Optimization norms

Aditya Ghose

Decision Systems Lab

School of Computer Science and

Software Engineering

University of Wollongong

Decision Systems Lab

University of Wollongong

What drives the normative design of

institutions?

Optimization norms1 Norms are conventionally Boolean e.g. “reduce

carbon emissions by N¨

• N is arbitrary

– Difficult market conditions would lead to a lower N

– Affordable “green” technologies would lead to a

higher N.

• Regulations/legislation revised infrequently – hence

any choice of N will be quickly obsolete!

(1) A. K. Ghose and B. T. R. Savarimuthu. Norms as Objectives: Revisiting

Compliance Management in Multi-Agent Systems. In Proc. of the 14th

International Workshop on Coordination, Organisations, Institutions and Norms

(COIN 2012) held in conjunction with AAMAS-2012, Valencia, Spain, June, 2012.

Optimization norms

• The choice of an appropriate N is highly contextual –

e.g., setting a carbon price

– Cannot be too high – else business environment

becomes unviable

– Cannot be too low – else no incentive to change

behaviour

• The underpinning norm: minimize carbon footprint

The key ideas (1/2) • The dichotomy between:

– Boolean norms

– Optimization norms

• Compliance with optimization norms

• Defining consistency

• Inconsistency resolution

– Connnections with non-monotonic inference

The key ideas (2/2)

• An algebraic formulation of optimization norms

– Graded compliance

– Supports formalization of contrary-to-duty obligations

– Graded sanctions

• A scheme for sanction decomposition in multi-agent

settings

– Requires the algebraic preference combination operator to be

idempotent

• Some implementations in agent programming languages

Optimization objectives

• An optimization problem consists of:

– A set of decision variables (the signature)

– A set of constraints defined over the signature

– An objective function (also defined over the signature)

that we want to minimize or maximize

• The set of feasible solutions to an optimization

problem is the set of value assignments to

variables in the signature that satisfy all of the

constraints

• An optimal solution to a maximization (resp.

minimization) problem is the feasible solution that

maximizes (resp. minimizes) the value of the

objective function

What is different?

• Optimization objectives differ from

classical assertions:

– They do not admit boolean valuation

– They can only be represented via preference

relations in specific contexts (given a specific

set of alternatives)

– A notion of consistency is not straightforward

• A pair of objectives might induce a consistent pair

of preference orderings in one context and an

inconsistent pair in another

Objective functions in context: An example

• Obj1: max x/y

• Obj2: max x.y

Context 1 (Objectives are consistent)

Soln 1: <x=3, y=1> Soln 2: <x=2, y=1>

Context 2 (Objectives are inconsistent)

Soln 1: <x=4, y=2> Soln 2: <x=4, y=3>

Norm compliance: The carbon tribunal

thought experiment

• In a carbon tribunal of the future, a company

stands charged with having violated the carbon

footprint minimization norm

• How does the accused launch a defence?

– Take a log of critical decisions over an audit period

• A record of the available choices and the selected

choice

– Note that every choice was optimal

– Requires agreement with the tribunal on the set of

available choices in each instance

• More on this…..

Agreeing on best effort • In the carbon tribunal example, compliance with an

optimization norm reduces to:

– Establishing that an optimal choice was made in each

instance amongst available options

– Establishing that the listed options were the only feasible

ones

– Establishing that the constraints solved to obtain the

feasible options were sound and complete relative to the

context

• Compliance checking reduces to:

– Determining the soundness and completeness of a set of

constraints

– Applying (standard) optimization machinery

Compliance

Do optimization norms underpin all norms?

• Prohibitions → minimization

• Obligations → maximization

• Fairness norms → load-balancing

objectives

– Translating between norms and objectives is

easy

Identifying Optimization Norms

Norm Conflict (1)

Norm Conflict (2)

• Choice consistency

– Let S1 be the set of best solutions according

to o1

– Let S2 be the set of best solutions according

to o2

– o1 and o2 conflict iff S1 ∩ S2 = ᶲ

• Consistency with boolean norms

Amartya Sen’s thought experiment:

Norm conflict resolution

• X, Y and Z are slices of cake

• I prefer bigger slices of cake

• X > Y > Z

• Choose amongst {X, Y, Z}: Y

• Choose amongst (Y, Z): Z

• I prefer to not pick the largest slice on the plate

(the “politeness” norm)

Norm conflict resolution

• Few solutions in the literature – some inadequate solutions

from computational social choice theory

• Conflict resolution concepts in the Operations Research

literature:

– Pareto-optimality

– Lexicographic orderings

– Weighted sums

• Many critical social processes involve conflict resolution

amongst optimization norms

– Policy formulation

– Budget formulation (maximize investment in health, or in

education, or in R&D?)

• Sometimes we need to resolve conflicts between boolean

(and optim.) norms, in the spirit of non-monotonic inference

AN ALGEBRAIC

FORMALIZATION

C-semirings

(Bistarelli, Montanari and Rossi, 1996)

C-semiring instances

• Boolean: <{T, F}, , , F, T> • Fuzzy: <[0, 1], max, min, 0, 1> • Cost/time:< Z+, min, +, +, 0 > • Quality: <{HIGH, MEDIUM, LOW}, , , LOW, HIGH>

Valued Optimization Norm

Sanction decomposition

• The “strength” of a sanction is related to the extent to

which an optimization norm is satisfied

• In many multi-agent settings, norms have to be

decomposed.

– When a contract incurs a penalty because of an outcome that is

the joint responsibility of multiple contractors: How should the

penalty (and blame) be apportioned across the sub-contractors?

• When the algebraic combination operator for preference

values is idempotent (a a = a), the analysis is easy.

– For non-idempotent operators, more complex causal analysis is

required.

Sanction decomposition

• Consider a set of abstract preference values {v1, v2, v3}

such that: v1 ≥ v2 ≥ v3 (v1 v2 = v1, v2 v3 = v2

etc…)

• Let the final outcome be assessed at preference value

v2

• Assume that:

– subcontractor agent 1 delivers its part at preference value v1

– Subcontractor agents 2 and 3 deliver their parts at preference

value v2

• Then the final penalty should be shared equally between

agents 2 and 3

Sanction decomposition

SOME IMPLEMENTATIONS

QUESTIONS?