# Resource Allocation with Answer-Set Programming

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3 João Leite, José Alferes, Belopeta Mito Resource Allocation with Answer-Set Programming where wmax Π and lmax Π denote the maximum weight and maximum level over the weak constraints in Π, respectively; Ni Π (A) denotes the set of the weak constraints in level i that are violated by A, and weight(w) denotes the weight of the weak constraint w. Intuitively, the function f Π handles priority levels. It guarantees that the violation of a single constraint of priority level i is more expensive than the violation of all weak constraints of the lower levels (i.e., all levels <i). For a DLV program Π (possibly with weak constraints), a set A is an (optimal) answer set of Π if and only if: 1) A is an answer set of Rules(Π) and 2) H P (A) is minimal over all the answer sets of Rules(Π). Let AS (Π) denote the set of (optimal) answer sets of Π. DLV also allows for aggregate functions whose syntax is defined as follows. A (DLV) symbolic set is a pair {Vars: Conj} where Varsis a list of variables and Conj is a conjunction of standard literals. Intuitively, a symbolic set {X:a(X,Y),p(Y)} stands for the set of X-values making a(x,y),p(y) true. An aggregate function is of the form f (S) where S is a symbolic set, and f is a function name among #count, #min, #max, #sum, #times. An aggregate atom is Lg 1 f (S) 2 Rg where f (S) is an aggregate function, 1, 2 {=,<,,>, }and Lg and Rg are terms. One of Lg and Rg can be omitted. Rules can also include aggregate atoms in their bodies. As an example, consider a predicate person(name,age,salary). The aggregate atom 19<#min{A:person(N,A,S)} 30 then evaluates to true iff the age of the youngest person is in the range [20..30]. The aggregate function #sum{s,n:person(n,a,s)}=t returns the sum of the salaries of all the employees 1. The formal semantics of aggregates can be found in [7]. 3. RESOURCE ALLOCATION In this Section we take a closer look at resources, agents, allocations and preference representations, presenting, as we proceed, ASP encodings of these concepts that serve as the building blocks of the ASP programs used to solve the different MARA problems. Before we proceed with a detailed account of the relevant concepts related to MARA we set forth a more formal definition both of the input and the solution space. A Multi-Agent Resource Allocation Setting (MARA Setting) is a triple A, R, U where A = {a 1,a 2,..., a n} is a set of n agents, R = {r 1,r 2,..., r m} is a set of m resources and U = {ur a1,ur a2,..., ur an } is a set of preference representations of the agents. An allocation is a mapping M : A 2 R from agents to a subset of resources R such that M (a i) M (a j)= for any two agents a i a j. If Θ= A, R, U is a MARA Setting, then, Ag (Θ) = A, Res (Θ) = R, Pref (Θ) = U and M Θ denotes the set of all possible allocations. 3.1 Resources, Agents and Allocations Resources play a central role in any resource allocation problem. We can categorise resources according to their nature and the way they are treated during allocation. They are usually categorised as continuous or discrete, indivisible or divisible, sharable or non sharable and static or non static (c.f. [5] for explanation). In this paper we deal with the allocation of discrete, indivisible, non sharable and static resources, each represented by a fact in ASP. Likewise, agents will be represented by simple facts in ASP. Let Θ be a MARA Setting. Then Π Θ AR contains: ag(a). a Ag (Θ). res(r). r Res (Θ). 1 The sum is formed over the first variable. The second, N, ensures that one value per person is summed. Otherwise, two people with the same salary would only be summed once. Before we proceed, we introduce the ASP encoding of an allocation. Let M i : A 2 R be an allocation. Then Π Mi contains: has(a,r,i). a, r : r M i (A). al(i). 3.2 Preference Representation Agents preferences represent the relative or the absolute agent s satisfaction with a particular allocation, and are necessary for defining the quality aspects of a solution to a resource allocation problem. A preference structure represents the preferences of an agent over a possible set of alternatives. We will consider two preference structures: cardinal and ordinal. The cardinal preference structure consists of a utility function u :2 R Valwhere 2 R is the set of alternative bundles (sets of resources), and Valis a set of numeric values or a totally ordered scale of qualitative values (e.g. very bad, bad, average, good...). The former is called quantitative while the latter alternatives qualitative preference structure. In this paper we do not consider the qualitative case as it can be reduced to the quantitative one. The ordinal preference structure consist of binary relations between alternatives x y denoting that the bundle x is at least as preferred as the bundle y. Strict preference is denoted by x y. Other kinds not considered here include the binary preference structure which partitions the set of alternatives into good ones and bad ones, and fuzzy preference structure which is a fuzzy relation over the set of alternatives 2 R, i.e. a function f :2 R 2 R [0, 1] defining the degree to which one alternative is preferred over the other. Since such preferences have not been extensively studied and are rarely used in practice, they are not further considered here. Preferences must be properly represented when used in practice. Since the number of alternatives is exponential wrt. the number of resources, an explicit representation (listing all alternatives with their utility values for cardinal preference, or the full relation for ordinal preferences) is exponentially large. For this reason, languages for succinct preference representation are important. In what follows, we present some common languages, first for the quantitative cardinal preference representation and then for the ordinal preference representation, together with their ASP encodings Bundle Form Preferences The bundle form is the most basic approach for representing quantitative preferences. It simply enumerates all bundles for which the agent has a non-zero utility. Thus, a preference representation in the bundle form is a set of pairs B,u(B) for all bundles B such that u (B) 0. In ASP, we will encode quantitative preferences through the definition of a predicate ut(a i,v,a j,x)which represents the utility V agent A i assigns to the set of resources (bundle) given to agent A j in allocation X. Let Θ be a MARA Setting with maximum utility u max and preferences in the bundle form 2. Then Π Θ BFS contains the rules: utg(0..u max ). ut(a,u,a,x):-al(x),ag(a), has(a,r 1,X),...,has(A,r i,x), not has(a,r j,x),...,not has(a,r n,x). B,u ur a : B = {r 1,..., r i} Res (Θ) B = {r j,..., r n} and ur a Pref (Θ). The first two predicates in the body of the rule, al(x),ag(a), are required to make the rules safe. Similar predicates will be used in other ASP encodings. 2 u max is the highest utility expressed by an agent, rather than the highest utility expressible by an agent, hence easy to determine. 651

5 João Leite, José Alferes, Belopeta Mito Resource Allocation with Answer-Set Programming not hasmin(a,a,x 1 ). Π BOb =Π BOc together with the rules: pref_b(a,a 1,A 2,X):-al(X),ag(A,A 1 ;A 2 ),A 1!=A 2, M 2<=M 1,#min{R 1 :nsat(a,a 1,R 1,X)}=M 1, #min{r 2 :nsat(a,a 2,R 2,X)}=M 2. pref_b(a,a 1,A 2,X):-al(X),ag(A,A 1 ;A 2 ),A 1!=A 2, not hasmin(a,a 1,X). Descrimin ordering: An allocation M 2 is descrimin preferred over an allocation M 1 if the minimal rank of goal(s) not satisfied in M 1 but satisfied in M 2 is lower than the minimal rank of goal(s) not satisfied in M 2 but satisfied in M 1, or if the sets of goals satisfied in each allocation are equal. Formally, let d (M 1,M 2) = min{r (G i):m 1 = G i M 2 = G i}. Then M 1 do GB M 2 iff d (M 1,M 2) <d(m 2,M 1) or {G i : M 1 = G i} = {G i : M 2 = G i}. The encodings in ASP are: Π DOal contains the rules: ald(a,x 1,X 2,D):-al(X 1 ;X 2 ),X 1!=X 2,ag(A), g(a,g,d),not sat(a,g,a,x 1 ),sat(a,g,a,x 2 ). hasd(a,x 1,X 2 ):-al(x 1 ;X 2 ),X 1!=X 2,ag(A), ald(a,x 1,X 2,D). G 2<G 1,#min{D 1 :ald(a,x 1,X 2,D 1 )}=G 1, #min{d 2 :ald(a,x 2,X 1,D 2 )}=G 2. not hasd(a,x 1,X 2 ). Π DOb contains the rules: bd(a,a 1,A 2,X,D):-al(X),ag(A;A 1 ;A 2 ),g(a,g,d), not sat(a,g,a 1,X),sat(A,G,A 2,X). hasbd(a,a 1,A 2,X):-al(X),ag(A;A 1 ;A 2 ), bd(a,a 1,A 2,X,D). pref_b(a,a 1,A 2,X):-al(X),ag(A,A 1 ;A 2 ),G 2 <G 1, #min{d 1 :bd(a,a 1,A 2,X,D 1 )}=G 1, #min{d 2 :bd(a,a 2,A 1,X,D 2 )}=G 2. pref_b(a,a 1,A 2,X):-al(X),ag(A,A 1 ;A 2 ), not hasbd(a,a 1,A 2,X). Leximin ordering: An allocation M 2 is leximin preferred over an allocation M 1 if it satisfies more goals of certain rank i, and an equal number of goals for all ranks which are smaller than i. Formally, let d k (M) = {G i : M = G i r (G i)=k}. Then, M 1 lo GB M 2 iff k : d k (M 1) < d k (M 2) and j < k : d j (M 1) = d j (M 2). M 1 lo GB M 2 iff M 1 lo GB M 2 or j : d j (M 1)=d j (M 2). The encodings in ASP are: Π LOc contains the rules: dk(a,a 2,R,DK,X):-al(X),ag(A;A 2 ),r(r), #count{g:sat(a,g,a 2,X),goal(A,G,R)}=DK. Π LOal =Π LOc together with the rules: difalp(a,x 1,X 2,R):-al(X 1 ;X 2 ),X 1!=X 2,ag(A), r(r;r 2 ),R>R 2,dk(A,A,R 2,DK 1,X 1 ), dk(a,a,r 2,DK 2,X 2 ),DK 1!=DK 2. r(r),dk(a,a,r,dk 1,X 1 ),dk(a,a,r,dk 2,X 2 ), DK 1>DK 2,not difalp(a,x 1,X 2,R). not difalp(a,x 1,X 2,r max +1). Π LOb =Π LOc together with the rules: difbr(a,a 1,A 2,X,R):-ag(A;A 1 ;A 2 ),A 1!=A 2,al(X), r(r;r 2 ),R>R 2,dk(A,A 2,R 2,DK 2,X), dk(a,a 1,R 2,DK 1,X),DK 1!=DK 2. pref_b(a,a 1,A 2,X):-al(X),ag(A;A 1 ;A 2 ),A 1!=A 2, r(r),dk(a,a 1,R,DK 1,X),dk(A,A 2,R,DK 2,X), DK 1>DK 2,not difbr(a,a 1,A 2,X,R). pref_b(a,a 1,A 2,X):-al(X),ag(A;A 1 ;A 2 ),A 1!=A 2, not difbr(a,a 1,A 2,X,r max +1). 3.3 Qualitative Issues In general, the quality of a solution to a resource allocation problem is defined by a metric that depends, in one way or another, on the preferences of the agent. The aggregation of the individual preferences in such metric is often modeled using the notion of social welfare as studied in Welfare Economics and Social Choice Theory. Depending on the system s goals, different social welfare functions can be used for measuring the quality of an allocation [1, 15]. The most fundamental quality criterion for a solution is Pareto Optimality. Given a MARA Setting Θ, an allocation M is paretodominated by another allocation Q if and only if: 1) a i Ag (Θ) : M (a i) ai N (a i) and 2) a i Ag (Θ) : M (a i) ai N (a i). In the case of utility functions, we simply replace M (a i) ai N (a i) and M (a i) ai N (a i) with u i (M (a i)) u i (N (a i)) and u i (M (a i)) <u i (N (a i)). An allocation is pareto-optimal if and only if it is not pareto-dominated by any other allocation. The ASP encodings of pareto dominance define a predicate pd(x 1,X 2) indicating that allocation X 2 is dominated by allocation X 1. They are, for the cardinal and ordinal preferences respectively: Π POc contains the rules: pd(x 1,X 2 ):-al(x 1 ;X 2 ),dm(x 1,X 2 ),not dm(x 2,X 1 ). dm(x 1,X 2 ):-al(x 1 ;X 2 ),ag(a),ut(a,u 1,A,X 1 ), ut(a,u 2,A,X 2 ),U 1 >U 2. Π POo contains the rules: pd(x 1,X 2 ):-al(x 1 ;X 2 ),dm(x 1,X 2 ),not dm(x 2,X 1 ). dm(x 1,X 2 ):-al(x 1 ;X 2 ),ag(a),pref_al(a,x 1,X 2 ), not pref_al(a,x 2,X 1 ). Envy freeness is also a purely ordinal quality criterion. An allocation is envy-free when every agent is at least as satisfied with the bundle allocated to it as with the bundles allocated to other agents. Given a MARA Setting Θ, an allocation M is envy-free if and only if a i,a j Ag (Θ) : M (a j) ai M (a i) for the case of ordinal preferences, and a i,a j Ag (Θ) : u i (M (a j)) u i (M (a i)). The ASP encodings of envy-freeness define a predicate ef(x) indicating that allocation X is not envy-free. They are, for the cardinal and ordinal preferences respectively: Π EFc contains the rules: ef(x):-al(x),ag(a 1 ;A 2 ),A 1!=A 2, ut(a 1,U 1,A 1,X),ut(A 1,U 2,A 2,X),U 2 >U 1. Π EFo contains the rules: ef(x):-al(x),ag(a 1 ;A 2 ),A 1!=A 2, pref_b(a 1,A 2,A 1,X),not pref_b(a 1,A 1,A 2,X). In some cases, when no envy-free allocation exists, we may be interested in minimizing some measure of the degree of envyness. Given an allocation, the degree of envyness of some agent wrt. another agent is given by the difference between its utility and the other agent s utility, in case its own is smaller. This is encoded in ASP as follows: Π EFod contains the rules: envy(a 1,A 2,D,X):-al(X),ag(A 1 ;A 2 ),A 1!=A 2, ut(a 1,U 1,A 1,X),ut(A 1,U 2,A 2,X),U 2 >U 1,U 2 -U 1=D. Given a MARA Setting Θ and an allocation M, envy (M) denotes the sum of the degree of envyness of all pairs of agents Social Welfare When the agents have their preferences expressed with a utility function, then every allocation M gives rise to the utility vector u 1 (M),..., u n (M). A collective utility function (CUF) is a mapping from such a vector to a numeric value. Since every allocation determines a utility vector, CUF can also be considered as a function from allocations to numeric values which represent the social welfare of the corresponding allocation. Thus, given a CUF sw, an allocation M is socially preferred to allocation N if and 653

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