Consider a Variant of the ultimatum game we studied in class in which players have Fairness considerations. The timing of the game is usual. First, Player 1 proposes the split (100-X², X) OF a hundred dollars to player 2, where XE [0,100]. Player 2 observes split & decides whether to accept (in which case they recieve money according to proposed Split) or reject (in which case they both get žero dollars). But now player i's utility equals to her monetary Utility minus the disutility From unFairness proportional to the differenc in Monetary outcomes. That is, given a Final Split (m., m₂) let u₁ (m₁, m₂) = m₁ -P. (m₁-m₂) ² U₁ (m₁, m₂) = m₁ - P₂ (m₁-m.)² where P₁, P₂ are parameters of the game indicating how strongly Players care about Fairness. Note that the case we considered corresponds to B₁ = P₂ = 0 (a) Let B₁ = T₁, P₂ = 0. Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies For player 2. (b) Let B₁ = 10₁ $₂=0. For each sequentially rational strategy OF player 2 you identified in part (a), either describe which proposal Maximizes player 1's continuation valve or explain why it does not exist (C) Let B₁ = 1 0 ₁ P₂=0. Describe all SPNE of the game.
Consider a Variant of the ultimatum game we studied in class in which players have Fairness considerations. The timing of the game is usual. First, Player 1 proposes the split (100-X², X) OF a hundred dollars to player 2, where XE [0,100]. Player 2 observes split & decides whether to accept (in which case they recieve money according to proposed Split) or reject (in which case they both get žero dollars). But now player i's utility equals to her monetary Utility minus the disutility From unFairness proportional to the differenc in Monetary outcomes. That is, given a Final Split (m., m₂) let u₁ (m₁, m₂) = m₁ -P. (m₁-m₂) ² U₁ (m₁, m₂) = m₁ - P₂ (m₁-m.)² where P₁, P₂ are parameters of the game indicating how strongly Players care about Fairness. Note that the case we considered corresponds to B₁ = P₂ = 0 (a) Let B₁ = T₁, P₂ = 0. Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies For player 2. (b) Let B₁ = 10₁ $₂=0. For each sequentially rational strategy OF player 2 you identified in part (a), either describe which proposal Maximizes player 1's continuation valve or explain why it does not exist (C) Let B₁ = 1 0 ₁ P₂=0. Describe all SPNE of the game.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Consider a Variant oF the ultimatum qame We Studicd inclass
in which players have Fairness considerations . The timing
OF the qame is vSual. First , Player 1 propasar the split
(100 -x", x) OF a hundred dellars to player 2,Where
XE [0,100]. Player 2 observes split & decides whether to accept
(in which case they recieve Money accor ding ti proposed Split)
or reject (in Which Case they both get žero dollars).But
now player i's Utility equals to her monetary Vtility minus
the disutility From unFairneas proportional tO the differene
in Monetary OutcCOMeS . That is, qiven a Final Split (m. ,m.) ket
u, (m. ,m.) = m, -P. (m, - m2)"
%3D
U. (m. ,m2)
= m,- P2 (m,-m.)
Where PP are parametens of the game indicating how strongly
Players care a bout Fairness. Note that the case we
considered corres pends to B, = B2 = 0
(a) Let B, = To ,P2 =
alcept ? revect ? Describe all seguentially rational Strategies
For player 2.
b) Let B, = 10 ,$=0. For each sequentially rational Strategy
of player 2 yoU identiFied in part (a) ;tither descripě
Whi ch propasa) Maxi mizes player 1s continuation.
valve or explaih Why it does not exist
©) Let B, - =0. Describe all SPNE OFthe game.
= 0. which oFfers will player 2 deFihitely](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4c0753e-a375-4dbe-a41d-9a101b34842c%2Ffff245d4-d972-429d-8e07-538f3bb64bd0%2F65q6x8vv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a Variant oF the ultimatum qame We Studicd inclass
in which players have Fairness considerations . The timing
OF the qame is vSual. First , Player 1 propasar the split
(100 -x", x) OF a hundred dellars to player 2,Where
XE [0,100]. Player 2 observes split & decides whether to accept
(in which case they recieve Money accor ding ti proposed Split)
or reject (in Which Case they both get žero dollars).But
now player i's Utility equals to her monetary Vtility minus
the disutility From unFairneas proportional tO the differene
in Monetary OutcCOMeS . That is, qiven a Final Split (m. ,m.) ket
u, (m. ,m.) = m, -P. (m, - m2)"
%3D
U. (m. ,m2)
= m,- P2 (m,-m.)
Where PP are parametens of the game indicating how strongly
Players care a bout Fairness. Note that the case we
considered corres pends to B, = B2 = 0
(a) Let B, = To ,P2 =
alcept ? revect ? Describe all seguentially rational Strategies
For player 2.
b) Let B, = 10 ,$=0. For each sequentially rational Strategy
of player 2 yoU identiFied in part (a) ;tither descripě
Whi ch propasa) Maxi mizes player 1s continuation.
valve or explaih Why it does not exist
©) Let B, - =0. Describe all SPNE OFthe game.
= 0. which oFfers will player 2 deFihitely
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