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A note on equivalent comparisons of information channels

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Abstract

Nakata (Theory Decis 71:559–574, 2011) presents a model of acquisition of information where the agent does not know what pieces of information she is missing. In this note, we point out some technical problems in a few of Nakata’s results and show how to correct them.

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Notes

  1. 1.

    Nakata (2011) follows DLR and defines \(\mathcal{B }(\Delta (C))\) to be the class of all subsets of \( \Delta (C)\). We restrict ourselves to closed subsets of \(\Delta (C)\) to simplify the exposition.

  2. 2.

    That is, there exists a non-constant function \(u:C\rightarrow \mathbb{R }\) such that, for every \(p\in \Delta (C), U(p,m_{j})=\sum _{x\in C}p(x)u(x)\).

  3. 3.

    That is, if, for every \(i,j\in \mathcal{J }\) and \(A,B\in \mathcal{B }\left( \Delta \left( C\right) \right) \), we have \((i,A)\succsim (j,B)\) iff \( v(i,A)\ge v(j,B)\).

  4. 4.

    That is, every \(\varphi _{j}\) is strictly increasing on \(\{\left\langle \max _{p\in A}U(p,m_{j})\right\rangle _{m_{j}\in \mathcal{M }_{j}}:A\in \mathcal{B }\left( \Delta \left( C\right) \right) \}\).

  5. 5.

    Notation. For any lottery \(p\in \Delta (C), \mathbb E _{p}(u):=p(x)u(x)+p(y)u(y)\).

  6. 6.

    Notation. By \(\lambda x\oplus (1-\lambda )y\) we mean the lottery that pays prize \(x\) with probability \(\lambda \) and prize \(y\) with probability \(1-\lambda \).

  7. 7.

    Since we did not include the requirement that all the information nodes in the representation be relevant, we cannot exclude the possibility that \( (j,A)\sim (j,B)\) for all \(A\) and \(B\) in \(\mathcal{B }(\Delta (C))\), for some \( j\in \mathcal{J }\). This is the reason why Axiom 2 is not part of the postulates in Theorem 1.

  8. 8.

    For any pair of menus \(A\) and \(B\), and any \(\lambda \in [0,1]\), the convex combination of \(A\) and \(B\) is defined by \(\lambda A+(1-\lambda )B:=\{\lambda p+(1-\lambda )q:p\in A\) and \(q\in B\}\).

  9. 9.

    Now, the requirement that all the aggregators \(\{\varphi _{j}\}_{j\in \mathcal{J }}\) are strictly increasing implies that Axiom 2 is satisfied. This is the reason why this postulate is present in the statement of Theorem 2.

  10. 10.

    Here we are using the standard abuse of notation of writing \(x\) and \(y\) to represent the degenerate lotteries that assign probability one to the alternative \(x\) and \(y\), respectively.

  11. 11.

    For example, pick a representation \(((\mathcal{M }_{n+1},\Sigma _{n+1}),\mu _{n+1},\tilde{U})\) of the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \) where \(\mu _{n+1}\) is a probability measure and define \(\ U^{\prime }:=\tilde{U}-\int _{\mathcal{M } _{n+1}}\max _{p\in \underline{A}_{n+1}}\tilde{U}(p,m_{n+1})\mu _{n+1}(dm_{n+1})+V(j,\Delta (C))\) and replace \(\tilde{U}\) by \(U^{\prime }\) in the representation.

  12. 12.

    For example, pick a representation \(((\mathcal{M }_{n+1},\Sigma _{n+1}),\mu _{n+1},\tilde{U})\) of the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \) where \(\mu _{n+1}\) is a probability measure and define \(U^{\prime }:=\tilde{U}-\int _{\mathcal{M } _{n+1}}\max _{p\in \underline{A}_{n+1}}\tilde{U}(p,m_{n+1})\mu _{n+1}(dm_{n+1})+\max _{j\in \mathcal J\setminus \{ n+1\}}V(j,\Delta (C))+1\) and replace \(\tilde{U}\) by \(U^{\prime }\) in the representation.

  13. 13.

    To show the uniqueness part of this claim,suppose that \(\hat{W}\) is an affine extension of \(W\) and fix \(A\in \mathcal{B }\left( \Delta \left( C\right) \right) \). Pick \(\lambda \in (0,1)\) large enough so that \(\lambda \underline{A}_{n+1}+(1-\lambda )A\in \mathcal{B }^{*}\). By construction, \( \lambda \hat{W}(\underline{A}_{n+1})+(1-\lambda )\hat{W}(A)=\hat{W}(\lambda \underline{A}_{n+1}+(1-\lambda )A)=W(\lambda \underline{A}_{n+1}+(1-\lambda )A)\). Since \(\hat{W}(\underline{A}_{n+1})=W(\underline{A}_{n+1})\), this implies that \(\hat{W}(A)=\frac{W(\lambda \underline{A}_{n+1}+(1-\lambda )A)-\lambda W(\underline{A}_{n+1})}{1-\lambda }\), which shows that \(\hat{W}\) is unique.

  14. 14.

    By \(\left\| q-p\right\| \) we mean \(\sqrt{\sum _{x\in C}(q(x)-p(x))^{2}}\).

References

  1. Dekel, E., Lipman, B. L., & Rustichini, A. (2001). Representing preferences with a unique subjective state space. Econometrica, 69(4), 891–934.

  2. Dekel, E., Lipman, B. L., & Rustichini, A. (2009). Temptation-driven preferences. Review of Economic Studies, 76(3), 937–971.

  3. Dekel, E., Lipman, B. L., Rustichini, A., & Sarver, T. (2007). Representing preferences with a unique subjective state space: A corrigendum. Econometrica, 75(2), 591–600.

  4. Kopylov, I. (2009). Finite additive utility representations for preferences over menus. Journal of Economic Theory, 144, 354–374.

  5. Nakata, H. (2011). Equivalent comparisons of information channels. Theory and Decision, 71, 559–574.

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Acknowledgments

We thank José Heleno Faro, Rogerio Mazali, Hiroyuki Nakata, Leandro Nascimento and José Guilherme de Lara Resende for helpful discussions and suggestions.

Author information

Correspondence to Gil Riella.

Appendix: Proofs

Appendix: Proofs

Proof of Theorem 1

The proof that the axioms are necessary for the representation is left to the reader. We will show only that the axioms are sufficient for \(\succsim \) to have a Weak EU representation. By Axiom 4, \(\succsim \) is a continuous preference relation on the compact (therefore separable) metric space \(\mathcal J\times B (\Delta (C))\). This allows us to invoke Debreu’s utility representation theorem to find a continuous function \(v: \mathcal{J }\times \mathcal{B }\left( \Delta \left( C\right) \right) \longrightarrow \mathbb{R } \) such that, for every pair \(\left( i,A\right) ,(j,B)\in \mathcal{J }\times \mathcal{B }\left( \Delta \left( C\right) \right) \),

$$\begin{aligned} \left( i,A\right) \succsim (j,B)\Longleftrightarrow v\left( i,A\right) \ge v(j,B). \end{aligned}$$

Now fix \(j\in \mathcal{J }\) and define the relation \(\succsim _{j}\subseteq \mathcal{B }\left( \Delta \left( C\right) \right) \times \mathcal{B }\left( \Delta \left( C\right) \right) \) by \(A\succsim _{j}B\) if and only if \( (j,A)\succsim (j,B)\). Theorem 1 in DLR implies that there exits a set \( \mathcal{M }_{j}\), a state dependent function \(U_{j}:\Delta \left( C\right) \times \mathcal{M }_{j}\rightarrow \mathbb{R } \) and an agregator \(\gamma _{j}: \mathbb{R } ^{\mathcal{M }_{j}}\rightarrow \mathbb{R } \) such that, for every \(A,B\in \mathcal{B }\left( \Delta \left( C\right) \right) \),

$$\begin{aligned} A\succsim _{j}B\iff \gamma _{j}\left( \left\langle \max _{p\in A}U_{j}(p,m_{j})\right\rangle _{m_{j}\in \mathcal{M }_{j}}\right) \ge \gamma _{j}\left( \left\langle \max _{p\in B}U_{j}(p,m_{j})\right\rangle _{m_{j}\in \mathcal{M }_{j}}\right) , \end{aligned}$$

and, for every \(m_{j}\in \mathcal{M }_{j}, U_{j}(.,m_{j})\) is a nontrivial expected utility function. Moreover, since \(\succsim _{j}\) agrees with \( \succsim \) on \(\{j\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \) , it is clear that whenever \(\gamma _{j}\left( \langle \max _{p\in A}U(p,m_{j})\rangle _{m_{j}\in \mathcal{M }_{j}}\right) =\gamma _{j}\left( \langle \max _{p\in B}U(p,m_{j})\rangle _{m_{j}\in \mathcal{M }_{j}}\right) \) for some \(A,B\in \mathcal{B }\left( \Delta \left( C\right) \right) \), we have \(v\left( j,A\right) =v(j,B)\). We can also assume, without loss of generality, that for any distinct \(i\) and \(j\) in \(\mathcal{J }, \mathcal{M } _{i}\cap \mathcal{M }_{j}=\emptyset \). This now allows us to define \(\varphi _{j}: \mathbb{R } ^{\mathcal{M }_{j}}\rightarrow \mathbb{R } \) by \(\varphi _{j}(\xi ):=v(j,A)\), if there exists \(A\in \mathcal{B }\left( \Delta \left( C\right) \right) \) with \(\left\langle \max _{p\in A}U_{j}(p,m_{j})\right\rangle _{m_{j}\in \mathcal{M }_{j}}=\xi \) and \(\varphi _{j}(\xi ):=0\) otherwise. Finally, define \(U:\Delta \left( C\right) \times \cup _{j\in \mathcal{J }}\mathcal{M }_{j}\longrightarrow \mathbb{R } \) by \(U(p,m_{j}):=U_{j}(p,m_{j})\) for every \(j\in \mathcal{J }\) and \(m_{j}\in \mathcal{M }_{j}\). Since, for every \(j\in \mathcal{J }\) and \(A\in \mathcal{B } \left( \Delta \left( C\right) \right) , V(j,A)=\varphi _{j}(\langle \max _{p\in A}U(p,m_{j})\rangle _{m_{j}\in \mathcal{M }_{j}}), (\{\mathcal{M } _{j},\varphi _{j}\}_{j\in \mathcal{J }},U)\) is a Weak EU representation of \(\succsim \). \(\square \)

Proof of Theorem 2

Again, we leave the proof that the axioms are necessary for the representation to the reader. To show that the axioms are sufficient for the representation, as we did in the proof of Theorem 1, we can first use Debreu’s utility representation theorem to find a continuous function \(v:\mathcal{J }\times \mathcal{B }\left( \Delta \left( C\right) \right) \longrightarrow \mathbb{R } \) such that, for every pair \(\left( i,A\right) ,(j,B)\in \mathcal{J }\times \mathcal{B }\left( \Delta \left( C\right) \right) \),

$$\begin{aligned} \left( i,A\right) \succsim (j,B)\Longleftrightarrow v\left( i,A\right) \ge v(j,B). \end{aligned}$$

Now fix \(j\in \mathcal{J }\) and define the relation \(\succsim _{j}\subseteq \mathcal{B }\left( \Delta \left( C\right) \right) \times \mathcal{B }\left( \Delta \left( C\right) \right) \) by \(A\succsim _{j}B\) if and only if \( v\left( j,A\right) \ge v(j,B)\). Since \(v\) represents \(\succsim , \succsim _{j}\) satisfies all the axioms in DLR’s Theorem 3. This implies that there exits a set \(\mathcal{M }_{j}\), a state dependent function \(U_{j}:\Delta \left( C\right) \rightarrow \mathbb{R } \) and a strictly increasing agregator (on the relevant domain) \(\gamma _{j}: \mathbb{R } ^{\mathcal{M }_{j}}\rightarrow \mathbb{R } \) such that, for every \(A,B\in \mathcal{B }\left( \Delta \left( C\right) \right) \),

$$\begin{aligned} A\succsim _{j}B\iff \gamma _{j}\left( \left\langle \max _{p\in A}U_{j}(p,m_{j})\right\rangle _{m_{j}\in \mathcal{M }_{j}}\right) \ge \gamma _{j}\left( \left\langle \max _{p\in B}U_{j}(p,m_{j})\right\rangle _{m_{j}\in \mathcal{M }_{j}}\right) , \end{aligned}$$

and, for every \(m_{j}\in \mathcal{M }_{j}, U_{j}(.,m_{j})\) is an expected utility function. Again, we can assume, without loss of generality, that for any distinct \(i\) and \(j\) in \(\mathcal{J }, \mathcal{M }_{i}\cap \mathcal{M } _{j}=\emptyset \). Repeating the steps in the proof of Theorem 1, we obtain a Weak EU representation, \((\{\mathcal{M }_{j},\varphi _{j}\}_{j\in \mathcal{J }},U)\) of \(\succsim \), with the additional restriction that all the aggregators \(\varphi _{j}\) are strictly increasing on their relevant domain. That is, \((\{\mathcal{M }_{j},\varphi _{j}\}_{j\in \mathcal{J }},U)\) is an Ordinal EU representation of \(\succsim \). \(\square \)

Proof of Theorem 3

Once more, we will leave the proof that the axioms are necessary for the representation to the reader. We will show by induction on the number of elements of \(\mathcal{J }\) that they are also sufficient. If \(\left| \mathcal{J }\right| =1\), then we are in DLR’s setup and the representation comes from Theorem 2 in Dekel et al. (2007). Suppose now that the characterization is true when \(\left| \mathcal{J }\right| =n\) and consider a setup with \(\left| \mathcal{J }\right| =n+1\). Without loss of generality, we can assume that \(\mathcal{J }:=\{1,...,n+1\}\). By Continuity and the compactness of \(\mathcal{B }\left( \Delta \left( C\right) \right) \), we know that for each \(j\in \mathcal{J }\) there exists \(\underline{ A}_{j}\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that \( (j,B)\succsim (j,\underline{A}_{j})\) for every \(B\in \mathcal{B }\left( \Delta \left( C\right) \right) \). Order the elements of \(J\) so that \( \underline{A}_{n+1}\) satisfies \((n+1,\underline{A}_{n+1})\succsim (j, \underline{A}_{j})\) for every \(j\in \mathcal{J }\). Now consider the restriction of \(\succsim \) to \((\mathcal{J }\setminus \{n+1\})\times \mathcal B \left( \Delta \left( C\right) \right) \). By the induction hypothesis, this restriction admits a Monotone Additive EU representation \((\{(\mathcal{M } _{j},\Sigma _{j}),\mu _{j}\}_{j\in (\mathcal{J }\setminus \{n+1\})},U)\). For each \(j\in \mathcal{J }\setminus \{n+1\}\) and \(A\in \mathcal{B }\left( \Delta \left( C\right) \right) \), let

$$\begin{aligned} V(j,A):=\int \limits _{\mathcal{M }_{j}}\max _{p\in A}U(p,m_{j})\mu _{j}(dm_{j})\text {.} \end{aligned}$$

Suppose first that \((n+1,\underline{A}_{n+1})\succsim (j,\Delta (C))\) for every \(j\in \mathcal J\setminus \{ n+1\mathcal \} \). If there exists \(j\in \mathcal{J }\setminus \{n+1\}\) such that \((n+1,\underline{A}_{n+1})\sim (j,\Delta (C))\), then take any Monotone Additive EU representation \((( \mathcal{M }_{n+1},\Sigma _{n+1}),\mu _{n+1},\tilde{U})\) of the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \) and normalize \(\tilde{U}\) so thatFootnote 11

$$\begin{aligned} \tilde{V}(n+1,\underline{A}_{n+1}):=\int \limits _{\mathcal{M }_{n+1}}\max _{p\in \underline{A}_{n+1}}\tilde{U}(p,m_{n+1})\mu _{n+1}(dm_{n+1})=V(j,\Delta (C)). \end{aligned}$$

If \((n+1,\underline{A}_{n+1})\succ (j,\Delta (C))\) for every \(j\in \mathcal{J }\setminus \{n+1\}\), then take any Monotone Additive EU representation \(((\mathcal{M }_{n+1},\Sigma _{n+1}),\mu _{n+1}, \tilde{U})\) of the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B } \left( \Delta \left( C\right) \right) \) and normalize \(\tilde{U}\) so that

$$\begin{aligned} \tilde{V}(n+1,\underline{A}_{n+1}):=\int \limits _{\mathcal{M }_{n+1}}\max _{p\in \underline{A}_{n+1}}\tilde{U}(p,m_{n+1})\mu _{n+1}(dm_{n+1})>V(j,\Delta (C)) \end{aligned}$$

for every \(j\in \mathcal{J }\).Footnote 12 Now define \( \hat{U}:\Delta (C)\times \mathcal \cup _{j\in \mathcal{J }}\mathcal{M } _{j}\rightarrow \mathbb{R } \) by \(\hat{U}(.,j):=U(.,j)\) if \(j\in \mathcal{J }\setminus \{n+1\}\) and \(\hat{ U}(.,n+1):=\tilde{U}\). It is clear that \((\{(\mathcal{M }_{j},\Sigma _{j}),\mu _{j}\}_{j\in \mathcal{J }},\hat{U})\) is a Monotone Additive EU representation of \(\succsim \).

Now suppose that there exists \(j\in \mathcal{J }\setminus \{n+1\}\) such that \( (j,\Delta (C))\succ (n+1,\underline{A}_{n+1})\). Let \(\mathcal I :=\{j\in \mathcal{J }\setminus \{n+1\}:(j,\Delta (C))\succ (n+1,\underline{A}_{n+1})\}\). Since \(\mathcal{J }\) is a finite set, Continuity II and Nontriviality guarantee that there exists \(A^{*}\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that \((j,\Delta (C))\succ (n+1,A^{*})\succ (n+1, \underline{A}_{n+1})\succsim (j,\underline{A}_{j})\) for every \(j\in \mathcal I \). Now define \(\mathcal{B }^{*}:=\{A\in \mathcal{B }\left( \Delta \left( C\right) \right) :(n+1,A^{*})\succsim (n+1,A)\}\). We note that the fact that the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \) admits a Monotone Additive EU representation implies that \(\mathcal{B }^{*}\) is a convex set. Now, for every \(A\in \mathcal{B }^{*}\) let \(W(A):=V(j,B)\) for any \(j\in \mathcal{J } \setminus \{n+1\}\) and any \(B\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that \((n+1,A)\sim (j,B)\). A standard argument using Axiom 4 guarantees that, for any \(j\in \mathcal I \), such a \(B\) exists. We proceed through several claims.

Claim 1

\(W\) is affine in \(\mathcal{B }^{*}\).

Proof of Claim

Fix \(A,B\in \mathcal{B }^{*}, \lambda \in (0,1)\) and \(j\in \mathcal I \). By Axiom 4, there exist \(A^{\prime },B^{\prime }\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that \((n+1,A)\sim (j,A^{\prime })\) and \((n+1,B)\sim (j,B^{\prime })\). By definition, \( W(A)=V(j,A^{\prime })\) and \(W(B)=V(j,B^{\prime })\). By Axiom 8, we also have that \((n+1,\lambda A+(1-\lambda )B)\sim (j,\lambda A^{\prime }+(1-\lambda )B^{\prime })\). Since \(\mathcal{B } ^{*}\) is convex, \(\lambda A+(1-\lambda )B\in \mathcal{B }^{*}\). Since \(V(j,.)\) is affine, by the definition of \(W\), we have \(W(\lambda A+(1-\lambda )B)=V(j,\lambda A^{\prime }+(1-\lambda )B^{\prime })=\lambda V(j,A^{\prime })+(1-\lambda )V(j,B^{\prime })=\lambda W(A)+(1-\lambda )W(B)\).

\(\square \)

It turns out that \(W\) has a unique affine extension to \(\mathcal{B }\left( \Delta \left( C\right) \right) \).Footnote 13 Let’s abuse notation and call this extension \(W\) again. We need the following three claims.

Claim 2

\(W\) represents the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B } \left( \Delta \left( C\right) \right) \).

Proof of Claim

Fix \(A,B\in \mathcal{B }\left( \Delta \left( C\right) \right) \) and pick \( \lambda \) large enough so that \(\lambda \underline{A}_{n+1}+(1-\lambda )A\) and \(\lambda \underline{A}_{n+1}+(1-\lambda )B\) belong to \(\mathcal{B }^{*}\). By the definition of \(W\) and its affinity, we have that \((n+1,\lambda \underline{A}_{n+1}+(1-\lambda )A)\succsim (n+1,\lambda \underline{A} _{n+1}+(1-\lambda )B)\) iff \(W(\lambda \underline{A}_{n+1}+(1-\lambda )A)\ge W(\lambda \underline{A}_{n+1}+(1-\lambda )B)\) iff \(W(A)\ge W(B)\). Now, since the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \) admits a Finite Monotone Additive EU representation, we also have that \((n+1,A)\succsim (n+1,B)\) iff \( (n+1,\lambda \underline{A}_{n+1}+(1-\lambda )A)\succsim (n+1,\lambda \underline{A}_{n+1}+(1-\lambda )B)\), which completes the proof of the claim. \(\square \)

Claim 3

Fix \(A\in \mathcal{B }\left( \Delta \left( C\right) \right) \). If there exists \(j\in \mathcal{J }\setminus \{n+1\}\) and \(B\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that \((n+1,A)\sim (j,B)\), then \( W(A)=V(j,B)\). If \((n+1,A)\succ (j,B)\) for every \(j\in \mathcal{J }\setminus \{n+1\}\) and \(B\in \mathcal{B }\left( \Delta \left( C\right) \right) \), then \( W(A)>V(j,B)\) for every \(j\in \mathcal{J }\setminus \{n+1\}\) and \(B\in \mathcal{B }\left( \Delta \left( C\right) \right) \).

Proof of Claim

The claim is immediate if \(A\in \mathcal{B }^{*}\). Suppose, thus, that \( A\notin \mathcal{B }^{*}\) and \((n+1,A)\sim (j,B)\) for some \(j\in \mathcal J \setminus \{n+1\}\) and some \(B\in \mathcal{B }\left( \Delta \left( C\right) \right) \). Axiom 4 implies that there exists \( \underline{B}\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that \((n+1,\underline{A}_{n+1})\sim (j,\underline{B})\). Now pick \(\lambda \in (0,1)\) large enough so that \(\lambda \underline{A}_{n+1}+(1-\lambda )A\in \mathcal{B }^{*}\). By Axiom 8, we have \( (n+1,\lambda \underline{A}_{n+1}+(1-\lambda )A)\sim (j,\lambda \underline{B} +(1-\lambda )B)\), which implies that \(W(\lambda \underline{A} _{n+1}+(1-\lambda )A)=V(j,\lambda \underline{B}+(1-\lambda )B)\). Since both \( W\) and \(V(j,.)\) are affine, and \(W(\underline{A}_{n+1})=V(\underline{B})\), we get that \(W(A)=V(j,B)\). Finally, suppose that \((n+1,A)\succ (j,B)\) for every \(j\in \mathcal{J }\setminus \{n+1\}\) and \(B\in \mathcal{B }\left( \Delta \left( C\right) \right) \). Fix \(j^{*}\in \mathcal{J }\setminus \{n+1\}\) and \(B^{*}\in \mathcal{B }\left( \Delta \left( C\right) \right) \). If there exists \(A^{*}\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that \((n+1,A^{*})\sim (j,B^{*})\), then, by the previous claim and the previous observation, we have \(W(A)>W(A^{*})=V(j,B^{*})\). Otherwise, we have \(W(A)>W(\underline{A}_{n+1})>V(j,B^{*})\). \(\square \)

Claim 4

There exists a Monotone Additive EU representation \(((\mathcal{M } _{n+1},\Sigma _{n+1}),\mu _{n+1},\tilde{U})\) of the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \) such that

$$\begin{aligned} W(A):=\int \limits _{\mathcal{M }_{n+1}}\max _{p\in A}\tilde{U}(p,m_{n+1})\mu _{n+1}(dm_{n+1}), \end{aligned}$$

for every \(A\in \mathcal{B }\left( \Delta \left( C\right) \right) \).

Proof of Claim

By Theorem 2 in Dekel et al. (2007), we know that the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \) admits a Monotone Additive EU representation \(((\mathcal{M }_{n+1},\Sigma _{n+1}),\mu _{n+1},\hat{U})\). Moreover, we can choose this representation so that \(\mu _{n+1}\) is a probability measure. Let \(\hat{V}:\mathcal{B }\left( \Delta \left( C\right) \right) \rightarrow \mathbb{R } \) be defined by

$$\begin{aligned} \hat{V}(A):=\int \limits _{\mathcal{M }_{n+1}}\max _{p\in A}\hat{U}(p,m_{n+1})\mu _{n+1}(dm_{n+1})\text {, for every }A\in \mathcal{B }\left( \Delta \left( C\right) \right) \text {.} \end{aligned}$$

Now notice that both \(W\) and \(\hat{V}\) are affine representations of the restriction of \(\succsim \) to \(\{n+1\}\times \mathcal{B }\left( \Delta \left( C\right) \right) \). The uniqueness properties of such representations imply that there exists \(\alpha \in \mathbb{R } _{++}\) and \(\beta \in \mathbb{R } \) such that \(W(A)=\alpha \hat{V}(A)+\beta \) for every \(A\in \mathcal{B } \left( \Delta \left( C\right) \right) \). Define \(\tilde{U}:\Delta (C)\rightarrow \mathbb{R } \) by \(\tilde{U}(p):=\alpha \hat{U}(p)+\beta \) for every \(p\in \Delta (C)\). Notice that

$$\begin{aligned} W(A)=\int \limits _{\mathcal{M }_{n+1}}\max _{p\in A}\tilde{U}(p,m_{n+1})\mu _{n+1}(dm_{n+1})\text {, for every }A\in \mathcal{B }\left( \Delta \left( C\right) \right) \text {.} \end{aligned}$$

This concludes the proof of the claim. \(\square \)

Now define \(\tilde{V}:\mathcal{J }\times \mathcal{B }\left( \Delta \left( C\right) \right) \rightarrow \mathbb{R } \) by \(\tilde{V}(j,.):=V(j,.)\) if \(j\in \mathcal{J }\setminus \{n+1\}\) and \( \tilde{V}(n+1,.):=W\). The previous three claims imply that \(\tilde{V}\) represents \(\succsim \) and that it admits a Monotone Additive EU representation. \(\square \)

Proof of Proposition 1

Suppose first that \(\succsim \) admits an Ordinal EU representation \((\{ \mathcal{M }_{j},\varphi _{j}\}_{j\in \mathcal{J }},U)\). We can assume, without loss of generality, that the sets \(\{\mathcal{M }_{j}\}_{j\in \mathcal{J }}\) have no redundant nodes. That is, for every \(j\in \mathcal{J }\) and every distinct \(m_{j}\) and \(m_{j}^{\prime }\) in \(\mathcal{M }_{j}, U(.,m_{j})\) and \(U(.,m_{j}^{\prime })\) represent different nontrivial expected utility preferences. We now show that Finiteness implies that all the sets \(\mathcal{M }_{j}\) are finite. Fix \(j\in \mathcal{J }\) and pick any sphere \(A\in \mathcal{B }\left( \Delta \left( C\right) \right) \). That is, pick \(A\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that there exists \(p\in \Delta (C)\) and \(\delta >0\) such that \(A:=\{q\in \mathbb{R } ^{C}:\left\| q-p\right\| \le \delta \) and \(\sum _{c\in C}q(c)=1\}\).Footnote 14 Because \(A\) is a sphere, for every pair \(m_{j}\) and \(m_{j}^{\prime }\) in \( \mathcal{M }_{j}\) we have that \(\arg \max _{p\in A}U(p,m_{j})\ne \arg \max _{p\in A}U(p,m_{j}^{\prime })\) and both \(\arg \max _{p\in A}U(p,m_{j})\) and \(\arg \max _{p\in A}U(p,m_{j}^{\prime })\) are singletons. Now suppose that \(\mathcal{M }_{j}\) is not a finite set for some \(j\in \mathcal{J }\) and pick any finite \(B\in \mathcal{B }\left( \Delta \left( C\right) \right) \) such that \(B\subseteq A\). By our previous observation, there exists \( m_{j}\in \mathcal{M }_{j}\) such that \(\max _{p\in B}U(p,m_{j})<\max _{p\in A}U(p,m_{j})\). Since \(\varphi _{j}\) is strictly increasing and \((\{\mathcal{M }_{j},\varphi _{j}\}_{j\in \mathcal{J }},U)\) represents \(\succsim \), this implies that \((j,A)\succ (j,B)\). That is, for no finite subset \(B\) of \(A\) we can have \((j,A)\sim (j,B)\), which contradicts Finiteness. We conclude that \( \mathcal{M }_{j}\) must be a finite set.

Now suppose that the preference relation \(\succsim \) admits a Monotone Additive EU representation \((\{(\mathcal{M }_{j},\Sigma _{j}),\mu _{j}\}_{j\in \mathcal{J }},U)\). From Dekel et al. (2007), we know that \((\{( \mathcal{M }_{j},\Sigma _{j}),\mu _{j}\}_{j\in \mathcal{J }},U)\) can be chosen so that, for each \(j\in \mathcal{J }, \mathcal{M }_{j}\) is a metric space and has no redundant nodes, \( \Sigma _{j}\) is the collection of Borel subsets of \(\mathcal{M }_{j}, \mu _{j}\) is a probability measure on \((\mathcal{M }_{j},\Sigma _{j})\) whose support is exactly \(\mathcal{M }_{j}\) and, for each menu \(A\in \mathcal{B } \left( \Delta \left( C\right) \right) \), the function \(\sigma _{A}^{j}: \mathcal{M }_{j}\rightarrow \mathbb{R } \) defined by \(\sigma _{A}^{j}(m_{j}):=\max _{p\in A}U(p,m_{j})\) for every \( m_{j}\in \mathcal{M }_{j}\), is continuous. Now, for each \(j\in \mathcal{J }\), define \(\varphi _{j}: \mathbb{R } ^{\mathcal{M }_{j}}\rightarrow \mathbb{R } \) by

$$\begin{aligned} \varphi _{j}(\xi ):=\int \limits _{\mathcal{M }_{j}}\xi (m_{j})\mu _{j}(dm_{j}), \end{aligned}$$

if there exists \(A\in \mathcal{B }\left( \Delta \left( C\right) \right) \) with \(\left\langle \max _{p\in A}U_{j}(p,m_{j})\right\rangle _{m_{j}\in \mathcal{M }_{j}}=\xi \) and \(\varphi _{j}(\xi ):=0\) otherwise. Given our choice of Monotone Additive EU representation for \(\succsim \), it turns out that \((\{\mathcal{M }_{j},\varphi _{j}\}_{j\in \mathcal{J }},U)\) is an Ordinal EU representation of \(\succsim \). Now the first part of this proof implies that all the sets in \(\{\mathcal{M }_{j}\}_{j\in \mathcal{J }}\) must be finite. \(\square \)

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Barbosa, L.F.B., Riella, G. A note on equivalent comparisons of information channels. Theory Decis 78, 33–44 (2015). https://doi.org/10.1007/s11238-013-9399-7

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Keywords

  • Information channels
  • Preferences over menus
  • Weak expected utility representation
  • Ordinal expected utility representation
  • Additive expected utility representation