Complex collective choices Luigi Marengo 1 Dept. of Management, LUISS University, Roma, lmarengo@luiss.it Based on joint work with G. Amendola, G.Dosi, C. Pasquali and S. Settepanella
Complex collective decisions ◮ we consider “complex” multidimensional decisions, in the sense that: ◮ they involve several items (features) ◮ there are non-separabilities and non-monotonicities (interdependencies) among such items
A simple example: “What shall we do tonight?” ◮ C = { movie, theater, restaurant, stay home, . . . } ◮ the object “going to the movies” is defined by: ◮ with whom ◮ which movie ◮ which theater ◮ what time ◮ . . . ◮ the object “stay home” is defined by: ◮ with whom ◮ to do what ◮ e.g. watch TV, or have a drink put on a nice record and see what happens . . . ◮ which show ◮ which movie ◮ what we eat ◮ . . .
Some obvious non-standard properties 1. objects typically do not partition the set of traits/features 2. in general there are obvious non-separabilities and non-monotonicities (interdependencies) among traits ◮ e.g. I might prefer Franc ¸oise to Corrado as instance of the “with whom” if associated to “staying at home” and “tˆ ete-` a-tˆ ete dinner”, but Corrado to Franc ¸oise as an instance of the “with whom” if associated to “going to the football match” and “with ten more male friends”
The general question ◮ how does the aggregation of items/features into objects determines collective outcomes
Two families of models 1. a committee where a group of people choose (e.g. by pairwise majority voting) a value for all items according only to their preferences 2. an organization where decision rights are divided are delegated to individual agents and outcomes have some “objective” value notions of authority and power: ◮ in the committee model: power of object construction (putting items together to form an object of choice) and power of agenda ◮ in the organization model: power of allocating decisions (delegation) and power of vetoing and overruling decisions
The Committee Model I ◮ Choices are made over a set of N elements or features F = { f 1 , f 2 , . . . , f N } , each of which takes on a value out of a finite set possibilities. ◮ Simplifying assumption: such a set is the same for all elements and contains two values labelled 0 and 1: f i ∈ { 0 , 1 } . ◮ The space of possibilities is given by 2 N possible choice configurations : X = { x 1 , x 2 , . . . , x 2 N } .
The Model II ◮ There exist h individual agents A = { a 1 , a 2 , . . . , a h } , each characterized by a (weak) ordering on the set of choice configurations ◮ We call this ranking agent k ’s individual decision surface Ω k .
The Model III ◮ Given a status quo x i and an alternative x j agents sincerely vote according to their preferences. ◮ A majority rule is used to aggregate their preferences: ℜ : (Ω 1 , Ω 2 , . . . , Ω h ) �→ Ω .
The Model IV ◮ Given an initial configuration and a social decision rule ℜ this process defines a walk on the social decision surface which can either: 1. end up on a social optimum 2. cycle forever among a subset of alternatives.
Objects (Modules) Let I = { 1 , 2 , . . . , N } be the set of indexes. An object (decision module) C i ⊆ I The size of object C i is its cardinality | C i | . An object scheme is a set of modules: C = { C 1 , C 2 , . . . , C k } k � such that C i = I i = 1 (. . . but not necessarily a partition)
Agendas An agenda α = C α 1 C α 2 . . . C α k over the object set C is a permutation of the set of objects which states the order according to which objects are examined.
Voting procedure We use the following algorithmic implementation of majority voting: 1. repeat for all initial conditions x = x 1 , x 2 , . . . , x 2 N 2. repeat for all objects C α i = C α 1 , C α 2 , . . . , C α k until a cycle or a local optimum is found; 3. repeat for j=1 to 2 | C α i | ◮ generate an object-configuration C j α i of object C α i ◮ vote between x and x ′ = C j α i ∨ x ( C − α i ) ◮ if x ′ � ℜ x then x ′ becomes the new current configuration
Stopping rule We consider two possibilities: 1. objects which have already been settled cannot be re-examined 2. objects which have already been settled can be re-examined and if new social improvements have become possible
Walking on social decision surfaces Given an object scheme C = { C 1 , C 2 , . . . , C k } , we say that a configuration x i is a preferred neighbor of configuration x j with respect to an object C h ∈ C if the following three conditions hold: 1. x i � ℜ x j ν = x j 2. x i ν ∀ ν / ∈ C h 3. x i � = x j We call H i ( x , C i ) the set of neighbors of a configuration x for object C i . A path P ( x i , C ) from a configuration x i and for an object scheme C is a sequence, starting from x i , of preferred neighbors: P ( x i , C ) = x i , x i + 1 , x i + 2 , . . . with x i + m + 1 ∈ H ( x i + m , C ) A configuration x j is reachable from another configuration x i and for decomposition C if there exist a path P ( x i , C ) such that x j ∈ P ( x i , C ) .
Social outcomes ◮ A configuration x is a local optimum for the decomposition scheme C if there does not exist a configuration y such that y ∈ H ( x , C ) and y ≻ ℜ x . 0 , . . . , x j ◮ A cycle is a set X 0 = { x 1 0 , x 2 0 } of configurations 0 ≻ ℜ x 2 0 ≻ ℜ . . . ≻ ℜ x j 0 ≻ ℜ x 1 such that x 1 0 and that for all x ∈ X 0 , if x has a preferred neighbor y ∈ H ( x , C ) then necessarily y ∈ X 0 .
The relevance of objects I ◮ object construction mechanisms forego and constrain choices. ◮ Influence of the generative mechanism: 1. define the sequence of voting; 2. define which subset of alternatives undergoes examination.
The relevance of objects II ◮ Different sets of objects may generate different social outcomes. ◮ Social optima do – in general – change when objects are different both because: 1. the subset of generated alternatives is different (and some social optima may not belong to many of these subsets) 2. the agenda is different (and this may determine different outcomes). ◮ Framing power appears therefore as a more general phenomenon than agenda power.
Results in a nutshell ◮ Under general conditions (notably if preferences are not fully separable) the answer to the previous question is entirely dependent upon decision modules. ◮ We show algorithmically that, given a set of individual preferences: 1. by appropriate modifications of the decision modules it is possible to obtain different social outcomes. 2. cycles ` a la Condorcet-Arrow may also appear and disappear by appropriately modifying the decision modules. 3. the median voter theorem is also dependent upon the set of alternatives (median voter may be transformed into outright loser) ◮ trade-off decidability-manipulability: “finer” objects make cycles disappear but generate many local optima (social outcome will depend on initial status quo) and simplify the pairwise voting process
Results I ◮ Social outcomes are, in general, dependent upon the objects scheme ◮ Consider a very simple example in which 5 agents have a common most preferred choice. ◮ By appropriately modifying the objects scheme one can obtain different social outcomes or even the appearance/disappearance of intransitive limit cycles.
Results II Rank Agent1 Agent2 Agent3 Agent4 Agent5 1st 011 011 011 011 011 2nd 111 000 010 101 111 3rd 000 001 001 111 000 4th 010 110 101 110 010 5th 100 010 000 100 001 6th 110 111 110 001 101 7th 101 101 111 010 110 8th 001 100 100 000 100
Results III ◮ With C = {{ f 1 , f 2 , f 3 }} the only local optimum is the global one 011 whose basin of attraction is the entire set X . ◮ With C = {{ f 1 } , { f 2 } , { f 3 }} we have the appearance of multiple local optima and agenda-dependence. ◮ With C = {{ f 1 , f 2 } , { f 3 }} multiple local optima but agenda-independence.
Object-dependent cycles I Redefining modules can make path dependence disappear. ◮ Consider the case of three agents and three objects with individual preferences expressed by: Order Agent 1 Agent 2 Agent 3 1st x y z 2nd y z x 3rd z x y
Object-dependent cycles II ◮ Social preferences expressed through majority rule are intransitive and cycle among the three objects: x ≻ ℜ y and y ≻ ℜ z , but z ≻ ℜ x . ◮ Imagine that x,y,z are three-features objects which we encode according to the following mapping: x �→ 000 , y �→ 100 , z �→ 010
Object-dependent cycles III ◮ Suppose that individual preferences are given by: Order Agent 1 Agent 2 Agent 3 1st 000 100 010 2nd 100 010 000 3th 010 000 100 4th 110 110 110 5th 001 001 001 6th 101 101 101 7th 011 011 011 8th 111 111 111
Object-dependent cycles IV 1. With C = {{ f 1 , f 2 , f 3 }} the voting process always ends up in the limit cycle among x,y and z . 2. The same happens is each feature is a separate object: C = {{ f 1 } , { f 2 } , { f 3 }} . 3. However, with: C = {{ f 1 } , { f 2 , f 3 }} or with: C = {{ f 1 , f 3 } , { f 2 }} Voting always produces the unique global social optimum 010 in both cases.
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