Contribution
It turns the opinion pool, a weighted average Σπᵢ Fᵢ of the members’ distributions whose weights earlier writers had no principled way to choose, into the limit of a process. Each member chooses weights on the others’ opinions once. Iterating those averages either converges to a common pool or does not, and Markov-chain theory says which and gives the pool’s weights. Of the three advantages the paper claims, the second and third are the substance: “It presents simple conditions for determining whether it is possible for the group to reach a consensus” and “When a consensus can be reached, the weights to be used in this consensus can be explicitly and simply calculated” (§1).
Key insight
The weight matrix P is row-stochastic, so it “can be regarded as the one-step transition probability matrix of a Markov chain with k states” (§5). Repeated averaging is then the chain’s n-step matrix Pⁿ applied to the opinions. A consensus is the chain forgetting where it started, so that every row of Pⁿ tends to the same vector π. Who ends up counting in the consensus is decided by who is listened to, indirectly as well as directly, and not by how much each member trusts himself or by any member’s accuracy.
Assumptions
- k members. Each holds a subjective distribution Fᵢ for a parameter θ on an abstract measurable space Ω, or, in §7, a point in a fixed convex set of an arbitrary linear space, such as a real-valued estimate.
- Each member i fixes weights p_ij ≥ 0 with Σⱼ p_ij = 1 before seeing the others’ opinions. The first revision is F_i1 = Σⱼ p_ij Fⱼ (2.1).
- The weights stay fixed at every stage. The justification is consistency: if member i “still wishes to form his subjective distribution by assigning the weight p_ij to the distribution of individual j—and there does not seem to be any basis for his changing these weights at this time—then in order to remain consistent with this principle he must again revise” (§3). Hence F⁽ⁿ⁾ = PF⁽ⁿ⁻¹⁾ = PⁿF (3.3).
- Revisions go on “indefinitely or until F⁽ⁿ⁺¹⁾ = F⁽ⁿ⁾” (§3).
- No outside information: “there is no possibility of learning whether the opinion of one individual is closer to the truth than that of another” (§8).
Key results
- Definition (§4). “A consensus is reached if and only if all k components of F⁽ⁿ⁾ converge to the same limit as n → ∞” (4.1). From (3.3), “a consensus is reached if and only if there exists a vector π” with lim p_ij⁽ⁿ⁾ = π_j for all i, j (4.2). The common distribution is then Σπᵢ Fᵢ.
- Theorem 1 (§5), “adapted from a result given by Doob”: “If there exists a positive integer n such that every element in at least one column of the matrix Pⁿ is positive, then a consensus is reached.”
- Theorem 2 (§5). “If all the recurrent states of the Markov chain communicate with each other and are aperiodic, then a consensus is reached.” This is weaker than irreducibility: transient members are allowed.
- The negative cases (§5). “On the other hand, if the states of the chain form at least two disjoint closed sets of communicating states or if the communicating states in a single closed set are periodic, then a consensus is not reached. These results are discussed in texts such as Feller [5], Karlin [6], or Parzen [10].” So the periodic case is named, and named as a failure.
- Theorem 3 (§6). “Suppose that a consensus is reached and let Σπᵢ Fᵢ denote the common subjective distribution that is reached in the consensus. Then π = (π₁, ···, π_k) is the unique stationary probability vector.” It is found from πP = π and Σπᵢ = 1. A transient member gets πᵢ = 0: “no weight will be assigned in the consensus to the prior distribution Fᵢ of individual i.”
- Examples (§6.1). P = [[½, ½], [¼, ¾]] gives π = (⅓, ⅔). Adding a third member who weights everyone ⅓, with no one weighting him, gives (⅓, ⅔, 0): he is transient. P = I gives no consensus. A four-member P of two ½-blocks gives two internal consensuses after one revision and none overall. “In general any individuals for whom the corresponding states of the Markov chain form a closed communicating, aperiodic class will reach a consensus among themselves.”
- Delphi (§8). Self-weight p_ii ∈ (0, 1) and equal weights (1 − p_ii)/(k − 1) on the rest always lead to a consensus by Theorem 1. The model is offered as “a formalization of the Delphi technique, a formalization in which a consensus is reached.”
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Iterated fixed-weight averaging of opinions reaches a consensus if some power of P has a strictly positive column | strong | Theorem 1, cited from Doob, no proof given; re-derived in THEORY-960 |
| C2 | It reaches one if all recurrent states communicate and are aperiodic | strong | Theorem 2, stated as “analogous”, no proof; equivalent to C1 for a finite chain (re-derived, not remarked in the paper) |
| C3 | Two closed classes, or one periodic closed class, give no consensus | strong as a statement about consensus from every start; it needs that quantifier | stated, with textbook references; checked |
| C4 | When a consensus is reached its weights are the unique stationary vector, zero on transient members | strong | Theorem 3, stated as “well known”; re-derived |
| C5 | The model applies unchanged to point estimates in any convex subset of a linear space | strong | §7; the iteration is linear and convex |
| C6 | It formalizes the Delphi technique, of which it is a special case under equal weights to others | moderate | §8, argued, no data |
Concepts
- Consensus: all members’ opinions converge to one common limit (4.1). Not agreement at a finite stage, though it may happen at one, as in the four-member example.
- Opinion pool: a convex combination Σπᵢ Fᵢ of the members’ distributions. The term is Stone’s (1961), as the paper says.
- Stationary probability vector: π ≥ 0, Σπᵢ = 1, πP = π (§6).
- Recurrent, transient, closed class, periodic: the Markov-chain terms, used without definition and referred to Feller, Karlin and Parzen.
Connections
It cites earlier work on pooling subjective probabilities: Eisenberg and Gale (1959) and Norvig (1967) on consensus and pari-mutuel methods, Stone (1961) on the opinion pool, Madansky (1964) on externally Bayesian groups, Morris (1971), Winkler (1968) and Savage (1971). The Markov-chain results are from Doob (1953), Feller (1968), Karlin (1966) and Parzen (1962). The Delphi technique is cited through Dalkey et al. (1972). The record reads its descendants: Golub and Jackson’s naïve learning (LIT-999, THEORY-321), which calls the same iteration DeGroot updating and takes the stationary vector as influence; the DeGroot–Friedkin model of Jia et al. (LIT-1146); and Friedkin, Jia and Bullo (LIT-1482), who let the self-weights change, the possibility DeGroot names in §8 and leaves out.
Bearing on the record
- THEORY-960 is filed from this reading: the conditions for consensus and the weights it uses, as DeGroot states them, re-derived.
- THEORY-321 already uses this model through Golub and Jackson. Its “strongly connected” narrowing is stronger than DeGroot’s Theorem 2 needs for a consensus to exist: one closed class plus aperiodicity suffices, with transient members getting zero weight. Wisdom is Golub and Jackson’s question, not DeGroot’s, so nothing in THEORY-321 changes.
- CASE-160. Its ℓ, the left eigenvector of a row-stochastic W with ℓᵀW = ℓᵀ and Σℓᵢ = 1, is DeGroot’s π for P = W, by the same equation (Theorem 3). Its discrete-time amendment, that x ← Wx settles only if W is aperiodic, is DeGroot’s own negative statement for a periodic closed class (§5). Its exclusion of “a reducible W” is broader than needed: with one closed class and transient members, ℓ is unique, zero off the class, and the closure still holds (checked, in THEORY-960). The roles differ. DeGroot’s π weights the initial opinions in a limit. The case’s ℓ weights a running coordinate that the coupling conserves. They are the same invariant, πPⁿF = πF.
- CLAIM-453 lists “a strongly connected population” among its conditions. DeGroot’s condition is one closed class (with aperiodicity in discrete time).
- CLAIM-262 and the owner’s exchange: the U36 reply’s “an emergent summary of who is listened to, much as in classical consensus and DeGroot models” is borne out. In DeGroot the weights are fixed and chosen, so “who is listened to” is literal, and accuracy plays no part.
The proposed edits to those entries are written for the consolidator, not made here. No instruction for machine-learning practice, so nothing for the anthology.
Limitations
- No proofs. All three theorems are cited from the Markov-chain literature. The paper’s contribution is the reading of opinion pooling as a chain, not new mathematics.
- The “if and only if” in §4 needs a quantifier. Convergence of every Pⁿ row to π is necessary for a consensus from every initial F, not from a given one. With P = [[0, 1], [1, 0]] the opinions never converge for F₁ ≠ F₂, but two members who start equal stay equal. The negative cases in §5 hold in the same sense: no consensus for a generic start.
- Theorems 1 and 2 are equivalent for a finite chain, and with the negative cases they make a complete characterization: a consensus from every start exists iff the chain has exactly one closed class and it is aperiodic. The paper presents them as “analogous” results and does not say this.
- Fixed weights and no truth. The weights never change, and no observation enters, so the model says nothing about whether the consensus is right. That question is Golub and Jackson’s.
- Discrete time only. The periodic failure is a discrete-time one: in continuous time, ẋ = (P − I)x, a periodic P converges (checked).
- A typo. In §6 the normalization reads Σᵢ₌₁ⁿ πᵢ = 1 for Σᵢ₌₁ᵏ.
Open questions
- What happens when members revise their weights after seeing how others move? §8 names this and leaves it. Friedkin, Jia and Bullo (LIT-1482) and the DeGroot–Friedkin model (LIT-1146) take it up.
- How fast is the consensus reached? The paper says nothing about rates. Golub and Jackson treat speed separately from correctness (THEORY-321).