The preference lattice

The preference lattice

Gregorio Curello Ludvig Sinander University of Bonn Northwestern University

6 December 2018

Preference comparisons

Preference comparisons arise in many settings: – choice under risk/uncertainty:

0 is more risk-/ambiguity-averse than  – monotone comparative statics:

0 _{takes larger actions than }

– dynamic problems:

0 is more delay-averse/impatient than 

All are special cases of single-crossing dominance.

Outline

We study the lattice structure of single-crossing dominance: characterisation, existence and uniqueness results for the

minimum upper bounds of arbitrary sets of preferences. Some applications:

Environment

The abstract environment is (X ,_&): – a non-empty set X of alternatives. . . – equipped with a partial order _&.

Notation: P denotes the set of all preferences on X .

Definition.1

For preferences , 0 ∈ P,

0 single-crossing dominates , written 0 S , iff for any_{&-comparable pair of alternatives x & y,} x () y implies x 0(0) y.

In a word: 0 is more aligned with_{& than is .} (Note: definition of S depends on_&.)

1

Milgrom and Shannon (1994).

Choice under uncertainty

– states of the world Ω – monetary prizes Π ⊆ R

– a set X of acts, meaning functions X : Ω → Π – the subset of constant acts is denoted C ⊆ X

Notation: P are all preferences (no axioms) on X .

Definition.2

For preferences , 0 ∈ P over acts, 0 is more ambiguity-averse than ,

iff for any act X ∈ X and constant act C ∈ C, C () X implies C 0(0) X.

‘More ambiguity-averse than’ as single-crossing

Definition.

For preferences , 0 ∈ P over acts, 0 is more ambiguity-averse than ,

iff for any act X ∈ X and constant act C ∈ C, C () X implies C 0(0) X.

Define_{& on X as follows:}

for acts X, Y ∈ X , X_{& Y iff either} (i) X is constant and Y is not, or (ii) X = Y .

‘More ambiguity-averse than’ is precisely single-crossing dominance S as induced by_&.

Upper bounds

Let P ⊆ P be a set of preferences.

0 ∈ P is an upper bound of P iff 0 S  for every  ∈ P . If in addition 00S 0 for every (other) upper bound 00 of P , then 0 is a minimum upper bound.

(= ‘join’ = ‘supremum’)

Choice under uncertainty:

Lattice structure

We study the lattice structure of (P, S): (1) characterisation theorem:

characterisation of the minimum upper bounds of any set P ⊆ P of preferences.

(2) existence theorem:

necessary and sufficient condition on _&

for every set P ⊆ P to possess ≥ 1 minimum upper bound. (The condition: _{& contains no crowns or diamonds.)} (3) uniqueness proposition (not today):

necessary and sufficient condition on _&

for every set P ⊆ P to possess = 1 minimum upper bound. (The condition: _{& is complete.)}

Applications

Choice under uncertainty:

– study generalised maxmin preferences:

those represented by X 7→ inf∈P c(, X) for some P ⊆ P.

– characterisation: ? admits maxmin representation P iff it a MUB of P w.r.t. ‘more ambiguity-averse than’ – representation: ? admits a maxmin representation iff

it satisfies certain axioms

Monotone comparative statics: – group with preferences P

P -chains

Definition.

For alternatives x_{& y,} a P -chain from x to y is a finite sequence (wk)Kk=1 such that

(1) w₁ = x and w_K = y (2) w_{k& wk+1}, ∀k < K

(3) w_k w_k+1 for some  ∈ P , ∀k < K

Strict P -chain: wk  wk+1 for some  ∈ P , ∃k < K.

Example.

X = {x, y, z} with x > y > z. P = {1, 2}, where

z 1x 1 y and y 2 z 2x.

P -chains, all strict: (x, y), (y, z) and (x, y, z). Note: (x, z) is not a P -chain.

Characterisation theorem

Characterisation theorem.

For a preference ? ∈ P and a set P ⊆ P, TFAE: (1) ? is a minimum upper bound of P .

Proof of (2) implies (1)

Characterisation theorem.

For a preference 0 ∈ P and a set P ⊆ P, TFAE: (1) ? is a minimum upper bound of P .

(2) ? satisfies: for any_{&-comparable x, y ∈ X , wlog x & y,} (?) x ?_{y iff there is a P -chain from x to y, and}

(??) y ? x iff there is no strict P -chain from x to y. (2) =⇒ (1), upper bound: WTS ?S  for every  ∈ P :

if x_{& y and x  y, then x }? y.

Holds by (?) because (x, y) is a P -chain.

(2) =⇒ (1), minimum: WTS 0S ? for every UB 0 of P : if x_{& y and x }? y, then x 0 y.

By (?), ∃ P -chain (wk)Kk=1 from x to y:

∀k < K, w_{k& wk+1} and w_k  w_k+1 for some  ∈ P =⇒ w_k0 _w

k+1 because 0 is an UB of P

Failure of existence

Example: X = {x, y, z, w} with the following partial order_&: x y z w Let P = {₁, 2} ⊆ P, where w 1x 1y 1 z and y 2 z 2w 2x.

∃ strict P -chain x → y and z → w =⇒ x ? _{y and z}? _w

Crowns

The same idea applies whenever _{& contains a crown:} a1 a2 a3 a4 (a) A 4-crown. a1 a2 a3 a4 a5 a6 (b) A 6-crown. a1 a2 a3 a4 a5 a6 a7 a8 (c) An 8-crown.

A K-crown (K ≥ 4 even) is a sequence (a_k)K_k=1 in X s.t. – a_k−1> ak< ak+1 for 1 < k ≤ K even (aK+1 ≡ a1)

– non-adjacent ak, ak0 are incomparable.

Diamonds

Existence also fails when_{& contains a diamond:} x

y z

w

A diamond is (x, y, z, w) such that x > y > w and x > z > w, but y, z are incomparable.

Existence theorem

Existence theorem.

The following are equivalent:

(1) Every set of preferences has ≥ 1 minimum upper bound. (2) _{& is crown- and diamond-free.}

Comments:

– (2) holds whenever there are ≤ 3 alternatives

– (2) fails for any lattice that isn’t a chain (=totally ordered), as (x ∧ y, x, y, x ∨ y) is a diamond for any incomparable x, y ¬(2) =⇒ ¬(1): by counter-example.

(2) =⇒ (1): quite difficult.

(Relies on Suzumura’s extension theorem.)

Choice under uncertainty: failure of existence

minimum upper bound w.r.t. ‘more ambiguity-averse than’? Recall: ‘more ambiguity-averse than’ is S as induced by_&, where X _{& Y iff either}

(i) X is constant and Y is not, or (ii) X = Y .

& contains crowns!

C

X

D

Y

Existence

Let’s restrict attention to monotone preferences: Preference  ∈ P is monotone

iff for any constant acts C, D ∈ C, C  D iff C ≥ D. Augment the definition of_{&: X &}0 Y iff either

(i) X is constant and Y is not, (ii) X = Y , or

(iii) X, Y are constant and X ≥ Y .

All monotone preferences agree with _&0 on pairs of type (iii). Thus: for monotone preferences, ‘more ambiguity-averse than’ coincides with S as induced by_&0.

And _&0 is crown- and diamond-free.

=⇒ every set of monotone preferences has

≥ 1 minimum upper bound w.r.t. ‘more ambiguity-averse than’.

Solvability

A certainty equivalent for  ∈ P of an act X ∈ X is a prize c(, X) ∈ Π such that X  c(, X)  X.

A preference with a certainty equivalent for every act is called solvable.

Maxmin representations

Definition.

A set P ⊆ P of monotone and solvable preferences is a maxmin representation of a preference ?∈ P iff

X 7→ inf

∈Pc(, X)

ordinally represents ?.

Maxmin expected utility3 is a special case: P a set of expected-utility preferences with

the same (strictly increasing) u but different beliefs µ.

X 7→ inf ∈Pc(, X) = inf∈Pu −1Z Ω [u ◦ X]dµ  = u−1  inf ∈P Z Ω [u ◦ X]dµ  3

Gilboa and Schmeidler (1989).

Maxmin–join equivalence

Proposition.

For a preference ? ∈ P and a set P ⊆ P of monotone and solvable preferences over acts, TFAE:

(1) P is a maxmin representation of ?. (2) ? is a minimum upper bound of P

w.r.t. ‘more ambiguity-averse than’.

(Trivial) representation theorem

Proposition.

A preference over acts admits a maxmin representation iff it is monotone and solvable.

⇐=: if ? _{is monotone & solvable}

then {?} is a maxmin representation.

=⇒ : suppose ? admits maxmin representation P . Solvable: certainty equivalent of X is inf∈P c(, X).

Monotone: on the constant acts C, ? is represented by C 7→ inf

∈Pc(, C)_{| {z }} =C

= C.

For interesting representation th’m, need more structure on P .

Monotone comparative statics

Let X ⊆ R be a set of actions, ordered by inequality ≥. The argmax of a preference  ∈ P is

X() := {x ∈ X : x  y for every y ∈ X }. The consensus among a group with preferences P ⊆ P is

C(P ) := \

∈P

X().

Comparative statics question:

Standard theory

For X, X0 ⊆ X ,

X0 dominates X in the (≥-induced) strong set order iff for any x ∈ X and x0 ∈ X0_,

the meet (join) of {x, x0} lies in X (in X0).

Theorem.4

For , 0∈ P, if 0 _{S ,}

then X(0) dominates X() in the (≥-induced) strong set order.

4

Milgrom and Shannon (1994), LiCalzi and Veinott (1992).

Consensus comparative statics

=⇒ every set of preferences has ≥ 1 meet and join. In fact, = 1.

For P, P0 ⊆ P,

P0 dominates P in the (S-induced) strong set order iff for any  ∈ P and 0 ∈ P0_,

the meet (join) of {, 0} lies in P (in P0_).

Proposition.

For P, P0 ⊆ P,

if P0 dominates P in the (S-induced) strong set order,

Proof

Take x ∈ C(P ) and x0 ∈ C(P0_);

WTS x ∧ x0∈ C(P ) and x ∨ x0 ∈ C(P0). Take arbitrary  ∈ P and 0∈ P0_.

Note x ∈ C(P ) ⊆ X().

By existence theorem, ∃ minimum upper bound ? of {, 0}. Since P0 dominates P in the SSO, ? lies in P0.

=⇒ x0 ∈ C(P0_{) ⊆ X(}?_).

Since ? S , X(?) dominates X() in the SSO by the standard theorem on the previous slide.

=⇒ x ∧ x0 ∈ X(). Since  ∈ P was arbitrary,

=⇒ x ∧ x0 ∈T

Thank you!

Failure of uniqueness

Observe that 0S  holds for any , 0 ∈ P: ‘for any _{&-comparable pair of alternatives x & y,}

x () y implies x 0(0) y.’

Holds vacuously since no alternatives are_{&-comparable.} So every preference  ∈ P is a minimum upper bound of every set P ⊆ P.

Uniqueness

Uniqueness proposition.

The following are equivalent:

Failure of existence for diamonds

y z

w

Existence fails for P = {1, 2} ⊆ P, where

y 1 w 1 z 1 x and w 2z 2 x 2 y.

∃ strict P -chain x → w (viz. (x, y, w)) @ P -chain z → w or x → z

=⇒ x ? w ? z ? x. Not a preference! ( /∈ P) (back to slide 15)

Proof of maxmin–join equivalence

∃ (strict) P -chain from X to Y iff inf

∈Pc(, X) ≥(>) inf∈Pc(, Y ).

X = C constant, Y not: TFAE: – ∃ (strict) P -chain from C to Y .

– C 0(0) Y for some preference 0 ∈ P . – inf∈P c(, C) ≥(>) inf∈Pc(, Y ).

X = C, Y = D both constant: TFAE: – ∃ (strict) P -chain from C to D. – C ≥(>) D.

References

Epstein, L. G. (1999). A definition of uncertainty aversion. Review of Economic Studies, 6(3), 579–608.

doi:10.1111/1467-937X.00099

Ghirardato, P., & Marinacci, M. (2002). Ambiguity made precise: A comparative foundation. Journal of Economic Theory, 102(2), 251–289. doi:10.1006/jeth.2001.2815

Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18(2), 141–153. doi:10.1016/0304-4068(89)90018-9

LiCalzi, M., & Veinott, A. F. (1992). Subextremal functions and lattice programming. working paper.

doi:10.2139/ssrn.877266

Milgrom, P., & Shannon, C. (1994). Monotone comparative statics. Econometrica, 62(1), 157–180. doi:10.2307/2951479