Mathematical Foundations of AI Lecture 10 – Fair Allocation (Indivisible goods)

Content Allocation of indivisible goods via deals Social welfare Envy freeness Back to cake-cutting

Model A finite set of items, that cannot be divided G = {g1,g2,…,gk} A set of agents N = {1,2,…,n} Every agent has an evaluation function, which assigns a value for every subset of items Vi : 2G  R

Allocations An allocation A is a partition of the set G, so that every agent gets several items (perhaps none) A = {A1,…,An} , such that: For all i≠j, Ai ∩ Aj is empty All items in G are allocated

Social Welfare We want an “objective” measure to the goodness of an allocation We simply sum the utility of all agents: SW(A) = ΣVi(Ai) Allocation A is called efficient, if it maximizes the social welfare Note that efficiency does not imply fairness

Example 3 2 10 6 1 7 value 3 6 9 SW:

Example 3 2 10 6 1 7 value 10 SW:

Deals and Payments One approach is an offline allocation of the goods by a central authority Maximizing SW is NP-hard in the general case We are interested is a situation where each agent makes its own decisions To facilitate such decisions, we introduce “money” into the system

Deals and Payments (2) A deal (A,A’) is transaction of items between agents so that the allocation changes from A to A’ A payment vector P=(P1,…Pn) determines how much each agent “pays” The sum of payments is always 0.

Deals and Payments (3) A deal (A,A’) is individually rational, if there is a payment vector P such that Vi(A’i)-Vi(Ai) ≥ Pi for every agent, and the inequality is strict for at least one agent. This means each agent gains at least what he is paying.

Deals - Example 2 3 7 8 value 7 SW:

Deals - Example No rational deal for this allocation 2 3 7 8 value 2 3 5 No rational deal for this allocation SW: Agent 1 has to “pay” at most 2-7 = -5, i.e. to get at least 5 Agent 2 has to pay at most 3-0 = 3

Deals - Example 2 3 7 8 value 8 SW: 8 SW: P1 = -7.5 ; P2 = 7.5 The social welfare increased after the IR deal!

Rationality and Social Welfare Lemma 1: a deal (A,A’) is IR iff it strictly increases the social welfare Lemma 2: any sequence of IR deals is finite Theorem 3: any sequence of IR deals, from any initial allocation, will eventually reach an efficient allocation

Envy freeness Without payments, envy-free allocations may not be possible E.g. : we have only 1 item, wanted by 2 agents We define a state as a pair S=(π,A) where π is a payment vector

Envy freeness (2) A state S is envy-free, if no agent wants to switch places with another, i.e. if for all i,j Vi(Si)- πi ≥ Vi(Sj) – πj Example: V1 = 3 V2 = 2 π1 = 1 π2 = -1 V1-π1 = 2 ≥ 1 =0-π2 0-π2 = 1 ≥ V2-π1

Envy freeness (2) A state S is envy-free, if no agent wants to switch places with another, i.e. if for all i,j Vi(Si)- πi ≥ Vi(Sj) – πj Example: V1 = 3 V2 = 2 π1 = -? π2 = ? -π1 ≥ 3 - π2 > 2 – π2 ≥ -π1

EEF We want to reach a state that is both Efficient and Envy-free (EEF) There is always an EEF We will prove for a less general case

Super additivity An evaluation function V is super additive, if for any distinct sets A,B V(A U B) ≥ V(A)+V(B) Theorem 4: if all agents have super additive evaluations, then there is an EEF state We take an efficient allocation A*, and add the payments πi = Vi(A*) – SW(A*)/n

Reaching EEF states Consider the following initial state: V1 = 3 There is only IR deal, and it will make agent 1 pay at least 2, thus making it envy But the state wasn’t fair in the first place! V1 – π1 = 0 V2 - π2 = 2 V1 – π1 ≤ 3-2 =1 V2 - π2 ≥ 2

Globally uniform payments For every initial allocation A0, we set the initial payment to π0i = Vi(A0) – SW(A0)/n The equitability condition For every deal (A,A’), we define a globally uniform payment function (GUPF) Pi = Vi(A’)-Vi(A) – (SW(A’)-SW(A))/n These payments combine the private gain and the common gain. An agent will pay more if the deal helps him But less if this deal helps the society

Reaching EEF states (2) Theorem 5: If all Vi are super additive, and we start from any initial allocation A0, with initial payments π0, then every sequence of IR deals with GUPF will reach an EEF state. From Theorem 3 we know that efficiency is guaranteed, so we only need to prove envy freeness. We simply show by induction that equitability still holds after each deal.

Approximate envy freeness Suppose we cannot use payments for some reason Legal Practical etc. We may still want to “minimize” the envy that agents may feel towards others Let eij(A) the “envyness” between i,j, in A We want to minimize e(A) = max(eij(A))

Example 3 2 10 6 1 7 3 7 eij(A)=max{0, Vi(Aj)-Vi(Ai)} e(A) = 7

Example 3 2 10 6 1 7 2 5 eij(A)=max{0, Vi(Aj)-Vi(Ai)} e(A) = 5

Marginal utility The marginal utility of an item, is the value it adds to a set of items already owned Depends on the agent and the set The maximum marginal utility, is the maximal increase in utility of an agent, than can occur from getting one more item α = max [ Vi(G’ U {g}) – Vi(G’) ] The maximum is taken over all agents, all items g and all subsets of items G’

Example 3 2 1 10 6 7 α = V2({fork,spoon}) – V2({spoon}) = 10-1 = 9

Bounding the envy Theorem 6: There is an allocation A, such that the total envy of A is bounded: e(A) ≤ α Further, there is a polynomial algorithm that finds such an allocation

Back to cake cutting We want a division whose total envy as at most ε We can use the following protocol: Each agent cuts the cake to pieces of size ≤ε We now have O(n/ε) parts. The marginal value of each part is at most ε, thus α≤ε We use the algorithm from theorem 6 to allocate the parts. We get that e(A)≤α≤ε Two types of queries are needed