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Negotiating Socially Optimal Allocations of Resources. U. Endriss, N. Maudet, F. Sadri, and F. Toni Presented by: Marcus Shea. Introduction. Consider a society of independent agents Agents have an initial allocation of indivisible resources - PowerPoint PPT Presentation
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Negotiating Socially Optimal Allocations of Resources
U. Endriss, N. Maudet, F. Sadri, and F. Toni
Presented by: Marcus Shea
Introduction
• Consider a society of independent agents • Agents have an initial allocation of
indivisible resources• Agents can make deals with one another
in order to increase their utility
Introduction
• We will examine different classes of deals– Identify necessary and sufficient classes that
will allow our society to converge to an optimal allocation
Introduction
• We will examine different classes of deals– Identify necessary and sufficient classes that
will allow our society to converge to an optimal allocation
• Examples– 1-deals without side payments– Multilateral deals with side payments
Introduction
• We will consider at different measures of social welfare– Changes definition of an ‘optimal’ allocation
Introduction
• We will consider at different measures of social welfare– Changes definition of an ‘optimal’ allocation
• Examples– Measure social welfare based on average
utility of a system– Measure social welfare based on lowest utility
of a system
Introduction
• Distributed approach to multiagent resource allocation– Local negotiation
• Compare to the centralized approach– Single entity decides on final allocation based
on agents preferences over all allocations– Combinatorial auctions– May be difficult to find an ‘auctioneer’
Outline
• Preliminaries• Rational Negotiation with Side Payments• Rational Negotiation without Side
Payments• Egalitarian Agent Societies• Conclusions
Negotiation Framework
• Finite set of agents A • Finite set of resources R• Each agent i in A has a utility function ui that
maps every set of resources to a real number
Allocation of Resources
An allocation of resources is a function A from A to subsets of R such that A(i)∩A(j) = for i ≠ j
• An allocation of resources is just a partition of resources amongst the agents
Deals
A deal is a pair δ = (A,A’) where A and A’ are distinct allocations of resources– ‘old’ allocation and ‘new’ allocation
The set of agents involved in a deal δ = (A,A’) is given by Aδ = { i in A : A(i) ≠ A’(i) }- everyone whose set of resources has changed
The composition of two deals δ1 = (A,A’) and δ2 = (A’,A’’) is δ1◦δ2 = (A,A’’) - two deals are processed simultaneously
Independently Decomposable
A deal δ is independently decomposable if there exist deals δ1 and δ2 such that δ= δ1◦δ2 and Aδ1∩Aδ2 =
• δ is made up of two subdeals concerning disjoint sets of agents
δ =
Independently Decomposable
A deal δ is independently decomposable if there exist deals δ1 and δ2 such that δ= δ1◦δ2 and Aδ1∩Aδ2 =
• δ is made up of two subdeals concerning disjoint sets of agents
δ =
δ1δ2
δ = δ1◦δ2
Utility Functions
• We may restrict our attention to utility functions ui with particular properties:– Monotonic: for all R1,R2 R– Additive: for all R R
– 0-1 Function: Additive and for all r in R – Dichotomous: for all R R
1 2 i 1 i 2R R u (R ) u (R )
i ir R
u (R) u ({r})
iu ({r}) {0,1}iu (R) {0,1}
Utility Functions
• We may restrict our attention to utility functions ui with particular properties:– Monotonic: for all R1,R2 R– Additive: for all R R
– 0-1 Function: Additive and for all r in R – Dichotomous: for all R R
• An agent’s utility of an allocation is just the utility of his set of resources ui(A) = ui(A(i))
1 2 i 1 i 2R R u (R ) u (R )
i ir R
u (R) u ({r})
iu ({r}) {0,1}iu (R) {0,1}
Rational Negotiation with Side Payments
• We consider the scenario where agents can exchange money as well as resources
• We define a payment function as a function p from agents to real numbers that, when summed over agents, equals zero:
i
p(i) 0A
Rational Negotiation with Side Payments
Our goal is to maximize utilitarian social welfare
• Utilitarian social welfare is just the sum of all agents utility– Maximizing is equivalent to maximizing average utility– Useful in any market where agents act individually
u ii
sw (A) u (A)A
Individually Rational
• We assume our agents are rational
• We say a deal is individually rational if there exists a payment function so that every involved agent’s increase in utility is strictly greater than their payment
• Formally: deal δ = (A,A’) is individually rational if there exists a payment function p such that ui(A’) – ui(A) > p(i) for all agents i, except possibly p(i) = 0 for agents with A(i) = A’(i)
1-deals
A 1-deal is a deal involving reallocation of exactly one resource
• Question: If (rational) agents are permitted to perform 1-deals only, will we eventually reach an optimal allocation?
1-deals
• Consider a system with two agents and two resources, r1 and r2
• We specify the utility functions:
• Initial allocation A: Agent 1 has both resources
u1({}) = 0 u2({}) = 0
u1({r1}) = 2 u2({r1}) = 3
u1({r2}) = 3 u2({r2}) = 3
u1({r1,r2}) = 7 u2({r1,r2}) = 8
1-deals
• Consider a system with two agents and two resources, r1 and r2
• We specify the utility functions:
• Initial allocation A: Agent 1 has both resources– swu(A) = 7, optimal allocation has value 8
– 1-deals are not sufficient to get to an optimal allocation
u1({}) = 0 u2({}) = 0
u1({r1}) = 2 u2({r1}) = 3
u1({r2}) = 3 u2({r2}) = 3
u1({r1,r2}) = 7 u2({r1,r2}) = 8
First Result
• We are going to move toward showing that if we allow our agents to perform arbitrary individually rational deals, then we will reach an optimal allocation through negotiation
Lemma 1
Lemma 1: A deal δ = (A,A’) is individually rational iff swu(A) < swu(A’)
• Intuition: If an entire society gets a strict
increase in utility, then those profiting can payoff those who are losing so that everyone shares the gain
Thm 1: Maximal Utilitarian Social Welfare
Theorem 1: Any sequence of individually rational deals will eventually result in an allocation A that maximizes swu(A)
Thm 1: Maximal Utilitarian Social Welfare
Theorem 1: Any sequence of individually rational deals will eventually result in an allocation A that maximizes swu(A)
Proof:
Thm 1: Maximal Utilitarian Social Welfare
Theorem 1: Any sequence of individually rational deals will eventually result in an allocation A that maximizes swu(A)
Proof:• Termination Argument
– A and R finite means that there are only finitely many allocations
– Lemma 1 gives that any individually rational deal strictly increases social welfare
Thm 1: Maximal Utilitarian Social Welfare
Theorem 1: Any sequence of individually rational deals will eventually result in an allocation A that maximizes swu(A)
Proof:
Thm 1: Maximal Utilitarian Social Welfare
Theorem 1: Any sequence of individually rational deals will eventually result in an allocation A that maximizes swu(A)
Proof:• Suppose terminal allocation A is such that
swu(A) < swu(A’) for some A’
Thm 1: Maximal Utilitarian Social Welfare
Theorem 1: Any sequence of individually rational deals will eventually result in an allocation A that maximizes swu(A)
Proof:• Suppose terminal allocation A is such that
swu(A) < swu(A’) for some A’ ≠ A• Then deal δ = (A,A’) increases social
welfare, and thus is individually rational by Lemma 1, contradicting termination
Thm 1: Maximal Utilitarian Social Welfare
• Implications of Theorem 1– Not really surprising
• Class of individually rational deals allows for any number of resources to be moved between any number of agents
Thm 1: Maximal Utilitarian Social Welfare
• Implications of Theorem 1– Not really surprising
• Class of individually rational deals allows for any number of resources to be moved between any number of agents
– Difficulty in actually finding an individually rational deal
Thm 1: Maximal Utilitarian Social Welfare
• Implications of Theorem 1– Not really surprising
• Class of individually rational deals allows for any number of resources to be moved between any number of agents
– Difficulty in actually finding an individually rational deal
– We will not get stuck in a local optimum, any sequence will bring us to optimum allocation
Thm 1: Maximal Utilitarian Social Welfare
• Implications of Theorem 1– Not really surprising
• Class of individually rational deals allows for any number of resources to be moved between any number of agents
– Difficulty in actually finding an individually rational deal
– We will not get stuck in a local optimal, any sequence will bring us to optimum allocation
– This sequence could, however, be very long
Do we need the entire class of individually rational deals to guarantee that negotiation
will eventually reach a socially optimal allocation?
Thm 2: Necessary Deals w/ Side Payments
Theorem 2: Fix A, R. For every deal δ that is not independently decomposable, there exist utility functions and an initial allocation so that any sequence of individually rational deals leading to an optimal allocation must include δ.
Thm 2: Necessary Deals w/ Side Payments
Theorem 2: Fix A, R. For every deal δ that is not independently decomposable, there exist utility functions and an initial allocation so that any sequence of individually rational deals leading to an optimal allocation must include δ.
• This remains true if we restrict utility functions to be monotonic, or dichotomous
Thm 2: Necessary Deals w/ Side Payments
Theorem 2: Fix A, R. For every deal δ that is not independently decomposable, there exist utility functions and an initial allocation so that any sequence of individually rational deals leading to an optimal allocation must include δ.
• This remains true if we restrict utility functions to be monotonic, or dichotomous
Proof: Carefully define utility functions and initial allocation so that δ is the only improving deal
Thm 2: Necessary Deals w/ Side Payments
• Implications of Theorem 2– Any negotiation protocol that puts restrictions
on the structural complexity of deals will fail to guarantee optimal outcomes if the class of utility functions is unrestricted, monotone, or dichotomous
Thm 2: Necessary Deals w/ Side Payments
• Implications of Theorem 2– Any negotiation protocol that puts restrictions
on the structural complexity of deals will fail to guarantee optimal outcomes if the class of utility functions is unrestricted, monotone, or dichotomous
• What can we do?
Thm 2: Necessary Deals w/ Side Payments
• Implications of Theorem 2– Any negotiation protocol that puts restrictions
on the structural complexity of deals will fail to guarantee optimal outcomes if the class of utility functions is unrestricted, monotone, or dichotomous
• What can we do?– Restrict utility functions– Change notion of social welfare
Additive Scenario
• Consider the scenario where utility functions are additive (no synergy effects)
• Will we be able to reach an optimal allocation without needing such a broad class of deals?
Thm 3: Additive Scenario
Theorem 3: In additive scenarios, any sequence of individually rational 1-deals will eventually result in an allocation with maximal utilitarian social welfare
Thm 3: Additive Scenario
Theorem 3: In additive scenarios, any sequence of individually rational 1-deals will eventually result in an allocation with maximal utilitarian social welfare
Proof:
Thm 3: Additive Scenario
Theorem 3: In additive scenarios, any sequence of individually rational 1-deals will eventually result in an allocation with maximal utilitarian social welfare
Proof:• We get termination since we are looking
at individually rational deals
Thm 3: Additive Scenario
Proof:
R
For any allocation , define the function R A to
simply tell us the agent that holds resource in allocation
That is,
We can write since utility functions arA
A
A
u f (r )r
A f :
r A
f (r) i r A(i)
sw (A) u ({r})
e additive
Thm 3: Additive Scenario
Proof:
R
For any allocation , define the function R A to
simply tell us the agent that holds resource in allocation
That is,
We can write since utility functions arA
A
A
u f (r )r
A f :
r A
f (r) i r A(i)
sw (A) u ({r})
e additive
Suppose negotiation terminates with allocation but another
allocation with higher social welfare,
By the function above, some resource R must be generating
higher utility un
u u
u
A
A ' sw (A ') sw (A)
sw r
der that new allocation than under allocation
That is, A ' Af (r ) f (r )
A ' A
u ({r}) u ({r})
Thm 3: Additive Scenario
Proof:
R
For any allocation , define the function R A to
simply tell us the agent that holds resource in allocation
That is,
We can write since utility functions arA
A
A
u f (r )r
A f :
r A
f (r) i r A(i)
sw (A) u ({r})
e additive
Suppose negotiation terminates with allocation but another
allocation with higher social welfare,
By the function above, some resource R must be generating
higher utility un
u u
u
A
A ' sw (A ') sw (A)
sw r
der that new allocation than under allocation
That is, A ' Af (r ) f (r )
A ' A
u ({r}) u ({r})
Then the 1-deal passing from agent to increases social welfare
By Lemma 1, is an individually rational 1-deal, contradicting terminationA A 'r f (r) f (r)
Thm 3: Additive Scenario
• Are 1-deals necessary to achieve an optimal allocation in the additive scenario?
Thm 3: Additive Scenario
• Are 1-deals necessary to achieve an optimal allocation in the additive scenario?– Paper does not address this question
Thm 3: Additive Scenario
• Are 1-deals necessary to achieve an optimal allocation in the additive scenario?– Paper does not address this question– Easy to see that they are necessary:
• Let δ be a 1-deal that moves resource r1 from agent i to agent j
• Give all resources to agent j, except r1 to agent i• Set uk({r}) = 0 for every resource r, every agent k≠j• Set uj({r}) = 1 for every resource r• Only individually rational deal is 1-deal δ
Class of DealsSidePayments
Utility Functions
Measure of Social Welfare
Nature of Optimality
Necessary / Sufficient
Individually Rational Deals Yes
UnrestrictedMonotonicDichotomous Utilitarian
Global Maximum
Sufficient[1] & Necessary[2]
Individually Rational 1-deals Yes Additive Utilitarian
Global Maximum
Sufficient[3] & Necessary
Cooperatively Rational Deals No
UnrestrictedMonotonicDichotomous Utilitarian
Pareto Optimal
Sufficient[4] & Necessary[5]
Cooperatively Rational 1-deals No 0-1 Functions Utilitarian
Global Maximum
Sufficient[6]& Necessary
Equitable Deals NoUnrestrictedDichotomous Egalitarian
Global Maximum
Sufficient[7] & Necessary[8]
Simple Pareto-Pigou-Dalton Deals No 0-1 Functions Mixed
Lorenz Optimal Sufficient[9]
Summary of Results
Rational Negotiation w/o Side Payments
• Now we consider the scenario where there are no side payments made
Rational Negotiation w/o Side Payments
• Now we consider the scenario where there are no side payments made
• The class of individually rational deals no longer allows us to achieve optimal social welfare:– Agent 1 has sole resource r
u1({}) = 0 u2({}) = 0
u1({r}) = 1 u2({r}) = 2
Rational Negotiation w/o Side Payments
• Maximizing social welfare is no longer possible in general
• We will instead see if a Pareto optimal outcome is possible, and what types of deals are sufficient to guarantee this outcome
Pareto Optimal
• A Pareto optimal allocation is one in which there is no other allocation with higher social welfare that would be no worse for any of the agents in the system
Formally: Allocation A is Pareto optimal if there is no allocation A’ such that swu(A) < swu(A’) and ui(A) ≤ ui(A’) for all agents i
Pareto Optimal
• Recall our previous example – Agent 1 has sole resource r
– This is Pareto optimal since agent 1 is worse off by giving resource r to agent 2, even though it would increase social welfare
u1({}) = 0 u2({}) = 0
u1({r}) = 1 u2({r}) = 2
Cooperative Rationality
We say a deal is cooperatively rational if no agent’s utility decreases, but at least one agent’s utility strictly increases
Formally: We say a deal δ = (A,A’) is cooperatively rational if ui(A) ≤ ui(A’) for all agents i and there is an agent j such that uj(A) < uj(A’)
• We examine the class of cooperatively rational deals for the scenario without side payments
Thm 4: Pareto Optimal Outcomes
Theorem 4: Any sequence of cooperatively rational deals will eventually result in a Pareto optimal allocation of resources
• Very similar proof to Theorem 1
Thm 5: Necessary deals w/o side payments
Theorem 5: Fix A, R. Then for every deal δ that is not independently decomposable, there exist utility functions and an initial allocation such that any sequence of cooperatively rational deals leading to a Pareto optimal allocation would have to include δ
Thm 5: Necessary deals w/o side payments
Theorem 5: Fix A, R. Then for every deal δ that is not independently decomposable, there exist utility functions and an initial allocation such that any sequence of cooperatively rational deals leading to a Pareto optimal allocation would have to include δ
• Still holds if utility functions are restricted to be monotonic or dichotomous
Thm 5: Necessary deals w/o side payments
• Analogously to Theorem 3, we can restrict our utility functions to get a positive result about converging to an optimal solution under the class of cooperatively rational 1-deals
Thm 6: 0-1 Scenarios
Theorem 6: If utility functions are 0-1 functions (additive and ui({r}) = 0 or 1), any sequence of cooperatively rational 1-deals will eventually result in an allocation with maximal utilitarian social welfare
Thm 6: 0-1 Scenarios
Theorem 6: If utility functions are 0-1 functions (additive and ui({r}) = 0 or 1), any sequence of cooperatively rational 1-deals will eventually result in an allocation with maximal utilitarian social welfare
• Note that we actually get optimal social welfare in this case, not just Pareto optimal!
Thm 6: 0-1 Scenarios
Theorem 6: If utility functions are 0-1 functions (additive and ui({r}) = 0 or 1), any sequence of cooperatively rational 1-deals will eventually result in an allocation with maximal utilitarian social welfare
• Note that we actually get optimal social welfare in this case, not just Pareto optimal!
• Proof is simple– If A is not optimal, must have a agents i and j and
resource r where r is in A(i), ui({r}) = 0 and uj({r}) = 1– That 1-deal is cooperatively rational
Class of DealsSidePayments
Utility Functions
Measure of Social Welfare
Nature of Optimality
Necessary / Sufficient
Individually Rational Deals Yes
UnrestrictedMonotonicDichotomous Utilitarian
Global Maximum
Sufficient[1] & Necessary[2]
Individually Rational 1-deals Yes Additive Utilitarian
Global Maximum
Sufficient[3] & Necessary
Cooperatively Rational Deals No
UnrestrictedMonotonicDichotomous Utilitarian
Pareto Optimal
Sufficient[4] & Necessary[5]
Cooperatively Rational 1-deals No 0-1 Functions Utilitarian
Global Maximum
Sufficient[6]& Necessary
Equitable Deals NoUnrestrictedDichotomous Egalitarian
Global Maximum
Sufficient[7] & Necessary[8]
Simple Pareto-Pigou-Dalton Deals No 0-1 Functions Mixed
Lorenz Optimal Sufficient[9]
Summary of Results
Egalitarian Social Welfare
Consider a new measure of social welfare called egalitarian social welfare
Ae isw (A) min{u (A) | i }
Egalitarian Social Welfare
Consider a new measure of social welfare called egalitarian social welfare
• Measures the utility of the ‘weakest/poorest’ member of the society
Ae isw (A) min{u (A) | i }
Egalitarian Social Welfare
Consider a new measure of social welfare called egalitarian social welfare
• Measures the utility of the ‘weakest/poorest’ member of the society
• Makes sense when the society is working together or trying to be fair with one another– Recall: Earth Observation Satellite Access
Ae isw (A) min{u (A) | i }
Equitable Deals
A deal δ = (A,A’) is equitable if
min{ ui(A) | i in Aδ } < min{ ui(A’) | i in Aδ}
• Lowest utility of all agents involved in a deal increases
Equitable Deals
A deal δ = (A,A’) is equitable if
min{ ui(A) | i in Aδ } < min{ ui(A’) | i in Aδ}
• Lowest utility of all agents involved in a deal increases• Note: we do not need the weakest member of society to
improve– Would not be a local condition
Equitable Deals
A deal δ = (A,A’) is equitable if
min{ ui(A) | i in Aδ } < min{ ui(A’) | i in Aδ }
• Lowest utility of all agents involved in a deal increases• Note: we do not need the weakest member of society to
improve– Would not be a local condition
Lemma 2: If A and A’ are allocations with swe(A) < swe(A’), then δ = (A,A’) is equitable
Thm 7: Maximal Egalitarian Social Welfare
Theorem 7: Any sequence of equitable deals will eventually result in an allocation of resources with maximal egalitarian social welfare
Thm 7: Maximal Egalitarian Social Welfare
Theorem 7: Any sequence of equitable deals will eventually result in an allocation of resources with maximal egalitarian social welfare
• Only difficulty of proof is showing termination, the rest comes from the definition of equitable
Thm 8: Necessary Deals in Egalitarian Systems
Theorem 8: Fix A, R. Then for every deal δ that is not independently decomposable, there exist utility functions and an initial allocation such that any sequence of equitable deals leading to an allocation with maximal egalitarian social welfare would have to include δ.
Thm 8: Necessary Deals in Egalitarian Systems
Theorem 8: Fix A, R. Then for every deal δ that is not independently decomposable, there exist utility functions and an initial allocation such that any sequence of equitable deals leading to an allocation with maximal egalitarian social welfare would have to include δ.
• Still holds if utility functions are restricted to be dichotomous
Class of DealsSidePayments
Utility Functions
Measure of Social Welfare
Nature of Optimality
Necessary / Sufficient
Individually Rational Deals Yes
UnrestrictedMonotonicDichotomous Utilitarian
Global Maximum
Sufficient[1] & Necessary[2]
Individually Rational 1-deals Yes Additive Utilitarian
Global Maximum
Sufficient[3] & Necessary
Cooperatively Rational Deals No
UnrestrictedMonotonicDichotomous Utilitarian
Pareto Optimal
Sufficient[4] & Necessary[5]
Cooperatively Rational 1-deals No 0-1 Functions Utilitarian
Global Maximum
Sufficient[6] & Necessary
Equitable Deals NoUnrestrictedDichotomous Egalitarian
Global Maximum
Sufficient[7] & Necessary[8]
Simple Pareto-Pigou-Dalton Deals No 0-1 Functions Mixed
Lorenz Optimal Sufficient[9]
Summary of Results
Class of DealsSidePayments
Utility Functions
Measure of Social Welfare
Nature of Optimality
Necessary / Sufficient
Individually Rational Deals Yes
UnrestrictedMonotonicDichotomous Utilitarian
Global Maximum
Sufficient[1] & Necessary[2]
Individually Rational 1-deals Yes Additive Utilitarian
Global Maximum
Sufficient[3] & Necessary
Cooperatively Rational Deals No
UnrestrictedMonotonicDichotomous Utilitarian
Pareto Optimal
Sufficient[4] & Necessary[5]
Cooperatively Rational 1-deals No 0-1 Functions Utilitarian
Global Maximum
Sufficient[6] & Necessary
Equitable Deals NoUnrestrictedDichotomous Egalitarian
Global Maximum
Sufficient[7] & Necessary[8]
Simple Pareto-Pigou-Dalton Deals No 0-1 Functions Mixed
Lorenz Optimal Sufficient[9]
Summary of Results
Conclusions
• We studied an abstract negotiation framework where members of an agent society arrange multilateral deals to exchange bundles of indivisible resources
Conclusions
• We studied an abstract negotiation framework where members of an agent society arrange multilateral deals to exchange bundles of indivisible resources
• We analyzed how the resulting changes in resource distribution affect society with respect to different social welfare orderings
Conclusions
• We see that convergence to an optimal allocation depends on:– the class of allowable deals
Conclusions
• We see that convergence to an optimal allocation depends on:– the class of allowable deals– the notion of optimality being considered
Conclusions
• We see that convergence to an optimal allocation depends on:– the class of allowable deals– the notion of optimality being considered– the restrictions on utility functions
Conclusions
• We see that convergence to an optimal allocation depends on:– the class of allowable deals– the notion of optimality being considered– the restrictions on utility functions– the availability of side payments
Conclusions
• We see that convergence to an optimal allocation depends on:– the class of allowable deals– the notion of optimality being considered– the restrictions on utility functions– the availability of side payments
• Natural question: Complexity results– How fast do we converge to the optimal
allocation? [Endriss and Maudet (2005)]
Conclusions
• Authors are looking at welfare engineering– Application-driven choice of a social welfare
ordering– Design of agent behaviour profiles and
negotiation mechanisms that permit socially optimal outcomes
Lorenz Domination
• Let A, A’ be allocations for a society with n agents. Then A is Lorenz dominated by A’ if
and furthermore, that inequality is strict for at least one k.
• k = 1 gives egalitarian social welfare• k = n gives utilitarian social welfare
k k
i ii 1 i 1
u (A) u (A ') k 1,...,n
############################
Pigou-Dalton Transfer
• A deal δ = (A,A’) is called a Pigou-Dalton transfer if it satisfies:– 2 agents involved– Mean-preserving: ui(A) + uj(A) = ui(A’) + uj(A’)– Reduces inequality:
|ui(A’) – uj(A’)| < |ui(A) – uj(A)|• A simple Pareto-Pigou-Dalton deal is a 1-
deal that is either cooperatively rational, or a Pigou-Dalton transfer
Class of DealsSidePayments
Utility Functions
Measure of Social Welfare
Nature of Optimality
Necessary / Sufficient
Individually Rational Deals Yes
UnrestrictedMonotonicDichotomous Utilitarian
Global Maximum
Sufficient[1] & Necessary[2]
Individually Rational 1-deals Yes Additive Utilitarian
Global Maximum
Sufficient[3] & Necessary
Cooperatively Rational Deals No
UnrestrictedMonotonicDichotomous Utilitarian
Pareto Optimal
Sufficient[4] & Necessary[5]
Cooperatively Rational 1-deals No 0-1 Functions Utilitarian
Global Maximum
Sufficient[6] & Necessary
Equitable Deals NoUnrestrictedDichotomous Egalitarian
Global Maximum
Sufficient[7] & Necessary[8]
Simple Pareto-Pigou-Dalton Deals No 0-1 Functions Mixed
Lorenz Optimal Sufficient[9]
Summary of Results
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