Microeconomics
21st Edition
ISBN: 9781259915727
Author: Campbell R. McConnell, Stanley L. Brue, Sean Masaki Flynn Dr.
Publisher: McGraw-Hill Education
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Question
Chapter 8, Problem 5RQ
To determine
Ethics of a proposer.
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4. Consider a three-player bargaining, where the players are negotiating over
a surplus of one unit of utility. The game begins with player 1 proposing
a three-way split of the surplus. Then player 2 must decide whether to
accept the proposal or to substitute for player 1's proposal his own alternative
proposal. Finally, player 3 must decide whether to accept or reject current
proposal (it is player 1's if player 2 accepts or player 2's if player 2 offer a
new one). If he accepts, then the players obtain the specified shares of the
surplus. If player 3 rejects, then the players each get 0.
(a) Draw the extensive form game of this perfect-information game.
(b) Determine the subgame perfect NE.
4. Consider the following variant of the Prisoner's Dilemma game: Player 1 is unsure
whether Player 2 is "nice" or "selfish", while Player 2 knows Player 1's preferences.
Further suppose that Player 1's preferences depend on whether Player 2 is nice
or selfish. Specifically, suppose that there is a probability p that Player 2 is "selfish",
in which case the game is given as follows.
Game with Selfish Player 2
Player1/Player 2
Cooperate (C)
Don't Cooperate (D)
Cooperate (C)
4, 4
0, 6
Don't Cooperate (D)
6, 0
2, 2
And Player 2 is "nice" with probability 1-p, in which case the following game
results.
Game with Nice Player 2
Player1/Player 2
Cooperate (C)
Don't Cooperate
(D)
2, 4
Cooperate (C)
6, 6
Don't Cooperate (D)
4, 0
0, 2
[Note that C = cooperate (with each other) and D = don't cooperate or defect).
a) Write the extensive form of this game. How many strategies does each
player have in this game?
b) For what values of p (if any) is it a Bayes-Nash equilibrium for Player 1 to
play D in…
Ex. 5-
- Agenda Setting
An agenda-setting game is described as follows. The "issue space" (set of feasible poli-
cies) is an interval X = [0,5]. An agenda setter (player 1) proposes an alternative x = X
against the status quo q = 4. After player 1 proposes x, the legislator (player 2) ob-
serves the proposal and selects between the proposal x and the status quo q. Player 1's
most-preferred policy is 1, and for any final policy y = X his payoff is given by
v1(y) = 10|y — 1|,
-
where |y1| denotes the absolute value of (y-1). Player 2's most preferred policy is 3,
any final policy y Є X he payoff is given by
and for
v2(y) = 10|y3|.
That is, each player prefers policies that are closer to his/her most-preferred policy.
(1) Write down the strategic-form representation of the dynamic game.
(2) Find a subgame perfect Nash equilibrium. Is it unique?
Knowledge Booster
Similar questions
- David wants to auction a painting, and there are two potential buyers. The value for eachbuyer is either 0 or 10, each value equally likely. Suppose he offers to sell the object for $6, and the two buyers simultaneously accept or reject. If exactly one buyer accepts, the object sold to that person for $6. If both accept, the object is allocated randomly to the buyers, also for $6. If neither accepts, the object is allocated randomly to the bidders for $0. (a) Identify the type space and strategy space for each buyer. (b) Show that there is an equilibrium in which buyers with value 10 always accept. (c) Show that there is an equilibrium in which buyers with value 10 always reject.arrow_forward2. Consider the ultimatum game, where player 1 makes a take-it-or-leave-it offer to player 2, but suppose that player 2 is one of two types: with probability p, player 2 is a "standard" type, who maximizes his utility as usual; with probability 1 - p, player 2 is a "crazy" type, who accepts an offer if and only if the offer gives player 2 at least fraction q of the surplus. Player 1 does not know the type of player 2. In this game, a strategy profile must specify strategies for both player 1 and the standard type of player 2 (note that the strategy of the crazy type of player 2 is given). For every pair (p, q) E (0,1] x (0,1], find all strategy profiles where player 1 and the standard type of player 2 are both maximizing their expected utility, given their opponent's strategy. Hint: First, figure out the best responses for the standard type of player 2. Then, to find player l's best response to these strategies for various values of (p, q), start by writing down the possible expected…arrow_forwardWe have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does Jerry have any strictly dominant strategy? Does Elaine?arrow_forward
- We have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does this game have a Nash Equilibrium? (If not, why not? If yes, what is the Nash Equilibrium?)arrow_forwardIs it possible that Step. 3 contains a contradiction? As "If Player 1 plays Ball (B): Player 2 meets (M) if the expected payoff of meeting is greater than the expected payoff of avoiding." and then, "If Player 1 has Dinner (D): Player 2 meets (M) if the expected payoff of meeting is greater than the expected payoff of avoiding." The same arises in the one for player 2. Is for both the second sentence meant to be "smaller than"? If not, please elaborate.arrow_forwardRita is playing a game of chance in which she tosses a dart into a rotating dartboard with 8 equal-sized slices numbered 1 through 8. The dart lands on a numbered slice at random. This game is this: Rita tosses the dart once. She wins $1 if the dart lands in slice 1, $2 if the dart lands in slice 2, $5 if the dart lands in slice 3, and $8 if the dart lands in slice 4. She loses $3 if the dart lands in slices 5, 6, 7, or 8. (If necessary, consult a list of formulas.) (a) Find the expected value of playing the game. | dollars (b) What can Rita expect in the long run, after playing the game many times? O Rita can expect to gain money. She can expect to win dollars per toss. Rita can expect to lose money. She can expect to lose dollars per toss. O Rita can expect to break even (neither gain nor lose money).arrow_forward
- 10. Player A and Player B are playing a game . First , Player A chooses to either " Keep " or " Pass " . Second , Player B observes A's choice and Player B then chooses to either Keep or Pass . This process continues which creates the sequential game below . Please mark decisions that rational and selfish players will choose at every decision node ( 3 decisions by player A and 3 decisions by player B ) - mark them on the Figure . What is the equilibrium of this game ?arrow_forward8. Two states, A and B, have signed an arms-control agreement. This agreementcommits them to refrain from building certain types of weapons. The agreement is supposed tohold for an indefinite length of time. However, A and B remain potential enemies who wouldprefer to be able to cheat and build more weapons than the other. The payoff table for A (player1, the row player) and B (player 2, the column player) in each period after signing thisagreement is below. a) First assume that each state uses Tit-for-Tat (TFT) as a strategy in this repeated game.The rate of return is r. For what values of r would it be worth it for player A to cheat bybuilding additional weapons just once against TFT? b) For what values of r would it be worth deviating from the agreement forever to buildweapons? c) Convert both values you found in parts a and b to the equivalent discount factor dusing the formula given in lecture and section. d) Use the answers you find to discuss the relationship between d and r:…arrow_forwardConsider a variant of the ultimatum game we studied in class in which players have fairness considerations. The timing of the game is as usual. First, player 1 proposes the split (100 – x, x) of a hundred dollars to player 2, where x € [0, 100]. Player 2 observes the split and decides whether to accept (in which case they receive money according to the proposed split) or reject (in which case they both get 0 dollars). But now player i's utility equals to her monetary utility minus the disutility from unfairness proportional to the difference in the monetary outcomes. That is, given a final split (m1, m2), let u1(m1, m2) = m1 – B1(m1 -– m2)² u1(m1, m2) = m2 - B2(m1 – m2)², where B1, B2 are parameters of the game indicating how strongly players care about fairness. Note that the case we considered in class corresponds to ß1 = B2 = 0.arrow_forward
- Suppose that the proposer in the ultimatum game may not propose fractional amounts, and therefore must propose $0, $1, $2, ..., or $10 (see figure below). As always, the responder must Accept (A) or Reject (R). Suppose that this game is played by two egoists who care only about their money. Which of the following statements are true? (Multiple Choice) XXX $0 $1 $2 $3 $4 $5 $6 $7 || $8 $9 $10 A R ($6,$4) ($0,$0) A. Player 2's threat to reject a low offer is not credible; player 1 anticipates this and offers nothing. B. The Nash equilibrium is: Player 2 accepts no offers and Player 1 offers nothing. C. The Nash equilibrium is: Player 2 accepts all offers and Player 1 offers nothing. D. When played with real-world subjects, few people actually play the Nash strategy.arrow_forwardTwo players are bargaining over a three period bargaining model as discussed in class with player 1 making offers in rounds 1 and 3. Player 2 makes an offer in round 2 only. Each player has a common discount factor delta. The two players are bargaining to split $20. They have three time periods available to them for their bargaining game. At the end of round 3, if no agreement has been reached then player 1 receives $2 and player 2 receives $1 and the rest of the money is destroyed. Find the subgame perfect Nash equilibrium outcome in the finite horizon model in which the game ends after period 3.arrow_forward4 Consider an extensive game where player 1 starts with choosing of two actions, A or B. Player 2 observes player 1’s move and makes her move; if the move by player 1 is A, then player 2 can take three actions, X, Y or Z, if the move by player 1 is B, then player 2 can take of of two actions, U or V. Write down all teminal histories, proper subhistories, the player function and strategies of players in this game.arrow_forward
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