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## Negotiating Socially Optimal Allocations of Resources

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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**What class of deals will encourage our system to eventually**reach a socially optimal state?**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**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**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 =**δ = δ1◦δ2 δ2 δ1 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**Utility Functions**• We may restrict our attention to utility functions ui with particular properties: • Monotonic: for all R1,R2R • Additive: for all R R • 0-1 Function: Additive and for all r in R • Dichotomous: for all R R**Utility Functions**• We may restrict our attention to utility functions ui with particular properties: • Monotonic: for all R1,R2R • 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))**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:**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**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**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**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:**Thm 3: Additive Scenario**Proof: