Proof of Theorem 2.1

In this subsection we prove our baseline representation result. The proof of the Theorem proceeds through several lemmas. First consider the following remark.

Remark 2.3. Our axioms (weak order, bi-continuity and comparable independence)

imply that _≽𝑧 satisfies the three axioms of the expected utility theorem; namely

∙ Weak Order: _≽𝑧 is complete and transitive.

∙ : vNM Continuity: For any 𝑝𝐴, 𝑝′𝐴, 𝑝′′𝐴 ∈ Δ𝐴, if 𝑝𝐴 ≻𝑧 𝑝′𝐴 ≻𝑧 𝑝′′𝐴, then there

𝜆𝑝𝐴 + (1−𝜆)𝑝′′𝐴 ≻𝑧 𝑝′𝐴 ≻𝑧 𝜆′𝑝𝐴 + (1−𝜆′)𝑝′′𝐴

∙ vNM Independence: For any𝑝𝐴, 𝑝′𝐴 ∈ Δ𝐴, if𝑝𝐴 ≻𝑧 𝑝′𝐴, then for any𝑝

′′ 𝐴 ∈ Δ𝐴, 𝜆 ∈ (0,1] 𝜆𝑝𝐴 + (1−𝜆)𝑝𝐴′′ ≻𝑧 𝜆𝑝′𝐴 + (1−𝜆 ′_)𝑝′′ 𝐴

Hence _≽𝑧 can be represented by a von Neumann-Morgenstern utility function. That

is, _≽𝑧 admits a representation 𝑣𝑧 : Δ𝐴 → ℝthat satisfies: ∀𝑝𝐴, 𝑞𝐴 ∈Δ𝐴, 𝜆 ∈[0,1],

𝑣𝑧(𝜆𝑝𝐴 + (1−𝜆)𝑞𝐴) = 𝜆𝑣𝑧(𝑝𝐴) + (1−𝜆)𝑣𝑧(𝑞𝐴)

In the remainder of this proof, we will therefore refer to any _≽𝑧 as a von Neumann-

Morgenstern preference (vNM preference, for short).

We next define a family of mixture set structures on Δ𝐴/∼𝑧 and [Δ𝐴/∼𝑧]2.

Mixture Sets: Consider any _≽𝑧. For any [𝑝𝐴]𝑧, [𝑞𝐴]𝑧 ∈ Δ𝐴/∼𝑧, and 𝜆 ∈ [0,1], we

define a unique element 𝜆[𝑝𝐴]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑞𝐴]𝑧 ∈ Δ𝐴/∼𝑧 as follows:4

𝜆[𝑝𝐴]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑞𝐴]𝑧 = [𝜆𝑝𝐴 + (1−𝜆)𝑞𝐴]𝑧

Note that ⊕ˆ𝑧 is well defined since ≽𝑧 is a vNM preference, and so if we take any 𝑝′𝐴

∈ [𝑝𝐴]𝑧 and 𝑞′𝐴 ∈ [𝑞𝐴]𝑧, then 𝜆𝑝′𝐴 + (1−𝜆)𝑞

′

𝐴 ∈ [𝜆𝑝𝐴 + (1−𝜆)𝑞𝐴]𝑧.

Further, for any ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧), ([𝑝′𝐴]𝑧, [𝑞′𝐴]𝑧) ∈ [Δ𝐴/∼𝑧]2, and 𝜆 ∈ [0,1], we define

a unique element 𝜆([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝 ′ 𝐴]𝑧, [𝑞′𝐴]𝑧) ∈ [Δ𝐴/∼𝑧]2 as follows:5 𝜆([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝 ′ 𝐴]𝑧, [𝑞′𝐴]𝑧) = (𝜆[𝑝𝐴]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑝 ′ 𝐴]𝑧,𝜆[𝑞𝐴]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑞_𝐴′ ]𝑧) 4_Formally, ˆ ⊕𝑧 : Δ𝐴/∼𝑧 ×Δ𝐴/∼𝑧 ×[0,1]→Δ𝐴/∼𝑧. 5_Formally, ˆ

⊕𝑧 : [Δ𝐴/∼𝑧]2 × [Δ𝐴/∼𝑧]2 × [0,1] → [Δ𝐴/∼𝑧]2. Observe that we are abusing

notation here by using the same notation⊕ˆ𝑧to denote ‘mixture operations’ on the sets Δ𝐴/∼𝑧 and

[Δ𝐴/∼𝑧]2. We do so because this should not cause any confusion, and it allows us to economize on

That is,

𝜆([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝′𝐴]𝑧, [𝑞𝐴′ ]𝑧) = ([𝜆𝑝𝐴 + (1−𝜆)𝑝′𝐴]𝑧, [𝜆𝑞𝐴 + (1−𝜆)𝑞𝐴′ ]𝑧)

Any subset of [Δ𝐴/∼𝑧]2 that is itself a mixture set shall be referred to as a mixture

subset of [Δ𝐴/∼𝑧]2. Note that because≽𝑧 satisfies the vN-M independence condition,

any cube 𝐶 ⊆ [Δ𝐴/∼𝑧]2 is a mixture subset of [Δ𝐴/∼𝑧]2. In addition note the

following result about mixture subsets of [Δ𝐴/∼𝑧]2. (The proof is standard, and

hence omitted).

Lemma 2.1. Every mixture subset of [Δ𝐴/∼𝑧]2, in particular [Δ𝐴/∼𝑧]2 itself, is

connected.

We shall now collect some useful notation to aid the exposition of the subsequent results. We shall denote the restriction of ≽𝑧ˆ to any set Ω in [Δˆ _𝐴/∼_𝑧]2 by (≽𝑧ˆ )_Ωˆ.

Further, let

𝑖𝑛𝑡(Δ𝐴/∼𝑧) = {[𝑝𝐴]𝑧 ∈ Δ𝐴/∼𝑧 : [𝑝𝐴]𝑧 ∕= [𝑞𝐴∗∗]𝑧, [𝑞𝐴∗]𝑧}

𝐷∗ = {([𝑞𝐴]𝑧, [𝑞𝐴]𝑧) ∈ [Δ𝐴/∼𝑧]2 : [𝑞𝐴]𝑧 ∈ Δ𝐴/∼𝑧}

𝐷 ={([𝑞𝐴]𝑧, [𝑞𝐴]𝑧)∈ [Δ𝐴/∼𝑧]2 : [𝑞𝐴]𝑧 ∈ 𝑖𝑛𝑡(Δ𝐴/∼𝑧)}

Ω = Δ𝐴/∼𝑧 × 𝑖𝑛𝑡(Δ𝐴/∼𝑧), and Ω∗ = Ω ∪ 𝐷∗.

Note that if ≽𝑧ˆ does not have any extremal elements then, Δ𝐴/∼𝑧 = 𝑖𝑛𝑡(Δ𝐴/∼𝑧)

and 𝐷∗ = 𝐷. In that case 𝐷∗ ⊆𝑖𝑛𝑡(Δ𝐴/∼𝑧)× 𝑖𝑛𝑡(Δ𝐴/∼𝑧) = Ω and so Ω∗ = Ω.

Lemma 2.2. Let≻𝑧 ∕=∅. For any([𝑝𝐴]𝑧, [𝑞𝐴]𝑧)∈ Ωthere exists a cube𝐶 containing

([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) such that ≽𝑧ˆ restricted to 𝐶 (denoted (≽𝑧ˆ )𝐶), satisfies the following.

2. vNM Continuity: Let([𝑝𝐴]𝑧,[𝑞𝐴]𝑧), ([𝑝′𝐴]𝑧, [𝑞𝐴′ ]𝑧), ([𝑝′′𝐴]𝑧, [𝑞𝐴′′]𝑧)∈𝐶 be such that

([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ≻ˆ𝑧 ([𝑝

′

𝐴]𝑧, [𝑞𝐴′ ]𝑧) ≻ˆ𝑧 ([𝑝

′′

𝐴]𝑧, [𝑞𝐴′′]𝑧). Then there exists 𝜆, 𝜆′ ∈ (0,1)

such that 𝜆([𝑝𝐴]𝑧,[𝑞𝐴]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝 ′′ 𝐴]𝑧,[𝑞′′𝐴]𝑧) ≻ˆ𝑧 ([𝑝 ′ 𝐴]𝑧,[𝑞𝐴′ ]𝑧) ≻ˆ𝑧 𝜆 ′_([𝑝 𝐴]𝑧,[𝑞𝐴]𝑧) ⊕ˆ𝑧 (1− 𝜆′)([𝑝′′_𝐴]𝑧,[𝑞𝐴′′]𝑧).

3. vNM Independence: Let ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧), ([𝑝′𝐴]𝑧, [𝑞′𝐴]𝑧) ∈ 𝐶 be such that ([𝑝𝐴]𝑧,

[𝑞𝐴]𝑧) ≻ˆ𝑧 ([𝑝

′

𝐴]𝑧, [𝑞𝐴′ ]𝑧). Then for any([𝑝′′𝐴]𝑧, [𝑞′′𝐴]𝑧) ∈ 𝐶, 𝜆 ∈ (0,1],

𝜆([𝑝𝐴]𝑧,[𝑞𝐴]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝′′𝐴]𝑧,[𝑞𝐴′′]𝑧) ≻ˆ𝑧 𝜆([𝑝

′

𝐴]𝑧,[𝑞𝐴′ ]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝′′𝐴]𝑧,[𝑞𝐴′′]𝑧).

4. Monotonicity: for any ([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧), ([𝑝′′𝐴]𝑧, [𝑞𝐴′′]𝑧) ∈ 𝐶,

[𝑝′_𝐴]𝑧 ≽𝑧ˆ [𝑝′′_𝐴]𝑧 and [𝑞′𝐴]𝑧 ≽𝑧ˆ [𝑞_𝐴′′]𝑧 ⇒ ([𝑝′𝐴]𝑧, [𝑞𝐴′ ]𝑧) ≽𝑧ˆ ([𝑝′′_𝐴]𝑧, [𝑞𝐴′′]𝑧).

5. Non Degeneracy: ≻ˆ𝑧 ∕= ∅.

Proof. We first consider the case of ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧)∈Ω for which [𝑝𝐴]𝑧 ∕= [𝑞𝐴∗∗]𝑧 or [𝑞∗𝐴]𝑧.

∙ (≽𝑧ˆ )𝐶 is complete and transitive, for an appropriately defined cube 𝐶.

Pick any ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ∈ Ω. There may be two possibilities. First, 𝑝𝐴 ≁𝑧 𝑞𝐴, and

second 𝑝𝐴 ∼𝑧 𝑞𝐴. For the first case assume without loss of generality that𝑝𝐴 ≻𝑧 𝑞𝐴.

We can then find 𝑎,𝑎′ ∈𝐴 such that𝑎_∼𝑧 𝑝𝐴≻𝑧 𝑞𝐴 ≻𝑧 𝑎′. The fact that we may find

𝑎 as specified follows from the fact that any lottery in Δ𝐴 has a certainty equivalent

with respect to_≽𝑧. On the other hand𝑎′ exists as specified because ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧)∈Ω

and so 𝑞𝐴 ∈/ [𝑞∗𝐴]𝑧. Further, since ≽𝑧 is a vNM preference, it follows that there exists

𝜆∗ ∈ (0,1) such that,

Now consider the case where, 𝑝𝐴 ∼𝑧 𝑞𝐴. In this case pick 𝑎, 𝑎′ ∈ [𝑞𝐴]𝑧 (It is possible

that 𝑎 = 𝑎′). Then for any 𝜆 ∈ [0,1], since_≽𝑧 is a vNM preference, we have that

[𝑎, 𝜆;𝑎′,1−𝜆]_∼𝑧 𝑞𝐴

In this case take any 𝜆∗ ∈ (0,1). In either case therefore we can find 𝑎, 𝑎′ ∈ 𝐴, and some 𝜆∗ ∈ (0,1) such that the above preference indifference condition holds. Henceforth, without loss of generality, we shall consider 𝑞𝐴 = [𝑎, 𝜆∗;𝑎′,1−𝜆∗].

We know by the assumption ofcontingent values that there exists_≽𝑧′ ∕=≽𝑧, with ≻𝑧′ ∕=∅, such that for an appropriate choice of 𝑎, 𝑎′, there exists 𝑎, 𝑎 and 𝑎′, 𝑎′ that

satisfy,

𝑎 _∼𝑧′ 𝑎 _∼𝑧′ 𝑎 and 𝑎 ≻_𝑧 𝑎 ≻_𝑧 𝑎,

𝑎′ _∼𝑧′ 𝑎′ _∼𝑧′ 𝑎′ and 𝑎′ ≻_𝑧 𝑎′ ≻_𝑧 𝑎′

In particular, bi-continuity allows us to choose 𝑎, 𝑎 and 𝑎′,𝑎′ in such a way that: 𝑞_𝐴 ≡[𝑎, 𝜆∗; 𝑎′,1−𝜆∗] ≻𝑧 𝑞𝐴 ≻𝑧 [𝑎, 𝜆∗;𝑎′,1−𝜆∗]≡ 𝑞_𝐴.

We can now define the cube 𝐶 ⊆ Ω that the statement of the lemma requires us to do. Define, 𝐶 = ][𝑎]𝑧,[𝑎]𝑧[ × ][𝑞_𝐴]𝑧,[𝑞𝐴]𝑧[ Further, let, 𝐼𝑎 ={ˆ𝑎 ∈ [𝑎]𝑧′ : 𝑎 ≽𝑧 ˆ𝑎 ≽𝑧 𝑎}, & 𝐼𝑎′ = {ˆ𝑎 ′ _{∈ [𝑎}′_] 𝑧′ : 𝑎′ ≽_𝑧 ˆ 𝑎′ _≽𝑧 𝑎′}.

Define a subset 𝑀 of Δ as follows:

𝑀 ={[(𝑧,_ˆ𝑎), 𝜆∗; (𝑧′,_ˆ𝑎′),1−𝜆∗] ∈ Δ :_ˆ𝑎 ∈ 𝐼𝑎,ˆ𝑎

′ _{∈ 𝐼}

Consider any 𝑝′ = [(𝑧,_ˆ𝑎), 𝜆∗; (𝑧′,_ˆ𝑎′),1−𝜆∗] ∈ 𝑀. Since, _ˆ𝑎 ∈ 𝐼𝑎 ⊆ [𝑎]𝑧′,

ˆ𝑎

′ _{∈ 𝐼}

𝑎′ ⊆

[𝑎′]𝑧′ and ≽𝑧′ is a vNM preference, it follows that

𝑝′_𝐴 = [_ˆ𝑎, 𝜆∗;_ˆ𝑎′,1−𝜆∗] _∼𝑧′ [𝑎, 𝜆∗;𝑎′,1−𝜆∗] = 𝑞_𝐴.

Therefore, for any𝑝′, 𝑝′′ ∈ 𝑀,

[𝑎(𝑧′, 𝑝′)]𝑧′ = [𝑎(𝑧′, 𝑝′′)]_𝑧′ = [𝑎′]_𝑧′ and [𝑝′

𝐴]𝑧′ = [𝑝′′

𝐴]𝑧′ = [𝑞_𝐴]_𝑧′.

That is, any 𝑝′, 𝑝′′ ∈ 𝑀 are comparable at 𝑧, and accordingly if 𝑝′ ≻ 𝑝′′, then the procedure-contingent outcome (𝑧, 𝑎(𝑧, 𝑝′), 𝑝′_𝐴) is revealed strictly better than the procedure-contingent outcome (𝑧, 𝑎(𝑧, 𝑝′′), 𝑝′′_𝐴), and if 𝑝′ _∼ 𝑝′′, then (𝑧, 𝑎(𝑧, 𝑝′), 𝑝′_𝐴) is

revealed indifferent to (𝑧, 𝑎(𝑧, 𝑝′′), 𝑝′′_𝐴). Hence,

𝑝′ ≻ 𝑝′′ ⇒ ([𝑎(𝑧, 𝑝′)]𝑧,[𝑝′𝐴]𝑧) ≻ˆ𝑧 ([𝑎(𝑧, 𝑝 ′′_)] 𝑧,[𝑝′′𝐴]𝑧) 𝑝′ _∼ 𝑝′′ ⇒ ([𝑎(𝑧, 𝑝′)]𝑧,[𝑝′𝐴]𝑧) ∼ˆ𝑧 ([𝑎(𝑧, 𝑝 ′′_)] 𝑧,[𝑝′′𝐴]𝑧)

Consider any ([𝑝_ˆ𝐴]𝑧,[𝑞ˆ𝐴]𝑧) ∈ 𝐶. Since,𝑎 ≻𝑧 𝑝ˆ𝐴 ≻𝑧 𝑎, it follows that there exists ˆ𝑎 ∈

𝐼𝑎 such thatˆ𝑎 ∼𝑧 𝑝ˆ𝐴.

6 _{Further, since}

≽𝑧 is a vNM preference, it follows that

[_ˆ𝑎, 𝜆∗;𝑎′,1−𝜆∗] ≻𝑧 𝑞𝐴 ≻𝑧 ˆ𝑞𝐴 ≻𝑧 𝑞_𝐴 ≻𝑧 [ˆ𝑎, 𝜆

∗_;𝑎′_,1−𝜆∗_].

It follows from bi-continuity that there exists _ˆ𝑎′ ∈ 𝐼𝑎′ such that

ˆ

𝑞𝐴 ∼𝑧 [ˆ𝑎, 𝜆

∗_;

ˆ

𝑎′,1−𝜆∗]. That is for any ([𝑝_ˆ𝐴]𝑧, [𝑞ˆ𝐴]𝑧) ∈ 𝐶, there exists

𝑝′ = [(𝑧,_ˆ𝑎), 𝜆∗; (𝑧′,_ˆ𝑎′),1−𝜆∗] ∈ 𝑀 such that 𝑝_ˆ𝐴 ∼𝑧 𝑎(𝑧, 𝑝′) = ˆ𝑎 and ˆ𝑞𝐴 ∼𝑧 𝑝

′ 𝐴 = [ˆ𝑎, 𝜆 ∗_; ˆ 𝑎′,1−𝜆∗]. Accordingly, ≽𝑧ˆ is a weak order on 𝐶.

6_{This follows since [𝑎]}

𝑧′ is a connected subset of𝐴. Note that 𝑊1 ={_˜𝑎∈[𝑎]𝑧′ :

˜

𝑎_≽𝑧𝑝_ˆ𝐴}, &𝑊2 ={_˜𝑎∈[𝑎]𝑧′ :

ˆ

𝑝𝐴 ≽𝑧_˜𝑎}

∙ (≽𝑧ˆ )𝐶 satisfies vNM Continuity.

First we establish that for any 𝑝_˜, _ˆ𝑝∈ 𝑀, any 𝜆 ∈ [0,1], 𝜆𝑝_˜⊕(1−𝜆)𝑝_ˆ∈𝑀. Let

˜

𝑝 = [(𝑧,˜𝑎), 𝜆∗; (𝑧′,𝑎˜′), 1−𝜆∗] & 𝑝_ˆ= [(𝑧,𝑎ˆ),𝜆∗; (𝑧′,ˆ𝑎′), 1−𝜆∗] Further let 𝑎𝜆 ∈ 𝐼𝑎 be such that,

𝑎𝜆 ∼𝑧 [˜𝑎, 𝜆; ˆ𝑎,1−𝜆].

Let,

˜

𝑞𝐴 = 𝜆[˜𝑎, 𝜆∗; ˜𝑎′, 1−𝜆∗] + (1−𝜆)[ˆ𝑎, 𝜆∗; ˆ𝑎′, 1−𝜆∗]

Since _≽𝑧 is a vNM preference, it follows that

[𝑎𝜆, 𝜆∗; 𝑎′, 1−𝜆∗] ≽𝑧 𝑞𝐴 ≽𝑧 ˜𝑞𝐴 ≽𝑧 𝑞_𝐴 ≽𝑧 [𝑎𝜆, 𝜆

∗_{; 𝑎}′_{, 1−𝜆}∗_],

with strict preference holding at least somewhere. Bi-ontinuity in conjunction with the fact the [𝑎′]𝑧′ is a connected subset of 𝐴 implies that there exists, 𝑎′

𝜆 ∈ 𝐼𝑎′, such that [𝑎𝜆, 𝜆∗; 𝑎′𝜆, 1−𝜆 ∗_] ∼𝑧 𝑞˜𝐴. Hence, [(𝑧, 𝑎𝜆), 𝜆∗; (𝑧′, 𝑎′𝜆), 1−𝜆 ∗_{] = 𝜆} ˜ 𝑝 ⊕ (1−𝜆)𝑝_ˆ.

We now establish that (≽𝑧ˆ )𝐶 satisfies the vN-M Continuity axiom. Note that this is

equivalent to proving the following: For any 𝑝, 𝑝′, 𝑝′′ ∈ 𝑀 such that 𝑝 ≻ 𝑝′ ≻ 𝑝′′, there exists 𝜆,𝜆′ ∈(0,1), such that:

𝜆𝑝 ⊕ (1−𝜆)𝑝′′ ≻𝑝′ ≻ 𝜆′𝑝 ⊕ (1−𝜆′)𝑝′′

Suppose otherwise – say that𝑝′ _≽𝜆𝑝 ⊕(1−𝜆)𝑝′′ for all𝜆 ∈(0,1). We proved above that for all 𝜆 ∈ [0,1] there exists 𝑎𝜆 ∈ 𝐼𝑎, 𝑎′𝜆 ∈ 𝐼𝑎′ such that,

[(𝑧, 𝑎𝜆), 𝜆∗; (𝑧′, 𝑎′𝜆); 1−𝜆

∗_{] = 𝜆𝑝 ⊕ (1−𝜆)𝑝}′′_.

Denote,

𝑝 = [(𝑧,˜𝑎), 𝜆∗; (𝑧′,˜𝑎′), 1−𝜆∗] We may then construct a sequence (𝑎𝜆𝑘, 𝑎

′

𝜆𝑘)𝑘∈ℤ+ ⊆ 𝐼𝑎 × 𝐼𝑎′ converging to (˜𝑎, ˜𝑎

′_{) ∈} 𝐼𝑎 × 𝐼𝑎′, such that for all 𝑘 ∈ _ℤ+,

𝑝′ _≽ 𝜆𝑘𝑝⊕ (1−𝜆𝑘)𝑝′′ = [(𝑧, 𝑎𝜆𝑘),𝜆 ∗_{; (𝑧}′_{, 𝑎}′ 𝜆𝑘); 1−𝜆 ∗_] Let Ξ = {(𝑎𝜆𝑘, 𝑎 ′ 𝜆𝑘) ∈ 𝐼𝑎 × 𝐼𝑎′ :𝑝 ′ ≽ [(𝑧, 𝑎𝜆𝑘),𝜆 ∗_{; (𝑧}′_{, 𝑎}′ 𝜆𝑘); 1−𝜆 ∗_]}

ByBi-continuity the set Ξ is closed in𝐼𝑎 ×𝐼𝑎′. It then follows that (˜𝑎, ˜𝑎′)∈ Ξ, that

is 𝑝′ _≽𝑝 = [(𝑧,˜𝑎), 𝜆∗; (𝑧′,˜𝑎′), 1−𝜆∗], which is absurd.

∙ (≽𝑧ˆ )𝐶 satisfies vNM Independence.

Now, let ([𝑝1

𝐴]𝑧, [𝑞𝐴1]𝑧), ([𝑝2𝐴]𝑧, [𝑞2𝐴]𝑧), ([𝑝3𝐴]𝑧, [𝑞𝐴3]𝑧) ∈ 𝐶 be such that ([𝑝𝐴1]𝑧, [𝑞1𝐴]𝑧) ≻ˆ𝑧

([𝑝2

𝐴]𝑧, [𝑞𝐴2]𝑧). We need to establish that for any𝜆 ∈ (0,1],

𝜆([𝑝1

𝐴]𝑧,[𝑞𝐴1]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝3𝐴]𝑧,[𝑞𝐴3]𝑧) ≻ˆ𝑧 𝜆([𝑝

2

𝐴]𝑧,[𝑞2𝐴]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝3𝐴]𝑧,[𝑞𝐴3]𝑧).

We know from the analysis above that there exists𝑝,𝑝′, 𝑝′′ ∈𝑀 such that ([𝑎(𝑧, 𝑝)]𝑧,

[𝑝𝐴]𝑧) = ([𝑝1𝐴]𝑧, [𝑞1𝐴]𝑧), ([𝑎(𝑧, 𝑝′)]𝑧, [𝑝′𝐴]𝑧) = ([𝑝2𝐴]𝑧, [𝑞2𝐴]𝑧), ([𝑎(𝑧, 𝑝′′)]𝑧 ,[𝑝′′𝐴]𝑧) = ([𝑝3𝐴]𝑧,

[𝑞3_𝐴]𝑧), and 𝑝≻ 𝑝′. By comparable independence, it follows that for any 𝜆 ∈ (0,1]

𝜆𝑝⊕(1−𝜆)𝑝′′ ≻𝜆𝑝′⊕(1−𝜆)𝑝′′

Accordingly, it follows that

([𝑎(𝑧, 𝜆𝑝 ⊕(1−𝜆)𝑝′′)]𝑧,[(𝜆𝑝 ⊕ (1−𝜆)𝑝′′)𝐴]𝑧) ≻ˆ𝑧

Or, ([𝜆𝑎(𝑧, 𝑝) + (1−𝜆)𝑎(𝑧, 𝑝′′)]𝑧,[𝜆𝑝𝐴 + (1−𝜆)𝑝′′𝐴]𝑧)≻ˆ𝑧 ([𝜆𝑎(𝑧, 𝑝′) + (1−𝜆)𝑎(𝑧, 𝑝′′)]𝑧, [𝜆𝑝′𝐴 + (1−𝜆)𝑝 ′′ 𝐴]𝑧). That is, (𝜆[𝑎(𝑧, 𝑝)]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑎(𝑧, 𝑝′′)]𝑧,𝜆[𝑝𝐴]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑝′′𝐴]𝑧) ≻ˆ𝑧 (𝜆[𝑎(𝑧, 𝑝′)]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑎(𝑧, 𝑝′′)]𝑧, 𝜆[𝑝′𝐴]𝑧 ⊕ˆ𝑧 [𝑝′′𝐴]𝑧). Or, (𝜆[𝑝1 𝐴]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑝3𝐴]𝑧, 𝜆[𝑞1𝐴]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑞𝐴3]𝑧) ≻ˆ𝑧 (𝜆[𝑝2_𝐴]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑝3𝐴]𝑧, 𝜆[𝑞𝐴2]𝑧 ⊕ˆ𝑧 (1−𝜆)[𝑞3𝐴]𝑧). Or, 𝜆([𝑝1_𝐴]𝑧,[𝑞𝐴1]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝3𝐴]𝑧,[𝑞𝐴3]𝑧) ≻ˆ𝑧 𝜆([𝑝 2 𝐴]𝑧,[𝑞2𝐴]𝑧) ⊕ˆ𝑧 (1−𝜆)([𝑝3𝐴]𝑧,[𝑞𝐴3]𝑧).

Hence, (≽𝑧ˆ )𝐶 satisfies the vN-M Independence axiom.

∙ (≽𝑧ˆ )𝐶 satisfies Monotonicity.

This follows immediately from conditional monotonicity

∙ (≽𝑧ˆ )𝐶 is Non Degenerate.

This follows immediately from the assumption made in the lemma that≻𝑧 ∕= ∅.

The proof for the case when [𝑝𝐴]𝑧 is equal to either [𝑞∗∗𝐴]𝑧, [𝑞𝐴∗]𝑧 is exactly along

similar lines. When [𝑝𝐴]𝑧 = [𝑞∗∗𝐴]𝑧, take𝑎=𝑎 in the above proof, and define the cube

𝐶 as follows:

The rest of the details are exactly identical. Similarly, when [𝑝𝐴]𝑧 = [𝑞𝐴∗]𝑧, take 𝑎 =

𝑎 in the above proof, and define

𝐶 = [[𝑎]𝑧, [𝑎]𝑧[ × ][𝑞_𝐴]𝑧, [𝑞𝐴]𝑧[ .

Lemma 2.3. (≽𝑧ˆ )Ω∗ is a weak order. Further, there exists

(i) a function 𝑣𝑧 : Δ𝐴 → ℝ that represents ≽𝑧 and satisfies: for all 𝜆 ∈ [0,1], 𝑝𝐴, 𝑞𝐴

∈ Δ𝐴,

𝑣𝑧(𝜆𝑝𝐴 + (1− 𝜆)𝑞𝐴) = 𝜆𝑣𝑧(𝑝𝐴) + (1− 𝜆)𝑣𝑧(𝑞𝐴), and

(ii) a constant 𝜎𝑧 ∈ [0, 1],

such that the function 𝑉𝑧 : Ω∗ → ℝ given by

𝑉𝑧([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) = (1−𝜎𝑧)𝑣𝑧(𝑝𝐴) + 𝜎𝑧𝑣𝑧(𝑞𝐴)

represents(≽𝑧ˆ )Ω∗. Further, another pair(

˜

𝑣𝑧,𝜎˜𝑧)represents(≽𝑧ˆ )Ω∗ in the above sense

iff _˜𝑣𝑧 is a positive affine transformation of 𝑣𝑧 and 𝜎˜𝑧 = 𝜎𝑧, for all 𝑧 ∈ 𝑍 such that

≻𝑧 ∕= ∅.

Proof. First consider those𝑧 ∈ 𝑍 for which≻𝑧 ∕=∅. From Lemma 2.2 it follows that

for any ([𝑞𝐴]𝑧, [𝑞𝐴]𝑧) ∈ 𝐷, there exists a cube containing ([𝑞𝐴]𝑧, [𝑞𝐴]𝑧), which we can

take to be

𝐶[𝑞𝐴]𝑧 = ][𝑞_𝐴]𝑧, [𝑞𝐴]𝑧[ × ][𝑞_𝐴]𝑧, [𝑞𝐴]𝑧[ ⊆ [Δ𝐴/∼𝑧]2

such that ≽𝑧ˆ restricted to 𝐶[𝑞𝐴]𝑧 satisfies the five axioms of the Anscombe Aumann

Theorem (for finite states) –weak order,vN-M continuity,vN-M independence,mono- tonicity andnon-degeneracy. It follows that there exists a function𝑣𝑞𝐴

𝑧 : ][𝑞_𝐴]𝑧, [𝑞𝐴]𝑧[

→ _ℝ that is unique up to positive affine transformation, and a constant 𝜎𝑞𝐴

𝑧 ∈ [0, 1]

that is unique, such that the function 𝑉𝑞𝐴

𝑉𝑞𝐴 𝑧 ([𝑝 ′ 𝐴]𝑧, [𝑞𝐴′ ]𝑧) = (1−𝜎𝑧𝑞𝐴)𝑣𝑧𝑞𝐴(𝑝 ′ 𝐴) + 𝜎𝑧𝑞𝐴𝑣𝑧𝑞𝐴(𝑞 ′ 𝐴)

represents (≽𝑧ˆ )𝐶[_𝑞𝐴]𝑧. Thats is for all ([𝑝

′ 𝐴]𝑧, [𝑞𝐴′ ]𝑧), ([𝑝′′𝐴]𝑧, [𝑞𝐴′′]𝑧)∈ 𝐶[𝑞𝐴]𝑧, ([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧)≽𝑧ˆ ([𝑝′′_𝐴]𝑧, [𝑞′′𝐴]𝑧) if and only if𝑉𝑧𝑞𝐴([𝑝 ′ 𝐴]𝑧, [𝑞𝐴′ ]𝑧) ≥ 𝑉𝑧𝑞𝐴([𝑝 ′′ 𝐴]𝑧, [𝑞′′𝐴]𝑧)

Further note that the function𝑣𝑧 satisfies: for all𝜆∈[0,1], [𝑝𝐴]𝑧, [𝑝′𝐴]𝑧 ∈][𝑞𝐴]𝑧, [𝑞𝐴]𝑧[,

𝑣𝑧(𝜆[𝑝𝐴]𝑧 ⊕ˆ𝑧 (1− 𝜆)[𝑝′𝐴]𝑧) = 𝜆𝑣𝑧([𝑝𝐴]𝑧) + (1 −𝜆)𝑣𝑧([𝑝′𝐴]𝑧).

In addition, for any ([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧) ∈ 𝐶[𝑞𝐴]𝑧, there exists [𝑞ˆ𝐴]𝑧 ∈ ][𝑞_𝐴]𝑧, [𝑞𝐴]𝑧[ such that

([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧)∼ˆ𝑧 ([𝑞ˆ𝐴]𝑧, [ˆ𝑞𝐴]𝑧).

Note that ≽𝑧ˆ restricted to 𝐷∗ is complete. This follows since, any two degenerate

lotteries like [(𝑧, 𝑎),1] and [(𝑧, 𝑎′),1] are comparable at 𝑧, and accordingly ([𝑎]𝑧, [𝑎]𝑧) ≻ˆ𝑧 ([𝑎 ′_] 𝑧, [𝑎′]𝑧) if (𝑧, 𝑎) ≻𝑧 (𝑧, 𝑎′), or, ([𝑎]𝑧, [𝑎]𝑧)∼ˆ𝑧 ([𝑎 ′_] 𝑧, [𝑎′]𝑧) if (𝑧, 𝑎) ∼𝑧 (𝑧, 𝑎′). 7 Now define𝑂 = (∪[𝑞𝐴]𝑧∈𝐷 𝐶[𝑞𝐴]𝑧)∪ 𝐷 ∗

. We will next show that ≽𝑧ˆ restricted to 𝑂 is

a weak order. Pick any ([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧) ∈ 𝐶[𝑞𝐴]𝑧, ([𝑝

′′

𝐴]𝑧, [𝑞𝐴′′]𝑧) ∈ 𝐶[𝑝𝐴]𝑧. We know that

there exists [𝑞_ˆ𝐴]𝑧, [𝑝ˆ𝐴]𝑧 ∈ Δ𝐴/∼𝑧 such that

([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧)∼ˆ𝑧 ([𝑞ˆ𝐴]𝑧, [ˆ𝑞𝐴]𝑧) and ([𝑝

′′

𝐴]𝑧, [𝑞′′𝐴]𝑧) ∼ˆ𝑧 ([𝑝ˆ𝐴]𝑧, [𝑝ˆ𝐴]𝑧).

Accordingly, it follows that ([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧)≻ˆ𝑧 ([𝑝 ′′ 𝐴]𝑧, [𝑞𝐴′′]𝑧) if ([𝑞ˆ𝐴]𝑧, [𝑞ˆ𝐴]𝑧) ≻ˆ𝑧 ([𝑝ˆ𝐴]𝑧, [𝑝ˆ𝐴]𝑧), or, ([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧)∼ˆ𝑧 ([𝑝 ′′ 𝐴]𝑧, [𝑞𝐴′′]𝑧) if ([𝑞ˆ𝐴]𝑧, [𝑞ˆ𝐴]𝑧) ∼ˆ𝑧 ([𝑝ˆ𝐴]𝑧, [𝑝ˆ𝐴]𝑧). 7_{Note that{[𝑎]} 𝑧 ∈Δ𝐴/∼𝑧 :𝑎∈𝐴} = Δ𝐴/∼𝑧.

Hence, (≽𝑧ˆ )𝑂 is a weak order.

Now consider any two cubes 𝐶[𝑞𝐴]𝑧 and 𝐶[𝑝𝐴]𝑧 that intersect. Pick ([𝑞

′

𝐴]𝑧, [𝑞𝐴′ ]𝑧),

([𝑞_𝐴′′]𝑧, [𝑞𝐴′′]𝑧)∈𝐶[𝑞𝐴]𝑧 ∩𝐶[𝑝𝐴]𝑧, [𝑞

′

𝐴]𝑧∕= [𝑞′′𝐴]𝑧, and recalibrate the function𝑣𝑧𝑝𝐴 by setting

𝑣𝑝𝐴

𝑧 ([𝑞𝐴′ ]𝑧) = 𝑣𝑧𝑞𝐴([𝑞𝐴′ ]𝑧) and 𝑣𝑧𝑝𝐴([𝑞′′𝐴]𝑧) = 𝑣𝑧𝑞𝐴([𝑞𝐴′′]𝑧)

Note that by the uniqueness result of the Anscombe Aumann Theorem, the pair (𝑣𝑝𝐴

𝑧 , 𝜎𝑧𝑝𝐴) continues to represent (≽𝑧ˆ )𝐶[_𝑝𝐴]. Further, 𝑣

𝑝𝐴

𝑧 = 𝑣𝑞𝑧𝐴 on ][𝑝_𝐴]𝑧, [𝑝𝐴]𝑧[ ∩

][𝑞

𝐴]𝑧, [𝑞𝐴]𝑧[. Hence it follows that 𝜎 𝑝𝐴

𝑧 = 𝜎𝑧𝑞𝐴. Next consider [𝑞𝐴]𝑧, [𝑝𝐴]𝑧 such that

cubes 𝐶[𝑞𝐴]𝑧 and 𝐶[𝑝𝐴]𝑧 do not intersect. Since the set 𝐷 is connected, ([𝑞𝐴]𝑧, [𝑞𝐴]𝑧)

and ([𝑝𝐴]𝑧, [𝑝𝐴]𝑧) can be linked by finitely many cubes; that is there are finitely

many cubes 𝐶_[𝑝1 𝐴]𝑧, . . . , 𝐶[𝑝 𝑚 𝐴]𝑧, such that 𝐶[𝑝1𝐴]𝑧 = 𝐶[𝑞𝐴]𝑧, 𝐶[𝑝 𝑚 𝐴]𝑧 = 𝐶[𝑝𝐴]𝑧, and

each subsequent pairs of 𝐶_[𝑝𝑗

𝐴]𝑧’s intersect. Further, we can take 𝐶[𝑝 𝑗

𝐴]𝑧 ∩ 𝐶[𝑝 𝑗−𝑘 𝐴 ]𝑧 =

∅ for every 𝑘 ≥ 2. We can then repeat the above re-calibration exercise over pairs of intersecting cubes in the link. This exercise allows us to define a function 𝑣𝑧 on

𝑖𝑛𝑡(Δ𝐴/ ∼𝑧), as well as establish 𝜎𝑧𝑞𝐴 = 𝜎𝑧𝑝𝐴 = 𝜎𝑧, for all 𝑞𝐴 ∕= 𝑝𝐴, [𝑞𝐴]𝑧, [𝑝𝐴]𝑧 ∈

𝑖𝑛𝑡(Δ𝐴/∼𝑧). Finally, for [𝑝𝐴]𝑧 = [𝑞𝐴∗∗]𝑧, or [𝑞𝐴∗]𝑧 define

𝑣𝑧([𝑝𝐴]𝑧) = lim

𝜆→1𝑣𝑧(𝜆[𝑝𝐴]𝑧⊕ˆ𝑧(1−𝜆)[𝑝𝐴]𝑧), where [𝑝𝐴]𝑧 is any element of𝑖𝑛𝑡(Δ𝐴/∼𝑧).

We next establish the following claim: for any ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ∈ Ω∗ there exists

([𝑝′_𝐴]𝑧, [𝑝′𝐴]𝑧) in 𝐷∗ such that ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ∼ˆ𝑧 ([𝑝

′

𝐴]𝑧, [𝑝′𝐴]𝑧). To that end, define the

function 𝑉𝑧 : Ω∗ → ℝ by

where 𝑣𝑧 and 𝜎𝑧 are as defined above. For any [ˆ𝑞𝐴]𝑧 ∈𝑖𝑛𝑡(Δ𝐴/∼𝑧), let

𝐽 ˆ

𝑞𝐴 ={([𝑝𝐴]𝑧,[𝑞𝐴]𝑧)∈Ω :𝑉𝑧([𝑝𝐴]𝑧,[𝑞𝐴]𝑧) = 𝑉𝑧([ˆ𝑞𝐴]𝑧,[ˆ𝑞𝐴]𝑧)}

We claim that for all ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧), ([𝑝′𝐴]𝑧, [𝑞𝐴′ ]𝑧)∈𝐽ˆ𝑞𝐴, ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧)∼ˆ𝑧 ([𝑝 ′

𝐴]𝑧, [𝑞𝐴′ ]𝑧).

To see this note that, Lemma 2.2 guarantees that for any ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ∈ 𝐽𝑞ˆ𝐴, there

exists a cube𝐶 containing ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) such that (≽𝑧ˆ )𝐶 satisfies the three vN-M ax-

ioms of Weak Order, vNM Continuity and Independence on the mixture set (𝐶,⊕ˆ𝑧).

Accordingly (≽𝑧ˆ )𝐶 can be represented by a von Neumann-Morgenstern utility func-

tion. Consider two such cubes 𝐶1 and 𝐶2 that intersect. Because of the axiom of

revealed consistency, it follows that for any ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧), ([𝑝′𝐴]𝑧, [𝑞𝐴′ ]𝑧) ∈ 𝐶1 ∩ 𝐶2,

([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) (≽𝑧ˆ )𝐶1 ([𝑝

′

𝐴]𝑧, [𝑞𝐴′ ]𝑧) iff ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) (≽𝑧ˆ )𝐶2 ([𝑝

′

𝐴]𝑧, [𝑞𝐴′ ]𝑧).

Further note that if𝑉𝐶1 and 𝑉𝐶2 are two vN-M utility functions that represent (≽𝑧ˆ )𝐶1

and (≽𝑧ˆ )𝐶2 respectively, these functions can be re-calibrated (in a manner similar to

that used in Step 2) and set equal on𝐶1 ∩ 𝐶2.

Now, consider the cube 𝐶[𝑞ˆ𝐴]𝑧 around ([𝑞ˆ𝐴]𝑧, [𝑞ˆ𝐴]𝑧). We have already established

above that (≽𝑧ˆ )𝐶[ˆ𝑞𝐴]𝑧

is represented by the function 𝑉𝑧. Further, 𝐽𝑞ˆ𝐴 is connected.

Accordingly, ([_ˆ𝑞𝐴]𝑧, [𝑞ˆ𝐴]𝑧) can be linked to any ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ∈ 𝐽𝑞ˆ𝐴 using a finite

number of cubes. On each pair of intersecting cubes ≽𝑧ˆ must coincide as suggested

in the last paragraph. Furthermore the vN-M representations of ≽𝑧ˆ on these cubes

can be re-calibrated and brought in line with 𝑉𝑧. Hence, we may conclude that for

all ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧), ([𝑝′𝐴]𝑧, [𝑞′𝐴]𝑧) ∈𝐽𝑞ˆ𝐴, ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ∼ˆ𝑧 ([𝑝 ′

𝐴]𝑧, [𝑞′𝐴]𝑧).

Note that if𝜎𝑧 ∕= 1, or if [𝑞𝐴∗∗]𝑧 and [𝑞𝐴∗]𝑧 do not exist, then we are done establishing

our claim. However, if 𝜎𝑧 = 1, and either [𝑞𝐴∗∗]𝑧 or [𝑞∗𝐴]𝑧 exists then members of the

𝐵 ={([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ∈Ω : [𝑝𝐴]𝑧 = [𝑞∗∗𝐴]𝑧 or [𝑞∗𝐴]𝑧}

are not indifferent to any element of 𝐷. In this case it is straightforward to verify that for any ([𝑞_𝐴∗∗]𝑧, [𝑞𝐴]𝑧) ∈ 𝐵, ([𝑞𝐴∗∗]𝑧, [𝑞𝐴]𝑧) ∼ˆ𝑧 ([𝑞

∗∗

𝐴]𝑧, [𝑞𝐴∗∗]𝑧). Similarly, for any

([𝑞_𝐴∗]𝑧, [𝑞𝐴]𝑧) ∈ 𝐵, ([𝑞𝐴∗]𝑧, [𝑞𝐴]𝑧) ∼ˆ𝑧 ([𝑞

∗

𝐴]𝑧, [𝑞𝐴∗]𝑧).

Now consider any ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧), ([𝑝𝐴′ ]𝑧, [𝑞𝐴′ ]𝑧)∈Ω∗. From the argument just made,

we know that there exists ([𝑞_ˆ𝐴]𝑧, [𝑞ˆ𝐴]𝑧), ([˜𝑞𝐴]𝑧, [˜𝑞𝐴]𝑧) ∈ 𝐷

∗_{, such that ([𝑝} 𝐴]𝑧, [𝑞𝐴]𝑧) ˆ ∼𝑧 ([ˆ𝑞𝐴]𝑧, [𝑞ˆ𝐴]𝑧) and ([𝑝 ′ 𝐴]𝑧, [𝑞𝐴′ ]𝑧)∼ˆ𝑧 ([𝑞˜𝐴]𝑧, [𝑞˜𝐴]𝑧). Hence, ([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ≽𝑧ˆ ([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧) iff ([ˆ𝑞𝐴]𝑧, [𝑞ˆ𝐴]𝑧) ≽𝑧ˆ ([𝑞˜𝐴]𝑧, [˜𝑞𝐴]𝑧).

Clearly it also follows that,

([𝑝𝐴]𝑧, [𝑞𝐴]𝑧) ≽𝑧ˆ ([𝑝′_𝐴]𝑧, [𝑞𝐴′ ]𝑧) ⇔ 𝑉𝑧(([𝑝𝐴]𝑧,[𝑞𝐴]𝑧))≥ 𝑉𝑧(([𝑝′𝐴]𝑧,[𝑞′𝐴]𝑧)).

Note that we may (with an abuse of notation) define the function𝑣𝑧 on Δ𝐴by simply

giving all elements of an equivalence class, say [𝑝𝐴]𝑧, the value 𝑣𝑧([𝑝𝐴]𝑧). It then

follows that for all 𝜆 ∈ [0,1], 𝑝𝐴, 𝑞𝐴 ∈ Δ𝐴,

𝑣𝑧(𝜆𝑝𝐴 + (1− 𝜆)𝑞𝐴) = 𝜆𝑣𝑧(𝑝𝐴) + (1 − 𝜆)𝑣𝑧(𝑞𝐴).

The uniqueness statement is simply a re-statement of the essential uniqueness result in the first half of the proof. This then completes the proof for those𝑧 ∈ 𝑍 for which

≻𝑧 ∕=∅.

The proof for those 𝑧 ∈ 𝑍 for which ≻𝑧 = ∅ is trivial. Note that for this case

[Δ𝐴/∼𝑧 × Δ𝐴/∼𝑧] is a singleton. We can take 𝑣𝑧 to be any constant function, and

𝜎𝑧 to be any number in [0, 1].

Remark 2.4. Note that the function 𝑣𝑧 : Δ𝐴 → ℝ in a von-Neumann Morgenstern

utility representation of the preference relation _≽𝑧. We know that if there is some

and 𝑣ˆ𝑧 must be positive affine transformations of one another. This is a fact that we

shall draw on below.

We next establish that any procedure-contingent outcome (𝑧, 𝑎, 𝑝𝐴) is revealed

indifferent to a procedure contingent outcome (𝑧,_˜𝑎,_˜𝑎) in which the outcome and pro- cedure are the same. The result follows immediately when we combine the conclusion of the last lemma with the following one.

Lemma 2.4. Suppose_≽𝑧 satisfies stochastic dominance and vN-M continuity. If ≻𝑧

∕

= ∅, then there does not exist 𝑝 ∈ Δsuch that (a) 𝑝𝐴 ≻𝑧 𝑝𝐴,𝑧 and 𝑝𝐴 is the maximal

element of _≽𝑧, or (b) 𝑝𝐴,𝑧 ≻𝑧 𝑝𝐴 and 𝑝𝐴 is the minimal element of ≽𝑧.

Proof. Consider any𝑞𝐴 ∈ Δ𝐴. Note the following mutually exclusive possibilities:

[A] 𝑎 _∼𝑧 𝑎′ for all 𝑎, 𝑎′ in the support of𝑞𝐴: In this case it must be that𝑞𝐴 ∼𝑧 𝑎 for

any 𝑎 in the support of𝑞𝐴. To see this assume otherwise – say 𝑞𝐴 ≻𝑧 𝑎. Given that

≻𝑧 ∕=∅, it follows that there exists𝑎′′ satisfying 𝑎′′ ≻𝑧 𝑎 or𝑎 ≻𝑧 𝑎′′. Assume it is the

former.8 _{Note that 𝑎}′′ _{first order stochastically dominates 𝑞}

𝐴; hence we have: 𝑎′′ ≻

𝑞𝐴 ≻𝑧 𝑎. But then by vN-M Continuity, there exists 𝜆 ∈ (0,1), such that 𝑞𝐴 ≻𝑧 𝜆𝑎′′

+ (1−𝜆)𝑎′. But the lottery𝜆𝑎′′ + (1−𝜆)𝑎′ first order stochastically dominates 𝑞𝐴!

[B] 𝑎 ≻𝑧 𝑎′ for some 𝑎, 𝑎′ in the support of 𝑞𝐴: In this case there exists 𝑎, 𝑎 in

the support of 𝑞𝐴 such that 𝑎 ≻𝑧 𝑞𝐴 ≻𝑧 𝑎. This claim is again easily established by

appealing to the fact that _≽𝑧 satisfies stochastic dominance; so we omit the details here.

Now we proceed to prove the Lemma. Suppose there exists 𝑝 ∈ Δ such that 𝑝𝐴 ≻𝑧

𝑝𝐴,𝑧 and 𝑝𝐴 is the maximal element of ≽𝑧. Then we know from above that there

exists 𝑎 in the support of 𝑝𝐴,𝑧 such that 𝑝𝐴,𝑧 ≽𝑧 𝑎. Hence, 𝑝𝐴 ≻𝑧 𝑎. Clearly, 𝑎 is

In document Other-Regarding Preferences and Consequentialism (Page 40-60)