Chapter 10: Recommendation Systems and Algorithmic Curation

A content platform recommends items to a flowing audience, and the recommendations themselves change which users come back, what they engage with, and how niche their tastes become. This chapter applies the Model Specification Protocol of Chapter 9 to that loop and builds a model large enough to make filter-bubble configurations visible in the phase space. The earlier running examples did not have that structure: the Chapter 0 two-state model had no separated clusters, and the Section 2.8 routing chain had no competing traffic patterns. The conclusion is deliberately limited: heavily specialized curation policies appear as boundary equilibria of the coupled actor-critic-law system, while a diverse-engagement operating point appears as an interior equilibrium. The compact attractor contains these equilibria, but containment in the attractor is not the same as attraction from nearby interior states; that stability question is decided by the transverse sign calculation below and, in higher-dimensional slices, by numerical basin computation.

A content platform recommends items to users. At each moment, the platform maintains a curation policy (the actor), a running estimate of how valuable different recommendations are (the critic), and a picture of the audience composition: how many users are browsing diverse content, how many are engaged with niche cluster A, and how many are deep in niche cluster B.

These three quantities form the familiar feedback loop. The curation policy shapes which content users see, which changes their browsing behavior and shifts the audience composition, which changes the rewards the platform observes, which feeds back into the curation policy. The loop closes on itself.

In the Chapter 0 model the same feedback loop was present, but with only two user types the platform could only distinguish “prefers action 1” from “prefers action 2,” leaving no room for a genuine content cluster to form. A filter bubble requires at least three ingredients: a diverse starting state where users are not yet specialized, a niche discovery state where users begin engaging with a specific content type, and a deep-engagement state where users are locked into that content type with low probability of returning to diverse browsing. Two states cannot represent this progression.

We therefore build a model with five user-browsing states arranged in a Y-shaped graph. The root of the Y is a diverse-browsing state. Two branches extend from the root, each consisting of a niche-discovery state and a deep-engagement state. The platform’s curation policy can push users deeper into a niche (curate) or pull them back toward diverse browsing (diversify).

The Y-graph captures the essential filter-bubble dynamics: users who start in diverse browsing can be drawn into either niche branch by curation, and once in deep engagement they are hard to pull back. The five states are few enough for explicit computation but enough for genuinely competing attractor branches.

We execute Steps 1—4 of the Model Specification Protocol.

Step 1. State space. Five user-browsing states on a Y-graph:

• State 1: Diverse browsing (the Y-root). Users browse a mix of content types.
• State 2: Niche-A discovery. Users have begun engaging with content cluster A (e.g., political commentary) but have not committed.
• State 3: Niche-A deep engagement. Users consume cluster-A content almost exclusively and rarely return to diverse browsing.
• State 4: Niche-B discovery. Users have begun engaging with content cluster B (e.g., entertainment) but have not committed.
• State 5: Niche-B deep engagement. Users consume cluster-B content almost exclusively.

The graph structure is on one branch and on the other, with no direct edges between the two branches.

Step 2. Action space. Two curation modes at each state:

• (Curate): recommend content aligned with the user’s current cluster. At state 1, this has no strong effect (diverse content is not a cluster). In niche states, this pushes users deeper into the niche.
• (Diversify): recommend content from outside the user’s current cluster. At state 1, this maintains diverse browsing. In niche states, this pulls users back toward the root.

Step 3. Actor features. Two-dimensional actor () with block-diagonal features, as previewed in Chapter 9.3. The first coordinate controls curation in the A-branch, and the second coordinate controls curation in the B-branch. At the diverse root (state 1), the features are zero: the policy is uniform regardless of .

• State 1: and .
• State 2: and .
• State 3: and .
• State 4: and .
• State 5: and .

When , the policy at states 2 and 3 favors Curate (deeper niche-A engagement). When , it favors Diversify. The second coordinate does the same for the B-branch, independently.

At state 1, the features are zero, so and for all . The platform’s policy at the diverse root is always a coin flip between curating and diversifying. This is a modeling choice, not a necessity: it says the platform does not have a curation lever that applies specifically to already-diverse users. The policy acts only in the niche branches.

Step 4. Critic features. Two-dimensional critic () with the same block-diagonal structure:

• State 1: and .
• State 2: and .
• State 3: and .
• State 4: and .
• State 5: and .

The critic tracks two reward signals, one per branch. The first component estimates the curation advantage in branch A, and estimates it in branch B.

Step 5. Rewards. The reward table encodes the core tension of algorithmic curation: per-interaction engagement is higher in niche states under curation, but diverse browsing has its own value.

• State 1: and .
• State 2: and .
• State 3: and .
• State 4: and .
• State 5: and .

At the diverse root (state 1), diversification pays more than curation: . Users in diverse browsing engage more when shown a mix of content. In niche states, curation pays progressively more: at discovery states, ; at deep engagement, . Users deep in a niche click on aligned content at very high rates and barely respond to diverse recommendations.

The tension is between short-horizon and long-horizon outcomes. Curating in niche states earns immediate reward, but it pushes users deeper into the niche, making future diversification harder and future engagement dependent on a narrow content supply. Diversifying earns less per interaction but pulls users toward the diverse root, where the audience is more flexible.

The rewards are symmetric between the two branches: , , and so on. This is a deliberate simplification. A real platform would have different engagement rates for different content types. Breaking the branch symmetry would split the two bubble equilibria into attractors of different sizes–an interesting variation that Exercise 10.5 explores.

Step 6. Generator construction. We build two action-conditioned generators (Curate) and (Diversify) on the Y-graph. The graph topology constrains the sparsity: transitions are only possible along Y-graph edges (, , , ). No direct transitions between the two branches are possible.

Curate generator : Under curation, users are pushed deeper into their current branch. Users at the diverse root are drawn into either niche with moderate rates. Users in discovery states advance into deep engagement. Users in deep engagement have very low return rates.

The entries read as follows. From the diverse root (row 1), users transition to niche-A discovery (state 2) at rate 0.6 and to niche-B discovery (state 4) at rate 0.6. From niche-A discovery (row 2), users advance to deep engagement (state 3) at rate 0.6 or return to the root at rate 0.2. From deep engagement (row 3), users return to discovery at rate 0.1–a slow leak. The B-branch rates are symmetric.

The key structural feature: under curation, the deep-engagement states (3 and 5) are nearly absorbing. The exit rate is only 0.1, so once a user enters deep engagement under sustained curation, the expected residence time is 10 units. This is the generator-level encoding of the filter-bubble phenomenon: curation creates sticky niche states.

Diversify generator : Under diversification, users are pulled back toward the root. Discovery users return to the root at high rates. Deep engagement users are pulled to discovery at moderate rates.

Under diversification, the return rate from deep engagement to discovery is 0.8 (row 3, column 2), and the return rate from discovery to the root is 1.0 (row 2, column 1). Both are much higher than under curation. The root itself has low exit rates under diversification (total rate 0.4 versus 1.2 under curation), so diverse users tend to stay diverse.

Verification of the generator property. Both and have nonnegative off-diagonal entries (all rates are nonnegative) and rows that sum to zero by construction. The softmax mixture is built row-by-row, with each row of the convex combination of the corresponding rows of and with weights and ,

matching the generator-construction template of Chapters 2 and 6. The mixture inherits the generator property because each row is a convex combination of generator rows.

Reference-state structure. State 1 (the diverse root) is the reference state for minorization. From every state, there is a path to state 1 of length at most 2:

• From state 2: (one step, rate under either action).
• From state 3: (two steps, minimum rates and ).
• From state 4: (one step, rate ).
• From state 5: (two steps, minimum rates and ).

All rates are bounded below by positive constants independent of , since the softmax assigns positive weight to both actions. The bridge theorem of Chapter 6 (Corollary 6.5, via Theorems 6.3 and 6.4) consumes Assumption 6.2, uniform exponential mixing of the frozen chain. The minorization criterion that produces it, Proposition 6.6, postulates a uniform direct jump rate into the reference state from every other state, and the Y-graph fails that literal hypothesis: the deep-engagement states 3 and 5 have no direct edge to the root, under both actions. The two-step structure substitutes for it at the level of the propagator. For fixed and every state , the entry is bounded below by a constant independent of , because mass moves from to the intermediate state at rate at least on a window of length and from there to the root at rate at least on the remaining window; Exercise 10.9 carries this computation for the worst case (state 5 is symmetric), states 2 and 4 have a direct edge and need only the one-step estimate from the proof of Proposition 6.6, and the root retains mass by the Gronwall bound of that proof’s first stage. Column domination of a fixed-time propagator is exactly the input to the contraction stage of the same proof, which uses only the kernel bound and not the rate hypothesis, so that stage yields Assumption 6.2 with and . Theorems 6.3 and 6.4 then give the unique frozen invariant law and the Lipschitz map , so the bridge theorem applies. Chapter 11 (Lemma 11.1) states and proves the same two-step minorization with explicit constants for its hub-and-spoke chain.

We now assemble the recommendation ODE from the data of Sections 10.2—10.4.

Phase space. The system lives on , where with . The phase space dimension is (using four free coordinates on the five-state simplex).

Occupancy measure. For each state-action pair,

At the diverse root, for all , so . In the A-branch (states 2, 3), the policy depends only on : , where .

Critic coefficients. From Chapter 2, Sections 2.3—2.4:

The block-diagonal feature pattern collapses the rank-one outer products to coordinate projectors:

since and similarly for the B-branch features. Summing weighted by yields a diagonal :

where is the total occupancy at A-branch states and is the total occupancy at B-branch states. The critic equation decouples into two scalar equations:

Actor drift. The raw drift has two components. The first component receives contributions only from A-branch states (the features at state 1 are zero, and is zero at state 1 since is identically zero in ; the features at B-branch states have zero first component):

Following the Chapter 2.4 decomposition, we split the integrand into three pieces–reward advantage, critic correction, and entropy regularization–and write with

where is the entropy of a probability vector , and the second equality in the last line uses together with . The softmax entropy is maximized at the uniform policy and decreases as , so the entropy gradient is positive at and negative at for . Section 10.6 invokes this sign pattern when locating the interior zero.

The second component has the same structure with B-branch states and , and the same three-piece decomposition . The two-dimensional damping is

and the damped drift is .

The full system. Setting and and :

This is a system of the form in Chapter 2, Definition 2.10, with the distribution equation in controlled-chain generator form rather than prescribed-closure relaxation form, so the distribution tracks the chain’s generator dynamics instead of relaxing toward a prescribed closure.

Standing constants. From the data tables: , , . The critic forcing bound from Chapter 4 is . The Chapter 4 absorbing set is with .

We now compute the equilibria of the recommendation system and identify them in domain terms.

We label the equilibria (balanced curation, interior), (A-bubble), (B-bubble), and (dual bubble), written without subscripts to keep them distinct from the entropy components of the drift decomposition in Section 10.5. The interpretive labels are domain readings of the equilibrium structure rather than theorem outputs. Proposition 10.1 below makes a purely mathematical statement about the existence and location of the four equilibria; the identification of with a filter bubble in branch A is a modeling reading, valid under the specific feature, reward, and generator choices of Sections 10.2—10.4.

Boundary equilibria. The damping vanishes when for some coordinate , so on each boundary face the damping suppresses the actor drift regardless of the raw gradient. At fixed , the critic and distribution equations each have a unique equilibrium: the critic converges to by coercivity, and the distribution converges to the invariant law of the generator by exponential mixing.

Some boundary equilibria have direct domain meaning. Locating them needs a little care: with a two-dimensional actor, saturating a single coordinate (, say) makes only the -drift vanish through the damping factor . The other coordinate is still interior, so its equation forces at any equilibrium. A boundary equilibrium therefore lives either at a corner, where both coordinates saturate and the damping kills both drifts outright, or on an open face, where one coordinate saturates and the raw drift in the other coordinate happens to vanish.

The same damping mechanism also creates mathematically possible boundary candidates that are not the three curation configurations emphasized below. Faces with correspond to anti-curation or aggressive diversification in that branch, and mixed-sign corners correspond to asymmetric policies that curate one branch while diversifying the other. We therefore do not claim that the four equilibria listed below exhaust the boundary. They are the domain-relevant candidates selected by the symmetric reward and feature design; additional face or corner equilibria must be checked by the same raw-drift equations on each face.

E2: The A-bubble. Saturate the A-branch by setting , and let be a zero of the B-branch raw drift on that face, with . Such a zero exists by the same intermediate-value argument used below for the interior point, applied to the continuous map restricted to the face, with the endpoint signs supplied by Hypothesis (H’) recorded below (numerically ). The policy at A-branch states strongly favors Curate (), so the generator pushes users into A-branch deep engagement and lets them leak slowly back, and the frozen invariant law concentrates mass on states 2 and 3. In domain terms, the platform curates heavily in niche A and the audience becomes dominated by users deep in that content cluster. This is the branch-A filter-bubble configuration.

The A-branch raw drift is nonzero at this point; for the reward, feature, and generator data of Sections 10.2—10.4 it is in fact negative (Exercise 10.3 asks the reader to compute its sign), because the entropy-regularization piece dominates the reward-advantage and critic-correction pieces at the saturated edge. The equilibrium persists because the damping suppresses this drift entirely, not because the incentive signal vanishes.

The sign also determines the transverse stability of the saturated face. Write for a nearby interior point with . Then

Thus, at the face, a negative saturated raw drift makes : nearby interior points move away from the face. For the baseline reward and entropy parameters used here, the pure curation face is therefore transversely repelling from the interior, not a lock-in attractor. True attracting lock-in at this boundary would require the opposite transverse sign, for example after changing the reward scale or the regularization strength.

E3: The B-bubble. By the branch symmetry of the model, saturating the B-branch with and taking the matching zero of the A-branch raw drift on that face produces an analogous equilibrium with mass concentrated on states 4 and 5. The platform curates heavily in niche B.

E4: The dual bubble. Set , . At this corner the damping vanishes in both coordinates, so is an equilibrium outright, with no condition on the raw drift. The policy strongly curates in both branches simultaneously, and the audience splits between the two niche clusters with little mass at the diverse root. This is the worst-case scenario from a diversity perspective: the platform has specialized in both directions, and only users who arrive in diverse browsing (a shrinking population under this policy) see mixed content.

Interior equilibrium. An interior equilibrium requires and simultaneously, with in the interior of . We exploit the branch symmetry of the model to reduce the simultaneous 2-D problem to a 1-D one. Rewards, features, and generators are all invariant under the swap ; Section 10.3 records the reward symmetry, and the generator and feature symmetries are visible in Sections 10.2 and 10.4.

Equivariance under implies the diagonal is invariant under the flow, and on the two raw drifts coincide: . Restricting to the reduced manifold inside , where , , and the diagonal symmetry holds, it suffices to locate a single zero of the scalar function

We check the signs of at the box edges using the three-piece decomposition of Section 10.5.

Sign at s = -1$: The policy at A-branch states strongly favors Diversify, with . A direct computation using the niche-state rewards and the score functions , at gives . The entropy piece is positive at because on , with the entropy increasing as moves toward zero from the box edge, and evaluates to . The critic-correction piece is , so . A sufficient condition for is , comfortably inside ; we adopt at as part of Hypothesis (H) below.

Sign at s = +1$: The policy at A-branch states strongly favors Curate. The entropy piece satisfies , negative because on , and the reward-advantage and critic-correction pieces evaluate to and by the same kind of direct computation as at . The damping does not enter the raw drift, so the sign of is decided by

This is negative whenever the coefficient on is negative, , and the coefficient on is negative, ; the binding constraint is . We collect both endpoint conditions as a single sign hypothesis on the frozen critic along the diagonal:

Both bounds sit well inside the absorbing-set range . The face equilibria and use the same edge sign pattern on the boundary faces, where the moving coordinate is governed by the other critic component, so we record the branch-symmetric analogue:

and symmetrically at on the face . As on the diagonal, the lower bound is binding at the endpoint (it produces the positive sign) and the upper bound at the endpoint (the negative sign). Exercise 10.2 verifies an analogous bound at the boundary by direct computation of , and the parallel computation at and follows the same route. Under Hypothesis (H), and .

The intermediate value theorem applied to the continuous function on yields with . By -equivariance, jointly solves on the reduced manifold, and the corresponding inherit the diagonal symmetry. The result is an interior equilibrium distinct from the three boundary equilibria , each of which has at least one coordinate equal to .

In domain terms, is a balanced curation level: the platform partially curates in each branch, but not aggressively enough to concentrate users in a single branch. The audience distribution has substantial mass at the diverse root, and the critic’s value estimate reflects the moderate engagement rates of a mixed audience.

Proposition 10.1. Under Hypotheses (H) and (H’), the recommendation system with the Y-graph data of Sections 10.2—10.4 has at least four equilibria: three boundary equilibria (A-bubble E2$, B-bubble E3$, dual bubble E4$) and one interior equilibrium E1$ where the advantage signal vanishes in both coordinates.

Proof. At frozen the critic and distribution equations have unique fixed points (by coercivity of and exponential mixing of ), so an equilibrium is determined by the actor equation for . The dual-bubble corner is an equilibrium outright: the damping vanishes in both coordinates, so both actor equations hold regardless of the raw drift. The A-bubble and B-bubble are face equilibria: one coordinate saturates (killing its drift through the damping), while the other coordinate is interior, so its raw drift must vanish. On the face the continuous map is positive at and negative at (the same edge sign pattern established for below, with the branch roles exchanged, using Hypothesis (H’) for the face critic component), so the intermediate value theorem yields with ; the B-bubble is symmetric. For the interior equilibrium , the branch symmetry of the model (rewards, features, generators all invariant under the swap defined above) makes the diagonal invariant under the flow, and on the raw drifts satisfy . Restricting to the reduced manifold intersected with , the scalar function is continuous on , with and under Hypothesis (H) above (positive at from the joint contribution of reward-advantage and entropy pieces, with the bounded critic-correction term controlled; negative at from the entropy piece dominating the reward-advantage and critic-correction pieces, with damping playing no role in the raw drift). The intermediate value theorem yields with ; by -equivariance, jointly solves , and is interior. The four points are distinct because has both coordinates in while each of , , and has at least one coordinate at .

Corollary 10.2. The recommendation system has a compact global attractor \mathcal{A}$ containing all four equilibria.

Proof. The model data satisfy Assumption 2.7 (finite-state, positive temperature, positive regularization) and Assumption 2.8 for the generator family constructed in Section 10.4. The law equation is in generator form, so we invoke the exact-system theorem of Chapter 7, Section 7.1, applied with : the system generates a continuous semiflow with compact forward invariant absorbing set , and hence has a unique compact global attractor . Equilibria are invariant points, hence contained in .

The equilibrium structure of the recommendation system gives a precise geometric vocabulary for a phenomenon that is usually discussed only in qualitative terms. We now extract the domain-level conclusions, keeping the existence theorem separate from stability claims.

Filter bubbles are boundary equilibria. The A-bubble equilibrium and the B-bubble equilibrium owe their existence to the saturated coordinate, where the damping suppresses the actor drift, rather than to a vanishing advantage signal in that coordinate. The saturated-branch raw drift at these points is nonzero–for the data of Sections 10.2—10.4 it points inward, back toward the box interior, because the entropy-regularization pull toward a uniform policy outweighs the reward and critic incentives at the edge (Exercise 10.3). The bubble configuration is thus held in place by the geometry of the parameter box, not by a raw gradient that has been clipped while still pushing outward. With the baseline sign just computed, it is transversely repelling from nearby interior points in the saturated coordinate.

This observation has a concrete implication. A platform operator who observes the system near a bubble equilibrium should not conclude that the curation policy is a genuine optimum: the advantage signal in the saturated branch has not vanished, so the equilibrium is an artifact of the parameter limit and the audience dynamics rather than a stationary point of the unconstrained policy gradient. Because the saturated-branch drift here points inward, enlarging the actor box would let relax back toward the interior rather than push the bubble further out. The boundary point exists because of the damping geometry; whether it attracts requires a separate transverse-stability and basin calculation.

Diverse engagement is the true gradient zero. The interior equilibrium is the only point where the advantage signal genuinely vanishes. In domain terms this is the operating point where the curation level is self-consistent: the policy, the critic’s value estimate, and the audience composition are mutually compatible without parameter saturation. A platform at has no incentive to change its policy, not because it has hit a wall but because the current policy is locally optimal given the current audience.

Basin structure is a numerical question here. The theory proves that the equilibria lie in a compact global attractor; it does not classify which equilibria attract interior trajectories. For the baseline parameters, the transverse calculation above already rules out interpreting the pure curation faces as attracting lock-in from the interior. If a modified reward or regularization regime reverses that transverse sign, the next question is basin geometry: which initial conditions converge to the interior operating point and which converge to a boundary configuration?

Those basin boundaries are subsets of the eight-dimensional phase space; whether they are smooth manifolds depends on the hyperbolicity structure of the equilibria, which we do not analyze here. Appendix B describes how to compute lower-dimensional cross-sections in the -plane at fixed critic and distribution values; those computations are evidence, not part of Proposition 10.1 or Corollary 10.2.

Design levers. The reward table and the generator structure are the platform designer’s instruments. Changing the diversity reward changes the strength of the incentive to diversify at the root. Changing the return rates in changes how quickly users leave deep engagement. Both changes feed directly into the raw-drift signs and the basin computation; one should recompute the transverse signs and the numerical cross-sections after changing them rather than infer lock-in from the existence of boundary equilibria alone.

These are domain-actionable conclusions. The attractor framework does not tell the platform what it should do–that is a normative question–but it shows which structural quantities must be recomputed when those choices change.

The fast-slow reading. The audience equation displayed in Section 10.5 reads , with rates of and that are ; this is the regime, where the audience and the policy update on the same time scale. The fast-slow regime of Chapter 7 introduces a separation parameter by writing the audience equation as (equivalently rescaling time by ), so small corresponds to faster user response than policy update at the generator scale of . In this regime, the audience distribution tracks the frozen invariant law on finite time horizons. The reduced system–the actor-critic ODE on the invariant-law graph–is four-dimensional () and governs the long-run curation outcome. By Corollary 7.5 (Chapter 7), the attractor of the exact eight-dimensional system is close to the attractor of this reduced system as .

In domain terms, when users adapt quickly to content changes, the platform designer can analyze the reduced system to understand the long-run behavior. The frozen-audience analysis is a good proxy because the audience never falls far behind the policy.

This chapter executed the Model Specification Protocol on a content recommendation platform and built a model with enough structure to represent filter-bubble configurations as boundary equilibria. The five-state Y-graph, the block-diagonal features, and the engagement-diversity reward tension produced a system with at least four equilibria: a diverse-engagement interior equilibrium and three boundary equilibria corresponding to filter bubbles.

The attractor framework gave three domain-level insights that are not available from convergence-rate analysis alone: (1) bubble equilibria are boundary artifacts, not gradient zeros; (2) their transverse stability must be checked before calling them lock-in attractors; (3) the reward table and generator structure are design levers because they change the raw-drift signs and the basin computation.

Chapter 11 applies the same protocol to a network routing problem, where the competing attractor branches correspond to hub-heavy versus spoke-heavy traffic configurations rather than diverse versus niche audience states. Appendix B describes how to compute the corresponding phase portraits and parameter sweeps for both models.

Exercise 10.1 (Verify: frozen invariant law at the root). Compute the frozen invariant law for the Y-graph chain at by solving the linear system with the simplex constraint . What fraction of the audience is in diverse browsing (state 1)? Describe the distribution in domain terms.

Exercise 10.2 (Compute: the A-bubble equilibrium). At the A-bubble boundary with :

(a) Compute the softmax policy for each state-action pair.

(b) Form the mixture generator and solve for the invariant law .

(c) Compute the critic equilibrium at .

(d) What fraction of the audience is in deep niche-A engagement (state 3)?

Exercise 10.3 (Verify: nonzero raw drift at the bubble). At the saturated A-branch edge , compute the A-branch raw actor drift from the three-piece decomposition of Section 10.5. Confirm that it is nonzero and state its sign. (You should find it negative: the entropy piece outweighs the reward and critic pieces.) Since the drift points inward, explain why enlarging the actor box to would let relax back toward the interior rather than push it further out–so the A-bubble is held in place by the chain and damping geometry, not by an outward-pushing gradient.

Exercise 10.4 (Break: removing block-diagonal structure). Replace the block-diagonal features with uniform features: set and for all states (keep state 1 features at zero). Now the actor has two dimensions but only one effective degree of freedom , since the policy depends on only through . Show that the maximal-curation policy inside the box is reached at the corner , where the damping vanishes in both coordinates, so the two separate A- and B-bubbles collapse to a single “generic bubble” at that corner. (Note that the point is interior to the box, with damping , so it cannot be a damping-created boundary equilibrium.) Why does the block-diagonal structure matter for distinguishing the two bubbles?

Exercise 10.5 (Compute: asymmetric branch rewards). This exercise replaces the scaffolded sweep over with a qualitative branch-asymmetry study, since the asymmetric variant illustrates how the interior equilibrium loses its diagonal symmetry. Break the branch symmetry by changing the B-branch rewards to , , , (niche B is less engaging than niche A). Describe qualitatively how the interior equilibrium shifts: does increase or decrease relative to the symmetric case? What about ?

Exercise 10.6 (Connect: fast-slow reading). Starting from the -scaled audience equation of the Section 10.7 fast-slow paragraph and using the fast-slow reduction of Chapter 7, write the reduced actor-critic system that governs the recommendation platform when the audience responds instantaneously (). The reduced system is four-dimensional () with at all times. Explain why a platform that updates slowly (small actor learning rate) is more likely to track the reduced system’s trajectory than one that updates aggressively.

Exercise 10.7 (Extend: the trident model). Add a third content cluster with states 6 (discovery) and 7 (deep engagement), forming a “trident” Y-graph with three branches from the root. What is the actor dimension? How many boundary equilibria does the model have? State (without proof) the expected number of equilibria.

Exercise 10.8 (Connect: design lever for bubble prevention). A platform operator proposes adding an “exploration bonus” to the diversification reward at all deep-engagement states: replace and with and . Using the attractor framework, explain qualitatively how this changes the basin boundaries. For large enough , can the bubble equilibria disappear? Why or why not?

Exercise 10.9 (Verify: two-step minorization). Check that state 1 supplies the two-step propagator minorization used in Section 10.4 for the generator at all . Specifically, show that the minimum two-step transition probability from state 3 to state 1 is bounded below by a positive constant that does not depend on .