Frédéric Koessler
email: koessler@hec.fr

Marie Laclau
email: laclau@hec.fr

Tristan Tomala
email: tomala@hec.fr

A belief-based approach to signaling *

Keywords: belief-based approach, cheap talk, information transmission, incomplete information, signaling

In this paper, we provide a geometric characterization of the set of interim equilibrium payoff vectors in a general class of signaling games. To obtain a tractable characterization, we use the belief based approach found in the literature on repeated games with incomplete information, cheap talk and Bayesian persuasion. This approach avoids to specify the prior, the strategies of the sender and receiver, and the belief system. The key ingredient is to consider Bayes-plausible belief distributions that are incentive-compatible for the sender. Geometrically, this leads to a constrained convexification of the graphs of the non-revealing interim payoff correspondences. Our characterization extends the analogous result for sender-receiver cheap talk games. We illustrate the results with some classical signaling games. We derive the best equilibrium payoff for the sender when his preferences are type-independent. For zero-sum preferences, we obtain an explicit formula for the ex-ante equilibrium payoff and establish a simple condition for the uniqueness of interim equilibrium payoffs.

Introduction

Signaling games represent an archetypical class of strategic and dynamic interactions involving one party possessing private information (the sender) and one uninformed decision-maker (the receiver) observing a costly signal from the sender. These games exhibit their relevance across numerous domains in economics, including instances like job market signaling, where candidates communicate their qualifications to potential employers; reputation analysis, where individuals build and manage their reputations in various social and professional contexts; and within financial markets, where firms signal their financial well-being to entice investors. Additionally, signaling games play a critical role in comprehending advertising strategies within the business realm, where companies use signals to convey product quality to consumers. Beyond economics, signaling games have also found an application in biology, notably exemplified by the handicap principle, where individuals signal their fitness through costly traits.

In this paper, we provide a geometric characterization of (perfect Bayesian) interim equilibrium payoffs for a general class of signaling games. We consider a finite set of types for the sender, but apart from that, we do not make specific assumptions on utility functions, such as single-peakness or single-crossing, and we do not assume that actions, signals, or types are one-dimensional. Within 1 this framework, the sender can utilize cheap talk messages in addition to costly signals. We employ a tractable belief-based approach, similar to that used in the literature on repeated games with incomplete information, cheap talk games, and Bayesian persuasion. By using this approach, we eliminate the need to specify the strategies of the sender and receiver, as well as the associated belief system.

Unlike models of cheap talk and Bayesian persuasion, information is conveyed through signals with payoff consequences, so the sender's interim payoff depends not only on the induced belief of the receiver, but also on the specific signal. Considering the collection of correspondences of non-revealing interim payoffs indexed by the sender's signal, a splitting of the prior belief (i.e., a Bayes-plausible distribution of beliefs) must associate each posterior belief with one of these payoff correspondences. Geometrically, this implies that the set of attainable interim payoff vectors of the sender lies in the convex hull of the graphs of those correspondences. Taking into account the sender's equilibrium conditions, our method is based on incentive-compatible splittings of the prior type distribution, resulting in a constrained convexification of the graphs of the non-revealing payoff correspondences. Notably, our characterization extends the existing equilibrium description for sender-receiver cheap talk games, regardless of any assumptions about the sender's preferences such as state-independent payoffs.

Our characterization implies that the set of equilibrium payoff vectors does not depend on the set of cheap talk messages, provided it is at least as large as the set of types of the sender: we show that regardless of the sizes of the action, signal and message sets, any equilibrium payoff vector of the sender can be obtained with a sender's strategy which utilizes no more pairs of signals and messages, than the number of types. We deduce the existence of a perfect Bayesian equilibrium in signaling games with such sets of cheap talk messages. Then, we consider two applications, a simplified poker game and the classical job market signaling game of [START_REF] Spence | Job Market Signaling[END_REF]. Our geometric characterization enables the explicit calculation of all equilibrium payoffs for all possible priors, including for partially revealing equilibria. Finally, we consider two special cases. First, in signaling games with transparent motives [START_REF] Lipnowski | Cheap talk with transparent motives[END_REF] where the sender's utility is type independent, we characterize the maximal equilibrium payoff of the sender via the quasi-concave enveloppe of the best non-revealing sender's payoff. Second, for zero-sum signaling games, we characterize the ex-ante equilibrium payoff through the concave closure of the maxmin value of the non-revealing game; we show that the interim equilibrium payoff vector is unique when this concave closure is differentiable. Furthermore, in this case, cheap talk messages can be fully dispensed with.

Related literature

The literature on signaling games, initiated by [START_REF] Spence | Job Market Signaling[END_REF] in economics and by [START_REF] Zahavi | Mate selection -A selection for a handicap[END_REF] and [START_REF] Grafen | Biological Signals as Handicaps[END_REF] in biology, has led to significant research in applied work across various fields and in the analysis of equilibrium refinements in games. For a literature review, see, e.g., [START_REF] Kreps | Signalling[END_REF] and [START_REF] Sobel | Signaling games[END_REF].

Our belief-based approach to signaling games shares significant connections with the approach employed in the literature on repeated games with incomplete information, cheap talk games, and Bayesian persuasion. Roughly, the idea revolves around viewing the informed player's strategy as a means to split the prior belief of the uninformed player into a convex combination of posteriors beliefs. These splittings enable the informed player to generate new payoffs within the convex hull of the graphs of the non-revealing payoff correspondences. For a comprehensive review and comparison of how this technique is applied across these areas, refer to Forges (2020).

In the context of zero-sum repeated games where only one player possesses private information, Aumann andMaschler (1966, 1995) and [START_REF] Ponssard | Zero-sum sequential games with incomplete information[END_REF] showed that the equilibrium payoff for the informed player is given by the concave closure (or concavification) of the minmax value of the one-shot game with no information and common belief, evaluated at the prior. The minmax theorem for those zero-sum games implies that the informed player's commitment to his strategy is irrelevant.

The approach can be applied to one-shot non-zero-sum scenarios when the informed player is able to commit to an information disclosure strategy. For instance, in the Bayesian persuasion setting introduced by [START_REF] Kamenica | Bayesian Persuasion[END_REF], the sender commits to his messaging strategy before learning the state. The maximum ex-ante expected payoff he can obtain is given by the concave closure cav w(p 0 ), where w(p) is the highest ex-ante expected payoff for the sender when the uninformed receiver selects an optimal action based on belief p. [START_REF] Hart | Nonzero-Sum Two-Person Repeated Games with Incomplete Information[END_REF] expands upon Aumann and Maschler's analysis to encompass non-zero-sum repeated games and provides a geometric characterization of all Nash equilibrium payoffs in undiscounted games. Since no commitment is assumed, the characterization needs to account for the informed player's incentive-compatible conditions, and it keeps track of the vector of interim payoffs for each type of the informed player. These ideas have been adapted and extended by [START_REF] Forges | Note on Nash equilibria in repeated games with incomplete information[END_REF]Forges ( , 1990) ) and [START_REF] Aumann | Long Cheap Talk[END_REF] for characterizing equilibrium payoffs in cheap talk games where both players send messages over multiple periods before taking action with payoff consequences. Similar techniques have been employed by [START_REF] Chakraborty | Persuasion by Cheap Talk[END_REF] and [START_REF] Lipnowski | Cheap talk with transparent motives[END_REF] to characterize equilibrium payoffs in cheap talk games with transparent motives.

Unlike the cheap talk literature, our focus centers on signals that are payoff-relevant either for the sender, for the receiver, or both. Technically, this implies that the sender can split the prior belief and the corresponding payoff vector on multiple graphs of non-revealing payoff correspondences, one for each signal. In cases where all signals of the sender are payoff-irrelevant, the characterization reduces to the one mentioned above for sender-receiver cheap talk games. Our characterization is also related to the one for disclosure games with hard information in [START_REF] Forges | Communication Equilibria with Partially Verifiable Types[END_REF], by considering signals that are payoff-irrelevant for the receiver, and either payoff-irrelevant or very costly for the sender, depending on his type.

Lastly, our characterization is related to the recent contribution by [START_REF] Boleslavsky | Signaling With Commitment[END_REF], who extend the Bayesian persuasion model to include payoff-relevant signals. They characterize the maximal ex-ante expected payoff the sender can achieve in signaling games in which the sender ex-ante commit to his signaling strategy. Like our work, their approach revolves around the graphs of non-revealing payoff correspondences for each signal. However, because we do not assume commitment, in line with the cheap talk literature we keep track of interim payoff vectors and maintain incentivecompatible conditions for the sender.

Model

Setup There are two players, a sender and a receiver. The nonempty and finite set of types of the sender is denoted by T . The sender is privately informed of his type, while the receiver is uninformed. The prior probability distribution is denoted by p 0 ∈ ∆(T ) and is assumed to have full support. The set of actions of the receiver is denoted by A.1 The sender has a set of costly signals S and a set of costless (cheap talk) messages M . We assume that A, S, and M are nonempty compact metric spaces. Except stated otherwise, we assume that M has at least |T | elements. Given an action a ∈ A, a signal s ∈ S, and a type t ∈ T , the utility of the sender is u(a, s, t), and the utility of the receiver is u R (a, s, t). The utilities u(a, s, t) and u R (a, s, t) are continuous on A × S × T . Utility functions are extended to the set of mixed actions ∆(A)2 in the usual way.

Signaling game

The timing of the signaling game is as follows:

1. The sender's type t is drawn from T according to p 0 .

2. The sender observes t ∈ T , chooses a signal s ∈ S and a cheap talk message m ∈ M .

3. The receiver observes the signal s and the cheap talk message m, then chooses an action a ∈ A.

A strategy for the sender is σ : T → ∆(S × M ), and a strategy for the receiver is a Borel measurable τ : S × M → ∆(A). The interim payoff vector for the sender induced by the strategy profile (σ, τ ) is given by v = (v t ) t∈T with v t = S×M u(τ (s, m), s, t)dσ(s, m | t), for every t ∈ T .3

A belief system is given by a Borel measurable µ : S × M → ∆(T ), where µ(t | s, m) is the belief assigned by the receiver to type t when he observes (s, m).

Perfect Bayesian equilibrium For every p ∈ ∆(T ) and s ∈ S, let Y (s, p) be the set of mixed actions that are optimal for the receiver given signal s and belief p. That is:

Y (s, p) := arg max y∈∆(A) u R (y, s, p),
where, with some abuse of notation, u R (y, s, p) = t p(t)u R (y, s, t) = t p(t) A u R (a, s, t)dy(a).

A strategy profile (σ, τ ), with payoff vector v = (v t ) t∈T , is a perfect Bayesian equilibrium (PBE) of the signaling game if there exists a belief system µ such that the following three conditions are satisfied. First, for each type, the sender cannot increase his interim expected payoff by deviating to any pair of signal and message. Second, for every pair of signal and message, the receiver chooses optimal actions given the signal and his belief about the sender's type. Finally, the receiver's belief is computed using Bayes' rule whenever possible. Formally: (i) Sequential rationality for the sender: for every t ∈ T and (s, m) ∈ S × M , v t ≥ u(τ (s, m), s, t);

(ii) Sequential rationality for the receiver: for every (s, m) ∈ S × M , τ (s, m) ∈ Y (s, µ(s, m));

(iii) Belief consistency: for every t ∈ T and every Borel set B ⊆ S × M ,

p 0 (t)σ(B | t) = t∈T p 0 ( t) B µ(t | s, m)dσ(s, m | t).
Such a triple (σ, τ, µ) is called a PBE assessment. We say that a payoff vector for the sender v ∈ R T is a PBE payoff vector if it is induced by some PBE strategy profile.

Geometric characterization of equilibrium payoffs

For every s ∈ S and p ∈ ∆(T ), let E(s, p) be the set of all feasible vectors of interim payoffs of the sender when the receiver chooses an optimal (mixed) action given s and p:

E(s, p) := v ∈ R T : ∃ y ∈ Y (s, p) s.t. v = (u(y, s, t)) t .
We call this the set of non-revealing payoffs for the sender at p given s.

Observe that E(s, p) depends on the signal s through two distinct channels: through Y (s, p), when the receiver's optimal action varies with s for a given posterior p, and through the direct impact of s on the utility of the sender for a given action of the receiver. We say that the signal s is cheap talk if it does not affect the utilities of the sender and of the receiver; in this case, E(s, p) is independent of s, as in [START_REF] Forges | Note on Nash equilibria in repeated games with incomplete information[END_REF]Forges ( , 1990Forges ( , 1994) ) and [START_REF] Aumann | Long Cheap Talk[END_REF].

We define then the modified set of non-revealing payoffs for the sender at p given s as:

E + (s, p) := v ∈ R T : ∃ y ∈ Y (s, p) s.t. ∀t, v t ≥ u(y, s, t) and v t = u(y, s, t) if p(t) > 0 .
To obtain the modified non-revealing payoff correspondence, consider for any p in ∆(T ), the nonrevealing payoffs given the signal s and, at the boundary of the simplex, adjust these payoffs by granting higher payoffs to the sender's types with vanishing probabilities. Notice that E(s, p) ⊆ E + (s, p), and E(s, p) = E + (s, p) if p has full support.

The graph of the modified non-revealing payoff correspondence of the sender given s is denoted by:

G s := gr E + (s, •) := (v, p) ∈ R T × ∆(T ) : v ∈ E + (s, p) .
We denote by G = s∈S G s the union of the graphs of the non-revealing payoff correspondences for all s, that is,

G = {(v, p) ∈ R T × ∆(T ) : ∃s ∈ S, y ∈ Y (s, p) s.t. ∀t, v t ≥ u(y, s, t) and v t = u(y, s, t) if p(t) > 0}.
Let co v (G) be the set of payoff vectors obtained by convexifying G with respect to p, when the vector payoff v is kept constant. Formally this is: Next, we introduce an individual rationality condition. Let INTIR s be the set of all payoff vectors v ∈ R T such that there exists p ∈ ∆(T ) and y ∈ Y (s, p) satisfying v t ≥ u(y, s, t) for every t ∈ T . Finally, let

co v (G) := (v, |T | k=1 λ k p k ) ∈ R T × ∆(T ) : (λ k ) |T | k=1 ∈ ∆({1, . . . , |T |}), (v, p k ) ∈ G ∀ k .
INTIR := s∈S INTIR s ,
be the set of all payoff vectors v ∈ R T such that for every s ∈ S there exists p ∈ ∆(T ) and y ∈ Y (s, p) such that v t ≥ u(y, s, t) for every t ∈ T . The payoff vectors in INTIR are called interim individually rational. 4 Observe that INTIR is directly obtained from G, this is the set of payoff vectors which are coordinate-wise above some point in the projection of G on the set of payoff vectors R T :

INTIR = proj R T (G) + R T + .
Our main result is that the interim individual rationality condition INTIR and the constrained convex hull of modified non-revealing payoffs co v (G), fully pin down the set of PBE payoff vectors. This is stated in the following theorem.

Theorem 1. The payoff vector v ∈ R T is a PBE payoff vector of the sender in the signaling game if and only if v ∈ INTIR and (v, p 0 ) ∈ co v (G).

Proof.

(i) From constrained convexification to equilibrium.

Let v ∈ INTIR be such that (v, p 0 ) ∈ co v (G), we construct a PBE assessment (σ, τ, µ) with payoff vector v, where σ has finite support. Since (v, p 0 ) ∈ co v (G), there exists K ∈ {1, . . . , |T |}, (λ k ) K k=1 ∈ ∆({1, . . . , K}), λ k > 0, s k ∈ S and y k ∈ Y (s k , p k )
for every k such that:

p 0 = K k=1 λ k p k , (1)
and for every k = 1, . . . K:

v t ≥ u(y k , s k , t) and v t = u(y k , s k , t) if p k (t) > 0. (2) Consider a profile of messages (m k ) k with m k ∈ M and m k = m k ′ for every k = k ′ , this is possible as |M | ≥ |T | ≥ K. Since v ∈ INTIR, for every s ∈ S, there exists q s ∈ ∆(T ) and ȳs ∈ Y (q s , s), (3)
such that:

v t ≥ u(ȳ s , s, t), for every t ∈ T . (4)
Consider the following strategies σ for the sender and τ for the receiver:

σ(s k , m k | t) = p k (t)λ k p 0 (t) , for every k, (5)
σ(s, m | t) = 0, if (s, m) = (s k , m k ) for every k, (6)
τ (s, m) = y k if (s, m) = (s k , m k ) for some k, ȳs otherwise. ( 7
)
Consider the following belief system:

µ(t | s, m) = p k if (s, m) = (s k , m k ) for some k, q s otherwise. ( 8
)
For the sender of type t, the interim payoff induced by (σ, τ ) is indeed v t because:

S×M u(τ (s, m), s, t)dσ(s, m | t) = K k=1 p k (t)λ k p 0 (t) u(y k , s k , t)
, by ( 5) and ( 7), = v t , by ( 2) and (1).

Then, sequential rationality for the sender (PBE condition (i)) follows directly from (2), (4), and ( 7), sequential rationality for the receiver (PBE condition (ii)) follows directly from (3), ( 7), and (8). Finally, to verify belief consistency (PBE condition (iii)), it remains to show that the probability of t conditional on (s k , m k ), given by ( 8), is equal to p k (t) for every k and t. From Bayes' rule and (5), this conditional probability is:

σ(s k , m k | t)p 0 (t) t∈T σ(s k , m k | t)p 0 ( t) = p k (t)λ k p 0 (t) p 0 (t) t∈T p k ( t)λ k p 0 ( t) p 0 ( t) = p k (t)λ k t∈T p k ( t)λ k = p k (t).
This completes the "if" part of the theorem.

(ii) From equilibrium to constrained convexification.

Consider a PBE assessment (σ, τ, µ) with payoff vector v. First, we show that v ∈ INTIR. For every s ∈ S, consider an arbitrary pair of signal and message (s, m) ∈ S × M and let y = τ (s, m).

Since τ is sequentially rational for the receiver, we have y ∈ Y (s, p) for p = µ(s, m) ∈ ∆(T ). Since σ is sequentially rational for the sender, we have v t ≥ u(y, s, t) for every t. We conclude that for every signal s,

we have v ∈ INTIR s , therefore v ∈ INTIR. Second, we show that (v, p 0 ) ∈ co v (G). Let λ(•) := t∈T p 0 (t)σ(•|t) ∈ ∆(S × M ) denote the marginal distribution on S × M induced by p 0 and σ. To prove that (v, p 0 ) ∈ co v (G), it suffices to show that there exists X ⊆ S × M with λ( X) = 1 such that (a) v ∈ E + (s, µ(s, m)) for every (s, m) ∈ X, and (b) p 0 ∈ co({µ(s, m) : (s, m) ∈ X}).
The PBE condition for the sender:

v t ≥ u(τ (s, m), s, t) ∀(s, m) ∈ S × M, (9) implies:
t∈T

(v t -u(τ (s, m), s, t)) µ(t | s, m) ≥ 0, for every (s, m) ∈ S × M. (10)
We also have:

E λ t (v t -u(τ (s, m), s, t)) µ(t | s, m) = S×M t (v t -u(τ (s, m), s, t)) µ(t | s, m)dλ(s, m) = t S×M (v t -u(τ (s, m), s, t)) µ(t | s, m)dλ(s, m) = t S×M (v t -u(τ (s, m), s, t)) p 0 (t)dσ(s, m | t) = t v t p 0 (t) S×M dσ(s, m | t) =1 - t p 0 (t) S×M u(τ (s, m), s, t)dσ(s, m | t) =vt = 0.
Hence, combined with (10), the previous equality implies that, λ-almost surely:

t (v t -u(τ (s, m), s, t)) µ(t | s, m) = 0.
Thus, from (9), there exists a Borel subset X ⊆ S × M with λ( X) = 1 such that:

v t = u(τ (s, m), s, t), for all (s, m) ∈ X and t ∈ T s.t. µ(t | s, m) > 0. ( 11
)
The PBE condition for the receiver implies:

τ (s, m) ∈ Y (s, µ(s, m)), for all (s, m) ∈ X. (12)
Putting ( 9), ( 11), and ( 12) together, we get v ∈ E + (s, µ(s, m)) for every (s, m) ∈ X, which proves (a). Now, from belief consistency with B = X we have:

p 0 = X µ(s, m)dλ(s, m).
Hence, property (b) directly follows from the following lemma.

Lemma 1. Let p 0 ∈ ∆(T ), σ : T → ∆(S×M ), and denote by λ(•)

:= t∈T p 0 (t)σ(•|t) ∈ ∆(S×M )
the marginal distribution on S × M induced by p 0 and σ. Let X be a Borel subset of S × M such that λ( X) = 1, and let µ : S × M → ∆(T ) be a Borel measurable function satisfying:

p 0 (t) = X µ(t | x)dλ(x). ( 13
)
Then,

p 0 ∈ co({µ(x) : x ∈ X}).
Notice that the conclusion that p 0 belongs to the convex hull of the beliefs {µ(x) : x ∈ X}, is stronger than requiring that it merely belongs to the closure of the convex hull. Proof. Let Z ⊆ X be a Borel subset of X such that λ(Z) = 1 and such that the dimension d(Z) of the convex hull of {µ(x) :

x ∈ Z} is minimal among the d(Z ′ ), for all Borel subsets Z ′ ⊆ X with λ(Z ′ ) = 1. We have: ∀t ∈ T, p 0 (t) = Z µ(t|x)dλ(x).
Suppose that p 0 / ∈ co({µ(x) : x ∈ Z}). From the separation theorem, there exists ψ ∈ R T such that:

∀x ∈ Z,

t p 0 (t)ψ t ≥ t µ(t|x)ψ t ,
where the separating vector ψ is not 0 and belongs to the d(Z)-dimensional affine hull of {µ(x) :

x ∈ Z}. Integrating this inequality gives:

t p 0 (t)ψ t ≥ Z t µ(t|x)ψ t dλ(x) = t Z µ(t|x)dλ(x)ψ t = t p 0 (t)ψ t .
It follows that t µ(t|x)ψ t = t p 0 (t)ψ t , λ-almost surely, that is:

λ x ∈ Z : t µ(t|x)ψ t = t p 0 (t)ψ t = 1. Therefore, Z ′ = {x ∈ Z : t µ(t|x)ψ t = t p 0 (t)ψ t } satisfies λ(Z ′ ) = 1 and d(Z ′ ) < d(Z)
which is a contradiction. We conclude that p 0 ∈ co({µ(x) : x ∈ Z}) and therefore p 0 ∈ co({µ(x) : x ∈ X}).

This completes the "only if" part of the theorem and ends the proof.

Consequences As illustrated in Section 4, this geometric characterization enables the explicit calculation of all equilibrium payoffs, for all possible priors, in various examples and applications. Furthermore, it has several interesting consequences. Corollary 1 illustrates a first direct consequence of Theorem 1: to characterize all PBE payoff vectors of the signaling game with enough cheap talk messages, it suffices to focus on the sender's strategies with finite support, of cardinality at most |T |, regardless of the sizes of the action, signal and message spaces.5 This result also implies that the set of PBE payoff vectors does not depend on the message space M , provided that M contains at least |T | elements.

Corollary 1. The payoff vector v ∈ R T is a PBE payoff vector of the sender in the signaling game if and only if v is a payoff vector of a PBE in which at most |T | pairs of signal and message are used with positive probability.

What about the existence of PBE? If the game is finite (i.e., both sets of signals S and of actions A are finite), then there exists at least one sequential equilibrium [START_REF] Kreps | Sequential Equilibria[END_REF], Proposition 1), and any sequential equilibrium is a PBE. Under the compactness and continuity assumptions of our model, [START_REF] Manelli | Cheap talk and sequential equilibria in signaling games[END_REF] shows that the signaling game has a PBE as long as the message space is large enough, precisely for M = ∆(A). 6 Corollary 1 shows that for any such PBE, the same payoff vector can be obtained through a PBE of the signaling game with any message set of cardinality at least |T |. Hence, due to our maintained assumption that M contains at least |T | elements, Corollary 2 in [START_REF] Manelli | Cheap talk and sequential equilibria in signaling games[END_REF] and Corollary 1 imply the existence of a PBE in our signaling game. Summarizing:

Corollary 2. The signaling game has at least one PBE and the set of payoff vectors:

v ∈ R T : v ∈ INTIR and (v, p 0 ) ∈ co v (G) , is non-empty.
A cheap talk game is a signaling game where signals are payoff-irrelevant, i.e., u(a, s, t) = u(a, s ′ , t) and u R (a, s, t) = u R (a, s ′ , t) for all signals s and s ′ . Theorem 1 characterizes PBE payoff vectors for a cheap talk game as a particular case. In such a game, we have G = G s for every s, and the condition v ∈ INTIR is irrelevant as it is straightforwardly implied by (v, p 0 ) ∈ co v (G). Therefore, we have the following corollary of Theorem 1.

Corollary 3. Assume that signals are payoff-irrelevant. The payoff vector v ∈ R T is a PBE payoff vector of the sender in the cheap talk game iff (v, p 0 ) ∈ co v (G).

Corollary 3 is well known in the cheap talk literature.7 Notice that in a cheap talk game, every nonrevealing payoff vector v ∈ E(s, p 0 ) is a "babbling" PBE payoff vector. However, in a signaling game with payoff-relevant messages, non-revealing payoff vectors may fail to be interim individually rational and thus a non-revealing PBE may not exist. That is, in a signaling game, the set E(s, p 0 ) ∩ INTIR may be empty for every s (see Section 4 for some examples). We deduce the following.

Corollary 4. The payoff vector v is a non-revealing PBE payoff vector if and only if there exists a signal s such that v ∈ E(s, p 0 ) ∩ INTIR. Thus, the signaling game has a non-revealing PBE if and only if

(∪ s E(s, p 0 )) ∩ INTIR = ∅.

Examples

Simple poker

Two risk-neutral individuals are engaged in a simplified poker game as follows. Initially, both players place 1 euro each on the table as the blind. Player 1 (the sender) randomly draws a card, which can be either high (t H ) or low (t L ), with probabilities p 0 (t H ) = p 0 and p 0 (t L ) = 1p 0 respectively. Only the sender has knowledge of the drawn card. Then, the sender chooses between multiple available signals: he can either fold (signal s = 0), in which case player 2 (the receiver) earns the 2 euros on the table, thereby concluding the game. Alternatively, the sender can opt to raise s times the blind, with s ∈ {1, 2, . . . , s}, necessitating an additional bet of s euros. The raise is bounded by s = s, corresponding to the "all in" scenario. Finally, it is the receiver's turn, who must choose between two options: he can fold (action a = f ), causing the sender to claim the 2 + s euros on the table; or the receiver can call (action a = c), which involves placing an additional bet of s euros to reveal the sender's card. If the card is high, the sender secures the 2 + 2s euros on the table; whereas if the card is low, the receiver claims the winnings.

Hence, the sender's (net) payoff as a function of the state t ∈ T = {t L , t H }, the signal s ∈ S = {0, 1, 2, . . . , s}, and the receiver's action a ∈ {f, c}, is given by:

u(a, 0, t) = -1, u(f, s, t) = 1, u(c, s, t H ) = 1 + s, and u(c, s, t L ) = -1 -s, for s > 0.
The payoff for the receiver is u R (a, s, t) = -u(a, s, t). For s > 0, the set of optimal mixed actions for the receiver as a function of p = p(t H ) ∈ [0, 1] is given by:

Y (s, p) =      {c} if p < 2+s 2+2s , ∆({c, f }) if p = 2+s 2+2s , {f } if p > 2+s 2+2s .
Observe that p(s) := 2+s 2+2s is strictly decreasing in s, with p(1) = 3 4 and lim s→∞ p(s) = 1 2 . For s = 0, the action of the receiver is irrelevant: Y (0, p) = ∆({c, f }) for every p.

The set of non-revealing payoffs for the sender at p given s > 0 is given by:

E(s, p) =      {(1 + s, -1 -s)} if p < p(s), co {(1 + s, -1 -s), (1, 1)} if p = p(s), {(1, 1)} if p > p(s),
and for s = 0, E(0, p) = {(-1, -1)}.

The interim individually rational payoff vectors are given by

INTIR 0 = {(v H , v L ) : v H ≥ -1, v L ≥ -1}, INTIR s = (v H , v L ) : v H ≥ 1, v L ≥ -1 -s, v L ≥ -2 -s s v H + 2 + 2s s .
Hence:

INTIR = s s=0 INTIR s = (v H , v L ) : v H ≥ 1, v L ≥ -1, v L ≥ -2 - s s v H + 2 + 2s s .
Figure 1 represents the projections of the graphs of E + (s, •) on the sender's vector payoffs for three possible signals: s = 0 (in green), s = 2 (in red), and s = s = 4 (in blue). The points at the north-east of these curves satisfy the INTIR condition, represented by the gray area. In the INTIR set, the curves intersect at (v H , v L ) = (1, 1) and (v H , v L ) = ( 73 , -1). The unique non-revealing payoff that belongs to INTIR is then (v H , v L ) = (1, 1), for every s > 0 and p 0 > p(s) = 2+s 2+2s . Indeed, the non-revealing payoff (v H , v L ) = (-1, -1) for s = 0 and p 0 ∈ (0, 1) does not belong to INTIR, so it is never a PBE payoff. Similarly, the non-revealing payoffs (v H , v L ) = (1 + s, -1s) for s > 0 and p 0 < p(s) = 2+s 2+2s do not belong to INTIR, so they are never PBE payoffs.

1 2 3 4 5 1 -2 -3 -4 -5 E + (0, p) E + (2, p) E + (4, p) p = 1 p = 0 p = 2 / 3 p = 3 / 5 NRE (p 0 > 3/5) PRE (p 0 < 3/5) v H v L Figure 1: Graphs of E + (s,
•) (projected on the space of the sender vector payoffs) in the simplified poker for s = 0 (in green), s = 2 (in red), and s = 4 (in blue).

Then, (v H , v L ) = ( 7 3 , -1) is the only candidate for an incentive compatible splitting. Such a partially revealing splitting, with beliefs p ′ = 0 (on the green curve) and p ′′ = p(4) = 3 5 (on the blue curve), is feasible for all priors p 0 ∈ (0, 3 5 ). In this equilibrium, the high type always raises s = s = 4 euros, and the low type randomizes between folding (s = 0) and raising s = 4 euros in such a way that the receiver's belief after seeing a raise of s = 4 euros is p = 3 5 (so the low type chooses s = 4 with probability 2p 0 3(1-p 0 ) , and s = 0 with the complementary probability). In this example where the raise is bounded by s = 4 euros, we conclude that the unique PBE payoff is (v H , v L ) = ( 73 , -1) if p 0 < 3 5 (represented by the orange circle PRE in the figure), and the unique PBE payoff is (v H , v L ) = (1, 1) if p 0 > 3 5 (represented by the orange circle NRE in the figure). The geometric characterization of equilibria easily extends to any bound s. If p 0 > p(s) = 2+s 2+2s , every PBE is non-revealing: the sender raises with the same amount s > 0, where p 0 > p(s) = 2+s 2+2s regardless of his type. The receiver always folds, and the unique PBE payoff is (v H , v L ) = (1, 1). If p 0 < p(s) = 2+s 2+2s , the PBE outcome is partially revealing and unique: the sender splits the prior p 0 into beliefs p ′ = 0 (with s = 0) and p ′′ = 2+s 2+2s (with s = s). The sender raises s euros when t = t H , and randomizes between s = s (with probability p 0 s

(1-p 0 )(2+s) ) and s = 0 (with the complementary probability) when t = t L . The corresponding payoff vector for the sender is ( 2+3s 2+s , -1). Notice that 2+3s 2+s is strictly increasing in s, so the value of the game is strictly increasing in s (and it only benefits the sender of type t H ), but it is bounded, with lim s→∞ 2+3s 2+s = 3. Therefore, in this simplified poker when p 0 < p(s) = 2+s 2+2s (and in particular, for p 0 = 1 2 ), it is never optimal to raise with an intermediate bet s ∈ (0, s): the informed player either folds (s = 0) or goes "all in" (s = s). The expected payoff of the informed player is strictly increasing in the amount s of chips the player has in his stack. In this example, the PBE outcomes are the same as the PBE outcomes of the signaling game without cheap talk.

Spence job market signaling

Consider the classical textbook job market signaling game (e.g., [START_REF] Mas-Colell | Microeconomic Theory[END_REF] where the sender is a worker with two possible types, T = {t L , t H }. Type t ∈ T corresponds to the worker's productivity and t H > t L ≥ 0. With some abuse of notation, we identify a probability distribution over T by the corresponding probability of t H , so p 0 = p 0 (t H ) ∈ (0, 1) denotes the prior probability that the worker's productivity is high. A signal s ≥ 0 is the amount of education obtained by the worker. The cost of obtaining education level s for a type t worker is given by a twice continuously differentiable function c(s, t), where c(0, t) = 0, c s (s, t) > 0, c t (s, t) < 0 for s > 0, c ss (s, t) > 0 and c st (s, t) < 0.

The receiver is an employer in a competitive market. His action is a wage offer y ≥ 0, and let's assume that a sequentially rational wage offer given belief p = p(t H ) is simply the expected productivity of the worker, y(p) = pt H + (1p)t L . Using the notations of our model, this can be obtained from the utility function of the receiver u R (y, s, t) = -(yt) 2 , whereby Y (s, p) = {y(p)} for every p and s. The utility function of the sender is given by: u(y, s, t) = f (y)c(s, t), where f (y) is continuous and strictly increasing. Without loss of generality, we normalize t H and f (1) to 1, and t L and f (0) to 0, so y(p) = p and f (y(p)) = f (p) is strictly increasing from 0 to 1.

It is immediate to show that INTIR

= R 2 + , E(s, p) = {(f (p) -c(s, t H ), f (p) -c(s, t L )} and: E + (s, p) =      E(s, p) if p ∈ (0, 1), E(s, 0) + R + × {0} if p = 0, E(s, 1) + {0} × R + if p = 1.
Observe that:

• co v (G s ) = G s for every s, which implies that the equilibrium payoffs are the same with or without cheap talk;

• Since |T | = 2, it is without loss of generality in terms of sender equilibrium payoffs to consider equilibria in which at most two signals are used (and, by the observation above, only one cheap talk message). So the equilibrium is either pooling on some signal s, or it uses two signals s and s with s > s.

Figure 2 represents the projections of the graphs of E + (s, •) on the sender's vector payoffs for three possible signals, s = 0, s = s > 0 and s = s > s. The points in the positive orthant (i.e., satisfying the INTIR condition, represented by the gray area) in which the graphs cross each other are represented by orange dots and are candidates for incentive-compatible splittings.

1 -1 1 -1 E + 0 E + s E + s p = 1 p = 0 NRE FRE PRE v H v L
Figure 2: Graphs of E + (s, •) (projected on the space of the sender vector payoffs, denoted E + s ) in Spence job market signaling for s = 0 (in green), s = s (in blue) and s = s (in red).

Non-revealing equilibria Vector v is a non-revealing (pooling) equilibrium payoff iff v is positive (the INTIR condition) and (v, p 0 ) ∈ E s (p 0 ) for some s, in which case signal s is sent with probability one by every sender's type. Hence, there is a non-revealing equilibrium on signal s if and only if f (p 0 )c(s, t L ) ≥ 0, which can be written as s ≤ sNR (p 0 ), where sNR (p 0 ) > 0 is strictly increasing in p 0 and is the unique solution of f (p 0 )c(s N R (p 0 ), t L ) = 0. In Figure 2, the non-revealing equilibrium payoff candidates for some prior p 0 are represented by orange dashed lines. On the figure, we see that: there is no non-revealing equilibrium with signal s = s regardless of the prior; there is a range of high enough priors with a non-revealing equilibrium with signal s; and there is a non-revealing equilibrium with signal s = 0 for all priors.

Fully-revealing equilibria Vector v is a fully-revealing equilibrium payoff iff v ∈ R 2 + , (v, 0) ∈ E + (s L , 0) and (v, 1) ∈ E + (s H , 1) for some pair of signals (s L , s H ). Such an equilibrium corresponds to a splitting of p 0 on p = 0 with signal s L and p = 1 with signal s H . In Figure 2, such a fully revealing payoff vector is a point where an horizontal line (corresponding to E + s L ) crosses a vertical line (corresponding to E + s H ). Such a point exists only for s H > s L , and to satisfy the INTIR condition we must have s L = 0 (because for other signals the horizontal line is not in the positive orthant). In the figure, there is a fully revealing equilibrium with s H = s (represented by the filled orange dot) but not with s H = s. More generally, such a fully revealing equilibrium exists if and only if:

E(s H , 1) ⊆ R + × R -, that is: f (1) -c(s H , t H ) ≥ 0 and f (1) -c(s H , t L ) ≤ 0,
which we can rewrite: c(s H , t H ) ≤ f (1) ≤ c(s H , t L ). The best fully revealing equilibrium for the sender is obtained with signal s H = s * H where c(s * H , t L ) = f (1). The worst fully revealing equilibrium for the sender is obtained with signal s H = s # H where c(s # H , t H ) = f (1). Hence, a fully revealing equilibrium always exists, the set of fully revealing equilibria is characterized by the splittings on p = 0 with signal s = 0 and p = 1 with signal s = s H for every

s H ∈ [s * H , s # H ],
and v is a fully revealing equilibrium payoff vector if and only if 0

≤ v H ≤ f (1) -c(s * H , t H ) and v L = 0.
Partially-revealing equilibria Vector v is a partially-revealing equilibrium payoff iff v ∈ R 2 + , (v, p ′ ) ∈ E + (s ′ , p ′ ) and (v, p ′′ ) ∈ E + (s ′′ , p ′′ ) for some pair of signals (s ′ , s ′′ ) and some pair of beliefs (p ′ , p ′′ ) with p ′ < p 0 < p ′′ and either p ′ > 0 or p ′′ < 1 (if p ′ = 0 and p ′′ = 1 we get a fully revealing equilibrium as characterized in the previous paragraph). Such equilibrium candidates are represented in Figure 2 by the orange circles. As can be observed in the figure, there is no such equilibrium with 0 < p ′ < p ′′ < 1 because every such point is either on an horizontal line (with belief p = 0) or a vertical line (with belief p = 1). This means that one sender type randomizes on two signals and the other type does not randomize. Contrary to what we observed in a fully revealing equilibrium, it is possible that the two signals sent with strictly positive probability are both strictly positive. In the figure, this corresponds to the point where the red curve crosses the blue curve in the positive orthant, with payoff vector v = (0.6, 0.2). Such an equilibrium exists for all high enough priors, and corresponds to a splitting on p ′ ∈ (0, p 0 ) with signal s ′ = s and on p ′′ = 1 with signal s ′′ = s , meaning that type t H randomizes between signal s and s and type t L sends signal s with probability 1. Two other partially revealing equilibria with similar properties appear in the figure with signals (s ′ , s ′′ ) = (0, s) and (s ′ , s ′′ ) = (0, s) and they exist for all high enough priors. An alternative type of partially revealing equilibrium is represented by the crossing point of the green and blue curve with v L = 0. Such an equilibrium exists for all low enough priors, and corresponds to a splitting on p ′ = 0 with signal s ′ = 0 and on p ′′ ∈ (p 0 , 1) with signal s ′′ = s, meaning that type t H sends signal s with probability 1 and type t L randomizes between signal s and 0.

Sender preferred equilibria It is clear from the analysis above that the best non-revealing equilibrium for the sender is the one using signal s = 0, i.e., the worker pools on no education. Such an equilibrium also dominates (at least weakly) any partially revealing equilibrium for both sender types, and gives payoff

v N R L = v N R H = f (p 0 ). The best fully revealing equilibrium gives payoff v F R L = 0 to type t L and payoff v F R H = f (1) -c(s * H , t H ) to type t H . Letting p * be the unique prior satisfying f (p * ) = f (1) -c(s *
H , t H ), we get that when p 0 > p * both sender types prefer the best non-revealing equilibrium, and when p 0 < p * there is a conflict between type t L and t H because type t L prefers the pooling equilibrium and type t H prefers the fully revealing equilibrium.

5 Further comments and results

Comparison with commitment

If the sender can commit ex-ante to his messaging strategy, then the incentive compatibility conditions for the sender, as well as the interim individual rationality condition, are no longer relevant, and only the sequential rationality of the receiver matters. Thus, an interim payoff vector v is feasible in the signaling game with commitment if and only if (v, p 0 ) ∈ co(G), and the maximum ex-ante expected utility of the sender is given by the concave closure cav max s∈S w(s, p), where w(s, p) = max y∈Y (s,p) t p(t)u(y, s, t) for every s, and cav is the concavification operator, i.e., cav f is the lowest (pointwise) concave function above f . When signals are payoff-irrelevant, or when |S| = 1, we directly obtain the characterization of [START_REF] Kamenica | Bayesian Persuasion[END_REF].

In the signaling game with commitment and without cheap talk messages, an interim payoff vector v is feasible if and only if (v, p 0 ) belongs to the join of the family of sets {G s : s ∈ S} which are convex combinations (v, p 0 ) = s λ s (v s , p s ) where each point (v s , p s ) belongs to G s . This latter setting is analyzed by [START_REF] Boleslavsky | Signaling With Commitment[END_REF].

Transparent motives

In this section, we consider a signaling game with transparent motives whereby the sender's utility is type independent, and we simply denote it by u : A × S → R. The interim equilibrium payoff of the sender is then the same regardless of his type and we denote it v ∈ R. With some abuse of notation, the set G can be written as:

G = {(v, p) ∈ R × ∆(T ) : ∃s ∈ S, y ∈ Y (s, p) s.t. v = u(y, s)}.
Then, our conditions reformulate as follows:

• (v, p 0 ) ∈ co v (G) if and only if for some K ∈ {1, . . . , |T |}, there exist (p k ) K k=1 and (s k ) K k=1 , with p k ∈ ∆(T ) and s k ∈ S for every k, such that p 0 ∈ co {p 1 , . . . , p K }, y k ∈ Y (s k , p k ) and v = u(y k , s k ) for every k;

• v ∈ INTIR if and only if v ≥ max s∈S min p∈∆(T ) min y∈Y (s,p) u(y, s).

Hence, from Theorem 1, v is a PBE payoff of the signaling game if and only if:

v ≥ max s∈S min p∈∆(T ) min y∈Y (s,p) u(y, s),
and there exists a family

(p k , s k , y k ) K k=1 such that p 0 ∈ co {(p k ) K k=1 }, and for all k, y k ∈ Y (s k , p k ) and v = u(y k , s k ).

Maximal equilibrium payoff

For every s ∈ S and p ∈ ∆(T ), the best non-revealing payoff for the sender at p given signal s is denoted by: w(s, p) := max y∈Y (s,p)

u(y, s).

For every p ∈ ∆(T ), let w(p) := max s∈S w(s, p) be the best non-revealing payoff at p for the sender.

For every function f : ∆(T ) → R, we denote by qcav f the smallest (pointwise) quasi-concave function above f (the quasi-concavification of f ).

Theorem 2. With transparent motives, the maximal PBE payoff of the sender in the signaling game is qcav w(p 0 ).

If we further assume that signals are payoff-irrelevant, then the signaling game is a cheap talk game with transparent motives as in [START_REF] Lipnowski | Cheap talk with transparent motives[END_REF] and the characterization of Theorem 2 coincides with their Theorem 2. The proof of Theorem 2 can be found in Appendix A.

Signaling without cheap talk

A signaling game is called without cheap talk if |M | = 1.
Theorem 3. Under transparent motives, if for each s, p → w(s, p) is quasi-concave, then every PBE payoff v ≥ w(p 0 ) of the signaling game is also a PBE payoff of the signaling game without cheap talk.

The proof of Theorem 3 can be found in Appendix B.

Zero-sum signaling games

In this section, we consider the particular case of zero-sum signaling games where u R (a, s, t) = -u(a, s, t) for all (a, s, t) ∈ A × S × T . Then:

Y (s, p) = arg min y∈∆(A)
u(y, s, p).

Let w(p) := max s∈S min y∈∆(A) u(y, s, p) be the maxmin value of the zero-sum signaling game where the sender does not know the type and p is the common prior. The concave closure of w(•) is:

cav w(p) = max k λ k w(p k ) : λ k ≥ 0, p k ∈ ∆(T ), k λ k = 1, k λ k p k = p .
We say that a splitting (λ k , p k ) k of p 0 is optimal if it achieves cav w(p 0 ) in this maximisation problem.

Theorem 4. Assume that the game is zero-sum.

1. The vector v ∈ R T is a PBE payoff if and only if:

t p 0 (t)v t = cav w(p 0 ) and ∀p ∈ ∆(T ), t p(t)v t ≥ w(p). (14)
Condition ( 14) holds if and only if v belongs to the super-differential of cav w(•) at p 0 . Thus, the equilibrium payoff vector is unique if and only if cav w is differentiable at p 0 .

In each equilibrium: the sender chooses an optimal splitting (λ k , p k ) k of p 0 , and for each k, a signal s k that achieves the maximum in w(p k ); for each signal s, the receiver chooses a mixed action y s such that for all t, u(y s , s, t) ≤ v t .

2. Any PBE payoff vector is also a PBE payoff in the signaling game without cheap talk.

The first point is a generalization of the results of [START_REF] Ponssard | Zero-sum sequential games with incomplete information[END_REF]. The proof of Theorem 4 can be found in Appendix C.

Example: simple poker

Consider a version of the poker example of Section 4.1 where the sender has only two signals s = 0 or s = 2. Players' payoffs as a function of the state t ∈ T = {t L , t H }, the signal s ∈ S = {0, 2}, and the receiver's action a ∈ {f, c} are given by the following tables:

s = 0 t H -1, 1 t L -1, 1 s = 2 f c t H 1, -1 3, -3 t L 1, -1 -3, 3
We have

Y (2, p) =      {c} if p < 2 3 , ∆({c, f }) if p = 2 3 , {f } if p > 2 3 . It follows that: w(p) = max -1, min{1, 3p -3(1 -p)} .
As seen on Figure 3, the concave closure cav w is differentiable at all points but p = 2 3 . Thus for p < 2 3 , the unique equilibrium payoff vector is (v H , v L ) = (2, -1), for p > 2 3 , the unique equilibrium payoff vector is (v H , v L ) = (1, 1). For p = 2 3 any vector

( 3-v L 2 , v L ) for v L ∈ [-1, 1] is an equilibrium payoff. p = p(t H ) w(p) s = 0 s = 2 1 1 1 2 -3 -1 -1 0 1 2 2 3 v L 3-v L 2 cav w(p)
Figure 3: Simplified poker with s = 0, 2.

A Proof of Theorem 2

To prove Theorem 2, we first establish the next proposition, which is an analogue of Theorem 1 in [START_REF] Lipnowski | Cheap talk with transparent motives[END_REF] for signaling games with transparent motives.

Proposition 1. Let (p k ) K k=1 be a set of posteriors with p k ∈ ∆(T ) for every k = 1, . . . , K such that p 0 ∈ co {p 1 , . . . , p K } and min k w(p k ) ≥ w(p 0 ). Then, min k w(p k ) is a PBE payoff of the signaling game with transparent motives.

Proof of Proposition 1. Suppose that p 0 ∈ co {p 1 , . . . , p K }. First, we argue that, for every j = 1, . . . , K and every α ∈ [0, 1]:

p 0 ∈ co {(1 -α)p 0 + αp j , p k , k = j}.
To see this, note that:

p 0 = k λ k p k = λ j p j + (1 -λ j ) k =j λ k 1 -λ j p k := λ j p j + (1 -λ j )p j .
Consider now q = (1α)p 0 + αp j and assume 0 < α < 1 (the cases α = 0 or 1 are clear). It is easy to check that:

p 0 = λ j λ j + α(1 -λ j ) q + α(1 -λ j ) λ j + α(1 -λ j ) pj .
Now, fix (p k ) k such that p 0 ∈ co {p 1 , . . . , p K } and w * = min k w(p k ) ≥ w(p 0 ). Suppose that for some j, w(p j ) > w * . Lemma 2. There exists α j ∈ [0, 1], s j ∈ S and y j ∈ Y (s j , (1α j )p 0 + α j p j ) such that u(s j , (1α j )p 0 + α j p j ) = w * .

Proof of Lemma 2. For q ∈ ∆(T ), denote v(q) = max s∈S min y∈Y (s,q) u(y, s) and W (q) = u(y, s) : s ∈ S, y ∈ Y (s, q), u(y, s) ≥ v(q) . Claim 1. The correspondence W is Kakutani (i.e., u.h.c. with non-empty, convex, compact values).

Proof of Claim 1. W (q) is non-empty because w(q) ∈ W (q). Verification of u.h.c. is routine: Take q n → q, s n → s, y n ∈ Y (s n , q n ), y n → y and for each s ′ , there exists y ′n ∈ Y (s ′ , q n ) such that u(y n , s n ) ≥ u(y ′n , s ′ ). Since u(•, •) is continuous, from the maximum theorem, Y (•, •) is u.h.c. and thus y ∈ Y (s, q). For each s ′ , take a converging subsequence to make sure that y ′n → y ′ and then y ′ ∈ Y (s ′ , q). So for each s ′ , there exists y ′ ∈ Y (s ′ , q) such that u(y, s) ≥ u(y ′ , s ′ ), i.e., u(y, s) ≥ v(q).

We prove now that W (q) is convex. Observe that W (q) = ∪ s∈S W s (q) with:

W s (q) = u(y, s) : y ∈ Y (s, q), u(y, s) ≥ v(q) .
More precisely, W (q) = ∪ s∈S,Ws(q) =∅ W s (q). For each s, Y (s, q) is a convex set of mixed actions and y → u(y, s) is linear on this set, thus W s (q) is a compact interval of R. To prove that the union of those intervals is convex, it is enough to show that any pair of them has a non-empty intersection. Thus, take s 1 , s 2 such that W s 1 (q) and W s 2 (q) are non-empty and suppose by contradiction that W s 1 (q)∩ W s 2 (q) = ∅. Then, one of those intervals, say W s 2 (q), is "above" the other, that is: max W s 1 (q) < min W s 2 (q); or equivalently: max{u(y, s 1 ) : y ∈ Y (s 1 , q), u(y, s 1 ) ≥ v(q) < min{u(y, s 2 ) : y ∈ Y (s 2 , q), u(y, s 2 ) ≥ v(q) .

Consider then the interval:

W ′ s 2 (q) = {u(y, s 2 ) : y ∈ Y (s 2 , q) . We have W s 2 (q) ⊆ W ′ s 2 (q) and min W ′ s 2 (q) ≤ v(q). Then, either max W s 1 (q) < min W ′ s 2 (q), thus max W s 1 (q) < v(q) which contradicts that W s 1 (q) is non-empty. Or max W s 1 (q) ≥ min W ′ s 2 (q), but then W s 1 (q) ∩ W ′ s 2 (q) = ∅. This is also a contradiction because W s 1 (q) ∩ W ′ s 2 (q) ⊆ W s 1 (q) ∩ W s 2 (q). This concludes the proof of Claim 1. Now, it follows that the correspondence α → W ((1α)p 0 + αp j ) from [0, 1] to R is also Kakutani. From Lemma 3 in Lipnowski and Ravid, its image is an interval. This proves Lemma 2 because w(p j ) ∈ W (p j ), w(p 0 ) ∈ W (p 0 ) and w * ∈ [w(p 0 ), w(p j )].

We conclude the proof of Proposition 1. For each j such that w(p j ) > w * , we replace p j by (1α j )p 0 + α j p j given by Lemma 2. Together with the family of s j , y j , this defines an equilibrium with payoff w * .

Consider now the maximal equilibrium payoff u * (p 0 ) of the sender at p 0 . Observe that:

u * (p 0 ) = max v ∈ R : ∃(p k ) k , p 0 ∈ co {(p k ) k } and ∀k, ∃s k ∈ S, ∃y k ∈ Y (s k , p k ), v = u(y k , s k ) .
The right-hand-side of this equality is greater or equal than w(p 0 ), by considering the non-revealing splitting (∀k, p k = p 0 ) and w(p 0 ) ≥ max s∈S min p,y∈Y (s,p) u(y, s). Thus the INTIR condition is superfluous in defining u * (p 0 ). Also, u * (p 0 ) ≥ w(p 0 ). We now derive two consequences of Proposition 1. The previous corollaries lead to the following proposition.

Proposition 2. The function p 0 → u * (p 0 ) is quasi-concave.

Proof of Proposition 2. We prove now that for any family (p

k ) k such that p 0 ∈ co {(p k ) k }, u * (p 0 ) ≥ min k u * (p k ).
For each k, let (p * kj , s * kj , y * kj ) j be an optimal family in the maximisation problem defining u

* (p k ). Since p 0 ∈ co {(p k ) k } and p k ∈ co {(p * kj ) j }, then p 0 ∈ co {(p * kj ) k,j }. Then from Corollary 5, u * (p 0 ) ≥ min k min j w(p * kj )
and from Corollary 6, min j w(p * kj ) = u * (p k ). Thus, u * is quasi-concave.

Proof of Theorem 2. We are now in position to prove the theorem. We know that the maximal equilibrium payoff u * (p 0 ) of the sender is a quasi-concave function of p 0 which is greater than or equal to w(p 0 ). We argue that this is the smallest such quasi-concave function: if f is quasi-concave and f (p 0 ) ≥ w(p 0 ) for all p 0 , then f (p 0 ) ≥ u * (p 0 ) for all p 0 . To see this, let (p * k , s * k , y * k ) k be an optimal family in the maximisation problem defining u * (p 0 ). From Corollary 6, u * (p 0 ) = min k w(p * k ). Then,

u * (p 0 ) = min k w(p * k ) ≤ min k f (p * k ) ≤ f (p 0 ),
since f is quasi-concave. This concludes the proof.

B Proof of Theorem 3

Proof. Let v ≥ w(p 0 ) be a PBE payoff of the signaling game. We construct a PBE with payoff v such that k, k ′ , p k = p k ′ and s k = s k ′ . In words, all the information transmitted by the equilibrium is contained in the signal: if two signals are the same, then the posterior beliefs are also the same.

Since v is a PBE payoff of the signaling game, there exists a finite family (p k , s k , y k ) K k=1 such that p 0 ∈ co {(p k ) K k=1 } and for all k, y k ∈ Y (s k , p k ) and v = u(y k , s k ). Since the statement is clear for v = w(p 0 ), take v > w(p 0 ). For each s ∈ S ′ = {s k : k = 1, . . . , K}, denote K s = {k : s k = s}. For some convex combination coefficients λ k > 0, we have: Since w s is quasi-concave, D s is convex, it is thus a non-empty convex and compact set (since w s is u.s.c). From the equilibrium condition, for all s ∈ S ′ and all k ∈ K s , u(y k , s k ) = v, thus p k ∈ D s . Since D s is convex, we also have p s ∈ D s for each s.

p 0 = k λ k p k = s∈S ′ λ s k∈Ks λ k λ s p k = s λ s p s , with λ s = k∈Ks λ k and p s = k∈Ks λ k λs p k , so p 0 ∈ co {(p s ) s∈S ′ }.
Thus for each s we have:

w s (p 0 ) ≤ w(p 0 ) < v ≤ w s (p s ).
Consider then for α ∈ [0, 1],

F s (α) = {u(y, s) : y ∈ Y (s, (1 -α)p 0 + αp s )}.
Similarly to the proof of Lemma 2, this correspondance is Kakutani, thus its image is an interval. It follows that there exists α s ∈ [0, 1] and y s ∈ Y (s, (1-α)p 0 +αp s ) such that v = u(y s , s). We modify then the equilibrium by replacing all the (y k , p k ) for k ∈ K s , by (y s , p s ). This gives the desired conclusion.

C Proof of Theorem 4

Proof.

(1) Let v ∈ R T be such that Condition (14) holds.

Consider an optimal splitting (λ k , p k ) k of p 0 such that cav w(p 0 ) = k λ k w(p k ) and let the sender choose a strategy that achieves this splitting. Precisely, there is a finite subset of signals and messages (s k , m k ) k and let σ : T → ∆(S × M ) be such that σ(s k , m k |t) = p k (t)λ k /p 0 (t) and s k achieves the maximum in w(p k ). For any strategy of the receiver playing action a k after observing (s k , m k ), the subsequent payoff u(a k , s k , p k ) is greater or equal to min a u(a, s k , p k ) and therefore it is greater or equal to max s min a u(a, s, p k ) = w(p k ) from optimality of s k . Thus, the ex-ante payoff is at least k λ k w(p k ) = cav w(p 0 ). This shows that the sender can guarantee cav w(p 0 ) which is greater or equal to the maxmin value. where the second equality follows from the minmax Theorem (if A is not finite, see [START_REF] Sion | On general minimax theorems[END_REF]). Therefore, for all s, there exists y s such that for all p, u(y s , s, p) ≤ t p(t)v t and in particular for all t, u(y s , s, t) ≤ v t . Consider the strategy of the receiver that plays y s after signal s. Since u(y s , s, t) ≤ v t for all t, the ex-ante payoff is no more than t p 0 (t)v t = cav w(p 0 ), for any strategy of the sender. Thus, cav w(p 0 ) is less or equal to the minmax value.

It follows that the minmax theorem holds and that the equilibria are given by the minmax and maxmin strategies described above.

Suppose now that v ∈ R T is an equilibrium payoff vector. Then from the arguments above, its ex-ante value t p 0 (t)v t has to be equal to cav w(p 0 ) since a zero-sum game has a unique ex-ante equilibrium payoff. Now, v must belong to INTIR, so we have: ∀s, ∃y, ∀t, v t ≥ u(y, s, t).

Therefore, for all s, there exists y such that for all p, t p(t)v t ≥ u(y, s, p). It follows that for each p, t p(t)v t ≥ max s min y u(y, s, p) = w(p). Hence Condition ( 14) is satisfied.

Notice that since t p(t)v t is a linear function of p, the inequalities ∀p ∈ ∆(T ), t p(t)v t ≥ w(p) are equivalent to ∀p ∈ ∆(T ), t p(t)v t ≥ cav w(p). Condition ( 14) is thus equivalent to v belonging to the super-differential of cav w at p 0 .

(2) Next, we prove that cheap talk messages are not necessary. Take an equilibrium of the game given by a family (λ k , p k , s k , y k ) k with cardinality K, and with payoff vector v = (v t ) t , and suppose that there are two indices j = ℓ such that s j = s ℓ := s, λ j > 0, λ ℓ > 0. We will merge j and ℓ together replacing p j and p ℓ by p = λ j λ j +λ ℓ p j + λ ℓ λ j +λ ℓ p ℓ . We will choose ȳ in such a way that the family:

{(λ k , p k , s k , y k ) k =j,ℓ } ∪ {(λ j + λ ℓ , p, s, ȳ)},
is an equilibrium with the same payoff vector v.

We start by oberving that w(p) must be linear on the segment [p j , p ℓ ], that is:

∀µ ∈ [0, 1], w(µp j + (1µ)p ℓ ) = µw(p j ) + (1µ)w(p ℓ ).

To see this, first assume that w(µp j + (1µ)p ℓ ) > µw(p j ) + (1µ)w(p ℓ ) for some µ ∈ [0, 1]. This contradicts the optimality of the splitting because one could replace p j and p ℓ by µp j + (1µ)p ℓ and improve the value of k λ k w(p k ). Thus, w(µp j + (1µ)p ℓ ) ≤ µw(p j ) + (1µ)w(p ℓ ). On the other hand:

w(µp j + (1µ)p ℓ ) = max The second inequality follows from concavity of the minimum of linear functions and the last equality from the optimality of s at p j and p ℓ . This implies that s is optimal all along the segment [p j , p ℓ ] and thus min a u(a, s, µp j + (1µ)p ℓ ) is an affine function of µ. From this, we deduce that the set of optimal actions: A * µ := arg min a u(a, s, µp j + (1µ)p ℓ ), is the same set A * for all µ ∈ (0, 1) and that all actions in A * are also optimal at p j and at p ℓ (µ = 0, 1). Now, from the equilibrium conditions, for all t such that p j (t) > 0, we have u(y j , s, t) ≥ u(y ℓ , s, t), and thus u(y j , s, p j ) ≥ u(y ℓ , s, p j ). Further, it must be the case that u(y j , s, t) = u(y ℓ , s, t) for all t such that p j (t) > 0. Otherwise, we would have that u(y j , s, t) > u(y ℓ , s, t) for some t such that p j (t) > 0, implying u(y j , s, p j ) > u(y ℓ , s, p j ) and contradicting the optimality of y j at p j . We conclude that: ∀t s.t. p j (t) > 0, u(y j , s, t) = u(y ℓ , s, t) = u(a, s, t), ∀a ∈ A * .

By a symmetrical argument: ∀t s.t. p ℓ (t) > 0, u(y ℓ , s, t) = u(y j , s, t) = u(a, s, t), ∀a ∈ A * .

The set of types t such that p j (t) > 0 or p ℓ (t) > 0 is precisely the set of types that send signals s j , s ℓ with positive probability. Since s j = s ℓ , the receiver can play some ȳ ∈ A * for both signals. This construction forms an equilibrium with a family of cardinality K -1 and the same vector of interim payoffs. Iterating the argument, we find that this payoff vector can be obtained with the property that if j = ℓ then s j = s ℓ .

  Since p lies in the (|T | -1)-dimensional simplex, only convex combination of cardinality at most |T | are considered, thanks to Carathéodory's theorem. A family (λ k , p k ) |T | k=1 such that p 0 = |T | k=1 λ k p k and (v, p k ) belongs to G for all k, will be called an incentive compatible splitting of p 0 .

Corollary 5 .

 5 For any family (p k ) k such that p 0 ∈ co {(p k ) k }, we have u * (p 0 ) ≥ min k w(p k ). Proof of Corollary 5. If min k w(p k ) ≤ w(p 0 ), this follows form u * (p 0 ) ≥ w(p 0 ). If min k w(p k ) > w(p 0 ), from Proposition 1, min k w(p k ) is an equilibrium payoff, thus less or equal to u * (p 0 ). Corollary 6. Let (p * k , s * k , y * k ) k be an optimal family in the maximisation problem defining u * (p 0 ). Then u * (p 0 ) = min k w(p * k ). Proof of Corollary 6. For each k, u * (p 0 ) = u(y * k , s * k ) ≤ w(p * k ), thus u * (p 0 ) ≤ min k w(p * k ). On the other hand, min k w(p * k ) in an equilibrium payoff from Proposition 1, thus it is less or equal to u * (p 0 ).

  For each signal s, denote: D s = {p ∈ ∆(T ) : ∃y ∈ Y (s, p), u(y, s) ≥ v} = {p ∈ ∆(T ) : w s (p) ≥ v}.

  s, µp j + (1µ)p ℓ ) ≥ min a u(a, s, µp j + (1µ)p ℓ ) ≥ µ min a u(a, s, p j ) + (1µ) min a u(a, s, p ℓ ) = µw(p j ) + (1µ)w(p ℓ ).

We can allow the set of actions of the receiver to depend on the signal, i.e., let A(s) denote the set of actions of the receiver given s.

For a metric compact X, ∆(X) denotes the set of Borel probability measures over X.

We focus on the interim payoff vectors of the sender for simplicity, the expected payoff of the receiver can be easily included in the analysis.

This individual rationality condition requires the receiver to choose a mixed action which is optimal for the receiver for some belief. A weaker condition requires that for every s ∈ S, there exists y ∈ ∆(A) such that vt ≥ u(y, s, t) for every t ∈ T . This weaker form is the relevant condition for the study of Nash equilibria instead of PBE of the signaling game.

Taking into account the receiver's ex ante expected payoff would require |T + 1| pairs of signal and message instead of |T |.

A PBE may not exist for continuous sets of signals if the message space is too small, see the examples in[START_REF] Manelli | Cheap talk and sequential equilibria in signaling games[END_REF].

The characterization of Corollary 3 is known for Nash equilibria of cheap talk games where PBE and Nash equilibrium payoffs coincide.

The authors gratefully acknowledge the support of the ANR through the program Investissements d'Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047).


Se ´bastien Hentz
email: sebastien.hentz@inpg.fr

Laurent Daudeville
email: laurent.daudeville@inpg.fr

Fre ´de ´ric

V Donze

Identification and Validation of a Discrete Element Model for Concrete

Keywords: Discrete elements, Concrete, Strain rate, Dynamic loads, Models

The use of a three-dimensional discrete element method ͑DEM͒ is proposed to study concrete structures submitted to dynamic loading. The aim of this paper is to validate the model first in the quasistatic domain, and second in dynamic compression, at the sample scale. A particular growing technique is used to set a densely packed assembly of arbitrarily sized spherical particles interacting together, representing concrete. An important difference from classical DEMs where only contact interactions are considered, is the use of an interaction range. First, the correct identification of parameters of the DEM model to simulate elastic and nonlinear deformation including damage and rupture is made through quasistatic uniaxial compression and tension tests. The influence of the packing is shown. The model produces a quantitative match of strength and deformation characteristics of concrete in terms of Young's modulus, Poisson's coefficient, and compressive and tensile strengths. Then, its validity is extended through dynamic tests. The simulations exhibit complex macroscopic behaviors of concrete, such as strain softening, fractures that arise from extensive microcracking throughout the assembly, and strain rate dependency.

Introduction

The design of concrete safety structures is a big challenge for engineers; for example, some structures present in mountainous areas are dedicated to protection against natural hazards, such as avalanches, rock falls, etc. and thus may be submitted to impact loads and high deformation. Despite their geometry which is usually massive, with an extremely high fraction of reinforcement, and of course a design satisfying usual building standards, some are found to be totally damaged. This inconsistency demands further investigation and understanding of the failure mechanisms.

Different approaches have been used to model fracture of brittle geomaterials, such as concrete. Some introduce damage in the constitutive relations of the material, and most often are used for engineering purposes, whereas others deal directly with numerous microscopic cracks; the latter are often used to understand the damage mechanisms of the materials.

The use of a three-dimensional ͑3D͒ distinct element method ͑DEM͒͑Cundall and Strack 1979 ;[START_REF] Cundall | Formulation of a three-dimensional distinct element model. I: Scheme to detect and represent contacts in a system composed of many polyhedral blocks[END_REF] is proposed here to study the fracture and the fragmentation of a complete concrete structure, which is now made feasible thanks to ever-increasing computing power. This method does not rely upon any assumption about where and how a crack or several cracks occur and propagate as the medium is naturally discontinuous, and is very well adapted to dynamic problems. Although numerous authors, ͑Meguro and Hakuno 1989; [START_REF] Potyondy | Modeling rock using bonded assemblies of circular particles[END_REF] have used similar two-dimensional approaches to model cohesive geomaterials, few have thus modeled concrete, and even fewer have modeled complete 3D structures. This requires some modifications of the usual DEM models. Unlike continuous methods, the parameters of the material behavior model are not defined at the macroscopic scale; in addition, access to physical interaction quantities, such as friction coefficient and stiffnesses is not allowed, unlike real granular materials, for which information at the grain scale may be obtained. Therefore, a calibration of the parameters at the local scale is needed. This paper deals first with the identification of the constitutive equations parameters of the discrete element model thanks to elementary tests, such as quasistatic uniaxial compression and tension tests. Second, once parameters are known, the validation of the model will be extended through split Hopkinson pressure bar ͑SHPB͒ tests.

Discrete Element Method Model Used

The present numerical model has been implemented within the ''spherical discrete element code'' ͑SDEC͒͑ Donze ´2000͒. It uses discrete spherical elements of individual radius and mass. These elements represent a polydisperse assembly with a size distribution obtained by using a particular growing technique ͑Donze 2002͒. Once the assembly has been set, pairs of initially interacting discrete elements are identified. These interactions have been chosen to represent, as well and as simply as possible, the elastic and cohesive nature of a certain class of geomaterials, such as concrete. To do this, elastic forces with a local rupture criterion are applied between interacting elements.

Using the constitutive equations for each interaction, the numerical model solves the equations of motion with an algorithm similar to those used in molecular dynamics ͑Allen and Tildesley 1987͒. The explicit time integration of the laws of motion will provide the new displacement and velocity for each discrete element.

As time proceeds during the evolution of the system, changes in the packing of discrete elements may occur and new interactions be created. One of the features of this numerical model will then be to determine the interacting neighbors of a given element. This will be achieved by defining an interaction range and identifying all elements within it which are interacting. Depending on the spatial distribution of the discrete elements, two different methods are available to identify the interacting neighbors. If the distribution is rather compact with little fluctuations in size, a grid subdivision method is used ͑Allen and Tildesley 1987, [START_REF] Magnier | Discrete element project, rapport d'activite ´s[END_REF]. If, however, the distribution is dispersive with large size fluctuations then the previous method will be costly and a spatial sorting method is used ͑Mu ¨ller 1996; O'Connor 1996; [START_REF] Magnier | Discrete element project, rapport d'activite ´s[END_REF].

Interaction Range

The overall behavior of a material can be reproduced by means of this model by associating a simple constitutive law to each interaction. An interaction between elements a and b of radius R a and R b respectively, is defined within an interaction range ␥ and does not necessarily imply that two elements are in contact. Then, these elements will interact if,

␥͑R a ϩR b ͒уD a,b (1)
where D a,b ϭdistance between the centroids of elements a and b and ␥у1. This is an important difference from classical DEMs which use spherical elements ͑Cundall and Strack 1979͒ where only contact interactions are considered ͑␥ϭ1͒. This choice was made so that the method could simulate materials other than simple granular materials in particular those which involve a matrix as found in concretes. To take into account the effects of this matrix which may cement two aggregates which are not themselves in contact, the interaction range ␥ is set to be greater than 1 when the assembly of elements is initially built. However, this long-range interaction is limited to nearest neighbors. The location of the interaction point is given by

x a,b ϭx a ϩ͑R a Ϫ0.5͑͑R a ϩR b ͒ϪD a,b ͒n (2)
where nϭunit vector pointing from element a to element b; and x a ϭposition vector of element a.

Interaction Forces

The interaction force vector F, which represents the action of element a on element b, may be decomposed into a normal and a shear vector F n and F s , respectively, so that,

FϭF n ϩF s (3)
where

F n ϭK n ͑ D eq a,b ϪD a,b ͒n (4)
where D eq a,b ϭequilibrium distance between the two elements a and b which was set when the interaction was created.

The normal force vector may be expressed such that it accounts for possible inelastic deformations.

The shear vector force F s is computed incrementally and was given by other authors ͑Hart et al. 1988͒. The incremental force is given by ⌬F s ϭϪK s ⌬U s

(5)

where ⌬U s ϭshear displacement vector increment between the locations of the interacting points of the two elements over a timestep ⌬t.

Elastic Properties

The strain energy stored in a given interaction cannot be assumed to be independent of the size of the interacting elements. Therefore, interaction stiffnesses are not identical over the sample, but follow a certain distribution. A ˜int

K n ͩ 0.825K n ϩ2.65K s 2.5K n ϩK s ͪ (6) ϭ K n ϪK s 2.5K n ϩK s (7)
where A ˜int ϭsurface where the interaction is defined, with

A ˜int ϭ•min͑R a ,R b ͒ 2 (8)
The distribution of the local stiffnesses over the sample obtained through the use of this interaction surface is another important particularity of the SDEC model. The adequacy of the relations ͑6͒ and ͑7͒ will be discussed further.

Strength Properties

Before Rupture A modified Mohr-Coulomb rupture criterion is used. Thus, for a given interaction, a maximum tensile strength T ͑with TϾ0) is given and a maximum normal force F max n is defined such that

F max n ϭϪTA ˜int (9)
A maximum interaction distance D max is defined such that,

D max ϭD eq ϩ͑␤ϩ1 ͒ ͩ ͉F max n ͉ K n ͪ (10)
where D eq ϭequilibrium distance; and ␤ϭsoftening factor for this interaction with ␤Ͼ0. Note that in all these developments, the indices a and b which represented the interacting elements, have been dropped to alleviate the formulations. Two cases may occur: 1.

F n ϽF max n and DϽD max ⇒F n ϭ K n ␤ ͑ DϪD max ͒ (11) F n ϽF max n and DуD max ⇒ ͭ F n ϭ0 F s ϭ0 (12)
The cohesive interaction is broken: Rupture occurs. Fig. 1 summarizes the behavior of the normal force for two different values of ␤ which illustrates the difference between a brittle elastic behavior ͑dotted line͒ and a quasibrittle behavior ͑i.e., presence of softening, dashed line͒. As long as F n уF max n the normal force follows the solid line. As soon as F n ϽF max n and depending on the value chosen for ␤, different paths can be followed. If ␤→0, then D max →D max 1 and rupture occurs ͑dotted line in Fig. 1͒.I f␤ϭ2 then D max ϭD max 2 and before rupture, the normal force follows the dashed line.

However, a global softening may still be induced at the macroscopic scale while the local softening factor tends to zero if the size distribution of the elements is large enough. This is because, if there exists a large distribution of radii for the elements, then according to Eq. ͑9͒ there will be a large distribution of maximum normal forces which will in turn introduce softening effects ͑Herrmann et al. 1989͒. Therefore, some caution must be exercised before using a local softening factor.

The maximum shear force can be calculated as

F max s ϭcA ˜int ϩF n tan i ( 13
)
where cϭcohesion; and i ϭ''internal'' friction angle. If the absolute value of the shear force is

͉F s ͉ϭ͑F s "F s ͒ 1/2 (14)
which is greater than ͉F max s ͉, then the shear force is reduced to the limiting value and written as

F reduced s ϭF s ͓F max s /͉F s ͉͔ (15)
After Rupture After initial interactions have broken, new ones are identified, which are not cohesive any more: They are merely ''contact'' interactions, and cannot undergo any tension force. Then, a new maximum shear force can be calculated as

F max s ϭF n tan c ( 16
)
where cϭcohesion; and c ϭcontact friction angle, which may be different from i , the internal friction angle. If the absolute value of the shear force is

͉F s ͉ϭ͑F s "F s ͒ 1/2 (17)
which is greater than ͉F max s ͉, then the shear force is reduced to the limiting value and written as

F reduced s ϭF s ͓F max s /͉F s ͉͔ (18)
Again, the use of A ˜int induces a certain strength properties distribution over the sample, which is very consistent with a real material behavior.

Fig. 2 summarizes the rupture criteria used in the model.

Irreversible Deformations

The model is able to take into account the loss of rigidity due to damage after a compression phase, where irreversible deformations occur. To do so, the normal force vector may be expressed differently. The irreversibility will be initiated after a compressional phase (D a,b ϽD eq a,b , where D eq a,b is the equilibrium distance between the two elements a and b which was set when the interaction was created͒. In that case, a different loading and unloading path can be considered, using a coefficient ␣. ␣ϭ1 means the unloading path is unchanged, and ␣→ϱ means that the unloading path is vertical. Fig. 3 summarizes the behavior of the normal force.

Homogenized Quantities

It was considered important to be able to look at the state of stress and strain in the specimen. The computation of a homogenized stress and strain has then been implemented within SDEC. As far as the stress is concerned, a technique based on an analogy with the continuous media has been chosen ͑Bardet 1997; Moreau 1997͒, the average stress in the volume V is defined as the weighted average of the average stress in each particle a in V, and is given by:

ij ϭ 1 V ͚ aV ͚ bI a F i b→a R a n j ab (19)
where I a ϭset of particle b in contact with particle a; F b→a ϭcontact force of particle b on particle a; and n ab ϭunit vector from the centroid of particle a to the contact point with particle b. The computation of the homogenized strain leads to a more complex relation: It follows the best fit method, fully described in Liao et al. ͑1997͒.

Quasistatic Simulations

Calibration of the model parameters is performed by adjusting the properties of the material represented by the assembly of discrete elements to the real geomaterial properties, a particular type of concrete. For this purpose, we have established a quasistatic uniaxial compression/traction procedure.

Sample Preparation

This is a step of particular importance, as the properties of the ''discrete'' material strongly depend on the packing. The samples used are standard specimens, cylinders with a height of 0.032 m and a diameter of 0.016 m. The procedure must produce isotropic packings, with a high compacity and a given size distribution. It consists in several points: 1. First, an initial set of elements, all of identical radius R, are distributed according to the most compact geometry possible ͑e.g., a face-centered-cubic lattice͒ in a cylinder shaped volume, as well as two platens, and a hollow cylinder surrounding the sample. 2. The second step introduces a certain disorder in the packing, in terms of the sample element positions, and of their radii, inside the hollow cylinder and between the bottom and top platens, which remain unchanged. This is performed thanks to an original growing technique ͑Donze ´2002͒, based on an algorithm described in Jodrey and Tory ͑1985͒; this technique is completely geometrical, unlike many others which are dynamical. It allows fine isotropy and compacity ͑of roughly 0.67 for the samples used͒. Fig. 4 shows a typical size distribution obtained. 3. The final step consists in removing the hollow cylinder around the sample in its final state.

Monitoring

The load is applied to the sample by moving the top wall downward and keeping the bottom wall at a fixed position. One difficult point is to assure the quasistatic aspect of the simulation, as the method finds the new element positions through the integration of Newton's equation, and at the same time the computational cost must stay reasonable. Therefore, attention has to be paid to the way the load is applied ͑low speed and low accelera-tion͒, and to the damping of the elastic waves propagating in the medium. The displacement law of the top platen consists in three phases: constant acceleration, then constant speed, then constant position. Equilibrium is checked by comparing the total forces applied by the sample to both platens, and looking at the total kinetic energy in the sample.

Results: Elasticity

First, simulations have been run with strength properties set to very high values, so the material remains elastic. The stress is computed by dividing the total axial force applied by the plate to the sample by the surface of the sample; the axial and radial deformations are computed via geometrical measures. We deduce from these quantities Young's modulus, E and Poisson's coefficient, .

At this point, some remarks can be made: As usual, the DEM gives good qualitative results; the study of the homogenized stress in the medium shows a real 3D stress state and it is easy to obtain a good order of magnitude of Young's modulus and Poisson's coefficient using Eqs. ͑6͒ and ͑7͒.

However, in order to study the influence of the disorder, 20 different samples have been generated, with the same geometry, and a nearly identical size distribution ͑see Fig. 4͒, but only differing through the random aspect of their generation. They all contain roughly 2000 elements and the interaction range ␥ has been set to 1.55. Figs. 5 and6 show the results. E is the sample Young's modulus, E m ϭ28 MPa is its mean value, is the Poisson's coefficient, and m ϭ0.252 its mean value.

Dispersions of Young's modulus ͑respectively of Poisson's co-efficient͒ are 28% ͑16%͒, and standard deviation of the ratio E/E m ͑respectively / m ) are 0.07 ͑0.05͒. In addition, there is no obvious convergence of the elastic properties when the number of elements increases. Considering our objective, which is to model a complete structure, these results are somehow awkward and demand the modification of the macro-micro relations ͑6͒ and ͑7͒. This is discussed in the next section.

Energy Criterion

The strain energy E sd stored in the assembly may be computed in the following way: where (n,m)ϭinteraction between the nth and the mth element; U គ n ϭdisplacement of the nth element; and F គ n→m ϭforce applied by the nth element on the mth element, proportional to the interaction surface, A nm . It appears that the strain energy, and thus the elastic properties, are strongly dependent on the definition of the interaction surface. This remark leads one to plot the Young's modulus versus the total sum of the interaction surfaces in the assembly, as in Fig. 7, which shows an obvious relation between these two quantities.

E sd ϭ ͵ ⍀ d tr͑ d • d = ͒d⍀ d ϭ ͚ ͑ n,m ͒ F គ n→m •͑U គ m ϪU គ n ͒ ϭ ͚ ͑ n,m ͒ A nm •T គ n→m •͑U គ m ϪU គ n ͒ (20)
The ratio eϭE sc /E sd was computed for each assembly, where E sd is the strain energy we wish to store in the real concrete sample.

It is easy to show that

eϭ E sc E sd ϭ E E app • S S app • ᐉ app ᐉ ( 21
)
where the indice app corresponds to the apparent properties of the discrete sample.

It appears that e differs from assembly to assembly and is not equal to one. The decision was then made to use as new interaction surfaces A nmЈ ϭe•A nm , and to run the same tests. Results show the new dispersions of Young's modulus ͑respectively of Poisson's coefficient͒ are 12% ͑10%͒, and standard deviation of the ratio E/E m ͑respectively / m ) are 0.038 ͑0.03͒. This is an encouraging result, as dispersions are divided by roughly 2. With the energy criterion, the influence of the random aspect of the assembly generation, as well as the influence of the number, size, and local configuration of the elements, is highly reduced.

Results: Rupture

Parameters of the model are here set to study the fracture of the sample, and to obtain strength properties of a typical concrete. Properties of interest in this section are the compression and tension strengths c and t , and their associated strains c and t , as well as the fracture energy G f . First, concrete behavior is qualitatively very well represented, and the simulations exhibit a very similar response to observations during a laboratory uniaxial compression test.

However, the high value of the ratio c / t for concrete, makes its behavior very specific, and has to be numerically matched. If unchanged, the SDEC model can only reach a ratio of around 3 or 4 ͓Huang ͑1999͒ obtained the same value with a similar model͔. This is due to the fact that a single interaction between two elements does not transmit any moment: Relative rotation without sliding is free. Therefore, rotations are prevented for the following simulations, which led to a very correct ratio c / t around 8. Then, the use of the softening factor led to Gf In order to estimate how the values of interest are affected by the random aspect of the assembly generation, tests have been run with the 20 different assemblies already used in elasticity, and with the corrected interaction surfaces. Results show that dispersions of compressive strength ͑respectively of tension strength͒ are 19% ͑31%͒, and standard deviation of the ratio c / cm ͑respectively t / tm ) are 0.047 ͑0.083͒, where cm ͑respectively tm ) are the mean compression ͑tension͒ strengths, 36 MPa ͑4.4 MPa͒. Compared with the elasticity dispersions, these values are higher, but not so far from what could be obtained with real concrete failure tests, which makes these results very satisfactory.

In addition, failure patterns show a good match with experimental results, see Figs. 10 and 11. In compression, fretting cones appear, as the elements are ''glued'' to the platens, which is consistent with experimental observations. In tension, a macrocrack perpendicular to the loading direction appears, which was expected. The study of the homogenized strain shows a clear strain localization in this macrocrack, in the peak region.

Dynamic Simulations: Split Hopkinson Pressure Bar Experiments

Keeping in mind our objective, which is to model a concrete structure submitted to impact loading, it is of importance to validate the model through dynamic simulations. Now, simulations have shown that the method is able to reproduce the quasistatic behavior of concrete, the writers propose here to simulate dynamic compression tests.

Uniaxial compressive or tensile strengths of concrete increase with an increase in strain rate. At low strain rates, ( ˙Ͻ1s Ϫ1 ), this effect is relatively well explained by the presence of free water in

Dynamic Compression

A typical SHPB experimental setup ͑Zhao and Gary 1996͒ is shown in Fig. 12. It consists in two long aligned metallic bars and a short concrete specimen between them. A projectile impacts the free end of the input bar thus leading to the development of a compressive longitudinal incident wave i (t). Once it arrives at the bar-specimen interface, it splits into a reflected wave r (t) which travels in the input bar and a transmitted wave t (t) which travels in the output bar. These three waves are recorded by gauges which have been cemented on each bar. They are then artificially shifted so as to record them at the bar-specimen interfaces, so that the forces and velocities measured on both faces of the specimen ͑Zhao and Gary 1996͒ are given by

V input ͑ t ͒ϭC" i ͑ t ͒Ϫ r ͑ t ͒… (22) V output ͑ t ͒ϭC t ͑ t ͒ (23) F input ͑ t ͒ϭS B E" i ͑ t ͒ϩ r ͑ t ͒… (24) F output ͑ t ͒ϭS B E t ͑ t ͒ (25)
where CϭͱE/ϭcelerity of the medium; EϭYoung's modulus; is the density; and S B ϭcross-sectional area of the bars. Once these forces and velocities are obtained, a so-called three-waves formula gives the average strain rate and the average stress imposed on the specimen, so that,

˙s͑t ͒ϭ V output ͑ t ͒ϪV input ͑ t ͒ l s (26) s ͑ t ͒ϭ F input ͑ t ͒ϩF output ͑ t ͒ 2S s ( 27
)
where ᐉ s and S s denote respectively the length and the crosssectional area of the specimen. The complete data set then consists of F input , F output , V input , and V output . If the correct constitutive behavior has been used in a model, then given one of these pairs, the other data pair should be obtained. In the following simulations, the velocities will be used as the input data and the forces will thus be computed numerically and compared to the real experimental data forces.

Experimental Data Set SHPB tests on concrete specimens have been carried out by Gary ͑1990͒ in the framework of the ''GRECO ge ´omate ´riaux project'' and data obtained from these different tests were made widely available.

The concrete specimens that are used are cylinders with a height of 0.036 m and a diameter of 0.036 m. The density is 2500 kg/m 3 and the average compressive wave velocity is 3865 m/s ͑Toutlemonde 1995͒.

Three loading experiments, ms2b, ms3b, and ms5b, respectively at 350 s Ϫ1 , 500 s Ϫ1 , and 700 s Ϫ1 strain rate have been run and for each of these runs both the input and output velocities and forces are recorded. Some problems in these recordings have to be noted: Arrival times of the output velocities vary in a large range, and so do the Young's moduli which are much lower than what was expected from the static value ͑Toutlemonde 1995͒ which is more than 30 GPa. The Young's moduli are obtained from the slopes of the stress strain curves of Fig. 13. This plot is deduced from Eqs. ͑26͒ and ͑27͒, and shows best the strain rate dependency.

Such fluctuations in the data set are not surprising when considering the very high strain rates at which these experiments are run and may come from the error bar on the applied correction of the travel times of the compressive waves, but it makes it extremely difficult to define a precise reference material that could be used in the numerical model. Despite these difficulties, local parameters were calibrated using the quasistatic procedure already discussed to obtain the expected concrete behavior: Density 2,500 kg/m 3 , Young's modulus 30 GPa, Poisson's coefficient 0.2, compressive strength 50 MPa, and tension strength 5 MPa. Table 1 shows the values of the local parameters.

Numerical Setup

Up to 6,200 spherical discrete elements with sizes ranging from 9.10 Ϫ4 m to 47.10 Ϫ4 m, have been used to build the numerical concrete sample, and the shape of the size distribution is similar to Fig. 4.

The experimental input and output velocities are applied to the platens. The radial displacement values are assumed to be zero on the input and output surfaces. The resulting input and output forces are computed by summing all the forces applied on the platens. Given the experimental velocity histories, at each time step, the applied input and output velocities are updated leading to the computation of numerical force histories. No damping has been used, and rotations are still prevented.

Results

Fig. 14 shows force histories for both the experiment and the simulation at 350 s Ϫ1 , as well as the number of broken interactions during the simulation. The global wave form of the signal, as well as the amplitude are well reproduced, which is comforting.

In particular, the elastic phase fits very well, for both input and output phases. However, fractures arise a little too soon, and thus the amplitude of the input force is slightly too low; at the same time, the amplitude of the numerical output force exceeds that of the experiment, meaning that the material is not damaged enough. When the force is at its peak, more than one-third of the interactions have ruptured.

The oscillation amplitude of the postpeak region is slightly too low as far as the input force is concerned, but quite well reproduced for the output force. It seems then that the numerical behavior around the peak is too ductile.

At the end of the simulation, numerical and experimental samples show very similar behaviors, except in the last 30 s. Fractures keep occurring, spreading all over the sample.

Fig. 15 shows stress/strain curves obtained for the three tests ͓stress and strain are deduced from the computationed forces and velocities by Eqs. ͑26͒ and ͑27͔͒. The experimental curves are very well fitted in the prepeak region, but this is less often the case after the peak, except for ms2b. The behavior seems too brittle in the last part of the simulation. Maximum amplitudes of computationed stress fits the experimental stress quite well, despite the recording problems of the tests. Fig. 16 shows the damage state of the sample for a given stress of 50 MPa in the elastic phase, for the slowest and the fastest test. As the damage increases, the color darkens. It can clearly be seen that as the strain rate increases, the extent of the damage lessens. Moreover, this damage, very diffuse, is not homogeneous and tends to propagate inwardly from the lateral free surfaces of the specimen thus forming a contact cone as seen in real experiments.

The model proves able to quantitatively reproduce the increase of compressive strength with the increase of loading rate, and this, without requiring the use of viscosity in the model, or of any characteristic time. An inertia-based hypothesis, given by Janach ͑1976͒, might then explain the strain rate dependency: The compressive stress wave C in the brittle material generates a dilation in the radial direction because of the opening of cracks. Thus, the material exhibits a reduced unloading modulus in the radial direction which allows the damage zone to propagate inwardly from the free surface with an unloading velocity C f which can be less than the compressive wave velocity. If this is the case, then the unloading could be sufficiently slow such that the central core of the specimen would not have had time to unload which means that this specimen will have a greater load-carrying capacity ͑see Fig. 17͒. Because of the fretting condition on the boundary, the unloading front from the free surface is not parallel to it, explaining the conelike shape of the damage distribution. All of these points are confirmed by the results of the tests simulations with the discrete element model.

Conclusions

A 3D DEM has proved its capability to model concrete. Quasistatic simulations of uniaxial compression and tension tests have been run to calibrate the model parameters, and to identify the numerical sample behavior.

First, the identification of the elastic behavior has been studied, and the results are qualitatively as well as quantitatively good. The stress state in the sample is consistent with laboratory results as well. Nonetheless, perturbations in the packing disorder have been found to strongly influence the values of Young's modulus and Poisson's coefficient, and high dispersions are observed. In order to reduce these dispersions, an energetic criterion has been introduced to modify the macro-micro constitutive equations of the model, which gives satisfactory results.

The method has proved its ability to capture the other characteristic properties of concrete: Quantitative results like peak strengths under both compression and tension and their associated strains, and fracture energy, as well as qualitative results like softening stress-strain relation in the postpeak region, and failure patterns, are in good accordance with laboratory results. Dispersions are higher than for elastic properties but are of the same order of magnitude as what experiments have shown.

Despite the great difficulty to reproduce such tests, SHPB dynamic compression experiments have been satisfactorily simulated, thus increasing the capability of the model. With the 3D discrete element model, the transient specimen state of damage and stress may be investigated and quite accurately represented. What is more, it proves able to quantitatively reproduce the increase of compressive strength with the increase of strain rate, and this, without requiring the use of viscosity in the model, or of any characteristic time. This confirms the inertia-based hypothesis: In other words, the increase of the dynamic strength in this range of strain rates is merely apparent and seems to be a structural effect.

SHPB tension tests are now being simulated to study the strain rate dependency under this type of loading. Then, steel-reinforced concrete structures under dynamic loading should be modeled.

Notation

The following symbols are used in this paper: A ˜int ϭ average surface where an interaction is defined;

A ˜nm ϭ average surface where an interaction between particles n and m; C ϭ celerity of the specimen's medium; C f ϭ unloading velocity; c ϭ cohesion; D a,b ϭ D distance between centroids of two discrete elements a and b; ϭ correction distance for unloading; D eq a,b ϭ D eq equilibrium distance between two discrete elements a and b; D max ϭ maximum interaction distance; E ϭ Young's modulus; E app ϭ apparent Young's modulus; E m ϭ average Young's modulus; E sc ϭ strain energy stored in the equivalent continuous medium; E sd ϭ strain energy stored in the discrete assembly; e ϭ interaction coefficient; F ϭ interaction force vector; F a→b ϭ force from particle a on particle b;

F n ϭ normal interaction force vector; F s ϭ shear interaction force vector; F reduced s ϭ updated shear force vector during slip; F input ϭ input force at the bar-specimen interface; F output ϭ output force at the bar-specimen interface; F max n ϭ maximum normal force; F max s ϭ maximum shear force; G f ϭ fracturation energy of the material; I a ϭ set of particles in contact with particle a; K n ϭ interaction normal stiffness; K s ϭ interaction shear stiffness;

ᐉ ϭ length of the real specimen; ᐉ app ϭ apparent length of the numerical specimen; n ϭ unit interaction vector; n ab ϭ unit interaction vector pointing from element a to element b; R a ϭ R radius of a discrete element a; S ϭ cross-sectional area of the real specimen; S app ϭ apparent cross-sectional area of the numerical specimen; T ϭ maximum tensile strength; U គ n ϭ displacement of the element n; V ϭ homogenization volume; V input ϭ input velocity at the bar-specimen interface; V output ϭ output velocity at the bar-specimen interface;

x a ϭ x position vector of a discrete element a; x a,b ϭ vector location of the interaction point between a and b; x ˙ϭ velocity vector of a discrete element; x ¨ϭ acceleration vector of a discrete element; x a ϭ component of the position vector of a discrete element a; ␣ ϭ unloading factor; ␤ ϭ softening factor; ␥ ϭ interaction range; ⌬F s ϭ incremental interaction shear force vector; ⌬U s ϭ shear displacement increment vector; ⌬t ϭ integration time step; i ϭ strain of the longitudinal incident wave; r ϭ strain of the longitudinal reflected wave; t ϭ strain of the longitudinal transmitted wave; c = ϭ strain matrix in the continuous medium; d = ϭ strain matrix in the discrete medium; ˙ϭ strain rate; ˙s ϭ average strain rate imposed on the specimen; ϭ Poisson's ratio; m ϭ average Poisson's ratio; ϭ density of the specimen; c ϭ compressive strength; cm ϭ average compressive strength;

s ϭ average stress imposed on the specimen; t ϭ tensile strength; tm ϭ average tensile strength; = ϭ stress matrix in the continuous medium; d = ϭ stress matrix in the discrete medium; c ϭ contact friction angle; i ϭ internal friction angle; and ⍀ d ϭ volume occupied by the assembly.

Fig. 1 .Fig. 3 .

 13 Fig. 1. Strength properties of normal force Fig. 2. Rupture criteria used in the model

Fig. 4 .

 4 Fig. 4. Size distribution of discrete elements in the sample

Fig. 5 .Fig. 6 .

 56 Fig. 5. Values of the ratio E/E m

Fig. 8 .

 8 Fig. 8. Stress/strain plot in compression

Fig. 10 .

 10 Fig. 10. Radial displacement field in an axial cut after failure in compression

Fig. 12 .

 12 Fig. 12. Setup for split Hopkinson pressure bar experiment

Fig. 14 .

 14 Fig. 14. Input and output forces for ms2b ͑350 s Ϫ1 ͒. Dotted lines are original experimental curves; solid lines are computed numerical results.

Fig. 15 .Fig. 16 .

 1516 Fig. 15. From the left-to right-hand side: Stress/strain curves for ms2b, ms3b, and ms5b

Fig. 17 .

 17 Fig. 17. Simplified lateral unloading process proposed for cylindrical specimen that fails by brittle fracture in split Hopkinson pressure bar compressional tests

Table 1 .

 1 Local Parameter Values

	Parameters	Values
	␥	1.4
	␣	1
	␤	1
	T (MPa)	7
	c (MPa)	3.5
	i (degrees)	4
	c (degrees)	40