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Anchored Sequential Deliberation

Sijing Tu, Ashish Goel

Year: 2026Venue: arXiv preprintArea: cs.GTType: PreprintEmbeddings: 124

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%

Last extracted: 9/16/2026, 7:30:50 AM

Summary

The paper introduces 'Anchored Sequential Deliberation,' a model for collective decision-making that extends existing sequential deliberation frameworks by incorporating an anchoring bias. In this model, participants' expressed stances shift toward the previous outcome (status quo) with a strength parameter λ before Nash bargaining. The authors analyze the trade-off between convergence speed and stability, proving that stronger anchoring slows the mixing time (convergence) but reduces stationary social cost (distortion) and concentrates the outcome around a unique deliberative fixed point. Theoretical bounds and simulations for uniform and Beta distributions support these findings.

Entities (11)

Anchored Sequential Deliberation · model · 98%Anchoring Bias · concept · 95%Lambda · parameter · 95%Nash Bargaining · mechanism · 95%Deliberative Fixed Point · concept · 90%Distortion · metric · 90%Fain et al. · researcher · 90%Social Cost · metric · 90%Beta Population · distribution · 85%Uniform Population · distribution · 85%Wasserstein Contraction · mathematical-property · 85%

Relation Signals (7)

Anchoring Strength → decreases → Stationary Social Cost

confidence 95% · stationary social cost weakly decreases with λ

Anchored Sequential Deliberation → extends → Sequential Deliberation

confidence 95% · We introduce anchored sequential deliberation... Existing theory by Fain et al. treats the current outcome solely as the disagreement alternative... We add an individual-behavior perspective... anchoring bias

Anchoring Strength → slows → Mixing Time

confidence 95% · stronger anchoring slows mixing... tmix(W1)(ε) increases as λ increases

Anchored Sequential Deliberation → appliesto → One-dimensional Decision Space

confidence 90% · We introduce anchored sequential deliberation on a one-dimensional decision space.

Anchoring Strength → concentrates → Deliberative Fixed Point

confidence 90% · stationary distribution concentrates around it [the fixed point] as λ → 1

Nash Bargaining → determines → Deliberative Outcome

confidence 90% · then Nash-bargain using O_{t-1} as the disagreement alternative... O_t = Median(U~, V~, O_{t-1})

Anchoring Bias → influences → Expressed Stances

confidence 90% · participants... shift their positions toward the previous outcome... with anchoring strength λ

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Abstract

Abstract:Sequential deliberation is a mechanism for collective decision making: at each round, a uniformly randomly selected pair is asked to revise a collective outcome, which then becomes the reference point for the next round. Existing theory by Fain et al.~\cite{fain2017sequential} treats the current outcome solely as the disagreement alternative in bargaining. Yet an existing draft, policy, or proposal might carry social influence and anchor participants' expressed positions toward the status quo. We introduce anchored sequential deliberation on a one-dimensional decision space. In each round, two participants with bliss points $U$ and $V$ shift their positions toward the previous outcome $O_{t-1}$ with anchoring strength $\lambda$, then Nash-bargain using $O_{t-1}$ as the disagreement alternative. The update simplifies to $O_t=(1-\lambda)\mathsf{Median}\{U,V,O_{t-1}\}+\lambda O_{t-1}$. We establish a convergence--stability trade-off. For every population distribution and $\lambda<1$, the process has a unique stationary distribution. A monotone coupling yields a $1$-Wasserstein contraction factor of at most $\frac{1+\lambda}{2}$ and at least $\lambda$; thus, stronger anchoring slows mixing. On the other hand, stationary social cost weakly decreases with $\lambda$, although the worst-case distortion remains $\frac{1+\sqrt{2}}{2}$. We also identify a unique \emph{deliberative fixed point}, where the expected unanchored movement is zero, and prove that the stationary distribution concentrates around it as $\lambda \to 1$. For the uniform population, stationary distortion lies between $1+\frac{1-\lambda}{9+7\lambda}$ and $1+\frac{1-\lambda}{6(1+\lambda)}$, with both bounds approaching $1$ as $\lambda\to1$. Simulations for uniform and Beta populations show that stronger anchoring slows mixing, concentrates the stationary distribution, and lowers stationary distortion in these instances.

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Anchored Sequential Deliberation Sijing Tu Note: sijingtu@stanford.edu Affiliation: Management Science and Engineering, Stanford University Ashish Goel Note: ashishg@stanford.edu Affiliation: Management Science and Engineering, Stanford University Abstract Sequential deliberation is a mechanism for collective decision making: at each round, a uniformly randomly selected pair is asked to revise a collective outcome, which then becomes the reference point for the next round. Existing theory by Fain et al. [10] treats the current outcome solely as the disagreement alternative in bargaining. Yet an existing draft, policy, or proposal might carry social influence and anchor participants’ expressed positions toward the status quo. We introduce anchored sequential deliberation on a one-dimensional decision space. In each round, two participants with bliss points U and V shift their positions toward the previous outcome Ot−1O_t-1 with anchoring strength λ, then Nash-bargain using Ot−1O_t-1 as the disagreement alternative. The update simplifies to Ot=(1−λ)Median⁡U,V,Ot−1+λOt−1O_t=(1-λ) \U,V,O_t-1\+λ O_t-1. We establish a convergence–stability trade-off. For every population distribution and λ<1λ<1, the process has a unique stationary distribution. A monotone coupling yields a 11-Wasserstein contraction factor of at most 1+λ2 1+λ2 and at least λ; thus, stronger anchoring slows mixing. On the other hand, stationary social cost weakly decreases with λ, although the worst-case distortion remains 1+22 1+ 22. We also identify a unique deliberative fixed point, where the expected unanchored movement is zero, and prove that the stationary distribution concentrates around it as λ→1λ→ 1. For the uniform population, stationary distortion lies between 1+1−λ9+7λ1+ 1-λ9+7λ and 1+1−λ6(1+λ)1+ 1-λ6(1+λ), with both bounds approaching 11 as λ→1λ→ 1. Simulations for uniform and Beta populations show that stronger anchoring slows mixing, concentrates the stationary distribution, and lowers stationary distortion in these instances. 1 Introduction Public collective decision making has recently attracted growing attention in the computer science literature. A significant line of work models collective choice as a social choice problem, where participants’ preferences are aggregated through a voting rule [6, 1, 15, 21, 22, 27]. In many public decision-making processes, however, voting is only the final step and should not replace deliberation [2]. Deliberation helps participants become better informed about the issues and can improve the quality of collective decisions. This importance has motivated practical platforms for public deliberation, including the Online Deliberation Platform [30], Polis [31], and Decidim [8]. A practical mechanism for deliberation should be simple to implement, decentralized, scalable to large populations, and should not require participants to reason over the entire decision space [10]. Motivated by these principles, recent work has proposed mechanisms based on small-group deliberation [17, 10, 16]. These mechanisms are evaluated using distortion, defined as the ratio between the social cost of the chosen outcome and the optimal social cost [32]. This line of research shows that integrating deliberation into a voting rule can produce higher-quality outcomes with lower distortion. In this paper, we build on the sequential deliberation framework of Fain et al. [10]: in each round, a pair of participants is drawn from the population to bargain over the deliberative outcome from the previous round, which serves as the disagreement alternative. We add an individual-behavior perspective that is natural in deliberation but absent from the theoretical model. When participants deliberate over an existing policy, draft proposal, or budgeting arrangement, they might not interpret the current outcome only as a disagreement alternative. It can also act as a source of social influence that shapes what they are willing to propose. Changing an existing outcome can require effort, justification, and social risk. We model this tendency to stay close to the current outcome as an anchoring bias [37]. We use a research lab writing a shared lab rule as an illustrative example. Example 1.1. A research lab wants to write a shared lab rule. The professor writes the first draft. In each round, two randomly selected students revise the current draft by mutual agreement. Each student may have an ideal version of the rule: some prefer more supervision, while others prefer working independently. Yet large edits can be effortful or confrontational, especially when the current text already looks legitimate. As a result, the pair might propose an incremental revision that reflects their most important shared concerns, rather than an outcome obtained by bargaining directly from their ideal positions. Thus, a single round of deliberation has two components: anchoring pulls participants’ expressed stances toward the previous outcome, and bargaining turns those stances into the next outcome. We now introduce a model of anchored sequential deliberation that incorporates both components. 1.1 Our model: anchored sequential deliberation We formulate anchored sequential deliberation within the framework of Fain et al. [10]; the process is shown in Process 1. In each round, participants first observe the outcome from the previous round and shift their stances toward it, as in Equation 1; they then bargain using that previous outcome as the disagreement alternative, as in Equation 2. Process 1 Sequential Deliberation Process 1: Participants provide their preferred outcomes; an arbitrary outcome is selected as O0O_0. 2: for t=1,2,…,Tt=1,2,…,T do 3: Select uniformly at random a pair of participants u and v with bliss points U and V, respectively. 4: Provide u and v with the current deliberation outcome Ot−1O_t-1. 5: The participants form anchored stances: U~=Anchorλ⁡(U,Ot−1)V~=Anchorλ⁡(V,Ot−1) U= _λ(U,O_t-1) V= _λ(V,O_t-1) (1) 6: Participants u and v bargain using Ot−1O_t-1 as the disagreement alternative. Ot=Bargain⁡(U~,V~,Ot−1).O_t= ( U, V,O_t-1). (2) 7: return OTO_T The function Anchorλ⁡(⋅) _λ(·) takes a participant’s bliss point and the previous outcome Ot−1O_t-1 as inputs and returns the participant’s anchored stance. The function Bargain⁡(⋅) (·) then takes the anchored stances and the previous deliberation outcome and returns the deliberative outcome for the current round. 1.2 Analytical assumptions For our analysis, the deliberation space is the interval [0,1][0,1]. Each agent has a bliss point, representing the agent’s ideal outcome, drawn from a cumulative distribution function G on [0,1][0,1]. The anchoring parameter λ. Let λ∈[0,1)λ∈[0,1) measure the strength of anchoring: the extent to which agents move their expressed stances toward the previous deliberative outcome before bargaining. Larger λ means stronger anchoring, so expressed stances lie closer to the current outcome. We use the following simple linear update rule: U~=(1−λ)U+λOt−1,V~=(1−λ)V+λOt−1 U=(1-λ)U+λ O_t-1, V=(1-λ)V+λ O_t-1 (3) This linear update is motivated by the empirical study of Lieder et al. [24], which finds that anchoring bias grows linearly with the distance between the anchor and the correct value. It is also consistent with social-influence models in which people revise their opinions by weighting and combining their own positions with others’ positions, including the DeGroot model [9] and the Friedkin–Johnsen model [12]. Nash bargaining. For tractability, we assume that the two selected individuals use Nash bargaining. Given the disagreement alternative Ot−1O_t-1 and the anchored stances U~ U and V~ V, their utility gains from a proposed outcome OtO_t are d⁡(Ot−1,U~)−d⁡(Ot,U~)d(O_t-1, U)-d(O_t, U) and d⁡(Ot−1,V~)−d⁡(Ot,V~)d(O_t-1, V)-d(O_t, V), respectively. The Nash bargaining solution maximizes the Nash product: arg⁡maxOt∈[0,1]⁡(d⁡(Ot−1,U~)−d⁡(Ot,U~))(d⁡(Ot−1,V~)−d⁡(Ot,V~)), _O_t∈[0,1] (d(O_t-1, U)-d(O_t, U) ) (d(O_t-1, V)-d(O_t, V) ), where both utility gains are non-negative. By Fain et al. [10, Lemma 1], the line is a special case of a median graph, and the Nash bargaining solution is the median of the two anchored stances and the disagreement alternative. We therefore write the bargaining outcome as Ot=Med⁡(U~,V~,Ot−1),O_t= ( U, V,O_t-1), (4) where the initial outcome O0∼GO_0 G is drawn from the population distribution. Social cost and distortion. For an outcome O, the social cost is the expected distance from the agents’ bliss points to O: ⁡(O)≔⁡[∫01|O−x|G(x)] SC(O) [ _0^1|O-x|dG(x)]. Let O∗O^* minimize social cost, so ⁡(O∗)=minO⁡(O) SC(O^*)= _O SC(O). The distortion of O is ⁡(O)≔⁡[⁡(O)]⁡(O∗) Distortion(O) E[ SC(O)] SC(O^*). 1.3 Our results Does anchoring lead to better deliberative results? Our analysis shows a trade-off between convergence and stability. Stronger anchoring makes each round stay closer to the previous outcome, so the process takes longer to forget its starting point. At the same time, once the process reaches its long-run distribution, stronger anchoring can make the outcome more stable. This stability occurs around a deliberative fixed point, the point where the expected unanchored movement is zero. In fact, the stationary distortion weakly decreases as the anchor effect strengthens; however, in the worst-case, the distortion remains the same as in the unanchored one. We therefore study more restricted settings where the fixed point is socially optimal: for symmetric populations, stronger anchoring reduces stationary variance around the socially optimal median; for the uniform case, this gives explicit distortion bounds. We also use simulations on several Beta distributions to show the same convergence–distortion trade-off beyond the cases where we have closed-form bounds. Convergence. For every population distribution G on [0,1][0,1] and every λ<1λ<1, the anchored process has a unique stationary distribution. Its 11-Wasserstein mixing time satisfies tmix(W1)(ε)≤⌈log⁡(1/ε)log⁡(2/(1+λ))⌉,t_mix^(W_1)( )≤ (1/ ) (2/(1+λ)) , and, when 0<λ<10<λ<1, it also has the lower bound tmix(W1)(ε)≥⌈log⁡(1/(2ε))log⁡(1/λ)⌉t_mix^(W_1)( )≥ (1/(2 )) (1/λ) for 0<ε<1/20< <1/2 (Theorem 2.1). Both bounds increase as λ increases. We also show that the upper-bound rate is worst-case tight up to a factor of 22 by constructing a symmetric two-cluster population at the endpoints (Lemma 2.2). Thus stronger anchoring slows convergence. The proof uses a monotone coupling. First, the one-step update can be written as Ot=(1−λ)Med⁡(U,V,Ot−1)+λOt−1,O_t=(1-λ) (U,V,O_t-1)+λ O_t-1, (5) so the anchored step is a convex combination of the unanchored median step and the previous outcome. We then run two copies of the chain with the same sampled bliss points in every round. If Xt≤YtX_t≤ Y_t, the order is preserved, and the new gap is the old anchored gap plus the part of the interval [Xt,Yt][X_t,Y_t] covered by the two sampled bliss points. Taking expectations gives ⁡[Yt+1−Xt+1]≤1+λ2[Yt−Xt],E[Y_t+1-X_t+1]≤ 1+λ2E[Y_t-X_t], which gives the W1W_1 upper bound and uniqueness of the stationary distribution. For the lower bound, the same coupling from the endpoints keeps the gap at least λtλ^t; Kantorovich–Rubinstein duality (Theorem A.2) and the triangle inequality then give the stated mixing-time lower bound. Stationary distortion bound. We next compare the stationary social cost under different anchoring parameters. For every population distribution G, let 0≤λ2≤λ1<10≤ _2≤ _1<1 and the stationary distribution Xi,∞∼πλi,GX_i,∞ _ _i,G, the social cost of stationary outcome weakly decreases as λ increases, i.e., (Theorem 3.1): X1,∞∼πλ1,G[⁡(X1,∞)]≤X2,∞∼πλ2,G[⁡(X2,∞)].E_X_1,∞ _ _1,G[ SC(X_1,∞)] _X_2,∞ _ _2,G[ SC(X_2,∞)]. We obtain a stationary distortion upper bound immediatly as the unanchored process: (1+2)/2(1+ 2)/2. Actually, this bound is tight: a two-cluster population with suitable masses attains the same distortion for all anchoring parameters (Lemma 3.3). The proof of Theorem 3.1 uses two ingredients: first, we find such convex function φ:[0,1]→ℝ :[0,1] that ⁡[⁡(X1,∞)]−⁡[⁡(X2,∞)]=⁡[φ⁡(X1,∞)]−⁡[φ⁡(M1)],E[ SC(X_1,∞)]-E[ SC(X_2,∞)]=E[ (X_1,∞)]-E[ (M_1)], where U,V∼i.i.d.GU,V .i.d. G and M1=(1−λ2)⋅Med(X1,∞,U,V)+λ2X1,∞M_1=(1- _2)·Med(X_1,∞,U,V)+ _2X_1,∞. By the convexity of φ and the linearity of expectation, we can show that ⁡[φ⁡(X1,∞)]−⁡[φ⁡(M1)]≤0.E[ (X_1,∞)]-E[ (M_1)]≤ 0. Concentration around the deliberative fixed point. Every population distribution has a unique deliberative fixed point θG _G, defined as the point where the expected unanchored movement is zero (Lemma 4.1). As λ increases, the stationary distribution concentrates around this point (Theorem 4.2). In particular, ⁡[(O∞−θG)2]≤1−λ.E[(O_∞- _G)^2]≤ 1-λ. The proof uses the drift function bG(x)=⁡[Med⁡(x,U,V)]−xb_G(x)=E[ (x,U,V)]-x. An integral formula for bGb_G shows that it is strictly decreasing, so it has a unique zero. Substituting Equation (5) into the stationary second moment ⁡[(O∞−θG)2]E[(O_∞- _G)^2] and applying the slope bound on bGb_G gives the bound above. Symmetric distributions. This concentration improves distortion when the deliberative fixed point is also socially optimal. For symmetric population distributions, the deliberative fixed point is the population median, which minimizes social cost on the line. For the uniform distribution, the stationary distortion is at most 1+1−λ6(1+λ)1+ 1-λ6(1+λ) and at least 1+1−λ9+7λ1+ 1-λ9+7λ for every 0≤λ<10≤λ<1; when λ=0λ=0, the exact stationary distortion is π−2π-2 (Theorem 4.4). Both bounds decrease with λ and approach 11 as λ approaches 11. Thus, in the uniform case, stronger anchoring tends to improve the stationary distortion guarantee. More generally, for centered symmetric distributions, [|O∞−12|2]≤2(1−λ)1+λVar(U)E\! [ |O_∞- 12 |^2 ]≤ 2(1-λ)1+λ (U) (Theorem 4.8). The proof reduces distortion to variance around the median. In the uniform case, distortion is exactly 1+4[(O∞−12)2]1+4E[(O_∞- 12)^2], and a one-step second-moment recursion gives the displayed upper and lower bounds. For general centered symmetric distributions, the same recursion is bounded using symmetry and the variance of a fresh bliss point. Simulations. We complement the theoretical analysis with simulations for several Beta distributions in Section 5 and Appendix F. The simulations show the same message: stronger anchoring slows convergence, concentrates the stationary distribution, and lowers stationary distortion in the simulated instances. Overall, anchoring trades convergence speed for stationary stability: it slows mixing, but it can improve long-run outcome quality for natural distributions. 1.4 Further related work Anchoring bias. Anchoring was first studied by Tversky and Kahneman [37]: when people estimate a quantity after being shown an arbitrary value, their judgments remain biased toward that value. The effect is robust across domains, tasks, expertise, motivation, and cognitive load [13], and arises in both individual [33] and cooperative group [39] decision making. Anchors can stem from advice taking [3, 40], social influence [9, 12, 26], or negotiation, where manipulated listing prices cause bias even for expert real-estate agents [29] and first offers may shape final agreements [14]. Most relevant to our model, Lieder et al. [24] empirically show that anchoring bias grows linearly with the distance between the anchor and the correct value, motivating our linear update rule. For deliberation specifically, Hartmann and Rafiee Rad [19] and Braun et al. [5] model deliberation as preference updating and show that outcomes can disproportionately resemble initial speakers’ preferences, exhibiting anchoring as a structural bias that arises even among rational participants. Nash bargaining. Bargaining is a fundamental concept in cooperative game theory. Well-known solutions include the Nash bargaining solution [28], the Kalai–Smorodinsky solution [20], and the egalitarian and utilitarian solutions [25]. Nash bargaining has been widely adopted in theoretical analysis, as it is the only solution satisfying all of the following properties: (1) invariance to the origins and units of the utilities, (2) Pareto optimality, (3) symmetry, and (4) independence of irrelevant alternatives [28]. Sequential deliberation and distortion. Fain et al. [10] introduced sequential deliberation as a simple and decentralized mechanism for social choice on median graphs. Related work studies large-scale deliberative decision making and metric deliberation mechanisms [17, 16]. Our work keeps the pairwise bargaining structure of sequential deliberation but adds anchoring as a behavioral feature of the current outcome. The notion of distortion was introduced by Procaccia and Rosenschein [32] in the voting setting to quantify the loss from choosing based on limited preference information rather than full utilities. Metric distortion asks for similar guarantees when voters and alternatives lie in an unknown metric space [1]. A long line of work has improved the best achievable distortion bounds for deterministic and randomized voting rules [1, 27, 15, 21, 22, 6]. Conceptually close to our framework is the model of reality-aware social choice [34, 35] proposed by Shapiro and Talmon, where both models study how participants’ preferences can be aggregated to update a status quo, but they use different aggregation methods. In particular, our model does not require the system to know all participants’ explicit positions in the space, whereas Shapiro and Talmon’s framework requires the disclosure of each participant’s ideal element and the proposal’s explicit position to obtain a generalized median. 1.5 Notation Let G be a population cumulative distribution on [0,1][0,1], and let U,V∼i.i.d.GU,V .i.d. G. Fix an anchoring parameter λ∈[0,1)λ∈[0,1). Conditioned on the previous outcome Ot−1O_t-1, we define the anchored stances U~≔(1−λ)U+λOt−1 U (1-λ)U+λ O_t-1 and V~≔(1−λ)V+λOt−1 V (1-λ)V+λ O_t-1. When we specify the previous outcome Ot−1=xO_t-1=x, we write the corresponding stances as U~x U_x and V~x V_x. Let Kλ,GK_λ,G denote the one-step transition kernel Kλ,G(x,A)≔Pr⁡((1−λ)Med⁡(U,V,x)+λx∈A)K_λ,G(x,A) ((1-λ) (U,V,x)+λ x∈ A ) for every Borel set A⊆[0,1]A [0,1]. When there is no ambiguity, we write K for Kλ,GK_λ,G and KtK^t for the t-step transition kernel. We use δxK _xK, equivalently K⁡(x,⋅)K(x,·), for the one-step distribution starting from x. Because of space constraints, we omit the definitions and properties of the 11-Wasserstein distance and total-variation distance and the mixing-time definitions and refer to Appendix A; we omit the one-step transition law and refer to Appendix B. All the omitted proofs are presented in the appendix. 2 Contraction analysis Our main result in this section is the 1-Wasserstein mixing-time bounds in Theorem 2.1; we first state the theorem and then present the coupling argument that proves it. We put omitted proofs in Appendix C, where we also present self-contained results for the uniform distribution, including the one-step minorization argument. Theorem 2.1. Let G be any population distribution on [0,1][0,1], and let 0≤λ<10≤λ<1. The chain, i.e., Equation (4), has a unique stationary distribution πλ,G _λ,G, and for every 0<ε<10< <1, tmix(W1)(ε)≤⌈log⁡(1/ε)log⁡(2/(1+λ))⌉.t_mix^(W_1)( )≤ (1/ ) (2/(1+λ)) . If 0<λ<10<λ<1, then for every 0<ε<120< < 12, ⌈log⁡(1/(2ε))log⁡(1/λ)⌉≤tmix(W1)(ε). (1/(2 )) (1/λ) ≤ t_mix^(W_1)( ). Next, we show that there exists a population distribution G such that the W1W_1-mixing-time upper bound is tight up to a factor of 22. The construction is a symmetric two-cluster population at the endpoints. Lemma 2.2. For every 0≤λ<10≤λ<1, there exists a population distribution G such that, for every t≥0t≥ 0, supx∈[0,1]W1(δxKt,πλ,G)≥12(1+λ2)t. _x∈[0,1]W_1( _xK^t, _λ,G)≥ 12 ( 1+λ2 )^t. Consequently, tmix(W1)(ε)≥⌈log⁡(1/(2ε))log⁡(2/(1+λ))⌉t_mix^(W_1)( )≥ (1/(2 )) (2/(1+λ)) for every 0<ε<120< < 12. 0011Case 1: same side outsideZt+1=λzZ_t+1=λ zXtX_tYtY_t0011Case 2: one insideZt+1=(1−λ)(R−Xt)+λzZ_t+1=(1-λ)(R-X_t)+λ zXtX_tYtY_t0011Case 3: both insideZt+1=(1−λ)(R−L)+λzZ_t+1=(1-λ)(R-L)+λ zXtX_tYtY_t0011Case 4: opposite sidesZt+1=zZ_t+1=z XtX_tYtY_tLLRRXt+1X_t+1Yt+1Y_t+1LLRRXt+1=XtX_t+1=X_tYt+1Y_t+1LLRRXt+1X_t+1Yt+1Y_t+1LLRRXt+1=XtX_t+1=X_tYt+1=YtY_t+1=Y_t Figure 1: The four cases in the coupling argument. The black points XtX_t and YtY_t are fixed in each row, while U and V are sampled from the same distribution G; we let L=min⁡U,VL= \U,V\ and R=max⁡U,VR= \U,V\, and demonstrate the four cases with respect to the relative positions of L and R with respect to XtX_t and YtY_t. The diamonds show the corresponding next states Xt+1X_t+1 and Yt+1Y_t+1, we use the red and blue arrows to show the selected medians, and how they are updated and become Xt+1X_t+1 and Yt+1Y_t+1. The λ is set to be 0.40.4 in this figure. The remaining part of this section is devoted to the proof of Theorem 2.1. We start with the coupling argument. The update rule in Equation (4) can be rewritten as a linear combination of the previous outcome and the median of the two bliss points with the previous outcome: Ot O_t =Med⁡(U~,V~,Ot−1) = (\ U, V,O_t-1\) (6) =Med⁡((1−λ)U+λOt−1,(1−λ)V+λOt−1,Ot−1) = (\(1-λ)U+λ O_t-1,\;(1-λ)V+λ O_t-1,\;O_t-1\ ) =(1−λ)⋅Med⁡(U,V,Ot−1)+λ⋅Ot−1. =(1-λ)· (U,V,O_t-1 )+λ· O_t-1. Couple two copies of the Markov chain, (Xt,Yt)(X_t,Y_t), initialized with X0≤Y0X_0≤ Y_0. The extremal initialization X0=0X_0=0 and Y0=1Y_0=1 gives the largest possible initial gap. At each step, update both chains with the same pair of bliss points Ut,Vt∼i.i.d.GU_t,V_t .i.d. G; the order of XtX_t and YtY_t remains. At one step of the coupling, write L=min⁡U,VL= \U,V\ and R=max⁡U,VR= \U,V\. Then Yt+1−Xt+1 Y_t+1-X_t+1 =(1−λ)⋅(Med⁡(U,V,Yt)−Med⁡(U,V,Xt))+λ⋅(Yt−Xt) =(1-λ)·( (U,V,Y_t )- (U,V,X_t ))+λ·(Y_t-X_t) (7) =(1−λ)|[L,R]∩[Xt,Yt]|+λ⁡(Yt−Xt), =(1-λ) |[L,R]∩[X_t,Y_t] |+λ(Y_t-X_t), where |⋅| |· | denotes interval length. Figure 1 illustrates the four cases behind this identity; the uniform case is treated in more detail in Appendix C.5. The distance between the coupled chains never increases, because |[L,R]∩[Xt,Yt]|≤Yt−Xt |[L,R]∩[X_t,Y_t] |≤ Y_t-X_t. This also preserves the order of the coupled chains. These geometric properties make the coupling well suited to a 11-Wasserstein convergence analysis. The crucial step is to express the overlap length |[L,R]∩[Xt,Yt]| |[L,R]∩[X_t,Y_t] | as an integral: |[L,R]∩[Xt,Yt]|=∫XtYtL≤r≤Rr. |[L,R]∩[X_t,Y_t] |= _X_t^Y_t1_L≤ r≤ R\,dr. (8) Taking conditional expectations in Equation (7) therefore reduces the computation of the overlap length to the integral of Pr⁡(L≤r≤R) (L≤ r≤ R) over [Xt,Yt][X_t,Y_t]. Lemma 2.3. Let G be any population distribution on [0,1][0,1]. Let (Xt,Yt)(X_t,Y_t) be the coupling obtained by using the same pair Ut,Vt∼i.i.d.GU_t,V_t .i.d. G in both updates. Let Zt≔|Xt−Yt|Z_t |X_t-Y_t |. If X0≤Y0X_0≤ Y_0, then Xt≤YtX_t≤ Y_t for every t, and [Zt+1∣Xt,Yt]=λZt+(1−λ)∫XtYt2G(r)(1−G(r))dr.E[Z_t+1 X_t,Y_t]=λ Z_t+(1-λ) _X_t^Y_t2G(r) (1-G(r) )\,dr. Consequently, [Zt+1∣Xt,Yt]≤1+λ2ZtE[Z_t+1 X_t,Y_t]≤ 1+λ2Z_t. The [Zt+1∣Xt,Yt]≤1+λ2ZtE[Z_t+1 X_t,Y_t]≤ 1+λ2Z_t follows immediately from 2G(r)(1−G⁡(r))≤122G(r)(1-G(r))≤ 12 for all r∈[0,1]r∈[0,1]. Taking expectations gives ⁡[Zt+1]≤λ[Zt]+1−λ2[Zt]=1+λ2[Zt].E[Z_t+1]≤ [Z_t]+ 1-λ2E[Z_t]= 1+λ2E[Z_t]. This coupling controls the expected distance between two copies of the chain, and hence gives a 11-Wasserstein bound. It does not directly imply a total-variation bound, because the coupled chains become closer but their distances never become zero purely through this specific coupling analysis. Corollary 2.4. For every 0≤λ<10≤λ<1 and every pair of probability measures μ,νμ,ν on [0,1][0,1], W1(μK,νK)≤1+λ2W1(μ,ν).W_1(μ K,ν K)≤ 1+λ2\,W_1(μ,ν). The Markov chain admits a unique stationary distribution πλ,G _λ,G. Iterating this contraction gives the upper bound in Theorem 2.1. The lower bound uses the same monotone coupling from the extremal initial states 00 and 11: the gap satisfies Yt−Xt≥λtY_t-X_t≥λ^t, so the Kantorovich–Rubinstein duality theorem (Theorem A.2) implies W1(δ0Kt,δ1Kt)≥|⁡[Yt]−⁡[Xt]|≥λt.W_1( _0K^t, _1K^t)≥ |E[Y_t]-E[X_t] |≥λ^t. By applying the triangle inequality, we get the lower bound in Theorem 2.1. 3 Stationary distortion bound We now compare the stationary distortions of the anchored processes under different anchoring parameters. For any population distribution, the stationary social cost weakly decreases as the the anchoring parameter λ increases. Nevertheless, the same worst-case distortion bound 1+22 1+ 22 remains tight for every fixed λ<1λ<1. Theorem 3.1. Let G be any population distribution on [0,1][0,1]. For every 0≤λ2≤λ1<10≤ _2≤ _1<1, let the outcome of stationary distribution with anchoring parameter λi _i be Xi,∞∼πλi,GX_i,∞ _ _i,G, it holds that ⁡[⁡(X1,∞)]≤⁡[⁡(X2,∞)]E\! [ SC(X_1,∞) ] \! [ SC(X_2,∞) ]. The next lemma restates the unanchored stationary distortion bound on. It is consistent with Theorem 1 of Fain et al. [10]. Lemma 3.2. Let X0,∞∼π0,GX_0,∞ _0,G be the stationary outcome of the unanchored process. Then ⁡[⁡(X0,∞)]≤1+22mina⁡(a).E[ SC(X_0,∞)]≤ 1+ 22 _a SC(a). This bound is tight in the worst case. In particular, for every fixed anchoring parameter, there are instances whose distortion approaches 1+22 1+ 22. The construction uses a two-point population with mass p at 00 and mass q=1−pq=1-p at 11; when p=22p= 22 and q=1−22q=1- 22, the distortion is exactly 1+22 1+ 22 for every λ<1λ<1. Lemma 3.3. There exists a population distribution G such that for every 0≤λ<10≤λ<1, the distortion is exactly 1+22 1+ 22. We outline the proof of Lemma 3.2 and devote the remaining of the section to the proof of Theorem 3.1. We first apply Lemma 3.4 that rewrites social cost as an integral over thresholds. It uses the identity |O−X|=∫01(X≤r<O+O≤r<X)r|O-X|= _0^1 (1_X≤ r<O+1_O≤ r<X )\,dr and the fact that G⁡(r)≤1−G⁡(r)G(r)≤ 1-G(r) to the left of a population median, while G⁡(r)≥1−G⁡(r)G(r)≥ 1-G(r) to its right. This also identifies population medians as social-cost minimizers. Next, we observe that Lemma 3.5 gives the stationary CDF of the unanchored chain. If Ft(r)=Pr⁡(Xt≤r)F_t(r)= (X_t≤ r), stationarity and the unanchored transition imply Ft(r)=G(r)2+2G(r)(1−G⁡(r))Ft−1(r)F_t(r)=G(r)^2+2G(r)(1-G(r))F_t-1(r). Hereby, Lemma 3.2 is obtained through pluging the stationary CDF from Lemma 3.5 into the threshold representation of social cost in Lemma 3.4 and uses max0≤p≤1/2⁡1−p2+(1−p)2≤1+22. _0≤ p≤ 1/2 1-pp^2+(1-p)^2≤ 1+ 22. This leads to the same upper bound on the distortion as Fain et al. [10]. The role of a threshold r here is analogous to a coordinate cut in the median-graph argument of Fain et al.; the proof bounds the contribution of each threshold cut and then integrates over r. Lemma 3.4. Let G be any population distribution on [0,1][0,1], and let O be any random outcome. Then ⁡[⁡(O)]=∫01(G⁡(r)Pr⁡(O>r)+(1−G⁡(r))Pr⁡(O≤r))r.E[ SC(O)]= _0^1 (G(r) (O>r)+(1-G(r)) (O≤ r) )d\,r. (9) Moreover, for any deterministic a∈[0,1]a∈[0,1], ⁡(a)=∫0aG⁡(r)r+∫a1(1−G⁡(r))r. SC(a)= _0^aG(r)\,dr+ _a^1(1-G(r))\,dr. If mGm_G is any population median, then mina⁡(a)=⁡(mG)=∫01min⁡G⁡(r),1−G⁡(r)r. _a SC(a)= SC(m_G)= _0^1 \G(r),1-G(r)\d\,r. Lemma 3.5. Let G be any cumulative distribution function on [0,1][0,1], and Ut,Vt∼i.i.d.GU_t,V_t .i.d. G. Consider the unanchored chain when λ=0λ=0, Xt+1=Med⁡Xt,Ut,VtX_t+1= \X_t,U_t,V_t\. The stationary CDF is ΠG(z)=G(z)2G(z)2+(1−G⁡(z))2, _G(z)= G(z)^2G(z)^2+ (1-G(z) )^2, for every z∈[0,1]z∈[0,1]. Poof of Theorem 3.1. Proof. Let X1,∞∼πλ1,GX_1,∞ _ _1,G and X2,∞∼πλ2,GX_2,∞ _ _2,G, and let U,V∼i.i.d.GU,V .i.d. G. We define the function φnλ(x)=(1−λ)[∑i=0n−1⁡(Yi)∣Y0=x], _n^λ(x)=(1-λ)E[ _i=0^n-1 SC(Y_i) Y_0=x], where Yii=0∞\Y_i\_i=0^∞ is the outcome sequence with an anchoring parameter λ. Let M1=(1−λ2)⋅Med(X1,∞,U,V)+λ2X1,∞M_1=(1- _2)·Med(X_1,∞,U,V)+ _2X_1,∞. Thus, we can write ⁡[⁡(X1,∞)]−⁡[⁡(X2,∞)]E[ SC(X_1,∞)]-E[ SC(X_2,∞)] as a telescoping identity, i.e., ⁡[⁡(X1,∞)]−⁡[⁡(X2,∞)]=11−λ2limn→∞⁡[φnλ2(X1)]−⁡[φnλ2(M1)].E[ SC(X_1,∞)]-E[ SC(X_2,∞)]= 11- _2 _n→∞\E[ _n _2(X_1)]-E[ _n _2(M_1)]\. This is due to the telescoping property and the stationay distribution of the Markov chain when n→∞n→∞, in particular, limn→∞⁡[φnλ2(X1,∞)]−⁡[φnλ2(M1)] _n→∞\E[ _2_n(X_1,∞)]-E[ _2_n(M_1)]\ =(a)limn→∞(1−λ2)∑i=0n[[(Yi)∣Y0=X1,∞]] (a)= _n→∞\(1- _2) _i=0^nE[E[ SC(Y_i) Y_0=X_1,∞]] (10) −(1−λ2)∑i=1n+1[[(Yi)∣Y0=X1,∞]] -(1- _2) _i=1^n+1E[E[ SC(Y_i) Y_0=X_1,∞]]\ =(b)(1−λ2)(⁡[⁡(X1,∞)]−⁡[⁡(X2,∞)]), (b)=(1- _2)(E[ SC(X_1,∞)]-E[ SC(X_2,∞)]), where (a)(a) holds by the definition of M1M_1 and (b)(b) holds as limn→∞⁡[⁡(Yi)∣Y0=X1,∞]=⁡[⁡(X2,∞)] _n→∞E[ SC(Y_i) Y_0=X_1,∞]=E[ SC(X_2,∞)] for any setting of Y0Y_0. Next, we exhibit the convexity of φnλ(x) _n^λ(x) and that it is bounded. Lemma 3.6 (convexity of φnλ(x) _n^λ(x)). For every n≥0n≥ 0 and λ∈[0,1)λ∈[0,1), the function φnλ _n^λ is convex and 11-Lipschitz on [0,1][0,1]. Proof. We use induction on n. The base function with n=1n=1 is φ1λ(x)=(1−λ)[⁡(Z0)∣Z0=x]=(1−λ)(x). _1^λ(x)=(1-λ)E[ SC(Z_0) Z_0=x]=(1-λ) SC(x). Since ⁡(x) SC(x) is convex and 11-Lipschitz and 0<1−λ≤10<1-λ≤ 1, it follows directly that φ1λ ^λ_1 is convex and (1−λ)(1-λ)-Lipschitz, and therefore 11-Lipschitz. We now prove the induction implication. Suppose that, for some n≥1n≥ 1, φnλ ^λ_n is convex and 11-Lipschitz. Conditioning on the sampled pair U and V gives φn+1λ(x)=(1−λ)(x)+⁡[φnλ(λx+(1−λ)Med⁡(x,U,V))]. ^λ_n+1(x)=(1-λ) SC(x)+E [ ^λ_n\! (λ x+(1-λ) (x,U,V) ) ]. Since U∼GU G and V∼GV G, it holds by the definition of social cost that ⁡(x)=⁡[|x−U|] SC(x)=E[ |x-U |] and that ⁡(x)=⁡[|x−V|] SC(x)=E[ |x-V |]; puting together, we get (1−λ)(x)=⁡[1−λ2(|x−U|+|x−V|)].(1-λ) SC(x)=E\! [ 1-λ2 (|x-U|+|x-V| ) ]. We condition on a fixed pair U,VU,V. Let L=min⁡U,VL= \U,V\ and R=max⁡U,VR= \U,V\. For this pair, we consider BL,R(x)=1−λ2(|x−L|+|x−R|)+φnλ(λx+(1−λ)Med⁡(x,L,R)).B_L,R(x)= 1-λ2 (|x-L|+|x-R| )+ ^λ_n\! (λ x+(1-λ) (x,L,R) ). If x<Lx<L, then Med⁡(x,L,R)=L (x,L,R)=L, and the slope of BL,RB_L,R has the form −(1−λ)+λ(φnλ)′.-(1-λ)+λ( ^λ_n) . If L<x<RL<x<R, then Med⁡(x,L,R)=x (x,L,R)=x, and the slope has the form (φnλ)′.( ^λ_n) . If x>Rx>R, then Med⁡(x,L,R)=R (x,L,R)=R, and the slope has the form (1−λ)+λ(φnλ)′.(1-λ)+λ( ^λ_n) . Since φnλ ^λ_n is convex and 11-Lipschitz, its slopes within these three regimes are non-decreasing and lie in [−1,1][-1,1]. Therefore, all three slopes lie in [−1,1][-1,1]. It remains to check that the slope does not decrease when x crosses L or R. At L, the increase from the left slope to the right slope is not smaller than (1−λ)+(1−λ)(φn,−λ(L))′≥0(1-λ)+(1-λ)( ^λ_n,-(L)) ≥ 0; at R, the increase is not smaller than (1−λ)−(1−λ)(φn,+λ(R))′≥0(1-λ)-(1-λ)( ^λ_n,+(R)) ≥ 0. Both inequalities follow because φnλ ^λ_n is convex and its slopes lie in [−1,1][-1,1]. Here, (φn,−λ)′( ^λ_n,-) means taking the derivative from the left side and (φn,+λ)′( ^λ_n,+) means taking the derivative from the right side. Thus BL,RB_L,R is convex and 11-Lipschitz for every fixed pair L,RL,R. Taking expectation over U,VU,V we can show that φn+1λ ^λ_n+1 is convex and 11-Lipschitz, and we are done. ∎ Using again U,VU,V, we next define X~1=(1−λ1)⋅Med(X1,∞,U,V)+λ1⋅X1,∞ X_1=(1- _1)·Med(X_1,∞,U,V)+ _1· X_1,∞. By stationarity, X~1∼πλ1,G X_1 _ _1,G; hence ⁡[φnλ2(X1,∞)]=⁡[φnλ2(X~1)]E[ _n _2(X_1,∞)]=E[ _n _2( X_1)] Relating X~1 X_1, X1,∞X_1,∞, and M1M_1, we have X~1=(1−1−λ11−λ2)X1,∞+1−λ11−λ2M1, X_1= (1- 1- _11- _2 )X_1,∞+ 1- _11- _2M_1, which implies ⁡[φnλ2(X1,∞)]=⁡[φnλ2((1−1−λ11−λ2)X1,∞+1−λ11−λ2M1)]E[ _n _2(X_1,∞)]=E[ _n _2( (1- 1- _11- _2 )X_1,∞+ 1- _11- _2M_1)]. By the convexity of φnλ2 _n _2 and the linearity of expectation, we obtain ⁡[φnλ2(X1,∞)]−⁡[φnλ2(M1)]≤0.E[ _n _2(X_1,∞)]-E[ _n _2(M_1)]≤ 0. Combining this inequality with the formula above completes the proof. ∎ 4 The outcome distribution We now study how the stationary outcome distribution changes with the anchoring parameter. We show that, when λ approaches 11, the stationary distribution concentrates around a fixed point at which the expected one-step movement is zero. We call this point the deliberative fixed point. In particular, for symmetric population distributions, this fixed point coincides with the population median; in the uniform case this yields an explicit stationary distortion bound. When λ approaches 11. We show that, as λ approaches 11, the stationary distribution concentrates around a fixed point. For a current disagreement alternative x, define the expected unanchored movement bG(x)≔⁡[Med⁡(x,U,V)]−x,b_G(x) [ (x,U,V)]-x, where U,V∼i.i.d.GU,V .i.d. G. The deliberative fixed point θG _G is the unique point satisfying bG(θG)=0b_G( _G)=0. Starting from θG _G, the median step has no expected movement to the left or to the right. Uniqueness follows from two facts: bGb_G is continuous and strictly decreasing, while bG(0)=⁡[Med⁡(0,U,V)]≥0b_G(0)=E[ (0,U,V)]≥ 0 and bG(1)=⁡[Med⁡(1,U,V)]−1≤0b_G(1)=E[ (1,U,V)]-1≤ 0. Lemma 4.1. Let bG(x)≔⁡[Med⁡(x,U,V)]−xb_G(x) [ (x,U,V)]-x, where U,V∼i.i.d.GU,V .i.d. G. Then there exists a unique point θG∈[0,1] _G∈[0,1] such that bG(θG)=0b_G( _G)=0. Equivalently, bG(x)=∫x1(1−G⁡(r))2r−∫0xG(r)2r.b_G(x)= _x^1(1-G(r))^2\,dr- _0^xG(r)^2\,dr. This follows by writing Med⁡(x,U,V)−x=∫x1Med⁡(x,U,V)≥rr−∫0xMed⁡(x,U,V)≤rr (x,U,V)-x= _x^11_ (x,U,V)≥ r\,dr- _0^x1_ (x,U,V)≤ r\,dr and then taking expectations; for a CDF with atoms, the choice of strict or weak threshold inequalities affects only countably many values of r and therefore does not change the integrals. Thus, for 0≤x<y≤10≤ x<y≤ 1, bG(y)−bG(x)=−∫xy(G(r)2+(1−G(r))2)dr≤−12(y−x).b_G(y)-b_G(x)=- _x^y (G(r)^2+(1-G(r))^2 )\,dr≤- 12(y-x). (11) This monotonicity is a crucial component of the concentration result below. The next theorem shows that concentration around the deliberative fixed point holds for every population distribution. Theorem 4.2. Let 0≤λ<10≤λ<1 and O∼πλ,GO _λ,G. Then ⁡[(O−θG)2]≤1−λE[(O- _G)^2]≤ 1-λ. The proof expands (Ot+1−θG)2(O_t+1- _G)^2 as ((Ot−θG)+(1−λ)(Mt−Ot))2((O_t- _G)+(1-λ)(M_t-O_t))^2 using Equation (6), where Mt=Med⁡(Ot,Ut,Vt)M_t= (O_t,U_t,V_t). The cross term is controlled by the drift identity ⁡[Mt−Ot∣Ot]=bG(Ot)E[M_t-O_t O_t]=b_G(O_t). Since bG(θG)=0b_G( _G)=0, Equation (11) implies (x−θG)bG(x)≤−12(x−θG)2(x- _G)b_G(x)≤- 12(x- _G)^2 for every x∈[0,1]x∈[0,1]. The stationary distribution concentrates around the deliberative fixed point as λ increases; however, this does not necessarily translate to smaller distortion, because the deliberative fixed point need not be the population median. This concentration is most useful for reducing distortion when the two points coincide. We therefore turn to symmetric population distributions. We say that G is centered if G⁡(12+s)=1−G⁡(12−s)G\! ( 12+s )=1-G\! ( 12-s ) for 0≤s≤120≤ s≤ 12. Lemma 4.3. If G is centered, then the deliberative fixed point is θG=12 _G= 12. Thus, for centered populations, anchoring concentrates the stationary distribution around the social optimum. In this case, anchoring reduces the stationary second moment around the median, which gives sharper distortion bounds. 4.1 Uniform distribution In the uniform case, define t≔|Ot−12|Q_t |O_t- 12 |, the signed distance from the current outcome to the population median. We show that the stationary distortion follows an upper bound of 1+1−λ6(1+λ)1+ 1-λ6(1+λ) and a lower bound of 1+1−λ9+7λ1+ 1-λ9+7λ. Theorem 4.4. When G is uniform and 0≤λ<10≤λ<1, if O∞∼πλ,GO_∞ _λ,G, then 1+1−λ9+7λ≤⁡[⁡(O∞)]⁡(1/2)≤1+1−λ6(1+λ);1+ 1-λ9+7λ≤ E[ SC(O_∞)] SC(1/2)≤ 1+ 1-λ6(1+λ); when λ=0λ=0, the distortion is exactly π−2π-2. The proof reduces distortion to a second-moment bound. Lemma 4.5 expresses uniform distortion exactly in terms of ⁡[t2]E[Q_t^2]. Lemma 4.6 then computes the one-step conditional second moment, and Corollary 4.7 iterates the resulting upper bound. Lemma 4.5. When G is uniform, the distortion after t steps is ⁡[⁡(Ot)]⁡(1/2)=1+4[t2]. E[ SC(O_t)] SC(1/2)=1+4E[Q_t^2]. For the uniform distribution, we compute ⁡[t+12∣Ot=x]E[Q_t+1^2 O_t=x] by applying the transition density in Equation (20), the uniform special case of Proposition B.3. This gives ⁡[t+12∣Ot=x] [Q_t+1^2 O_t=x] =2x(1−x)(x−12)2 =2x(1-x) (x- 12 )^2 (12) +2(1−λ)2∫λx(z−12)2(z−λx)dz + 2(1-λ)^2 _λ x^x (z- 12 )^2(z-λ x)\,dz +2(1−λ)2∫x1−λ+λx(z−12)2(1−λ+λx−z)dz. + 2(1-λ)^2 _x^1-λ+λ x (z- 12 )^2(1-λ+λ x-z)\,dz. Evaluating the integrals and collecting terms yields the following identity. Lemma 4.6. Let t≔Ot−12Q_t O_t- 12. Then ⁡[t+12∣t]=(1−λ)248+1+λ22t2−(1−λ)(3+λ)3t4.E[Q_t+1^2 _t]= (1-λ)^248+ 1+λ^22Q_t^2- (1-λ)(3+λ)3Q_t^4. (13) Dropping the negative fourth-moment term in Lemma 4.6 gives a linear recursion for the second moment. Iterating that recursion gives the following upper bound. The lower bound follows by observing that t4≤14t2Q_t^4≤ 14Q_t^2. Corollary 4.7. For any t≥1t≥ 1, (1−λ)248+7λ2+2λ+312[t2]≤⁡[t+12]≤(1−λ)248+1+λ22[t2]. (1-λ)^248+ 7λ^2+2λ+312E[Q_t^2] [Q_t+1^2]≤ (1-λ)^248+ 1+λ^22E[Q_t^2]. As t→∞t→∞, 1−λ4(9+7λ)≤⁡[∞2]≤1−λ24(1+λ) 1-λ4(9+7λ) [Q_∞^2]≤ 1-λ24(1+λ). The limiting upper bound 1−λ24(1+λ) 1-λ24(1+λ) decreases monotonically in λ. Together with Lemma 4.5, this proves Theorem 4.4. 4.2 Symmetric distributions We now extend the second-moment analysis to population distributions that are symmetric around 1/21/2. Throughout this subsection, write again t=|Ot−12|Q_t= |O_t- 12 |. Assume further that G is absolutely continuous on (0,1)(0,1) with density g satisfying g⁡(12+s)=g⁡(12−s)g\! ( 12+s )=g\! ( 12-s ) for s∈[0,1/2)s∈[0,1/2). Under these assumptions, the main symmetric-distribution bound is the following. Theorem 4.8. Assume that G is centered and satisfies the absolute-continuity and symmetric-density conditions stated above. Let U∼GU G. For every t≥0t≥ 0, [t2]≤14(1+λ22)t+2(1−λ)1+λVar(U).E[Q_t^2]≤ 14 ( 1+λ^22 )^t+ 2(1-λ)1+λ (U). At stationarity, [∞2]≤2(1−λ)1+λVar(U).E[Q_∞^2]≤ 2(1-λ)1+λ (U). The proof uses the following one-step analogue of Lemma 4.6. Lemma 4.9. Assume that G is centered and satisfies the absolute-continuity and symmetric-density conditions stated above. ⁡[t+12] [Q_t+1^2] ≤(1−λ)24∫01/2(12−u)2G(u)g(u)du+1+λ22[t2]. ≤(1-λ)^24 _0^1/2 ( 12-u )^2G(u)g(u)\,du+ 1+λ^22E[Q_t^2]. (14) The proof follows the same strategy as Lemma 4.6: write ⁡[t+12∣t]E[Q_t+1^2 _t] using the transition density from Proposition B.3, integrate the squared distance to the median, and then use symmetry to simplify the expressions in the computation. The computation itself is more involved because it requires multiple applications of symmetries. Because G⁡(u)≤12G(u)≤ 12 for u≤12u≤ 12 and g is symmetric, 4∫01/2(12−u)2G(u)g(u)u≤2∫01/2(12−u)2g(u)u≤Var(U).4 _0^1/2 ( 12-u )^2G(u)g(u)\,du≤ 2 _0^1/2 ( 12-u )^2g(u)\,du≤ (U). Therefore ⁡[t+12] [Q_t+1^2] ≤(1−λ)2Var(U)+1+λ22[t2], ≤(1-λ)^2 (U)+ 1+λ^22E[Q_t^2], (15) and Theorem 4.8 follows by iterating this recursion. 5 Simulation results We approximate the anchored deliberation Markov chain by discretizing [0,1][0,1] into 50005000 equal bins and constructing the transition matrix from the conditional CDF of the one-step update. For each population distribution and each λ∈0.00,0.01,…,0.99λ∈\0.00,0.01,…,0.99\, we compute a stationary approximation by iterating the transition matrix from the uniform distribution up to the 11-Wasserstein mixing-time upper bound, using tolerance εmix=1/5000 _mix=1/5000. The empirical mixing time is the first step at which the discretized 11-Wasserstein distance to this stationary approximation is at most εmix _mix. Further implementation details and additional Beta instances are reported in Appendix F. The simulations illustrate the same trade-off as the theoretical results. Figure 2 gives a representative example for Beta⁡(0.5,0.7)Beta(0.5,0.7). As λ increases, empirical mixing becomes slower, the stationary median moves toward the deliberative fixed point, and the stationary distribution becomes more concentrated around that point. Figure 3 shows that anchoring reduces stationary distortion: in the uniform case the empirical curve lies between the theoretical bounds, and the same decreasing pattern appears across five Beta population laws. (a) Mixing time (b) Medians and fixed point (c) Stationary density Figure 2: Representative simulation results for Beta⁡(0.5,0.7)Beta(0.5,0.7). Empirical mixing time increases with λ and is plotted on a logarithmic scale against the theoretical bounds. The stationary median moves toward the deliberative fixed point, and stronger anchoring concentrates the stationary distribution. (a) Uniform population (b) Selected Beta populations Figure 3: Stationary distortion under anchored sequential deliberation. In the uniform case, the empirical distortion is compared with the theoretical upper and lower bounds. Across the selected Beta populations, stationary distortion decreases as anchoring becomes stronger. 6 Conclusion This paper studies how anchoring changes sequential deliberation. Anchoring slows convergence but it can improve the long-run quality of the decision. At stationarity, the anchored process has social cost no larger than the unanchored process for every fixed population distribution, and stronger anchoring concentrates outcomes around a unique deliberative fixed point. Several extensions remain open. First, we currently assume that λ is the same constant for all individuals. This assumption may not hold in practice, since anchoring can vary across people; future work could allow different individuals or groups to have different values of λ. Second, the model could be extended from the line to median graphs, as in Fain et al. [10]. The main technical difficulty is defining anchored stances on the median graph: unlike on the line, an interpolation between a bliss point and the current outcome may not be unique. Third, future work could study the use of AI tools. Recent work incorporates AI tools into group decision-making processes [36, 11, 4, 7], and a natural question is whether such tools can be integrated into sequential deliberation to improve deliberative outcomes. Acknowledgments We used ChatGPT with the GPT-5.6 model to assist with simulation code and parts of the theoretical analysis. 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Here C⁡(μ,ν)C(μ,ν) is the set of all joint probability measures on [0,1]×[0,1][0,1]×[0,1] whose marginals are μ and ν. We also use the Kantorovich–Rubinstein duality theorem, adapted from [38, Remark 6.5]. Theorem A.2 (Kantorovich–Rubinstein duality). Let μ,ν∈⁡([0,1])μ,ν ([0,1]). Then the 11-Wasserstein distance satisfies the dual representation W1(μ,ν)=supf∈Lip1([0,1])∫[0,1]f⁡(x)μ(x)−∫[0,1]f⁡(x)ν(x).W_1(μ,ν)= _f _1([0,1]) \ _[0,1]f(x)\,dμ(x)- _[0,1]f(x)\,dν(x) \. Here Lip1([0,1])Lip_1([0,1]) is the set of all Lipschitz functions on [0,1][0,1] with Lipschitz constant 11. Plugging either f⁡(x)=xf(x)=x or f⁡(x)=−xf(x)=-x into Theorem A.2 gives the following corollary. Corollary A.3. For any μ,ν∈⁡([0,1])μ,ν ([0,1]), if X∼μX μ and Y∼νY ν, then W1(μ,ν)≥|⁡[X]−⁡[Y]|.W_1(μ,ν)≥ |E[X]-E[Y] |. The total-variation distance is adapted from [23, Proposition 4.7]. Definition A.4. Let ⁡([0,1])P([0,1]) be the set of probability measures on the interval [0,1][0,1]. The total-variation distance between μ, ν∈⁡([0,1])ν ([0,1]) is ‖μ−ν‖TV \|μ-ν\|_TV ≔supA⊆[0,1]|μ⁡(A)−ν⁡(A)| _A [0,1] |μ(A)-ν(A) | (17) =infPr⁡(X≠Y),where (X,Y) is a coupling of μ and ν. = (X≠ Y),where $(X,Y)$ is a coupling of $μ$ and $ν$. For any stationary distribution πλ,G _λ,G, define the 11-Wasserstein mixing time by tmix(W1)(ε)≔min⁡t:supx∈[0,1]W1(δxKt,πλ,G)≤ε.t_mix^(W_1)( ) \t: _x∈[0,1]W_1( _xK^t, _λ,G)≤ \. The total-variation mixing time is defined as follows. tmix(TV)(ε)≔min⁡t:supx∈[0,1]‖δxKt−πλ,G‖TV≤ε.t_mix^(TV)( ) \t: _x∈[0,1]\| _xK^t- _λ,G\|_TV≤ \. Appendix B One-step transition law We first compute the conditional CDF of an anchored stance. The same formula applies to U~x U_x and V~x V_x. Proposition B.1. For every x,z∈[0,1]x,z∈[0,1], we denote the conditional CDF of U~x U_x (as well as V~x V_x) as Fx(z)F_x(z). Fx(z) F_x(z) ≔Pr⁡(U~x≤z)=0,z<λx,G⁡(z−λx1−λ),λx≤z≤λx+(1−λ),1,z>λx+(1−λ). ( U_x≤ z)= cases0,&z<λ x,\\[5.69054pt] G\! ( z-λ x1-λ ),&λ x≤ z≤λ x+(1-λ),\\[8.53581pt] 1,&z>λ x+(1-λ). cases We next compute the conditional CDF of the updated outcome OtO_t. The median structure gives the two cases. If z<xz<x, then Med⁡U~x,V~x,x≤z \ U_x, V_x,x\≤ z only when both anchored stances are at most z. If z≥xz≥ x, then the complementary event Ot>zO_t>z occurs only when both anchored stances exceed z. The atom at x is the jump of this CDF. Proposition B.2. For every x,z∈[0,1]x,z∈[0,1], Pr⁡(Ot≤z∣Ot−1=x)=Fx(z)2,z<x,1−(1−Fx(z))2,z≥x. (O_t≤ z O_t-1=x)= casesF_x(z)^2,&z<x,\\[5.69054pt] 1- (1-F_x(z) )^2,&z≥ x. cases If G is continuous at x, Pr⁡(Ot=x∣Ot−1=x)=2G(x)(1−G⁡(x)). (O_t=x O_t-1=x)=2G(x) (1-G(x) ). Proposition B.3. Assume that G is absolutely continuous with density g. Conditioned on Ot−1=xO_t-1=x, the transition kernel has an atom of mass 2G(x)(1−G⁡(x))2G(x)(1-G(x)) at x. Away from this atom, its density is kx(z)=21−λFx(z)g(z−λx1−λ),λx<z<x,21−λ(1−Fx(z))g(z−λx1−λ),x<z<λx+(1−λ),0,otherwise.k_x(z)= cases 21-λF_x(z)\,g\! ( z-λ x1-λ ),&λ x<z<x,\\[8.53581pt] 21-λ (1-F_x(z) )\,g\! ( z-λ x1-λ ),&x<z<λ x+(1-λ),\\[8.53581pt] 0,&otherwise. cases Proof of Proposition B.1 Proof. Condition on Ot−1=xO_t-1=x. If z<λxz<λ x, then (1−λ)U+λx≤z(1-λ)U+λ x≤ z is impossible, so Fx(z)=0F_x(z)=0. If z>λx+(1−λ)z>λ x+(1-λ), then the event is certain, so Fx(z)=1F_x(z)=1. For λx≤z≤λx+(1−λ)λ x≤ z≤λ x+(1-λ), the event (1−λ)U+λx≤z(1-λ)U+λ x≤ z is equivalent to U≤(z−λx)/(1−λ)U≤(z-λ x)/(1-λ). Hence Fx(z)=G(z−λx1−λ),F_x(z)=G\! ( z-λ x1-λ ), which proves the stated formula. ∎ Proof of Proposition B.2 Proof. Condition on Ot−1=xO_t-1=x and write Ot=Med⁡(U~x,V~x,x)O_t= ( U_x, V_x,x), where U~x U_x and V~x V_x are independent with CDF FxF_x. If z<xz<x, then Ot≤zO_t≤ z if and only if both anchored stances are at most z. Hence Pr⁡(Ot≤z∣Ot−1=x)=Fx(z)2. (O_t≤ z O_t-1=x)=F_x(z)^2. If z≥xz≥ x, then Ot>zO_t>z if and only if both anchored stances are larger than z. Therefore Pr⁡(Ot≤z∣Ot−1=x)=1−Pr⁡(U~x>z,V~x>z)=1−(1−Fx(z))2. (O_t≤ z O_t-1=x)=1- ( U_x>z, V_x>z)=1- (1-F_x(z) )^2. These two identities prove the stated CDF. If G is continuous at x, then FxF_x is continuous at x and Fx(x)=G(x)F_x(x)=G(x). The atom at x is the jump of the CDF: Pr⁡(Ot=x∣Ot−1=x) (O_t=x O_t-1=x) =Pr⁡(Ot≤x∣Ot−1=x)−Pr⁡(Ot<x∣Ot−1=x) = (O_t≤ x O_t-1=x)- (O_t<x O_t-1=x) =[1−(1−Fx(x))2]−Fx(x)2 = [1- (1-F_x(x) )^2 ]-F_x(x)^2 =2Fx(x)(1−Fx(x))=2G(x)(1−G⁡(x)). =2F_x(x) (1-F_x(x) )=2G(x) (1-G(x) ). ∎ Proof of Proposition B.3 Proof. Differentiate the two continuous branches of Proposition B.2. By the chain rule, Fx′(z)=11−λg(z−λx1−λ)F_x (z)= 11-λ\,g\! ( z-λ x1-λ ) on the interior of [λx,λx+(1−λ)][λ x,λ x+(1-λ)]. For z<xz<x, dzFx(z)2=2Fx(z)Fx′(z), ddzF_x(z)^2=2F_x(z)F_x (z), and for z>xz>x, dz(1−(1−Fx(z))2)=2(1−Fx(z))Fx′(z). ddz (1- (1-F_x(z) )^2 )=2 (1-F_x(z) )F_x (z). Substituting the expression for Fx′(z)F_x (z) gives the stated density away from the atom at x. ∎ Appendix C Omitted proofs from Section 2 This appendix contains the convergence details omitted from Section 2. We put the proofs of Theorem 2.1 after its supporting lemmas. The material specific to the uniform distribution, including the one-step minorization argument, is presented in Appendix C.5. C.1 Proof of Lemma 2.2 Proof. We consider a distribution with mass 12 12 at 00 and 12 12 at 11. Conditional on the current state Ot=xO_t=x, the next state is Ot+1=λx,with probability 1/4,x,with probability 1/2,λx+(1−λ),with probability 1/4.O_t+1= casesλ x,&with probability 1/4,\\ x,&with probability 1/2,\\ λ x+(1-λ),&with probability 1/4. cases Hence ⁡[Ot+1∣Ot=x]=1+λ2x+1−λ4.E[O_t+1 O_t=x]= 1+λ2x+ 1-λ4. If O∼πλ,GO _λ,G is stationary, then ⁡[O]=1+λ2[O]+1−λ4,E[O]= 1+λ2E[O]+ 1-λ4, so ⁡[O]=12E[O]= 12. Starting from O0=1O_0=1, the mean recursion gives ⁡[Ot]−⁡[O]=12(1+λ2)t.E[O_t]-E[O]= 12 ( 1+λ2 )^t. By Corollary A.3, W1(δ1Kt,πλ,G)≥|⁡[Ot]−⁡[O]|=12(1+λ2)t.W_1( _1K^t, _λ,G)≥ |E[O_t]-E[O] |= 12 ( 1+λ2 )^t. The stated mixing-time lower bound follows by solving 12((1+λ)/2)t≤ε 12((1+λ)/2)^t≤ for t. ∎ C.2 Proof of Lemma 2.3 Proof. Write L=min⁡U,VL= \U,V\ and R=max⁡U,VR= \U,V\. Assume first that Xt≤YtX_t≤ Y_t; the case Xt=YtX_t=Y_t is immediate. By the transition rule in Equation (6), Yt+1−Xt+1 Y_t+1-X_t+1 =(1−λ)(Med⁡(U,V,Yt)−Med⁡(U,V,Xt))+λ⁡(Yt−Xt) =(1-λ) ( (U,V,Y_t)- (U,V,X_t) )+λ(Y_t-X_t) =(1−λ)|[L,R]∩[Xt,Yt]|+λ⁡(Yt−Xt). =(1-λ) |[L,R]∩[X_t,Y_t] |+λ(Y_t-X_t). Since the overlap length is nonnegative, Yt+1≥Xt+1Y_t+1≥ X_t+1. Since the overlap length is at most Yt−XtY_t-X_t, we also have Yt+1−Xt+1≤Yt−XtY_t+1-X_t+1≤ Y_t-X_t. Thus the order is preserved and the gap is non-increasing. Conditional on Xt,YtX_t,Y_t, Fubini’s theorem gives [|[L,R]∩[Xt,Yt]|∣Xt,Yt] \! [ |[L,R]∩[X_t,Y_t] | X_t,Y_t ] =[∫XtYtL≤r≤Rdr|Xt,Yt] =E\! [ _X_t^Y_t1_L≤ r≤ R\,dr\, |\,X_t,Y_t ] =∫XtYtPr⁡(L≤r≤R)dr. = _X_t^Y_t (L≤ r≤ R)\,dr. For an arbitrary CDF G, Pr⁡(L≤r≤R)=1−(1−G⁡(r))2−G(r−)2. (L≤ r≤ R)=1-(1-G(r))^2-G(r^-)^2. This equals 2G(r)(1−G⁡(r))2G(r)(1-G(r)) at every continuity point of G. A CDF has at most countably many discontinuities, so the equality holds for Lebesgue-a.e. r and hence after integration. Therefore [Zt+1∣Xt,Yt]=λZt+(1−λ)∫XtYt2G(r)(1−G(r))dr.E[Z_t+1 X_t,Y_t]=λ Z_t+(1-λ) _X_t^Y_t2G(r)(1-G(r))\,dr. The same argument applies when Xt>YtX_t>Y_t after swapping the labels. Since 2G(r)(1−G⁡(r))≤1/22G(r)(1-G(r))≤ 1/2, the final inequality follows. ∎ C.3 Proof of Corollary 2.4 Proof. Let c=(1+λ)/2<1c=(1+λ)/2<1. For deterministic x,y∈[0,1]x,y∈[0,1], update two copies from x and y using the same pair U,V∼GU,V G. Lemma 2.3, with the labels ordered if necessary, gives ⁡[|T⁡(x,U,V)−T⁡(y,U,V)|]≤c|x−y|,E\! [ |T(x;U,V)-T(y;U,V) | ]≤ c |x-y |, where T⁡(x,U,V)=λx+(1−λ)Med⁡(x,U,V)T(x;U,V)=λ x+(1-λ) (x,U,V). Now take any coupling (X0,Y0)(X_0,Y_0) of μ and ν and update both coordinates with the same U,VU,V. Then ⁡[|X1−Y1|]≤c[|X0−Y0|].E[ |X_1-Y_1 |]≤ c\,E[ |X_0-Y_0 |]. Taking the infimum over all couplings of μ and ν gives W1(μK,νK)≤cW1(μ,ν).W_1(μ K,ν K)≤ c\,W_1(μ,ν). Since [0,1][0,1] is compact, (⁡([0,1]),W1)(P([0,1]),W_1) is complete. Thus μ↦μKμ μ K is a contraction on a complete metric space, so Banach’s fixed-point theorem [18, Theorem 1.1] gives a unique fixed point πλ,G _λ,G, equivalently a unique stationary distribution. ∎ C.4 Proof of Theorem 2.1 Proof. Upper bound. Let c=(1+λ)/2c=(1+λ)/2. By Corollary 2.4, for every x∈[0,1]x∈[0,1], W1(δxKt,πλ,G)=W1(δxKt,πλ,GKt)≤ctW1(δx,πλ,G)≤ct,W_1( _xK^t, _λ,G)=W_1( _xK^t, _λ,GK^t)≤ c^tW_1( _x, _λ,G)≤ c^t, because the diameter of [0,1][0,1] is one. Therefore tmix(W1)(ε)≤⌈log⁡(1/ε)log⁡(2/(1+λ))⌉.t_mix^(W_1)( )≤ (1/ ) (2/(1+λ)) . Lower bound. Assume 0<λ<10<λ<1. Couple two copies of the chain by using the same sampled pair of bliss points at every time step. Let one copy start from X0=0X_0=0 and the other from Y0=1Y_0=1. Since the coupling preserves order, Yt+1−Xt+1≥λ⁡(Yt−Xt).Y_t+1-X_t+1≥λ(Y_t-X_t). Iterating gives Yt−Xt≥λtY_t-X_t≥λ^t. Hence ⁡[Yt]−⁡[Xt]=⁡[Yt−Xt]≥λt.E[Y_t]-E[X_t]=E[Y_t-X_t]≥λ^t. The laws of XtX_t and YtY_t are respectively δ0Kt _0K^t and δ1Kt _1K^t. By Corollary A.3, W1(δ0Kt,δ1Kt)≥λt.W_1( _0K^t, _1K^t)≥λ^t. The triangle inequality gives W1(δ0Kt,δ1Kt)≤W1(δ0Kt,πλ,G)+W1(πλ,G,δ1Kt)≤2supx∈[0,1]W1(δxKt,πλ,G).W_1( _0K^t, _1K^t)≤ W_1( _0K^t, _λ,G)+W_1( _λ,G, _1K^t)≤ 2 _x∈[0,1]W_1( _xK^t, _λ,G). Thus any t with supxW1(δxKt,πλ,G)≤ε _xW_1( _xK^t, _λ,G)≤ must satisfy λt≤2ελ^t≤ 2 . Solving this inequality gives tmix(W1)(ε)≥⌈log⁡(1/(2ε))log⁡(1/λ)⌉t_mix^(W_1)( )≥ (1/(2 )) (1/λ) for every 0<ε<1/20< <1/2. ∎ C.5 Uniform distribution We now specialize the transition law to the case where the population distribution is uniform on [0,1][0,1]. Thus the two sampled bliss points are independent draws U,V∼Unif⁡[0,1]U,V [0,1]. Conditionally on Ot−1=xO_t-1=x, the anchored stances U~ U and V~ V are i.i.d. uniform on [λx,λx+(1−λ)][λ x,λ x+(1-λ)] with density 1/(1−λ)1/(1-λ). For any real value z, the conditional CDF of either anchored stance is Pr⁡(U~≤z∣Ot−1=x)=Pr⁡(V~≤z∣Ot−1=x)=0,z<λx,z−λx1−λ,λx≤z<λx+(1−λ),1,z≥λx+(1−λ). ( U≤ z O_t-1=x)= ( V≤ z O_t-1=x)= cases0,&z<λ x,\\ z-λ x1-λ,&λ x≤ z<λ x+(1-λ),\\ 1,&z≥λ x+(1-λ). cases (18) Applying Proposition B.2 gives the transition CDF Pr⁡(Ot≤z∣Ot−1=x)=0,z<λx,(z−λx1−λ)2,λx≤z<x,1−(1−z−λx1−λ)2,x≤z<λx+(1−λ),1,z≥λx+(1−λ). (O_t≤ z O_t-1=x)= cases0,&z<λ x,\\ ( z-λ x1-λ )^2,&λ x≤ z<x,\\ 1- (1- z-λ x1-λ )^2,&x≤ z<λ x+(1-λ),\\ 1,&z≥λ x+(1-λ). cases (19) Differentiating away from the atom at x gives the density kx(z)=2(z−λx)(1−λ)2,λx<z<x,2(1−λ+λx−z)(1−λ)2,x<z<λx+(1−λ),0,otherwise,k_x(z)= cases 2(z-λ x)(1-λ)^2,&λ x<z<x,\\[5.69054pt] 2(1-λ+λ x-z)(1-λ)^2,&x<z<λ x+(1-λ),\\[5.69054pt] 0,&otherwise, cases (20) plus an atom of mass 2x(1−x)2x(1-x) at x. We record two refinements for the uniform case. The first decomposes the one-step coupling by the position of the sampled bliss points relative to the current gap. The second is a one-step minorization that gives total-variation contraction when λ<1/2λ<1/2. Let (Xt,Yt)(X_t,Y_t) be the coupling obtained by using the same pair Ut,Vt∼i.i.d.Unif⁡[0,1]U_t,V_t .i.d. Unif[0,1] in both updates. Let Zt≔|Xt−Yt|Z_t |X_t-Y_t |. Lemma C.1. Condition on Xt≤YtX_t≤ Y_t and Zt=zZ_t=z. The order of the coupled chains is preserved and ZtZ_t is non-increasing. Moreover, the one-step gap satisfies the following four-case decomposition: with probability at least (1−z)22 (1-z)^22, Zt+1=λzZ_t+1=λ z; with probability 2z(1−z)2z(1-z), [Zt+1∣Zt=z,Case 2]=1+λ2zE[Z_t+1 Z_t=z,Case 2]= 1+λ2z; with probability z2z^2, [Zt+1∣Zt=z,Case 3]=1+2λ3zE[Z_t+1 Z_t=z,Case 3]= 1+2λ3z; and with probability at most (1−z)22 (1-z)^22, Zt+1=zZ_t+1=z. Corollary C.2. For every z∈[0,1]z∈[0,1], ⁡[Zt+1∣Zt=z]≤1+λ2z.E[Z_t+1 Z_t=z]≤ 1+λ2z. Consequently, ⁡[Zt]≤(1+λ2)t[Z0].E[Z_t]≤ ( 1+λ2 )^tE[Z_0]. Proof of Lemma C.1 Proof. Write L=min⁡U,VL= \U,V\ and R=max⁡U,VR= \U,V\. Condition on Xt≤YtX_t≤ Y_t and Zt=Yt−Xt=zZ_t=Y_t-X_t=z. Equation (6) gives Yt+1−Xt+1 Y_t+1-X_t+1 =(1−λ)(Med⁡(U,V,Yt)−Med⁡(U,V,Xt))+λ⁡(Yt−Xt) =(1-λ) ( (U,V,Y_t)- (U,V,X_t) )+λ(Y_t-X_t) (21) =(1−λ)|[L,R]∩[Xt,Yt]|+λ⁡(Yt−Xt). =(1-λ) |[L,R]∩[X_t,Y_t] |+λ(Y_t-X_t). The overlap length is between 00 and z, so 0≤Yt+1−Xt+1≤z0≤ Y_t+1-X_t+1≤ z. Hence the order is preserved and the gap is non-increasing. Case 1. Both sampled points are outside [Xt,Yt][X_t,Y_t] and on the same side, i.e., R<XtR<X_t or L>YtL>Y_t. Let α=Xt/(1−z)α=X_t/(1-z) and β=(1−Yt)/(1−z)β=(1-Y_t)/(1-z) when z<1z<1; then α,β≥0α,β≥ 0 and α+β=1α+β=1. The probability of this case is Pr⁡(R<Xt or L>Yt) (R<X_t or L>Y_t) ≥minα,β≥0,α+β=1⁡((1−z)α)2+((1−z)β)2 ≥ _α,β≥ 0,α+β=1((1-z)α)^2+((1-z)β)^2 (22) =minα,β≥0,α+β=1⁡(1−z)2(α2+β2) = _α,β≥ 0,α+β=1(1-z)^2(α^2+β^2) ≥(1−z)2(α+β2)2⋅2 ≥(1-z)^2( α+β2)^2· 2 =(1−z)212. =(1-z)^2 12. When z=1z=1, the same lower bound is 00 and is immediate. In this case the overlap length is 00, so Zt+1=λzZ_t+1=λ z. Case 2. Exactly one sampled point lies in [Xt,Yt][X_t,Y_t]. This has probability 2z(1−z)2z(1-z). Let γ=|[L,R]∩[Xt,Yt]|γ= |[L,R]∩[X_t,Y_t] |. Then Zt+1=(1−λ)γ+λzZ_t+1=(1-λ)γ+λ z, and the contribution to the expectation is [Zt+1∣Zt=z,Case 2]⋅Pr[Case 2] [Z_t+1 Z_t=z,Case 2]· [Case 2] =∫0z2⋅(1−z)⋅((1−λ)γ+λz)γ = _0^z2·(1-z)·((1-λ)γ+λ z)dγ (23) =2⋅(1−z)⋅∫0z((1−λ)γ+λz)γ =2·(1-z)· _0^z((1-λ)γ+λ z)dγ =2⋅(1−z)⋅(λz2+(1−λ)12z2) =2·(1-z)·(λ z^2+(1-λ) 12z^2) Dividing by Pr⁡[Case 2]=2z(1−z) [Case 2]=2z(1-z) gives [Zt+1∣Zt=z,Case 2]=λz+12(1−λ)z=(1+λ)2⋅z. [Z_t+1 Z_t=z,Case 2]=λ z+ 12(1-λ)z= (1+λ)2· z. (24) Case 3. Both sampled points lie in [Xt,Yt][X_t,Y_t]. This has probability z2z^2. Let γ1=d⁡(Xt,U) _1=d(X_t,U) and γ2=d⁡(Yt,V) _2=d(Y_t,V). Then d⁡(U,V)=|z−γ1−γ2|d(U,V)= |z- _1- _2 | and Zt+1=(1−λ)d(U,V)+λzZ_t+1=(1-λ)d(U,V)+λ z. The contribution of d⁡(U,V)d(U,V) is [d(U,V)∣Zt=z,Case 3]Pr[Case 3] [d(U,V) Z_t=z,Case 3] [Case 3] =∫0z∫0z|z−γ1−γ2|dγ1dγ2 = _0^z _0^z |z- _1- _2 |\,d _1d _2 (25) =∫0z(∫0z−γ1(z−γ1−γ2)dγ2+∫z−γ1z(γ1+γ2−z)dγ2)dγ1 = _0^z ( _0^z- _1(z- _1- _2)d _2+ _z- _1^z( _1+ _2-z)d _2 )d _1 =∫0z(12(z−γ1)2+12(γ1)2)dγ1 = _0^z ( 12(z- _1)^2+ 12( _1)^2 )d _1 =∫0z(12z2+γ12−zγ1)dγ1 = _0^z ( 12z^2+ _1^2-z _1 )d _1 =12z3+13z3−12z3=13z3. = 12z^3+ 13z^3- 12z^3= 13z^3. Since Pr⁡[Case 3]=z2 [Case 3]=z^2, [d(U,V)∣Zt=z,Case 3]=13z. [d(U,V) Z_t=z,Case 3]= 13z. (26) Applying Equation (21), [Zt+1∣Zt=z,Case 3] [Z_t+1 Z_t=z,Case 3] =(1−λ)[d(U,V)∣Zt=z,Case 3]+λz=13z+23λz. =(1-λ)E[d(U,V) Z_t=z,Case 3]+λ z= 13z+ 23λ z. (27) Case 4. Both sampled points are outside [Xt,Yt][X_t,Y_t] but on opposite sides. This has probability (1−z)2−Pr⁡[Case 1]≤(1−z)2/2(1-z)^2- [Case 1]≤(1-z)^2/2. In this case the overlap length is z, so Zt+1=zZ_t+1=z. ∎ zzdensity103 10359 59000.4=λ0.4= 1/21/20.6=1−λ0.6=1- 11 fX1(z)f_X_1(z) from X0=0X_0=0fY1(z)f_Y_1(z) from Y0=1Y_0=1overlapm0.4(z)m_0.4(z)Iλ=[λ,1−λ]I_λ=[λ,1-λ] Figure 4: One-step minorization for the uniform chain with λ=0.4λ=0.4, starting from X0=0X_0=0 and Y0=1Y_0=1. The two one-step densities are fX1(z)=2(0.6−z)/0.62f_X_1(z)=2(0.6-z)/0.6^2 on [0,0.6][0,0.6] and fY1(z)=2(z−0.4)/0.62f_Y_1(z)=2(z-0.4)/0.6^2 on [0.4,1][0.4,1]. The shaded triangle is their common lower density on Iλ=[0.4,0.6]I_λ=[0.4,0.6]; its area is β⁡(0.4)=1/18β(0.4)=1/18, which is the minorization mass in Lemma C.3. Contraction analysis based on one-step minorization. We now give a sharper but less general argument for total-variation distance. When G is uniform and λ<12λ< 12, every starting point has a common chance of moving into the interval [λ,1−λ][λ,1-λ] in one step. Figure 4 illustrates this common overlap. This overlap yields a one-step minorization and hence total-variation contraction. Lemma C.3. Assume G is uniform on [0,1][0,1] and 0≤λ<120≤λ< 12. Then there exists a probability measure νλ _λ on [0,1][0,1] such that K⁡(x,⋅)≥β⁡(λ)νλ(⋅)K(x,·)≥β(λ) _λ(·) for every x∈[0,1]x∈[0,1], where β⁡(λ)=(1−2λ)22(1−λ)2β(λ)= (1-2λ)^22(1-λ)^2. Consequently, for every pair of probability measures μ,νμ,ν on [0,1][0,1], ‖μK−νK‖TV≤(1−β⁡(λ))‖μ−ν‖TV. \|μ K-ν K \|_TV≤(1-β(λ))\|μ-ν\|_TV. Corollary C.4. Assume that G is uniform on [0,1][0,1], and let 0≤λ<120≤λ< 12. Then tmixTV(ε)≤⌈1β⁡(λ)log⁡1ε⌉.t^TV_mix( )≤ 1β(λ) 1 . Proof of Corollary C.2 Proof. Combine the four cases in Lemma C.1. The combined contribution of Cases 1 and 4 is at most [Zt+1∣Zt=z,Case 1]Pr[Case 1]+[Zt+1∣Zt=z,Case 4]Pr[Case 4] [Z_t+1 Z_t=z,Case 1] [Case 1]+E[Z_t+1 Z_t=z,Case 4] [Case 4] (28) ≤(a)(1−z)212λz+(1−z)212z (a)≤(1-z)^2 12λ z+(1-z)^2 12z =(1−z)22⋅(λz+z) = (1-z)^22·(λ z+z) because Zt+1=λzZ_t+1=λ z in Case 1, Zt+1=zZ_t+1=z in Case 4, Pr⁡[Case 1]+Pr⁡[Case 4]=(1−z)2 [Case 1]+ [Case 4]=(1-z)^2, and Pr⁡[Case 1]≥(1−z)2/2 [Case 1]≥(1-z)^2/2. [Zt+1∣Zt=z]=∑i=14[Zt+1∣Zt=z,Case i]Pr[Case i] [Z_t+1 Z_t=z]= _i=1^4E[Z_t+1 Z_t=z,Case $i$] [Case $i$] (29) ≤(1−z)22⋅(λz+z)+2⋅(1−z)⋅(λz2+(1−λ)12z2)+(1+2λ3z)z2 ≤ (1-z)^22·(λ z+z)+2·(1-z)·(λ z^2+(1-λ) 12z^2)+ ( 1+2λ3z )z^2 =1+λ2z+λ−16z3≤1+λ2z. = 1+λ2z+ λ-16z^3≤ 1+λ2z. ∎ Proof of Lemma C.3 Proof. We use the uniform transition density in Equation (20). When λ<12λ< 12, for every z∈Iλ=[λ,1−λ]z∈ I_λ=[λ,1-λ], z belongs to the support interval [λx,λx+(1−λ)][λ x,λ x+(1-λ)] for every x∈[0,1]x∈[0,1]. If x>zx>z, then kx(z)=2(z−λx)(1−λ)2≥2(z−λ)(1−λ)2,k_x(z)= 2(z-λ x)(1-λ)^2≥ 2(z-λ)(1-λ)^2, and if x<zx<z, then kx(z)=2(1−λ+λx−z)(1−λ)2≥2(1−λ−z)(1−λ)2.k_x(z)= 2(1-λ+λ x-z)(1-λ)^2≥ 2(1-λ-z)(1-λ)^2. Hence, for every x, kx(z)≥mλ(z)≔2(1−λ)2min⁡z−λ, 1−λ−z[λ,1−λ](z).k_x(z)≥ m_λ(z) 2(1-λ)^2 \z-λ,\ 1-λ-z\1_[λ,1-λ](z). The mass of this common lower density is β⁡(λ)=∫01mλ(z)z=(1−2λ)22(1−λ)2.β(λ)= _0^1m_λ(z)\,dz= (1-2λ)^22(1-λ)^2. Let νλ _λ be the probability measure with density mλ(z)/β(λ)m_λ(z)/β(λ). Then K⁡(x,⋅)≥β⁡(λ)νλ(⋅)K(x,·)≥β(λ) _λ(·) for every x. Write K⁡(x,⋅)=β⁡(λ)νλ(⋅)+(1−β⁡(λ))K~(x,⋅)K(x,·)=β(λ) _λ(·)+(1-β(λ)) K(x,·). Since K~ K is a Markov kernel, for every probability measures μ,νμ,ν, ‖μK−νK‖TV=(1−β⁡(λ))‖μK~−νK~‖TV≤(1−β⁡(λ))‖μ−ν‖TV.\|μ K-ν K\|_TV=(1-β(λ))\|μ K-ν K\|_TV≤(1-β(λ))\|μ-ν\|_TV. ∎ Proof of Corollary C.4 Proof. Lemma C.3 gives one-step total-variation contraction by the factor 1−β⁡(λ)1-β(λ). Iterating it and using stationarity of πλ,G _λ,G gives supx∈[0,1]‖Kt(x,⋅)−πλ,G‖TV≤(1−β⁡(λ))t. _x∈[0,1]\|K^t(x,·)- _λ,G\|_TV≤(1-β(λ))^t. Since (1−β)t≤e−βt(1-β)^t≤ e^-β t for β∈(0,1)β∈(0,1), the right-hand side is at most ε whenever t≥β(λ)−1log⁡(1/ε)t≥β(λ)^-1 (1/ ). This proves the stated bound. ∎ Appendix D Omitted proofs from Section 3 This section first proves Lemma 3.3; next it provides the ingredients for Lemma 3.2. The proof of Lemma 3.2 is put to the end. D.1 Proof of Lemma 3.3 Proof. Consider the population with mass p at 00 and mass q=1−pq=1-p at 11. Let Mt=Med⁡(Ot,Ut,Vt)M_t= (O_t,U_t,V_t), so Ot+1=(1−λ)Mt+λOt.O_t+1=(1-λ)M_t+λ O_t. With probability p2p^2, both sampled bliss points are 00 and Mt=0M_t=0. With probability q2q^2, both are 11 and Mt=1M_t=1. With probability 2pq2pq, the two sampled bliss points are split and Mt=OtM_t=O_t. Taking expectations at stationarity gives ⁡[O∞]=p2λ[O∞]+q2((1−λ)+λ[O∞])+2pq[O∞].E[O_∞]=p^2 [O_∞]+q^2 ((1-λ)+ [O_∞] )+2pqE[O_∞]. Since λ<1λ<1, this simplifies to ⁡[O∞]=q2p2+q2,E[O_∞]= q^2p^2+q^2, which is independent of λ. The social cost is ⁡[⁡(O∞)] [ SC(O_∞)] =⁡[pO∞+q⁡(1−O∞)] =E[pO_∞+q(1-O_∞)] (30) =q+(p−q)[O∞]=pqp2+q2. =q+(p-q)E[O_∞]= pqp^2+q^2. Without loss of generality, assume q≤pq≤ p, so the optimal outcome is 00 and its social cost is q. The stationary distortion is therefore ⁡[⁡(O∞)]⁡(0)=p2+q2. E[ SC(O_∞)] SC(0)= pp^2+q^2. Maximizing this expression over p∈[1/2,1]p∈[1/2,1] gives p=22p= 22 and q=1−22q=1- 22, where the value is 1+22 1+ 22. This value is independent of λ. ∎ D.2 Proof of Lemma 3.4 Proof. Let Y∼GY G be independent of O. Since |O−Y|=∫01(Y≤r<O+O≤r<Y)r, |O-Y |= _0^1 (1_Y≤ r<O+1_O≤ r<Y )\,dr, Fubini’s theorem gives ⁡[⁡(O)] [ SC(O)] =⁡[|O−Y|] =E[ |O-Y |] =∫01(Pr⁡(Y≤r<O)+Pr⁡(O≤r<Y))r = _0^1 ( (Y≤ r<O)+ (O≤ r<Y) )\,dr =∫01(G⁡(r)Pr⁡(O>r)+(1−G⁡(r))Pr⁡(O≤r))dr. = _0^1 (G(r) (O>r)+(1-G(r)) (O≤ r) )\,dr. For deterministic a, this formulation becomes ⁡(a)=∫0aG⁡(r)r+∫a1(1−G⁡(r))r. SC(a)= _0^aG(r)\,dr+ _a^1(1-G(r))\,dr. Let mGm_G be a population median, so G⁡(r)≤1/2G(r)≤ 1/2 for r<mGr<m_G and G⁡(r)≥1/2G(r)≥ 1/2 for r>mGr>m_G, except possibly on a median interval where either choice gives the same value. For any a≥mGa≥ m_G, ⁡(a)−⁡(mG)=∫mGa(2G(r)−1)r≥0, SC(a)- SC(m_G)= _m_G^a(2G(r)-1)\,dr≥ 0, and for any a≤mGa≤ m_G, ⁡(a)−⁡(mG)=∫amG(1−2G(r))r≥0. SC(a)- SC(m_G)= _a^m_G(1-2G(r))\,dr≥ 0. Thus the minimum deterministic social cost is attained at any population median. Finally, ⁡(mG)=∫0mGG⁡(r)r+∫mG1(1−G⁡(r))r=∫01min⁡G⁡(r),1−G⁡(r)r. SC(m_G)= _0^m_GG(r)\,dr+ _m_G^1(1-G(r))\,dr= _0^1 \G(r),1-G(r)\\,dr. ∎ D.3 Proof of Lemma 3.5 Proof. The transition CDF is proved by the same argument as in Proposition B.2. For every x,z∈[0,1]x,z∈[0,1], Pr⁡(Xt+1≤z∣Xt=x)=G(z)2,z<x,1−(1−G⁡(z))2,z≥x. (X_t+1≤ z X_t=x)= casesG(z)^2,&z<x,\\[5.69054pt] 1- (1-G(z) )^2,&z≥ x. cases For the stationary CDF, fix z∈[0,1]z∈[0,1] and write Ft(z)=Pr⁡(Xt≤z)F_t(z)= (X_t≤ z). Conditioning on whether Xt≤zX_t≤ z gives Ft+1(z) F_t+1(z) =Ft(z)[1−(1−G⁡(z))2]+(1−Ft(z))G(z)2 =F_t(z) [1- (1-G(z) )^2 ]+(1-F_t(z))G(z)^2 =G(z)2+2G(z)(1−G⁡(z))Ft(z). =G(z)^2+2G(z)(1-G(z))F_t(z). At stationarity, Ft+1(z)=Ft(z)=ΠG(z)F_t+1(z)=F_t(z)= _G(z), which implies ΠG(z)=G(z)21−2G(z)(1−G⁡(z))=G(z)2G(z)2+(1−G⁡(z))2. _G(z)= G(z)^21-2G(z)(1-G(z))= G(z)^2G(z)^2+(1-G(z))^2. ∎ D.4 Proof of Lemma 3.2 Proof. By Lemma 3.4, the social cost of the unanchored stationary distribution can be written as ⁡[⁡(X∞)] [ SC(X_∞)] =∫01[G⁡(r)Pr⁡(X∞>r)+(1−G⁡(r))Pr⁡(X∞≤r)]r = _0^1 [G(r) (X_∞>r)+(1-G(r)) (X_∞≤ r) ]d\,r (31) =(a)∫01G(r)(1−G(r))G(r)2+(1−G⁡(r))2r (a)= _0^1 G(r)(1-G(r))G(r)^2+(1-G(r))^2d\,r ≤(b)∫012+12min⁡G⁡(r),1−G⁡(r)r (b)≤ _0^1 2+12 \G(r),1-G(r)\d\,r =(c)2+12(mG). (c)= 2+12 SC(m_G). Step (a)(a) substitutes Pr⁡(X∞≤r)=ΠG(r) (X_∞≤ r)= _G(r) and Pr⁡(X∞>r)=1−ΠG(r) (X_∞>r)=1- _G(r) using Lemma 3.5. Step (b)(b) follows from G(r)(1−G(r))G(r)2+(1−G⁡(r))2≤2+12min⁡G⁡(r),1−G⁡(r), G(r)(1-G(r))G(r)^2+(1-G(r))^2≤ 2+12 \G(r),1-G(r)\, which is equivalent to max0≤p≤1/2⁡(1−p)/(p2+(1−p)2)≤(2+1)/2 _0≤ p≤ 1/2(1-p)/(p^2+(1-p)^2)≤( 2+1)/2. Step (c)(c) applies Lemma 3.4, which gives ⁡(mG)=∫01min⁡G⁡(r),1−G⁡(r)r SC(m_G)= _0^1 \G(r),1-G(r)\\,dr for any population median mGm_G. Since every population median minimizes deterministic social cost, the lemma follows. ∎ Appendix E Omitted proofs from Section 4 E.1 Proof of Lemma 4.1 Proof. When U and V lie on different sides of x, Med⁡(x,U,V)=x (x,U,V)=x. Thus only pairs on the same side of x contribute to the expected movement. We can write bG(x)b_G(x) as bG(x) b_G(x) =⁡[Med⁡(x,U,V)−x] =E[ (x,U,V)-x] (32) =(a)⁡[(Med⁡(x,U,V)−x)+]−⁡[(x−Med⁡(x,U,V))+] (a)=E[( (x,U,V)-x)_+]-E[(x- (x,U,V))_+] =⁡[∫x1Med⁡(x,U,V)≥rdr]−⁡[∫0xMed⁡(x,U,V)≤rdr] =E[ _x^11_ (x,U,V)≥ rd\,r]-E[ _0^x1_ (x,U,V)≤ rd\,r] =∫x1Pr⁡(U>r,V>r)r−∫0xPr⁡(U<r,V<r)r = _x^1 (U>r,V>r)d\,r- _0^x (U<r,V<r)d\,r =∫x1(1−G⁡(r))2dr−∫0xG(r)2dr. = _x^1(1-G(r))^2d\,r- _0^xG(r)^2d\,r. Here (⋅)+=max⁡⋅,0(·)_+= \·,0\, and (a)(a) uses the identity y=y+−(−y)+y=y_+-(-y)_+. The last equality is unaffected by the choice of strict or weak threshold inequalities, because the left limits of G differ from G only at countably many points. For 0≤x<y≤10≤ x<y≤ 1, bG(y)−bG(x)=−∫xy(G(r)2+(1−G(r))2)dr≤−12(y−x).b_G(y)-b_G(x)=- _x^y (G(r)^2+(1-G(r))^2 )\,dr≤- 12(y-x). Thus bGb_G is continuous and strictly decreasing. Moreover, bG(0)=∫01(1−G(r))2dr≥0,bG(1)=−∫01G(r)2dr≤0.b_G(0)= _0^1(1-G(r))^2\,dr≥ 0, b_G(1)=- _0^1G(r)^2\,dr≤ 0. By the intermediate value theorem, there exists θG∈[0,1] _G∈[0,1] with bG(θG)=0b_G( _G)=0. Strict monotonicity gives uniqueness. ∎ E.2 Proof of Theorem 4.2 Proof. Let Ox,λO_x,λ be the one-step outcome from Ot=xO_t=x with anchoring parameter λ. Then Ox,λ=x+(1−λ)(Ox,0−x)O_x,λ=x+(1-λ)(O_x,0-x). We bound the conditional second moment around θG _G. ⁡[(Ot+1−θG)2∣Ot=x] [(O_t+1- _G)^2 O_t=x] =⁡[((x−θG)+(1−λ)(Ox,0−x))2] =E[ ((x- _G)+(1-λ)(O_x,0-x) )^2] (33) =(a)(x−θG)2+2(1−λ)(x−θG)bG(x)+(1−λ)2[(Ox,0−x)2] (a)=(x- _G)^2+2(1-λ)(x- _G)b_G(x)+(1-λ)^2E[(O_x,0-x)^2] ≤(b)(x−θG)2−(1−λ)(x−θG)2+(1−λ)2 (b)≤(x- _G)^2-(1-λ)(x- _G)^2+(1-λ)^2 =λ(x−θG)2+(1−λ)2. =λ(x- _G)^2+(1-λ)^2. where (a)(a) follows from linearity of expectation and bG(x)=⁡[Ox,0−x]b_G(x)=E[O_x,0-x]. For (b)(b), Lemma 4.1 gives bG(x)−bG(θG)=−∫θGx(G(r)2+(1−G(r))2)dr.b_G(x)-b_G( _G)=- _ _G^x (G(r)^2+(1-G(r))^2 )\,dr. Since bG(θG)=0b_G( _G)=0, this implies (x−θG)bG(x)≤−12(x−θG)2.(x- _G)b_G(x)≤- 12(x- _G)^2. We also use (Ox,0−x)2≤1(O_x,0-x)^2≤ 1. Taking expectations under the stationary distribution gives ⁡[(Oλ−θG)2]≤λ[(Oλ−θG)2]+(1−λ)2.E[(O_λ- _G)^2]≤ [(O_λ- _G)^2]+(1-λ)^2. Solving yields ⁡[(Oλ−θG)2]≤1−λE[(O_λ- _G)^2]≤ 1-λ, where Oλ∼πλ,GO_λ _λ,G. ∎ E.3 Proof of Lemma 4.3 Proof. By the integral formula for bGb_G from Lemma 4.1, bG(12)=∫1/21(1−G⁡(r))2r−∫01/2G(r)2r.b_G\! ( 12 )= _1/2^1(1-G(r))^2\,dr- _0^1/2G(r)^2\,dr. In the first integral, substitute r=12+sr= 12+s. Since G is centered, 1−G⁡(12+s)=G⁡(12−s)1-G( 12+s)=G( 12-s). Therefore ∫1/21(1−G⁡(r))2r=∫01/2G(12−s)2s=∫01/2G(u)2u. _1/2^1(1-G(r))^2\,dr= _0^1/2G\! ( 12-s )^2\,ds= _0^1/2G(u)^2\,du. Hence bG(1/2)=0b_G(1/2)=0. Lemma 4.1 gives uniqueness, so θG=1/2 _G=1/2. ∎ E.4 Proof of Theorem 4.4 Proof. The upper and lower bounds follow directly from Corollary 4.7. We now compute the exact value when λ=0λ=0. By Lemma 3.5, the stationary CDF for the uniform unanchored chain is Pr⁡(O∞≤r)=r2r2+(1−r)2. (O_∞≤ r)= r^2r^2+(1-r)^2. Applying Lemma 3.4 with G⁡(r)=rG(r)=r gives ⁡[⁡(O∞)] [ SC(O_∞)] =∫01(rPr⁡(O∞>r)+(1−r)Pr⁡(O∞≤r))r = _0^1 (r (O_∞>r)+(1-r) (O_∞≤ r) )\,dr =∫01r⁡(1−r)r2+(1−r)2r = _0^1 r(1-r)r^2+(1-r)^2\,dr =14∫−111−s21+s2ds=π−24, = 14 _-1^1 1-s^21+s^2\,ds= π-24, where the penultimate equality uses the substitution s=2r−1s=2r-1. Since ⁡(1/2)=1/4 SC(1/2)=1/4, we have the distortion π−2π-2. ∎ E.5 Proof of Lemma 4.5 Proof. When the distribution is uniform, for any fixed outcome Ot∈[0,1]O_t∈[0,1] we have ⁡(Ot)=∫0Ot(Ot−x)x+∫Ot1(x−Ot)x=Ot22+(1−Ot)22=14+(Ot−12)2. SC(O_t)= _0^O_t(O_t-x)\,dx+ _O_t^1(x-O_t)\,dx= O_t^22+ (1-O_t)^22= 14+ (O_t- 12 )^2. Since ⁡(O∗)=⁡(1/2)=1/4 SC(O^*)= SC(1/2)=1/4, substituting Ot=1/2+tO_t=1/2+Q_t into the above equation gives ⁡[⁡(Ot)]⁡(O∗)=1+4[t2]. E[ SC(O_t)] SC(O^*)=1+4E[Q_t^2]. (34) ∎ E.6 Proof of Lemma 4.6 Proof. We first recall the transition density for the uniform distribution from Equation (20): kx(z)=2(z−λx)(1−λ)2,λx<z<x,2(1−λ+λx−z)(1−λ)2,x<z<λx+(1−λ),0,otherwise,k_x(z)= cases 2(z-λ x)(1-λ)^2,&λ x<z<x,\\[5.69054pt] 2(1-λ+λ x-z)(1-λ)^2,&x<z<λ x+(1-λ),\\[5.69054pt] 0,&otherwise, cases plus the atom 2x(1−x)δx2x(1-x) _x, where δx _x is the Dirac mass at x. Fix x∈[0,1]x∈[0,1]. By the transition density, ⁡[t+12∣Ot=x] [Q_t+1^2 O_t=x] =2x(1−x)(x−12)2 =2x(1-x) (x- 12 )^2 (35) +2(1−λ)2∫λx(z−12)2(z−λx)dz + 2(1-λ)^2 _λ x^x (z- 12 )^2(z-λ x)\,dz +2(1−λ)2∫x1−λ+λx(z−12)2(1−λ+λx−z)dz. + 2(1-λ)^2 _x^1-λ+λ x (z- 12 )^2(1-λ+λ x-z)\,dz. Substituting x=12+yx= 12+y gives ⁡[t+12∣t=y] [Q_t+1^2 _t=y] =(12y2−2y4) = ( 12y^2-2y^4 ) (36) +2(1−λ)2∫λ2+λy12+y(z−12)2(z−λ2−λy)dz + 2(1-λ)^2 _ λ2+λ y 12+y (z- 12 )^2 (z- λ2-λ y )\,dz +2(1−λ)2∫12+y 1−λ2+λy(z−12)2(1−λ2+λy−z)dz. + 2(1-λ)^2 _ 12+y^\,1- λ2+λ y (z- 12 )^2 (1- λ2+λ y-z )\,dz. Now set u=z−12u=z- 12, so z=u+12z=u+ 12. The integration limits become u=yu=y when z=12+yz= 12+y, u=−1−λ2+λyu=- 1-λ2+λ y when z=λ2+λyz= λ2+λ y, and u=1−λ2+λyu= 1-λ2+λ y when z=1−λ2+λyz=1- λ2+λ y. ⁡[t+12∣t=y] [Q_t+1^2 _t=y] =12y2−2y4 = 12y^2-2y^4 +2(1−λ)2∫−1−λ2+λyyu2(u+1−λ2−λy)du + 2(1-λ)^2 _- 1-λ2+λ y^yu^2 (u+ 1-λ2-λ y )\,du +2(1−λ)2∫y1−λ2+λyu2(1−λ2+λy−u)du. + 2(1-λ)^2 _y 1-λ2+λ yu^2 ( 1-λ2+λ y-u )\,du. We now compute the two integral terms. ∫−1−λ2+λyyu2(u+1−λ2−λy)u=(14u4+13(1−λ2−λy)u3)|−1−λ2+λy _- 1-λ2+λ y^yu^2 (u+ 1-λ2-λ y )\,du= ( 14u^4+ 13( 1-λ2-λ y)u^3 ) |_- 1-λ2+λ y^y (37) =(14y4+13(1−λ2−λy)y3)−(14(−1−λ2+λy)4+13(1−λ2−λy)(−1−λ2+λy)3) = ( 14y^4+ 13( 1-λ2-λ y)y^3 )- ( 14(- 1-λ2+λ y)^4+ 13( 1-λ2-λ y)(- 1-λ2+λ y)^3 ) =14y4+13(1−λ2−λy)y3+112(−1−λ2+λy)4 = 14y^4+ 13( 1-λ2-λ y)y^3+ 112(- 1-λ2+λ y)^4 ∫y1−λ2+λyu2(1−λ2+λy−u)u=(13u3(1−λ2+λy)−14u4)|y1−λ2+λy _y 1-λ2+λ yu^2 ( 1-λ2+λ y-u )\,du= ( 13u^3( 1-λ2+λ y)- 14u^4 ) |_y 1-λ2+λ y (38) =(13(1−λ2+λy)3(1−λ2+λy)−14(1−λ2+λy)4)−(13y3(1−λ2+λy)−14y4) = ( 13( 1-λ2+λ y)^3( 1-λ2+λ y)- 14( 1-λ2+λ y)^4 )- ( 13y^3( 1-λ2+λ y)- 14y^4 ) =14y4−13y3(1−λ2+λy)+112(1−λ2+λy)4 = 14y^4- 13y^3( 1-λ2+λ y)+ 112( 1-λ2+λ y)^4 Adding Equations (37) and (38) gives 12y4−2λ3y4+112((−1−λ2+λy)4+(1−λ2+λy)4) 12y^4- 2λ3y^4+ 112 ((- 1-λ2+λ y)^4+( 1-λ2+λ y)^4 ) (39) =(12−2λ3)y4+112(2(1−λ2)4+12(1−λ2)2(λy)2+2(λy)4) =( 12- 2λ3)y^4+ 112 (2( 1-λ2)^4+12( 1-λ2)^2(λ y)^2+2(λ y)^4 ) =(12−2λ3+16λ4)y4+(1−λ)24λ2y2+16(1−λ2)4 =( 12- 2λ3+ 16λ^4)y^4+ (1-λ)^24λ^2y^2+ 16( 1-λ2)^4 Combining these terms, ⁡[t+12∣t=y] [Q_t+1^2 _t=y] =(12y2−2y4)+2(1−λ)2((12−2λ3+16λ4)y4+(1−λ)24λ2y2+16(1−λ2)4) = ( 12y^2-2y^4 )+ 2(1-λ)^2 (( 12- 2λ3+ 16λ^4)y^4+ (1-λ)^24λ^2y^2+ 16( 1-λ2)^4 ) (40) =(1−λ)248+1+λ22y2−(1−λ)(λ+3)3y4 = (1-λ)^248+ 1+λ^22y^2- (1-λ)(λ+3)3y^4 ∎ E.7 Proof of Corollary 4.7 Proof. Upper bound. Taking expectations in Lemma 4.6 gives ⁡[t+12]≤(1−λ)248+1+λ22[t2]E[Q_t+1^2]≤ (1-λ)^248+ 1+λ^22E[Q_t^2]. Let a≔(1−λ)248a (1-λ)^248 and b=1+λ22b= 1+λ^22. Iterating the recursion gives ⁡[t+12] [Q_t+1^2] ≤a+b[t2] ≤ a+bE[Q_t^2] (41) =a⁡(1+b+…+bt−1)+bt[12] =a(1+b+…+b^t-1)+b^tE[Q_1^2] =a⋅1−bt1−b+bt[12] =a· 1-b^t1-b+b^tE[Q_1^2] =a1−b+bt(⁡[12]−a1−b) = a1-b+b^t (E[Q_1^2]- a1-b ) =1−λ24(1+λ)+(1+λ22)t(⁡[12]−1−λ24(1+λ)). = 1-λ24(1+λ)+ ( 1+λ^22 )^t (E[Q_1^2]- 1-λ24(1+λ) ). Since ⁡[12]≤14E[Q_1^2]≤ 14, ⁡[12]−1−λ24(1+λ)≤5+7λ24(1+λ)E[Q_1^2]- 1-λ24(1+λ)≤ 5+7λ24(1+λ) This implies the stated finite-time upper bound; taking t→∞t→∞ gives the stationary upper bound. Lower bound. Since t4≤14t2Q_t^4≤ 14Q_t^2, Lemma 4.6 also gives ⁡[t+12]≥(1−λ)248+7λ2+2λ+312[t2]E[Q_t+1^2]≥ (1-λ)^248+ 7λ^2+2λ+312E[Q_t^2]. Let a≔(1−λ)248a (1-λ)^248 and b=7λ2+2λ+312b= 7λ^2+2λ+312. Since ⁡[12]≥0E[Q_1^2]≥ 0, iterating gives ⁡[t+12] [Q_t+1^2] ≥a+b[t2] ≥ a+bE[Q_t^2] (42) =a⁡(1+b+…+bt−1)+bt[12] =a(1+b+…+b^t-1)+b^tE[Q_1^2] =a⋅1−bt1−b+bt[12] =a· 1-b^t1-b+b^tE[Q_1^2] ≥a1−b+bt(−a1−b) ≥ a1-b+b^t (- a1-b ) =1−λ4(9+7λ)[1−(7λ2+2λ+312)t] = 1-λ4(9+7λ) [1- ( 7λ^2+2λ+312 )^t ] ∎ E.8 Proof of Lemma 4.9 Proof. By symmetry, ⁡[t+12∣t=q]=⁡[t+12∣t=−q]E[Q_t+1^2 _t=q]=E[Q_t+1^2 _t=-q], so it is enough to prove the bound for q≥0q≥ 0. Write xq=12+qx_q= 12+q and condition on Ot=xqO_t=x_q. By Proposition B.3, the conditional distribution has an atom at OtO_t and a continuous part on the two sides of OtO_t. For the display below, set uz=(z−λxq)/(1−λ)u_z=(z-λ x_q)/(1-λ). Therefore ⁡[t+12∣Ot=xq] [Q_t+1^2 O_t=x_q] =2G(xq)(1−G⁡(xq))q2 =2G(x_q) (1-G(x_q) )q^2 +21−λ∫λxqxq(z−12)2G(uz)g(uz)dz + 21-λ _λ x_q^x_q (z- 12 )^2G(u_z)g(u_z)\,dz +21−λ∫xq1−λ+λxq(z−12)2(1−G(uz))g(uz)dz. + 21-λ _x_q^1-λ+λ x_q (z- 12 )^2 (1-G(u_z) )g(u_z)\,dz. Substitute u=z−λxq1−λu= z-λ x_q1-λ, so z−12=λq+(1−λ)(u−12)z- 12=λ q+(1-λ) (u- 12 ). Then ⁡[t+12∣t=q] [Q_t+1^2 _t=q] =2G(12+q)(1−G⁡(12+q))q2 =2G\! ( 12+q ) (1-G\! ( 12+q ) )q^2 (43) +2∫012+q[λq+(1−λ)(u−12)]2G(u)g(u)du +2 _0 12+q [λ q+(1-λ) (u- 12 ) ]^2G(u)g(u)\,du +2∫12+q1[λq+(1−λ)(u−12)]2(1−G(u))g(u)du. +2 _ 12+q^1 [λ q+(1-λ) (u- 12 ) ]^2 (1-G(u) )g(u)\,du. For the first integral, split at 1/21/2 and substitute u=12−su= 12-s on the left half and u=12+vu= 12+v on the right half. 2∫012+q[λq+(1−λ)(u−12)]2G(u)g(u)u 2 _0 12+q [λ q+(1-λ) (u- 12 ) ]^2G(u)g(u)\,du (44) =2∫012[λq+(1−λ)(u−12)]2G(u)g(u)u+2∫1212+q[λq+(1−λ)(u−12)]2G(u)g(u)u =2 _0 12 [λ q+(1-λ) (u- 12 ) ]^2G(u)g(u)\,du+2 _ 12 12+q [λ q+(1-λ) (u- 12 ) ]^2G(u)g(u)\,du =2∫012[λq−(1−λ)s]2G(12−s)g(12−s)ds+2∫0q[λq+(1−λ)v]2G(12+v)g(12+v)dv. =2 _0 12 [λ q-(1-λ)s ]^2G( 12-s)g( 12-s)\,ds+2 _0^q [λ q+(1-λ)v ]^2G( 12+v)g( 12+v)\,dv. For the second integral, we substitute u=12+su= 12+s. Using the symmetry of G, 1−G⁡(12+s)=G⁡(12−s)1-G( 12+s)=G( 12-s). Therefore 2∫12+q1[λq+(1−λ)(u−12)]2(1−G⁡(u))g(u)u 2 _ 12+q^1 [λ q+(1-λ) (u- 12 ) ]^2 (1-G(u) )g(u)\,du (45) =2∫q1/2(λq+(1−λ)s)2G(12−s)g(12+s)ds. =2 _q^1/2 (λ q+(1-λ)s )^2G\! ( 12-s )g\! ( 12+s )\,ds. Since g is symmetric around 1/21/2, g⁡(12−s)=g⁡(12+s)g( 12-s)=g( 12+s). We split the integral into the intervals [0,q][0,q] and [q,1/2][q,1/2]. Putting these pieces together, we have ⁡[t+12∣t=q] [Q_t+1^2 _t=q] =2G(12+q)(1−G⁡(12+q))q2 =2G\! ( 12+q ) (1-G\! ( 12+q ) )q^2 (46) +2∫0q(λq−(1−λ)s)2G(12−s)g(12+s)ds +2 _0^q (λ q-(1-λ)s )^2G\! ( 12-s )g\! ( 12+s )\,ds +2∫q1/2(λq−(1−λ)s)2G(12−s)g(12+s)ds +2 _q^1/2 (λ q-(1-λ)s )^2G\! ( 12-s )g\! ( 12+s )\,ds +2∫0q(λq+(1−λ)s)2G(12+s)g(12+s)ds +2 _0^q (λ q+(1-λ)s )^2G\! ( 12+s )g\! ( 12+s )\,ds +2∫q1/2(λq+(1−λ)s)2G(12−s)g(12+s)ds. +2 _q^1/2 (λ q+(1-λ)s )^2G\! ( 12-s )g\! ( 12+s )\,ds. Expanding the four squared terms, and grouping the λ2q2λ^2q^2, (1−λ)2s2(1-λ)^2s^2, and λ⁡(1−λ)qsλ(1-λ)qs parts, gives ⁡[t+12∣t=q] [Q_t+1^2 _t=q] =2G(12+q)(1−G⁡(12+q))q2 =2G\! ( 12+q ) (1-G\! ( 12+q ) )q^2 +2λ2q2[∫0qg⁡(12+s)ds+2∫q1/2G⁡(12−s)g(12+s)ds] +2λ^2q^2 [ _0^qg\! ( 12+s )\,ds+2 _q^1/2G\! ( 12-s )g\! ( 12+s )\,ds ] +2(1−λ)2[∫0qs2g(12+s)ds+2∫q1/2s2G(12−s)g(12+s)ds] +2(1-λ)^2 [ _0^qs^2g\! ( 12+s )\,ds+2 _q^1/2s^2G\! ( 12-s )g\! ( 12+s )\,ds ] +4λ(1−λ)q∫0qs[G(12+s)−G(12−s)]g(12+s)ds. +4λ(1-λ)q _0^qs [G\! ( 12+s )-G\! ( 12-s ) ]g\! ( 12+s )\,ds. Here we used G⁡(12−s)+G⁡(12+s)=1G( 12-s)+G( 12+s)=1 on the interval [0,q][0,q], and the cross terms on [q,1/2][q,1/2] cancel. Next, we simplify the terms in each group. A. 2G(12+q)(1−G⁡(12+q))q22G\! ( 12+q ) (1-G\! ( 12+q ) )q^2. 2G(12+q)(1−G⁡(12+q))q2 2G\! ( 12+q ) (1-G\! ( 12+q ) )q^2 =2(14−(G⁡(12+q)−12)2)q2 =2 ( 14- (G\! ( 12+q )- 12 )^2 )q^2 (47) =12q2−2q2(G⁡(12+q)−12)2 = 12q^2-2q^2 (G\! ( 12+q )- 12 )^2 B. The 2λ2q22λ^2q^2 group. We again apply the symmetry property of g to simplify the integral. We have ∫0qg⁡(12+s)s=G⁡(12+q)−12 _0^qg\! ( 12+s )\,ds=G\! ( 12+q )- 12 and, using g⁡(12+s)=g⁡(12−s)g( 12+s)=g( 12-s), ∫q1/2G⁡(12−s)g(12+s)s=12(1−G⁡(12+q))2 _q^1/2G\! ( 12-s )g\! ( 12+s )\,ds= 12 (1-G\! ( 12+q ) )^2. 2λ2q2[∫0qg⁡(12+s)ds+2∫q1/2G⁡(12−s)g(12+s)ds] 2λ^2q^2 [ _0^qg\! ( 12+s )\,ds+2 _q^1/2G\! ( 12-s )g\! ( 12+s )\,ds ] (48) =2λ2q2[G⁡(12+q)−12+(1−G⁡(12+q))2] =2λ^2q^2 [G\! ( 12+q )- 12+ (1-G\! ( 12+q ) )^2 ] =2λ2q2[G⁡(12+q)−12+1+G(12+q)2−2G(12+q)] =2λ^2q^2 [G( 12+q)- 12+1+G( 12+q)^2-2G( 12+q) ] =2λ2q2[(G⁡(12+q)−12)2+14] =2λ^2q^2 [(G( 12+q)- 12)^2+ 14 ] =2λ2q2(G⁡(12+q)−12)2+12λ2q2 =2λ^2q^2(G( 12+q)- 12)^2+ 12λ^2q^2 C. The (1−λ)2s2(1-λ)^2s^2 group. We use s≤qs≤ q on [0,q][0,q] to derive an upper bound. 2(1−λ)2[∫0qs2g(12+s)ds+2∫q1/2s2G(12−s)g(12+s)ds] 2(1-λ)^2 [ _0^qs^2g\! ( 12+s )\,ds+2 _q^1/2s^2G\! ( 12-s )g\! ( 12+s )\,ds ] (49) =2(1−λ)2[∫0qs2(1−2G(12−s))g(12+s)ds+2∫01/2s2G(12−s)g(12+s)ds] =2(1-λ)^2 [ _0^qs^2(1-2G( 12-s))g\! ( 12+s )\,ds+2 _0^1/2s^2G\! ( 12-s )g\! ( 12+s )\,ds ] =4(1−λ)2∫01/2s2G(12−s)g(12+s)s+2(1−λ)2∫0qs2(1−2G(12−s))g(12+s)s =4(1-λ)^2 _0^1/2s^2G\! ( 12-s )g\! ( 12+s )\,ds+2(1-λ)^2 _0^qs^2(1-2G( 12-s))g\! ( 12+s )\,ds =4(1−λ)2∫01/2(12−u)2G(u)g(u)u+2(1−λ)2∫0qs2(1−2G(12−s))g(12+s)s =4(1-λ)^2 _0^1/2 ( 12-u )^2G(u)g(u)\,du+2(1-λ)^2 _0^qs^2(1-2G( 12-s))g\! ( 12+s )\,ds ≤4(1−λ)2∫01/2(12−u)2G(u)g(u)u+4q2(1−λ)2∫0q(G⁡(12+s)−12)g(12+s)s ≤ 4(1-λ)^2 _0^1/2 ( 12-u )^2G(u)g(u)\,du+4q^2(1-λ)^2 _0^q(G( 12+s)- 12)g\! ( 12+s )\,ds ≤4(1−λ)2∫01/2(12−u)2G(u)g(u)du+2q2(1−λ)2(G⁡(12+q)−12)2 ≤ 4(1-λ)^2 _0^1/2 ( 12-u )^2G(u)g(u)\,du+2q^2(1-λ)^2 (G( 12+q)- 12 )^2 D. The 4λ(1−λ)q4λ(1-λ)q group. Again using s≤qs≤ q on [0,q][0,q], 4λ(1−λ)q∫0qs⁡[G⁡(12+s)−G⁡(12−s)]g(12+s)s 4λ(1-λ)q _0^qs [G\! ( 12+s )-G\! ( 12-s ) ]g\! ( 12+s )\,ds (50) =8λ(1−λ)q∫0qs⁡[G⁡(12+s)−12]g(12+s)s =8λ(1-λ)q _0^qs [G\! ( 12+s )- 12 ]g\! ( 12+s )\,ds ≤8λ(1−λ)q2∫12q+12(G⁡(u)−12)g(u)u ≤ 8λ(1-λ)q^2 _ 12^q+ 12 (G(u)- 12 )g(u)\,du =4λ(1−λ)q2(G⁡(12+q)−12)2 =4λ(1-λ)q^2 (G( 12+q)- 12 )^2 Adding the four groups A–D and reorganizing the terms, we obtain ⁡[t+12∣t=q] [Q_t+1^2 _t=q] ≤12q2−2q2(G⁡(12+q)−12)2 ≤ 12q^2-2q^2 (G\! ( 12+q )- 12 )^2 (51) +2λ2q2(G⁡(12+q)−12)2+12λ2q2 +2λ^2q^2(G( 12+q)- 12)^2+ 12λ^2q^2 +4(1−λ)2∫01/2(12−u)2G(u)g(u)du+2q2(1−λ)2(G(12+q)−12)2 +4(1-λ)^2 _0^1/2 ( 12-u )^2G(u)g(u)\,du+2q^2(1-λ)^2 (G( 12+q)- 12 )^2 +4λ(1−λ)q2(G⁡(12+q)−12)2 +4λ(1-λ)q^2 (G( 12+q)- 12 )^2 =(a)4(1−λ)2∫01/2(12−u)2G(u)g(u)du+1+λ22q2 (a)=4(1-λ)^2 _0^1/2 ( 12-u )^2G(u)g(u)\,du+ 1+λ^22q^2 +(−2q2+2λ2q2+2q2(1−λ)2+4λ(1−λ)q2)(G⁡(12+q)−12)2 + (-2q^2+2λ^2q^2+2q^2(1-λ)^2+4λ(1-λ)q^2 ) (G( 12+q)- 12 )^2 =4(1−λ)2∫01/2(12−u)2G(u)g(u)du+1+λ22q2 =4(1-λ)^2 _0^1/2 ( 12-u )^2G(u)g(u)\,du+ 1+λ^22q^2 +2q2(−1+(λ+(1−λ))2)(G⁡(12+q)−12)2 +2q^2 (-1+(λ+(1-λ))^2 ) (G( 12+q)- 12 )^2 =4(1−λ)2∫01/2(12−u)2G(u)g(u)du+1+λ22q2. =4(1-λ)^2 _0^1/2 ( 12-u )^2G(u)g(u)\,du+ 1+λ^22q^2. Step (a)(a) is just a rearrangement of terms. This proves the upper bound for q≥0q≥ 0, and the case q<0q<0 follows by symmetry. Taking expectations on both sides completes the proof. ∎ E.9 Proof of Theorem 4.8 Proof. Since G⁡(u)≤12G(u)≤ 12 for u∈[0,12]u∈[0, 12], 4∫01/2(12−u)2G(u)g(u)u 4 _0^1/2 ( 12-u )^2G(u)g(u)\,du ≤2∫01/2(12−u)2g(u)u ≤ 2 _0^1/2 ( 12-u )^2g(u)\,du (52) =Var(U) = (U) by symmetry around 1/21/2. Let c=(1+λ2)/2c=(1+λ^2)/2 and At=⁡[t2]A_t=E[Q_t^2]. Lemma 4.9 and the preceding bound give At+1≤(1−λ)2Var(U)+cAt.A_t+1≤(1-λ)^2 (U)+cA_t. Iterating the recursion and using A0≤1/4A_0≤ 1/4 gives At A_t ≤ctA0+(1−λ)2Var(U)∑j=0t−1cj ≤ c^tA_0+(1-λ)^2 (U) _j=0^t-1c^j ≤14ct+(1−λ)2Var(U)1−ct1−c ≤ 14c^t+(1-λ)^2 (U) 1-c^t1-c =14(1+λ22)t+2(1−λ)1+λVar(U)(1−(1+λ22)t), = 14 ( 1+λ^22 )^t+ 2(1-λ)1+λ (U) (1- ( 1+λ^22 )^t ), because 1−c=(1−λ)(1+λ)/21-c=(1-λ)(1+λ)/2. Taking t→∞t→∞ gives the stationary bound. ∎ Appendix F Simulation setups and more results Here we provide details of our simulations and more results. We discretize the Markov chain by partitioning the interval [0,1][0,1] into 50005000 equal bins, yielding 50005000 representative nodes in the discretized chain. For each value of λ, we construct the discretized transition matrix from the conditional CDF of the one-step update: the probability of moving from node xix_i to bin [aj,aj+1][a_j,a_j+1] is Pr⁡[Ot+1∈[aj,aj+1]∣Ot=xi]=Fλ,G(aj+1∣xi)−Fλ,G(aj∣xi), [O_t+1∈[a_j,a_j+1] O_t=x_i]=F_λ,G(a_j+1 x_i)-F_λ,G(a_j x_i), where Fλ,G(⋅∣xi)F_λ,G(· x_i) is the conditional CDF induced by drawing two independent voters from the corresponding Beta distribution, applying the anchored update with parameter λ, and taking the median with the current outcome xix_i. We evaluate 100100 equally spaced anchoring parameters λ∈0.00,0.01,…,0.99λ∈\0.00,0.01,…,0.99\. We compute the stationary distribution by power iteration, using the mixing time as the stopping criterion: following Theorem 2.1, we set the mixing-time tolerance to εmix=15000 _mix= 15000. More precisely, for each λ we start from the uniform distribution on the 50005000 nodes and iterate the transition matrix for the mixing-time upper bound ⌈log⁡(1/εmix)log⁡(2/(1+λ))⌉ (1/ _mix) (2/(1+λ)) steps; the resulting distribution is used as the stationary approximation. The empirical mixing time reported in the figures is then computed from the same trajectory as the first time t for which the discretized 11-Wasserstein distance from the time-t distribution to this stationary approximation is at most εmix _mix. For two binned distributions μ and ν on grid midpoints x1,…,xnx_1,…,x_n, this distance is computed as W1(n)(μ,ν)=∑i=1n−1|∑j=1i(μj−νj)|(xi+1−xi).W_1^(n)(μ,ν)= _i=1^n-1 | _j=1^i( _j- _j) |(x_i+1-x_i). When computing the theoretical lower bound according to Theorem 2.1, we take its value to be 00 at λ=0λ=0. The additional Beta distributions below show the same qualitative pattern as in the main text: stronger anchoring increases the empirical mixing time, while the stationary distributions become more concentrated around the deliberative fixed point. (a) (b) (c) (d) Figure 5: Combined results on Beta⁡(0.5,0.7)Beta(0.5,0.7). (a) The distribution of Beta⁡(0.5,0.7)Beta(0.5,0.7). (b) The theoretical mixing time and the estimated mixing time, we use the logarithm scale for the mixing time. (c) The population median, stationary median, and deliberative fixed point for different values of λ. (d) Larger anchoring concentrates the stationary distribution. (a) (b) (c) (d) Figure 6: Combined results on Beta⁡(0.5,5)Beta(0.5,5). (a) The distribution of Beta⁡(0.5,5)Beta(0.5,5). (b) The theoretical mixing time and the estimated mixing time, we use the logarithm scale for the mixing time. (c) The population median, stationary median, and deliberative fixed point for different values of λ. (d) Larger anchoring concentrates the stationary distribution. (a) (b) (c) (d) Figure 7: Combined results on Beta⁡(1,2)Beta(1,2). (a) The distribution of Beta⁡(1,2)Beta(1,2). (b) The theoretical mixing time and the estimated mixing time, we use the logarithm scale for the mixing time. (c) The population median, stationary median, and deliberative fixed point for different values of λ. (d) Larger anchoring concentrates the stationary distribution. (a) (b) (c) (d) Figure 8: Combined results on Beta⁡(2,5)Beta(2,5). (a) The distribution of Beta⁡(2,5)Beta(2,5). (b) The theoretical mixing time and the estimated mixing time, we use the logarithm scale for the mixing time. (c) The population median, stationary median, and deliberative fixed point for different values of λ. (d) Larger anchoring concentrates the stationary distribution. (a) (b) (c) (d) Figure 9: Combined results on Beta⁡(5,5)Beta(5,5). (a) The distribution of Beta⁡(5,5)Beta(5,5). (b) The theoretical mixing time and the estimated mixing time, we use the logarithm scale for the mixing time. (c) The population median, stationary median, and deliberative fixed point for different values of λ. (d) Larger anchoring concentrates the stationary distribution.

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