Negotiating Socially Optimal Allocations of Resources

1. Negotiating Socially Optimal Allocations of Resources U. Endriss, N. Maudet, F. Sadri, and F. Toni Presented by: Marcus Shea

2. 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

3. What class of deals will encourage our system to eventually reach a socially optimal state?

4. Introduction • We will examine different classes of deals • Identify necessary and sufficient classes that will allow our society to converge to an optimal allocation

5. 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

6. Introduction • We will consider at different measures of social welfare • Changes definition of an ‘optimal’ allocation

7. 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

8. Introduction • Distributed approach to multiagent resource allocation • Local negotiation

9. 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’

10. Outline • Preliminaries • Rational Negotiation with Side Payments • Rational Negotiation without Side Payments • Egalitarian Agent Societies • Conclusions

11. Preliminaries

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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))

19. Rational Negotiation with Side Payments

20. 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:

21. 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

22. 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)

23. 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?

24. 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

25. 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

26. 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

27. 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

28. 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)

29. 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:

30. 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

31. 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:

32. 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’

33. 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

34. 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

35. 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

36. 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

37. 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

38. Do we need the entire class of individually rational deals to guarantee that negotiation will eventually reach a socially optimal allocation?

39. 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.

40. 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

41. 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

42. 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

43. 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?

44. 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

45. 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?

46. 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

47. 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:

48. 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

49. Thm 3: Additive Scenario Proof:

50. Thm 3: Additive Scenario Proof: