Contribution
It turns the looking-glass self into a rule for updating an influence network. Cooley’s and Mead’s premise, that self-appraisal follows others’ appraisals, is formalized as one equation: a member’s openness to influence on the next issue reflects how much of the last issue’s outcome was theirs. Put in the Friedkin–Johnsen model of a single issue, the rule drives almost every group to a single leader. The paper calls this a near “iron law”, gives a laboratory test of the rule (not of the end state), and offers it as a micro-level route to Michels’s oligarchy that needs no bureaucracy.
Key insight
If those whose positions carry become more closed and those whose positions do not become more open, the inequality compounds across issues. The leader’s control in the end runs through the network of the others, who stay connected to each other and fully open: the end state is not everyone deferring to one person directly (form (a)) but one self-weight of 1 with an accommodative mass whose interpersonal weights carry it on (form (b)).
Assumptions
- Influence network. Each member i allots weights w_ij ≥ 0, Σ_j w_ij = 1; w_ii is the self-weight. W_s = A_s C + I − A_s (9), with C the relative interpersonal weights c_ij = w_ij/(1 − w_ii), fixed across issues, and A_s = diag(1 − w_sii).
- One issue (Friedkin–Johnsen). y(t+1) = AWy(t) + (I − A)y(1) (2): an ongoing anchorage on initial positions. At equilibrium y(∞) = Vy(1), with V = (I − AW)⁻¹(I − A) (6) when I − AW is nonsingular.
- Reflected appraisal across issues. w_(s+1)ii = (1/n) Σ_j v_sji (10): the self-weight becomes the member’s share of all total influence in V_s. Appraisals are assumed accurate, “like the assumptions of perfect information and rational action”, and members are assumed keenly interested in the group’s outcomes.
- Scope. W_1 strongly connected, with an equilibrium V_1 that shows some inequality; one issue domain; fixed membership and fixed C.
Note on (2) and (9) together: since W already holds I − A on its diagonal, AW gives the self-weight a second role, so a member’s own initial position enters both through the anchorage and through their own diagonal weight. This is the specification the paper states; it is not the DeGroot step that LIT-1146 analyses.
Key results
- Experiment. 50 tetrads (25 all-female, 25 all-male), each through five issues (three risky-shift, two monetary awards), in five communication networks, by telephone, with self-weights derived from observed position changes by (12). Multilevel regression of the next self-weight on prior control and prior self-weight, 800 cases (50 groups × 4 members × 4 transitions): control on prior issue .199 (SE .079, p = .011); self-weight on prior issue −.124 (SE .054, p = .021); constant .198 (SE .021). Prior control raises the next self-weight; extreme self-weights regress.
- Simulation (Table 2). 10,000 random systems for each of two types and five sizes (5–25): Type 1 complete, Type 2 with random densities and arrangements of zero weights. The share that evolves to a single leader is 0.9996–1.0000 for Type 1 and 0.9900–0.9987 for Type 2. The rest end bilateral, w_kk + w_mm = 1, and in those one member holds on average 0.70–1.00 of the control (panel c). The member with the most initial control becomes the leader in about 0.56 of five-person groups and 0.22–0.26 of 25-person ones (panel b).
- Not established. The end state itself is shown only in simulation; the experiment tests one step of the rule, over five issues.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A member’s control over one issue’s outcome raises their self-weight on the next | moderate | experiment, 50 groups, one design, Table 1 |
| C2 | Under rule (10) in the Friedkin–Johnsen model, almost every strongly connected group evolves to a single leader | strong, within the model | simulation of 100,000 systems; re-run here for complete networks |
| C3 | The leader’s control is exercised through intermediaries, not by direct deference | strong, within the model | the form of the end state |
| C4 | The tendency toward concentrated power is independent of bureaucracy and may hold in any enduring group | weak | inference from C2 to real groups, with no test |
| C5 | The tendency disappears if members resist becoming accommodative or if accorded influence is regularly disturbed | weak | argued from the model, not simulated |
Method
- Fix C and an initial self-weight vector, giving W_1.
- For issue s, form W_s from (9) and compute V_s from (6).
- Set each member’s next self-weight to the column mean of V_s, (10).
- Repeat across issues; report the end state.
The behavioural self-weight (12) inverts the model on each member’s observed movement, so that the experiment can be scored.
Concepts
- Self-weight (w_ii): openness or closure to influence on an issue; “a social construction, as opposed to fixed personality characteristics”.
- Control matrix V: the total direct and indirect contribution of each member’s initial position to each member’s settled one.
- Reflected appraisal: as Eq. (10), the self-weight set to one’s appraised share of control.
- Self-appropriation: Cooley’s premise, as Friedkin reads it, that a member accorded influence keeps it rather than passing it on.
Connections
The DeGroot–Friedkin model of Jia, MirTabatabaei, Friedkin and Bullo (LIT-1146, NOTE-1135, THEORY-916) is the same reflected-appraisal rule on a different single-issue step: DeGroot averaging to consensus, so that control is the dominant left eigenvector of W, with no anchorage on initial positions. Its result is not this one. Under it, self-weights go from every non-autocratic start to one equilibrium ordered by eigenvector centrality, autocratic exactly for a star. The re-run below puts the two side by side. The paper cites French (1956), Harary (1959) and DeGroot (1974) as the single-issue tradition, Michels and Mayhew and Levinger for oligarchy, and Cooley (Human Nature and the Social Order, LIT-1379), Mead and Gecas and Schwalbe for reflected appraisal.
Bearing on the record
- The re-run. Rule (10) with (6) and (9) as printed, on 300 random complete networks each of sizes 5, 10 and 20 (scratch script, 2026-10-29): every one ended with one self-weight above 0.99 and the rest below 0.01, which matches Table 2’s Type 1. On 300 random strongly connected networks of size 10 and density 0.4, 93% ended with a single leader and the rest bilateral, a larger bilateral share than Table 2’s Type 2 (under 1%). The Type 2 densities were random, so the two are not the same population. On 100 random complete networks of sizes 5 and 10, the DeGroot–Friedkin update gave a median largest self-weight of 0.29 and 0.13, and never above 0.40, where this paper’s update gave 1.0. The concentration depends on the single-issue model. With Friedkin–Johnsen anchorage, as here, power concentrates almost always; with DeGroot consensus, as in LIT-1146, it does not. Filed as THEORY-1161 (Proposed).
- CASE-092’s note of 2026-10-23 says, from the abstract, that reflected appraisal “generates transitions of the influence network toward concentrated power” and treats LIT-1404 and LIT-1146 as one baseline. The reading says they are two baselines with different predictions, and that a coupled meaning–influence model compared with “DeGroot–Friedkin” should say which single-issue step it is compared with. A proposed dated note is in the coordinator’s report.
- THEORY-916 states the DeGroot–Friedkin result. This paper is not a counter-example to it, since its model differs, but a reader of THEORY-916 should know that the same rule gives autocracy under the Friedkin–Johnsen step.
- TERM-075 (attributed self). Friedkin’s formalization is of the sense in which a person takes up what others take them to be: the self-weight is set by “appraisals of others” of one’s control, “assumed accurate”. It formalizes the held and appropriated senses only when the imagined and held appraisals agree, and the paper assumes they do.
- CLAIM-297 (an operator’s authority updated by how well it performs on the interpretation) has the same shape: authority on the next round set by success on the last. This paper is a worked case in which that update concentrates authority.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- The experiment tests one step of the rule over five issues in four-person groups; the iron law is the simulation’s.
- Appraisals are assumed accurate and C fixed; the paper says other specifications “are possible” and gives none.
- Footnote 6 says the construct gives 0 ≤ w_ii < 1, which “excludes the extreme case of a completely self-weighted individual”, yet the end state is “a single completely self-weighted, inner-directed individual”. The end state is a limit the construct approaches and does not reach; the paper does not say so.
- Errors and slips. The reference to Krackhardt (1990) gives Administrative Science Quarterly “50: 505–535”; the 1990 volume of ASQ is 35, and Crossref gives the paper as 35, first page 342 (doi:10.2307/2393394, checked 2026-10-29). “what has been referred to as a ”becalming’ (Zald and Ash, 1966)” lacks a closing quotation mark. “a concentrated form of power exits or not” for “exists”. In the extracted text two displays carry the number (8), the vector of settled positions and the matrix C.
Open questions
- Which single-issue step do real groups follow, anchored or consensual? The two predict opposite end states under the same reflected appraisal, and the paper’s experiment, scored by the anchored model, cannot tell them apart.