Information disclosure with many alternatives

Abstract

We consider two-stage collective decision problems where some agents have private information about alternatives and others don’t. In the first stage informed agents (experts) may or may not disclose their private information, thus eventually influencing the preferences of those initially uninformed. In the second stage the resulting preferences of all agents after disclosure are aggregated by a social choice function. We provide general conditions on social choice functions guaranteeing that the collective outcome will be the same that would obtain if all agents shared all the information available in society. Experts should be granted a coalitional veto power: changes in the social outcome that are due to changes in the preferences of other agents after information disclosure should not harm all the experts at the same time. We then specialize our general results. When the set of experts is a priori determined, we characterize those strategy-proof rules defined on single-peaked or separable preference domains that ensure that desired level of information disclosure. We also prove that, when the set of experts is unknown, no voting rule can fully achieve this goal, but majority voting provides a unique second best solution when preference profiles are single-peaked.

Appendix

Proof of Theorem 1

Suppose f attributes coalitional veto power to the set E. If $E=\{i\}$, then for all pairs $R ,R^{ \prime } \in {\mathcal {R}}$ with $R_{i} =R_{i}^{ \prime }$ $f (R) =f (R^{ \prime })$ and f trivially satisfies full outcome-relevant information disclosure. If $|E|\ge 2$, then fix any $\theta \in \Theta \text {.}$ Let $\bar{m}$ be a strategy profile such that $\bar{m}_{i} = {\mathcal {I}}$ for all $i \in E\text {.}$ It is immediate to check that $\bar{m}$ is a Nash equilibrium of the game and $f(\theta _{E} ({\mathcal {I}}) ,\theta _{N} (g (\bar{m})) =f (\theta ({\mathcal {I}}))$ . To prove that there are no other Nash equilibrium outcomes, suppose m be a Nash equilibrium involving partial or no disclosure and $f (\theta _{E} ({\mathcal {I}}) ,\theta _{N} (g (m))) \ne f (\theta ({\mathcal {I}}))\text {.}$ By assumption there is an expert $i \in E$ such that $f (\theta ({\mathcal {I}})) P_{i} f (\theta _{E} ( {\mathcal {I}}) ,\theta _{N} (g (m)))$. Expert i can profitably deviate by announcing $m_{i}^{ \prime } ={\mathcal {I}}:$ in fact $g (m_{i}^{ \prime } ,m_{ -i}) ={\mathcal {I}}$ and therefore $f (\theta _{E} ({\mathcal {I}}) ,\theta _{N} (g (m_{i}^{ \prime } ,m_{ -i}))) P_{i} f (\theta _{E} ({\mathcal {I}}) ,\theta _{N} (g (m)))$. $\square$

Proof of Theorem 2

If f does not attribute coalitional veto power to the set $E\text {,}$ then there exists a pair of preference profiles $(R_{E} ,R_{N})\text {,}$ $(R_{E} ,R_{N}^{ \prime })$ such that $f (R_{E} ,R_{N}) \ne f (R_{E} ,R_{N}^{ \prime })$ and $f (R_{E} ,R_{N}) R_{i} f (R_{E} ,R_{N}^{ \prime })$ for all $i \in E\text {.}$ Consider a type $\theta _{j} \in \Theta _{j}$ such that $\theta _{j} (I) =R_{j}$ for all $I \subset {\mathcal {I}}$ and $\theta _{j} ({\mathcal {I}}) =R_{j}^{ \prime }\text {.}$ This type exists because $\Theta _{j}$ is rich for every $j \in A .$ Consider a strategy profile $m^{ *}$ such that for all $i \in E$ , $m_{i}^{ *} =\hat{I}$ for some $\hat{I} \subset \mathcal { I}\text {.}$ The strategy profile $m^{ *}$ is a Nash equilibrium of the $\theta$-game. In fact any deviation $m_{i}^{ \prime } \ne {\mathcal {I}}$ is irrelevant because it does not modify non-experts’ preferences and, consequently, the final outcome, while the deviation $m_{i}^{ \prime } ={\mathcal {I}}$ is not profitable for any expert i, because $\theta _{j} ({\mathcal {I}}) =R_{j}^{ \prime }$ for all $j \in N\text {, }$ and $f (R_{E} ,R_{N}) R_{i} f (R_{E} ,R_{N}^{ \prime })$ for all $i \in E.$ $\square$

Proof of Proposition 1

Sufficiency. Consider any generalized median voter social choice function with a left coalition system W such that for all $x \in X,\text {}$ i) there exists $c_{x} \subseteq E$ $\text {and }ii)\text {}\text {}\text {}$ $c \in W (x)$ only if there exists $i \in E \cap c$. Fix an arbitrary single-peaked preference profile ${R}_{1},\ldots ,{R}_{n}\in \hat{{\mathcal {R}}}^{n}$, where $\hat{{\mathcal {R}}}^{n}$ denote the set of all single-peaked preference profiles. Let $l \in E$ be such that for all $j \in E,$ $B\text {} (R_{l}) \le B (R_{j})$ and let $r \in E$ be such that for all $j \in E\text {,}$ $B (R_{r}) \ge B (R_{j})\text {.}$ By (ii) $f (R) \ge B (R_{l})$ and by $(i)\; f (R) \le B (R_{r})$. Consider now any $R^{ \prime } \in \hat{{\mathcal {R}}}^{n}$ such that for all $j \in E$, $R_{j} =R_{j}^{ \prime }\text {.}$ For the same arguments as above $B (R_{l}) \le f (R^{ \prime }) \le B (R_{r})\text {.}$ Suppose $f (R^{ \prime }) \ne f (R)\text {,}$ and without loss of generality, suppose that $f (R^{ \prime }) <f (R)\text {.}$ It follows that $f (R^{ \prime }) P_{l} f (R)$ and $f (R) P_{r} f (R^{ \prime })$ and coalitional veto power is satisfied.

Necessity. Consider any generalized median voter social choice function f. Since the type set $\Theta _{i}$ is rich in $\hat{{\mathcal {R}}}_{i}$ for each $i \in A$, then by Theorem 2 the generalized median voter rule should satisfy coalitional veto power relative to $E\text {.}$ Let W be its associated left coalition system and $X =[a ,b]\text {.}$ Suppose first that there exists $x <b$ such that for each coalition $c \in W (x)\text {,}$ a member of c is a non-expert. Consider $R \in \hat{{\mathcal {R}}}^{n}$ such that for all $i \in E\text {,}$ $B (R_{i}) =x\;$and for all $j \in N\text {,}$ $B (R_{j}) =b$; it follows that $f (R) >x\text {.}$ Let $R_{j}^{ \prime }\in \hat{{\mathcal {R}}}_{j}$ be a preference such that $B (R_{j}^{ \prime }) =x\text {.}$ By unanimity $f (R_{E} ,R_{N}^{ \prime }) =x$. Consider a profile $\theta$ such that for all $i \in A$, $\theta _{i} (I) =R_{i}^{ \prime }$ for all $I \subset {\mathcal {I}}$ and $\theta _{i} ({\mathcal {I}}) =R_{i}\text {.}$ It follows that there exists a Nash equilibrium of the $\theta$-game such that for all $i \in E$, $m_{i}^{ *} =\hat{I}$ for some $\hat{I} \subset {\mathcal {I}}$ and $f (\theta _{E} ({\mathcal {I}}) ,\theta _{N} (g (m^{ *}))) \ne f (\theta ({\mathcal {I}}))\text {.}$ Therefore, the social choice function does not ensure full outcome-relevant information disclosure. Suppose now that there exists $x <b$ and $c \in W (x)$ such that $E \cap c =\varnothing \text {.}$ Let $R_{i}^{ \prime } ,{\bar{R}}_{i} \in \hat{{\mathcal {R}}}_{i}$ be a pair of preference such that $B (R_{i}^{ \prime }) =x$ and $B ({\bar{R}}_{i}) =b\text {.}$ At preference profile $({\bar{R}}_{E} ,R_{N}^{ \prime })$ we have $f ({\bar{R}}_{E} ,R_{N}^{ \prime }) \le x\text {.}$ By unanimity $f({\bar{R}}_{E},{\bar{R}}_{N})=b$. Consider a type profile $\theta$ such that for all $j \in N$, $\theta _{j} (I) ={\bar{R}}_{j}$ for all $I \subset {\mathcal {I}}$ and $\theta _{j} ({\mathcal {I}}) =R_{j}^{ \prime },$ and for all $i \in E,$ $\theta _{i} ({\mathcal {I}}) ={\bar{R}}_{i}\text {.}$ It follows that there exists a Nash equilibrium of the $\theta$-game such that for all $i \in E$, $m_{i}^{ *} =\hat{I}$ for some $\hat{I} \subset {\mathcal {I}}$ and $f (\theta _{E} ({\mathcal {I}}) ,\theta _{N} (g (m^{ *}))) =f ({\bar{R}}) \ne f ({\bar{R}}_{E} ,R_{N}^{ \prime }) =f (\theta ({\mathcal {I}})),$ and the social choice function does not ensure full outcome-relevant information disclosure. $\square$

Proof of Proposition 2

By Moulin (1980), we know that a social choice function is anonymous, unanimous and strategy-proof if the left coalition system satisfies this additional condition: for any $x\in X$ if $c\in W_{x}$ then $c'\in W_{x},$ for all $c'\in 2^{A}$ with $|c'|\ge |c|.$ By Theorem 2 and Corollary 2, when $\left| E\right| <\frac{n +1}{2}$, it follows immediately that no anonymous, unanimous and strategy-proof social choice function may ensure full outcome-relevant information disclosure. By Corollary 3 we know that the median rule ensures information disclosure when $\left| E\right| \ge \frac{n +1}{2}$ . To conclude the proof, we show that every anonymous, unanimous and strategy-proof social choice function different than the median rule does not ensure full outcome-relevant information disclosure when $\left| E\right| =\frac{n +1}{2}$. Let M be an arbitrary set of agents with $\left| M\right| = \frac{n +1}{2}\text {.}$ Consider any anonymous, unanimous and strategy-proof voting rule f such that for some $z<b$ $c \in C (z)$ if and only if $\left| c\right| \ge k$ with $k \ne \frac{n +1}{2}\text {.}$ Suppose first $k <\frac{n +1}{2}\text {.}$ Consider a pair of preference profiles $R^{0} ,R^{1}\in \hat{{\mathcal {R}}}^{n}$ with $R_{M}^{1} =R_{M}^{0} =R_{M}\text {,}$ $B (R_{i}) =b$ for all $i \in M\text {,}$ and $B (R_{j}^{0}) =b$, $B (R_{j}^{1}) =z$ for all $j \notin M.$ Let $\theta$ be a type profile such that for all $i \in M ,\theta _{i} ({\mathcal {I}}) =R^{0}_{i}$ and for all $j \notin M$ $\theta _{j} (I) =R_{j}^{0}$ for all $I \subset {\mathcal {I}}$ and $\theta _{j} ({\mathcal {I}}) =R_{j}^{1}.$ Consider a $\theta$-game such that $M =E.$ It is immediate to check that there exists a Nash equilibrium $m^{ *}$ such that $m_{i}^{ *} =I$ for some $I\subset \mathcal {I}$ and for all $i \in E$ and $f (\theta _{E} ,\theta _{N} (g (m^{ *}))) =z \ne f (\theta ({\mathcal {I}})) =b\text {.}$ The proof for the case $k >\frac{n +1}{2}$ is analogous: consider a pair of preference profiles ${\bar{R}}^{0} ,{\bar{R}}^{1}\in \hat{{\mathcal {R}}}^{n}$ with ${\bar{R}}_{M}^{1} ={\bar{R}}_{M}^{0} = {\bar{R}}_{M},$ $B ({\bar{R}}_{i}) =z$ for all $i \in M$, $B (\bar{ R}_{j}^{0}) =z$, $B ({\bar{R}}_{j}^{1}) =b$ for all $j \notin M$ and a type-profile $\theta$ such that for all $j \notin M,$ $\theta _{j} (I) ={\bar{R}}_{j}^{0}$ for all $I \subset {\mathcal {I}}$ and $\theta _{j} (\mathcal {I}) ={\bar{R}}_{j}^{1}.$ $\square$

Proof of Proposition 3

Sufficiency. We prove that a social choice function based on voting by committees that satisfies requirements (a) and (b) ensures full outcome-relevant information disclosure. Suppose by contradiction that there exists a type profile $\theta$ and a Nash equilibrium $m^{*}$ of the $\theta$-game such that $m^{*}=I\subset {\mathcal {I}}$ and $f\left( \theta _{E}({\mathcal {I}}),\theta _{N}(g(m^{*})\right) )=f(P)\ne f\left( \theta ({\mathcal {I}})\right) .$ Since every winning coalition contains an expert (requirement (b)) and for every alternative there exists a winning coalition, formed only by experts (requirement (a)), then $x\in \bigcup G(P_i)$ implies that x is selected both at f(P) and at $f\left( \theta ({\mathcal {I}})\right)$ and $x\notin G(P_i)$ for every $i\in E$ implies that x is selected neither at f(P) nor at $f\left( \theta ({\mathcal {I}})\right)$. Therefore (i) for every $x\notin f(P)$ but $x\in f(\theta ({\mathcal {I}}))$ there exists an expert i such that $x\in G(P_i)$ and (ii) for every $x\in f(P)$ but $x\notin f(\theta ({\mathcal {I}}))$, there exists an expert i such that $x\notin G(P_i)$.

Consider first case (i). Since $x\in f(\theta ({\mathcal {I}}))$ then by separability of the type set $f\left( \theta _{E}({\mathcal {I}}),\theta _{N}(g(m^{*}\cup \textit{I}_x)\right) )=f(P)\cup x$ and therefore the strategy $m_i=m^{*}\cup \textit{I}_x$ is a profitable deviation for expert i. Case (ii) can be proved in an analogous way: since $x\notin f(\theta ({\mathcal {I}}))$ then by separability of the type set $f\left( \theta _{E}({\mathcal {I}}),\theta _{N}(g(m^{*}\cup \textit{I}_x)\right) )=f(P)\backslash x$ and therefore the strategy $m_i=m^{*}\cup \textit{I}_x$ is a profitable deviation for expert i.

Necessity. Without loss of generality let $X=\{x,y\}$ . Suppose first that there exists an alternative x such that it has an associated winning coalition formed only by non experts. Consider a pair of preference $P,P'$ such that $G(P)=\{x,y\}$ and $G(P')=\{x\}$. Consider a separable type profile $\theta$ such that

• (i) for all $i \in N$, $\theta _{i}(I_{x}) =P^{'}$ and $\theta _{i} (I) =P$, for all $I\supset I_{x}$;

• (ii) for all $j \in E$, $\theta _{j} ({\mathcal {I}}) =P^{'}$.

It follows that there exists a Nash equilibrium of the $\theta -$game such that in equilibrium every expert $j\in E$ discloses information $m_j^{*}=I_{x}$ and therefore full outcome-relevant information it is not disclosed.

Suppose now that there exists an alternative x such that no willing coalition is formed only by experts, that is for every $M\in {\mathcal {W}}_{x}$, $M\cap N\ne \emptyset$. Consider a separable type profile $\theta ^{'}$ such that

• (i) for all $i \in N$, $\theta ^{'} _{i}(I_{x}) =P$ and $\theta ^{'} _{i} (I) =P^{'}$, for all $I\supset I_{x}$;

• (ii) for all $j \in E$, $\theta ^{'} _{j} ({\mathcal {I}}) =P$.

It follows that there exists a Nash equilibrium of the $\theta ^{'}-$game such that in equilibrium every expert $j\in E$ discloses information $m_j^{*}=I_{x}$ and therefore full outcome-relevant information it is not disclosed. $\square$

Proof of Theorem 3

To simplify notation we do not distinguish anymore the preferences of the experts and non-experts when writing a preference profile at stage 2 after information g(m) has been disclosed and we simply write $\theta (g(m))$, recalling that experts’ preference are fixed and do not change with information disclosure. Suppose f attributes coalitional veto power to every set $E_k \in \mathcal {Q}$. Fix any $\theta \in \Theta \text {.}$ Let $\bar{m}$ be a strategy profile such that $\bar{m}_{i} =I_i$ for all $i \in E\text {.}$ It is immediate to check that $\bar{m}$ is a Nash equilibrium of the game since for every $E_k \in \mathcal {Q}$, $|E_k|\ge 2$ and $f(\theta (g (\bar{m})) =f (\theta ({\mathcal {I}}))$. To prove that there are no other Nash equilibrium outcomes, suppose m be a Nash equilibrium and $f (\theta (g (m)) \ne f (\theta ({\mathcal {I}}))\text {.}$ Since $g(m)\ne {\mathcal {I}}$, there is at least one expert who did not fully disclose her outcome-relevant private information. Let $i \in E_k$ be an expert who did not fully disclose her private information. Consider the strategy $m_i^{\prime }=I_k$: either $f(\theta (g(m_{-i},m_{i^{\prime }}))\ne f(\theta (g(m)))$, meaning that the disclosure of some additional information by agent i affects the outcome chosen by the social choice function, or $f(\theta (g(m_{-i},m_{i^{\prime }}))= f(\theta (g(m)))$. Consider the former case. By assumption the social choice function grants veto power to the subset $E_k$, and therefore there is an expert $j \in E_k$ such that $f(\theta (g(m_{i^{\prime }}, m_{-i})) P_{j} f (\theta ((g (m)))$. It follows that expert j can profitably deviate at m by announcing $m_{j}^{ \prime } =I_k$. Consider the latter case. It follows that there is some other expert $j \in E_{k'}$ who did not fully disclose her information: $m_j \subset I_{k'}$. Consider then strategy profile $(m'_i=I_k, m'_j=I_{k'},m_{-i,j})$. Again two cases may occur. Either $f(g(m'_i, m'_j,m_{-i,j})) \ne f(g(m'_i,m_{-i}))$ or $f(g(m'_i, m'_j,m_{-i,j})) =f(g(m'_i,m_{-i}))$. In the former case the same argument as before applies. Since the social choice function grants veto power to the set of agents in $E_{k'}$ there is at least one agent in this group who has a profitable deviation. In the latter case, it must exist some other expert in another set $E_{k''}$ who did not fully disclose the information. Since $f (\theta (g(m)))\ne f(\theta ({\mathcal {I}}))$, there must exist an agent who by fully disclosing her information affects the final outcome. It follows that there exist an expert $l\in E_{k''}$ and a strategy profile $\hat{m}$ with $\hat{m}_l \ne I_k''$ such that $f (\theta (g(\hat{m}_l,\hat{m}_{-l}))) \ne f (\theta (g(m_l=I_{k''}. \hat{m}_{-l}))$. By assumption the social choice function satisfies coalitional veto power and therefore there exists an expert $h\in E_{k''}$ who can profitably deviate at $\hat{m}$ announcing $m_h=I_{k''}$. $\square$