1% ============================================================================2% Theory: valuation as decentralised information aggregation (WP 26)3% Author: Simon-Pierre Boucher — simon-pierre.boucher@uqo.ca4% Proofs: appendix/proofs.tex. Simulations: scripts/41_theory_sim.py5% ============================================================================6% Theorem environments are declared here only if main.tex has not done so.7\makeatletter8\@ifundefined{proposition}{\theoremstyle{plain}\newtheorem{proposition}{Proposition}}{}9\@ifundefined{lemma}{\theoremstyle{plain}\newtheorem{lemma}{Lemma}}{}10\@ifundefined{corollary}{\theoremstyle{plain}\newtheorem{corollary}{Corollary}}{}11\@ifundefined{assumption}{\theoremstyle{definition}\newtheorem{assumption}{Assumption}}{}12\@ifundefined{definition}{\theoremstyle{definition}\newtheorem{definition}{Definition}}{}13\@ifundefined{remark}{\theoremstyle{remark}\newtheorem{remark}{Remark}}{}14\makeatother15\theoremstyle{plain}16\providecommand{\figdir}{../figures}17\providecommand{\E}{\mathbb{E}}18\providecommand{\Var}{\operatorname{Var}}19\providecommand{\Cov}{\operatorname{Cov}}20\providecommand{\MSE}{\operatorname{MSE}}21\providecommand{\one}{\mathbf{1}}22\providecommand{\med}{\operatorname{med}}2324\section{Theory: Valuation as Decentralised Information Aggregation}25\label{sec:theory}2627This section formalises the object that every arm of the experiment is asked to estimate, and derives what a committee of partially informed, possibly correlated agents can and cannot achieve by deliberating. The model is deliberately simple: linear signals, Gaussian errors, and a fixed-weight revision rule. Its purpose is not realism but discipline. It delivers four results that the empirical design is built to test: the pooled error of a committee has a floor that no number of agents removes when their errors share a common component; opinion-only deliberation manufactures consensus without information; evidence exchange lowers the pooled error only when it redistributes weight towards evidence that the committee under-uses; and a role permutation that leaves the evidence partition untouched does not change the expected error. Proofs are collected in Appendix~\ref{app:proofs}. Table~\ref{tab:notation} fixes the notation.2829\begin{table}[t]30\centering\footnotesize31\caption{Notation.}32\label{tab:notation}33\begin{tabular}{@{}lp{10.6cm}@{}}34\toprule35$P,\ p=\log P$ & transaction price of the subject property and its logarithm \\36$\mathcal{I}_t$ & information available at the valuation date $t$ (cutoff enforced in every arm) \\37$D$ & sale indicator; $D=1$ if the property transacts in the evaluation window \\38$v^{\ast}=\E[p\mid\mathcal{I}_t]$ & latent market-consistent log value \\39$\tilde v=\E[p\mid\mathcal{I}_t,D=1]$ & selected market value, the operational estimation target; $\kappa=\tilde v-v^{\ast}$ \\40$\varepsilon=p-\tilde v$ & transaction noise (bargaining, urgency, liquidity) \\41$V^{F},\ R_t,\ u_t$ & structural fundamental value, net rent (housing-service flow), user cost of capital \\42$K$ & number of agents; $\one$ is the $K$-vector of ones \\43$s_k=v+b_k+\eta_k$ & signal (round-0 estimate) of agent $k$; bias $b_k$, error $\eta_k$ \\44$\Sigma,\ \sigma^{2},\ \rho$ & error covariance; common variance and pairwise correlation in the exchangeable case \\45$w,\ \hat y(w)=w^{\prime}s$ & linear pool weights ($w^{\prime}\one=1$) and pooled estimate \\46$x^{(r)},\ \lambda,\ T$ & vector of estimates after round $r$; anchoring weight; committee statistic (mean or median) \\47$W,\ \pi$ & influence matrix of a linear deliberation and its consensus weights \\48$\zeta,\ \Gamma$ & revision term carrying new evidence or persuasion; its covariance \\49$z_j=v+c+u_j$ & evidence item $j=1,\dots,J$; common evidence error $c$ (variance $\sigma_c^{2}$), item error $u_j$ (variance $\sigma_u^{2}$) \\50$A_k,\ a_k,\ \omega$ & evidence view of agent $k$, its within-view weights, implied committee weights $\omega=K^{-1}\sum_k a_k$ \\51$g_m,\ e_k$ & provider-common error ($m=1,\dots,M$; variance $\sigma_g^{2}$) and idiosyncratic reasoning error (variance $\tau_k^{2}$) \\52\bottomrule53\end{tabular}54\end{table}5556\subsection{Prices, values and the estimation target}57\label{subsec:values}5859\begin{definition}[Transaction price]60The transaction price $P_{it}$ is the consideration at which property $i$ transfers between unrelated parties at date $t$. All empirical metrics in this paper are computed on $p_{it}=\log P_{it}$ for arm's-length residential sales.61\end{definition}6263\begin{definition}[Market-consistent value and the selected target]64Given the information set $\mathcal{I}_t$, the latent market-consistent log value is $v^{\ast}_{it}=\E[p_{it}\mid\mathcal{I}_t]$, the conditional expectation of the price the property would fetch at $t$. Because prices are observed only for properties that sell, the quantity any price-trained method actually targets is the selected market value $\tilde v_{it}=\E[p_{it}\mid\mathcal{I}_t,D_{it}=1]$, and we write $\kappa_{it}=\tilde v_{it}-v^{\ast}_{it}$ for the selection term.65\end{definition}6667\begin{definition}[Appraisal estimate and investment value]68An appraisal estimate is any $\mathcal{I}_t$-measurable function $\hat y_{it}=\hat y(\mathcal{I}_t)$, whether produced by a hedonic regression, a gradient-boosted model, a single language model or a committee of agents. An investment value is the present value of the property to a particular holder under that holder's discount rate, tax position and horizon; it is neither the target nor the benchmark of any arm.69\end{definition}7071\begin{definition}[Structural fundamental value]72Let $R_t$ denote the net flow of housing services (the net rent the property commands or would command), $i_t$ the nominal cost of mortgage funds, $\tau_p$ the property-tax rate, $\delta$ maintenance and depreciation, $\gamma_t$ a risk and liquidity premium and $\pi^{e}_t=\E_t[P_{t+1}/P_t]-1$ the expected one-period capital gain. The one-period user-cost condition of \citet{poterba1984tax}, in the form used by \citet{himmelberg2005assessing}, states that a marginal household is indifferent between owning and renting when73\begin{equation}74R_t = P_t\,u_t,\qquad u_t = i_t+\tau_p+\delta+\gamma_t-\pi^{e}_t ,75\label{eq:usercost}76\end{equation}77where the Canadian tax treatment (no deductibility of mortgage interest or property taxes for owner-occupiers, no tax on imputed rent and an exemption of principal-residence gains) removes the marginal-tax-rate terms of the original United States formulation. Writing $k_t=i_t+\tau_p+\delta+\gamma_t$ and iterating \eqref{eq:usercost} forward,78\begin{equation}79P_t=\sum_{j=0}^{H-1}\E_t\!\left[\frac{R_{t+j}}{\prod_{h=0}^{j}(1+k_{t+h})}\right]+\E_t\!\left[\frac{P_{t+H}}{\prod_{h=0}^{H-1}(1+k_{t+h})}\right].80\label{eq:pv}81\end{equation}82The structural fundamental value $V^{F}_t$ is the first term of \eqref{eq:pv} as $H\to\infty$ under the transversality condition that the second term vanishes; the gap $P_t-V^{F}_t$ is a rational bubble or a persistent expectational component.83\end{definition}8485Three objects therefore coexist: the price $P$, the market-consistent value $V^{\ast}=\exp(v^{\ast})$ that an unbiased appraisal targets, and the structural value $V^{F}$ anchored in the service flow. They differ by economically meaningful terms. The price differs from $\tilde v$ by transaction noise $\varepsilon$, which collects bargaining outcomes, seller urgency, time on market and the idiosyncratic match between buyer and dwelling; $\tilde v$ differs from $v^{\ast}$ by the selection term $\kappa$, which is non-zero whenever the decision to sell is correlated with the unobserved component of the price, as the loss-aversion evidence of \citet{genesove2001loss} implies; and $v^{\ast}$ differs from $\log V^{F}$ by expectational and liquidity components that \citet{stein1995prices} shows can be large when down-payment constraints bind. The following lemma states what price-based evaluation can and cannot rank.8687\begin{lemma}[Price-based evaluation ranks estimators against the selected value]88\label{lem:proxy}89For any $\mathcal{I}_t$-measurable estimate $\hat y$,90\begin{equation}91\E\!\left[(\hat y-p)^{2}\mid D=1\right]=\E\!\left[(\hat y-\tilde v)^{2}\mid D=1\right]+\Var(p\mid\mathcal{I}_t,D=1).92\label{eq:proxy}93\end{equation}94The second term is common to every estimator. Hence the ranking of estimators by mean-squared error against realised prices equals their ranking by distance to $\tilde v$, and nothing in \eqref{eq:proxy} identifies the distance to $v^{\ast}$ or to $\log V^{F}$.95\end{lemma}9697\begin{proof}98Expand $(\hat y-p)^{2}=(\hat y-\tilde v)^{2}+(p-\tilde v)^{2}+2(\hat y-\tilde v)(p-\tilde v)$. Conditioning the cross term on $(\mathcal{I}_t,D=1)$, where $\hat y-\tilde v$ is fixed, gives $\E[p-\tilde v\mid\mathcal{I}_t,D=1]=0$ by the definition of $\tilde v$. Taking expectations conditional on $D=1$ yields \eqref{eq:proxy}.99\end{proof}100101Lemma~\ref{lem:proxy} licenses the empirical comparisons of Sections~\ref{sec:results} and \ref{sec:ablations}: differences in price-based error between arms are differences in distance to a well-defined target, the selected market value. It also delimits them. A method can rank first against $\tilde v$ while being uninformative about $V^{F}$, and the triangulation of Section~\ref{subsec:fundamental} is required to say anything about fundamentals. In what follows we write $v$ for the target of the agents and treat it as fixed, so that every error is measured relative to $\tilde v$.102103\subsection{One-shot aggregation of correlated opinions}104\label{subsec:static}105106\begin{assumption}[Linear-Gaussian signals]107\label{ass:lg}108Each agent $k=1,\dots,K$ reports $s_k=v+b_k+\eta_k$, where $b=(b_1,\dots,b_K)^{\prime}$ is a vector of biases and $\eta=(\eta_1,\dots,\eta_K)^{\prime}\sim N(0,\Sigma)$ with $\Sigma$ positive definite. $K$ is finite. Where optimal weights are discussed, $\Sigma$ is treated as known.109\end{assumption}110111A linear pool with weights $w$, $w^{\prime}\one=1$, produces $\hat y(w)=w^{\prime}s$ with error $w^{\prime}b+w^{\prime}\eta$ and112\begin{equation}113\MSE(w)=(w^{\prime}b)^{2}+w^{\prime}\Sigma w.114\label{eq:msew}115\end{equation}116The two pools of empirical interest are the equal-weight mean, $w=\one/K$, and the minimum-variance pool of \citet{bates1969combination}. The exchangeable case, in which every agent is equally accurate and every pair of agents is equally correlated, is the natural model of a committee of interchangeable language models that read the same dossier.117118\begin{proposition}[Correlated opinions]119\label{prop:exchangeable}120Let $b=0$ and $\Sigma=\sigma^{2}[(1-\rho)I+\rho\one\one^{\prime}]$ with $-1/(K-1)\le\rho<1$. Then the equal-weight mean has121\begin{equation}122\MSE_{\mathrm{mean}}(K,\rho)=\sigma^{2}\Big[\frac{1-\rho}{K}+\rho\Big],123\label{eq:msemean}124\end{equation}125equal weights are the optimal linear pool, and for $\rho>0$ the pooled error is bounded below by $\rho\sigma^{2}$ for every $K$. Adding agents removes only the idiosyncratic component $(1-\rho)\sigma^{2}/K$.126\end{proposition}127128Figure~\ref{fig:theory-rho} plots \eqref{eq:msemean} against its Monte Carlo counterpart. The floor $\rho\sigma^{2}$ is the central quantity of the paper: it is the part of the committee's error that reflects shared evidence, a shared model of the market, or a shared provider, and it is invisible in the cross-sectional dispersion of the estimates, which measures only $(1-\rho)\sigma^{2}$. With $\rho=0.6$, the pooled error falls from $0.73\sigma^{2}$ with three agents to $0.63\sigma^{2}$ with twelve, and no committee size brings it below $0.60\sigma^{2}$.129130\begin{figure}[t]131\centering132\includegraphics[width=0.72\textwidth]{\figdir/theory_mse_vs_rho.pdf}133\caption{Mean-squared error of the equal-weight mean of $K$ exchangeable signals as a function of the pairwise error correlation $\rho$ (lines: equation~\eqref{eq:msemean}; dots: Monte Carlo, $20{,}000$ draws; $\sigma^{2}=1$). The dashed line is the floor $\rho\sigma^{2}$, which no committee size removes. Source: \texttt{scripts/41\_theory\_sim.py}.}134\label{fig:theory-rho}135\end{figure}136137\begin{proposition}[General covariance, bias and optimal weights]138\label{prop:general}139Under Assumption~\ref{ass:lg} with $b=0$, the minimum-variance pool is $w^{\ast}=\Sigma^{-1}\one/(\one^{\prime}\Sigma^{-1}\one)$ with140\begin{equation}141\MSE^{\ast}=\frac{1}{\one^{\prime}\Sigma^{-1}\one}\ \le\ \MSE_{\mathrm{eq}}=\frac{\one^{\prime}\Sigma\one}{K^{2}},142\label{eq:optimal}143\end{equation}144with equality if and only if every row of $\Sigma$ has the same sum, that is $\Sigma\one\propto\one$. If $b\neq0$ the minimum of \eqref{eq:msew} is attained at $w\propto(\Sigma+bb^{\prime})^{-1}\one$; when the bias is common to all agents, $b=\beta\one$, it enters every pool as $\beta^{2}$ and no weighting removes it.145\end{proposition}146147The equality condition explains why inverse-variance weights estimated on a validation sample, as in Section~\ref{sec:design}, can be expected to help only when agents differ in accuracy or in correlation structure; with interchangeable agents they add estimation noise without any gain, the forecast-combination puzzle documented by \citet{clemen1989combining} and \citet{timmermann2006forecast}. Trimmed means and medians \citep{jose2014trimmed} trade a small loss under Gaussian errors for protection against the heavy-tailed hallucinations that language-model committees produce; the paper reports them but the theory below is stated for the mean.148149\begin{remark}[Anchoring on a public signal]150Suppose every agent shrinks towards a public value $a=v+\xi$ with weight $\beta$, so that $s_k=(1-\beta)(v+\eta_k)+\beta a$, with $\xi$ independent of $\eta$ and of variance $\sigma_\xi^{2}$. The equal-weight pool has error $(1-\beta)\bar\eta+\beta\xi$ and mean-squared error $f(\beta)=(1-\beta)^{2}\MSE_{\mathrm{eq}}+\beta^{2}\sigma_\xi^{2}$, minimised at $\beta^{\ast}=\MSE_{\mathrm{eq}}/(\MSE_{\mathrm{eq}}+\sigma_\xi^{2})$ and worse than no anchoring if and only if $\beta>2\beta^{\ast}$. Anchoring on the assessment-roll value is therefore not a pathology in itself; it is harmful when the weight placed on a public signal exceeds twice its optimal weight, the over-reaction to public information analysed by \citet{morris2002social}. The same logic applies to the committee median, which becomes a public signal in every deliberation round.151\end{remark}152153The exchangeable case hides where the correlation comes from. A nested structure makes the sources explicit and shows which of them the committee's composition can address.154155\begin{proposition}[Common-model dependence]156\label{prop:nested}157Let $\eta_k=c+g_{m(k)}+e_k$, where $c\sim N(0,\sigma_c^{2})$ is common to all agents, $g_m\sim N(0,\sigma_g^{2})$ is common to the $n_m$ agents served by provider $m\in\{1,\dots,M\}$, $e_k\sim N(0,\sigma_e^{2})$ is idiosyncratic, and all components are independent. Then158\begin{equation}159\MSE_{\mathrm{eq}}=\sigma_c^{2}+\sigma_g^{2}\sum_{m=1}^{M}\Big(\frac{n_m}{K}\Big)^{2}+\frac{\sigma_e^{2}}{K},160\label{eq:nested}161\end{equation}162which for a balanced committee ($n_m=K/M$) equals $\sigma_c^{2}+\sigma_g^{2}/M+\sigma_e^{2}/K$. At fixed $K$ the provider term is minimised by the balanced assignment, and $\sigma_c^{2}$ is unaffected by both $K$ and $M$.163\end{proposition}164165Equation~\eqref{eq:nested} is the formal content of the concern about algorithmic monoculture raised by \citet{kleinberg2021algorithmic} and \citet{bommasani2022picking}: a committee drawn from one provider cannot average away that provider's shared error, and a market in which every lender uses the same valuation model inherits $\sigma_c^{2}+\sigma_g^{2}$ as a common shock. It also motivates the provider-balanced committees of the design, which hold $K$ fixed and vary $M$.166167\subsection{Deliberation}168\label{subsec:deliberation}169170The protocol of Section~\ref{sec:design} has two ingredients. In every round each agent sees the anonymised table of the other estimates and the committee median, and may revise towards them; this is opinion exchange. Each agent also sees the identifiers of evidence cited by others that it had not been shown, within its permitted view; this is evidence exchange. The two have opposite implications for accuracy and identical implications for dispersion, which is why dispersion is not a measure of success. The machine-learning literature on multi-agent debate reports accuracy gains from rounds of mutual critique \citep{du2023improving}; the model specifies the conditions under which such gains can be genuine rather than an artefact of convergence.171172\begin{definition}[Opinion-only revision]173\label{def:opinion}174Let $x^{(0)}=s$ and, for $r\ge0$,175\begin{equation}176x^{(r+1)}=(1-\lambda)\,x^{(r)}+\lambda\,T\!\big(x^{(r)}\big)\one,\qquad \lambda\in(0,1],177\label{eq:revision}178\end{equation}179where the committee statistic $T:\mathbb{R}^{K}\to\mathbb{R}$ is translation- and positively scale-equivariant, $T(ax+b\one)=aT(x)+b$ for $a>0$. The mean, the median and every trimmed mean satisfy this condition.180\end{definition}181182\begin{proposition}[Consensus without information]183\label{prop:consensus}184Under Assumption~\ref{ass:lg} with $b=0$ and the revision \eqref{eq:revision}:185\begin{enumerate}[label=(\roman*)]186\item The committee statistic is invariant, $T(x^{(r)})=T(x^{(0)})$ for all $r$, and187\begin{equation}188x^{(r)}=(1-\lambda)^{r}x^{(0)}+\big[1-(1-\lambda)^{r}\big]\,T(x^{(0)})\,\one .189\label{eq:trajectory}190\end{equation}191Consequently $x^{(r)}_k-x^{(r)}_l=(1-\lambda)^{r}(x^{(0)}_k-x^{(0)}_l)$ for every pair, and every dispersion statistic that is translation invariant and homogeneous of degree one (standard deviation, interquartile range, range) equals $(1-\lambda)^{r}$ times its round-0 value, whatever $\Sigma$ is.192\item If $T$ is the mean, $\bar x^{(r)}=\bar x^{(0)}$ for all $r$: the equal-weight pool and its mean-squared error are unchanged by any number of rounds.193\item If $T$ is the median and $\Sigma$ is exchangeable with $0\le\rho<1$, then $\E[\bar x^{(r)}]=v$, the median-aggregated estimate is invariant, $\med(x^{(r)})=\med(x^{(0)})$, and194\begin{equation}195\MSE\big(\bar x^{(r)}\big)=\MSE\big(\bar x^{(0)}\big)+\big[1-(1-\lambda)^{r}\big]^{2}\Delta_K,\qquad196\Delta_K=\Var\big(\med(e)\big)-\Var(\bar e)\ \ge0,197\label{eq:medianpath}198\end{equation}199where $e$ is a vector of $K$ independent $N(0,(1-\rho)\sigma^{2})$ variables. The pooled error is non-decreasing in $r$ and converges to that of the round-0 median.200\end{enumerate}201\end{proposition}202203Part (i) is the mechanism identified experimentally by \citet{lorenz2011how}: social influence narrows the range of estimates at a geometric rate while leaving the location of the group estimate where it was. Part (ii) says that when agents anchor on the mean, the pooled estimate is literally the round-0 mean. Part (iii) says that when they anchor on the median, as in our protocol, the pooled mean drifts towards the round-0 median and, under Gaussian errors, loses accuracy by at most $\Delta_K$, which is modest (for $K=12$, $\Delta_K\approx0.41\,(1-\rho)\sigma^{2}/K$, the excess variance of the median over the mean) but of the wrong sign. In no case does opinion exchange reduce the pooled error. The dispersion-based stopping rule of the protocol and any confidence measure derived from the agreement of the agents are therefore uninformative about accuracy; they measure $\lambda$ and $r$.204205The revision \eqref{eq:revision} with the mean is a special case of the linear model of \citet{degroot1974reaching}, $x^{(r)}=W^{r}x^{(0)}$ with $W$ row-stochastic. Allowing heterogeneous weights and incomplete attention changes the limit but not the conclusion.206207\begin{proposition}[Linear deliberation conserves at most the round-0 information]208\label{prop:degroot}209Let $x^{(r)}=W^{r}x^{(0)}$ with $W$ row-stochastic and $b=0$.210\begin{enumerate}[label=(\roman*)]211\item For every $r$ and every pool $w$ with $w^{\prime}\one=1$ there is a round-0 pool $u=(W^{r})^{\prime}w$ with $u^{\prime}\one=1$ and $w^{\prime}x^{(r)}=u^{\prime}x^{(0)}$. Hence $\min_{w}\MSE(w^{\prime}x^{(r)})\ge\MSE^{\ast}$ for all $r$, with equality whenever $W^{r}$ is non-singular.212\item If $W$ is primitive, $x^{(r)}\to(\pi^{\prime}x^{(0)})\one$ where $\pi$ is the left Perron vector of $W$, and the limiting mean-squared error of every pool is $\pi^{\prime}\Sigma\pi\ge\MSE^{\ast}$, with equality if and only if $\pi=w^{\ast}$.213\item For the anonymised-table protocol with a common weight $\lambda$ and the mean as anchor, $W=(1-\lambda)I+\lambda\one\one^{\prime}/K$, $\pi=\one/K$, and the consensus is the round-0 equal-weight mean.214\end{enumerate}215\end{proposition}216217Part (ii) is the wisdom-of-crowds condition of \citet{golub2010naive}: consensus is accurate only if influence is spread in proportion to precision, a knife-edge that an anonymised table does not implement. The experimental evidence of \citet{becker2017network} that decentralised networks can improve group estimates is consistent with the proposition: improvement there comes from a $\pi$ that happens to weight accurate individuals more, not from communication as such.218219Evidence exchange is different in kind because it changes the error of each agent rather than mixing errors across agents. We model it as an additive revision and then specialise to a model in which the revision is generated by evidence.220221\begin{lemma}[Additive revisions]222\label{lem:additive}223Let the estimates $x$ with error $\eta$ be revised to $x+\zeta$, where $\zeta$ has mean zero and covariance $\Gamma$. For any pool $w$,224\begin{equation}225\MSE\big(w^{\prime}(x+\zeta)\big)-\MSE(w^{\prime}x)=w^{\prime}\Gamma w+2\,w^{\prime}\Cov(\zeta,\eta)\,w .226\label{eq:additive}227\end{equation}228If $\zeta$ is uncorrelated with $\eta$, every pool is weakly worse after the revision, strictly if $\Gamma$ is positive definite. A revision lowers the pooled error only if it is negatively correlated with the existing errors, which is what evidence does and what persuasion by narrative does not.229\end{lemma}230231\begin{assumption}[Evidence model]232\label{ass:evidence}233There are $J$ evidence items (comparable sales, roll data, market indices), $z_j=v+c+u_j$, with $c\sim N(0,\sigma_c^{2})$ common to all items and $u_j\sim N(0,\sigma_u^{2})$ independent across items. Agent $k$ is shown the view $A_k\subseteq\{1,\dots,J\}$ and reports234\begin{equation}235x^{(0)}_k=v+a_k^{\prime}(c\one+u)+g_{m(k)}+e_k,\qquad a_{kj}=\frac{\mathbf{1}\{j\in A_k\}}{|A_k|},236\label{eq:evidencesignal}237\end{equation}238where the uniform within-view weights are the minimum-variance weights given the view, $g_m$ and $e_k$ are the provider-common and idiosyncratic reasoning errors of Proposition~\ref{prop:nested} with $\Var(e_k)=\tau_k^{2}$, and all components are independent. An evidence exchange replaces the views by $A^{\prime}_k\supseteq A_k$ and the weights accordingly, leaving $c$, $u$, $g$ and $e$ unchanged. Let $\omega=K^{-1}\sum_k a_k$ and $\omega^{\prime}$ denote the implied evidence weights before and after the exchange.239\end{assumption}240241\begin{proposition}[Evidence exchange]242\label{prop:exchange}243Under Assumption~\ref{ass:evidence} the equal-weight pool has244\begin{equation}245\MSE_{\mathrm{eq}}=\sigma_c^{2}+\sigma_u^{2}\,\|\omega\|^{2}+\sigma_g^{2}\sum_{m}\Big(\frac{n_m}{K}\Big)^{2}+\frac{1}{K^{2}}\sum_{k}\tau_k^{2},246\label{eq:evidencemse}247\end{equation}248and an evidence exchange changes it by exactly $\sigma_u^{2}(\|\omega^{\prime}\|^{2}-\|\omega\|^{2})$. In particular:249\begin{enumerate}[label=(\roman*)]250\item exchange of items in the common view $\bigcap_k A_k$, and more generally any exchange that leaves $\omega$ unchanged, has no effect; the common evidence error $\sigma_c^{2}$ and the reasoning terms are never affected;251\item $\|\omega\|^{2}\ge1/|\bigcup_k A_k|$ with equality if and only if $\omega$ is uniform on the union of the views; the full exchange $A^{\prime}_k=\bigcup_l A_l$ attains this minimum, so that the committee's evidence error equals that of a single agent shown the union;252\item a partial exchange can raise the pooled error: with $J=2$, $K=3$, $A_1=\{1\}$, $A_2=A_3=\{2\}$, giving item 2 to agent 1 moves $\sigma_u^{-2}\MSE_{\mathrm{eq}}$ from $5/9$ to $13/18$.253\end{enumerate}254\end{proposition}255256The proposition identifies precisely when deliberation can add information. Exchange helps when it moves weight towards evidence that the committee under-uses, which happens when views are specialised and their union is rich; it is neutral when the cited items are already in everyone's dossier, which is the case when all roles work from the same comparable pool; and it hurts when it concentrates attention on items that are already popular, the herding on shared comparables that an information cascade produces \citep{bikhchandani1992theory}. Part (ii) also bounds what communication can achieve: after a full exchange the committee's evidence error equals that of one agent with the union view, so any residual advantage over the evidence-matched single agent of arm~E must come from the averaging of reasoning errors, $\sigma_g^{2}\sum_m(n_m/K)^{2}+K^{-2}\sum_k\tau_k^{2}$ against $\sigma_g^{2}+\tau^{2}$, which arm~F obtains without any communication. Figure~\ref{fig:theory-rounds} illustrates Propositions~\ref{prop:consensus} and \ref{prop:exchange}: every variant contracts dispersion at the same rate, only the exchange variant lowers the pooled error, and a persuasion term that is independent of the errors raises it, as Lemma~\ref{lem:additive} requires.257258\begin{figure}[t]259\centering260\includegraphics[width=\textwidth]{\figdir/theory_deliberation_rounds.pdf}261\caption{Four rounds of deliberation with $\lambda=0.5$ for $K=12$ agents in the evidence model (four comparables shared by all agents and two private ones per agent, $J=28$, $\sigma_c=0.05$, $\sigma_u=0.12$, $\sigma_g=0.04$, $\tau=0.10$; $40{,}000$ draws). Left: cross-sectional standard deviation of the estimates, which contracts by $(1-\lambda)$ per round under every mechanism (dotted line). Right: mean-squared error of the pooled mean; opinion-only revision leaves it unchanged (mean anchor) or raises it slightly (median anchor), evidence exchange at round~1 lowers it, and a zero-mean persuasion term raises it. Source: \texttt{scripts/41\_theory\_sim.py}.}262\label{fig:theory-rounds}263\end{figure}264265Deliberation also damages the aggregator's ability to exploit heterogeneity among agents, because mixing makes the agents' errors more alike and hides the precision differences that inverse-variance weights are meant to exploit.266267\begin{proposition}[Over-shrinkage of the variance-weighted pool]268\label{prop:overshrink}269Let round-0 errors be independent with variances $\sigma_1^{2},\dots,\sigma_K^{2}$ not all equal, and let agents revise towards the mean with a common $\lambda\in(0,1)$. Let the aggregator apply inverse-variance weights $w_k\propto1/\Var(x^{(r)}_k)$ to the round-$r$ estimates. At round 0 this pool is optimal, with $\MSE^{\ast}=(\sum_k\sigma_k^{-2})^{-1}$. For every $r\ge1$ it is strictly dominated by the round-0 pool, and as $r\to\infty$ its error converges to $\MSE_{\mathrm{eq}}=K^{-2}\sum_k\sigma_k^{2}>\MSE^{\ast}$.270\end{proposition}271272The practical implication is that aggregation weights estimated on validation data should be applied to round-0 estimates, or at least that the comparison between round-0 and round-$r$ pools must hold the weighting rule fixed; otherwise part of what looks like a deliberation effect is a change in the effective weights.273274\subsection{Placebo roles and what the role contrast identifies}275\label{subsec:placebo}276277Arm~I keeps the mechanics of arm~G and permutes the role prompts across agents. What that contrast identifies depends on whether the evidence view travels with the prompt.278279\begin{proposition}[Placebo roles]280\label{prop:placebo}281Let the committee be as in Assumption~\ref{ass:evidence} and let the role prompt $\pi_k$ affect the signal of slot $k$ only through the reasoning-error variance, $\tau_k^{2}=\tau^{2}(\pi_k)$ (additive separability of prompt and view). Let $\sigma$ be a permutation of $\{1,\dots,K\}$.282\begin{enumerate}[label=(\roman*)]283\item If prompts and views are permuted together, so that slot $k$ receives $(\pi_{\sigma(k)},A_{\sigma(k)})$, the vector of signals is a relabelling of the original and every permutation-invariant aggregator (mean, median, trimmed mean) has exactly the same error distribution as in arm~G.284\item If prompts alone are permuted and views stay with the slots, the equal-weight pool has the same mean-squared error as in arm~G for every $\sigma$, because $\sum_k\tau^{2}(\pi_{\sigma(k)})=\sum_k\tau^{2}(\pi_k)$ in \eqref{eq:evidencemse}.285\item If the reasoning-error variance depends on the match between prompt and view, $\tau_k^{2}=\tau^{2}(\pi_k,A_k)$, the expected difference between arms~I and G is $K^{-2}\sum_k[\tau^{2}(\pi_{\sigma(k)},A_k)-\tau^{2}(\pi_k,A_k)]$, which is non-zero only through this complementarity.286\end{enumerate}287\end{proposition}288289Under (i) the placebo is vacuous: arm~I is arm~G with relabelled agents and the contrast can only measure run-to-run noise, which is useful as a check but tests no hypothesis. Under (ii) and (iii) the contrast does not measure the value of role labels, nor the value of the evidence partition, which is held fixed; it measures the complementarity between a prompt and the evidence it is paired with. The value of the partition itself is identified by comparing committees that differ in their views with prompts held fixed, and the proposition shows that under separability a committee of role-prompted agents and a committee of generically prompted agents differ only through the average reasoning variance $K^{-1}\sum_k\tau^{2}(\pi_k)$, which is a prompt-quality effect and not a diversity effect in the sense of \citet{hong2004groups}. The preregistration therefore specifies arm~I as case (ii), views fixed to the slot and prompts permuted, and the paper reports the view-homogeneous committee as a secondary contrast.290291\subsection{Testable implications and the preregistered hypotheses}292\label{subsec:implications}293294The propositions translate into predictions that are matched one-to-one with the preregistered hypotheses of Section~\ref{sec:design}; Table~\ref{tab:hypmap} summarises the mapping.295296\begin{table}[t]297\centering\small298\caption{Mapping between the theory and the preregistered hypotheses.}299\label{tab:hypmap}300\begin{tabular}{@{}llp{8.6cm}@{}}301\toprule302Hypothesis & Contrast & Theoretical content \\303\midrule304H1 & G2 vs D & No prediction on the level; Propositions~\ref{prop:exchangeable} and \ref{prop:nested} imply that the committee cannot beat an AVM whose error lies below the floor $\sigma_c^{2}+\sigma_g^{2}/M$. \\305H2 & G2 vs F & Propositions~\ref{prop:consensus}--\ref{prop:exchange}: a gain requires evidence exchange with $\|\omega^{\prime}\|^{2}<\|\omega\|^{2}$; opinion exchange alone predicts no gain and a small loss with median anchoring. \\306H3 & G2 vs J & Equation~\eqref{eq:evidencemse}: the committee beats the compute-matched single agent only through provider diversification $\sigma_g^{2}(1-\sum_m(n_m/K)^{2})$ and the exchange term, net of the evidence-concentration loss $\sigma_u^{2}(\|\omega\|^{2}-1/J)$. \\307H4 & G2 vs I & Proposition~\ref{prop:placebo}: zero under separability; a non-zero contrast identifies prompt--evidence complementarity. \\308H5 & coverage & Proposition~\ref{prop:consensus}(i): agreement-based intervals contract by $(1-\lambda)^{r}$ while the pooled error does not; coverage falls with rounds unless agents' own quantiles are insensitive to the table; linear pools of quantiles require recalibration in any case \citep{ranjan2010combining}. \\309H6 & faithfulness & Lemma~\ref{lem:additive}: cited items that are not in the packet are persuasion, not evidence, and raise the pooled error; H6 is the precondition for the mechanism behind H2. \\310\bottomrule311\end{tabular}312\end{table}313314Under either mechanism the dispersion of the estimates falls with rounds at the rate $(1-\lambda)$, so a finding that the committee converges, that its interquartile range narrows, or that the agents report higher confidence after deliberation is not evidence for H2; this is the content of Proposition~\ref{prop:consensus}(i) and the reason the protocol records but never selects on convergence. The pooled error falls with rounds only through evidence exchange, and Proposition~\ref{prop:exchange} predicts that the gain is largest when roles have specialised views with a rich union and smallest when every role works from the same comparable pool; the paper tests this by measuring, for each property, the share of cited identifiers that lie outside the common view, and by interacting it with the G2--F contrast. The compute channel is isolated by arm~J: a single generalist sampled repeatedly draws its $e_k$ independently but shares one provider and one view, so that \eqref{eq:evidencemse} assigns it $\sigma_c^{2}+\sigma_u^{2}/J+\sigma_g^{2}+\tau^{2}/K$, and the G2--J contrast is predicted to be positive if and only if provider diversification and exchange outweigh the partition's concentration of evidence weight. The communication channel is isolated by arm~F, whose estimates are the round-0 vector of arm~G; by Proposition~\ref{prop:degroot}(i), no pool of round-$r$ estimates can beat the best pool of round-0 estimates unless evidence is exchanged, so a G2--F gain is attributable to the exchange term and to nothing else in the model. Finally, Proposition~\ref{prop:overshrink} prescribes that the stacking and inverse-variance weights be estimated and applied at the same round, and that the primary comparison use a weighting rule that is invariant to deliberation, the mean or the median.315316\subsection{Identifying fundamental value from repeat sales, rents and user cost}317\label{subsec:fundamental}318319Lemma~\ref{lem:proxy} shows that price-based accuracy is silent about $V^{F}$. Three proxies allow partial triangulation, each with its own selection problem. First, a subsequent resale of the same property at $t_2>t_1$ identifies the change in value net of quality, $p_{t_2}-p_{t_1}=(v^{\ast}_{t_2}-v^{\ast}_{t_1})+(\varepsilon_{t_2}-\varepsilon_{t_1})$, once the intervening market movement is removed by a repeat-sales index in the manner of \citet{case1989efficiency}; an appraisal at $t_1$ that is closer to the market-adjusted resale than the $t_1$ price was is evidence that it captured a component of $v^{\ast}$ that the first transaction's noise obscured. The selection problem is that properties that resell quickly are not a random sample: flips, forced sales, renovations and the loss-averse listing behaviour documented by \citet{genesove2001loss} all correlate the resale decision with $\varepsilon_{t_1}$, so that the resale-based error is informative only after conditioning on holding period and on observable changes to the property. Second, where a market rent $R_t$ exists for a comparable unit, \eqref{eq:usercost} delivers $V^{F}_t=R_t/u_t$ up to the user cost. The selection problems are that the rental stock differs from the owner-occupied stock in quality and location, that Quebec's rent regulation detaches contract rents from market rents for sitting tenants, and above all that $u_t$ contains the unobservable expected appreciation $\pi^{e}_t$, so that the price-to-rent diagnostic of \citet{himmelberg2005assessing} identifies $V^{F}$ only under a maintained assumption about expectations; we report it as a bound under a range of $\pi^{e}_t$ rather than as a point value. Third, independent professional appraisals, where legally accessible, are $\mathcal{I}_t$-measurable estimates like any other arm and are subject to the same lemma; their known tendency to confirm the contract price when one exists makes them a proxy for $\tilde v$ at best. A method that forecasts transaction prices well may therefore fail every one of these checks, and the converse is also possible; the paper keeps the two conclusions separate.320