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Quiz 7 :

Game Theory

Quiz 7 :

Game Theory

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Consider the following game: img a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions. c. Compute players' expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game.
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img img img img img img img img img img img img img img img

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The mixed-strategy Nash equilibrium in the Battle of the Sexes in Figure 8.3 may depend on the numerical values for the payoffs. To generalize this solution, assume that the payoff matrix for the game is given by img where K ? 1. Show how the mixed-strategy Nash equilibrium depends on the value of K. Reference: img
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The game of Chicken is played by two macho teens who speed toward each other on a single-lane road. The first to veer off is branded the chicken, whereas the one who does not veer gains peer-group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the Chicken game are provided in the following table. img a. Draw the extensive form. b. Find the pure-strategy Nash equilibrium or equilibria. c. Compute the mixed-strategy Nash equilibrium. As part of your answer, draw the best-response function diagram for the mixed strategies. d. Suppose the game is played sequentially, with teen 1 moving first and committing to this action by throwing away the steering wheel. What are teen 2's contingent strategies? Write down the normal and extensive forms for the sequential version of the game. e. Using the normal form for the sequential version of the game, solve for the Nash equilibria. f. Identify the proper subgames in the extensive form for the sequential version of the game. Use backward induction to solve for the subgame-perfect equilibrium. Explain why the other Nash equilibria of the sequential game are ''unreasonable.''
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The Nash equilibrium is a strategy profile in which each player plays his best response given other player's strategy and no unilateral transfer can have made any player better off.
The payoffs of the chicken game is given. The two teens are the players and their strategies are to veer off or does not veer off. Their respective payoffs are specified in the table.
a.
The extensive form of the game is used to model one shot game in a tree diagram where each player is choosing strategies simultaneously.
img b.
Pure strategy Nash equilibria refers to those payoffs which are best response of each player with respect to other player's strategy and no player has a reason to deviate from the Nash equilibrium strategy.
The best response for each player given other player's strategy is indicated by a line in the table given below and the outcome which has both payoffs underlined will give the pure strategy Nash equilibrium.
img From the table above, the pure strategy Nash equilibria are (does not veer, veer) and (does not veer, does not veer).
c.
Mixed strategy Nash equilibrium is a probability distribution which is chosen by each player which is best for him given the strategies used by the other players.
The expected payoffs of player one given mixed strategy of player two
img img For player one to be indifferent between the two strategies in equilibrium requires
img img Similarly, expected payoffs of player two can be derived given mixed strategy of player one
img and equating to derive equilibrium value of probability as done below:
img img To depict the best response functions diagrammatically, it is required to determine the relation between p and q based on their respective payoffs.
img Similarly,
img img From the above figure, it is clear that the mixed strategy Nash equilibrium is the point where best response curves of both players intersect. Both players have symmetric Nash equilibrium as
img .
d.
Suppose the game is played sequentially with player one making the first move and choosing his actions, the contingent strategy of player two will be seen as done below:
img The normal form for the sequential game can be seen below:
img The extensive form of the game can be seen diagrammatically as done below:
img e.
It is required to solve for Nash equilibria using the normal form. The normal form of the game is seen as done below:
img It is important to note that Does not veer can be written as DNV or DV. The best response for each player given other player's strategy is indicated by a line in the table given below and the outcome which has both payoffs underlined will give the pure strategy Nash equilibrium.
Hence, the sequential game reveals three pure strategy Nash equilibria as
1. Player one plays DOES NOT VEER, player two plays (V/V), (V/DNV).
2. Player one plays VEER, player two plays (DNV/V), (DNV/DNV)
3. Player one plays DOES NOT VEER, Player two plays (DNV/V), (DNV/V)
f.
Sub game perfect equilibrium is a Nash equilibrium for each subgame in a sequential game. Every finite sequential game pure strategy Nash equilibrium can be derived using backward induction. Backward induction is a process of reasoning backwards in time. From the end, to determine a sequence of optimal actions.
The sub games in the given chicken game can be seen as done below:
img Using backward induction, first determine the best response of player two in both the subgames as player two is playing in the last stage of the game. After getting the optimal actions, move to player one and determine his optimal action based on the given optimal strategy of his opponent.
img From backward induction, the subgame perfect equilibrium is Player one plays DOES NOT VEER, Player two plays (DNV/V), (DNV/V).
It rules out the empty threats and only shows the equilibrium path. Threats to play irrationally or choose something other than the best response is ruled out. There are credible threats which should considered and empty threats should be neglected when considering the equilibrium path.
The other two strategies which were part of normal extensive game pure strategy Nash equilibrium fall prey to empty threats and lie off the equilibrium path.

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Two neighboring homeowners, i = 1, 2, simultaneously choose how many hours li to spend maintaining a beautiful lawn. The average benefit per hour is img , and the (opportunity) cost per hour for each is 4. Homeowner i's average benefit is increasing in the hours neighbor j spends on his own lawn because the appearance of one's property depends in part on the beauty of the surrounding neighborhood. a. Compute the Nash equilibrium. b. Graph the best-response functions and indicate the Nash equilibrium on the graph. c. On the graph, show how the equilibrium would change if the intercept of one of the neighbor's average benefit functions fell from 10 to some smaller number.
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The Academy Award-winning movie A Beautiful Mind about the life of John Nash dramatizes Nash's scholarly contribution in a single scene: His equilibrium concept dawns on him while in a bar bantering with his fellow male graduate students. They notice several women, one blond and the rest brunette, and agree that the blond is more desirable than the brunettes. The Nash character views the situation as a game among the male graduate students, along the following lines. Suppose there are n males who simultaneously approach either the blond or one of the brunettes. If male i alone approaches the blond, then he is successful in getting a date with her and earns payoff a. If one or more other males approach the blond along with i, the competition causes them all to lose her, and i (as well as the others who approached her) earns a payoff of zero. On the other hand, male i earns a payoff of b 0 from approaching a brunette because there are more brunettes than males; therefore, i is certain to get a date with a brunette. The desirability of the blond implies a b. a. Argue that this game does not have a symmetric pure-strategy Nash equilibrium. b. Solve for the symmetric mixed-strategy equilibrium. That is, letting p be the probability that a male approaches the blond, find p * c. Show that the more males there are, the less likely it is in the equilibrium from part (b) that the blond is approached by at least one of them. Note: This paradoxical result was noted by S. Anderson and M. Engers in ''Participation Games: Market Entry, Coordination, and the Beautiful Blond,'' Journal of Economic Behavior Organization 63 (2007): 120-37.
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The following game is a version of the Prisoners' Dilemma, but the payoffs are slightly different than in Figure 8.1. img a. Verify that the Nash equilibrium is the usual one for the Prisoners' Dilemma and that both players have dominant strategies. b. Suppose the stage game is repeated infinitely many times. Compute the discount factor required for their suspects to be able to cooperate on silent each period. Outline the trigger strategies you are considering for them. Reference: img
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Return to the game with two neighbors in Problem 8.5. Continue to suppose that player i's average benefit per hour of work on landscaping is img Continue to suppose that player 2's opportunity cost of an hour of landscaping work is 4. Suppose that player 1's opportunity cost is either 3 or 5 with equal probability and that this cost is player 1's private information. a. Solve for the Bayesian-Nash equilibrium. b. Indicate the Bayesian-Nash equilibrium on a best-response function diagram. c. Which type of player 1 would like to send a truthful signal to player 2 if it could? Which type would like to hide his or her private information?
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In Blind Texan Poker, player 2 draws a card from a standard deck and places it against her forehead without looking at it but so player 1 can see it. Player 1 moves first, deciding whether to stay or fold. If player 1 folds, he must pay player 2 $50. If player 1 stays, the action goes to player 2. Player 2 can fold or call. If player 2 folds, she must pay player 1 $50. If player 2 calls, the card is examined. If it is a low card (2-8), player 2 pays player 1 $100. If it is a high card (9, 10, jack, queen, king, or ace), player 1 pays player 2 $100. a. Draw the extensive form for the game. b. Solve for the hybrid equilibrium. c. Compute the players' expected payoffs.
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Fairness in the Ultimatum Game Consider a simple version of the Ultimatum Game discussed in the text. The first mover proposes a division of $1. Let r be the share received by the other player in this proposal (so the first mover keeps 1 - r), where 0 ? r ? 1/2. Then the other player moves, responding by accepting or rejecting the proposal. If the responder accepts the proposal, the players are paid their shares; if the responder rejects it, both players receive nothing. Assume that if the responder is indifferent between accepting or rejecting a proposal, he or she accepts it. a. Suppose that players only care about monetary payoffs. Verify that the outcome mentioned in the text in fact occurs in the unique subgame-perfect equilibrium of the Ultimatum Game. b. Compare the outcome in the Ultimatum Game with the outcome in the Dictator Game (also discussed in the text), in which the proposer's surplus division is implemented regardless of whether the second mover accepts or rejects (so it is not much of a strategic game!). c. Now suppose that players care about fairness as well as money. Following the article by Fehr and Schmidt cited in the text, suppose these preferences are represented by the utility function img where x 1 is player 1's payoff and x 2 is player 2's (a symmetric function holds for player 2). The first term reflects the usual desire for more money. The second term reflects the desire for fairness, that the players' payoffs not be too unequal. The parameter a measures how intense the preference for fairness is relative to the desire for more money. Assume a 1/2. 1. Solve for the responder's equilibrium strategy in the Ultimatum Game. 2. Taking into account how the second mover will respond, solve for the proposer's equilibrium strategy r_ in the Ultimatum Game. (Hint: r * will be a corner solution, which depends on the value of a.) 3. Continuing with the fairness preferences, compare the outcome in the Ultimatum Game with that in the Dictator Game. Find cases that match the experimental results described in the text, in particular in which the split of the pot of money is more even in the Ultimatum Game than in the Dictator Game. Is there a limit to how even the split can be in the Ultimatum Game?
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In A Treatise on the Family (Cambridge, MA: Harvard University Press, 1981), Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child (player 1) and the child's parent (player 2). The child moves first, choosing an action r that affects his own income img and the income of the parent img . Later, the parent moves, leaving a monetary bequest L * to the child. The child cares only for his own utility, img but the parent maximizes img reflects the parent's altruism toward the child. Prove that, in a subgame-perfect equilibrium, the child will opt for the value of r that maximizes img even though he has no altruistic intentions. Hint: Apply backward induction to the parent's problem first, which will give a first-order condition that implicitly determines L * ; although an explicit solution for L * cannot be found, the derivative of L * with respect to r-required in the child's first-stage optimization problem-can be found using the implicit function rule.
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Alternatives to Grim Strategy Suppose that the Prisoners' Dilemma stage game (see Figure 8.1) is repeated for infinitely many periods. a. Can players support the cooperative outcome by using tit-for-tat strategies, punishing deviation in a past period by reverting to the stage-game Nash equilibrium for just one period and then returning to cooperation? Are two periods of punishment enough? b. Suppose players use strategies that punish deviation from cooperation by reverting to the stage-game Nash equilibrium for 10 periods before returning to cooperation. Compute the threshold discount factor above which cooperation is possible on the outcome that maximizes the joint payoffs. img
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Refinements of perfect Bayesian equilibrium Recall the job-market signaling game in Example 8.9. a. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the firm offers an uneducated worker a job. Be sure to specify beliefs as well as strategies. b. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the firm does not offer an uneducated worker a job. What is the lowest posterior belief that the worker is low-skilled conditional on obtaining an education consistent with this pooling equilibrium? Why is it more natural to think that a low-skilled worker would never deviate to E and thus an educated worker must be high-skilled? Cho and Kreps's intuitive criterion is one of a series of complicated refinements of perfect Bayesian equilibrium that rule out equilibria based on unreasonable posterior beliefs as identified in this part; see I. K. Cho and D. M. Kreps, ''Signalling Games and Stable Equilibria,'' Quarterly Journal of Economics 102 (1987): 179-221.
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