Multiagent Systems: Spring 2006

TL;DR: In this article, Endriss et al. consider the problem of multi-agent resource allocation in the context of social choice and welfare multiagent systems, where the objective is to find an agreement that is Pareto-optimal for each agent.

Abstract: ion 1: Allocations and Agreements • Our grand aim in this course is to better understand the multiagent resource allocation problem. • So, at the end of the day, we are interested in modelling individual and social preferences over alternative allocations. • Today we’ll just speak about agreements between agents. – This is more abstract, because allocations come with a certain structure: some potential solutions will not be feasible etc. – In particular, at this stage we shall not worry about the combinatorial explosion we face when representing preferences over the allocation of multiple goods. • We also say: decisions, outcomes, or simply alternatives. Ulle Endriss (ulle@illc.uva.nl) 4 Social Choice and Welfare Multiagent Systems 2006 Ordinal Preferences • The preference relation of agent i over alternative agreements: x i y ⇔ agreement x is not better than y (for agent i) • We shall also use the following notation: – x ≺i y iff x i y but not y i x (strict preference) – x ∼i y iff both x i y and y i x (indifference) • A preference relation i is usually required to be – transitive: if you prefer x over y and y over z, you should also prefer x over z; and – connected: for any two agreements x and y, you can decide which one you prefer (or whether you value them equally). • Discussion: useful model, but not without problems (humans cannot always assign rational preferences . . . ) Ulle Endriss (ulle@illc.uva.nl) 5 Social Choice and Welfare Multiagent Systems 2006 Utility Functions • Cardinal (as opposed to ordinal) preference structures can be expressed via utility functions . . . • A utility function ui (for agent i) is a mapping from the space of agreements to the reals. • Example: ui(x) = 10 means that agent i assigns a value of 10 to agreement x. • A utility function ui representing the preference relation i: x i y ⇔ ui(x) ≤ ui(y) • Discussion: utility functions are very useful, but they suffer from the same problems as ordinal preference relations — even more so (humans typically do not reason with numerical utilities . . . ) Ulle Endriss (ulle@illc.uva.nl) 6 Social Choice and Welfare Multiagent Systems 2006 The Unanimity Principle An agreement x is Pareto-dominated by another agreement y iff: • x i y for all members i of society; and • x ≺i y for at least one member i of society. An agreement is Pareto optimal (or Pareto efficient) iff it is not Pareto-dominated by any other feasible agreement (named so after Vilfredo Pareto, Italian economist, 1848–1923). The Unanimity Principle states that society should not select an agreement that is Pareto dominated by another feasible agreement. Ulle Endriss (ulle@illc.uva.nl) 7 Social Choice and Welfare Multiagent Systems 2006 The Equality Principle “All men are created equal . . . ” Equality is probably the most obvious fairness postulate. The Equality Principle states that the agreement selected by society should give equal utility to all agents. Ulle Endriss (ulle@illc.uva.nl) 8 Social Choice and Welfare Multiagent Systems 2006 The Equality-Efficiency Dilemma Of course, the Equality Principle may not always be satisfiable, namely if there exists no feasible agreement giving equal utility to everyone. But even when there are equal outcomes, they may not be compatible with the Unanimity Principle. Example: Ann and Bob need to divide four items between them: a piano, a precious vase, an oriental carpet, and a lawn-mower. Ann just wants the piano: she will assign utility 10 to any bundle containing the piano, and utility 0 to any other bundle. Bob only cares about how many items he receives: his utility will be 5 times the cardinality of the bundle he receives . . . Ulle Endriss (ulle@illc.uva.nl) 9 Social Choice and Welfare Multiagent Systems 2006 Minimising Inequality So the pure Equality Principle seems too strong . . . Instead, we could try to minimise inequality . In the case of two agents, a first idea would be to select the agreement x minimising |u1(x)− u2(x)| amongst all Pareto optimal agreements. Example: Suppose there are two feasible agreements x and y: u1(x) = 2 u1(y) = 8 u2(x) = 4 u2(y) = 3 Inequality is lower for x, but y seems “better” (if we swap utilities for y, we get an agreement that would be Pareto-superior to x) . . . I There are no easy solutions. We need a systematic approach . . . Ulle Endriss (ulle@illc.uva.nl) 10 Social Choice and Welfare Multiagent Systems 2006 Abstraction 2: Agreements and Utility Vectors • Let A = {1, . . . , n} be our agent society throughout. • An agreement x gives rise to a utility vector 〈u1(x), . . . , un(x)〉 • We are going to define social preference structures directly over utility vectors u = 〈u1, . . . , un〉 (elements of R), rather than speaking about the agreements generating them. • Example: The definition of Pareto-dominance is rephrased as follows. Let u, v ∈ R. Then u is Pareto-dominated by v iff: – ui ≤ vi for all i ∈ A; and – ui < vi for at least one i ∈ A.ion 2: Agreements and Utility Vectors • Let A = {1, . . . , n} be our agent society throughout. • An agreement x gives rise to a utility vector 〈u1(x), . . . , un(x)〉 • We are going to define social preference structures directly over utility vectors u = 〈u1, . . . , un〉 (elements of R), rather than speaking about the agreements generating them. • Example: The definition of Pareto-dominance is rephrased as follows. Let u, v ∈ R. Then u is Pareto-dominated by v iff: – ui ≤ vi for all i ∈ A; and – ui < vi for at least one i ∈ A. Ulle Endriss (ulle@illc.uva.nl) 11 Social Choice and Welfare Multiagent Systems 2006 Social Welfare Orderings A social welfare ordering (SWO) is a binary relation over R that is reflexive, transitive, and connected . Intuitively, if u, v ∈ R, then u v means that v is socially preferred over u (not necessarily strictly). We also use the following notation: • u ≺ v iff u v but not v u (strict social preference) • u ∼ v iff both u v and v u (social indifference) Terminology: In the (economics) literature, connectedness is usually referred to as “completeness”. Furthermore, many authors use the letters R, P and I instead of , ≺ and ∼. Ulle Endriss (ulle@illc.uva.nl) 12 Social Choice and Welfare Multiagent Systems 2006 Collective Utility Functions • A collective utility function (CUF) is a function W : R → R mapping utility vectors to the reals. • Intuitively, if u ∈ R, then W (u) is the utility derived from u by society as a whole. • Every CUF represents an SWO: u v ⇔ W (u) ≤W (v) • Discussion: often convenient to think of SWOs in terms of a CUFs, but not all SWOs are representable as CUFs (example to follow) Ulle Endriss (ulle@illc.uva.nl) 13 Social Choice and Welfare Multiagent Systems 2006 Utilitarian Social Welfare One approach to social welfare is to try to maximise overall profit across society. This is known as classical utilitarianism (advocated, amongst others, by Jeremy Bentham, British philosopher, 1748–1832). The utilitarian CUF is defined as follows: swu(u) = ∑