On the Aggregation of Preferences in Engineering Design Beth Allen University of Minnesota NSF DMI

INTRODUCTION Engineering design may require the aggregation of preferences of different decision makers (designers or customers, or society in general). QUESTION: How should this be done? (Different procedures can give different results on which design is chosen. These procedures can have different properties and some may be fundamentally flawed.)

LITERATURE SO FAR Many different procedures and tools have been advocated. NEGATIVE RESULT: Arrow Impossibility Theorem. (There is no aggregation rule that satisfies unrestricted domain, Pareto, decisiveness, symmetry, IIA, and nondictatorship. This applies to decision makers with preferences or ordinal utilities, classically over finite sets of alternatives.)

MAIN IDEA Cardinal utilities (also known as von Neumann-Morgenstern utilities, based on preferences over lotteries) are the appropriate framework for engineering design rather than ordinal utilities, which do not reflect risk aversion. Cardinal utilities lead to different desirable properties for group decisions and permit positive (possibility) results for group decision processes, based either on averaging or on axiomatic bargaining theory (part of cooperative game theory).

FRAMEWORK set of decision makers (finite; use for pictures). Preferences defined on or on lotteries on for cardinal utilities. Design space or set of feasible attributes is a subset of.

PREFERENCES set of binary relations on satisfying the following: (a)Reflexivity (b)Transitivity (c)Completeness (d)Continuity (e)Monotonicity Each can be represented by many ordinal utility functions which are continuous and monotone.

UTILITIES Ord = continuous monotone ordinal utilities. DOrd = subset of Ord (one for each equivalence class or one for each preference in Pref) equal to distance along diagonal; Card = cardinal utilities on To obtain a unique utility function for each equivalence class in Card, set and

CONTINUOUS AGGREGATION Put a distance function (or point set topology) on Pref or Ord. [This has been extensively studied.] RESULT: There is an aggregation rule (or function from n-tuples of preferences) to preferences satisfying the following axioms: (1)Continuity (weakened form of IIA) (2) Respect for unanimity (Paretian) (3)Anonymity (symmetry w.r.t.permutations)

In fact, taking utility representations in DOrd and averaging satisfies these axioms. [Proof in Allen, Social Choice and Welfare, 13 (1996).]

CARDINAL AGGREGATION First normalize cardinal utilities in Card to satisfy and (generalizes to common worst and best elements in the feasible set) and then average (Harsanyi’s possibility theorem or relative utilitarianism).

AXIOMATIC BARGAINING For n decision makers with cardinal utilities, given a subset G of of feasible payoffs and a status quo point, find a solution point in G according to some rule (bargaining solution) satisfying certain axioms.

UTILITARIAN Given a bargaining problem with where G is closed, comprehensive and bounded from above, the utilitarian solution is the only solution satisfying the following: Efficiency Symmetry (w.r.t. permutations of decision makers) Linearity

NASH BARGAINING Given a bargaining problem with where and G is convex, closed, and bounded from above, the Nash solution is unique and is the only solution satisfying the following properties: Individual Rationality Feasibility Efficiency Symmetry (w.r.t. permutations of decision makers when G is symmetric) IIA Scale Invariance (cardinal payoffs)

REMARKS There are nonsymmetric versions of the Nash and utilitarian bargaining solutions. Also, there’s an extensive literature on alternative axiomatic characterizations and additional bargaining solutions. It’s important to note that this approach is based on individual decision makers (where symmetry may be appropriate). Attributes cannot play the role of decision makers.