Françoise Forges
email: francoise.forges@gmail.com

Jérôme Renault

Anna Bogomolnaia

Ulrich Horst

Vincent Iehlé

Frédéric Koessler

Daniel Krähmer

Ehud Lehrer

Ronny Razin

Antoine Salomon

Roland Strausz

Bertrand Villeneuve

Strategic information transmission with sender's approval

We consider a sender-receiver game with an outside option for the sender. After the cheap talk phase, the receiver makes a proposal to the sender, which the latter can reject. We study situations in which the sender's approval is crucial to the receiver.

We show that a partitional, (perfect Bayesian Nash) equilibrium exists if the sender has only two types or if the receiver's preferences over decisions do not depend on the type of the sender as long as the latter participates. The result does not extend: we construct a counter-example (with three types for the sender and type-dependent affine utility functions) in which there is no mixed equilibrium. In the three type case, we provide a full characterization of (possibly mediated) equilibria.

Introduction

We consider a general model of sender-receiver games. The specific feature of our games is that the sender has an outside option. After the cheap talk phase, the receiver proposes a decision to the sender; if the sender approves it, the decision is made; otherwise, the sender chooses his outside option, which can be interpreted as "exit". Under complete information, the game reduces to an ultimatum game, in which one player makes a "take it or leave it" offer to the other. In our framework, this other player has private information and can send a costless message to the receiver before getting an offer.

We are interested in situations in which the sender's approval is crucial to the receiver. We thus assume that the receiver's utility in case of exit is very low, as compared to what he can expect if the sender accepts his proposal. It is not difficult to find examples in which a decision-maker consults with an informed party before making a proposal that can be ultimately rejected and in which rejection has unvaluable, damaging consequences for the decisionmaker. For instance, firms try to figure out workers' requirements in order to avoid strikes and boycotts. Governments discuss with kidnappers, hoping that hostages will not be killed. As a third example, analyzed in [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF], the U.S. Congress may worry about the President's veto.

We focus on equilibria in which the sender does not make uncredible threats at the approval stage, namely, accepts a proposal if and only if it gives him at least the utility of his outside option. 1 We ask whether our sender-receiver game has an equilibrium in which exit does not occur.

To answer this question, we introduce an auxiliary "limit game" Γ, in which equilibria are necessarily without exit. The equilibria of Γ are easily characterized by two sets of conditions: incentive compatibility and constrained optimization. Both sets of conditions are tractable but satisfying them jointly is demanding. Existence of an equilibrium in the game Γ is not obvious. For instance, as opposed to standard sender-receiver games, Γ may not have any nonrevealing equilibrium. This means that, in absence of information transmission, the receiver cannot make any decision that would give at least his reservation utility to the sender, whatever his type. We identify various assumptions which guarantee that, in a situation like this, the sender can credibly reveal some information to the receiver, in such a way that exit will never happen.

We maintain the following assumptions on the game Γ: the sender has finitely many types (which can be multidimensional, e.g., belong to R n 1 , for some n 1 ), the receiver has a compact set of decisions (typically, a closed, bounded set in R n2 , for some n 2 ) and both players' utility functions are continuous. 2 We also make the sine qua non assumption that under complete information, i.e., when the receiver knows the sender's type, there exists a decision that gives the sender at least his reservation utility. We then consider the subsets of all types such that there is a decision inducing them to participate and we call "participation structure" the maximal subsets (with respect to set inclusion). For instance, if the sender has only two possible types, 1 and 2, the participation structure is either {{1} , {2}} or {1, 2}.

We establish that the game Γ has a partitional equilibrium, namely, an equilibrium in which the sender's strategy is pure, in the following cases:

(i) the sender has two types;

(ii) the participation structure is a partition of the type set;

(iii) the decision set is a real interval and for every type, the sender's utility function is monotonic in the receiver's decision;

(iv) the receiver's utility function -when the sender participates -does not depend on the sender's type.

Cases (i) and (ii) are rather straightforward, with (ii) generalizing (i). Case (iii) applies in particular when the receiver has only two actions, over which he can randomize. Case (iv) has the most important scope. It applies as soon as the receiver knows his own preferences over decisions, but is eager to make a choice that will ensure the -type-dependent -informed player's participation. Existence of a partitional equilibrium under (iv) is the main result of the paper (Theorem 8).

The previous assumptions may look restrictive, but, without them, existence of an equilibrium in Γ cannot be guaranteed, even if the sender is allowed to use a mixed strategy. We indeed propose an example, in which the sender has three types, the receiver has three actions, the participation structure is not a partition and the utility functions are type-dependent. In this example, there is no mixed equilibrium. However, an equilibrium does exist if the information transmission stage is handled by a mediator.

Finally, we propose a complete analysis when the sender has three possible types. We identify two kinds of "participation structures" beyond the straightforward case (ii) above. The first one arises in the example mentioned in the previous paragraph. Existence of a mediated equilibrium can then be established. In the other case, another example shows that there may not be any partitional equilibrium. However we prove that there always exists a mixed equilibrium if the decision set is convex and the utility functions are affine (Proposition 9).

Here is a description of the paper. We discuss the related literature below. In Section 2, we make the sender-receiver game Γ and the solution concept fully precise. Propositions 3 and 4 (which are established in Section 6.1) allow us to argue that the game Γ is relevant to our study. In Section 3, we establish existence of a partitional equilibrium in Γ under assumptions (i), (ii), (iii) or (iv) above. Our main result, Theorem 8, is associated with case (iv). Section 4 is devoted to examples. Sections 4.1 and 4.2 propose a family of kidnapping games. Section 4.1 illustrates partitional equilibrium. Section 4.2 proposes a game that does not have any partitional equilibrium but has a mixed equilibrium. Section 4.3 goes on with a game that does not have any mixed equilibrium but has a mediated equilibrium. The three type case, including Proposition 9, is the topic of Section 5. Section 6 is an appendix containing the proofs of Propositions 3, 4 (Section 6.1) and 9 (Section 6.2).

Related papers

Shimizu (2013, 2017) adds an approval stage to Crawford and Sobel (1982)'s sender-receiver game, in the popular case where the prior is uniform over the unit interval and the utility functions are quadratic. He assumes, as we do, that exit is damaging for the receiver but the setup is otherwise quite different from ours. He shows that, in his particular model, credible exit possibilities can make cheap talk informative even when the players' conflict of interest is relatively large. [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF] studies a sender-receiver game motivated by a specific application, in which the sender is the U.S. President, the receiver is the Congress and the decision is about a practical matter, like the level of military expenditures. The President can veto the Congress' proposal. Preferences are unimodal, as in [START_REF] Shimizu | Cheap talk with an exit option: the case of a discrete action space[END_REF][START_REF] Shimizu | Cheap talk with an exit option: a model of exit and voice[END_REF], but the receiver's utility does not depend on the sender's type (as in the current paper, Theorem 8, Section 3.4). More importantly, in Matthews (1989)'s model, the sender's rejection leads to status quo, rather than to exit, and does not necessarily yield a very low utility to the receiver. Matthews (1989)'s point is to show that thanks to incomplete information on the President's type, veto can happen at equilibrium, i.e., without relying on uncredible threats.

Our model can be viewed as a principal-agent problem in which the principal -alias the receiver -cannot commit to a mechanism at the ex ante stage. This is an extreme case of [START_REF] Bester | Contracting with imperfect commitment and the revelation principle: the single agent case[END_REF]'s principal-agent problem with limited commitment. In this context, it makes sense to allow the agent -alias the sender -to veto the principal's decision. Under a mechanism design perspective, the principal looks for an equilibrium that gives him the best ex ante expected utility, which amounts to an equilibrium in which all types accept the principal's proposal if the principal's utility, when the agent chooses his outside option, is sufficiently low. This leads us to impose individual rationality conditions for the agent at the "posterior" stage, i.e., after the principal has made a proposal. The relevance of posterior individual rationality and its impact on incentive compatibility have been stressed in a number of papers, e.g., [START_REF] Gresik | Ex ante efficient, ex post individually rational trade[END_REF], Compte andJehiel (2007, 2009) [START_REF] Forges | Universal mechanisms[END_REF][START_REF] Forges | Ex post individually rational trading mechanisms[END_REF] and [START_REF] Matthews | Preplay communication in twoperson sealed-bid double auctions[END_REF].

Finally, Forges and Horst (2018)'s concept "talk and cooperate (perfect Bayesian) equilibrium" (TCE, Section 5.3) is motivated by the same questions as the present paper, but is defined in a different model: the sender also has to make a decision, which is relevant to his own payoff only. At a TCE, the receiver (who can be interpreted as a principal) proposes a joint decision, which the sender accepts whatever his type. Should player 1 reject player 2's proposal, both players would choose an action, independently of each other. By contrast, in the present paper, the sender just chooses an outside option. Forges and Horst (2018) establish an existence result for another solution concept -"cooperate and talk (perfect Bayesian) equilibrium" (CTE) -but just indicate that their methodology does not apply to TCE.

Model

Sender-receiver games

We start with a family of games Γ(v 0 ), v 0 ∈ R, between a sender (player 1) and a receiver (player 2). Γ(v 0 ) is described as follows:

• A type k ∈ K is chosen according to a prior probability p ∈ ∆(K).

• Player 1 is informed of k.

• Player 1 sends a message m ∈ M to player 2.

• Player 2 proposes a decision x ∈ X to player 1.

• If player 1 accepts player 2's proposal, the decision x is enforced, player 1 gets U k (x) and player 2 gets V k (x).

• If player 1 rejects player 2's proposal, player 1 chooses an outside option and gets u k 0 . Player 2 gets v 0 .

We assume that:

• The set of types K is finite3 and p k > 0 ∀k ∈ K.

• The set of messages M is finite, such that | M |≥| K |.

• The set of decisions X is compact. 4 As a typical example, X ⊂ R n 2 , for some n 2 ≥ 1; for instance, player 2 has a finite set of actions A and X = ∆(A) corresponds to the set of mixed strategies of player 2.

• The utility functions U k : X → R and V k : X → R are continuous; for instance, if X = ∆(A), U k and V k may correspond to expected utility.

We further assume that:

• For every k ∈ K, there exists x ∈ X such that U k (x) ≥ u k 0 .

• For every k ∈ K, for every x ∈ X, V k (x) ≥ v 0 , namely,

v 0 ≤ min k∈K min x∈X V k (x).
We are interested in situations in which the sender's approval is crucial to the receiver, namely, in which v 0 can be arbitrarily low. Let us denote as Γ the "limit game," in which v 0 = -∞. We will show that Γ is a tractable tool, which is appropriate to study Γ(v 0 ) when v 0 is small enough.

Let us set, for every

L ⊆ K X(L) = x ∈ X : U k (x) ≥ u k 0 , k ∈ L . ( 1
)
Given a subset of types L, X(L) is the set of decisions that are acceptable by all types in L. We write X(k) for X({k}), so that X(L) = k∈L X(k).

Equilibria

Our solution concept, in Γ(v 0 ) and Γ, is basically subgame perfect Nash equilibrium, but perfect Bayesian equilibrium would not be more demanding: as in standard sender-receiver games, finding beliefs rationalizing player 2's choices is not an issue. What is crucial here is to avoid non-credible threats from player 1. In the sequel, we simply refer to "equilibrium." At a subgame perfect Nash equilibrium, player 1 of type k accepts (resp., rejects) player 2's proposal x when U k (x) > u k 0 (resp., U k (x) < u k 0 ). We further assume that player 1 accepts the proposal when he is indifferent, which is consistent with our interest in situations in which player 2 strictly prefers that player 1 participates. By proceeding backwards, Γ(v 0 ) amounts to a standard sender-receiver game, with the following utility functions (in which I denotes the indicator function):

U k + (x) = U k (x)I(U k (x) ≥ u k 0 ) + u k 0 I(U k (x) < u k 0 ) = max U k (x), u k 0 (2)
for player 1 of type k and

W k (v 0 , x) = V k (x)I(U k (x) ≥ u k 0 ) + v 0 I(U k (x) < u k 0 ). ( 3
)
for player 2, when player 1 is of type k. 5 In the latter sender-receiver game, the receiver's utility function is not necessarily continuous, but it is uppersemi-continuous.

Lemma 1. For every k ∈ K and v 0 ∈ R, the utility function W k (v 0 , •) defined by ( 3) is upper-semi-continuous.

Proof: Let x n ∈ X, x n → x. The only possibly delicate case is when

U k (x n ) < u k 0 for every n and U k (x) = u k 0 . Then W k (v 0 , x n ) = v 0 ≤ V k (x) = W k (v 0 , x
), using our assumption.

Having determined player 1's behavior at the approval stage, we can define a strategy for player 1 (in Γ(v 0 ) and Γ) as a mapping σ : K → ∆(M). We interpret σ(k)(m) as the probability that player 1 sends message m when his type is k, and denote it as σ(m | k). We adopt the following notations:

For every m ∈ M, P σ (m) = k p k σ(m | k). (4)
For every k ∈ K and m ∈ M s.t.

P σ (m) > 0, p k m (σ) = p k σ(m | k) P σ (m) . (5)
p k m (σ) is thus the posterior probability of type k computed from p and σ; let p m (σ) = (p k m (σ)) k∈K denote the corresponding posterior probability distribution over K. We have

m P σ (m)p m (σ) = p.
We say that σ is nonrevealing if player 1 sends his message in a typeindependent way, namely, if

σ(m | k) = σ(m | k ′ ) for every m ∈ M, k, k ′ ∈ K. In this case, p m (σ) = p for every m s.t. P σ (m) > 0.
For player 2, a strategy is a mapping τ : M → X, namely, a "pure" strategy with respect to the set X (but as indicated above, X = ∆(A) for a finite set of actions A is a particular case). 6We say that (σ, τ ) is "without exit" if

U k (τ (m)) ≥ u k 0 ∀k ∈ K, ∀m ∈ M s.t. σ(m | k) > 0, (6)
namely, if

U k + (τ (m)) = U k (τ (m)) ∀k ∈ K, ∀m ∈ M s.t. σ(m | k) > 0.
Recalling (1) and denoting by supp q the support of a probability distribution q ∈ ∆(K), condition ( 6) is equivalent to

τ (m) ∈ X(supp p m (σ)) ∀m ∈ M s.t. P σ (m) > 0. ( 7
)
At an equilibrium of Γ, we require that player 2's expected utility be finite (> -∞). Hence an equilibrium of Γ is necessarily without exit. By contrast, an equilibrium of Γ(v 0 ) may involve exit of some types.

Proposition 2. For every v 0 ∈ R, the game Γ(v 0 ) has a nonrevealing equilibrium (possibly with exit). The game Γ has a nonrevealing equilibrium if and only if X(K) = ∅. Hence Γ may not have any nonrevealing equilibrium.

Proof:

The following strategies define a nonrevealing equilibrium in Γ(v 0 ): player 1 sends the same message m ∈ M whatever his type and then accepts x if and only if U k (x) ≥ u k 0 ; whatever the message, player 2 chooses x * ∈ X to maximize k p k W k (v 0 , x), which is well-defined thanks to Lemma 1.

In Γ, if X(K) = ∅, a nonrevealing equilibrium can be achieved as above, provided that player 2 chooses x * ∈ X to maximize k p k V k (x) subject to x ∈ X(K). If X(K) = ∅ and player 1's message is type-independent, condition ( 7) cannot be satisfied.

The next two propositions (which are established in Section 6.1) give us some foundations to study the equilibria of the limit game Γ by making precise relationships between the latter and the equilibria without exit of Γ(v 0 ). Proposition 3. Let (σ, τ ) be an equilibrium without exit in Γ(v 0 ), for some v 0 ∈ R. Then (σ, τ ) is an equilibrium without exit in Γ(z 0 ) for every z 0 ∈ R such that z 0 ≤ v 0 and is also an equilibrium in Γ, with the same interim expected utility as in Γ(v 0 ) for both players.

In other words, if Γ has no a equilibrium (which indeed may happen, see Section 4.3), then, whatever v 0 ∈ R, Γ(v 0 ) has no equilibrium without exit, that is, all equilibria of Γ(v 0 ) must involve non-participation of at least one type.

Proposition 4. Let (σ, τ ) be an equilibrium in Γ. Then there exists v 0 ∈ R such that, for every z 0 ≤ v 0 , (σ, τ ) is an equilibrium without exit of Γ(z 0 ), with the same interim expected utility as in Γ for both players.

The previous properties are useful under a mechanism design perspective. Assume player 2 is a "principal" who cannot commit to a mechanism µ : K → X but receives a message from the agent (player 1) and then, makes a decision in X subject to the agent's participation constraints. With this interpretation, which turns out to be an extreme case of [START_REF] Bester | Contracting with imperfect commitment and the revelation principle: the single agent case[END_REF]'s model, an optimal mechanism amounts to an equilibrium of Γ(v 0 ) which maximizes player 2's ex ante expected utility. At a given v 0 , the equilibrium of Γ(v 0 ) that is best for player 2 may involve the exit of some types of player 1. By contrast, player 2's best equilibrium payoff in the limit game Γ, when it exists, is achieved at an equilibrium (σ * , τ * ) without exit. Let us denote player 2's corresponding payoff as v * N E . By Proposition 4, if v 0 is small enough, (σ * , τ * ) is an equilibrium without exit in Γ(v 0 ), giving the same expected utility v * N E to player 2. By Proposition 3, v * N E is the best equilibrium payoff player 2 can achieve at an equilibrium without exit in Γ(v 0 ). Moreover, as we show in details in Section 6.1, if v 0 is sufficiently low, player 2 cannot expect an expected utility higher than v * N E at an equilibrium of Γ(v 0 ) with exit of some types. 7 Summing up, if the limit game Γ has an equilibrium, an optimal mechanism, when the principal's utility v 0 in case of exit is sufficiently low, can be found by maximizing player 2's utility over the equilibria of Γ, without worrying about the precise level v 0 .

Existence of a partitional equilibrium in Γ

In this section, we focus on the game Γ. We identify various sufficient conditions for the existence of a partitional equilibrium in Γ, in which player 1 uses a pure strategy, namely a mapping σ : K → M, to send his message to player 2. The strategy σ then induces the partition {K m , m ∈ σ(K)} of K, with K m = σ -1 (m) = {k ∈ K : σ(k) = m}. In this case, (4) and ( 5) become respectively:

For every m ∈ M, P σ (m) =

k∈Km p k . ( 8
)
For every k ∈ K and m ∈ M s.t.

P σ (m) > 0, p k m (σ) = p k I(k ∈ K m ) P σ (m) . (9)
As in Section 2, for player 2, we focus on strategies of the form τ : M → X. At an equilibrium of Γ, given player 1's strategy σ and the message m he receives, player 2 updates his belief over K into p m (σ). To avoid exit, we must have

X(supp p m (σ)) = ∅ (10) Player 2's strategy τ is then a best response to σ in Γ if and only if ∀m ∈ M s.t. P σ (m) > 0, τ (m) ∈ arg max x∈X(supp pm(σ)) k p k m (σ)V k (x). ( 11
)
We refer to these conditions as to constrained optimization. Player 1's equilibrium conditions reduce to incentive compatibility conditions expressing that given player 2's strategy τ , player 1 of type k prefers to send σ(k) than any other message m, namely,

U k (τ (σ(k)))) ≥ U k (τ (m)) for every k ∈ K and m ∈ M. ( 12
)

Two types

If only two types are possible, we show that either there is a decision giving both types at least their reservation utility or full revelation of information is credible and allows to avoid exit.

Proposition 5. Let us assume that | K |= 2. Then Γ has a partitional equilibrium.

Proof:

If X(1) ∩ X(2) = ∅, let x * ∈ arg max x∈X(1)∩X(2) p 1 V 1 (x) + p 2 V 2 (x)
and let m * be an arbitrary element of M. Then σ(1

) = σ(2) = m * and τ (m) = x * for every m ∈ M defines a nonrevealing equilibrium of Γ. Otherwise, if X(1) ∩ X(2) = ∅, let x k ∈ arg max x∈X(k) V k (x) k = 1, 2 and let m 1 = m 2 be two distinct elements of M. Then σ(k) = m k , τ (m k ) = x k , k = 1, 2
, defines a fully revealing equilibrium of Γ. Indeed, constrained optimization [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF] holds by construction; to see that incentive compatibility (12) also holds, observe that

x k ∈ X(k). Hence x k / ∈ X(ℓ) for ℓ = k, since X(1) ∩ X(2) = ∅. In other words, U ℓ (x k ) < u ℓ 0 ≤ U ℓ (x ℓ ) for ℓ = k.

Straightforward partitional equilibria

In this section, we propose an easy generalization of Proposition 5 when the sender has an arbitrary number of types. Recalling the definition of X(L) (see ( 1)), let us set

T = {∅ = L ⊆ K : X(L) = ∅}
and define T * as the set of maximal elements of T for set inclusion, namely,

T * = {L ∈ T : [L ′ ∈ T and L ⊆ L ′ ] ⇒ L ′ = L} . ( 13
)
We refer to T * as to the "participation structure" of the game Γ.

Proposition 6. If the participation structure of Γ is a partition of K, Γ has a partitional equilibrium.

Proof:

Let T * = {K r }. Consider the strategy of player 1 consisting of revealing the cell K r containing his type. Let x * r ∈ X(K r ) be an optimal decision of player 2 when he learns that player 1's type belongs K r , namely,

x * r ∈ arg max x∈X(Kr) k∈Kr p k j∈Kr p j V k (x).
Constrained optimization [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF] holds by construction. Incentive compatibility [START_REF] Shimizu | Cheap talk with an exit option: the case of a discrete action space[END_REF] is also immediate, because if k ∈ K r , x * r ∈ X(K r ) while for j = r, x * j / ∈ X(K r ).

Decision in a real interval and monotonic utility function for the sender

The next result holds in particular when player 2's decision can be interpreted as a probability distribution over two possible actions (i.e., X = ∆(A),| A |= 2) and the utility U k (x) of player 1 of type k is expected utility with respect to x.

Proposition 7. Let us assume that the decision set X is a real interval and every utility function U k , k ∈ K, is monotonic over X. Then Γ has a partitional equilibrium.

Proof:

Let us take, without loss of generality, X = [0, 1]; define

K -= k ∈ K : U k is
weakly decreasing and not constant

K + = k ∈ K : U k is weakly increasing or constant We can find x k 0 , k ∈ K, such that if k ∈ K -, U k (x) ≥ u k 0 ⇔ x ≤ x k 0 and if k ∈ K + , U k (x) ≥ u k 0 ⇔ x ≥ x k 0 . We define next x -= min k∈K -x k 0 if K -= ∅ ; x -= 1 if K -= ∅. x + = max k∈K + x k 0 if K + = ∅ ; x + = 0 if K + = ∅. If x + ≤ x -(in particular, if K -or K + = ∅), let x * ∈ arg max [x + ,x -] k∈K p k V k (x)
and let m * be an arbitrary element of M. Then σ(k) = m * for every k ∈ K and τ (m) = x * for every m ∈ M defines a nonrevealing equilibrium of Γ.

If x + > x -, let m * -= m * + be two distinct elements of M. Take σ(k) = m * -if k ∈ K -, σ(k) = m * + if k ∈ K + ,
namely, σ induces the partition {K -, K + }. Player 2's corresponding posterior probability distribution on K can be computed as in [START_REF] Gresik | Ex ante efficient, ex post individually rational trade[END_REF]:

p k m * -= p k I(k ∈ K -) j∈K -p j and p k m * + = p k I(k ∈ K + ) j∈K + p j . Let then x * -∈ arg max [0,x -] k∈K - p k m * -V k (x) and x * + ∈ arg max [x + ,1] k∈K + p k m * + V k (x).
Constrained optimization (11) holds by construction. There remains to check incentive compatibility [START_REF] Shimizu | Cheap talk with an exit option: the case of a discrete action space[END_REF]. Observe that

x * -< x * + ; for k ∈ K -, U k is decreasing, hence U k (x * -) ≥ U k (x * + ). Similarly for k ∈ K + , U k is increasing so that U k (x * -) ≤ U k (x * + ).

Type-independent utility function for the receiver

In this section, we assume that, when the sender accepts the receiver's proposal, the utility function of the receiver does not depend on the sender's type, namely, that V k (x) = V (x) for every k and x. This assumption is sometimes referred to as "private values" or "known-own payoff." [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF] formulates it in the context of a game of information transmission with sender's approval.

Theorem 8. Let us assume that player 2's utility function does not depend on player 1's type, namely, that there exists a continuous function V :

X → R such that V k = V for every k ∈ K.
Then Γ has a partitional equilibrium.

The proof consists of an algorithm, which constructs a partitional equilibrium that is as revealing as possible, given the incentive compatibility constraints to be fulfilled. More precisely, the initial candidate is the fully revealing equilibrium. Imagine that type k would envy type ℓ if one tried to implement fully revealing strategies, while type ℓ would not envy any type. By merging type ℓ and type k, one reduces the incentives problem. A key property is that, if player 2's utility function is independent of player 1's type, then player 2's optimal decision x ℓ when facing type ℓ remains optimal when facing type ℓ or type k. Before making use of it, we first show, by relying on the same kind of argument, that the envy relation cannot have any cycle.

Proof:

Let us fix, for every k ∈ K,

x k ∈ arg max x∈X(k) V (x). ( 14
)
The existence of such x k 's is guaranteed by our assumptions. If the previous optimization problem has several solutions, we take x k to maximize U k (x).

For every pair of types j, k ∈ K, we say that "type k envies type j"and write kRj -if

U k (x j ) > U k (x k ). An immediate property is that for every j, k ∈ K, kRj ⇒ V (x k ) > V (x j ). ( 15
)
To show this, observe that, by definition,

x k ∈ X(k), i.e., U k (x k ) ≥ u k 0 . Hence, if kRj, we must have U k (x j ) > u k 0 , which implies x j ∈ X(k) (so that x j ∈ X(j) ∩ X(k)) and V (x k ) ≥ V (x j ). But V (x k ) = V (x j ) cannot arise, be- cause U k (x j ) > U k (x k )
and, in case of multiple solutions to max x∈X(k) V (x), we choose x k to maximize U k (x).

The previous property implies that the envy relation R has no cycle. We will gradually construct a subset L ⊆ K of leader types which do not envy any other type in L and a subset F = K \ L of follower types which envy a type in L.

We start with L = F = ∅. Let us denote as

α 1 < • • • < α n the distinct values among V (x k ), k ∈ K. Necessarily, n ≤| K |. Define then K j = {k ∈ K : V (x k ) = α j } j = 1, ..., n. (16)
Step 1 Consider every type k ∈ K 1 : V (x k ) = α 1 is strictly below any other α j . By (15), type k cannot envy any other type. We modify L into L = K 1 , while F does not change (F = ∅).

Step 2 Consider every type k ∈ K 2 . If k does not envy any type, put k in L. Otherwise, again by (15), k can only envy a type in L (as defined at the end of step 1, namely, K 1 ), put k in F .

• • •

Step j Let L and F be the sets of leaders and followers constructed so far.

L ∪ F = K 1 ∪ • • • ∪ K j-1
so that by (15) and ( 16), types in L ∪ F cannot envy types in K j . Consider every such type k ∈ K j . If k envies a type in L, put k in F . Otherwise, put k in L. L and F are thus updated at the end of step j.

• • •

Step n Proceed as for step j. Deduce the final sets of leaders and followers.

For instance, if | K |= 3 and R is fully described by 3R2 and 2R1, the previous construction results in

K 1 = {1}, K 2 = {2}, K 3 = {3}, L = {1, 3}.
Using the x k 's defined by ( 14) and the set L, we construct an equilibrium (σ, τ ) of Γ. For simplicity, we rename the messages in M so that L ⊆ M. Player 1's strategy is such that σ(K) = L. More precisely, σ : K → L is defined by

σ(k) = k if k ∈ L arg max j∈L,kRj U k (x j ) if k ∈ K \ L.
In other words, leader types announce themselves, while non leader types report the leader type they most envy. Player 2's strategy is defined by τ : L → X : τ (ℓ) = x ℓ , with x ℓ defined by (14).

Incentive compatibility [START_REF] Shimizu | Cheap talk with an exit option: the case of a discrete action space[END_REF] follows from the fact that player 2's strategy τ restricts his decisions to the subset {x ℓ , ℓ ∈ L}. Hence types in L, who cannot envy any other type in L, are truthful. Types in K \ L behave as well as they can given the player 2's restricted decision set.

If player 1 follows σ, then, given message ℓ ∈ L, player 2 deduces that player 1's type k ∈ σ -1 (ℓ). The set σ -1 (ℓ) contains ℓ, x ℓ ∈ X(ℓ) by ( 14) and all other types in σ -1 (ℓ) envy ℓ, so that x ℓ ∈ k∈σ -1 (ℓ) X(k). Since x ℓ is a maximizer of V (x) over X(ℓ), it is also a maximizer of V (x) over k∈σ -1 (ℓ) X(k).

Remarks:

-A main feature of the proof of Theorem 8 is that, in the partitional equilibrium that is constructed, the receiver makes a decision in a subset of {x k , k ∈ K} where x k is the optimal decision he would make if he were sure to face type k. The receiver's private values guarantee that if type k envies type ℓ, then x ℓ , the receiver's optimal choice when he faces type ℓ for sure (i.e., under the constraint x ∈ X(ℓ)), is still optimal when he faces type k or type ℓ (i.e., under the constraint x ∈ X(ℓ) ∩X(k)). This property may no longer hold when player 2's utility is type-dependent.

-Theorem 8 does not depend on the underlying utility representation: the result holds if the receiver's von Neumann-Morgenstern preferences over X given type k are equivalent for every k ∈ K.

4 Examples (including a counter-example)

Partitional equilibrium

Let the informed player have three possible types, i.e., K = {1, 2, 3} and let the uninformed player's decision set be

X = {(x a , x b ) : x a ≥ 0, x b ≥ 0, x a + x b ≤ 100} .
Let the utility function and reservation utility of the informed player be

U 1 (x) = x a -x b u 1 0 = 30, U 2 (x) = x b -x a u 2 0 = 40, U 3 (x) = x a + 2x b u 3 0 = 20.
Let the uninformed player's utility function be type-independent:

V k (x) = V (x) = -(x a + x b ), k = 1, 2, 3.
There are two goods, a and b, X accounts for the decision-maker's resource constraints. Type 1 likes good a, dislikes good b; type 2 has symmetric preferences; type 3 likes both goods, and likes good b more than good a.

As a possible interpretation, player 1 is a kidnapper who can have political motivations (type 1), just look for a monetary ransom (type 2) or be opportunistic (type 3). Good a stands for political prisoners who can be released while good b stands for money. If player 1 does not accept player 2's offer, the hostage is killed, leading to an invaluable loss for player 2.

Recalling (1) and using "Co" for convex hull, we have here X({1}) = X({1, 3}) = Co {(30, 0), (100, 0), (65, 35)} , X({2}) = X({2, 3}) = Co {(0, 40), (0, 100), (30, 70)} , X({3}) = Co {(20, 0), (100, 0), (0, 100), (0, 10)} , X({1, 2}) = X({1, 2, 3}) = ∅.

Assume first that player 1's type k is known, namely, that p k = 1. Let then x * k be the uninformed player's optimal decision (in X) when he faces type k:

x * 1 = (30, 0), x * 2 = (0, 40), x * 3 = (0, 10).

Suppose next that only two types are possible. If p 1 = 0, given that X({2, 3}) = ∅, there is a nonrevealing equilibrium, x * 2 = (0, 40). Similarly for p 2 = 0, with x * 1 = (30, 0). If p 3 = 0, there is no way to satisfy type 1 and type 2 at the same time. But there is a completely revealing equilibrium:

x * 1 = (30, 0) to type 1, x * 2 = (0, 40) to type 2 is incentive compatible. This illustrates Proposition 5.

Let the three types be possible, namely p k > 0 for k = 1, 2, 3. There is no nonrevealing equilibrium, since X({1, 2, 3}) = ∅. There is no completely revealing equilibrium either: (17) implies that type 3 would pretend to be type 2 (type 3 envies type 1 and type 2 even more).

As expected from Theorem 8, there exists a partitional equilibrium. The informed player is invited to report whether his type is 1 or not. If he reports type 1, the uninformed player proposes x * 1 = (30, 0). If the informed player reports that his type is not 1, the uninformed player proposes arg min x∈X({2,3})

(x a + x b ) = x * 2 = (0, 40).
As in the proof of Theorem 8, incentive compatibility is ensured by the fact that the decision proposed to type 3 is the one he most envies among x * 1 and x * 2 .

Mixed equilibrium

Let us modify the uninformed player's utility function in the previous example, to make it depend on the informed player's type:

V 1 (x) = x a 3 , V 2 (x) = x b 3 , V 3 (x) = -(x a + x b ).
A possible interpretation is that the decision-maker is happy to pay when the kidnapper has sharp preferences. Let as above x * k denote the uninformed player's optimal decision (in X) when he faces type k; we have now

x * 1 = (100, 0), x * 2 = (0, 100), x * 3 = (0, 10). ( 18
)
Let us take p = ( 1 3 , 1 3 , 1 3 ). There is no nonrevealing equilibrium, since X({1, 2, 3}) = ∅. There is no completely revealing equilibrium: given (18), type 3 would pretend to be type 2. More generally, there is no partitional equilibrium. Given the above description of the sets X(L), two possible partitions must still be considered: {{1} , {2, 3}} and {{1, 3} , {2}}. {{1} , {2, 3}}: if the uninformed player believes he faces type 1 (posterior (1, 0, 0)), his optimal choice is x * 1 = (100, 0); if he believes he faces type 2 or type 3 (posterior (0, 1 2 , 1 2 )), his optimal choice is

x * 23 = arg min x∈X({2,3}) x a + 2 3 x b = (0, 40).
This cannot be incentive compatible for type 3:

100 = U 3 (x * 1 ) > U 3 (x * 23 ) = 80.
{{1, 3} , {2}}: if the uninformed player believes he faces type 2 (posterior (0, 1, 0)), his optimal choice is x * 2 = (0, 100); if he believes he faces type 1 or type 3 (posterior ( 12 , 0, 1 2 )), his optimal choice is

x * 13 = arg min x∈X({1,3}) 2 3 x a + x b = (100, 0).
Again, this cannot be incentive compatible for type 3:

200 = U 3 (x * 2 ) > U 3 (x * 13 ) = 100.
This illustrates that Theorem 8 does not extend to the case where player 2's utility function depends on player 1's type.

Let us show that if player 1 uses a mixed strategy, a partially revealing equilibrium exists in this example: type 1 reports that his type belongs to {1, 3}, type 2 reports that his type belongs to {2, 3}, type 3 reports that his type belongs to {1, 3} (resp., {2, 3}) with probability 1 3 (resp., 2 3 ). If the informed player follows this reporting strategy, the uninformed player's posterior upon receiving {1, 3} is ( 3 4 , 0, 1 4 ) while upon receiving {2, 3}, it is (0, 3 5 , 2 5 ). Given {1, 3}, the uninformed player's problem reduces to min x∈X({1,3}) x b . Every x = (x a , 0) with x a ∈ [30, 100] is optimal. Let us take x * 13 = (80, 0). Given {2, 3}, the uninformed player's optimal choice is

x * 23 = arg min x∈X({2,3}) [2x a + x b ] = (0, 40).
There remains to check incentive compatibility. Type 1 prefers x * 13 = (80, 0) to x * 23 = (0, 40), and vice-versa for type 2. Type 3 must be indifferent between sending {1, 3} or {2, 3}, because he must randomize between these two outcomes. Indeed we have U 3 (x * 13 ) = U 3 (x * 23 ) = 80. Proposition 9 in Section 5 states that the previous construction can be generalized.

No equilibrium at all

In the following example, none of the existence results of Section 3 can be applied. We will show that there is no equilibrium, even if player 1 makes use of a mixed strategy. The game is described by:| K |= 3, X = ∆(A), where A = {a, b, c}, u k 0 = 0, k = 1, 2, 3. The following payoff matrices describe (U k (α), V k (α)) for every α ∈ A:

a b c k = 1 0, 2 -2, 0 1, 1 k = 2 1, 1 0, 2 -2, 0 k = 3 -2, 0 1, 1 0, 2
The utility functions over X = ∆(A) are obtained as expected utilities with respect to x = (x a , x b , x c ).

If player 2 knows that he faces type k (i.e., p k = 1), he gets his first best by choosing

a if k = 1, b if k = 2, c if k = 3. But if p k > 0 for
every k, there is no nonrevealing equilibrium ( k U k (x) < 0 for every x ∈ ∆(A)) and no fully revealing equilibrium (incentive compatibility is violated).

Looking for a partially revealing equilibrium, we first check that there is a unique, nonrevealing equilibrium, as soon as only two types are possible. Take, e.g., p 3 = 0. Then

X({1, 2}) = {x ∈ X : -2x b + x c ≥ 0 and x a -2x c ≥ 0} = Co (1, 0, 0), ( 2 3 , 0, 1 3 ), ( 4 7 , 1 7 , 2 7 )
and player 2's optimization problem is:

max p 1 (2x a + x c ) + p 2 (x a + 2x b ) s.t. x ∈ X({1, 2}).
For every p such that p 1 > 0 and p 2 > 0, the unique solution is achieved at x = (1, 0, 0), namely, action a with probability 1. Similarly, action b is the only solution if p 2 > 0 and p 3 > 0, action c is the only solution if p 1 > 0 and p 3 > 0.

Let us start with p such that p k > 0 for every k. By sending his message according to a mixed strategy σ, player 1 "splits" the prior belief p into posteriors p m (σ) such that m P σ (m)p m (σ) = p (see [START_REF] Crawford | Strategic information transmission[END_REF]). Taking account of incentive compatibility, p cannot be split (only) extreme points, because there is no fully revealing equilibrium. At least one of the posteriors p m must be on an edge, say p 3 m = 0, so that τ (m) = a. There should be another posterior p m ′ with p 3 m ′ > 0, with τ (m ′ ) = b or c. To achieve the posteriors p m and p m ′ , message m must be sent with positive probability by types 1 and 2, while message m ′ must be sent with positive probability by at least type 3. If τ (m ′ ) = b, type 2 strictly prefers m ′ to m. If x(m ′ ) = c, type 1 strictly prefers m ′ to m. Hence there is no incentive compatible splitting and thus no equilibrium at all, even if player 1 can use a mixed strategy.

A mediated equilibrium, in which information transmission is monitored by a mediator, can nevertheless be achieved in the previous example. Consider the following three lotteries over A:

δ 1 = ( 1 2 , 0, 1 2 ), δ 2 = ( 1 2 , 1 2 , 0), δ 3 = (0, 1 2 , 1 2
). Assume that, instead of selecting a message by himself, player 1 can just choose among these three lotteries. If player 1 expects player 2 to pick the action selected by the lottery, player 1 prefers δ k over the other two lotteries when his type is k. Similarly, player 2 is happy to choose the action recommended by the lottery if he believes that player 1 reveals his type truthfully to the mediator. This procedure will be generalized in the next section. below that Γ always has a mediated equilibrium. In case 3, which has been illustrated in Section 4.2., player 2 can only guarantee the approval of two pairs of types. We will establish that under further assumptions, Γ always has then a mixed (possibly not pure) equilibrium.

T

* = {{1, 2} , {1, 3} , {2, 3}}
Let us enrich the description of Γ by adding a mediator who invites player 1 to report a type (in K) and then selects a decision (in X) that he recommends player 2. At a mediated equilibrium, player 1 truthfully reveals his type to the mediator, player 2 proposes to player 1 the decision x that is recommended by the mediator and finally, player 1 of type k accepts player 2's proposal x ′ provided that U k (x ′ ) ≥ u k 0 . Let us construct a mediated equilibrium in Γ when K = {1, 2, 3} and T * = {{1, 2} , {1, 3} , {2, 3}}. Recall that p k > 0 for every k. For every pair (j, k) of types, let x * jk ∈ X({j, k}) be an optimal decision for player 2 when he learns that player 1 is of type j or k, namely,

x * jk ∈ arg max p j p j + p k V j (x) + p k p j + p k V k (x) . (19)
Consider the following mediator: for every k = 1, 2, 3, if player 1 reports type k, he selects x * ik or x * jk , i, j = k, i = j, with equal probability 1 2 and recommends it to player 2. If player 1 reports his type k truthfully, player 2 learns, with equal probability, that player 1's type is in {i, k} or in {j, k} for i, j = k, i = j. Condition (19) guarantees that player 2 follows the mediator's recommendation.

For player 1, let us consider k = 1. By reporting his type truthfully, player 1 obtains

1 2 U 1 (x * 12 ) + 1 2 U 1 (x * 13 ).
If he lies by, say, pretending to be of type 2, he obtains8

U 1 2 = 1 2 max U 1 (x * 12 ), u 1 0 + 1 2 max U 1 (x * 23 ), u 1 0 . By construction, x * 12 ∈ X({1, 2}) and x * 13 ∈ X({1, 3}). Hence U 1 (x * 12 ) ≥ u 1 0 and U 1 (x * 13 ) ≥ u 1 0 . But x * 23 / ∈ X(1) because {2, 3} is maximal. Hence, U 1 (x * 23 ) < u 1 0 . U 1 2 = 1 2 U 1 (x * 12 ) + 1 2 u 1 0 ≤ 1 2 U 1 (x * 12 ) + 1 2 U 1 (x * 13 ).
The other incentive compatibility conditions of player 1 can be checked in a symmetric way.

5.2 T * = {{1, 3} , {2, 3}} Proposition 9. Let us assume that K = {1, 2, 3}, the participation structure T * = {{1, 3} , {2, 3}}, the decision set X is compact and convex and the utility functions U k and V k , k ∈ K, are affine. Then Γ has a partially revealing equilibrium, in which player 1 uses a mixed, possibly not pure, strategy.

Proof: See Section 6.2. We establish that there must exist an equilibrium in which type 1 reports that his type belongs to {1, 3}, type 2 reports that his type belongs to {2, 3} and type 3 reports that his type belongs to {1, 3} (resp., {2, 3}) with some probability δ ∈ (0, 1) (resp., 1δ). Incentive compatibility requires that type 3 be indifferent between reporting {1, 3} or {2, 3}. Such an equilibrium is shown to be the only possible one in the example of Section 4.2.

6 Appendix

Proof of Propositions 3 and 4

For the sake of completeness, we first explicitly recall the conditions to be satisfied by an equilibrium of Γ(v 0 ) whether they involve exit of some types on equilibrium path or not.

Let us fix a pair of strategies σ : K → ∆(M) and τ : M → X. Player 1's equilibrium conditions can be written as

U k + (τ (m)) ≥ U k + (τ (m ′ )) ∀k ∈ K, ∀m ∈ M s.t. σ(m | k) > 0, ∀m ′ ∈ M.
(20) Player 2's equilibrium conditions can be written as

k p k m (σ)W k (v 0 , τ (m)) ≥ k p k m (σ)W k (v 0 , x) ∀m ∈ M s.t. P σ (m) > 0, ∀x ∈ X.
We deduce that the necessary and sufficient conditions for (σ, τ ) to be an equilibrium without exit are:

For player 1:

U k (τ (m)) ≥ U k (τ (m ′ )) ∀k ∈ K, ∀m ∈ M s.t. σ(m | k) > 0, ∀m ′ ∈ M, (21) implying that U k (τ (m)) = U k (τ (m ′ )) ∀m, m ′ ∈ M s.t. σ(m | k) > 0 and σ(m ′ | k) > 0.
For player 2:

τ (m) ∈ arg max x∈X k p k m (σ)W k (v 0 , x) ∩ X(supp p m (σ)), (22)
implying constrained optimization, namely [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF].

Recall that by definition, an equilibrium of Γ cannot involve exit, so that the conditions for (σ, τ ) to be an equilibrium in Γ are thus just ( 21) and [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF].

Remarks on (21):

• As a refinement of subgame perfect equilibrium, [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF] strengthens (20) to (21) in the case of equilibria which typically involve exit on path.

• For an equilibrium without exit, player 1's equilibrium conditions take the simple form (21) because player 2's strategy τ is pure. In an equilibrium without exit, for every k ∈ K, player 1's equilibrium strategy consists of sending m, selected with probability σ(m | k), and then accept player 2's proposal, namely, τ (m), which is fully anticipated at the time to choose m. A deviation consists of sending m ′ ∈ M (possibly such that σ(m ′ | k) = 0) and then, accept or reject player 2's proposal τ (m ′ ). Player 1's equilibrium conditions thus take the form

U k (τ (m)) ≥ max U k (τ (m ′ )), u k 0 ∀k ∈ K, ∀m ∈ M s.t. σ(m | k) > 0, ∀m ′ ∈ M.
These are equivalent to (21), since [START_REF] Forges | Universal mechanisms[END_REF] holds at an equilibrium without exit. If player 2's strategy τ were mixed (τ : M → ∆(X)), we would have to write

9 x τ (x | m)U k (x) ≥ x τ (x | m ′ ) max U k (x)), u k 0 .
Proposition 3. Let (σ, τ ) be an equilibrium without exit in Γ(v 0 ), for some v 0 ∈ R. Then (σ, τ ) is an equilibrium without exit in Γ(z 0 ) for every z 0 ∈ R such that z 0 ≤ v 0 and is also an equilibrium in Γ, with the same interim expected utility as in Γ(v 0 ) for both players.

Proof: If (σ, τ ) satisfies (22) in Γ(v 0 ), the same holds in Γ(z 0 ) since for every x ∈ X,

k p k m (σ)W k (z 0 , x) ≤ k p k m (σ)W k (v 0 , x)
and for every x ∈ X(supp p m (σ)),

k p k m (σ)W k (z 0 , x) = k p k m (σ)W k (v 0 , x) = k p k m (σ)V k (x).
Furthermore, (11) must hold and player 1's equilibrium conditions (21) are the same in Γ(v 0 ) and Γ(z 0 ) or Γ as long as player 2's strategy remains unchanged.

Proposition 4. Let (σ, τ ) be an equilibrium in Γ. Then there exists v 0 ∈ R such that, for every z 0 ≤ v 0 , (σ, τ ) is an equilibrium without exit of Γ(z 0 ), with the same interim expected utility as in Γ for both players.

Proof: Let (σ, τ ) be an equilibrium without exit in Γ. By definition, constrained optimization [START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF] holds, so that in particular τ (m) ∈ X(supp p m (σ)) for every m such that P σ (m) > 0. Let us keep player 1's strategy, σ, fixed. Player 2's strategy τ remains a best reply to σ in Γ(v 0 ), with v 0 ≤ min k∈K min x∈X V k (x), provided that v 0 is such that optimality of no exit holds, namely, recalling (22),

∀m ∈ M s.t. P σ (m) > 0, k p k m (σ)V k (τ (m)) ≥ max x∈X k p k m (σ)W k (v 0 , x).
These can be viewed as finitely many inequalities over v 0 ,which have a solution in R, since the RHS are well-defined, for every v 0 ≤ min k∈K min x∈X V k (x), by Lemma 1. 10 Hence there exists v 0 such that (σ, τ ) is an equilibrium without exit of Γ(v 0 ) and from Proposition 3, in Γ(z 0 ), for every z 0 ≤ v 0 .

10 The RHS of the inequalities can be rewritten as max

L supp(pm(σ)) max x∈X(L)    k∈L p k m (σ)V k (x) + v 0 k∈supp(pm(σ)) L p k m (σ)    .

Application to a mechanism design problem

In Section 3, we have established that, under various reasonable assumptions, Γ has a partitional equilibrium (σ, τ ), in which both σ and τ are pure. In this case, we can easily compute the highest ex ante expected utility that player 2, interpreted here as the principal, can obtain at a partitional equilibrium of Γ. 11 As explained in Section 2, this application is inspired by a particular case of Bester and Strausz (2001)'s model.

Let us show that there exists v 0 ∈ R such that, for every z 0 ≤ v 0 , the highest ex ante expected utility player 2 can obtain at an arbitrary partitional equilibrium of Γ(z 0 ) (which can involve exit or not) is the same as in Γ. More precisely, there exists v 0 ∈ R such that, for every z 0 ≤ v 0 , the best partitional equilibrium for player 2 in Γ remains the best partitional equilibrium for player 2 in Γ(z 0 ).

Let v * N E be the highest ex ante expected utility player 2 can obtain at a partitional equilibrium of Γ. This number is well-defined if Γ has a partitional equilibrium. Let (σ * , τ * ) achieve the expected utility v * N E for player 2. Using Proposition 4, there exists v 0 sufficiently small such that for every z 0 ≤ v 0 , (σ * , τ * ) is an equilibrium without exit of Γ(z 0 ) with the same expected utility v * N E for player 2. By Proposition 3, for every such z 0 , there does not exist any equilibrium without exit giving a higher expected utility to player 2 (because such an equilibrium would still be an equilibrium of Γ, with the same expected utilities).

Let us consider the partitional equilibria (σ, τ ) of Γ(v 0 ) in which exit possibly occurs, i.e., in which the set

K E = k ∈ K : U k (τ • σ(k)) < u k 0 = ∅, i.e., p E = def k∈K E p k > 0.
The highest expected utility player 2 can achieve at such an equilibrium is

p E v 0 + (1 -p E )v where v = max k∈K max x∈X V k (x).
If v 0 is such that, for every p E that can arise given the prior p,

p E v 0 + (1 -p E )v ≤ v * N E , namely, v 0 ≤ 1 p E [v * N E -(1 -p E )v] (23)
then (σ * , τ * ) will guarantee the highest possible equilibrium utility to player 2, in every game Γ(z 0 ) with z 0 ≤ v 0 .

Let k be the type with the smallest prior probability, namely, such that p k = min p 1 , • • • , p K . The inequality (23) will hold at every p E that can arise given the prior p as soon as it holds at p E = p k : we just have to require

v 0 ≤ 1 p k v * N E -(1 -p k )v .
The previous result is quite intuitive: an upper bound on the receiver's expected utility at an equilibrium of Γ(v 0 ) with exit is obtained when the receiver's proposal is rejected by only the least likely type, while the best possible utility is achieved at all the other types. If v 0 is sufficiently low, the best equilibrium utility for the receiver in Γ(v 0 ) will be not be achieved at an equilibrium with exit, but rather at an equilibrium without exit, which is in turn is necessarily an equilibrium of Γ.

Proof of Proposition 9

Proposition 9. Let us assume that K = {1, 2, 3}, T * = {{1, 3}, {2, 3}} the decision set X is compact and convex and the utility functions U k and V k , k ∈ K, are affine. Then Γ has a partially revealing equilibrium.

For simplicity we assume here that u k 0 = 0 for each k. This is w.l.o.g. since we can translate the payoffs of each type of the sender. We start with preliminaries.

Mappings and multi-valued mappings

Define for each p in ∆(K):

f (p) = sup{ k∈K p k V k (x), x ∈ X(supp p)} ∈ IR ∪ {-∞}, Y (p) = Argmax x∈X(supp p) k p k V k (x) ⊂ X(supp p), and
Φ(p) = {(U k (x)) k∈K , x ∈ Y (p)} ⊂ IR K .
The sets Y (p) and Φ(p) are convex compact subsets of IR and IR K , respectively. If X(supp p) = ∅, then f (p) ∈ IR, Y (p) = ∅ and Φ(p) = ∅. For each u ∈ Φ(p), we have u k ≥ 0 for each k ∈ supp p. At an equilibrium of Γ, if the belief of the receiver (after having received the message of the sender) k p k n V k (x) for infinitely many n's, so lim sup n f (p n ) ≥ f (p) which is a contradiction. We have shown that there exists n 0 such that for n ≥ n 0 , Y (p) ∩ X(supp (p n )) = ∅. If x ∈ Y (p) and n ≥ n 0 , then x / ∈ X(supp p n ). So if u ∈ Φ(p) and n ≥ n 0 , there exists k ∈ supp p n \{supp p} such that u k < 0.

Lemma 12. Let F : [0, 1] ⇒ IR be a correspondence with non empty convex values and compact graph. If F (0) ⊂ {x ∈ IR, x < 0} and F (1) ⊂ {x ∈ IR, x > 0}, there exists t in (0, 1) such that 0 ∈ F (t).

Proof: The sets C + = {t ∈ [0, 1], F (t)∩IR + = ∅} and C -= {t ∈ [0, 1], F (t)∩ IR -= ∅} are closed because F is u.s.c

Existence of an equilibrium

For k = 2, 3, define δ k as the Dirac measure on the state k, and p -k as the conditional probability on K knowing the state is not k: δ 2 = (0, 1, 0), δ 3 = (0, 0, 1), p -2 = ( p 1 p 1 + p 3 , 0, p 3 p 1 + p 3

) and p -3 = ( p 1 p 1 + p 2 , p 2 p 1 + p 2 , 0).

Choose u 2 in Φ(δ 2 ), u 3 in Φ(δ 3 ), u 1,2 in Φ(p -3 ) and u 1,3 in Φ(p -2 ). These are vectors in IR 3 , and to simply notations we write: Here + means ≥ 0, andmeans < 0. We have u 2 2 ≥ 0, u 3 3 ≥ 0, c ≥ 0, u 2 1,2 ≥ 0, d ≥ 0 and u 3 1,3 ≥ 0 since for each p and u ∈ Φ(p), we have u k ≥ 0 for each k ∈ supp p. The subset {2, 3} is not in T , this gives u 3 2 < 0, u 2 3 < 0, u 3 1,2 < 0 and u 2 1,3 < 0. Suppose a ≤ d. Then a simple equilibrium exists. Player 1 uses the partition {{2}, {1, 3}} to communicate: he sends the message m = 2 if the state is 2, and the message m = {1, 3} if the state is 1 or 3. Player 2 proposes x 2 in Y (δ 2 ) such that u 2 = (U k (x 2 )) k∈K after receiving m = 2, and proposes The initial probability p is on the segment [δ 2 , p -2 ], and also on the segment [δ 3 , p -3 ]. For t ∈ [0, 1], define q t = tδ 2 + (1t)p -3 and q ′ t in [p -2 , δ 3 ] such that p belongs to the segment [q t , q ′ t ]. q ′ t is uniquely defined for each t, q ′ 0 = δ 3 and q ′ 1 = p -2 . We are going to construct an equilibrium with posteriors q t and q ′ t for some appropriate t. We need player 1 of type 1 to be indifferent between splitting to q t and q ′ t .

δ 2 δ 3 δ 1 p -3 p -2 q t q ′ t p 31
Define the correspondence F : [0, 1] ⇒ IR, with for each t in [0, 1]:

F (t) = {u 1 -v 1 , u ∈ Φ(q t ), v ∈ Φ(q ′ t )}.
F clearly has non empty convex compact values. We have seen that the restrictions of Φ to the segments [p -3 , δ 2 ] and [p -2 , δ 3 ] have closed graphs, moreover q t and q ′ t are continuous in t, hence F has a closed graph. F (0) = {u 1v 1 , u ∈ Φ(p -3 ), v ∈ Φ(δ 3 )}. If F (0) ∩ IR + = ∅, there exists a pure equilibrium where player 1 uses the partition {{3}, {1, 2}}, so we assume that F (0) is a subset of {x ∈ IR, x < 0}. Similarly, we assume that F (1) = {u 1v 1 , u ∈ Φ(δ 2 ), v ∈ Φ(p -2 )} is a subset of {x ∈ IR, x > 0} (otherwise there exists an equilibrium where player 1 uses the partition {{2}, {1, 3}}). Then by Lemma 12 we can find t * in [0, 1] such that 0 ∈ F (t * ).

We can now conclude the proof. We can find x in Y (q t * ), y in Y (q ′ t * ), u = (U k (x)) k ∈ Φ(q t * ) and u ′ = (U k (y)) k ∈ Φ(q ′ t * ) such that for some e ≥ 0:

u =   e + -   , u ′ =   e - +  
We have an equilibrium as follows. Player 1 sends a message so as to induce the posteriors q t * and q ′ t * (type 2 sends the message 2, type 3 sends the message 3, and type 1 randomizes between the messages 2 and 3 so that the posteriors are q t * after m = 2 and q ′ t * after m = 3). Player 2 then proposes x at q t * , and y at q ′ t * . Player 2 is in best reply by construction. Type 1 of player 1 is indifferent. If type 2 of player 1 deviates and sends q ′ t * , player 2 will propose y and type 2 will reject it, having the reserve payoff of 0, which is not better than the payoff without deviating. Similarly, player 1 of type 3 has no profitable deviation, and we have an equilibrium.

with a = u 1 2 ,

 2 b = u 1 3 , c = u 1 1,2 and d = u 1 1,3 .

x 1 , 3

 13 in Y (p -2 ) such that u 1,3 = (U k (x 1,3 )) k∈K after receiving m = {1, 3}. By definition of Y (δ 2 ) and Y (p -2 ), player 2 is in best reply. And no type of player 1 has an incentive to deviate, so we have an equilibrium where player 1 plays pure. If we suppose b ≤ c, we have a similar equilibrium where player 1 uses the partition {{3}, {1, 2}}. From now on, we assume that a > d ≥ 0 and b > c ≥ 0. Then a ≥ 0. Consider any sequence (p n ) converging to the Dirac measure on state 2 such that supp p n = {1, 2} for each n. By Lemma 11 part b), we must have lim sup n f (p n ) ≥ f (p), and since f is u.s.c., lim sup n f (p n ) = f (p). This being true for any such sequence, f (p n ) ---→ n→∞ f (p). That is, the restriction of f to the set {p, supp p ⊂ {1, 2}} is continuous at δ 2 . And by Lemma 11 part a), the restriction of Φ to the segment [p -3 , δ 2 ] has a closed graph. Similarly we have b ≥ 0, and we can prove that the restriction of f to the set {p, supp p ⊂ {1, 3}} is continuous at δ 3 , and the restriction of Φ to the segment [p -2 , δ 3 ] has a closed graph.

  . Since F has non empty values, C + and C -are non empty, andC + ∪ C -= [0, 1]. By connexity of [0, 1], one can find t in both sets, that is such that F (t) intersects both IR + and IR -. Since F (t) is convex, it contains 0.

Our solution concept is basically subgame perfect Nash equilibrium. Except for the approval stage, our model behaves as a standard cheap talk game, in which Perfect Bayesian equilibrium is not restrictive.

This covers the particular case where the receiver has finitely many actions, over which he can randomize.

We do not make any assumption beyond the fact that there are finitely many types; in particular, types can be "multidimensional," with K ⊂ R n1 , for some n 1 ≥ 1.

We will indicate explicitly when X will be required to be convex.

This observation is made in Chen, Nartik and Sobel (2008), in their account of[START_REF] Matthews | Veto threats: Rhetoric in a bargaining game[END_REF].

Restriction to pure strategies of player 2 is justified by the fact that these will be enough to establish existence of equilibria in Γ.

The observation that the principal's ex ante expected utility is maximized when all types of the agent participate is also made in[START_REF] Bester | Contracting with imperfect commitment and the revelation principle: the single agent case[END_REF], footnote

Equilibrium in the case of three typesIn this section, we propose a thorough analysis of the equilibria of Γ when player 1 has three possible types (| K |= 3). Recall that T * denotes the participation structure of Γ (see[START_REF] Bester | Contracting with imperfect commitment and the revelation principle: the single agent case[END_REF] and (13)).When | K |= 3, there are three typical cases:1. T * is a partition of K.2. T* = {{1, 2} , {1, 3} , {2, 3}}. 3. T * = {{1, 3} , {2, 3}}.In case 1, by Proposition

[START_REF] Forges | Universal mechanisms[END_REF], Γ has a partitional equilibrium. Case 2 means that player 2 is able to obtain the approval of every pair of types but cannot ensure the participation of the three types simultaneously. In this case, as illustrated in Section 4.3, Γ may have no mixed equilibrium. We will show

Player 1's incentive compatibility condition reflects the fact that this player can veto player 2's offer, namely, can lie about his type and/or reject player 2's proposal. This is a "veto-incentive compatibility condition" (see, e.g., Forges (1999)), which implies posterior individual rationality.

The expression is similar to "veto-incentive compatibility" (see, e.g.,[START_REF] Forges | Ex post individually rational trading mechanisms[END_REF]) and implies posterior individual rationality.

The number of pure equilibrium payoffs is finite, in the same way as the number of partitions of K. Hence as soon as there is a pure equilibrium in Γ, there is an equilibrium achieving the highest expected payoff for the receiver.

gratefully acknowledges funding from ANR-3IA Artificial and Natural Intelligence Toulouse Institute, grant ANR-17-EUR-0010 (Investissements d'Avenir program) and ANR MaSDOL.

is p, then he has to propose a decision in Y (p), inducing a vector payoff in Φ(p) for the different types of player 1.

We will use in the sequel the following three lemmas (Lemma 12 is a simple mean-value theorem for correspondences).

Lemma 10. The mapping f is u.s.c. and convex.

If

Proof: Suppose p n ---→ n→∞ p. Then for n large enough, supp p n ⊃ supp p so

, and taking the supremum for x in X(supp p) we get λf (p

Then there exists n 0 such that for each u ∈ Φ(p) and n ≥ n 0 , one can find k ∈ supp p n \{supp p} such that u k < 0.

Proof: a) Without loss of generality we assume that

By taking a converging subsequence we can assume that x n converges to some x in X. Since p n ---→ n→∞ p, for n large enough supp p n ⊃ supp p so x ∈ X(supp p). And

Then x belongs to Y (p), and u ∈ Φ(p). b) Assume that lim sup n f (p n ) < f (p). We first claim that for n large enough, Y (p) ∩ X(supp (p n )) = ∅. Otherwise, we can find x in Y (p) ∩ X(supp p n )) for infinitely many n's, we have f (p) = k p k V k (x) and f (p n ) ≥


Pauline Sebillaud
email: p.sebillaud@gmail.com

Liu Xiaoxi

Wang Xingsheng

Xing Chunguang

Wang Xinsheng

Revealing a Wall with RTK -A Non-destructive Investigation of a Chinese Medieval Walled Site

Keywords: non-invasive method, survey, GPS, RTK, walled site, China, Jilin

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

In October and November 2013, a joint team from the Jilin Province Archaeology Institute and the Cultural Institute of the Nong'an 农安 District carried out a salvage excavation on a gas pipeline between Harbin 哈 尔 滨 and Shenyang 沈阳, in Northeast China, outside of the Kujindui 库金堆 site.

Located northeast of the town of Helong 合隆, west of the Yitong 伊通 River and north of its east-west running tributary, the Kujindui site is near the modern village of Chengzili 城 子 里 (Figure 1; Guojia Wenwuju 1993: 41). It is surrounded by a large number of contemporary medieval settlements, most of them attributed to the "Liao-Jin" period. The Liao 辽 Dynasty (AD 907-1125) ruled Mongolia (today both the Mongolia Republic and Inner Mongolia Province in the PRC), a part of eastern Russia, and Northeast China (often called Manchuria). The Jin 金 Dynasty (1115-1234) took control over this region during the 10th to 13th centuries. This whole period is often called "Liao-Jin" in Chinese history.

A walled settlement had been previously described at Kujindui as:

An irregular square, oriented 30°, the east wall measuring 345 m, the west wall 330 m, the south wall 360 m, and the north wall 340 m, for a total perimeter of 1,345 m. The wall is built of rammed earth, the foundation of the south wall is preserved to a height of 5 m, only a height of 1 m exists for the west wall, and only fragments can be seen of the north and east walls. On each side there are three evenly spaced observation terraces: these are not very clear on the northern and eastern sides, but very distinct on the southern and western sides. The terraces have a top diameter of 2.5 m and a 10 m diameter at their base. There is a gate at the centres of the east and west walls, of 20 m width. Each of the four corners had a tower; the northeast and south-west ones are relatively well preserved, 3 m across at the top, 25 m across at the bottom, and 3.5 m in height. The northwest and southeast towers are only preserved 1 m above the ground level. Outside of the south wall, on the east and west sides, there is a terrace linking to each tower, the side of each measuring 10 m at the top, and 20 m at the foundation. Nowadays the eastern one is not visible any more, but the western one is well preserved (JSWB 1987: 136). The outline of the wall is not very clear on satellite photographs (Figure 2). Using the concepts of isoaxiality and isotopy we can make some hypotheses concerning the outline of the wall. Isoaxiality designates the fact that two elements, one ancient and vanished, and the other still visible in the planimetric patterns, should line up (Chouquer 2007: 259). Isotopy describes the fact that a lot of the landscape's natural or man-made boundaries remain at or around the same location (Chouquer 2000: 147).

Pedestrian survey showed that the remains of the wall are also not very visible on the ground surface, except for the south wall (Figure 3). The site might have been built on the edge of a natural terrace above a pre-existing marsh south of the site. One of the reasons for the choice of this location might have been due to its location on the edge of a terrace on the side of the largest ancient river bed belonging to the east-west tributary stream of the Yitong River. The longest previous description of the site fits into just one short page, and it is difficult to precisely locate the wall, as it has never been mapped or drawn before. Buried archaeological structures are often reflected in the overlying surface topography; the micro-reliefs are frequently a good sign of subsurface material. Could non-invasive topog-raphical survey give us more information about the remaining anomalies in the topography, thus revealing the outline of the medieval wall? Even if the landscape of corn fields is fairly open, the modern village of about 30 houses still makes it difficult to use a total station. In addition, one can only take a few hundred points a day with a total station, and would have to change measurement stations many times. Even if the error could have been minimal (a few centimetres), this method was not ideal for the conditions.

A location and altitude survey was carried out with the help of a RTK (Real Time Kinematic), which is a geopositioning technique that requires two specialized GPS receivers; one is set as a base station, so it can send the correction message to the moving receiver. Both receivers collect extra data from the GPS satellites. RTK permits maintaining centimetrelevel accuracy. This technique has been quite frequently used on non-invasive archaeological survey [START_REF] Bogacki | GPS RTK mapping, kite aerial photogrammetry, geophysical survey and GIS based analysis of surface artifact distribution at the pre-Hispanic site of the Castillo de Huarmey, North Coast of Peru[END_REF], and its use is developing in China (Wang 2009; Duan 2010).

RTK (GeoSun Inc., eFix R1) field survey benefited from the Geopac application used to mark out and to register the points with WGS84 projection of the UTM grid coordinates. After a day of pedestrian survey and preparation, 1221 points were taken in one day of fieldwork, in order to keep the highest level of accuracy. The DXF files were then processed with the CASS program and AutoCAD. Geophysical maps were produced with the Surfer 8.0 application.

Processing of the survey's results generated a contour plan for the entire site (Figure 4). The topography of the survey area is of very low relief with a maximum vertical relief of approximately 12 m (167 to 179 m) in the 19.62 ha of the survey boundary. The surface area surrounded by the wall is 7.95 ha. The wall is oriented 44° west, the east side is roughly 352 m long, the south side 228 m, the west side 305 m and the north side 253 m, for a total perimeter of 1140 m.

From the results of this survey, it is clear that the shape of the site today is different to that at the date of the national antiquity survey in the 1980's when the site was first described. The erosion is due to the agricultural ploughing of the fields for at least the past 60 years, particularly intense on the west side. But a contour map (Figure 5) and a raster 3D surface digital model with custom light positions and shading (Figure 6; Plate , Ⅴ 2) allow for further observations and hypotheses on the architectural features. They can be reported on a map compiling the 25 cm contour plan and satellite photographs of the site (Figure 7). The microrelief variations indicate the possible location for the corner towers, as well as two breaches on the west and east sides of the wall that could have been entrances and that were linked by a path that roughly follows a present day concrete road. The observation terraces are harder to locate, and might not have been evenly spaced. Three of them can be placed on the north and south walls, where the view is actually the widest.

From the distribution of the artifacts observed on the ground surface during the survey, pottery sherds are only distributed inside the city wall. Very few architectural materials are visible outside the outline of the wall, only next to the tombs on the west (bricks and sherds dug out for the construction of 20th century mound burials). Architectural materials were also visible outside the wall on the north side corner tower that could have collapsed both inward and outward. Thus, most of the inhabitants' activities might have taken place inside the enclosure, which had quite strong defensive features. In addition, inside Kujindui's wall there are only Liao-Jin and end of Qing (1644Qing ( -1911) ) to Republican period (early 20th century) diagnostic sherds. Was this settlement totally abandoned during the Ming 明 Dynasty (1368-1644)? It is possible that there was a continuation of the Liao-Jin material culture with very little exchange with Central China. Therefore, there was no import of characteristic Ming porcelain. The geolocated vector version of the Kujindui site map will be used as the basis for the site's analysis and for future research and preservation. By only investing two days of fieldwork, this method proved to be very efficient for this type of site. Thus, a settlement analysis of the hundreds of buried walled sites in the Manchuria Plain could greatly benefit from the development of the use of RTK topographical measurement.

Figure 1

 1 Figure 1 Location map of the Kujindui site and its environs in Jilin Province, northeast China.

Figure 2

 2 Figure 2 Satellite photograph of the Kujindui site and hypothesized wall outlines. Adapted from Google Earth.

Figure 3

 3 Figure 3 Ground surface survey photographs of the Kujindui wall: (top) east wall south section and (bottom) south wall middle section.

Figure 4

 4 Figure 4 Topographical map of the Kujindui site.

Figure 5

 5 Figure 5 Contour map of the Kujindui site.

Figure 6 Figure 7

 67 Figure 6 Raster 3D surface digital model.

Acknowledgments: We would like to thank Jilin Province

Archaeology Institute and its director Song Yubin 宋玉彬, and Prof. Wang Lixin 王立新 of the Research Centre for Chinese Frontier Archaeology of Jilin University for providing the occasion and equipment for this fieldwork. We are also grateful to James Williams for his helpful comments on a previous version of this paper.


Pauline Sebillaud

Xiaoxi Liu

Lixin Wang

The first salt production workshop discovered in the Manchuria Plain (Yinjiawopu site, Jilin, China)

Introduction

The heart of the Manchuria Plain is currently undergoing a dramatic desertification process. The environment is a patchwork of alkaline soil, cultivation zones and ancient marshes and lakes. Research on its past is vital for understanding both the ancient exploitation of marginal environments and the practices that are shaping the modern landscape (Figure 1). During 2012, a systematic regional survey in Da'an Municipality, Jilin Province, as part of an archaeological project around the Houtaomuga site (Wang et al. 2013), located a large quantity of Liao-Jin period (ninth-thirteenth century AD) sherds on the surface associated with mounds that constitute topographical anomalies in the flat landscape of Da'an (Figure 1). The site, Yinjiawopu, was once located on a peninsula in the estuary of an ancient river as it entered Xinhuang Lake, now the border between the cultivated land and alkaline soil zones (Figure 2). This region is composed of calcic chernozems, black aeolian sediments on thick layers of fine clay, which are characteristic of a dry continental climate with very cold winters and hot summers (FAOUN 2006). Today, the area surrounds a shallow body of water and as such has a high water table; it is also subject to evaporation and dramatic erosion. Thus, the soluble salts in the soil rise by capillary action and accumulate on the surface. Although salinisation is a dangerous desertification process, people still choose to settle in this region (Figure 3).

The salt production workshop

In 2014, an area of 300m² was excavated on Mound 6, revealing: one burial, two hearths, two channels, three water tank pits, fifteen refuse pits and five leaching pits.

Stratigraphically, the most recent feature is the simple shaft-pit burial of a man in a prone position (Figure 4). The bronze figurine discovered around his neck is characteristic of the Jin Dynasty (twelfth-thirteenth century AD). This provides a terminus ante quem for the chronology of all of the other excavated features that are best interpreted as a salt production workshop. Five leaching pits (LK1-5) share the same basic elements: a rectangular semi-subterranean pit built and lined with impermeable clay; wooden sticks laid across the bottom provide support for a filter comprised of a pierced wooden board, reeds and charcoal; a channel running through the centre of this shallow pit drains into a deeper rectangular one; the opening of this latter pit is protected by a wooden frame and is lined with wooden boards on the sides and a mat at the base (Figure 5). This rectangular pit receives brine from the filtration process, which is later boiled to obtain salt. Two channels (20-40 cm wide) were discovered in connection with these pits, as well as a number of pits square in plan with rounded corners used as water tanks. Wooden boards placed on the ground between the leaching pit and the water tanks served as a platform for workers to stand on.

The profile of the associated mound (Figure 6) contains a sequence of discrete soil dumps. The saline soil, once filtered through a leaching pit, was thrown to the side, forming an accumulation of fine clay mixed with tiny charcoal particles and a very small quantity of brick, tile and pottery fragments. Refuse pits were subsequently dug into the mound.

Discussion

The fragmentation ratio of the ceramic material recovered in the excavated features and mounds suggests that large containers used in the salt production process were rendered more friable by their prolonged contact with brine. The Tiangong kaiwu (Song 1637), a Chinese encyclopaedia from the seventeenth century, and the Aobotu (Tora 1993), a treaty of the fourteenth century on the boiling of seawater, describe leaching pits and other features similar to those excavated at Yinjiawopu. An ethnographical comparison with the filtration techniques of salty sand used by people in Niger (Gouletquer et al. 1994) or Benin [START_REF] Delbos | Production du sel marin sur les rives lagunaires du Bé nin et en mangrove de Guiné e[END_REF] supports the identification of the Yinjiawopu site as a salt production workshop. Distinct from the drilling of salt-water wells in Sichuan (Flad 2011) and sea-salt production in Shandong (Wang 2012), the salt at Yinjiawopu was obtained by scraping saline soil from the surface, which was then washed and filtered, and the brine boiled. We hope that future excavations will provide evidence for the large hearths used for heating the brine. Further, all the conditions at the site are ideal for the organisation of an experimental archaeology project to investigate each step of the salt production process.

Figure 1 .

 1 Figure 1. Map of the Manchuria Plain, showing the location of the Yinjiawopu site.

Figure 2 .

 2 Figure 2. Satellite photograph of Yinjiawopu, with the site boundaries determined by the survey shown in blue, the mounds measured with RTK in red, and the excavated area shown in green (© Pauline Sebillaud & Xiaoxi Liu).

Figure 3 .

 3 Figure 3. Saline soil surface (© Pauline Sebillaud & Xiaoxi Liu).

Figure 4 .

 4 Figure 4. Burial n.1 and ornaments (© Pauline Sebillaud & Xiaoxi Liu).

Figure 5 .

 5 Figure 5. Leaching pit n.1 with accompanying views of the detail (© Pauline Sebillaud & Xiaoxi Liu)

Figure 6 .

 6 Figure 6. Profile of mound n.6 (© Pauline Sebillaud & Xiaoxi Liu)

  FAOUN. Food and Agriculture Organization of the United Nations. 2006. World reference base for soil resources. Rome: ONU. FLAD, R. 2011. Salt production and social hierarchy in ancient China. Cambridge: Cambridge University Press. http://dx.doi.org/10.1017/CBO9780511921094 GOULETQUER, P., D. KLEINMANN & O. WELLER. 1994. Sels et techniques, in M.-Y. Daire (ed.) Le sel gaulois: 123-61. Saint-Malo: Dossier du Ce.R.A.A. SONG, YINGXING. 1637. The exploitation of the works of nature (Tiangong Kaiwu). World Digital Library (in Chinese). Available at: http://www.wdl.org/en/item/3021/ (accessed 23 December 2014) TORA, YOSHIDA. 1993. Salt production techniques in ancient China: the Aobo Tu. Translated by Hans Ulrich Vogel. Leiden: Brill. WANG, L., D.F. HUO, J.J. ZHAO & X.X. LIU. 2013. The Neolithic site of Houtaomuga, Da'an, Jilin, in Guojia Wenwuju (ed.) 2012 Important Archaeological Discoveries: 2-7. Beijing: Cultural Relics Publishing House. WANG, Q. 2012. Review and perspective on salt archaeology in Shandong. Huaxia archaeology 4: 59-69.

Acknowledgements

As part of the Houtaomuga project, the authors express their gratitude to the Research Center for Chinese Frontier Archaeology at Jilin University and to the Jilin Province Archaeological Institute for their financial and scientific support.