Answer: Fair runs can fail to reach aggregate quasi-quiescence. The worst-case revision mass is finite for each fixed instance but can be superpolynomial in the specified parameters.
Theorem 10 — class: algorithmic
PDF text-layer excerpt — equations and symbol order may be approximateSAMP-SB reaches a quasi-quiescent state after a number of bids bounded by a poly- nomial of the size of the network and the value of the maximum consumer value have been placed by consumers and active producers. Our previously mentioned quiescence-detection protocol (Wellman & Walsh, 2000) can also detect quasi-quiescence, and thus terminate negotiations when it is reached.
statement extracted from the paper’s text layer; consult the paper for the exact wording
Given a finite acyclic supply-chain network with \(\tau\) participant-role types, binary-rational masses \(\mu_t\) summing to one, and the paper's costs, values, and bid increments, model each participant as following SAMP-SB with its own local bid history. Auctions accept mass-valued offers and split marginal ties; schedules are fair when every sent message is eventually delivered. Does every run reach aggregate quasi-quiescence, and what is the worst-case bound in \(m,\tau,L,R\) on \(U=\sum_{t\in T}\int_{A_t}r(a)\,da\), where \(A_t\) is the mass of type \(t\) and \(r(a)\) counts that participant's bid revisions?
A society is given by its support: the list of the \(\tau\) participant-role types with positive binary-rational masses \(\mu_t\) summing to \(1\). Here \(\tau\) is the support size, not the number of possible types. Realize each type as an interval \(A_t\) of Lebesgue measure \(\mu_t\); participants retain individual bid histories.
The input is a finite acyclic bipartite network of \(m\) goods and these types. A consumer desires at most one unit of one good from a nonempty specified set, with nonnegative binary-rational values \(v_t(g)\). A producer has nonnegative binary-rational unit cost \(\kappa_t\), one output, and an explicitly listed multiset of complementary inputs, all required for production. Each good has positive potential buyer and seller mass, and there is at least one consumer. Let \(R=\max_{t,g}v_t(g)\), and let \(L\) be the total binary input length, including positive rational increments \(\delta_b,\delta_s\).
Each participant maintains one standing unit offer per relevant input or consumable good and one per output; an interval of participants therefore submits its measure in offers. Offers cannot be withdrawn. Revised buy and sell prices must increase by at least \(\delta_b\) and \(\delta_s\), respectively.
For each good, let \(S\) be total offered seller mass. Rank all buy and sell offers by decreasing price and select the top \(S\) mass. Selected buyers and unselected sellers win and exchange equal masses. The going price \(p\) is the essential supremum of prices in the unselected mass; the ask \(\alpha\) is the essential infimum in the selected mass. Preserve temporal precedence within each side; resolve remaining ties by a fixed ordering of participant and offer labels, cutting marginal tied mass as necessary. Trading offers submitted earlier take precedence over later offers on the same side.
Consumers initially offer zero on every desired good. A consumer winning any offer holds its bids; otherwise it bids \(p(g^*)+\delta_b\), where \(g^*\) maximizes \(v_t(g)-p(g)-\delta_b\), provided that maximum is nonnegative. Resolve equal maximizers by a fixed good ordering.
Producers initially bid zero for every input occurrence. An input offer increases by exactly \(\delta_b\) precisely when its latest quotes report that input lost and the output won. For each input occurrence set \(\widehat p=p\) when winning and \(\widehat p=\max\{\alpha,p+\delta_b\}\) otherwise. The first output offer is \(\kappa_t+\sum\widehat p\); when the perceived input-cost sum increases, replace the previous output price \(\beta\) by \(\max\{\beta+\delta_s,\kappa_t+\sum\widehat p\}\). Input and output revisions use the same current local quotes.
Initialize in topological order: auctions issue their first quotes only after receiving all initial offers. Subsequently each received revision generates quotes to all its bidders, recording its latest received bid identifiers. Participants process quotes in order and respond only when the relevant quotes acknowledge their latest sent bids. A run is a sequence of events acting on measurable participant subsets; fairness means every sent message is eventually delivered and processed.
Aggregate quasi-quiescence means that, almost everywhere among consumers and producers currently winning their output, all sent bids have arrived and no already received or pending quote would induce a revision. Let \(r(a)\) count all individual offer revisions before the first such state, or throughout the run if no such state occurs, and set \(U=\sum_t\int_{A_t}r(a)\,da\).
Target. Does every fair run reach aggregate quasi-quiescence, or can a fair run fail to reach it? What is the worst-case bound on \(U\): is it polynomial in \(m,\tau,L,R,\delta_b^{-1},\delta_s^{-1}\), or can it be superpolynomial or unbounded?
Nearest known results. For finite societies, quasi-quiescence is reached with polynomially many bids from consumers and active producers in network size and numerical maximum consumer value, with increments fixed; total bidding can be exponential in network size under asynchronous delivery. These are distinct counting guarantees ([original source](https://www.cs.cmu.edu/afs/cs/project/jair/pub/volume19/walsh03a.pdf)). No result settling the specified mass-valued dynamics was located.
A fair run can fail to reach aggregate quasi-quiescence. Nevertheless, the revision mass \(U\) is uniformly finite for each fixed instance.
More precisely, under the usual binary encoding of rational numbers:
Thus the proposed polynomial bound fails, but unbounded \(U\) on a fixed instance is impossible.
Here, “reaches” means that some state in the specified event sequence is aggregate quasi-quiescent. No additional state after infinitely many events is assumed.
We first establish properties used in the upper bound. All statements about individual participants are understood almost everywhere. Since a run is a sequence of events, the exceptional null sets can be discarded simultaneously.
For one good, regard its offers as a finite measure space, with one copy of a participant’s interval for each relevant offer occurrence. Write \(T\) for total offer mass and \(S\) for seller mass. The hypotheses give \(0<S<T\).
Let \(\nu\) denote offer measure and \(b\) the standing price function. The two boundary prices have the descriptions
\[ p=\inf\{x\in\mathbb R:\nu(\{b>x\})\leq S\}, \qquad \alpha=\sup\{x\in\mathbb R:\nu(\{b<x\})\leq T-S\}. \]
These formulas also cover ties and gaps between price levels.
Standing prices only increase. Consequently, for each fixed \(x\), the mass \(\nu(\{b>x\})\) cannot decrease, while \(\nu(\{b<x\})\) cannot increase. The displayed formulas therefore imply that both \(p\) and \(\alpha\) are nondecreasing.
A losing buy offer at price \(b\) belongs to the unselected mass. Therefore its quote satisfies \(p\geq b\), almost everywhere.
If all buy offers for a good have price at most \(D_0\), no positive mass of sellers offering above \(D_0\) can win. Indeed, positive winning seller mass entails equal selected buyer mass. Every selected offer has price at least every unselected offer, apart from null sets. A winning seller above \(D_0\) would therefore require selected buyers above \(D_0\), a contradiction.
Between revisions of its own standing bid, a buy offer cannot change from losing to winning. Other standing prices only increase; an offer ranked ahead of this fixed offer cannot move behind it. Temporal precedence and the fixed remaining tie ordering preserve this conclusion.
Thus, if one input offer is revised at most \(N\) times, its winning indicator changes at most \(2N+1\) times: at most one loss during each of its \(N+1\) fixed-bid periods, and at most one gain at each revision.
Let \(\beta\) be a producer’s latest sent output price, and let \(b_j\) be its latest sent bid for input occurrence \(j\).
The following invariant is important:
\[ \beta\geq b_j \quad\text{for every input occurrence }j. \]
Initially, every \(b_j=0\), so the invariant holds.
To verify preservation, consider a permitted response. Write \(c\) for the current perceived input-cost sum. The output rule maintains \(\beta\geq\kappa_t+c\) after the response: if \(c\) increases, the prescribed replacement enforces this inequality; if it decreases or stays unchanged, the previous output price already suffices.
If input occurrence \(j\) is revised, its acknowledged bid \(b_j\) is losing. Hence \(p\geq b_j\), and its perceived cost satisfies
\[ \widehat p_j=\max\{\alpha,p+\delta_b\}\geq b_j+\delta_b. \]
The new input bid is exactly \(b_j+\delta_b\). Because input and output revisions use the same local quotes, the resulting output price is at least this new input bid. Other input bids stay unchanged, and output prices never decrease. This proves the invariant.
Now suppose that all buyers of this producer’s output bid at most \(D_0\). An input revision requires a winning output quote acknowledging the latest sent output offer. By Section 1, this implies \(\beta\leq D_0\). The invariant gives \(b_j\leq D_0\), so the revised input bid is at most \(D_0+\delta_b\).
Acknowledgment is essential here: a stale winning quote for an earlier, cheaper output offer cannot authorize further input revisions.
Every producer has a directed path to a consumer. To see this, follow its output good to a potential buyer. If that buyer is a producer, continue through its output. Positive buyer mass prevents termination at a good, and acyclicity forces eventual termination at a consumer.
Define a producer’s consumer depth to be the maximum number of producer vertices on such a path, including itself. Its producer buyers have strictly smaller consumer depth. Induction from consumers, whose bids never exceed \(R\), therefore gives
\[ b_j\leq R+k\delta_b \]
for a producer of consumer depth \(k\). A directed path cannot repeat an output good, so \(k\leq m\).
Set
\[ B=R+m\delta_b, \qquad N=\left\lceil\frac{B}{\delta_b}\right\rceil. \]
Every individual buy offer is consequently revised at most \(N\) times. This applies to consumers as well, since their revised bids never exceed their corresponding values.
Let \(D\) be the maximum number of explicitly listed input occurrences of a producer, and set
\[ K=\max\{1,m,D\}, \qquad C_\kappa=\max_t\kappa_t. \]
Every participant has at most \(K\) buy offers, so its total buy revision count is at most \(KN\).
We now bound output revisions, including revisions by inactive producers.
Write all input rational numbers with integer denominators, and let \(Q\) be the product of those denominators. Set \(q=Q^{-1}\).
Every submitted price, and every quoted \(p\) and \(\alpha\), belongs to \(q\mathbb Z_{\geq0}\). Here is a finite-prefix justification. Initially the possible prices form a finite rational set. Given finitely many previous events, there are finitely many previous prices and quote values. The bidding rules form new prices using addition, maximum, and finite sums, preserving both a finite price set and membership in \(q\mathbb Z\). Essential boundary prices of a finite price set belong to that set. Topological initialization has the same property.
Give source producers, which have no inputs, supply depth zero. For a producer with inputs, its supply depth is one plus the maximum supply depth of any producer selling one of its inputs. Acyclicity makes this definition valid, and supply depth is at most \(m\).
Define bounds recursively:
\[ P_0=\max\{B,C_\kappa\}, \]
and, for \(1\leq h\leq m\),
\[ T_h=D\left(2\left\lceil\frac{P_{h-1}}q\right\rceil+2N+1\right), \]
\[ P_h=\max\left\{P_{h-1}, C_\kappa+D(P_{h-1}+\delta_b)+\delta_sT_h\right\}. \]
We prove that a producer of supply depth \(h\) makes at most \(T_h\) output revisions and never offers above \(P_h\). Source producers make no revisions and offer their fixed costs, establishing the base case.
For the inductive step, all sellers of this producer’s input goods offer at most \(P_{h-1}\), and all buyers offer at most \(B\leq P_{h-1}\). Thus the input quotes satisfy
\[ 0\leq p,\alpha\leq P_{h-1}. \]
Each quoted price is nondecreasing and belongs to the \(q\)-lattice. Therefore each changes strictly at most \(\lceil P_{h-1}/q\rceil\) times. For one input occurrence, its winning indicator changes at most \(2N+1\) times.
Its perceived cost is determined by these two prices and its winning indicator. Hence that perceived cost changes at most
\[ 2\left\lceil\frac{P_{h-1}}q\right\rceil+2N+1 \]
times. Summing over input occurrences bounds the number of changes in the perceived cost sum by \(T_h\).
Every output revision requires an increase of that sum. Such a revision can be charged to a component change since the preceding output revision or first output offer. Consequently, there are at most \(T_h\) output revisions.
At all times the target cost satisfies
\[ \kappa_t+\sum_j\widehat p_j \leq C_\kappa+D(P_{h-1}+\delta_b). \]
Starting from the first output offer, after \(r\) revisions the replacement rule therefore gives
\[ \beta\leq C_\kappa+D(P_{h-1}+\delta_b)+r\delta_s. \]
For \(r\leq T_h\), this is at most \(P_h\), completing the induction.
Since the sequence \(P_h\) is nondecreasing, every participant’s total revision count throughout the entire run is at most \(KN+T_m\). The participant masses sum to one, so
\[ U\leq KN+T_m<\infty. \]
This bound does not require fairness.
Under the usual binary encoding, \(Q\), the numerical input magnitudes, and the reciprocal increments are at most \(2^{O(L)}\), while \(m,D,K\leq O(L)\). The recurrence has coefficients bounded by \(2^{O(L)}\), and it has at most \(m\) stages. Thus
\[ U\leq 2^{O(mL)}. \]
Use one good and two types, each of mass \(1/2\):
Set \(\delta_b=\delta_s=1\).
After initialization, sellers offer \(2\), buyers offer \(0\), and all sellers are selected. Thus \(p=0\), \(\alpha=2\), and every consumer loses.
Process the first quote for the entire consumer interval. Every consumer sends a revision to price \(1\). Partition that interval into measurable sets \(E_n\), for \(n\geq1\), with
\[ \lambda(E_n)=2^{-n-1}, \]
whose union is the whole interval.
At stage \(n\), deliver the price-\(1\) revisions of \(E_n\). Immediately deliver and process every quote generated at that stage.
The sellers remain selected because \(2>1\). After the first delivery, \(p=1\). A consumer whose revision has arrived cannot make another revision because
\[ 1-p-\delta_b=-1<0. \]
A consumer whose revision has not arrived receives quotes acknowledging its old bid and cannot respond.
Every consumer’s revision arrives at a finite stage, and every generated quote is processed immediately. The run is fair. Nevertheless, after any finite stage, a positive consumer mass still has an outstanding sent bid. Therefore aggregate quasi-quiescence never occurs.
Every consumer revises exactly once, so this example has
\[ U=\frac12. \]
The obstruction is the absence of a common completion event for almost all participants, even though each participant completes individually.
We next construct a different family. Its run eventually reaches aggregate quasi-quiescence, but only after exponentially many mass-weighted output revisions.
Fix \(n\geq1\), let
\[ \tau=3n+5, \qquad \varepsilon=2^{-\lceil\log_2\tau\rceil}. \]
Every type except one designated consumer has mass \(\varepsilon\). That designated consumer has the remaining mass \(1-(\tau-1)\varepsilon\), which is positive. All masses are dyadic rationals.
Set \(\delta_b=\delta_s=1\).
Use goods \(g_0,\ldots,g_n\), goods \(a_i,b_i\) for \(1\leq i\leq n\), and an independent good \(h\).
For \(g_0\), include a source producer of cost \(1\) and a consumer of value \(1\).
At each level \(i\), include three zero-cost producers:
Include a consumer of \(g_n\) with value zero.
For \(h\), include a source producer of cost \(2\) and the designated consumer, of value \(1\).
The network is acyclic, and every good has positive buyer and seller mass. Its parameters satisfy
\[ m=3n+2,\qquad R=1,\qquad L=O(n\log n). \]
Initialize with all buy offers zero, and hold the two value-\(1\) consumers’ first quotes until initialization is complete.
Every auction in the diamond network selects its positive-price seller offers and leaves its zero-price buyers unselected. Thus \(p=0\), and \(\alpha\) equals its seller price.
Topological initialization gives
\[ \beta(F_i^a)=\beta(F_i^b)=2^{i-1}, \qquad \beta(M_i)=2^i. \]
All these producers lose their outputs.
Now process the designated consumer’s first quote on \(h\). It sends a bid of \(1\). Hold that bid in transit until the entire cascade described below is complete.
Throughout the cascade, this outstanding consumer bid prevents aggregate quasi-quiescence.
Process the value-\(1\) consumer’s first quote on \(g_0\), and deliver its revised bid of \(1\).
The root auction now has seller mass \(\varepsilon\) at price \(1\), consumer buyer mass \(\varepsilon\) at price \(1\), and both fork buyers at zero. Regardless of how the price-\(1\) tie is cut, selected and unselected mass both contain price-\(1\) offers. Therefore
\[ p(g_0)=\alpha(g_0)=1. \]
Both forks still lose their zero bids. Their perceived input costs rise from \(1\) to
\[ \max\{1,1+1\}=2. \]
Each consequently revises its output once, increasing its price by exactly one.
Whenever a producer sends an output revision, deliver it and process its acknowledgment before processing another input update for that producer.
Propagate each resulting price increase through the remaining network before processing its sibling branch. All relevant input bids remain zero and acknowledged. Every non-root seller remains selected, since its output price is positive and all its buyers remain at zero. Thus these producers stay inactive and never revise their input bids.
Each received input-price increase of one raises the recipient’s perceived sum by one. Its output replacement consequently raises its output price by exactly one.
One increase in the common input cost of a fork pair therefore produces:
The two merge increases are propagated separately downstream.
Induction over the levels now gives
\[ \text{output revisions of }F_i^a =\text{output revisions of }F_i^b =2^{i-1}, \]
\[ \text{output revisions of }M_i=2^i. \]
In particular, the entire mass \(\varepsilon\) of \(M_n\) revises \(2^n\) times.
After this finite cascade, deliver the designated consumer’s held bid and process all remaining messages in order. Its good still has seller price \(2\); the resulting going price is \(1\), so it makes no further revision. The root consumer also makes no further revision, and the terminal consumer’s value is zero. All propagated producer updates have been processed. Hence the run reaches aggregate quasi-quiescence and is fair.
All \(2^n\) revisions of \(M_n\) occurred before that first state, because the designated consumer’s bid remained outstanding. Therefore
\[ U\geq\varepsilon\,2^n \geq\frac{2^n}{2(3n+5)}. \]
Since \(m,\tau=O(n)\), \(L=O(n\log n)\), and \(R=\delta_b^{-1}=\delta_s^{-1}=1\), this exceeds every polynomial in the parameters proposed in the question.
The polynomial guarantee in [Walsh and Wellman, Theorem 10](https://www.cs.cmu.edu/afs/cs/project/jair/pub/volume19/walsh03a.pdf) counts bids by consumers and active producers. The lower-bound construction above counts revisions by inactive producers, while an outstanding consumer bid postpones quasi-quiescence. Its initialization and scheduling were verified directly under the rules in the question.
The two issues are therefore distinct: fairness does not ensure finite-event aggregate convergence, and counting all revisions permits superpolynomial growth even in runs that eventually converge.
⟨structured verdict parsed — rendered above⟩
The prover resolves the target as stated. The argument establishes that fair runs need not reach aggregate quasi-quiescence at any finite event, that \(U\) is uniformly bounded for every fixed instance, and that no polynomial in the proposed parameters bounds \(U\) across instances. I found no major or minor mathematical defects.
This conclusion uses the task’s stipulated ordered processing of quotes and its interpretation of a run as an event sequence without an additional state after infinitely many events.
The boundary-price formulas are correct. Since \(0<S<T\), both selected and unselected mass are positive. The displayed formulas recover the essential supremum of unselected prices and essential infimum of selected prices, including when a tied price level is split or a gap separates the two masses.
Price monotonicity follows. Raising standing offers increases, or preserves, the mass above each fixed threshold and decreases, or preserves, the mass below it. The directions of both claimed monotonicities are correct.
The losing-buyer inequality is valid almost everywhere. An unselected buyer’s price cannot exceed the essential supremum of unselected prices, except on a null set.
The winning-seller bound is valid. If positive seller mass above \(D_0\) were unselected, the rank cutoff would exclude every buyer priced at most \(D_0\). This contradicts the equality of winning seller mass and selected buyer mass.
The fixed-bid argument respects temporal precedence. An offer ahead of an unchanged buyer cannot fall behind it through a price increase. An unchanged tied offer retains its precedence. Thus a fixed buy offer cannot change from losing to winning, and the bound of \(2N+1\) winning-indicator changes follows from the stated division into fixed-bid periods.
The simultaneous removal of exceptional null sets is legitimate because the event sequence and the number of offer occurrences per participant are countable.
The invariant \(\beta\geq b_j\) is preserved. The output rule maintains \(\beta\geq\kappa_t+c\) for the perceived sum used in each permitted response. When an acknowledged input bid loses, its perceived cost is at least \(b_j+\delta_b\). Using the same quotes for input and output revisions therefore preserves the invariant for the revised bid; monotonicity preserves it for unchanged bids.
The acknowledgment condition is used correctly. A winning output quote acknowledging the latest sent output offer bounds that current offer by the maximum buyer bid. An obsolete winning quote cannot authorize further input revisions.
The downstream induction is justified. Positive buyer mass and acyclicity imply that every producer has a directed path ending at a consumer. Producer buyers have strictly smaller consumer depth. Consumer bids are bounded by \(R\), and the induction gives the stated bound \(R+k\delta_b\).
A path cannot repeat a good, so \(k\leq m\). Consequently, every buy offer is bounded by \(B=R+m\delta_b\). Since each revision raises its price by at least \(\delta_b\), the asserted bound \(N=\lceil B/\delta_b\rceil\) is valid. It also covers \(R=0\).
The common lattice argument is sound. Initial prices and all subsequent price operations preserve membership in \(q\mathbb Z_{\geq0}\). At a finite event prefix, only finitely many price values have occurred, so essential boundary prices belong to that finite set. The argument does not assume an attained boundary price at an infinite limit.
Supply-depth induction is well founded. Positive seller mass ensures that each input good has producer sellers. Acyclicity gives the stated recursion, with source producers as the base case and depth at most \(m\).
The quote-change count is valid. At supply depth \(h\), all relevant input-auction offers are bounded by \(P_{h-1}\). Therefore both boundary prices lie between zero and \(P_{h-1}\), and their strict changes are bounded by the lattice count. Ordered quote processing preserves this bound for each participant’s local quote history; acknowledgment filtering cannot add changes. Combining these counts with the winning-indicator bound gives the stated bound on perceived-cost changes.
The charging argument bounds output revisions. Every output revision requires a perceived-sum increase. Such an increase requires a component change, and successive revisions can be charged to distinct changes occurring between responses.
The output-price recurrence is correct. Given the target-cost bound
\[
C_\kappa+D(P_{h-1}+\delta_b),
\]
the replacement rule permits at most an additional \(r\delta_s\) after \(r\) revisions. This establishes the claimed \(P_h\) bound.
Consumers have no output revisions, and source producers have none after their initial offers. Thus \(KN+T_m\) bounds every participant’s revision count. Integrating over participant mass one correctly yields \(U\leq KN+T_m\).
The encoding estimate is consistent. Explicit listing bounds \(D\) by the input length. The denominator product, numerical magnitudes, and reciprocal increments are bounded by \(2^{O(L)}\). The recurrence has at most \(m\) stages and coefficients bounded by \(2^{O(L)}\), giving \(U\leq2^{O(mL)}\). Fairness is unnecessary for this bound.
The one-good construction satisfies all input requirements.
Initially, the price-\(2\) sellers are selected and the zero-price buyers lose, giving \(p=0\) and \(\alpha=2\). The consumer revision to price \(1\) is authorized.
The partition masses sum to the consumer mass:
\[
\sum_{n\geq1}2^{-n-1}=\frac12.
\]
Every individual revision is delivered at a finite stage, while after every finite stage positive consumer mass still has a sent bid in transit.
After the first delivery, \(p=1\), and acknowledged consumers cannot revise again. Quotes acknowledging an older bid cannot authorize a response from consumers whose revision remains outstanding. The described processing can satisfy the stipulated quote order.
Thus fairness holds, but aggregate quasi-quiescence never occurs at a finite event. Every consumer revises once, so \(U=1/2\). The construction correctly distinguishes individual eventual completion from aggregate completion.
The instance parameters check out. There are \(3n+5\) types and \(3n+2\) goods. The designated consumer’s remaining mass is positive, all masses are dyadic, and \(L=O(n\log n)\). The network is acyclic and every good has positive buyer and seller mass.
The initialization prices are correct. Positive-price sellers are selected and zero-price buyers are unselected. The stated perceived-cost rule gives the fork prices \(2^{i-1}\) and merge prices \(2^i\). These producers initially lose their outputs.
The root trigger is independent of cross-side tie ordering. There is price-\(1\) mass in both selected and unselected offers after the consumer revision, so \(p(g_0)=\alpha(g_0)=1\). Each fork’s perceived cost rises from \(1\) to \(2\).
The cascade respects the update rules. Every non-root seller remains selected because its price is positive and its buyers stay at zero. Those producers therefore remain inactive and do not revise input bids. Processing each output acknowledgment before the next input update allows separate downstream responses without violating quote order.
Each input-price increase of one produces one output-price increase of one. Propagating the sibling branches separately gives exactly the stated revision counts, including \(2^n\) revisions throughout the mass of \(M_n\).
The revisions precede the first aggregate quasi-quiescent state. The held consumer bid on \(h\) blocks quasi-quiescence throughout the cascade. Before that bid is sent, the held initial consumer quote would induce a revision, so initialization does not create an earlier qualifying state. After the cascade, processing all remaining messages produces the claimed fair completion.
Finally, \(\varepsilon\geq1/(2(3n+5))\), so the lower bound on \(U\) is correct and exceeds every polynomial in the proposed parameters.
The comparison with [Walsh and Wellman, Theorem 10](https://www.cs.cmu.edu/afs/cs/project/jair/pub/volume19/walsh03a.pdf) accurately identifies its count of bids by consumers and active producers. The present lower bound counts inactive producers’ revisions. The cited theorem is not used to establish the continuous result.
The output establishes termination and revision-count results for these dynamics. It does not establish computational hardness of a separate decision or optimization problem. Accordingly, the appropriate continuous-status classification is other.
No additional clarification or repair is required.
⟨structured verdict parsed — rendered above⟩
Tier: machine-proved, adversarially reviewed by an independent model. No human has checked this proof. Generated from pipeline artifacts by tools/atlas_build.py.