No-Forced-Sale Condition in Market Mechanisms

• No-forced-sale condition is a formal constraint that prevents agents from being compelled to liquidate or trade assets when prices violate reserve, budget, or collateral limits.
• It is applied across different models—assignment markets, mortgage contracts, repo systems, durable goods, and optimal liquidation—using domain-specific mechanisms.
• The condition balances state-dependent feasibility with strategic behavior, influencing auction design, pricing dynamics, and regulatory frameworks in financial markets.

Searching arXiv for recent and directly relevant papers to ground the article. arxiv_search.search(query="\"No-Forced-Sale\" OR \"forced sale\" market core budgets mortgage default repo liquidation", max_results=10) “No-Forced-Sale” (Editor’s term) denotes a family of formal conditions under which an agent is not compelled to liquidate an asset, consummate a trade, or default in a state where the relevant feasibility, reserve, collateral, or participation constraints are not voluntarily satisfied. The exact phrase often does not appear in the underlying papers. Instead, the idea is encoded through seller reserve prices and core constraints in assignment markets, elimination of selective underwater default in mortgage contracts, zero-liquidation feasibility in repo-fire-sale models, the boundary case $\theta_t=0$ in residual-supply pricing, period-by-period participation in durable-good mechanism design, and terminal-set plus gating constructions in optimal liquidation problems (Batziou et al., 2022, Kitapbayev et al., 2020, Bichuch et al., 2020, Wang, 29 May 2026, Doval et al., 2019, Aksu et al., 2023).

1. Terminological status and formal variants

The literature does not present a single canonical “No-Forced-Sale Condition.” Rather, it uses domain-specific objects that play the same operational role: they rule out compelled transactions at prices, balances, or inventories that violate the relevant side constraints. In assignment markets, the condition is expressed through seller outside options, reserve prices, budget feasibility, and the rule that only sold items may have a positive price. In mortgage design, it is expressed by eliminating the default boundary so that the outstanding balance never exceeds the house value. In repo-fire-sale models, it is a feasibility inequality ensuring obligations can be met with cash and borrowing alone. In residual-supply models, it is the zero-residual-inventory boundary. In durable-good mechanism design, it is embodied in the seller’s option not to sell and the buyer’s option to reject. In optimal liquidation, it is implemented by a measurable terminal set $S$ and a binary process $I$ that can stop trading (Batziou et al., 2022, Kitapbayev et al., 2020, Bichuch et al., 2020, Wang, 29 May 2026, Doval et al., 2019, Aksu et al., 2023).

Domain	Formal object	Operational meaning
Assignment markets	$$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$	No sale below reserve or above budget
Mortgage contracts	Default region empty; $D=0$	No selective underwater default
Repo/fire sales	$D_i-C_i \le g(0)X_i$	No liquidation required at $L=0$
Residual supply	$\theta_t=0$	No constrained inventory absorption
Durable goods	Participation and no-sale option	No forced trade for either side
Optimal liquidation	$\xi=-K1_{S^c}+\infty 1_S$, gated by $I$	No compulsory liquidation in bad-price states

This suggests that “No-Forced-Sale” is best treated as a comparative concept rather than a uniform theorem. The common structure is voluntary trade under state-dependent feasibility.

2. Assignment markets with budgets, reserves, and the core

In the assignment-market model with financially constrained buyers, buyers have unit-demand valuations $S$0, hard budgets $S$1, and sellers may have reserve prices $S$2. The formal statement corresponding to no forced sale is: if a sale occurs, then $S$3; if an item is unsold, the seller keeps it; and only sold items may have a positive price. In the $S$4 case, unsold items have price $S$5. The same framework defines buyer utility at prices $S$6 by

$S$7

A pair $S$8 blocks an outcome $S$9 if $I$0 and $I$1. Core-stability is the absence of such blocking pairs, and its coalitional version requires that no coalition of buyers and sellers can all strictly gain subject to budgets and reserves (Batziou et al., 2022).

In this setting, the no-forced-sale condition is not merely seller individual rationality. It is the conjunction of seller reserve feasibility, buyer budget feasibility, and core-stability. If no buyer can pay a price that clears the reserve while still respecting $I$2, the item remains unsold. The paper’s terminology also makes explicit that core outcomes under budgets need not be envy-free. A two-buyer, two-seller example shows that a welfare-maximizing assignment ignoring budgets is not core-stable once budgets bind, because a seller-buyer deviation can become profitable at a feasible price even when the original allocation has higher raw valuation.

The algorithmic treatment is an iterative generalization of the Demange–Gale–Sotomayor ascending auction. It uses only demand queries. At round $I$3, the auctioneer posts prices $I$4 and receives restricted demand sets $I$5. If there is a minimally overdemanded set $I$6, prices on that set are increased by $I$7. If a bidder becomes budget-tight, the auctioneer removes the newly unaffordable items from the bidder’s restricted set and rolls back prices to $I$8. When neither over- nor underdemand exists, an assignment $I$9 is chosen with $$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$0 and sold items exactly those with positive prices. The key ex-post condition is that whenever Step 3 is reached, $$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$1. Under appropriate auctioneer choices, and in particular if $$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$2 at every Step 3, the algorithm yields a welfare-maximizing core outcome (Batziou et al., 2022).

The same paper establishes sharp negative results. Without the additional conditions required by the ascending format, no mechanism that is incentive-compatible for budget-constrained unit-demand buyers can always terminate in a core-stable outcome. Moreover, the decision problem Maximum Welfare Budget Constrained Stable Bipartite Matching is NP-complete even with full value and budget queries. A plausible implication is that, in this domain, no-forced-sale is computationally intertwined with pricing feasibility rather than a purely normative side constraint.

3. Mortgage contracts and the elimination of underwater default

In mortgage design, the same umbrella condition is mapped to elimination of selective borrower default when the loan balance exceeds the house price. The paper studies perpetual fixed-rate mortgages (FRMs), Adjustable Balance Mortgages (ABMs), and Adjustable Payment Rate Mortgages with penalty (APRMs) in a continuous-time model with house-price dynamics

$$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$3

The borrower can default, prepay, or continue, and contract values are characterized as optimal stopping problems with free boundaries (Kitapbayev et al., 2020).

For FRMs, the worst-case valuation has two action boundaries: a default boundary $$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$4 and a prepayment boundary $$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$5. For ABMs, the outstanding balance is $$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$6 and the payment rate is $$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$7. For APRMs, the payment rate is $$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$8 and the balance is

$$x_{ij}=1 \Rightarrow r_j \le p_j \le \min(v_{ij}, B_i)$$9

Under the paper’s normalization $D=0$0, both ABM and APRM satisfy $D=0$1 for all $D=0$2, so the strategic default option value is zero and the default boundary disappears. In the paper’s terminology, these contracts remove the default incentive and eliminate the underwater effect (Kitapbayev et al., 2020).

This does not eliminate all strategic termination. ABM and APRM may still induce prepayment. For ABM, there is no low-state prepayment if $D=0$3; otherwise a low-state prepayment boundary $D=0$4 appears in addition to a high-state boundary $D=0$5. For APRM, the structure is richer and may include a low-state boundary and a high-state prepayment interval $D=0$6, depending on $D=0$7, $D=0$8, $D=0$9, and $D_i-C_i \le g(0)X_i$0. Thus, elimination of underwater default is not equivalent to elimination of termination incentives. It removes one region of the free-boundary problem while potentially leaving others.

The paper is also explicit that capital gain sharing features are ineffective as a remedy for the low-state problem. Prepayment penalties in high house-price states can virtually eliminate prepayment, but they do not contribute to removing low-state foreclosure incentives, which are already eliminated by automatic balance adjustment. Quantitatively, for observed foreclosure costs of $D_i-C_i \le g(0)X_i$1–$D_i-C_i \le g(0)X_i$2, contracts with automatic balance adjustments become preferable to traditional FRMs at mortgage rate spreads between $D_i-C_i \le g(0)X_i$3 and $D_i-C_i \le g(0)X_i$4 basis points. With $D_i-C_i \le g(0)X_i$5 and $D_i-C_i \le g(0)X_i$6, the ABM is preferable at spreads around $D_i-C_i \le g(0)X_i$7 basis points and the APRM at around $D_i-C_i \le g(0)X_i$8 basis points (Kitapbayev et al., 2020).

4. Fire sales, repo feasibility, and residual supply

In the repo-fire-sale model, each bank $D_i-C_i \le g(0)X_i$9 must cover short-term obligations $L=0$0 using initial cash $L=0$1, liquidation $L=0$2, and repo borrowing $L=0$3. Aggregate liquidation is $L=0$4, liquidation prices are determined either through a common VWAP closing price or through a limit order book, and the repo advance rate is $L=0$5. The bank-level no-forced-sale condition is the feasibility of zero liquidation:

$L=0$6

At $L=0$7, this reduces to

$L=0$8

System-wide, the same inequality must hold for all banks. At zero liquidation, both VWAP and LOB prices equal $L=0$9, so this feasibility condition is mechanism-independent at $\theta_t=0$0 (Bichuch et al., 2020).

The paper emphasizes that this is a feasibility condition, not a full equilibrium characterization. It guarantees that collateral constraints do not force sales, but it does not by itself imply that equilibrium liquidation is zero. With $\theta_t=0$1, banks may still liquidate to reduce borrowing costs; under $\theta_t=0$2, zero liquidation becomes a natural equilibrium if the condition holds. The model also exhibits strategic complementarities: higher aggregate liquidation lowers realized sale prices and reduces repo advance rates, tightening constraints and increasing incentives for others to sell. Theorems on existence, lattice structure, and uniqueness describe when zero liquidation is part of the clearing set and when it is the unique outcome (Bichuch et al., 2020).

A distinct but related formulation appears in the residual-supply model of risk absorption. There, the no-forced-sale benchmark is the boundary case in which normalized residual supply is zero:

$\theta_t=0$3

Equivalently, limited-capital absorbers do not have to carry inventory. If $\theta_t=0$4, $\theta_t=0$5, the capital constraint is slack, and the residual hedging term is zero, then the absorption component of expected returns vanishes. In the one-period benchmark, $\theta_t=0$6, so $\theta_t=0$7 implies $\theta_t=0$8 (Wang, 29 May 2026).

The empirical implementation maps mutual-fund flows through predetermined holdings. Forced-sale pressure predicts contemporaneous price declines and positive returns over the following one to six months. Moving from the bottom to the top forced-sale rank yields approximately $\theta_t=0$9 basis points at one month and $\xi=-K1_{S^c}+\infty 1_S$0 basis points at six months in the no-microcap sample, while the premium roughly doubles when aggregate absorption capacity is tight. The paper further states that the premium concentrates in stocks with thin investor bases and limited trading capacity, and that the pattern is not generated by mechanical return reversal (Wang, 29 May 2026). This suggests a dynamic asset-pricing analogue of no-forced-sale: when natural buyers or balance-sheet capacity are sufficient, residual inventory disappears and so does the absorption premium.

5. Durable-good mechanism design and two-sided voluntary trade

In the infinite-horizon durable-good model with limited commitment, one seller owns a perfectly durable good and one buyer has binary, fully persistent valuation $\xi=-K1_{S^c}+\infty 1_S$1 with $\xi=-K1_{S^c}+\infty 1_S$2. The no-forced-sale idea has two parts. First, the seller cannot be compelled to sell: in each period he chooses a mechanism and can always choose one with $\xi=-K1_{S^c}+\infty 1_S$3. Second, the buyer cannot be compelled to buy: if she rejects, then $\xi=-K1_{S^c}+\infty 1_S$4. In direct mechanisms, this becomes a period-by-period participation constraint

$\xi=-K1_{S^c}+\infty 1_S$5

The low type’s continuation utility satisfies $\xi=-K1_{S^c}+\infty 1_S$6 in any equilibrium, so the low-type participation constraint binds at zero (Doval et al., 2019).

Using the revelation principle in Doval and Skreta (2020), the paper shows that without loss of generality input messages can be identified with types and output messages with posteriors. The resulting equilibrium problem can be written as a virtual-surplus program over Bayes-plausible posterior distributions and allocation rules. The substantive result is that posted prices implement all Perfect Bayesian equilibrium outcomes of the mechanism-selection game. Mechanisms do no better and no worse than sequential posted prices, so the seller remains subject to Coase’s conjecture (Doval et al., 2019).

The no-forced-sale structure is essential to that equivalence. Buyer rejection preserves interim participation. Seller re-optimization preserves an outside option of “no sale now.” Together they imply declining beliefs after no trade and a posted-price characterization. The paper constructs thresholds $\xi=-K1_{S^c}+\infty 1_S$7 such that, for $\xi=-K1_{S^c}+\infty 1_S$8, sale to the low type occurs after exactly $\xi=-K1_{S^c}+\infty 1_S$9 periods and the posted price is

$I$0

A common misconception would be to treat the condition as a one-sided seller protection rule. In this model it is explicitly bilateral: neither party is forced into trade.

6. Optimal liquidation with minimum-price conditions

In the Almgren–Chriss liquidation problem, no forced sale is implemented directly through control design. The trader has inventory process $I$1, trading rate $I$2, market volume process $I$3, permanent impact parameter $I$4, and temporary impact cost $I$5. Two additional objects encode the condition: a binary process $I$6 that prescribes when trading takes place, and a measurable set $I$7 that prescribes when full liquidation is required. The terminal cost is

$I$8

On $I$9, liquidation is enforced because any nonzero $S$00 generates infinite cost. On $S$01, full liquidation is not required, and the remaining inventory receives a credit through the negative terminal term (Aksu et al., 2023).

The paper develops four examples of $S$02 and $S$03 based on a lower bound $S$04 for the risky price component $S$05: always trading but forcing liquidation only if $S$06; trading until first passage below $S$07; trading only above $S$08 with no restart near terminal time; and a buffered switching rule using $S$09 and $S$10. Economically, these constructions allow the trader to slow down or stop when prices are unattractive and to enforce liquidation only in favorable regions (Aksu et al., 2023).

The analytical difficulty is that the corresponding BSDE has singular terminal value and a driver that is convex but not monotone:

$S$11

with $S$12. Because the terminal cost can be negative on $S$13, the BSDE may explode to $S$14 backward in time. The paper therefore imposes a balancing assumption,

$S$15

or its uniform-bound variant, and derives a lower bound preventing backward blow-up. In the quadratic case $S$16, the minimal supersolution identifies the value function $S$17 and yields the feedback control

$S$18

In the non-quadratic Markovian case, the value function has the homogeneous form $S$19 and is characterized by HJB or PDE-system representations (Aksu et al., 2023).

The important conceptual point is that this formulation does not abolish liquidation constraints. It makes them state contingent. No-forced-sale holds on $S$20, while liquidation remains mandatory on $S$21.

7. Common structure, divergences, and recurrent misconceptions

Across these models, no-forced-sale is never just “no trade.” In assignment markets, items may be sold, but only at prices satisfying reserves, values, and budgets. In mortgages, default may disappear, while prepayment remains. In repo systems, zero liquidation may be feasible yet not privately optimal when the repo rate is positive. In residual-supply pricing, the condition is a boundary case $S$22, not a universal equilibrium property. In durable-good models, it is bilateral participation. In liquidation control, it is a regime-switching constraint rather than a permanent ban on selling (Batziou et al., 2022, Kitapbayev et al., 2020, Bichuch et al., 2020, Wang, 29 May 2026, Doval et al., 2019, Aksu et al., 2023).

Several non-equivalences recur. Core-stability under budgets does not imply envy-freeness in assignment markets. Elimination of the underwater effect does not imply elimination of prepayment incentives in mortgage contracts. Zero-liquidation feasibility does not by itself guarantee a no-sale Nash equilibrium in repo models when $S$23. High-state prepayment penalties do not enforce the low-state mortgage condition. In optimal liquidation, the existence of a no-forced-sale region depends on the joint design of $S$24, $S$25, liquidity, and permanent impact; without the balancing assumption, the BSDE can blow down to $S$26 (Batziou et al., 2022, Kitapbayev et al., 2020, Bichuch et al., 2020, Aksu et al., 2023).

A plausible implication is that no-forced-sale should be understood as a structural compatibility condition between state variables and feasible actions. Depending on the model, that compatibility is enforced by reserve inequalities, dominance of house value over balance, collateral capacity, elastic natural demand, participation options, or singular terminal penalties. The concept is therefore domain-general in interpretation but domain-specific in formalization.