Information revelation and coordination using cheap talk in a game with two-sided private information

We consider a version of the Battle of the Sexes with private information and allow cheap talk regarding the players’ types before the game. We show that a desirable type-coordination property is achieved at the unique fully revealing symmetric equilibrium (when it exists). Type-coordination is also obtained in a partially revealing equilibrium that exists when the fully revealing equilibrium does not. We further prove that truthfully revealed messages, followed by actions that depend meaningfully on these messages, are not equilibrium profiles with one-sided cheap talk. Finally, fully revealing equilibria do not exist under sequential communication either.


INTRODUCTION
Following the seminal paper by Crawford and Sobel (1982), much of the cheap talk literature has focused on the sender-receiver framework whereby one player has private information but takes no action and the other player is uninformed but is responsible for taking a payo¤-relevant decision.
There indeed is a small but growing literature on games where both players have private information and can send cheap talk messages to each other. 1 Our aim in this paper is to contribute to this literature by analysing symmetric cheap talk equilibria in a game with two-sided information and two-sided cheap talk.
We analyse a version of the Battle of the Sexes with two-sided private information using unmediated, cheap-talk.The complete information BoS has many economic applications (see the Introduction in Cabrales et al 2000); the corresponding game of incomplete information is not just a natural extension but is also relevant in many of these economic situations where the intensity of preference and its prior probability are important factors.BoS type games may be more complicated with incomplete information, where each player has private information about the "intensity of preference" for the other player's favorite outcome.Apart from its applications, the BoS with private information is clearly of interest to theorists and experimentalists.It is not obvious at all whether truthful revelation and thereby separation of players'types can be achieved in a cheap talk equilibrium for the BoS with private information; moreover, it is also not clear whether coordination using cheap talk, as in the theoretical and the experimental literature with the complete information BoS, 2 would extend to the BoS with private information.
To analyse the above two issues, namely truthful revelation and coordination, we use the simplest possible version of the BoS, as in Banks and Calvert (1992), with two types ("High" and "Low") for each player regarding the payo¤ from the other player's favorite outcome.The structure of the game we consider here has an in-built tension for each player between the desire to compromise in order to avoid miscoordination and the desire to force coordination on one's preferred Nash equilibrium 1 Examples of information transmission using two-sided cheap talk under two-sided incomplete information can be found in Farrell and Gibbons (1989), Matthews and Postlewaite (1989), Baliga and Morris (2002), Doraszelski et al (2003), Baliga and Sjöström (2004), Chen (2009), Goltsman and Pavlov (2014) and Horner et al (2015).Two-sided cheap talk using multiple stages of communication where only one of the players has incomplete information has also been studied by Aumann and Hart (2003) and Krishna and Morgan (2004). 2In a seminal paper, Farrell (1987) showed that rounds of cheap talk regarding the intended choice of play reduces the probability of miscoordination; the probability of coordination on one of the two pure Nash equilibria increases with the number of rounds of communication (although, at the limit, may be bounded away from 1). Park (2002) identi…ed conditions for achieving e¢ ciency and coordination in a similar game with three players.Parallel to the theory, the experimental literature also shows that cheap-talk and any pre-play non-binding communication can signi…cantly improve coordination in games like BoS (Cooper et  outcome.This contrasts with the Hawk-Dove game studied in Baliga and Sjöström (2012) and the Cournot game in Goltsman and Pavlov (2014) where a player's preference over the other player's action does not depend on his type or action.The question we ask is whether, in this game, players will (fully or partially) reveal their types in a direct cheap talk equilibrium and also coordinate on Nash equilibrium outcomes in di¤erent states of the world.With incomplete information, e¢ ciency and coordination do not necessarily go together; however, one might …nd it desirable to coordinate on the (ex-post) e¢ cient outcome when the two players are of di¤erent types, in which the compromise is made by the player who su¤ers a smaller loss in utility.In the game we consider here, it is not apriori clear at all whether either full information revelation or the desirable coordination can be achieved in equilibrium.
The primary focus of Banks and Calvert (1992) was to study communication in a similar game using an impartial mediator and the e¢ ciency implications of such mediated communication.Although incentive compatible mediated mechanisms (as in Banks and Calvert, 1992) inform us about all achievable possibilities with strategic communication, it might be impractical to conceive of or employ an impartial mediator in a real-world situation.For instance, in a market entry game (as in Dixit and Shapiro, 1985) or in the adoption of product compatibility standards (as in Farrell and Saloner 1988), it is not clear how a mechanism involving an impartial mediator can be implemented.However, we know that …rms talk to each other and/or make public announcements from time to time.We believe that direct cheap talk communication among players might occur more naturally in a strategic situation and this is the motivation for studying cheap talk equilibria in this paper.
Banks and Calvert (1992) also studied unmediated communication in a similar game allowing more general message spaces (i.e., not restricted to only two types) in the communication phase.Banks and Calvert identi…ed conditions (Proposition 2, Section 4 in their paper) under which the outcome of an ex-ante e¢ cient incentive compatible mediated mechanism can be achieved as the equilibrium of an unmediated communication process.In contrast, the focus of our current paper is to identify conditions under which (full) revelation occurs at the cheap talk stage and some form of coordination property holds.Obviously, these objectives are di¤erent from those studied in Banks and Calvert (1992).Ganguly and Ray (2009) have analysed this game to see if a truthful cheap talk equilibrium, in which the players reveal their types truthfully before playing, exists at all3 and compared it with the mediated equilibrium of Banks and Calvert (1992).
The main contributions of this paper are thus two-fold.We …rst prove that there exists a unique fully revealing symmetric cheap talk equilibrium of this game in which the players announce their types truthfully (Theorem 1). 4 Theorem 1 also suggests that full revelation is not a cheap talk equilibrium when the probability of a player being High-type is too high or too low; the allowable range of the prior probability of the High-type for the fully revealing equilibrium to exist in Theorem 1 has to be moderately low with the upper bound being strictly less than 1 2 .Secondly, we note that our unique fully revealing cheap talk equilibrium has the desirable type-coordination property: when the players' types are di¤erent, it fully coordinates on the ex-post e¢ cient pure Nash equilibrium.
We then consider partially revealing equilibria particularly for situations when the fully revealing equilibrium does not exist.Keeping the spirit of the fully revealing equilibrium, we characterise a class of partially revealing cheap talk equilibria in which only the High-type is not truthful, while the Low-type is truthful.We analyse this particular type of partial revelation because in the fully revealing equilibrium, the High-type is expected to compromise and coordinate on his less preferred outcome when the other player claims to be of Low-type.We identify the unique partially revealing cheap talk equilibrium with the type-coordination property in this set of equilibria 5 and prove its existence based on the prior probability of the High-type being within a range that turns out to be non-overlapping and higher than that for the fully revealing equilibrium (Theorem 2).We illustrate all these results using a numerical example.
We also consider the scenario when only one of the players is allowed to talk in our game, to understand the di¤erence between one-sided and two-sided cheap talk.We …nd (Theorem 3) that one sided truthfulness is not possible in any meaningful equilibrium, that is, there is no non-babbling equilibrium followed by truthful cheap talk by one player only.
Finally, we also consider non-canonical message spaces at the cheap-talk stage; we identify a new equilibrium with a bigger message space (including the types) and …nd (Theorem 4) that, with the help of more messages, truthfulness and the desirable coordination together may be achieved even when the direct truthful cheap talk equilibrium does not exist.

The Game
We consider a version of the BoS with incomplete information as given below, in which each of the two players has a set of strategies S i containing two pure strategies, namely, A and B, i.e., S i = fA; Bg, i = 1; 2. Let S = S 1 S 2 denote the set of strategy combinations of the two players.The payo¤s are as in the following table, in which the value of t i 2 T i is the private information of player i, i = 1; 2, identical symmetric situation ex-ante, we study (type and player) symmetric cheap talk equilibria, following the tradition in the literature (as in Farrell 1987 and Banks and Calvert 1992). 5We also characterise the complete set of partially revealing cheap talk equilibria in which only the High-type is not truthful while the Low-type is truthful (Proposition 2 in this paper).
with 0 < t i < 1.We assume that t i is a discrete random variable that takes only two values L and H, where, 0 < L < H < 1, whose realisation is only observed by player i.So, T i = fL; Hg, i = 1; 2. For i = 1; 2, we henceforth refer to the values of t i as player i's type (Low, High).We further assume that each player's type is independently drawn from the set fL; Hg according to a probability distribution with P rob(t i = H) = p (H) = p 2 [0; 1].Also, the payo¤s to both players from the miscoordinated outcome is normalised to 0, while the payo¤ to player 1 (player 2) from (A; A) ((B; B)) is normalised to 1.
These payo¤s will also formally be denoted by the players'utility functions u i : S T i !R; i = 1; 2.
Note that player i's utility depends here on own type t i only and not on the other player's private information t j .
The unique symmetric Bayesian-Nash equilibrium 6 of this game 7 can be characterised by i (s i jt i ), the probability that player i of type t i plays the pure strategy s i .

Proposition 1
The unique symmetric Bayesian Nash equilibrium of the BoS with incomplete information is given by the following strategy for player 1 (player 2's strategy is symmetric and is given by The proof is straightforward and hence has been omitted here.

Cheap Talk
We study an extended game in which the players are …rst allowed to have a round of simultaneous canonical cheap talk intending to reveal their private information before they play the above BoS.In the …rst (cheap talk) stage of this extended game, each player i simultaneously chooses a costless and nonbinding announcement i from the set T i = fL; Hg.Then, given a pair of announcements ( 1 ; 2 ), 6 The corresponding game with complete information with commonly known values t 1 and t 2 , has two pure Nash equilibria, (A; A) and (B; B), and a mixed Nash equilibrium in which player 1 plays A with probability 1 . 7In a similar game, Banks and Calvert (1992) also provided a similar characterisation. in the second (action) stage of this extended game, each player i simultaneously chooses an action s i from the set S i .
An announcement strategy in the …rst stage for player i is a function a i : T i !(T i ), where (T i ) is the set of probability distributions over T i .We write a i (H jt i ) for the probability that strategy a i (t i ) of player i with type t i assigns to the announcement H. Thus, the announcement i of player i with type t i is a random variable drawn from T i according to the probability distribution with Beliefs for player i are given by e p i : T j !(T i ); i; j = 1; 2: We will denote i's posterior belief by e p i (H j j ) = P rob(t j = H j j ).
In the second (action) stage, a strategy for player i is a function i : where ) is the set of probability distributions over S i .We write i (A jt i ; 1 ; 2 ) for the probability that strategy i (t i ; 1 ; 2 ) of player i with type t i assigns to the action A when the …rst stage announcements are ( 1 ; 2 ).Thus, player i with type t i 's action choice s i is a random variable drawn from fA; Bg according to a probability distribution with P rob( . Given a pair of realised action choices (s 1 ; s 2 ) 2 S 1 S 2 , the corresponding outcome is generated.Thus, given a strategy pro…le ((a 1 ; 1 ); (a 2 ; 2 )), one can …nd the players' actual payo¤s from the induced outcomes in the type-speci…c payo¤ matrix of the BoS and hence, the (ex-ante) expected payo¤s.As the game is symmetric, in our analysis, we maintain the following notion of symmetry in the strategies, for the rest of the paper.
De…nition 1 A strategy pro…le ((a 1 ; 1 ); (a 2 ; 2 )) is called announcement-symmetric (in the announce- , for all t; 1 ; 2 .A strategy pro…le is called symmetric if it is both announcement-symmetric and action-symmetric. Note that De…nition 1 preserves symmetry for both players and the types for each player.We consider the following standard notion of Perfect Bayesian Equilibrium (PBE) 8 in this two-stage cheap talk game.
De…nition 2 A symmetric strategy pro…le ((a 1 ; 1 ); (a 2 ; 2 )) together with beliefs ( e p 1 ; e p 2 ) is called a symmetric cheap talk equilibrium if it is a Perfect Bayesian Equilibrium (PBE) of the game with cheap talk, i.e., each player is playing optimally at all his information sets given the strategy of the other player and the beliefs are updated according to the Bayes rule whenever possible.
Formally, ((a 1 ; 1 ); (a 2 ; 2 )) and ( e p 1 ; e p 2 ) is a PBE of the game with cheap talk if (1) 8t i ; 8i 6 = j; i; j = 1; 2 a j ( j t 0 j )p t 0 j > 0 and e p i (: a j ( j t 0 j )p t 0 j = 0: In De…nition 2 above, Condition (1) ensures optimality at the cheap talk stage.For example, if the announcement strategy a i (: jt i ) is completely mixed, then condition (1) implies that both the messages ( i = H and i = L) should provide the same expected payo¤ to player i with type t i .
If a i (: jt i ) is a pure strategy, then condition (1) will yield a weak inequality whereby the expected payo¤ from the chosen pure strategy announcement and then following the equilibrium strategy at the action stage is at least as high as the expected payo¤ from choosing the other message and subsequently using the optimal strategy at the action stage (which could possibly be a deviation from the prescribed equilibrium pro…le).Similarly, Condition (2) ensures optimality at the action stage whereby a completely mixed action strategy yields an equality constraint for expected payo¤s (using suitable posterior beliefs) and a pure strategy yields an inequality constraint.Finally, Condition ensures that posterior beliefs are derived using Bayes rule.
De…nition 2 suggests that a symmetric cheap talk equilibrium can be characterised by a set of (symmetric) equilibrium constraints (2 for the announcement stage and another possible 8 for the action stage).

MAIN RESULTS
The main purpose of our paper is to …nd, if exists, an equilibrium with truthful talk.We thus …rst consider the possibility of full revelation of the types as a result of our canonical cheap talk.
Subsequently, we present and analyse some other cheap talk equilibria in this section.

Fully Revealing Equilibrium
We consider a speci…c class of strategies in this subsection where we impose the property that the cheap talk announcement should be fully revealing.
De…nition 3 A symmetric strategy pro…le ((a 1 ; 1 ); (a 2 ; 2 )) is called fully revealing if the announcement strategy a i reveals the true types with certainty, i.e., a i (H jH ) = 1 and a i (H jL ) = 0.
We now characterise the fully revealing symmetric cheap talk equilibrium.We …rst consider a spe-ci…c fully revealing (separating) strategy pro…le that we call S separating , in ‡uenced by the equilibrium action pro…le in Farrell (1987) for the complete information version of this game.In this strategy pro…le, the players announce their types truthfully and then in the action stage, they play the mixed Nash equilibrium strategies of the complete information BoS when both players' types are identical and they play (B; B) ((A; A)), when only player 1's type is H (L).
We state our …rst result below.9 Theorem 1 S separating is the unique fully revealing symmetric cheap talk equilibrium and it exists only for Before proving Theorem 1, we …rst observe the following fact that follows from De…nitions 2 and

3.
In a fully revealing symmetric cheap talk equilibrium ((a 1 ; 1 ); (a 2 ; 2 )), the players'strategies in the action phase must constitute a (pure or mixed) Nash equilibrium of the corresponding complete information BoS, that is, ( 1 (t 1 ; t 2 ); 2 (t 1 ; t 2 )) is a (pure or mixed) Nash equilibrium of the BoS with values t 1 and t 2 , 8t 1 ; t 2 2 fH; Lg.Thus, in a fully revealing symmetric cheap talk equilibrium ((a 1 ; 1 ); (a 2 ; 2 )), conditional on the announcement pro…le (H; H) or (L; L), the strategy pro…le in the action phase must be the mixed strategy Nash equilibrium of the corresponding complete information BoS, that is, whenever t 1 = t 2 , ( 1 (t 1 ; t 2 ); 2 (t 1 ; t 2 )) is the mixed Nash equilibrium of the BoS with values t 1 = t 2 .
Based on the above fact, one can easily identify all the candidate equilibrium strategy pro…les of the extended game that are fully revealing and symmetric.It implies that these pro…les are di¤erentiated only by the actions played when t 1 6 = t 2 , that is, when the players'types are (H; L) and (L; H).
The proof of Theorem 1 may now be completed easily; we have postponed the details of the rest of the proof to the Appendix of this paper.
The following couple of claims illustrate some features of the equilibrium S separating .The claims are easy to establish and hence we have omitted the formal proofs for them.

Claim 1
The ex-ante expected payo¤ for any player from S separating is given by EU separating = 1+L , which is increasing over the range of p where it exists.

Claim 2
The upper bound for p in Theorem 1, To understand why p must lie in such a low range for this equilibrium to exist (as stated in Theorem 1 and Claim 2), consider the incentives for deviations by player 1 at the announcement stage.
By deviating and claiming to be an L-type, player 1(H-type) gains 1 1+H when player 2 is a H-type (with probability p) and loses 10 H > 0 , the gain from the deviation when playing against player 2(H-type) is bigger than the loss when playing against player 2(L-type).If a H-type is equally or more likely than a L-type, then player 1(H-type) will obviously deviate and truthful revelation will not be an equilibrium.So, p must be <1 2 .Indeed, p needs to be small enough to make the above deviation unattractive and the precise value of p for which this holds is HL+H2 L 1+L+HL+H 2 L or less.However, p cannot be too close to 0 either.This is because of incentives for deviations by player 1(L-type).By deviating and claiming to be an H-type, player 1(L-type) gains 1+L when player 2 is a L-type (with probability 1 p) and loses 1 > 0 , the loss from the deviation when playing against player 2(H-type) is bigger than the gain when playing against player 2(L-type).The expected gain will outweigh the expected loss only if a L-type is much more likely than a H-type (and L is bigger than 0).Hence, player 1(L-type) would deviate at the cheap talk stage only if p is too close to 0.

Partially Revealing Equilibrium
The fully revealing symmetric cheap talk equilibrium exists only for a moderately low range of the prior probability p.One may now ask what sort of equilibria, if any, exists for any given p outside this range.Understandably, it is not easy to characterise all possible equilibria for this game.
In this subsection, we aim to characterise a speci…c class of partially revealing symmetric equilibria of the above cheap talk game.As noted earlier, in the fully revealing equilibrium, the H-type is expected to compromise and coordinate on his less preferred outcome when the other player claims to be of L-type.We thus …nd it natural to analyse the type of partial revelation in which only the L-type truthfully reveals while the H-type does not.
Formally, we consider a symmetric announcement strategy pro…le in which the H-type of player i announces H with probability r and L with probability (1 r) and the L-type of player i announces L with probability 1, i.e., a i (H jH ) = r and a i (H jL ) = 0. Clearly, after the cheap talk phase, the possible message pro…les ( 1 ; 2 ) that the H-type of player 1 may receive are (H; H), (H; L), (L; H) or (L; L) while the L-type of player 1 may receive either (L; H) or (L; L).
First note that, on receiving the message pro…le (H; H), the players know the true types and hence in any such partially revealing symmetric cheap talk equilibrium, q 0 has to correspond to the mixed Nash equilibrium of the complete information BoS with values H and H. Thus, q 0 = 1 1+H .One may indeed characterise the whole set of partially revealing symmetric cheap talk equilibria in this set up (in which only the L-type is truthful), by characterising the equilibrium values of (r; q 1 ; q 2 ; q 3 ; q 4 ; q 5 ), using the following equilibrium conditions.
If q 1 ; q 2 ; q 3 ; q 4 and q 5 correspond to completely mixed strategies in the action stage, then we must have the following …ve conditions for player 1 to be indi¤erent between playing A and B (where the LHS in each equation is the expected payo¤ from A and the RHS in each equation is the expected payo¤ from B): Player 1(H-type) receiving the message pro…le (L; H): Player 1(H-type) receiving the message pro…le (H; L): Player 1(H-type) receiving the message pro…le (L; L): Player 1(L-type) receiving the message pro…le (L; L): Player 1(L-type) receiving the message pro…le (L; H): Also, in the cheap talk phase, player 1(H-type), who is using a completely mixed strategy, should be indi¤erent between announcing H and L, which implies = p (r (q 2 (1 q 1 ) + (1 q 2 ) q 1 H) + (1 r) (q 3 (1 q 3 ) + (1 q 3 ) q 3 H)) where the LHS of (6) is the expected payo¤ from announcing H and the RHS is the expected payo¤ from announcing L.

M ax
x The LHS of (7) is the expected payo¤ from announcing L and following the equilibrium strategy thereafter while the RHS is the expected payo¤ from deviating and announcing H and then choosing the optimal strategy in the action phase given the deviation in the cheap talk phase.So, x is the optimal probability of playing A in the action phase after player 1(L-type) deviates and announces H and receives the message pro…le (H; L).Note that the RHS of this last inequality constraint (7) allows player 1(L-type) to deviate in both stages of the game and hence, (7) checks the player's incentives against the best possible deviation.
By virtue of symmetry, the equilibrium conditions for player 2 are identical to the above.Using these equilibrium conditions, one can prove that certain pro…les as listed in the proposition below constitute this equilibrium set.
We are not presenting the details of the proof of Proposition 2 which can be found in a previous discussion paper version of our work (Ganguly and Ray 2013).

Coordination
One may note that S separating features a speci…c form of coordination in which the players play (B; B) ((A; A)) when only player 1's type is H (L), that is, when the players'types are di¤erent, players fully coordinate on a pure Nash equilibrium outcome that generate the ex-post e¢ cient payo¤s of 1 and H.
We call this property "type-coordination".
De…nition 4 A strategy pro…le is said to have the type-coordination property if the induced outcome is (A; A) and (B; B), when the players' true type pro…le is (L; H) and (H; L), respectively.
Clearly, the type-coordination property can be achieved in other kinds of equilibria.Indeed, the Bayesian Nash equilibrium of the game without the cheap talk (as mentioned in Proposition 1) also satis…es type-coordination property when the prior p is between L 1+L and H 1+H (for example, between 1 4 and2 5 for the parameters L =1 3 , H = 2 3 ).Although the type-coordination property can be obtained in the fully revealing cheap talk equilibrium or in the Bayesian Nash equilibrium, one might still ask whether it is possible to obtain typecoordination where players do not reveal their types truthfully.This motivates us to check whether at all type-coordination can be obtained in any of the partially revealing equilibria, as described in Proposition 2.
In a partially revealing equilibrium in which only the L-type truthfully reveals while the H-type does not, following Proposition 2, note that for the type-coordination property to hold, we need pro…les satisfying q 1 = 0, q 3 = 0 and q 5 = 1.Using symmetry, for player 2(H-type), we then must have 2 (A jH; L; H ) = 1 q 1 = 1.This implies that in any such pro…le, q 2 = 1 and q 4 = 1.Thus, a candidate partially revealing equilibrium pro…le with the type-coordination property must have q 0 = 1 1+H , q 1 = 0, q 2 = 1, q 3 = 0, q 4 = 1 and q 5 = 1.We now state our second main result.The proof has been postponed to the Appendix.
Theorem 2 In a partially revealing symmetric cheap talk equilibrium, in which only the L-type is truthful, that satis…es the type-coordination property, r must be H+H 2 1+H+H 2 ; this equilibrium exists only when Note that the pro…le in Theorem 2 above is the same as that given in (iv) in Proposition 2. Let the equilibrium pro…le stated in Theorem 2 be called S pooling .
To understand why p must lie within such a range for S pooling to be an equilibrium (as stated in Theorem 2), consider the incentives for deviations by player 1 at the action stage.According to the above strategy pro…le, on receiving the message pro…le (L; L), player 1(H-type) needs to play B. Given that player 2(H-type) plays A and player 2(L-type) plays B, player 1(H-type) will indeed play B only if he believes that player 2 is more likely to be an L-type than an H-type.This means that player 1(H-type)'s posterior belief about player 2 being an H-type should not be too high.If we denote this posterior belief by p0, then p0 = P (t i = H j i = L ) = p rp 1 rp .Since this posterior p0 is an increasing function of the prior p ( @ @p ( p rp 1 rp ) = (1 r) (1 rp) 2 > 0), the constraint that p0 should not be too high implies that the prior p cannot be very high either. 11Hence, there is an upper bound for p that is strictly less than 1.Similarly, according to the above strategy pro…le, after receiving the message pro…le (L; L), player 1(L-type) needs to play A. Again, given that player 2(H-type) plays A and player 2(L-type) plays B, player 1(L-type) will play A only if the posterior p0 is not too small which explains the lower bound on p.
The following couple of claims illustrate some features of the equilibrium S pooling .These two claims are easy to establish and hence we have omitted the formal proofs for them.

Claim 3
The ex-ante expected payo¤ for any player from the equilibrium S pooling is given by .

Claim 4
The upper bound of the range for p in Theorem 2 does not involve L, is increasing in H and is bounded by 3 4 .

One-sided Talk
One-sided cheap talk with two-sided private information has also been studied in the literature (see, for example, Seidmann (1990) and more recently, Moreno de Barreda ( 2012)).One thus may be interested to know whether the properties of truthfulness and type-coordination of the two-sided cheap talk equilibria can be achieved with one-sided cheap talk in our game, when only one player (say, player 1) talks.
To do so, we assume that player 1 chooses a costless and nonbinding announcement 1 from the set T 1 = fL; Hg.We now write i (A jt i ; 1 ) for the probability that strategy i (t i ; 1 ) of player i with type t i assigns to the action A when the …rst stage announcement by player 1 is 1 .
We …rst consider below two speci…c strategy pro…les which we believe are closest to the two equilibrium strategy pro…les studied earlier with two-sided cheap talk and we show that these strategy pro…les are no longer equilibrium pro…les.
The …rst strategy pro…le we analyse concerns the situation where player 1 reveals his information truthfully.Consider the following strategy pro…le: in the cheap talk stage, player 1 reports his type truthfully, i.e., a 1 (H jH ) = 1 and a 1 (H jL ) = 0; in the action stage, player 1's strategy consists of any 0 1 (A jH; H ) 1 and 0 1 (A jL; L ) 1 and player 2's strategy is given by 2 (A jH; Call this strategy S onesided separating .It is easy to prove that S onesided separating is not an equilibrium in the game with one-sided cheap talk where only player 1 talks.To see this note that, in the action stage, player 1(H-type)'s expected payo¤ from playing A is p H 1+H whereas his expected payo¤ from playing B is p H 1+H + (1 p)H; hence, 1 (A jH; H ) = 0.But this implies that player 2(H-type) should play the pure strategy B after player 1(H-type) talks, i.e., The second strategy pro…le concerns the situation where player 1(H-type) partially reveals his information, while player 1(L-type) announces L truthfully.Formally, consider the following strategy pro…le: in the cheap talk stage, player 1 reveals his type partially, i.e., a 1 (H jH ) = r and a 1 (H jL ) = 0; in the action stage, player 1's strategy consists of any 0 1 (A jH; H ) 1; 0 1 (A jH; L ) 1 and 0 1 (A jL; L ) 1 and player 2's strategy is given by 2 (A jH; Call this strategy S onesided pooling .Following the same logic as in the case of S onesided separating , one can also show that S onesided pooling is not an equilibrium.We are now going to show a more general result, namely, that any strategy pro…le involving truthful revelation in the cheap talk stage does not lead to any meaningful equilibrium.To show this, we thus focus our attention only on equilibria where at least some of the actions in the second stage depend on the announcement from the …rst stage in a non-trivial manner. De…nition 5 A strategy pro…le in the game with one-sided cheap talk (by player 1) is called nonbabbling if at least one of the following holds: The following theorem con…rms that truthfully revealed messages followed by actions that depend meaningfully on the messages are no longer equilibrium pro…les when only one player (player 1) talks.
Theorem 3 With one-sided cheap talk where only player 1 talks, there does not exist an equilibrium with a non-babbling strategy pro…le where player 1 reports his type truthfully in the cheap talk stage, i.e., a 1 (H jH ) = 1 and a 1 (H jL ) = 0.
The proof of Theorem 3 has been postponed to the Appendix.

FURTHER REMARKS
We comment on several issues related to our results below.

Numerical Illustrations
Here we illustrate our main results in this paper by a speci…c numerical example.Let us consider the following version of our game in which the payo¤ of the H-type of player 1 from (B; B) is 2  3 , twice that of the L-type as described in the following table.
Types: HH Types: HL Types: LH Types: LL Also, take the (independent) prior probability of the H-type, P rob(t i = 2 3 ) to be 1 5 .Then, the unique symmetric Bayesian Nash equilibrium of this speci…c Bayesian game is given by the following symmetric strategy pro…le: player 1 plays A with probability 15  16 when the type is Low and plays the pure strategy B when the type is High (player 2's strategy is symmetric and is B with probability 15   16   when the type is Low and A when the type is High) which generates the following distribution over the outcomes for di¤erent type pro…les (states of the world).Types: HH Types: HL Types: LH Types: LL From the above distribution over outcomes, one may observe that the unique Bayesian Nash equilibrium for this speci…c game involves fair amount of miscoordination.
From Theorem 1, for these parameter values (L = 1 3 and H = 2 3 ), we know that the range of the prior p for which S separating exists is 5 41 (' 0:12) p 5 23 (' 0:22).For example, when p = 5 23 , one may check that the payo¤ from S separating is 241 529 (' 0:46) and that S separating generates the following distribution over the outcomes.

Comparison with Mediated Equilibria
Banks and Calvert (1992) characterised the (ex-ante) e¢ cient symmetric incentive compatible direct mechanism for a similar game so that the players are truthful and obedient to the mechanism (mediator).Following Banks and Calvert (1992), one may analyse (as in Ganguly and Ray 2009) a (direct) symmetric mediated equilibrium that provides the players with incentives (i) to truthfully reveal their types to the mediator and (ii) to follow the mediator's recommendations following their type-announcements.Clearly, our cheap talk equilibria can be achieved as outcomes of such mediated equilibria using incentive compatible mechanisms.Formally, one can easily prove that (i) the distribution over the outcomes generated by S separating can be achieved as a symmetric mediated equilibrium if and (ii) the distribution over the outcomes generated by S pooling can be achieved as a symmetric mediated equilibrium if 1+H+H 2 +H 3 .Not surprisingly, these ranges of p strictly contain the corresponding ranges for the cheap talk equilibria implying a larger range of p for which the corresponding mechanism is in equilibrium.Rather intuitively, this indicates that there are priors for which an outcome can be obtained as an equilibrium via a direct mechanism but not using the unmediated one-round cheap talk that only allows direct communication between players of di¤erent types.

Cheap Talk with More Messages
Finally, we consider the implications of the players using a richer message space.So far, the only messages that the players were allowed to use at the announcement stage were their own types, i.e.T i = fL; Hg.If instead, we allow the players to use more messages at the cheap talk stage, will that lead to new distinct equilibria that either have higher expected payo¤s compared to the previous equilibria and/or that exist for values of p where the previous equilibria do not exist?We explore this interesting question by modifying our model and expanding the message space of each player.
Let each player i now choose an announcement 0 i from the set T 0 i = fL; Hg fX; Y g.We modify the strategies accordingly.An announcement strategy in the …rst stage for player i is a function a i : where (T 0 i ) is the set of probability distributions over T 0 i .So, for example, a i (HX jt i ) stands for the probability that strategy a i (t i ) of player i with type t i assigns to the announcement HX.
Beliefs for player i are now based on the new expanded message spaces.In the second (action) stage, a strategy for player i is a function i : , where (S i ) is the set of probability distributions over S i .
We again restrict our attention to symmetric strategy pro…les.The de…nition of a symmetric cheap talk equilibrium is similar to De…nition 2 with the message spaces appropriately adjusted.
We consider a speci…c class of strategies where we impose the property that the cheap talk an-nouncement should be fully revealing about their types.We modify De…nition 3 in the following manner.
De…nition 6 A symmetric strategy pro…le ((a 1 ; 1 ); (a 2 ; 2 )) is called fully revealing if the announcement strategy a i reveals the true types with certainty, i.e., a i (HX jH )+a i (HY jH ) = 1 and a i (HX jL ) = a i (HY jL ) = 0.
We now look at equilibria using such message pro…les.The aim in studying such equilibria clearly should be to improve upon S separating , the unique fully revealing symmetric equilibrium that was obtained with fewer messages (described in subsection 3:1), in which the players play the mixed strategy equilibria in the action stage when their types coincide, resulting in miscoordination and low payo¤s.Using more messages may now help the players improve their coordination even when their types are the same.
In the new candidate equilibrium pro…le with more messages, we thus would like to keep the desirable type-coordination property when the types are di¤erent; for the type-coordination property to hold, we therefore restrict our attention to action-strategies where 1 (A jH; HX; LX ) = 1 (A jH; HX; LY ) = 1 (A jH; HY; LX ) = 1 (A jH; HY; LY ) = 0:This implies, again by symmetry, that 1 (A jL; LX; HX ) = 1 (A jL; LX; HY ) = 1 (A jL; LY; HX ) = 1 (A jL; LY; HY ) = 1: However, note that in such an equilibrium if the messages are identical, then we are bound to have the mixed strategy equilibria in the action stage; to be an equilibrium, in the second (action) stage, by symmetry, an action-strategy of player 1 must have: 1 (A jH; HX; HX ) = 1 (A jH; HY; HY ) = 1 1+H , . It is worth noting that the above partial speci…cation of part of the players' strategies mimics S separating .Let the class of strategy pro…les satisfying the above partial speci…cation be denoted by S 0 separating .
The following result shows that new and distinct equilibria emerge by virtue of the richer message spaces used by the players.
Theorem 4 Within the class of strategy pro…les described by S 0 separating , the following four pro…les constitute fully revealing symmetric cheap talk equilibria: Note that in each of the four pro…les in Theorem 4, players are able to coordinate in the action stage on one of the pure equilibrium outcomes of the BoS, even when their types are the same by using di¤erent messages (X or Y ).
In the Appendix, we provide the proof of (i).The other three cases are very similar and the proofs are therefore omitted.
We now compare the upper and lower bounds for p for the above equilibrium (S 0 separating ) and that of S separating .

Note that
This shows that the two intervals of p for the two equilibria (S separating and S 0 separating ) to exist are non-overlapping and the range for p for S 0 separating is strictly lower than the range for p for S separating .Theorem 4 establishes that enriching the message space enables fully revealing equilibria to exist for values of p that were not possible otherwise.

CONCLUSION
In this paper, we have analysed a simple 2 x 2 x 2 Bayesian game and studied the possibility of information revelation and desirable coordination using one round of direct cheap talk.The main takeaway of our paper is that the desirable type-coordination is achieved at the unique fully revealing equilibrium (when it exists) moreover, such a coordination may also be achievable with partial revelation when fully revealing cheap talk equilibrium does not exist.
We here have characterised the unique fully revealing symmetric cheap talk equilibrium in the BoS with private information.There are of course many fully revealing but asymmetric cheap talk equilibria of this game.Clearly, babbling equilibria exist in which the players ignore the communication and just play one of the Nash equilibria of the complete information BoS for all type-pro…les.There are other asymmetric equilibria as well, as Ganguly and Ray (2009) have already shown.
We are aware of many interesting open questions that come out of our analysis.For example, one may ask whether non-babbling cheap talk equilibrium always exist in our game for any given p or not.
We also do not characterise the general case where neither type reveals truthfully in the cheap talk phase.Finally, following Banks and Calvert (1992), one may also be interested in characterising the ex ante e¢ cient cheap talk equilibrium in our set up.We postpone all these issues for future research.
Proof of Theorem 4. For (i) to be an equilibrium, in the cheap talk phase, player 1(H-type) which will be equal when r 2 = L 2 1+L 2 .For player 1(H-type), the expected payo¤ from announcing HX must be greater than or equal to the expected payo¤ from announcing LX, which implies This will be satis…ed if p HL 3 (1+H+H 2 +H 3 ) 1+L+L 2 +L 3 +HL 3 +H 2 L 3 +H 3 L 3 +H 4 L 3 .For player 1(H-type), the expected payo¤ from announcing HX must also be greater than or equal to the expected payo¤ from announcing LY , which implies Note that = L H L L 2 +1 > 0. So, if the constraint (12) is satis…ed, then this constraint (13) will also be satis…ed.
For player 1(L-type), the expected payo¤ from announcing LX must be greater than or equal to the expected payo¤ from announcing HY , which implies This will be satis…ed if p L 4 (1+H+H 2 +H 3 ) 1+L+L 2 +L 3 +L 4 +HL 4 +H 2 L 4 +H 3 L 4 .
For player 1(L-type), the expected payo¤ from announcing LX must be greater than or equal to the expected payo¤ from announcing HX, which implies Note that = H L H 2 +1 < 0. So, if the constraint (14) is satis…ed, then this constraint (15) will also be satis…ed.

Following Theorem 2 ,
S pooling for these parameter values exists for p between 19 46 (' 0:41) and 38 65 (' 0:58); in S pooling , the L-type is truthful but the H-type partially reveals his true type with probability10  19 (' 0:53).When p = 38 65 , the payo¤ from such an equilibrium also turns out to be