of Kellogg's cereal. Sugar (grams) 5 15 12 11 8 6 7 4 7...
-
Background: SO MUCH CANDY DATA, SERIOUSLYCandy hierarchy data for 2017 Boing Boing Halloween candy hierarchy. This is survey data over the span of 4-years. The data is split into 4 separate files....
-
If the pressure drop in a H 2 O pipeline with a turbulent flow is 25 psi for a standard/ordinary steel pipe. What would the pressure drop be if the copper pipe was used? use the roughness...
-
Standard Costs Labor Materials Total Items per unit 2.00 hrs 1.00 Rate Per Item $ 25.00 $ 70.00 Variable Cost Per Unit Produced $ 50.00 $ 70.00 $ 120.00 During the period, the company produced 1,500...
-
Prob 2: Liabilities Equity Share Capital 12% Preference share capital General Reserve 16% debentures Trade payable Bank overdraft Rs. 2,00,000 Machinery 3,60,000 Investment Assets Rs. 5,92,000...
-
Condensed financial data are presented below for the Tulsa Corporationto answer the following questions: Show your calculations at the end 2021 2020 Accounts receivable $277,500 $230,000 Inventory...
-
The payment schedule from the lottery commission is $100,000 after taxes at the end of year one and 19 more payments of exactly $100,000 after taxes in equal annual end of the year deposits (i.e., 20...
-
Art-labeling Activity: Levels of Organization Arrange the items from the lowest to the highest level of organization. 1- Chemical or molecular level, 2- organ level, 3-cellular evel, 4- tissue level,...
-
Part: 2/3 Part 3 of 3 (c) Based on the confidence interval, is it reasonable to believe that the population standard deviation is less than 0.807 Explain. It (Choose one) reasonable to believe,...
-
Simplify the expression. Assume that all variables are positive. 23VI1 2 V44 8
-
(a) How many distinct paths are there from (- 1, 2, 0) to (1, 3, 7) in Euclidean three-space if each move is one of the following types? (H): (x, y, z) (x + 1, y, z); (V): (x, y, z) (x, y + l, z);...
-
a) How many two-factor unordered factorizations, where each factor is greater than 1, are there for 156,009? b) In how many ways can 156,009 be factored into two or more factors, each greater than 1,...
-
(a) For what sequence of numbers is g(x) = (1 - 2x) -5/2 the exponential generating function? (b) Find a and b so that (1 - ax)b is the exponential generating function for the sequence 1,7, 7 11, 7 ...
-
The comparative financial statements for CompuCircuit Corporation are in the Working Papers. The financial statements have been completed up to the financial analysis section. The following...
-
The comparative financial statements for Advanced Auto Technology, Inc., are in the Working Papers. The financial statements have been completed up to the financial analysis section. The following...
-
Corporate annual reports include a letter from the chief executive officer (CEO). The letter highlights achievements and positive results from the past year. If necessary, it delivers news of...
Study smarter with the SolutionInn App