Max Beaurepaire

Rachid Dris

Bruno Tassin

Johnny Gasperi

Microplastics

Microplastics in total atmospheric fallout: Monitoring in suburban and agricultural sites

• Deposition rates are lower in the agricultural site than in the suburban site. • The lockdown monitoring campaign in site A shows the lowest microplastic deposition rates (median of 5.36 p/m²/d).

• Important precipitation events seem to be followed by higher deposition rates.

• High precision comparison of microplastic deposition rates, precipitation and rain event characteristics. • Dynamic Atmospheric fallout monitoring during single rain events. • Comparison with earlier studies on the area. • This work was supported by the OPUR project. The PhD grant of Max Beaurepaire was provided by the ENS de Lyon institution. The FTIR used for data acquisition was provided by the OSU -EFLUVE.

• The authors certify that they have no affiliation with or involvement in any organization or entity with a financial or non-financial interest in the subject matter or materials discussed in this poster. Size repartition chart of all identified particles pooled together. Smaller microplastics appear more frequent, down to a point where analytical methods limit the analysis.

29.17

Sampling•

  Microplastic pollution is the source of an increasing scientific interest since 2004 [1]. • Microplastics were first described in the atmospheric compartment in 2015 by Dris et al. [2]. • Human activity and rain events are suspected to affect atmospheric microplastic concentrations and depositions . However, no clear result was obtained on the subject [3]. • Compare microplastic deposition in two different sites • Assess the effect of 2020 lockdown on microplastic deposition in a single suburban site. • Assess the effect of precipitation events on atmospheric deposition. 1m30 54 cm (0.3m²)

  Rachid Dris 1 , Bruno Tassin 1 and Johnny Gasperi 2 1 LEESU, Ecole des Ponts, Univ Paris Est Creteil, Marne-la-Vallee, France 2 LEE, Universite Gustave Eiffel, Nantes, France E-mail contact: max.beaurepaire@enpc.fr, rachid.dris@u-pec.fr

  rates over the three campaigns. Results are shown in particles settled per square meter per day (p/m²/d).

  in the three campaigns. The dominant polymer groups always were polypropylene followed by polyethylene and polystyrene groups.

5 10 15 20 03/04 18/04 03/05 18/05 02/06 17/06 02/07 17/07

					Preliminary results
	(p/m²/d)
	rate
	Deposition	0
		100
	(p/m²/d)	60 80
	rate	40
	Deposition	0 20 10/03	25/03	09/04	24/04	09/05	24/05	08/06	23/06	08/07	23/07

Microplastics in total atmospheric fallout: Monitoring in suburban and agricultural sites

	Context	Sampling sites and campaigns
		In total: 15 samples in 2020					Site A first Monitoring campaign
		campaign, suburban site	2020	Feb.	Mar.	Apr.	May	Jun.	Jul.
		(site A)
	Objectives	21 samples in 2021 campaign, suburban site			Site A second monitoring campaign
		(site A)		2021	Feb.	Mar.	Apr.	May	Jun.	Jul.
							Site B monitoring
		8 samples in agricultural site					campaign
		(site B)
		Treatment	FTIR mapping analysis
	•
					Discussion
				Further steps
				References

Passive sampling using a metallic funnel • Rinsing with filtered water and ethanol • New samples collected every 4 to 15 days depending on dry spells and rain events

	• [1] R. C. Thompson et al., 'Lost at Sea: Where Is All the Plastic?',
	Science, vol. 304, no. 5672, pp. 838-838, May 2004, doi:
	10.1126/science.1094559.
	• [2] R. Dris, J. Gasperi, V.

Rocher, S. Mohamed, and B. Tassin, 'Microplastic contamination in an urban area: A case study in Greater Paris', Environmental Chemistry, vol. 12, May 2015, doi: 10.1071/EN14167. • [3] M. Beaurepaire, R. Dris, J. Gasperi, and B. Tassin, 'Microplastics in the atmospheric compartment: a comprehensive review on methods, results on their occurrence and determining factors', Current Opinion in Food Science, vol. 41, pp. 159-168, Oct. 2021, doi: 10.1016/j.cofs.2021.04.010.

Acknowledgements


Stéphan Sémirat
email: stephan.semirat@univ-grenoble-alpes.fr

Françoise Forges
email: francoise.forges@gmail.com

Strategic information transmission with sender's approval: the single-crossing case

Keywords: discrete cheap talk, participation constraints, single-crossing JEL Classification: C72, D82

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

A standard sender-receiver game is played in two stages: first, the -informed -agent sends a message to the -uninformed -decision maker; then, the latter makes his decision. However, in practice, there is often a third stage: the agent can reject the decision in favor of an outside option.

As an example (to be developed further below), consider the interaction between patients and a public hospital. Patients are asked to report personal information (on previous diseases, allergies, etc.), physicians decide on medical treatments, but these cannot be implemented without the patients' approval.

In this paper, we consider three-stage sender-receiver games in which, given the sender's message, the receiver proposes a decision. If the sender accepts the proposal, it is implemented.

Otherwise, the sender "exits", and his outside option is implemented. The sender's utility (on decisions, but also on the outside option) depends on his type.

As in the most general version of [START_REF] Crawford | Strategic information transmission[END_REF], the receiver's decisions belong to a real interval and the type-dependent utility functions, over decisions, satisfy desirable properties, namely, they are concave, single-peaked and single-crossing; the sender is upwardly biased. As in Frug (2016), the sender has finitely many types, which are originally ordered to make sense of the previous properties.

By contrast with the previous papers, the sender's reservation utility matters in our model.

We do not assume that it varies in any specific way with the sender's type. For instance, many relationships are conceivable between a patient's type and his utility for his outside option (to get no treatment, to go a private hospital, etc.).

We show that the sender-receiver game with sender's approval has a perfect Bayesian equilibrium (PBE) in which exit does not happen. Such equilibria are meaningful if the receiver's utility in case of rejection is sufficiently low and the receiver is conceived as an expected utility maximizing principal.

Without the "no exit" restriction, existence of a PBE is not an issue, because our game always has a babbling equilibrium, in which the sender possibly exits, for some of his types.

A natural motivation for PBE without exit, under the assumption that the receiver's utility, when exit occurs, is sufficiently low, is that the maximal utility the receiver can expect, over all conceivable PBE, is achieved at a PBE without exit (provided there exists one, of course). 1In other words, if, as [START_REF] Bester | Contracting with imperfect commitment and the revelation principle: the single agent case[END_REF], we view the receiver as a principal who cannot commit to a mechanism, namely, must play a cheap talk game with the agent, but can choose among the equilibria of this game, a PBE without exit will emerge as the solution of the and the players' ideal decisions differ by a bias that is independent of the sender's type. We only restrict the sender's (type-dependent) reservation utility by assuming that, under complete information, the receiver would be able to make a decision inducing the sender's participation.

We establish that the limit game has a PBE in pure strategies, in which the sender accepts the receiver's proposal, whatever his type. Every such equilibrium corresponds to a partition of the sender's (finite) set of types satisfying two properties: individual rationality (IR) and incentive compatibility (IC). The idea is that the sender reveals the cell π of the partition containing his type and that, for every cell π of the partition, the receiver proposes a decision that maximizes his own utility, subject to the constraint that all types in π approve the decision.

In order to avoid exit, there must exist, for every cell π, a decision giving at least as much as the outside option to every sender's type in π. This is the (IR) property. If the partition is IR, let, for every cell π of the partition, y π be the best choice of the decision maker when the sender's type is in π. The (IC) property expresses that every sender's type in π prefers the decision y π associated with π to the decision y π associated with any other cell π . Our existence proof takes the form of a step by step algorithm, which starts with the finest partition of the sender's types (namely, the set of singletons) and gradually modifies it -by moving types from one cell to another and possibly merging cells -so as to eventually satisfy (IR) and (IC).

Let us go back to the principal-agent interpretation suggested above. The receiver cannot commit to a mechanism, so that the revelation principle does not hold. Hence to maximize his expected utility, the receiver can just choose the PBE to be played. Given the previous characterization of pure PBE without exit, the receiver has to select an IR and IC partition of the sender's types, with the understanding that the sender will report the partition cell containing his true type.

The paper is organized as follows: we first complete the introduction by indicating further links between this paper and related ones. Then, in Section 2, we introduce the formal model and, in Section 3, we state our main result. Section 4 contains some elementary examples (Subsection 4.1) and, as already announced, a counterexample, in Subsection 4.2. The algorithm is described in Section 5. The section starts with basic lemmas which are proved in an appendix. The different steps of the algorithm are detailed and its convergence is established.

A representative case is described in the body of the paper while the other cases are detailed in the appendix. Section 6 makes some suggestions for future research.

Related papers

As pointed out above, we add an approval stage to the cheap talk game considered by Frug (2016), namely, [START_REF] Crawford | Strategic information transmission[END_REF] model with finitely many types for the sender.

In other words, Frug's (2016) model can be viewed as a particular case of ours, when the participation constraints are never binding. Frug (2016) convincingly argues that finitely many types appear in relevant applications. Among other results, in the uniform quadratic case, he proposes a procedure to determine a specific IR and IC partition of the types, which corresponds to an ex ante optimal equilibrium. Matthews (1989) already considers a cheap talk game with an approval stage. He makes the same basic assumptions as [START_REF] Crawford | Strategic information transmission[END_REF]. In particular, the sender has a continuum of unidimensional types. In Matthews' (1989) game, rejection of the receiver's proposal leaves the players at the status quo, which is not particularly harmful to the receiver. As a consequence, in typical equilibria of the game studied by Matthews (1989), some sender's types do veto the receiver's proposal. Actually, one of Matthews' (1989) insights is that incomplete information (together with cheap talk) can explain rational exit in a framework where complete information would preclude it.

By contrast, equilibria without exit are thoroughly analyzed in Shimizu (2017). The model is again a direct extension of [START_REF] Crawford | Strategic information transmission[END_REF], with a focus on quadratic utility functions, uniformly distributed types and type-independent reservation utility levels. Shimizu's (2017) main point is that the sender's outside options can make cheap talk informative in spite of a substantial conflict of interest between the players. To this aim, the author shows rigorously how a parameter representing the credibility of exit is as important as the sender's bias in the standard uniform quadratic model (see also Section 4.1). Shimizu (2013) illustrates the same phenomenon in the case where the receiver has a binary decision.

Finally, [START_REF] Forges | Strategic information transmission with sender's approval[END_REF] can be viewed as a companion paper. The three-stage game and the solution concept are the same as in this paper. [START_REF] Forges | Strategic information transmission with sender's approval[END_REF] only assume that the receiver's decision set is compact and that the utility functions are continuous. In particular, the sender's types can be multidimensional, so that they cannot be ordered in a relevant way. [START_REF] Forges | Strategic information transmission with sender's approval[END_REF] show that a PBE without exit may fail to exist when the receiver has finitely many decisions, over which he can randomize. Their analysis indicates that to establish the existence of a PBE without exit, one has to solve constrained optimization problems for the receiver (to account for the IR conditions) together with (IC) conditions for the sender, which can reliably be formulated for finitely many types (but looks intractable with a continuum of types). They prove that a pure PBE without exit exists in three specific cases. First, when the sender has two types only. Second, when decisions belong to the real line (as in the current paper) and, for every type, the sender's utility function is monotonic in the receiver's decision (in sharp contrast with the current paper). Third, when the receiver's utility is type-independent. In the latter case, the proof proceeds by merging cells of the finest partition of the sender's types, which is one of the basic ingredients of the algorithm below. [START_REF] Forges | Strategic information transmission with sender's approval[END_REF] leave open the question of the existence of a PBE without exit in the popular case of single-crossing utility functions. The current paper provides an answer.

Model

We consider a sender-receiver game Γ, in which the receiver's decision cannot be implemented unless the sender approves it. The sender's set of types Θ is a finite ordered set, p denotes a probability distribution over Θ, such that p(θ) > 0 for every type θ. In the same way as types, messages belong to a finite set, denoted as M , such that |M | ≥ |Θ|. Finally, u θ 0 is type θ's reservation utility.

The timing of the game Γ is as follows:

• The sender's type θ is chosen in Θ according to p.

• The sender learns his type and sends a message in M to the receiver.

• The receiver proposes a decision x ∈ R to the sender.

• If the sender accepts the proposal, the sender's utility is U θ (x) and the receiver's utility is V θ (x). Otherwise, the sender's utility is u θ 0 and the receiver's utility is -∞.

At a perfect Bayesian equilibrium (PBE) of Γ, the sender accepts decision x if U θ (x) > u θ 0 and exits if U θ (x) < u θ 0 . We will focus on PBE without exit and such that, at equilibrium, x is accepted if and only if U θ (x) ≥ u θ 0 . This means that the effective utility function of the sender is max{U θ , u θ 0 }. At a PBE without exit, the receiver's utility is finite.

As pointed out in the Introduction, [START_REF] Forges | Strategic information transmission with sender's approval[END_REF] consider the more general case where the set of decisions is a compact subset of R n . They start by considering games Γ(v 0 ), in which the receiver's utility, when the sender exits, is a finite number v 0 . They show that the PBE without exit of the limit game Γ are relevant to those of Γ(v 0 ). Indeed, on the one hand, if Γ does not have a PBE without exit, then, the same happens in every Γ(v 0 ); on the other hand, if Γ has a PBE without exit, then every Γ(v 0 ) with a sufficiently low v 0 also has a PBE without exit.

In this paper, we directly concentrate on the game Γ, but we assume that the receiver's decision is a real number and that the functions (θ, x) → U θ (x) and (θ, x) → V θ (x) satisfy the following properties:

• Strict concavity:

For every θ ∈ Θ, functions x → U θ (x) and x → V θ (x) are twice continuously differentiable and for every

x ∈ R, ∂ 2 U θ (x) ∂x 2 < 0 and ∂ 2 V θ (x) ∂x 2 < 0; (A0)
• Single-crossing:

For every (θ 1 , θ 2 , x 1 , x 2 ) ∈ Θ 2 × R 2 , with θ 2 > θ 1 and x 2 > x 1 , if U θ 1 (x 2 ) -U θ 1 (x 1 ) ≥ 0, then U θ 2 (x 2 ) -U θ 2 (x 1 ) > 0, and if V θ 1 (x 2 ) -V θ 1 (x 1 ) ≥ 0, then V θ 2 (x 2 ) -V θ 2 (x 1 ) > 0; (A1)
• Existence of a unique maximizing argument:

For every θ ∈ Θ, there exists a unique x * (θ) ∈ R and a unique y

* (θ) ∈ R such that ∂U θ (x) ∂x x=x * (θ) = 0 and ∂V θ (x) ∂x x=y * (θ) = 0; (A2)
• The sender is right biased :

For every θ ∈ Θ, x * (θ) > y * (θ); (A3)

• For every type, the receiver has the opportunity to induce the sender's participation:

For every θ ∈ Θ, there exists x ∈ R such that U θ (x) > u θ 0 ;

(A4)

Note that if the functions (x, θ) → U θ (x) satisfy (A0)-(A3), then so do the functions (x, θ) → U θ (x) -u θ 0 . Therefore, we can assume w.l.o.g. that for each θ ∈ Θ, u θ 0 = 0.4 The effective utility function of the sender becomes max{U θ , 0}, which, of course, does not necessarily satisfy properties (A0) -(A3).

Existence of an equilibrium without exit

In Section 5, we establish the following proposition.

Proposition 1. The game Γ has a pure perfect Bayesian equilibrium without exit.

Set for every θ ∈ Θ,

X θ = {x ∈ R, U θ (x) ≥ 0}.
The set X θ contains the decisions that are approved by type θ. By assumption (A4), X θ = ∅.

We show below that X θ is a closed interval of R. Since X θ = ∅,

y θ = arg max x∈X θ V θ (x) is well-defined.
More generally, let, for every L ⊆ Θ and every x ∈ R,

X L = θ∈L X θ , V L (x) = θ∈L p(θ)V θ (x).
The set X L contains the decisions that are approved by all types in L. The function V L is proportional to the receiver's conditional expected utility, given that the sender's type belongs to L; it is concave and single-peaked on R. If X L = ∅,

y L = arg max x∈X L V L (x)
is well-defined. Note that y {θ} = y θ .

To prove Proposition 1, we construct a partition Π of Θ such that Π induces an equilibrium without exit of Γ.

Consider the following conditions:

(IR) Individual Rationality:

For every cell π ∈ Π, X π = ∅;
(IC) Incentive Compatibility:

If |Π| > 1, for every π, π ∈ Π and every θ ∈ π, U θ (y π ) ≥ U θ (y π ).
functions when exit does not occur. Hence, assuming a specific functional form, e.g., quadratic, for U θ together with type-independent reservation utilities, as in Shimizu ( 2017), may entail a loss of generality (see Subsection 4.1).

Let Π be a partition of Θ satisfying (IR) and (IC). 5 To construct strategies from Π, let us first associate a message m π to every cell π of Π. Let the sender's strategy σ associated to Π be:

for every θ ∈ Θ, σ(θ) = m π(θ) ,
where π(θ) is the cell that contains θ. Let the receiver's strategy τ associated to Π be:

for every m ∈ M , if m = m π for some π ∈ Π, τ (m π ) = y π , otherwise, τ (m) = y π 0 ,
where π 0 is an arbitrary cell of Π. It is straightforward to check that (σ, τ ) defines a PBE without exit of Γ.6

As a first particular case, the coarsest partition Π = {Θ} corresponds to a nonrevealing (aka pooling) strategy of the sender, for which (IC) is automatically satisfied. However, in our model, X Θ may be empty. In this case, (IR) cannot hold for Π = {Θ} and a nonrevealing PBE without exit does not exist. At the other extreme, the finest partition Π = {{θ} : θ ∈ Θ} corresponds to a fully revealing (aka separating) strategy for the sender, for which (IR) is satisfied as a consequence of assumption (A4). But in this case, (IC) is quite demanding.

Examples

Elementary examples

Let us consider quadratic utility functions, with type-independent bias b > 0 and typeindependent maximal utility c 2 for the sender7 :

U θ (x) = c 2 -(θ + b -x) 2 V θ (x) = -(θ -x) 2
In this case, assuming c > 0,

X θ = [θ + b -c, θ + b + c] x * (θ) = θ + b y * (θ) = θ Recalling y θ = arg max x∈X θ
V θ (x), we have here

y θ =    θ + b -c if c < b θ if c ≥ b.
This expression indicates that if c < b, the sender's participation constraint x ∈ X θ is binding, for every θ while for c ≥ b, it is not binding, for every θ. Furthermore, y θ increases with θ.

As next examples will show, these properties are not always satisfied in our model.

Let us show how we can construct a PBE without exit when c < b. Let us set Θ = {θ 1 , . . . , θ n }, with θ 1 < . . . < θ n . For every subset L = {θ j+1 , . . . , θ j+ } of consecutive types such that X L = ∅, the participation constraint, namely, x ∈ X L , is binding:

θ∈L p(θ)θ ≤ θ j+ ≤ θ j+ + b -c = min X L so that y L = θ j+ + b -c.
The partition Π 0 = {{θ i } , i = 1, . . . , n} satisfies (IR), but every type just receives his reservation value. Regarding (IC), if θ i < θ j , y θ i / ∈ X θ j so that type θ j cannot envy θ i . However, θ j envies θ k > θ j as soon as θ k < θ j + 2c. This suggests that the effect of the parameter c, which reflects the credibility of exit (since it determines the size of X θ ) is as important as the effect of the bias b, which reflects the players' conflict of interest. This is a key insight in Shimizu (2017).

A partition satisfying (IR) and (IC) can be gradually constructed from Π 0 . To see that, observe that the highest type θ n does not envy any type. Let us keep the cells as in Π 0 until we find θ k envying θ k+1 ; if this happens, we move θ k to the next cell to form {θ k , θ k+1 }. The receiver's constrained optimal decision over the new cell is

y {θ k ,θ k+1 } = θ k+1 + b -c = y θ k+1 .
Hence the move is helpful to fulfill type θ k 's (IC) condition but has no effect on the other types.

We can go on by identifying the next type, among θ 1 < . . . < θ k-1 , starting with θ k-1 , which envies the cell on its right, and so on to end up with a partition satisfying both (IR) and (IC).

These steps are consistent with the algorithm that we propose below and all correspond to case C. of Subsection 5.7.

In this particular example, the algorithm to go from partition Π 0 to an IR and IC partition is quite simple, because moving a type to a next cell does not generate a new decision for the receiver. As a result, checking (IC) from step to step is straightforward. Even with the utility functions above, the construction gets more intricate when c ≥ b, because the participation constraint x ∈ X L , which is not binding for L = {θ}, can become binding when L gets larger.

Note that when c is so large that the participation constraint is never binding, whatever the subset of types, we recover the standard case, in which there is no need to ask for the sender's approval.

Other examples are proposed in the sequel to illustrate the difficulties of a fully general algorithm. These examples are still generated by quadratic utility functions but both the bias and the maximum payoff of the sender depend on his type, namely

U θ (x) = c(θ) 2 -(θ + b(θ) -x) 2
with b(θ) > 0 and θ + b(θ) increasing with θ.

Counterexample

Existence of a partition Π that satisfies (IR) and (IC) is not guaranteed in general.8 Indeed, conditions (IR) and (IC) tend to exert opposite effects on the players' strategies:

• Condition (IC) directly reports on the sender's Nash equilibrium conditions. When condition (IR) is irrelevant (e.g. when the sender prefers the receiver's preferred action to his outside option), condition (IC) is obtained by pooling adjacent types, up to the situation in which players agree about the resulting pool-contingent actions.9

• Condition (IR) requires separating strategies for types whose participation constraints cannot be simultaneously satisfied.

Let us illustrate, in a setting close to ours, that conditions (IC) and (IR) may not be simultaneously satisfied. In the following example, we only relax the single-crossing assumption (A1), and show that it is not possible to pool and separate the types in such a way that both conditions (IR) and (IC) hold.

U θ 1 U θ 2 U θ 3 • y θ 1 • y {θ 1 ,θ 2 } • y θ 2 • y θ 3 = y {θ 1 ,θ 3 } • • • V θ 1 V θ 2 V θ 3
Example 1. There are three types θ 1 , θ 2 , θ 3 , with θ 1 < θ 2 < θ 3 . The utility functions and cellcontingent optimal actions are as depicted in Figure 1. In particular, assumptions (A0), (A2), (A3) and (A4) hold. Moreover, we have x * (θ 1 ) < x * (θ 2 ) < x * (θ 3 ) and y * (θ 1 ) < y * (θ 2 ) < y * (θ 3 ).

However, the single-crossing condition (A1) does not hold, because θ 2 > θ 1 and y θ 3 > y {θ 1 ,θ 2 } , but

U θ 1 (y θ 3 ) -U θ 1 (y {θ 1 ,θ 2 } ) > 0 and U θ 2 (y θ 3 ) -U θ 2 (y {θ 1 ,θ 2 } ) < 0.
In that context, there is no partition Π of Θ = {θ 1 , θ 2 , θ 3 } such that both (IR) and (IC) are satisfied:

• If Π = {{θ 1 , θ 2 , θ 3 }} (uninformative), or if Π = {{θ 1 }, {θ 2 , θ 3 }} (partially informative), from X {θ 1 ,θ 2 ,θ 3 } ⊂ X {θ 2 ,θ 3 } = X θ 2 ∩ X θ 3 = ∅, (IR) does not hold. • If Π = {{θ 1 , θ 2 }, {θ 3 }} (partially informative), then (IR) holds, but from U θ 1 (y {θ 1 ,θ 2 } ) < U θ 1 (y θ 3
), (IC) does not hold.

• If Π = {{θ 2 }, {θ 1 , θ 3 }} (partially informative), then (IR) holds, but from U θ 1 (y {θ 1 ,θ 3 } ) < U θ 1 (y θ 2 ), (IC) does not hold.

• If Π = {{θ 1 }, {θ 2 }, {θ 3 }} (fully informative), then (IR) holds, but from U θ 1 (y θ 1 ) < U θ 1 (y θ 2 ) or U θ 1 (y θ 1 ) < U θ 1 (y θ 3 ), (IC) does not hold.

Finally, note that assumption (A1) would hold if, in Figure 1, the alternative graph for utility function U θ 2 (the dashed one) was considered. In that case, X {θ 1 ,θ 2 ,θ 3 } = X θ 2 ∩ X θ 3 = ∅, y {θ 1 ,θ 2 ,θ 3 } = y θ 3 , and the uninformative partition Π = {{θ 1 , θ 2 , θ 3 }} does satisfy (IR) and (IC).

In the next section, we provide an algorithm showing that under assumptions (A0)-(A4), it is always possible to pool and separate the types in such a way that conditions (IR) and (IC) simultaneously hold.

Algorithm

We establish Proposition 1 by recursively defining a sequence of partitions (Π r ) r≥0 that ends on a partition Π of Θ satisfying (IR) and (IC).

Let us first state some lemmas about the receiver's constrained optimum (proofs are given in the appendix).

Preliminary lemmas

The first lemma states that X θ is a closed interval. 10

Lemma 1.

For every θ ∈ Θ, X θ = [min X θ , max X θ ].
The next two lemmas deal with the receiver's constrained optimal action in the complete information case, namely, y θ = arg max

x∈X θ V θ (x).
Lemma 2. For every θ ∈ Θ, y θ < x * (θ) < max X θ . In particular, U θ is increasing at x = y θ .

Lemma 3. For every θ ∈ Θ, y θ ≥ y * (θ), with equality iff y * (θ) ≥ min X θ . In particular, V θ is not increasing at x = y θ .

Summing up,

y θ =    y * (θ) if (x ∈ X θ ) is not binding, min X θ if (x ∈ X θ ) is binding.
10 The strict inequality in (A4) guarantees that, for every type θ, the set X θ has a nonempty interior. This is to avoid extra case by case analysis that, we believe, would not add much to the result.

Observe that, as expected from the single-crossing condition (A1), y * (θ) increases when θ increases according to the original order over Θ, but that this monotonicity does not necessarily hold for y θ (see Figure 2 below).

Finally, the next three lemmas identify relationships between the receiver's constrained optimal action in the partially informative case, namely, y L , L ⊂ Θ and y θ , θ ∈ L. Note that when L is not a singleton, y L is possibly achieved at both ends of X L :

y L = arg max x∈X L θ∈L p(θ)V θ (x) =    y * (L) if (x ∈ X L ) is not binding min X L or max X L if (x ∈ X L ) is binding where y * (L) = arg max x∈R θ∈L p(θ)V θ (x).
Lemma 4. Let L ⊆ Θ be such that X L = ∅. Let θ ∈ L be such that for every θ ∈ L, y θ ≤ y θ .

Then y θ ≤ y L .

Lemma 5. Let L ⊆ Θ be such that X L = ∅. Let θ ∈ L be such that for every θ ∈ L, y θ ≥ y θ .

Then

y θ ≥ y L . Lemma 6. Let L 1 , L 2 ⊆ Θ be such that X L 1 ∪L 2 = ∅. If y L 1 ≥ y L 2 then y L 1 ≥ y L 1 ∪L 2 ≥ y L 2 .

Initialization

Let us order the types in Θ as θ i 1 , . . . , θ in , so that

y θ i 1 < . . . < y θ in .
We start with the following partition Π 0 :

{π 1 0 = {θ i 1 }, . . . , π n 0 = {θ in }},
whose cells are ordered in the same way as the types. 11

Partition Π 0 clearly satisfies (IR). Moreover, since for each θ ∈ Θ, U θ is increasing at y θ (see Lemma 2), partition Π 0 is such that no type in a cell strictly prefers the action induced in a preceding cell. Hence partition Π = Π 0 also satisfies the following condition, which makes sense in any partition Π with ordered cells:

11 If there exist types θ and θ such that y θ = y θ , then it is straightforward to see that y {θ,θ } = y θ = y θ , so that such types can be merged into a unique cell, up to obtaining a partition Π0 that does satisfy y

π1 0 < y π2 0 < . . . < y π| Π0 | 0 .
In that case, the algorithm has to start from partition Π0 instead of Π 0 .

U θ 2 U θ 1 • y θ 2 • y θ 1 V θ 1 V θ 2 Figure 2: A situation in which θ 1 < θ 2 , but y θ 1 > y θ 2 .
(L-IC) Left-sided incentive compatibility:

If |Π| > 1, for every n 2 ∈ {2, . . . , |Π|}, for every θ ∈ π n 2 , for every n 1 ∈ {1, . . . , n 2 -1}, U θ (y π n 1 ) < U θ (y π n 2 ).
Condition (L-IC) can be seen as a half-way path to the more demanding condition (IC), which says that no type in a given cell strictly prefers the action induced in neither a preceding nor a succeeding cell. The recursive procedure detailed below will generate a sequence (Π r ) r≥0 in which every Π r is an ordered partition satisfying conditions (IR) and (L-IC), and such that, as r gets larger, Π r gradually fulfills the "right-sided incentive compatibility" (R-IC) constraint too. To achieve this goal, if Π r does not satisfy (R-IC), the idea is to construct Π r+1 from Π r by merging cells of Π r or by moving a type θ ∈ π n , with π n ∈ Π r , to some succeeding cell π n , n > n, of Π r . But to keep (IR) and (L-IC) all along the sequence, some care is needed in the way cells are ordered in every partition Π r . It turns out to be useful to identify further properties (namely, (P1) and (P2) below) to be satisfied by the partitions in (Π r ) r≥0 .

Orders on cells, types and induced actions

The type ordering used to define partition Π 0 , for which (L-IC) holds, might not be the original one, for which single-crossing holds. Indeed a lower type may have a greater outside option than a higher one. Then, as illustrated in Figure 2, in case of a large bias of the sender, the receiver's optimal type-contingent actions may be non monotonic in the type. 12 However, from the single-crossing condition, a partition that satisfies (IC) requires the receiver's optimization to associate types with actions increasingly. For instance, if, as in Figure 2, θ 1 < θ 2 and y θ 2 < y θ 1 , so that the ordered partition Π 0 = {{θ 2 }, {θ 1 }} associates types with actions decreasingly, then θ 1 necessarily prefers y θ 1 to y θ 2 , and from (A4), so does θ 2 . Thus partition Π 0 does not satisfy (IC).

As we shall see, an increasing association between types and actions from Π 0 can be achieved through merging adequately the types into cells. However, given that the action associated with a cell is a convex combination of the actions associated with the types that it contains (see Lemmas 4 and 5), the non monotonic association of type and type-contingent actions makes the task difficult. But we do have a monotonic association between types and type-contingent actions for at least a subset of the types. Indeed, as a consequence of the single-crossing condition on the receiver's utility function, according to which θ → y * (θ) increases, type and type-contingent actions are identically ordered at least for those types θs such that y θ = y * (θ),

i.e. for those types such that the receiver's constraint is not binding (such that y θ > min X θ ).

When considered in Π 0 , since y π k 0 = y θ i k , k ∈ {1, . . . , n}, the previous property can also be stated as a monotonic association between types and cell -contingent actions. In this way, the property can be stated for any partition Π = {π 1 , . . . , π |Π| } through the following two conditions:

(P1) Increasing cell-contingent actions: y π 1 < y π 2 < . . . < y π |Π| .

(P2) Increasing unconstrained type-contingent actions: If |Π| > 1, then for every n ∈ {1, ..., |Π|-1}, for every θ ∈ π n , if y θ > min X θ , then for every n ∈ {n+1, . . . , |Π|}, for every θ ∈ π n ,

y θ ≤ y θ . 13
It turns out that the two conditions are sufficient to deal with the way actions shift when, starting from Π 0 , we will recursively construct an (IR) and (IC) partition by merging types adequately. To this end, we shall keep conditions (P1) and (P2) at every step of the algorithm.

Moreover, as the next example illustrates, once a partition Π fully satisfies (P1), (IR) and (IC), property (P2) cannot be modified to include types whose receiver's constraint is binding (such that y θ = min X θ ). Thus, (P1) and (P2) can be seen as a minimal requirement concerning the relationship between the order of types and the order of cells, as induced by convex combinations of the associated type-contingent actions.

Example 2. There are three types θ 1 , θ 2 , θ 3 , with θ 1 < θ 2 < θ 3 . The utility functions and cell-contingent optimal actions are as depicted in Figure 3. The initial partition is Π

0 = U θ 2 U θ 1 U θ 3 • y θ 2 • y θ 1 = y {θ 1 ,θ 2 } • y θ 3 y {θ 2 ,θ 3 } V θ 1 V θ 2 V θ 3
Π = {π 1 , π 2 } with π 1 = {θ 1 }, π 2 = {θ 2 , θ 3 }. 14 This partition Π is such that y π 1 < y π 2 , but θ 1 ∈ π 1 (with binding receiver's constraint), θ 2 ∈ π 2 and y θ 1 > y θ 2 .

Recursive procedure

Proposition 1 results from the following claim, which we establish in Section 5.7.

Claim. Let Π r be a partition which satisfies (P1), (P2), (IR) and (L-IC). If Π r does not satisfy (IC), then there exists a partition Π r+1 , obtained by moving some type θ from some cell π n ∈ Π r to some succeeding cell π n , n > n, and possibly by merging cells, such that Π r+1 satisfies conditions (P1), (P2), (IR) and (L-IC).

The claim ultimately guarantees the existence of a partition satisfying (IR) and (IC) because the moves considered in the claim cannot be repeated indefinitely. Indeed, there are finitely many types and finitely many cells. Furthermore, every type move is always directed to a succeeding cell. In other words, starting from Π 0 , and constructing Π r+1 from Π r according to the claim, we necessarily reach a partition that satisfies (IR) and (IC).

14 When considering the fully revealing partition Π = {{θ 2 }, {θ 1 }, {θ 3 }}, or the partially revealing partition Π = {{θ 2 , θ 1 }, {θ 3 }}, type θ 2 prefers y θ3 to y θ2 or to y {θ2,θ1} respectively, and if we consider the non revealing partition Π = {{θ 2 , θ 1 , θ 3 }} or the partially revealing partition Π = {{θ 2 }, {θ 1 , θ 3 }}, the receiver may not satisfy the participation constraint of the sender.

As will be clear below, the algorithm used to establish the claim will be flexible regarding the choice of the cell of the type θ to be moved. Hence, different runs of the algorithm will be conceivable, possibly leading -a priori -to different IR and IC partitions. Given our primary goal, namely, to establish the existence of an equilibrium without exit, we do not constrain the algorithm unless further specification makes it simpler.

Choices for the moving type and the destination cell

According to the above procedure, we need to establish the existence of a partition Π r+1 that satisfies conditions (IR) and (L-IC), among the partitions that can be reached from Π r , by using the moves prescribed in the above claim.

Keeping (IR) satisfied from Π r to Π r+1 is easily obtained if only appropriate cells are merged (cells π and π such that X π ∩ X π = ∅), and if, as the following lemma (established in the appendix) states, type moves are restricted to moving a type to a cell that it prefers.

Lemma 7. Let L, L ⊂ Θ with X L = ∅, X L = ∅, and suppose that there exists θ ∈ L , θ / ∈ L,

such that U θ (y L ) > U θ (y L ). Then we have X L∪{θ} = ∅.
Keeping (L-IC) satisfied is more demanding because the considered moves from Π r to Π r+1 shift some induced actions, and thereby impact condition (L-IC) for every type. To that end, we need to carefully monitor every cell-contingent induced action y πr , π r ∈ Π r , during the process. As a first step, we will specify the way in which type θ is moved from its cell π n to a succeeding cell π n . Before this specification, let us introduce further notations.

Set, for every y ∈ R,

X θ (y) = {x ∈ R : U θ (x) ≥ U θ (y)}.
Since U θ is single-peaked, we have:

if y < x * (θ), then min X θ (y) = y; if y > x * (θ), then max X θ (y) = y; if y = x * (θ), then X θ (y) = {x * (θ)}.
Then we can write

X θ (y) = [x θ -(y), x θ + (y)],
with y = x θ -(y) or y = x θ + (y) conditional on y ≤ x * (θ) and y ≥ x * (θ) respectively. Note that from Lemma 2, for every θ ∈ Θ, x θ -(y θ ) = y θ .

Destination cell

Concerning the destination cell, if the chosen type θ to be moved is an element of π n , we show that we can w.l.o.g. choose π n with n = n + 1.

Let θ ∈ π n , for some n ∈ {1, . . . , |Π|-1}, that prefers the action induced in some succeeding cell π n , n > n, i.e. such that y π n ∈ X θ(y π n ). Since X θ(y π n ) is an interval, and since we also have y π n ∈ X θ(y π n ), we obtain [y π n , y π n ] ⊆ X θ(y π n ), where y π n < y π n results from (P1). From (P1), we also get y π n+1 ∈ [y π n , y π n ]. Therefore, we have y π n+1 ∈ X θ(y π n ), i.e. type θ prefers y π n+1 to y π n .

Moving type

We choose, among the types in π n who prefer y π n+1 , the type θ with the greatest associated action y θ . We set

θ = θ i ,
where we ordered

π n = {θ i 1 , . . . , θ i |π n | } in such a way that y θ i 1 ≤ . . . ≤ y θ i |π n | , and set = max{j : U θ j (y π n+1 ) > U θ j (y π n )}.
The next lemma allows us to characterize θ within π n .

Lemma 8. Either = |π n | or, if < |π n |, then for every j ∈ { + 1, . . . , |π n |}, y θ i j = min X θ i j .

In that latter case,

y π n = y θ i |π n | = min X θ i |π n | .

Action shifts

The choice of θ as a maximal element of π n and the choice of moving it to the next succeeding cell π n+1 allow us to characterize how the induced actions y π n and y π n+1 shift after the type move.

Lemma 9.

• If π n \ { θ} = ∅, then

y π n \{ θ} ≤ y π n , (1)
and, if moreover < |π n |,

y π n \{ θ} = y π n . (1a)
•

y θ ≤ y π n+1 ∪{ θ} ≤ y π n+1 (2) • If = |π n |, then y π n ≤ y π n+1 ∪{ θ} .
(3)

Partition Π r+1

Finally, we are now able to define partition Π r+1 from a partition Π = Π r that satisfies (IR), (L-IC), (P1) and (P2). We distinguish the following cases.

(A) If [π n \ { θ} = ∅ and y π n+1 ∪{ θ} ≤ y π n \{ θ} ], we set: Case C is the simplest one. It occurs for instance when π n is a cell of the initial partition Π 0 , in particular, if Π 0 does not satisfy (IC). We provide an illustration of the construction of Π r+1 in cases A and B in examples 3 and 4 respectively.

             π 1 r+1 = π 1 r , . . . , π n-1 r+1 = π n-1 r , π n r+1 = π n r ∪ π n+1 r , π n+1 r+1 = π n+2 r , . . . , π |Πr|-1 r+1 = π |Πr| r . ( 4
) (B) If [π n \ { θ} = ∅ and y π n+1 ∪{ θ} > y π n \{ θ} ], (B.1) if [n = 1 or y π n-1 r < y π n r \{ θ} ], we set                      π 1 r+1 = π 1 r , . . . , π n-1 r+1 = π n-1 r if n > 1, π n r+1 = π n r \ { θ}, π n+1 r+1 = π n+1 r ∪ { θ}, π n+2 r+1 = π n+2 r , . . . , π |Πr| r+1 = π |Πr| r . ( 5
) (B.2) if [n > 1 and y π n-1 r ≥ y π n r \{ θ} ], we set:                      π 1 r+1 = π 1 r , . . . , π n-2 r+1 = π n-2 r , π n-1 r+1 = π n-1 r ∪ π n r \ { θ} , π n r+1 = π n+1 r ∪ { θ}, π n+1 r+1 = π n+2 r , . . . , π |Πr|-1 r+1 = π |Πr| r . ( 6
) (C) If [π n \ { θ} = ∅]. we set:              π 1 r+1 = π 1 r , . . . , π n-1 r+1 = π n-1 r , π n r+1 = π n+1 r ∪ { θ}, π n+1 r+1 = π n+2 r , . . . , π |Πr|-1 r+1 = π |Πr| r . (7)
U θ 1 U θ 2 U θ 3 • y θ 1 = y {θ 2 ,θ 1 } = y {θ 2 ,θ 1 ,θ 3 } • y θ 2 • y θ 3 • y {θ 2 ,θ 3 } V θ 1 V θ 2 V θ 3
Example 3. There are three types θ 1 , θ 2 , θ 3 , with θ 1 < θ 2 < θ 3 . The utility functions and cell-contingent optimal actions are as depicted in Figure 4. In particular, y 2 < y θ 1 < y θ 3 .

Set Π 0 = {{θ 2 }, {θ 1 }, {θ 3 }} accordingly. Then type θ 2 prefers y θ 1 = y π 2 0 to y θ 2 = y π 1 0 . Set

Π 1 = {{θ 2 , θ 1 }, {θ 3 }}. Then type θ 2 prefers y θ 3 = y π 2 1 to y {θ 2 ,θ 1 } = y π 1 1 . When θ = θ 2 moves from π 1 1 = {θ 2 , θ 1 } to π 2 1 = {θ 3 }, we obtain y π 2 1 ∪{ θ} = y {θ 2 ,θ 3 } ≤ y θ 1 = y π 1 1
\{ θ} . This is case A. In words, moving a type from a cell to a succeeding cell may reverse the order of the shifted cell-contingent actions. Hence the resulting cell-contingent actions do not satisfy (P1). However, as we shall

show, in such a case it is always possible to merge the considered cells while keeping condition (IR) satisfied. In the present example, the next step is to set

Π 2 = {π 1 1 ∪ π 2 1 } = {{θ 2 , θ 1 , θ 3 }} which does satisfy (IR)

and (P1).

Example 4. There are four types θ 1 , θ 2 , θ 3 , θ 4 , with θ 1 < θ 2 < θ 3 < θ 4 . The utility functions and cell-contingent optimal actions are as depicted in Figure 5. In particular, y θ 2 < y θ 1 < y θ 3 < y θ 4 . Set Π 0 = {{θ 2 }, {θ 1 }, {θ 3 }, {θ 4 }} accordingly. Then type θ 2 prefers y θ 1 = y π 2 0 to

y θ 2 = y π 1 0 . Set Π 1 = {{θ 2 , θ 1 }, {θ 3 }, {θ 4 }}. Now type θ 2 prefers y θ 3 = y π 2 1 to y {θ 2 ,θ 1 } = y π 1 1 . Set Π 2 = {{θ 1 }, {θ 2 , θ 3 }, {θ 4 }}.
Then y θ 1 < y {θ 2 ,θ 3 } < y θ 4 , so that the order on the cell-contingent

U θ 1 U θ 2 U θ 3 U θ 4 • y θ 2 • y θ 1 = y {θ 2 ,θ 1 } • y θ 3 = y {θ 2 ,θ 3 } • y θ 4 = y {θ 3 ,θ 4 } V θ 1 V θ 2 V θ 3 V θ 4
prefers y θ 4 = y π 3 2 to y {θ ,θ 3 } = y π 2 2 . When θ = θ 3 moves from π 2 2 = {θ 2 , θ 3 } to π 3 2 = {θ 4 }, then y π 1 2 = y θ 1 ≥ y θ 2 = y π 2 2 \{ θ} . This is case B.2.
In words, moving a type from a cell π n to the next succeeding cell may reverse the order of the cell-contingent actions of the nth cell (which becomes π n \ { θ}) and its preceding cell π n-1 . In particular, the resulting cell-contingent actions do not satisfy (P1). However, as we shall show, in that case it is always possible to merge π n \ { θ} and π n-1 in such a way that the resulting partition satisfies (IR) and (P1). In the present example, the next step is to set Π

3 = {π 1 2 ∪ π 2 2 \ { θ}, π 3 2 ∪ { θ}} = {{θ 2 , θ 1 }, {θ 3 , θ 4 }}.
Once Π r+1 is constructed from Π r , we have to show that it also satisfies (IR), (L-IC), ( P1) and (P2). The construction is greatly simplified when y θ is increasing w.r.t. the initial order of types, a property that does not hold in examples 3 and 4. It holds when participation constraints are not binding for individual types (i.e., y θ = y * (θ)). Then only two cases (B.1 and C) can arise, the initial order of types is maintained all along the procedure and at every step, properties (P1) and (P2) are satisfied in a straightforward way. As an illustration, the proof of case C is given below. We give the proofs of cases A and B in the appendix.

The proof of case C. We suppose π n r \ { θ} = ∅, i.e. π n r = { θ}, and set

             π 1 r+1 = π 1 r , . . . , π n-1 r+1 = π n-1 r , π n r+1 = π n+1 r ∪ { θ}, π n+1 r+1 = π n+2 r , . . . , π |Πr|-1 r+1 = π |Πr| r . (7)
By construction, Π r+1 satisfies (IR).

We show that Π r+1 satisfies (P1). Since Π r satisfies (P1), we only have to show:

y π n-1 r < y π n+1 r ∪{ θ} < y π n+2 r .
From π n r = { θ}, y π n r = y θ. From y π n r < y π n+1 r , we get y θ < y π n+1 r and thus, from Lemma 6,

y θ ≤ y π n+1 r ∪{ θ} ≤ y π n+1 r
, that is: θ ∈ π n 1 r+1 . Suppose that there exists n 2 ≥ n 1 + 1 and θ ∈ π n 2 r+1 such that y θ < y θ . We show that necessarily y θ = min X θ .

y π n r ≤ y π n+1 r ∪{ θ} < y π n+1 r . (8
• If n 2 ≤ n -1, then n 1 ≤ n -2 and θ ∈ π n 1 r , θ ∈ π n 2 r with n 2 ≥ n 1 + 1, so that we have y θ = min X θ because Π r satisfies (P2).

• If n 2 = n, then either θ ∈ π n+1 r or θ = θ ∈ π n r . Moreover, n 1 ≤ n 2 -1 = n -1 implies θ ∈ π n 1
r with n 1 < n. The result also follows from (P2) applied to Π r .

• If n 2 ≥ n + 1, then θ ∈ π n 2 +1 r . -If n 1 ≥ n + 1 too, then θ ∈ π n 1 +1
r and the result follows by the same argument as above.

-If n 1 ≤ n, then either θ ∈ π n+1 r , or θ = θ ∈ π n r or θ ∈ π n 1 r . The result is similarly obtained in all three cases. Finally, we show that Π r+1 satisfies (L-IC). Let n 2 ∈ {2, . . . , |π r+1 |}, n 1 ∈ {1, . . . , n 2 -1} and θ ∈ π n 2 r+1 . We want to show that y π n 1 r+1 / ∈ X θ (y π n 2 r+1 ).

We first consider the case θ = θ. In that case, π

n 2 r+1 = π n+1 r
∪ { θ} and we have to show

y π n 1 r+1 / ∈ X θ(y π n+1 r ∪{ θ} ), i.e. U θ(y π n+1 r ∪{ θ} ) > U θ(y π n 1 r+1 ).
From n 1 < n 2 = n, we have y π n 1 r+1 = y π n 1 r . Since Π r+1 satisfies (P1), we have y π n 1 r < y π n r , and from (8), we obtain the inequalities ). Now by construction, we have U θ(y π n+1 r ) > U θ(y π n r ), and since Π r satisfies (L-IC), we also have U θ(y π n r ) > U θ(y π n 1 r ). This gives the result.

y π n 1 r < y π n+1 r ∪{ θ} ≤ y π n+1
Next we suppose θ = θ. In that case, we use the following lemma.

Lemma 10. Let Π = {π 1 , . . . , π |Π| } be a partition of Θ that satisfies (IR), ( P1) and (L-IC).

Let

n 2 ∈ {2, . . . , |Π|}, n 1 ∈ {1, . . . , n 2 -1} and let θ ∈ π n 2 . For every y 1 , y 2 ∈ R, if y 1 < y 2 , y 1 ≤ y π n 1 and y 2 ≤ y π n 2 , then y 1 / ∈ X θ (y 2 ).
According to that lemma, partition Π r+1 satisfies (L-IC) if y 1 = y π n 1 r+1 and y 2 = y π n 2 r+1 are such that y 1 < y 2 (guaranteed because Π r+1 satisfies (P1)) and such that their exist n 2 ∈ {2, . . . , |π r |}

and n 1 ∈ {1, . . . , n 2 -1} with θ 2 ∈ π n 2 r , y 1 ≤ y π n 1 r and y 2 ≤ y π n 2 r . The result is obtained by the following case by case analysis.

• If n 2 ≤ n -1, then θ ∈ π n 2 r+1 = π n 2 r = π n 2 r with n 2 = n 2 , so that y 2 = y π n 2 r = y π n 2 r . From n 1 ≤ n 2 -1 ≤ n -2, we also have y 1 = y π n 1 r = y π n 1 r
with n 1 = n 1 . This gives the result.

• If n 2 = n, then θ ∈ π n 2 r+1 = π n 2 r ∪ { θ}, and since θ = θ, we have θ ∈ π n 2 r = π n 2 r with n 2 = n 2 , and, according to (2), y 2 ≤ y π n 2 r = y π n 2 r . From n 1 ≤ n 2 -1 = n -1, we have y 1 = y π n 1 r = y π n 1 r
with n 1 = n 1 , which gives the result.

• If n 2 ≥ n + 1, then θ ∈ π n 2 r+1 = π n 2 +1 r = π n 2 r with n 2 = n 2 + 1, and then y 2 = y π n 2 +1 r = y π n 2 r with n 2 = n 2 + 1 ≥ n + 2. -If n 1 ≥ n + 1 or n 1 ≤ n -1, then π n 1 r+1 = π n 1 r = π n 1 r with n 1 = n 1 , and thus y 1 = y π n 1 r = y π n 1
r , which gives the result.

-

If n 1 = n, inequality (2) gives y 1 ≤ y π n+1 r = y π n 1 r
with n 1 = n + 1, and the result follows.

Concluding remarks

We have established the existence of a perfect Bayesian equilibrium without exit in a class of games of information transmission with sender's approval. To do so, we have proposed a step by step algorithm which achieves an IR and IC partition of the sender's type. In the equilibrium associated with the partition, the sender reveals the cell containing his type and for every cell, the receiver makes the optimal decision subject to participation of all types in the cell.

The algorithm starts by ordering the types so that no type envies a lower one (L-IC). The initial partition of singletons, which is trivially IR, is gradually modified into a not finer partition satisfying (L-IC), as long as some type envies a higher one. Our construction is applicable in every game of the class under consideration. As noted above, our algorithm allows for some flexibility, at every step r, to pick the cell π n ∈ Π r containing an envying type; then, a specific envying type θ is selected, namely, the maximal element of π n and θ is moved to the next cell π n+1 . As a consequence, it can happen that the algorithm has several distinct runs.

In any case, our algorithm guarantees that the sender-receiver game with sender's exit option has a partitional equilibrium without exit. Since, in our model, exit gives an infinitely low payoff to the receiver, our existence result implies that the problem of maximizing the equilibrium payoff of the receiver (interpreted as a principal) has a finite solution. A natural topic for future research would be to identify conditions guaranteeing that the receiver's best equilibrium is partitional, as may happen in models like the one we consider, with finitely many types for the sender, when participation constraints do not matter (see Frug ( 2016)).

Under such conditions, one might hope that (variants of) our algorithm converge to this best partitional equilibrium.

V L 2 (y L 2 ) ≥ V L 2 (y L 1 ). From that, we get that V L 2 is non increasing on [y L 2 , +∞). In particular, it is non increasing at x = y L 1 ≥ y L 2 . Therefore V L 1 ∪L 2 = V L 1 + V L 2 is non increasing at x = y L 1 . This implies y L 1 ∪L 2 ≤ y L 1 .

Lemma 7. Let L, L ⊂ Θ with X L = ∅, X L = ∅, and suppose that there exists θ ∈ L with θ / ∈ L. If U θ (y L ) > U θ (y L ), then X L∪{θ} = ∅.

Proof. By definition, y L ∈ X L , and since X L = X L \{θ} ∩ X θ ⊆ X θ , we have y L ∈ X θ .

Therefore U θ (y L ) ≥ 0, so that U θ (y L ) > U θ (y L ) implies U θ (y L ) > 0. Hence y L ∈ X θ . Since y L ∈ X L , we obtain y L ∈ X L ∩ X θ = X L∪{θ} , and in particular X L∪{θ} = ∅.

Lemma 8. Either = |π n | or, if < |π n |, then for every j ∈ { + 1, . . . , |π n |}, y θ i j = min X θ i j .

In that latter case, y π n = y

θ i |π n | = min X θ i |π n | .
Proof. If = |π n |, let j ∈ { + 1, ..., |π n |} = ∅. First, we show

y π n+1 ≥ x θ i j + (y π n ). (9)
Since j > , U θ i j (y π n+1 ) ≤ U θ i j (y π n ). This means that either (9) holds, or y π n+1 ≤ x θ i j -(y π n ). However this latter inequality cannot occur. Indeed, from (P1), we have y π n+1 > y π n . And from y π n ∈ X θ i j (y π n ), we have y π n ≥ x θ i j -(y π n ). We obtain y π n+1 > x θ i j -(y π n ). Second, we show

y π n = x θ i -(y π n ).
(10) If (10) does not hold, then y π n = x θ i + (y π n ). If this holds, then from U θ i (y π n+1 ) > U θ i (y π n ), we obtain y π n+1 < x θ i + (y π n ) = y π n . But from (P1), we have y π n < y π n+1 . Third, we show θ i j < θ i .

(11)

From ( 9), x θ i j + (y π n ) ≤ y π n+1 and from U θ i (y π n+1 ) > U θ i (y π n ), we have y π n+1 < x θ i + (y π n ). Then we obtain x θ i + (y π n ) > x θ i j + (y π n ). This implies x θ i + (y π n ) / ∈ X θ i j (y π n ). Since y π n ∈ X θ i j (y π n ), we get U θ i j (x

θ i + (y π n )) -U θ i j (y π n ) < 0. ( 12
)
From ( 10), U θ i (x θ i -(y π n )) -U θ i (y π n ) = 0, and since U θ i (x

θ i + (y π n )) = U θ i (x θ i
-(y π n )), we get

U θ i (x θ i + (y π n )) -U θ i (y π n ) = 0.
(P1), we have y π n 1 < y π n 2 ≤ x θ + (y π n 2 ), so that necessarily y π n 1 < x θ -(y π n 2 ). ( 16)

Let us first suppose y 2 ∈ X θ (y π n 2 ). This means U θ (y 2 ) ≥ U θ (y π n 2 ), which implies X θ (y 2 ) ⊆ X θ (y π n 2 ). In particular, x θ -(y π n 2 ) ≤ x θ -(y 2 ). This gives, with y 1 ≤ y π n 1 and ( 16),

y 1 ≤ y π n 1 < x θ -(y π n 2 ) ≤ x θ -(y 2 ).
In particular, y 1 / ∈ X θ (y 2 ). Now suppose y 2 / ∈ X θ (y π n 2 ). This means U θ (y 2 ) < U θ (y π n 2 ), so that y π n 2 ∈ X θ (y 2 ). In particular,

y π n 2 ≤ x θ + (y 2 ).
Let us consider the cases y π n 2 < x θ + (y 2 ) and y π n 2 = x θ + (y 2 ) respectively. If y π n 2 < x θ + (y 2 ), then from y 2 ≤ y π n 2 , we obtain y 2 < x θ + (y 2 ). Then necessarily y 2 = x θ -(y 2 ). Then from y 1 < y 2 we have y 1 < x θ -(y 2 ), which implies y 1 / ∈ X θ (y 2 ). If y π n 2 = x θ + (y 2 ) then X θ (y 2 ) = X θ (y π n 2 ), and in particular x θ -(y π n 2 ) = x θ -(y 2 ). This gives, with y 1 ≤ y π n 1 and ( 16),

y 1 ≤ y π n 1 1 < x θ -(y π n 2 ) = x θ -(y 2 ),
and thus y 1 / ∈ X θ (y 2 ).

B Proofs of cases A and B

Case A. Suppose π n r \ { θ} = ∅ and

y π n+1 r ∪{ θ} ≤ y π n r \{ θ} . (17)
In that case, partition Π r+1 is obtained by merging cells π n and π n+1 . From ( 17), (1) and ] ⊂ X θ . Then from (18), we have that for every θ ∈ π n+1 r ∪ { θ}, y π n r \{ θ} ∈ X θ . But this is also true for every θ ∈ π n r \ { θ}.

y π n r < y π n+1

Figure 1 :

 1 Figure 1: A situation in which (A1) does not hold, and in which (IR) and (IC) do not simultaneously hold.

Figure 3 :

 3 Figure 3: Non-monotonic receiver's constrained optimal actions

Figure 4 :

 4 Figure 4: Illustration of Case A.

Figure 5 :

 5 Figure 5: Illustration of Case B

  that Π r+1 satisfies (P2). Let n 1 ∈ {1, . . . , |π r+1 | -1 = |π r | -2} and let

r.

  If U θ increases at y π n+1 r ∪{ θ} , then the first inequality gives the result. Otherwise, the second inequality gives U θ(y π n+1 r ∪{ θ} ) ≥ U θ(y π n+1 r

  that for every θ ∈ π n+1 r ∪ { θ}, we have y π n+1 r ∪{ θ} , y π n+1 r ⊂ X θ . However, X θ is an interval. Therefore for every θ ∈ π n+1 r ∪ { θ}, we have [y π n+1 r ∪{ θ} , y π n+1 r

This quite intuitive result is established formally by[START_REF] Forges | Strategic information transmission with sender's approval[END_REF] in a model that is more general than the current one.

Replacing U θ by U θ -u θ 0 makes reservation utilities type-independent but of course modifies the utility

Note that single-crossing (i.e., (A1)) implies that a partition Π satisfying (IR) and (IC) is necessarily an interval partition, in which every cell consists of consecutive types, i.e., if θ 1 < θ 2 belong to the same cell π of Π, then every θ such that θ 1 < θ < θ 2 also belongs to π.

Note that conversely, every pure PBE without exit (σ, τ ) of Γ induces a partition Π satisfying (IR) and (IC).It is the partition whose cells contain the types that lead to the same action, for every action on the equilibrium path.

We have pointed out above that w.l.o.g., we can assume that the sender's reservation utility u θ 0 equals 0. However, imposing at the same time that U θ has a specific quadratic form is of course restrictive.

For instance,Forges and Renault (2021, Section 4.2 and 4.3) exhibit settings in which there is no pure strategy equilibrium without exit (Section 4.2), or in which there is no mixed strategy equilibrium without exit (Section 4.3).

The resulting set of pools possibly consists of the single set Θ. For instance, it is so in the leading example of[START_REF] Crawford | Strategic information transmission[END_REF], when the informed player's bias is sufficiently large.

This is in sharp contrast with the standard setting (e.g. as in[START_REF] Crawford | Strategic information transmission[END_REF]), for which the receiver's (unconstrained) optimal actions are ordered in the same way as the original order of types, whatever the sender's bias.

Note that property (P2) is meaningful as long as property (P1) is satisfied by partition Π.

Frug, Alexander, "A note on optimal cheap talk equilibria in a discrete state space," Games and Economic Behavior, 2016, 99, 180-185. Kreps, David M and Joel Sobel, "Signalling," Handbook of game theory with economic applications, 1994, 2, 849-867.

Matthews, Steven A, "Veto threats: Rhetoric in a bargaining game," The Quarterly Journal of Economics, 1989, 104 (2), 347-369.

Shimizu, Takashi, "Cheap talk with an exit option: The case of discrete action space," Economics Letters, 2013, 120 (3), 397-400.

, "Cheap talk with an exit option: a model of exit and voice," International Journal of Game Theory, 2017, 46 (4), 1071-1088.

Sobel, Joel, "Giving and receiving advice," Advances in economics and econometrics, 2013,

Proof. The proof is elementary analysis. Let us give the arguments. Function U θ is concave with respect to x, with a maximum value at x = x * (θ), where it is strictly positive. Since U θ (x * (θ)) > 0, there exists x < x * (θ) such that U θ (x) > 0. By concavity, on (-∞, x], the graph of x → U θ (x) is below its tangent line at x. Since x → U θ (x) is increasing on (-∞, x * (θ)), the tangent line has a negative slope. Therefore U θ (x) < 0 if x is sufficiently small. Since U θ is strictly positive at x, continuous and strictly increasing on (-∞, x], the Intermediate Value Theorem ensures the existence of a unique x(θ) < x at which U θ (x) vanishes. Then U θ (x) < 0 on (-∞, x(θ)), and U θ (x) > 0 on (x(θ), x * (θ)]. Similarly, we can define x(θ) that annihilates U θ and that is such that U θ (x) > 0 on (x * (θ), x(θ))), and U θ (x) < 0 on (x(θ), +∞). Then min

Lemma 2. For every θ ∈ Θ, y θ < x * (θ) < max X θ . In particular, U θ is increasing at x = y θ .

Proof. Let θ ∈ Θ. From y * (θ) < x * (θ), function x → V θ (x) decreases on [x * (θ) -ε, max X θ ] ⊂ X θ for some ε > 0. Therefore its maximal argument y θ on X θ is lower than x * (θ).

Lemma 3. For every θ ∈ Θ, y * (θ) ≤ y θ , with equality iff y * (θ) ≥ min X θ . In particular, V θ is not increasing at x = y θ .

Proof. Let θ ∈ Θ. Function x → V θ (x) increases on (-∞, y * (θ)) and decreases on (y * (θ), +∞).

From Lemma 2, V θ is not increasing at x = y θ . Therefore y * (θ) ≤ y θ . Now we show that y * (θ) = y θ iff y * (θ) ≥ min X θ . If y * (θ) < min X θ , then V θ decreases on X θ , so that its maximal argument on X θ , that is y θ , is min X θ , and thus y * (θ) < y θ . This shows that y * (θ) ≥ y θ (and in particular, y * (θ) = y θ ) implies y * (θ) ≥ min X θ . Conversely, if y * (θ) ≥ min X θ , then from x * (θ) < max X θ and y * (θ) < x * (θ) we obtain y * (θ) ∈ X θ . Since y * (θ) maximizes V θ on R, it maximizes V θ on X θ and therefore y * (θ) = y θ . Lemma 4. Let L ⊆ Θ be such that X L = ∅. Let θ ∈ L be such that for every θ ∈ L, y θ ≤ y θ . Then y θ ≤ y L .

Proof. We distinguish the four cases: (i) y θ = min X θ ; (ii) y θ > min X θ and y L = min X L ; (iii) y θ > min X θ and y L = max X L ; (iv) y θ > min X θ and min X L < y L < max X L .

(i) y θ = min X θ . Then y L ≥ min X L = max{min X θ , θ ∈ L} ≥ min X θ = y θ .

(ii) y θ > min X θ and y L = min X L . Let θ be such that min X L = min X θ . Then y θ = min X θ . Thus y L = y θ ≥ y θ (iii) y θ > min X θ and y L = max X L . Let θ be such that max X L = max X θ . Then y L = max X L = max X θ > y θ ≥ y θ , (iv) y θ > min X θ and min X L < y L < max X L . If the derivative of V L is positive at y θ , then the result follows. Otherwise, there is some θ ∈ L such that V θ is decreasing at x = y θ . Then V θ is also decreasing at x = y θ ≥ y θ . This implies y θ = min X θ . Then y L > min X L ≥ min X θ = y θ ≥ y θ . Lemma 5. Let L ⊆ Θ be such that X L = ∅. Let θ ∈ L be such that for every θ ∈ L, y θ ≥ y θ . Then y θ ≥ y L .

Proof. We distinguish the two cases: (i) y L = min X L ; and (ii) y L > min X L .

(i) y L = min X L . Let θ be such that min X L = min X θ . Then y θ ≥ min X θ = min X L = y L .

The result follows from y θ > y θ .

(ii) y L > min X L . Function V L vanishes at y L . For every θ ∈ L, V θ is decreasing on the right side of x = y θ . Thus it decreases at x = y θ . Therefore V L decreases at x = y θ . Hence

and thus so is

Therefore, according to (12),

Now from (10), x θ i + (y π n ) > y π n so that from the single-crossing condition (A1), (13) holds only if (11) holds.

Fourth, we show

Inequality (11) implies that either y θ i j = min X θ i j or y θ i = min X θ i , since otherwise, y θ i j > min X θ i j and y θ i > min X θ i , which implies, y * (θ i j ) = y θ i j and y * (θ i ) = y θ i . But in that case, from the single-crossing assumption on V θ (x), y * (θ i j ) < y * (θ i ) which gives y θ i j < y θ i , i.e. j < , a contradiction. Hence if y θ i > min X θ i then necessarily ( 14) holds. It remains to show that y θ i = min X θ i also implies (14). If y θ i = min X θ i then y * (θ i ) ≤ min X θ i . From ( 11) and the single-crossing condition, y * (θ i j ) < y * (θ i ) and thus y * (θ i j ) < min X θ i = y θ i . Then from y θ i ≤ y θ i j , we obtain y * (θ i j ) < y θ i j , which precisely implies (14).

Finally, consider j = |π n |. Since for every θ ∈ π n , y θ ≤ y θ i j , according to Lemma 5, we have, on the one hand,

On the other hand, from y π n ∈ θ∈π n X θ , we also have y π n ≥ min X θ i j . That is, according to (14), y π n ≥ y θ i j . Hence we obtain

and, if moreover < |π n |,

•

(2)

Proof of (1) and (1a). According to Lemma 8, either

Thus, we obtain (1a).

Proof of (2). According to Lemma 6, inequalities (2) hold whenever

holds. To show (2a), we use (P2).

Suppose that θ i is such that y

Then (2a) derives from Lemma 4.

Then from y π n+1 > y π n , we obtain y π n+1 > y θ i , which implies (2a).

Proof of (3). If = |π n |, then for every θ ∈ π n \ {θ i }, y θ ≤ y θ i . Then From Lemma 5, y π n \{θ i } ≤ y θ i , and then from Lemma 6,

Now suppose that θ i is such that y θ i > min X θ i . Then from (P2) we have: for every θ ∈ π n+1 , y θ i ≤ y θ . Then from Lemma 4, we have y θ i ≤ y π n+1 , and from Lemma 6, we deduce y θ i ≤ y π n+1 ∪{θ i } . Then from (15), we obtain y π n ≤ y θ i ≤ y π n+1 ∪{θ i } , so that (3) holds. If instead

which implies (3) using (15).

Lemma 10. Let Π = π 1 ∪ . . . ∪ π |Π| be a partition of Θ that satisfy (IR), (P1) and (L-IC).

Let

Thus y π n r ∪π n+1 r is well defined. Let us show that

But from Lemma 6, we also have

This gives (19).

We set

Partition Π r+1 satisfies (IR) by construction. Since Π r satisfies (P1), from ( 19), Π r+1 also satisfies (P1).

Let us show that it also satisfies (P2). Let n 1 ∈ {1, . . . , |Π r+1 | -1 = |Π r | -2} and let θ ∈ π n 1 r . Suppose that there exists n 2 ≥ n 1 + 1 and θ ∈ π n 2 r+1 such that y θ < y θ . Let us show that y θ = min X θ .

we then get y θ = min X θ .

and

Finally, we show that Π r+1 satisfies (L-IC). Let n 2 ∈ {2, . . . , |Π r+1 |}, n 1 ∈ {1, . . . , n 2 -1} and θ ∈ π n 2 r+1 . We want to show that y π n 1 r+1 / ∈ X θ (y π n 2 r+1 ).

• If n 1 = n and n 2 = n, then either π

r , and π

r ), and

), and

), so that y π n 1 r+1 / ∈ X θ (y π n 2 r+1 ) in the three cases.

• If n 1 = n, then from ( 19), y π n 1 r+1 = y π n r , and from

) and the result follows.

• If n 2 = n, then from ( 19), y π n 2 r+1 = y π n r , and from

-

Case B.1. Suppose that n = 1, or

In that case, the move of θ from π n r to π n+1 r keeps the order of the corresponding cell-contingent actions y π .

We set

.

(5)

Then Π r+1 satisfies (IR) by construction. Concerning (P1), we clearly have y π 1 r+1 < ... < y π n-1 r+1 (if n > 1) and y π n+2 r+1 < . . . < y π |Π r+1 | r+1

. From (20), we also have y π n r+1 = y π n \{ θ} < y π n+1 ∪{ θ} = y π n+1 r+1 . Moreover, from (2) we have

satisfies (P1) whenever y π n-1 r+1 < y π n r+1 , i.e. whenever y π n-1 r < y π n r \{ θ} , i.e. (21).

Now we show that it satisfies (P2). Let n 1 ∈ {1, . . . , |Π r+1 | -1 = |Π r | -1} and let θ ∈ π n 1 r+1 . Suppose that there exists n 2 ≥ n 1 +1 and θ ∈ π n 2 r+1 such that y θ < y θ . We show that necessarily y θ = min X θ .

• If θ = θ and θ = θ, then θ ∈ π n 1 r and θ ∈ π n 2 r with n 2 ≥ n 1 + 1, so that we have y θ = min X θ because Π r satisfies (P2).

Since n 2 ≥ n+2 > n, and since Π r satisfies (P2), we obtain y θ = min X θ .

• If θ = θ, then θ ∈ π n r . Moreover, n 2 = n + 1, and from n 2 ≥ n 1 + 1, we obtain n + 1 ≥ n 1 + 1, i.e. n 1 ≤ n. From θ = θ , we derive θ ∈ π n 1 r .

-If n 1 < n, since Π r satisfies (P2), we obtain y θ = min X θ .

-If n 1 = n, from y θ = y θ < y θ we have θ = θ i j for some j > . In that case, Lemma 8 gives y θ = min X θ .

Finally, we show that Π r+1 satisfies (L-IC). Let n 2 ∈ {2, . . . , |Π r+1 |}, n 1 ∈ {1, . . . , n 2 -1} and θ ∈ π n 2 r+1 . We want to show that y π n 1 r+1 / ∈ X θ (y π n 2 r+1 ).

We first suppose θ = θ. In that case, π n 2 r+1 = π n+1 r ∪ { θ} and we have to show ). By construction, we also have U θ(y π n+1 r ) > U θ(y π n r ). Since y π n r < y π n+1 r , it must be that U θ increases at y π n r . From (1), i.e. y π n r \{ θ} ≤ y π n r , we then get U θ(y π n r ) ≥ U θ(y π n r \{ θ} ), and therefore U θ(y π n+1 r ∪{ θ} ) > U θ(y π n r \{ θ} ).

In that case, since Π r satisfies (P1), and from ( 21) and ( 20) respectively,

The same argument as above gives the result.

Next we suppose θ = θ. Set y 1 = y π n 1 r+1 and y 2 = y π n 2 r+1 . Then according to Lemma 10, the result derives from y 1 < y 2 (guaranteed because Π r+1 satisfies (P1)), and the existence of

Setting

r . This gives the result.

r \ { θ}, and according to (1),

Setting

r . This gives the result.

r ∪ { θ}, and according to (2), y 2 ≤ y π n 2 r = y π n 2 r , with

r \ { θ}, and according to (1), y 1 ≤ y π n 1 r = y π n 1 r with n 1 = n 1 . This gives the result.

-

r with n 1 = n 1 , and then y 1 = y π n 1 r , which also gives the result.

r with n 2 = n 2 , and then y 2 = y π n 2 r .

-

which gives the result.

-

with n 1 = n 1 , and the result follows.

Case B.2. Suppose

In that case, when type θ is removed from π n r , the nth action y π n r in partition Π r moves to a rank which is possibly strictly lower than n. In that case, we merge π n r \ { θ} with π n-1 r .

We first show that necessarily = |π n r |. Indeed, if < |π n r |, then from Lemma 8,

cannot hold.

Next we show that X π n-1 r ∩ X π n r \{ θ} = ∅. From ( 22) and y π n r > y π n-1 r we get

, we obtain

This allows us to set

Next we show y π = y π n-1 r . Let θ ∈ π n r \ { θ} be such that for every θ ∈ π n r \ { θ}, y θ ≤ y θ . Then from Lemma 4 and from ( 22),

be such that for every θ ∈ π n-1 r , y θ ≥ y θ . Then from Lemma 5,

Therefore y θ ≥ y θ , where θ ∈ π n r and θ ∈ π n-1 r . Since partition Π r satisfies (P2), we get

r and y π n-1 r ≥ y θ , and thus

From Lemma 6 and y π n-1 r ≥ y π n r \{ θ} , we have y π n-1 r ≥ y π ≥ y π n r \{ θ} , and in particular y π ≤ min X θ . But θ ∈ π, and necessarily y π ≥ min X θ . Thus y π = min X θ = y π n-1 r .

We set: , which also holds by assumption (P1) on Π r , and if moreover and let θ ∈ π n 1 r+1 . Suppose that there exists n 2 ≥ n 1 + 1 and θ ∈ π n 2 r+1 such that y θ < y θ . We show that necessarily y θ = min X θ .

• If n 2 < n -1, then n 1 ≤ n 2 -1 < n -2, and then θ ∈ π n 1 r and θ ∈ π n 2 r . Since Π r satisfies (P2), y θ < y θ implies y θ = min X θ .

• If n 2 = n -1, then n 1 ≤ n 2 -1 = n -2, and then θ ∈ π n 1 r and either θ ∈ π n-1 r , with n -1 > n 1 , or θ ∈ π n r , with n ≥ n 1 + 2, and the same argument holds.

r with n 1 < n + 1 and n 1 < n respectively, and the same argument as above holds.

-

then we are done too. It remains the case in which θ ∈ π n r \ { θ} and θ = θ ∈ π n r . In that case, y θ < y θ guarantees that θ = θ i j for some j > , and Lemma 8 gives the result.

r . In each of the following cases, θ is in a cell of Π r whose rank is lower than n 2 + 1. Since Π r satisfies (P2), y θ < y θ implies y θ = min X θ .

-

Finally, we show that Π r+1 satisfies (L-IC). Let n 2 ∈ {2, . . . , |Π r+1 |}, n 1 ∈ {1, . . . , n 2 -1} and θ ∈ π n 2 r+1 . We want to show that y π n 1 r+1 / ∈ X θ (y π n 2 r+1 ).

We first consider the case θ = θ. In that case, π

∪ { θ} and we have to show

and from ( 23) and ( 2), we obtain the two inequalities ). Now by construction, we have U θ(y π n+1 r ) > U θ(y π n r ), and since Π r satisfies (L-IC), we also have U θ(y π n r ) > U θ(y π n r ). This gives the result.

Next we suppose θ = θ. As above, we set y 1 = y π n 1 r+1 and y 2 = y π n 2 r+1 . Then according to Lemma 10, the result derives from y 1 < y 2 (guaranteed because Π r+1 satisfies (P1)), and the existence of n 2 ∈ {2, . . . , |Π r |} and n 1 ∈ {1, . . . , n 2 -1} such that θ ∈ π n 2 r , y 1 ≤ y π n 1 r and y 2 ≤ y π n 2 r .

• If n 2 ≤ n -2, then θ ∈ π -If n 1 = n -1, then y 1 = y π = y π n-1 r = y π n 1 r with n 1 = n -1 gives the result.


Iana Sudreau

Mathilde Auxois

Marion Servel

Éric Lécolier

Sébastien Manneville

Thibaut Divoux

Residual stresses and shear-induced overaging in boehmite gels

Colloidal gels respond like soft solids at rest, whereas they flow like liquids under external shear. Starting from a fluidized state under an applied shear rate γp, abrupt flow cessation triggers a liquidto-solid transition during which the stress relaxes towards a so-called residual stress σres that tallies a macroscopic signature of previous shear history. Here, we report on the liquid-to-solid transition in gels of boehmite, an aluminum oxide, that shows a remarkable non-monotonic stress relaxation towards a residual stress σres( γp) characterized by a dual behavior relative to a critical value γc of the shear rate γp. Following shear at γp > γc, the gel obtained upon flow cessation is insensitive to shear history, and the residual stress is negligible. However, for γp < γc, the gel encodes some memory of the shear history, and σres increases for decreasing shear rate, directly contributing to reinforcing the gel viscoelastic properties. Moreover, we show that both σres and the gel viscoelastic properties increase logarithmically with the strain accumulated during the shear period preceding flow cessation. Such a shear-induced "overaging" phenomenon bears great potential for tuning the rheological properties of colloidal gels.

Introduction.-Soft Glassy Materials (SGMs) are viscoelastic soft solids ubiquitous in science and engineering [1][2][3]. These materials display a shear-induced solid-toliquid transition under large enough stresses, i.e., they flow like liquids above their yield stress σ y , whereas SGMs recover their solidlike properties upon flow cessation [4]. Such a liquid-to-amorphous-solid transition results in the partial relaxation of internal stresses, which usually follows a monotonic decay towards a so-called residual stress σ res < σ y [5,6]. Residual stresses are critical to the mechanical properties of a broad range of materials across hard and soft condensed matter, for they control both the linear and non-linear material responses [7,8]. In the case of SGMs, the residual stress encodes the sample memory of the plastic deformation accumulated prior to and during flow cessation [9,10]. As such, σ res depends on the shear rate γp applied prior to flow cessation, and σ res ( γp ) is usually reported to be a decreasing function, typically a weak power-law [11][12][13][14], or a logarithm of γp [15] such that residual stresses become negligible following strong enough shear.

From a microscopic point of view, the origin of the residual stress depends on the details of the SGMs microstructure. On the one hand, residual stresses in dense systems have been linked to the elastic contact forces in soft jammed particles [15][16][17], and to the so-called "supra-caging" effects in hard-sphere colloidal glasses [9]. On the other hand, residual stresses in dilute systems such as colloidal gels are associated with structural anisotropy frozen into the gel microstructure. Such anisotropy only emerges following low enough shear rates, i.e., low enough Peclet numbers, such that upon flow cessation, particles only move over short distances compared to the range of the inter-particle attraction, hence freezing their latest configuration [14,18]. Yet, the lack of in-formation on residual stresses in colloidal gels compared to denser systems calls for more experiments. Moreover, while residual stresses and, more generally, memory effects in gels are often reported to be driven by the shear rate [19][20][21], little is known about the impact of the total strain accumulated during the shear period prior to flow cessation, and disentangling the contribution of the shear rate from that of the strain to residual stresses and to final viscoelastic properties remains challenging.

In this Letter, we study the residual stress in acidinduced gels of boehmite, an aluminum oxide. We show that sudden flow cessation from a given shear rate γp yields an anomalous non-monotonic stress relaxation towards a residual stress that strongly depends on the shear rate γp applied before quenching the flow. We identify a critical shear rate γc , above which boehmite gels show negligible residual stresses, whereas, for γp < γc , they display significant residual stresses that increase when decreasing γp . Such a residual stress directly impacts the gel viscoelastic properties by reinforcing the elastic and viscous moduli, G ′ and G ′′ , respectively. In particular, for γp < γc , both G ′ and G ′′ are directly proportional to the residual stress. Moreover, we show that aside from the shear rate γp , the total strain γ p accumulated before flow cessation plays a crucial role: the residual stress σ res and the viscoelastic moduli grow logarithmically with γ p , with a prefactor that depends on γp . Our results are prototypical of shear-induced "overaging", and show that boehmite gels can be tuned by both shear-and straincontrolled memory effects in the limit of low enough shear rates. 24]. After 35 min of mixing, the suspension contains unbreakable aggregates of diameter 2R ag ≃ 100 nm, which further assemble into large clusters of typical diameter 2R cl ≃ 300-1200 nm, leading to a sol-gel transition after a couple of hours [25]. The sample is left at rest for at least seven days prior to any rheological test, until the pH stabilizes to pH = 3.5 [24,26]. About 10 mL of the gel sample is then transferred into a smooth cylindrical Couette cell of gap 1 mm connected to a stress-controlled rheometer (AR-G2, TA Instruments).

Materials

Results.-The gel is first rejuvenated by a constant shear rate γ = 1000 s -1 during 600 s into a liquid suspension that displays no residual stress. Then the flow is quenched to the shear rate of interest γp for a duration t p = 600 s, which is sufficient for the shear stress to reach a steady state whatever the value of γp in the range of 1 s -1 to 1000 s -1 . Finally, the flow cessation, which sets the origin of time, is induced by imposing γ = 0 s -1 , while we monitor the stress relaxation σ(t), as reported in Fig. 1 for shear rates γp ranging from γp = 3 s -1 to 500 s -1 . For all γp in the experimental range, the stress displays a remarkable non-monotonic evolution, which becomes more pronounced for decreasing values of γp . More precisely, the stress response shows a dual behavior relative to a critical shear rate γc ≃ 40 s -1 . For γp > γc , the stress response first shows a linear decay until t = t min ≃ 10 s, followed by a stress overshoot, before decreasing again towards a negligible stress value, below the rheometer resolution. In stark contrast, upon flow cessation from a shear rate γp < γc , the stress response decreases down to a minimum value reached at time t min ≲ 10 s, and then increases towards a final and steady residual stress σ res > 0 [see Fig. S1 in Supplemental Material [27] for the temporal evolution of the shear rate γ effectively applied, and Fig. S2 for longer stress relaxations and additional data recorded in a cone-andplate geometry]. In that case, both t min and σ res depend on the shear rate γp applied before flow cessation.

Figure 2(a) presents the residual stress σ res measured 2000 s after flow cessation as a function of γp over three decades [see also Fig. S3(a) in Supplemental Material [27] for t min ( γp )] and quantitatively confirms the dual behavior of boehmite gels: the residual stress is negligible for γp > γc , whereas it increases logarithmically with decreasing γp , up to σ res ≃ 9 Pa for γp = 1 s -1 . These results show that boehmite gels display little if no memory of shear history for γp > γc , whereas they build up some frustration while quenching the gel from γp < γc , hence freezing residual stresses. We further characterize the properties of these soft solids by measuring their steady-state linear viscoelastic properties at t = 3000 s after flow cessation and at a single frequency of 1 Hz. Such a choice of frequency does not impact the gener- ality of our results since the viscoelastic moduli of the present boehmite gels depend on the frequency as very weak power laws [25]. Consistently with σ res ( γp ), the elastic and viscous moduli, G ′ and G ′′ , both show a similar dual behavior [see also Fig. S3(b) in Supplemental Material [27] for tan δ = G ′′ /G ′ ]. Indeed, for γp > γc , the viscoelastic moduli are insensitive to shear history, with G ′ 0 = 444 ± 37 Pa and G ′′ 0 = 10 ± 2 Pa. In contrast, for γp < γc , G ′ and G ′′ increase logarithmically for decreasing γp , which suggests that γp affects both the skeletal backbone of the gel network, and its fractal microstructure, most likely through more anisotropy for smaller γp [28,29]. In summary, the shear rate γp applied before flow cessation appears to control the terminal viscoelastic properties of boehmite gels. However, one should keep in mind that so far, the shearing duration was kept constant for all values of γp so that the samples experienced different levels of strain prior to flow cessation.

= 1 s -1 (□), 2 s -1 (■), 3 s -1 (■), 5 s -1 (■), 7 s -1 (■), 10 s -1 (■), 15 s -1 (■),
In order to disentangle the impact of the shear rate γp from that of the accumulated strain γ prior to flow cessation, we vary the shearing duration t p for given values of γp , so as to generate gels with different levels of accumulated strain γ p = γp t p [see Fig. S4 in Supplemental Material [27] for corresponding stress relaxation data, which all display a non-monotonic evolution]. Note that for t p < 600 s, the stress does not reach a steady state for all levels of strain during the shearing step at γp . Figure 3 reveals the impact of both γ p and γp on σ res , G ′ and G ′′ . For a fixed shear rate γp < γc , all three quantities increase logarithmically for increasing strain γ p [see colored lines in Fig. 3]. This result holds over three decades of strain for various shear rates. Moreover, the prefac-tors of this logarithmic dependence increase weakly for decreasing γp [Fig. 3(d)-(f)], i.e., the lower the applied shear rate, the larger σ res , G ′ and G ′′ for a fixed accumulated strain γ p . In conclusion, the impact of both γp and γ p on the various observables of interest, X = σ res , G ′ , or G ′′ , is captured by: σ res , G ′ , and

G ′′ = α X ( γp ) log γ p + β X ( γp ), (1)
where both α X ( γp ) and β X ( γp ) are decreasing functions of γp . While the slope α X ( γp ) depends only weakly on γp , the intercept β X ( γp ) accounts for the strong sensitivity on γp reported in Fig. 2. Note that for γp > γc , neither the strain nor the shear rate have any impact on the residual stress or on the viscoelastic properties. Finally, the robustness of Eq. ( 1) prompts us to eliminate the strain and to directly link the gel viscoelastic properties to the residual stress. As pictured in Fig. 4, this allows us to collapse all data obtained for various shear histories, i.e., various shear rates γp applied over different durations t p , hence yielding various strains γ p , onto a master curve. Moreover, as expected from Eq. ( 1), the elastic and viscous moduli both increase proportionally to the residual stress, starting from the two reference values G ′ 0 and G ′′ 0 measured for γp > γc [see also Fig. S5 in the Supplemental Material [27] for the corresponding master curve for tan δ]. Note that such a collapse does not depend on frequency (here 1 Hz), for viscoelastic properties of boehmite gels show a very weak power-law frequency dependence [25]. Our results thus show that the viscoelastic properties of boehmite gels are reinforced by the residual stress, which grows due to an extended period of shear at low shear rates. Discussion and conclusion.-Let us now summarize and discuss the most prominent results of this Letter. First, we have identified a critical shear rate γc above which boehmite gels are insensitive to shear history. Such a critical shear rate is similar to the one reported in Ref. [25], although determined with a different rheological protocol. In particular, as a key difference with Ref. [25], here, we fully rejuvenate the gel under γ = 1000 s -1 before imposing the shear rate of interest γp . Still, in both cases, a clear transition is observed in the evolution of the elastic modulus with γp around the same values of γc ( γc = 30 ± 10 s -1 in Ref. [25]). Our results thus confirm the robustness of γc , which was interpreted as a critical Mason number comparing the strength of shear to the bond force between two particles at contact [25,[30][31][32][START_REF] Varga | Proc. Natl. Acad. Sci[END_REF]. The hypothetical scenario proposed in Ref. [25] to account for observations with γp < γc is as follows: low enough shear rates do not fully rejuvenate the gel, which builds up some shear-induced structural anisotropy. Here, we show that for γp < γc , boehmite gels also develop internal stresses, which bear some memory of the flow history.

Moreover, as reported in a broad variety of SGMs [START_REF] Derec | [END_REF][35][36], boehmite gels show spontaneous aging at rest characterized by a logarithmic time dependence of the viscoelastic moduli [25]. Such physical aging is a consequence of the system exploring its "energy landscape" by going from one metastable state to another under the effect of thermally activated dynamics [37,38]. In this framework, moderate shear was shown to act as a delicate driving force that relocates the system into deeper energy minima, thus inducing "overaging" in SGMs [39][40][41]. Here, the logarithmic increase of G ′ and G ′′ with the strain accumulated under low enough shear rate is strikingly reminiscent of aging and constitutes a clear instance of "overaging" in a colloidal gel. Finally, we note that for vanishingly low shear rates, our results may be also linked to the so-called "directed aging" recently achieved under a static load or strain in amorphous solids [42,43].

Yet, the microscopic process underpinning shearinduced "overaging" remains to be identified in the present boehmite gels. It most likely originates either from local compaction, i.e., the slow increase in local volume fraction turning the local mechanical response from that of gel to that of a glass [44,45], or from the strain-induced alignment of the microstructure [46][47][48]. Interestingly such a strain-controlled reinforcement process seems to reach a limit for large enough strains. Indeed, when imposing γp = 2 s -1 prior to flow cessation, the residual stress reaches a maximum at γ * p ≃ 2.10 3 before decreasing [see yellow symbols in Fig. 3(a)]. The same effect is visible beyond γ * p ≃ 2.10 4 for γp = 7 s -1 [see red symbols in Fig. 3(a)], suggesting that strain becomes detrimental beyond a certain point, breaking the strong microstructure that builds up at γ p < γ * p . Such a scenario remains to be investigated with numerical tools, including molecular dynamics (MD) simulations, which would allow one to assess whether plasticity is responsible for the drop in residual stress beyond γ * p , and whether it results from rearrangements within gel strands [49], or occurs at larger length scales, between clusters [32,50].

Another signature of shear-induced "overaging" on the present boehmite gels for γp < γc is their nonmonotonic stress relaxation following flow cessation. Such an anomalous relaxation, common to disordered solids [51,52], has been recently reported in viscoelastic liquids and attributed to two key ingredients, namely the alignment of the microstructure under flow, and the formation of bonds upon flow cessation due to interparticle attractive forces that locally deform the bulk material, hence inducing internal stresses [53]. This picture based on shear-induced alignment is consistent with the emergence of some anisotropy in the microstructure of boehmite gels as hypothesized in Ref. [25], and with the growth of residual stresses only observed for γp < γc in the present study. In that framework, boehmite gels display a striking analogy with supramolecular systems, despite their terminal relaxation, for the non-monotonic response in these systems falls within the physical network part of their viscoelastic spectrum [53]. Our results suggest that interparticle attractive forces are dominant for γp < γc , where residual stresses are observed. In contrast, hard-core repulsion takes over for γp > γc , accounting for the lack of residual stresses and the strong dampening of the non-monotonic stress relaxation. To our knowledge, non-monotonic relaxations in colloidal gels have not yet been captured by MD simulations [44,54].

Our data set thus constitutes a unique opportunity for future numerical work to identify the minimal ingredients necessary to reproduce such an anomalous relaxation. We anticipate that the anisotropy of boehmite particles and their noncentral interaction play a key role in the stress dynamics.

Finally, in cases where the stress relaxes to vanishingly small values, the late stages of the dynamics are consistently characterized by erratic fluctuations of σ(t) [see Fig. 1 and Fig. S4 in the Supplemental Material [27]]. Such fluctuations could be the signature of large-scale, heterogeneous dynamics occurring in the bulk or in the vicinity of the walls. Future experiments offering local insights into the sample microdynamics upon flow cessation should be undertaken to answer this open question.

Conclusion.-Our study unravels a deep connection between residual stresses and viscoelastic properties in colloidal gels: the elastic and viscous modulus of boehmite gels are reinforced proportionally to the residual stress trapped upon flow cessation. Interestingly, such a linear relationship was previously reported in gels of fractal-like particles experiencing van der Waals attractive interactions [11]. However, in contrast with the low shear rates explored here, the reinforcement in these fractal gels was observed in a shear-thickened state obtained only after a period of intense shearing, and following a monotonic stress relaxation. Such a discrepancy with our observations suggests that the linear dependence of the viscoelastic moduli with σ res is somewhat universal in colloidal gels, regardless of microscopic details. As such, our experimental results constitute a benchmark for future modeling of residual stresses at the particle scale [START_REF] Bouzid | Mechanics of soft gels: Linear and nonlinear response[END_REF], not only to rationalize their relation to elasticity and reinforcement, but also to understand their dependence with the accumulated strain, here reported to be logarithmic, as well as their temporal evolution after flow cessation.

Residual stresses and shear-induced overaging in boehmite gels

Supplemental Material

FULL TEMPORAL EVOLUTION OF RHEOLOGICAL OBSERVABLES Figure S1(a) displays the temporal evolution of the shear rate γ effectively applied (as measured by the rheometer) during the flow cessation step and normalized by γp . In all cases, it takes about 0.5 s for the shear rate to reach γ/ γp = 10 -3 . Therefore, we may assume the shear rate to be effectively zero for t > 1 s, which is the reason why we report the stress response starting at t = 1 s in Fig. 1 in the main text. For the sake of completeness, we also report in Fig. S1(b) the stress responses for t > 0.5 ms. for γp = 3 s -1 (■), 5 s -1 (■), 10 s -1 (■), 15 s -1 (■), 40 s -1 (■), 50 s -1 (■), 200 s -1 (■), and 500 s -1 (■).

ROBUSTNESS OF NON-MONOTONIC STRESS RELAXATION

Figure S2 reports additional stress responses following flow cessation as described in the main text. Using the same Couette geometry and the same rheometer as in the main text, we checked that the stress has reached a stationary value by monitoring σ(t) over 10 4 s instead of 2000 s (see square symbols). Moreover, measurements performed over 2000 s using a sandblasted cone-and-plate geometry connected to a different stress-controlled rheometer (MCR 302, Anton Paar) display a very similar non-monotonic evolution and quantitatively point to the same residual stresses (see dotted lines for γ = 2 and 5 s -1 ). These measurements emphasize the robustness of the anomalous stress relaxation towards significant residual stresses for γp < γc . Note that the stress relaxations do not strictly coincide in the two different geometries, which is not surprising as the details of the global stress dynamics following flow cessation must depend on the local deformation of the microstructure, which is itself likely to depend on the (geometry-dependent) local shear rate and shear stress fields. The colors code for γp = 2 s -1 (■), 5 s -1 (■), and 10 s -1 (■). The duration of the shear step at γp is tp = 600 s in all cases.

COMPLEMENTARY ANALYSIS OF THE STRESS RESPONSES

In Fig. S3, we show complementary observables extracted from Fig. 1 in the main text, as a function of the shear rate γp , namely the time t min at which the stress reaches a minimum [Fig. S3(a)], and the loss factor tan δ [Fig. S3(b)]. Both observables display a dual behavior depending on the position of γp relative to γc . For γp < γc , the data are well fitted by a logarithmic dependence in γp , while they become independent of γp above γc . Such a behavior is fully consistent with the trends observed in Fig. 2 in the main text for the residual stress and the viscoelastic moduli. Note that the stress minimum becomes hardly measurable after strong shearing, as the stress response flattens with γp , hence the large error bars on t min for the highest values of γp . Moreover, the fact that tan δ increases with decreasing γp indicates that the dissipative component increases more steeply than the elastic component as γp decreases.

INFLUENCE OF THE SHEARING DURATION ON THE STRESS RELAXATION

Figure S4 presents the stress responses measured after flow cessation following shearing at γp over different durations t p . It reveals that the accumulated strain γ p = γp t p has a crucial influence on the stress relaxation, and thus on the subsequent residual stress, as further analyzed in Fig. 3 in the main text. From the master curves for G ′ and G ′′ vs. σ res and the linear fits shown in Fig. 4 in the main text, it is straightforward that the loss tangent tan δ = G ′′ /G ′ should also follow a master curve as a function of the residual stress, independent of the shear rate γp and duration t p . This is checked in Fig. S5, where the ratio of the fits for G ′′ and G ′ (see black line) is shown to account well for the evolution of the loss tangent with σ res .

1 FIG. 1 .

 11 FIG.1. Stress response σ(t) of a boehmite gel following flow cessation induced by imposing a decreasing step of shear rate from γ = γp to γ = 0 at t = 0 s. The target shear rate γ = 0 is reached in less than 1 s. The colors code for γp = 3 s -1 (■), 5 s -1 (■), 10 s -1 (■), 15 s -1 (■), 40 s -1 (■), 50 s -1 (■), 200 s -1 (■), and 500 s -1 (■).

FIG. 2 .

 2 FIG. 2. (a) Residual stress σres, (b) elastic modulus G ′ , and (c) viscous modulus G ′′ vs. the shear rate γp applied prior to flow cessation. σres is measured at t = 2000 s after flow cessation, while G ′ and G ′′ are taken at t = 3000 s. The vertical dashed line highlights the critical shear rate γc = 40 s -1 below which the sample grows a residual stress, while G ′ and G ′′ are reinforced. Blue lines highlight σres = 0 Pa, G ′ 0 = 444 ± 35 Pa, and G ′′ 0 = 10 ± 2 Pa, and red lines are the best logarithmic fits of the data for γp < γc: X = aX log γp + bX with aσ = -4.5 Pa, a G ′ = -570 Pa, a G ′′ = -25 Pa, bσ = 6.8 Pa, b G ′ = 1315 Pa, and b G ′′ = 49 Pa. Error bars correspond to the standard deviations computed over 3 to 6 independent measurements.

FIG. 3 .

 3 FIG. 3. (a)Residual stress σres, (b) elastic modulus G ′ , and (c) viscous modulus G ′′ vs. the strain γp accumulated prior to flow cessation. Colors code for γp = 1 s -1 (□), 2 s -1 (■), 3 s -1 (■), 5 s -1 (■), 7 s -1 (■), 10 s -1 (■), 15 s -1 (■), and 50 s -1 (■) imposed over various durations tp in independent experiments. Colored lines correspond to the best logarithmic fits [see Eq. (1) in the main text]. Dashed and dotted gray lines highlight data measured for various shear rates γp applied over a fixed duration tp = 600 s (circles, same data as in Fig.2) and 2000 s (triangles), respectively. (d)-(f) slope αX of the logarithmic increase with γp of X = σres, G ′ , and G ′′ , respectively, vs. γp. The gray dashed lines are guides to the eye.

FIG. 4 .

 4 FIG. 4. Elastic modulus G ′ vs. residual stress σres for a boehmite gel submitted to various shear histories, i.e., various shear rates γp over different durations tp. Data replotted from Fig. 3(a)-(c). Inset: Viscous modulus G ′′ vs. σres. Black lines are the best linear fits for σres < 8.0 Pa, respectively G ′ = G ′ 0 (1 + λ ′ σres) and G ′′ = G ′′ 0 (1 + λ ′′ σres), with G ′ 0 = 444 ± 37 Pa, λ ′ = 0.35 ± 0.04, G ′′ 0 = 10 ± 2 Pa, and λ ′′ = 0.69 ± 0.06.

  FIG. S1. (a)Temporal evolution of the shear rate γ effectively applied and normalized by the shear rate γp imposed prior to the flow cessation step. The black horizontal dashed line highlights γ/ γp = 10 -3 . (b) Stress response σ(t) of a boehmite gel following flow cessation induced by imposing a decreasing step of shear rate from γ = γp to γ = 0 at t = 0 s. The colors code for γp = 3 s -1 (■), 5 s -1 (■), 10 s -1 (■), 15 s -1 (■), 40 s -1 (■), 50 s -1 (■), 200 s -1 (■), and 500 s -1 (■).

1 FIG

 1 FIG.S2. Stress response σ(t) of a boehmite gel following flow cessation induced by imposing a decreasing step of shear rate from γ = γp to γ = 0 at t = 0 s. Square symbols correspond to experiments performed in the same Couette geometry as in the main text but over a longer duration of 10 4 s, while dotted lines refer to experiments performed over 2000 s in a sandblasted coneand-plate geometry (angle of 2 • and diameter of 40 mm) connected to an MCR 302 (Anton Paar) stress-controlled rheometer. The colors code for γp = 2 s -1 (■), 5 s -1 (■), and 10 s -1 (■). The duration of the shear step at γp is tp = 600 s in all cases.

  FIG. S3. (a) Time tmin at which the stress reaches a minimum and (b) loss factor tan δ vs. the shear rate γp applied prior to flow cessation. The vertical dashed line highlights the critical shear rate γc = 40 s -1 below which the sample grows a residual stress. Blue lines highlight tmin = 11 ± 1 s and tan δ = 0.0228 ± 0.0005, and red lines are the best logarithmic fits of the data for γp < γc: X = aX log γp + bX with at = 4.45 s, a δ = -0.01, bt = 2.96 s, and b δ = 0.04.

  FIG. S4. Stress response σ(t) of a boehmite gel following a flow cessation induced by imposing a shear step from γ = γp to γ = 0 at t = 0 s. The colors code for (a) γp = 3 s -1 , (b) γp = 5 s -1 , (c) γp = 10 s -1 , and (d) γp = 15 s -1 , and for different durations tp as indicated in the graphs.

  FIG.S5. Loss factor tan δ vs. residual stress σres for a boehmite gel submitted to various shear histories. Colors code for γp = 2 s -1 (■), 3 s -1 (■), 5 s -1 (■), 7 s -1 (■), 10 s -1 (■), 15 s -1 (■), and 50 s -1 (■) imposed over various durations tp in independent experiments. The black line corresponds to the ratio of the linear fits of G ′′ and G ′ vs. σres shown in Fig.4in the main text, namelyG ′′ 0 (1 + λ ′′ σres)/[G ′ 0 (1 + λ ′ σres)],with the fit parameters given in the main text.

Acknowledgments.-We thank E. Del Gado, T. Gibaud, G. Petekidis, and V. Vasisht for fruitful discussions.