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Research Article
The Strategic Role of Nonbinding Communication
Luis A. Palacio,1Alexandra CortésAguilar,1and Manuel MuñozHerrera2
1Escuela de Econom´
ıa y Administraci´
on, Universidad Industrial de Santander, Calle 9 con 27, Bucaramanga, Colombia
2ICS, Faculty of Behavioural Social Sciences, University of Groningen, Grote Rozenstraat 31, 9712 TG Groningen, Netherlands
Correspondence should be addressed to Manuel Mu˜
nozHerrera; e.m.munoz.herrera@rug.nl
Received December ; Revised March ; Accepted March
Academic Editor: Jens Großer
Copyright © Luis A. Palacio et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
is paper studies the conditions that improve bargaining power using threats and promises. We develop a model of strategic
communication, based on the conict game with perfect information,inwhichanoisycommitment message is sent by a better
informed sender to a receiver who takes an action that determines the welfare of both. Our model capturesdierent levels of aligned
preferences, for which classical games such as stag hunt,hawkdove,andprisoner’s dilemma are particular cases. We characterise
the Bayesian perfect equilibrium with nonbinding messages under truthtelling beliefs and sender’s bargaining power assumptions.
rough our equilibrium selection we show that the less conict the game has, the more informative the equilibrium signal is and
less credibility is necessary to implement it.
1. Introduction
Bargaining power refers to the relative ability that a player has
in order to exert inuence upon others to improve her own
wellbeing. It is related also to idiosyncratic characteristics
such as patience, so that a player turns the nal outcome into
herfavourifshehasbetteroutsideoptionsorifsheismore
patient []. In addition, Schelling [] described bargaining
power as the chance to cheat and blu, the ability to set
thebestpriceforoneself.Forinstance,whentheunionsays
tothemanagementinarm,“wewillgoonstrikeifyou
do not meet our demands,” or when a nation announces
that any military provocation will be responded with nuclear
weapons, it is clear that communication has been used with a
strategic purpose, to gain bargaining power.
In bargaining theory, strategic moves are actions taken
prior to playing a subsequent game, with the aim of changing
the available strategies, information structure, or payo
functions. e aim is to change the opponent’s beliefs, making
it credible that the position is unchangeable. Following Selten
[],theformalnotionofcredibilityissubgameperfectness.
(Schelling developed the notion of credibility as the outcome
that survives iterated elimination of weakly dominated strate
gies. We know that, in the context of generic extensiveform
games with complete and perfect information, this procedure
does indeed work (see []).) Nevertheless, we argue that if
a message is subgame perfect, then it is neither a threat nor
a promise. Consider the following example: a union says to
management: “If you increase our salaries, we will be grate
ful.” In such case, credibility is not in doubt, but we could
hardly call this a promise or a threat. Schelling [] denotes
fully credible messages as warnings;andwefollowthis
dierentiation to threats and promises.
Commitment theory was proposed by Schelling [](for
a general revision of Schelling’s contribution to economic
theory, see Dixit []andMyerson[]), who introduced a
tactical approach for communication and credibility inside
game theory. Hirshliefer [,] and Klein and O’Flaherty []
worked on the analysis and characterisation of strategic
moves in the standard game theory framework. In the
same way, Crawford and Sobel []formallyshowedthat
an informed agent could reveal his information in order to
induce the uninformed agent to make a specic choice.
ere are three principal reasons for modelling pre
play communication: information disclosure (signalling),
coordination goals (cheaptalk), and strategic inuence (in
Schelling’s sense). Following Farrell [] and Farrell and
Rabin [], the main problem in modelling nonbinding mes
sagesisthe“babblingequilibrium,”wherestatementsmean
nothing. However, they showed that cheap talk can convey
Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2015, Article ID 910614, 11 pages
http://dx.doi.org/10.1155/2015/910614
Journal of Applied Mathematics
information in a general signalling environment, displaying
a particular equilibrium in which statements are meaningful.
In this line, Rabin []developedcredible message proles,
looking for a meaningful communication equilibrium in
cheaptalk games.
Our paper contributes to the strategic communication
literature in three ways. First, we propose a particular
characterisation of warnings,threats,andpromises in the
conict game with perfect information,asmutuallyexclusive
categories. For this aim, we rst dene a sequential protocol
in the 2×2conict game originally proposed by Baliga and
Sj¨
ostr¨
om []. is benchmark game is useful because it is
a stylised model that captures dierent levels of aligned
preferences, for which classical games such as stag hunt,
hawkdove,andprisoner’s dilemma are particular cases.
Second, we model strategic moves with nonbinding
messages, showing that choosing a particular message and its
credibility are related to the level of conict. In this way, the
conict game with nonbinding messages captures a bargaining
situation where people talk about their intentions, by simply
usingcheaptalk.Moreprecisely,weanalyseagamewhere
a second player (the sender) can communicate her action
plan to the rst mover (the receiver). (To avoid confusion
and gender bias, the sender will be denoted as “she,” and the
receiver as “he.”) In fact, the sender must decide aer she
observes the receiver’s choice, but the commitment message
is a preplay move.
ird, we introduce a simple parameterisation that can
be used as a baseline for experimental research. By means of
this model it is possible to study how, in a bargaining environ
ment, information and communication inuence the power
one of the parts may have. In other words, this addresses
the following: the logic supporting Nash equilibrium is that
each player is thinking, given what the other does, what
isthebesthecoulddo.Players,inarststance,cannot
inuence others’ behaviour. On the contrary, Schelling []
argues that players may consider what they can do, as a
preplay move, to inuence (i.e., manipulate) the behaviour
of their counterpart and turn their payos in their favour.
erefore, our behavioural model provides a framework
where it is possible to (experimentally) study the strategic
use of communication in order to inuence others, under
dierent levels of conict.
We analyse conceptually the importance of three essential
elements of commitment theory: (i) the choice of a response
rule, (ii) the announcement about future actions, and (iii) the
credibility of messages. We answer the following questions:
what is the motivation behind threats and promises? and can
binding messages improve the sender’s bargaining power? In
this paper, threats and promises are dened as a second mover
selfserving announcement, committing in advance how she
will play in all conceivable eventualities, as long as it species
at least one action that is not her best response (see [,]).
With this denition, we argue that binding messages improve
the sender’s bargaining power in the perfect information
conict game,evenwhenitisclearthatbyassumingbinding
messages we avoid the problem of credibility.
e next step is to show that credibility is related to
the probability that the sender fullls the action specied in
T : e 2×2conict game.
Player
Player , ,
,., .
the nonbinding message. For this, we highlight that players
shareacommonlanguage,andtheliteralmeaningmust
be used to evaluate whether a message is credible or not.
Hence,thereceiverhastobelieveintheliteralmeaningof
announcements if and only if it is highly probable to face the
truth. Technically, we capture this intuition in two axioms:
truthtelling beliefs and the sender’s bargaining power.Weask,
are nonbinding messages a mechanism to improve the sender’s
bargaining power? and how much credibility is necessary for a
strategicmovetobesuccessful?In equilibrium, we can prove
that nonbinding messages will convey private information
when the conict is low. On the other hand, if the conict
is high, there are too strong incentives to lie, and cheap talk
becomes meaningless. However, even in the worse situation,
the nonbinding messages can transmit some meaning in
equilibrium if the players focus on the possibility of fullling
threats and promises.
e paper is organised as follows. In Section ,theconict
game is described. In Section the conditioned messages
willbeanalysed,andthedenitionsofthreats and promises
are presented. Section presents the model with nonbind
ing messages, showing the importance of response rules,
messages, and credibility to improve the sender’s bargaining
power. Finally, Section concludes.
2. The 2×2Conflict Game
e 2×2conict game is a noncooperative symmetric
environment. ere are two decision makers in the set of
players, ={1,2}. (In this level of simplicity, players’ identity
is not relevant, but since the purpose is to model Schelling’s
strategic moves, in the following sections player is going to
be a sender of commitment messages.) Players must choose
an action ∈
={,},whererepresents being dove
(peaceful negotiator) and being hawk (aggressive negotia
tor). e utility function (1,2)for player is dened by the
payos matrix in Tab l e , where rows correspond to player
and columns correspond to player .
Note that both mutual cooperation and mutual defection
lead to equal payos, and the combination of strategies (,)
isalwaysParetooptimal.Inthesameway,thecombination
of strategies (,)is not optimal and can only be understood
as the disagreement point. Assuming that ≥,payosare
unequalwhenaplayerbehavesaggressivelyandtheother
cooperates, given that the player who plays aggressively has
an advantage over his/her opponent. In addition, we will
assume that = 0.25 and =1to avoid the multiplicity
of irrelevant equilibria. erefore, it will always be preferred
that the opponent chooses .Tohaveaparameterisationthat
serves as a baseline for experimental design, it is desirable
to x ∈ [0,0.5]and ∈ [0.5,1.5] within these intervals,
Journal of Applied Mathematics
T : Nash equilibria in the conict game.
(∗
1,∗
2)(
∗
1,∗
2)Pareto optimal
C (,) (, ) Yes
C (,) (, ) Yes
(,) (., .) No
C (,) (,) Ye s
(,) (,) Ye s
C (,) (., .) No
because if they are modelled as random variables with
uniform distribution we would have four games with the
same probability of occurring.
Under these assumptions, the 2×2conict game has
four particular cases that, according to Hirshliefer [], can be
ordered by their level of conict or anity in preferences:
() Level of conict 1 (C): if <1and > 0.25,there
is no conict in this game because cooperating is a
dominant strategy.
() Level of conict 2 (C): if <1and < 0.25,this
is the socalled stag hunt game, which formalises the
idea that lack of trust may lead to disagreements.
() Level of conict 3 (C): if >1and > 0.25,
depending on the history used to contextualise it, this
game is known as either hawkdove or chicken game.
Both anticipation and dissuasion are modelled here,
where fear of consequences makes one of the parts
give up.
() Level of conict 4 (C): if >1and < 0.25,thisisthe
classic prisoners dilemma, where individual incentives
lead to an inecient allocation of resources.
Based on the system of incentives, it is possible to explain
why these games are ordered according to their level of
conict, from lowest to highest (see Tab l e ). In the C
game the players’ preferences are well aligned and there is
no coordination problem because the Nash equilibrium is
unique in dominant strategies. erefore, a rational player
will always choose to cooperate , which will lead to the
outcome that is Pareto optimal. In the C game mutual
cooperation (,)is a Nash equilibrium, but it is not unique
in pure strategies. e problem lies in coordinating on either
a Pareto dominant equilibrium (,) or a risk dominant
equilibrium (,). In other words, negotiating as a dove
implies a higher risk and will only take place if a player
believes that the adversary will do the same. is is the reason
why it is possible to state that lack of trust between the parties
may lead to the disagreement point.
e C game portrays an environment with higher levels
of conict, since there are two equilibria with unequal payos.
In other words, players face two problems, a distributive and a
coordination one. If only one of the players chooses to behave
aggressively, this will turn the result in his/her favour, but it
is impossible to predict who will be aggressive and who will
cooperate. In this 2×2environment there is no clear criterion
to predict the nal outcome and therefore the behaviour.
T : e conict game: illustrative cases.
(a) C: −=0
, ., .
., . ., .
(b) C: −=0.5
, , .
., ., .
(c) C: −=1
, ., .
., . ., .
(d) C: −=1.5
, , .
., ., .
e last game is the classical social dilemma about the
limitations of rational behaviour to allocate resources e
ciently. e C game is classied as the most conictive one
because the players are faced with a context where the rational
choice clearly predicts that the disagreement point will be
reached. Additionally, we will argue along this document that
changing incentives to achieve mutual cooperation is not a
simple task in this bargaining environment.
Until this moment we have used equilibrium unicity and
its optimality to argue that the games are ordered by their
level of conict. However, it is possible to understand the
dierence in payos (−)as a proxy of the level of conict.
In other words, the dierence in payos between the player
who takes the advantage by playing aggressively and the
player who is exploited for cooperating is large, we can state
that the incentives lead players to a preference to behave
aggressively (see the illustrative cases in Ta ble ).
3. Response Rules and Commitment Messages
We consider now the conict game with a sequential decision
making protocol. e idea is to capture a richer set of
strategies that allows us to model threats and promises as
selfserving messages. In addition, the set of conditioned
strategies include the possibility of implementing ordinary
commitment, because a simple unconditional message is
always available for the sender.
Schelling [] distinguishes between two dierent types
of strategic moves: ordinary commitments and threats. An
ordinary commitment is the possibility of playing rst,
Journal of Applied Mathematics
dh
dhdh
1
1
x
y
y
x
0.25
0.25
P1
P2
P2
u1
u2
F : e conict game with perfect information.
announcing that a decision has already been made and it
is impossible to be changed, which forces the opponent to
make the nal choice. On the other hand, threats are second
player moves, where she convincingly pledges to respond to
the opponent’s choice in a specied contingent way (see []).
3.1. e Conict Game with Perfect Information. Suppose that
player 1moves rst and player 2observes the action made by
player and makes his choice. In theoretical terms, this is a
switch from the 2×2strategic game to the extensive game
with perfect information in Figure .Astrategyforplayer2is
a function that assigns an action 2∈{,}to each possible
action of player , 1∈{,}.us,thesetofstrategiesfor
player 2is 2={,,,},where2=22 represents
a possible reaction rule, such that the rst component 2
denotes the action that will be carried out if player 1plays
, and the second component 2 is the action in case that 1
plays . e set of strategies for player 1is 1={,}.
In this sequential game with perfect information a strat
egy prole is (1,2). erefore, the utility function (1,2)
is dened by (,22)=
(,2)and (,22)=
(,2),basedonthe2×2payo matrix presented before.
As the set of strategy proles becomes wider, the predictions
based on the Nash equilibrium are less relevant. us, in
the conict game with perfect information the applicable
equilibrium concept is the subgame perfect Nash equilibrium
(SPNE).
Denition 1 (SPNE). e strategy prole (∗
1,∗
2)is a SPNE
in the conict game with perfect information if and only if
2(1,∗
2)≥
2(1,2)for every 2∈
2and for every 1∈
1;
and 1(∗
1,∗
2)≥1(1,∗
2)for every 1∈1.
e strategy ∗
2=
∗
2∗
2 represents the best response
for player 2in every subgame. In the same way, the strategy
∗
1is the best response for player 1when player 2chooses
∗
2. By denition and using the payos assumptions, it is
clear that the strategy ∗
2=
∗
2∗
2 istheuniqueweakly
dominant strategy for player 2and, in consequence, the
reason for player to forecast his counterpart’s behaviour
basedonthecommonknowledgeofrationality.eforecast
T : SPNE in the conict game with perfect information.
(∗
1,∗
2)(
∗
1,∗
2)Pareto optimal
C (,) (, ) Yes
C (,) (, ) Yes
C (,) (,) Ye s
C (,) (., .) No
possibility leads to a rst mover advantage, as we can see in
Proposition .
Proposition 2 (rst mover advantage). If (∗
1,∗
2)is a SPNE in
the conict game with perfect information, then 1(∗
1,∗
2) =
and 2(∗
1,∗
2) =.
e intuition behind Proposition is that there is an
advantage related to the opportunity of playing rst, which is
the idea behind the ordinary commitment. In consequence,
the equilibrium that is reached is that in favour of Player ,
becausehealwaysobtainsatleastasmuchashisopponent.
isistrueexceptfortheCgame,becausethelevelof
conict is so high that regardless of what player chooses
he cannot improve his position. e SPNE for each game is
presented in Tab l e .
We can see that the possibility to play a response rule
is not enough to increase player ’s bargaining power. For
this reason, we now consider the case where player 2has the
possibility to announce the reaction rule she is going to play,
before player 1makes his decision.
3.2. reats and Promises as Binding Messages. Following
Schelling [], the sender’s bargaining power increases if she
is able to send a message about the action she is going to
play, since with premeditation other alternatives have been
rejected. For the receiver it must be clear that this is the
unique relevant option. is strategic move can be imple
mented if it is possible to send binding messages about second
mover’s future actions. With this kind of communication we
are going to show that there always exists a message that
allows player 2to reach an outcome at least as good as the
outcome in the SPNE. By notation, 2∈
2is a conditioned
message, where 2=
22.Fromnowon,player
represents the sender and player the receiver.
Denition 3 (commitment message). ∗
2∈
2is a commit
ment message if and only if 2(∗
1,∗
2)≥
2(∗
1,∗
2),where
1(∗
1,∗
2)≥
1(1,∗
2)for every 1∈
1.Itmeans∗
1 is
player best response given ∗
2.
e idea behind commitment messages is that player 2
wants to achieve an outcome at least as good as the one
without communication, given the receiver’s best response.
is condition only looks for compatibility of incentives,
since the receiver also makes his decisions in a rational way.
Following closely the formulations discussed in Schelling
[], Klein and O’Flaherty [], and Hirshliefer [], we classify
the commitment messages in three mutually exclusive cate
gories: warnings,threats,andpromises.
Journal of Applied Mathematics
T : Commitment messages.
Warning (∗
1,∗
2)reat (1,2)Promise (1,2)
C (,) (, ) (,) (, )
C (,) (, ) (,) (, )
C (,) (,) (,) (,) (,) (, )
C (,) (., .) (,) (, )
Denition 4 (warnings, threats, and promises). () e com
mitment message ∗
2∈2is a warning if and only if ∗
2=∗
2.
() e commitment message ∗
2∈
2is a threat if and
only if 2(,∗
2)=2(,∗
2)and 2(,∗
2)<2(,∗
2).
() e commitment message ∗
2∈2is a promise if and
only if 2(,∗
2)<2(,∗
2).
e purpose of a warning commitment is to conrm that
the sender will play her best response aer every possible
action of the receiver. Schelling does not consider warnings
as strategic moves, but we prefer to use it in this way
because the important characteristic of warnings is their full
credibility condition. If agents want to avoid miscoordination
related to the common knowledge of rationality, they could
communicateitandbelieveitaswell.Onthecontrary,
credibility is an inherent problem in threats and promises.
e second and third points in Denition show that at
least one action in the message is not the best response aer
observing the receiver’s choice. In threats, the sender does not
have any incentive to implement the punishment when the
receiver plays hawk. In promises, the sender does not have
any incentive to fulll the agreement when the receiver plays
dove.
e strategic goal in the conict game is to deter the oppo
nent of choosing hawk, because by assumption (,) >
(,). is is exactly the purpose of these binding messages,
as shown in Proposition .
Proposition 5 (second mover advantage). If
2is a threat or
a promise in the conict game with perfect information, then
∗
1
=.
e intuition behind Proposition is that, in Schelling’s
terms, if a player has the possibility to announce her inten
tions,shewillusethreatsorpromisestogainanadvantage
over the rst mover. at is, player uses these messages
because, if believed by player , she can make him cooperate.
Proposition species for which cases player inuences
player ’s choices by means of threats and promises. at is,
in which cases, when player has no incentives to cooperate,
messages can prompt a change in his behaviour.
Proposition 6 (message eectivity). ere exists a commit
ment message ∗
2such that 2(∗
1,∗
2)>
2(∗
1,∗
2)if and
only if >1.
erefore, threats and promises provide a material advan
tage upon the adversary only in cases with high conict (e.g.,
C and C). us, the condition >1is not satised
inCandCcases,wherethelevelofconictislow.
e implication is that mutual cooperation is achieved in
equilibrium and this outcome is the highest for both players.
e use of messages under these incentives only needs to
conrm the sender’s rational choice. If player 2plays ∗=
∗
2, receiver can anticipate this rational behaviour, which is
completely credible. is is exactly the essence of the subgame
perfect Nash equilibrium proposed by Selten [].
An essential element of commitments is to determine
underwhatconditionsthereceivermusttakeintoaccountthe
contentofamessage,giventhatthecommunicationpurpose
is to change the rival’s expectations. e characteristic of a
warning is to choose the weakly dominant strategy, but for
threats or promises at least one action is not a best response.
Proposition shows that in the C and C cases the sender’s
outcome is strictly higher if she can announce that she does
not follow the subgame perfect strategy. We summarise these
ndings in Tabl e .
Up to this point we have considered the rst two elements
of commitment theory. We started by illustrating that the
messages sent announce the intention the sender has to
execute a plan of action (i.e., the choice of a response rule).
Subsequently, we described for which cases messages are
eective (i.e., selfserving announcements). Now we inquire
aboutthecredibilityofthesestrategicmoves,becauseifthe
sender is announcing that she is going to play in an opposite
way to the game incentives, this message does not change the
receiver’s beliefs. e message is not enough to increase the
bargaining power. It is necessary that the specied action is
actually the one that will be played, or at least that the sender
believes it. e objective in the next section is to stress the
credibility condition. It is clear that binding messages imply
a degree of commitment at a % level, but this condition
is very restrictive, and it is not a useful way to analyse a
real bargaining situation. We are going to prove that for a
successful strategic move the degree of commitment must be
high enough, although it is not necessary to tell the truth with
a probability equal to .
4. The Conflict Game with
Nonbinding Messages
e credibility problem is related to how likely it is that the
message sent coincides with the actions chosen. e sender
announces her way of playing, but it could be a blu. In
other words, the receiver can believe in the message if it is
highly probable that the sender is telling the truth. In order
to model this problem the game now proceeds as follows. In
the rst stage Nature assigns a type to player 2following a
probability distribution. e sender’s type is her action plan;
her way of playing in case of observing each of the possible
Journal of Applied Mathematics
receiver’s action. In the second stage player 2observes her
type and sends a signal to player 1.esignalisthedisclosure
of her plan, and it can be seen as a noisy message, because it
is nonbinding. In the last stage, player 1,aerreceivingthe
signal information, chooses an action. is choice determines
the players’ payos together with the actual type of player 2.
Following the intuition behind credible message prole in
Rabin [], a commitment announcement can be considered
credible if it fullls the following conditions. (i) When
the receiver believes the literal meanings of the statements,
the types sending the messages obtain their best possible
payo; hence those types will send these messages. (ii) e
statements are truthful enough.eenough comes from the
fact that some types might lie to player 1by pooling with a
commitment message and the receiver knows it. However, the
probability of facing a lie is small enough that it does not aect
player ’s optimal response.
e objective of this section is to formalise these ideas
using our benchmark conict game.estrategiccredibility
problemisintrinsicallydynamic,anditmakessenseif
we consider threats and promises as nonbinding messages.
Bearing these considerations in mind, from now on the
messages are used to announce the sender’s intentions, but
they are cheap talk. Clearly, negotiators talk, and in most of
the cases it is free, but we show that this fact does not imply
that cheap talk is meaningless or irrelevant.
4.1. e Signalling Conict Game. Consider a setup in which
player 2moves rst; player 1observes a message from player
2but not her type. ey choose as follows: In the rst
stage Nature assigns a type 2to player 2as a function that
assigns an action 2∈{,}to each action 1∈{,}.
Player ’s type set is Θ2=
2= {,,,},where
2=
22.Nature chooses the sender’s type following a
probability distribution, where (2)>0is the probability
to choose the type 2,and∑2∈Θ2(2)=1.Inthesecond
stage, player 2observes her own type and chooses a message
2∈Θ
2.Atthenalstage,player1observes this message
and chooses an action from his set of strategies 1={,}.
e most important characteristic of this conict game with
nonbinding messages is that communication cannot change
the nal outcome. ough strategies are more complex in this
case, the 2×2payo matrix in the conict game is always the
way to determine the nal payos.
In order to characterise the utility function we need some
notation. A message prole 2=(
,,,)is a
function that assigns a message 2∈Θ
2to each type 2∈
Θ2.erstcomponent ∈
2is the message chosen in
case of observing the type 2=; the second component
∈2is the message chosen in case of observing the type
2=,andsoon.Bynotation,2=
22 is a specic
message sent by a player with type 2,and2=(2,−2)is
a generic message prole with emphasis on the message sent
by the player with type 2.
ere is imperfect information because the receiver can
observe the message, but the sender’s type is not observ
able. us, the receiver has four dierent information sets,
depending on the message he faces. A receiver’s strategy
1 =(
1,1,1,1)is a function that assigns
an action 1∈
1to each message 2∈
2,where1 is
the action chosen aer observing the message 2=,and
so on. In addition, 1 =(
1,1(−))is a receiver’s generic
strategy with emphasis on the message he faced. In this case,
the subindex is the way to highlight that the receiver’s
strategies are a prole of single actions. erefore, in the
conictgamewithnonbindingmessagesthe utility function
is (1𝜃2,1(−𝜃2),2,−2)=
(1,2)for 1𝜃2=
1and
2=2.
In this specication, messages are payo irrelevant and
what matters is the sender’s type. For this reason, it is
necessary to dene the receiver’s beliefs about who is the
sender when he observes a specic message. e receiver’s
belief 22≥0is the conditional probability of obtaining
the message from a sender of type 2,giventhatheobserved
the message 2.Naturally,∑2∈Θ222=1.
All the elements of the conict game with nonbinding
messages aresummarisedinFigure .emostsalientchar
acteristics are the four information sets in which the receiver
must choose and that messages are independent of payos.
For instance, the upper le path (blue) describes each possible
decision for the sender of type .Intherstplace,Nature
chooses the sender’s type; in this case 2=.Inthe
next node, must choose a message from the possible
reaction rules. We say that is telling the truth if she
chooses =, leading to the information set at the
top. We intentionally plot the game in a star shape in order
to highlight the receiver’s information sets. At the end, the
receiver chooses between and ,andcheap talk implies that
therearefeasiblepayos.
e signalling conict game has a great multiplicity of
Nash equilibria. For this particular setting, a characterisation
of this set is not our aim. Our interest lies on the character
isation of the communication equilibrium. For this reason
theappropriateconceptinthiscaseistheperfectBayesian
equilibrium.
Denition 7 (PBE). A perfect Bayesian equilibrium is a
sender’s message prole ∗
2=(
∗
,∗
,∗
,∗
),a
receiver’s strategy prole ∗
1 =(
∗
1,∗
1,∗
1,∗
1),anda
beliefs prole ∗
𝑠𝑠aer observing each message 2,ifthe
following conditions are satised:
() ∗
2is the argmax𝜃2∈Θ22(∗
1,2,−2),
() ∗
1 is the argmax𝜃2
1∈1∑2∈Θ222⋅1(1𝜃2,1(−𝜃2),
∗
2),
() ∗
22must be calculated following Bayes’ rule based
on the message prole ∗
2.Forall2who play the
message ∗
2,thebeliefsmustbecalculatedas2∗
2=
2/∑∗
2.
e conditions in this denition are incentive compatibil
ity for each player and Bayesian updating. e rst condition
requires message ∗
2to be optimal for type 2.esecond
requires strategy ∗
1 to be optimal given the beliefs prole
∗
22.Forthelastcondition,Bayesianupdating,thereceiver’s
Journal of Applied Mathematics
u1
u2
d
h
d
h
d
h
d
h
x
y
x
yx
y
x
y
N
1,1
1
1
1
1
1
1
1
1
1,1
x,y
x,y
0.25,0.25
0.25
0.25
0.25
0.25
0.25,0.25
𝛼hh
𝛼dh
𝛼dh
𝛼dh
𝛼dh
𝛼hd
𝛼hd
𝛼hd
𝛼hd
𝛼dd
𝛼dd
𝛼dd
𝛼dd
𝜃hh
𝜃dh
𝜃hd
𝜃dd
mhh
mdh
mhd
mdd
d
h
d
h
d
h
d
h
dhdhdhdh
dh
dhdh
dh
𝛼hh
𝛼hh
𝛼hh
u1,u2
y,x
y,x
1,1
1,1
x,y
x,y
0.25,0.25
0.25,0.25
u1,u2
y,x
y,x
u1
u2x
y
x
yx
y
x
y
1
1
1
1
0.25
0.25
0.25
0.25
F : Conict game with nonbinding messages.
beliefs must be derived via Bayes’ rule for each observed
message, given the equilibrium message prole ∗
2.
4.2. e Commitment Equilibrium Properties. ere are, in
general, several dierent equilibria in the conict game with
nonbinding messages.eobjectiveofthissectionistoshow
that a particular equilibrium that satises the following
properties leads to a coordination outcome, given it is both
salient and in favour of the sender. In what follows we will
present Axioms and which will be used to explain which
is the particular equilibrium that can be used as a theoretical
prediction in experimental games with dierent levels of
conict.
Axiom 1 (truthtelling beliefs). If the receiver faces a message
∗
2=
2,then
2
2>0.Ifthemessage2= 2is not part
of the messages prole ∗
2,then22=1.
Following Farrell and Rabin [] we assume that people in
real life do not seem to lie as much or question each other’s
statements as much, as the game theoretic predictions state.
Axiom captures the intuition that for people it is natural to
take seriously the literal meaning of a message. is does not
mean that they believe everything they hear. It rather states
that they use the meaning as a starting point and then assess
credibility, which involves questioning in the form of “why
would she want me to think that? Does she have incentives to
actually carry out what she says?”
More precisely, truthtelling beliefs emphasise that in
equilibrium when the receiver faces a particular message,
its literal meaning is that the sender has the intention of
playing in this way. us, the probability of facing truth
telling messages must be greater than zero. In the same
way, when the sender does not choose a particular message,
sheissignallingthattherearenoincentivestomakethe
receiver believe this, given that the receiver’s best response
is . erefore, we can assume that the receiver must fully
believe in the message, because both players understand that
the purpose of the strategic move is to induce the receiver
to play .Ifthesenderissignallingtheopposite,sheis
showing her true type by mistake; then the receiver believes
her with probability (see the column “belief of truthtelling”
in Table ).
Axiom 2 (senders’ bargaining power). If ∗
2is part of the
messages prole ∗
2,then∗
1𝜃2=.
Axiom captures the use of communication as a means
to inuence the receiver to play dove. at is, there is an
equilibrium where the only messages sent are those that
induce the receiver to cooperate. In order to characterise a
communication equilibrium such as the one described above,
we rst focus on the completely separating message prole,
when the sender is telling the truth. Naturally, 2is a truth
telling message if and only if
2=
2(see column “message
by type” in Ta ble ), and given the message the receiver’s best
Journal of Applied Mathematics
T : Perfect Bayesian equilibria that satisfy Axioms and .
Message by type Player ’s best resp. Belief of truthtelling
∗
,∗
,∗
,∗
∗
1,∗
1,∗
1,∗
1
∗
,∗
,∗
,∗

C (,,,)(
,,,)
+,1,1,1
C (,,,)(
,,,)
+,
+,1,1
C (,,,)(
,,,)1,
+,1,
+
C (,,,)(
,,,)1,
+ + + ,1,1
response will be to cooperate (see column “player ’s best
response” in Table ).
With this in mind, it is possible to stress that a contri
bution of our behavioural model is to develop experimental
designs that aim to unravel the strategic use of communi
cation to inuence (i.e., manipulate) others’ behaviour. at
is, the Nash equilibrium implies that players must take the
other players’ strategies as given and then they look for their
best response. However, commitment theory, in Schelling’s
sense, implies an additional step, where players recognise that
opponents are fully rational. Based on this fact, they evaluate
dierent techniques for turning the other’s behaviour into
their favour. In our case, the sender asks herself, “is is the
outcome I would like from this game; is there anything I can
do to bring it about?”
Proposition 8 (there is always a liar). e completely truth
telling messages prole 2=(,,,)cannot be part of
any PBE of the conict game with nonbinding messages.
Proposition shows that the completely truth telling
message prole is not an equilibrium in the conict game.
e problem lies in the sender type ,becauserevealingher
actual type is not incentive compatible and there always exists
at least one successful message to induce the counterpart to
play dove. For this reason, we can ask whether there exists
some message that induces the sender to reveal her actual
type but at the same time leads to a successful strategic move.
Denition is the bridge between nonbinding messages and
commitment messages presented in the previous section.
Denition 9 (selfcommitting message). Let
∗
2be a truth
telling message and
2
∗
2(
2)=1.
∗
2is a selfcom
mitting message if and only if
2(∗
1,
∗
2,
∗
−
2)≥
2(∗
1,
2,
∗
−
2),forevery
2∈Θ2.
We introduce t h e selfcommitting message property
becausewewanttostressthatastrategicmoveisatwo
stage process. Not only is communication useful in revealing
information, but also it can be used to manipulate others’
behaviour. e sender of a message must consider how the
receiver would react if he believes it and if that behaviour
works in her favour she will not have incentives to lie.
Amessageisselfcommitting and, if believed, it creates
incentives for the sender to fulll it []. e idea behind a
threat or a promise is to implement some risk for the opponent
in order to inuence him, but this implies a risk for the
sender too. is fact has led to associating strategic moves
with slightly rational behaviours, when actually, in order to
be executed, a very detailed evaluation of the consequences is
needed. Proposition and its corollary explain the relation
between the conditioned messages and the incentives to tell
the truth.
Proposition 10 (incentives to commit). Let
2=
∗
2
be a commitment message in the conict game with perfect
information. If ∗
1∗(
2)=,then
∗
2is a selfcommitting
message.
Corollary to Proposition 10.If
2is a threat or a promise in
the conict game with perfect information, then
∗
2=
2is a
selfcommitting message.
e intuition behind Proposition and its corollary is
that if a message induces the other to cooperate, then the
sender has incentives to tell the truth. Moreover, as illustrated
in Proposition ,threatsandpromisesalwaysinducethe
counterpart to cooperate; therefore, they endogenously give
the sender incentives to comply with what is announced.
As we can see in the conict game with perfect information
(for an illustration see Ta b l e ), in the C and C cases
the warning is the way to reach the best outcome. If we
consider the possibility to send nonbinding messages when
the sender’s type is equal to a warning strategy, then revealing
her type is selfcommitting. e problem in the C and C
cases is more complex given the warning message is not self
committing and the way to improve the bargaining power
is using a threat or a promise. is fact leads to a tradeo
between choosing a weakly dominant strategy that fails to
inducetheopponenttoplaydoveandastrategythatimproves
her bargaining power but implies a higher risk for both of
them.
e required elements for a perfect Bayesian equilibrium
at each game are shown in Tables and .Itisimportant
to bear in mind that the beliefs that appear in Table are
necessary conditions for implementing the PBE presented in
Table , given that they satisfy truthtelling beliefs and sender’s
bargaining power.
Journal of Applied Mathematics
T : Beliefs that support the perfect Bayesian equilibrium.
Warning reat Promise
C ∗
 ≥∗
 −
1− Tr u th
C ∗
 ≥∗
 (0.25−)
0.75 ∗
 ≥∗
 −
1−
C Lie ∗
 ≥∗
 −
(−0.25)∗
 ≥∗
(−1)
0.75
C Lie ∗
 ≥∗
 −1+∗
 −+∗
 (0.25−)
0.75
e problem of which message must be chosen is as
simple as follows in the next algorithm: rst, the sender tells
thetruth.Ifthetruthtellingmessageleadsthereceivertoplay
dove, then she does not have any incentive to lie. In the other
case, she must nd another message to induce the receiver
to play dove. If no message leads the receiver to play dove,
messages will lack any purpose, and she will be indierent
between them.
Table shows the messages; the receivers’ strategies and
theirbeliefprolesinaparticularequilibriumweargueis
themostsalient.Asweshowedabove,intheconictgame
(see Table ) the sender is always in favour of those messages
where the receiver’s best response is dove. In the C case
there are three dierent messages, in the C and C cases
therearetwomessages,andtheworstsituationistheCcase,
where every type of player sends the same message. is fact
leads to a rst result: if the conict is high, there are very
strong incentives to lie and communication leads to a pooling
equilibrium.
In addition, notice that Tab l e species which messages
will be used as commitment messages in the conict game
with binding communication illustrated in Figure .atis,
if credibility is exogenous the theoretical prediction would be
that such messages are sent. is means that messages are not
randomly sent, but there is a clear intention behind them, to
inducethereceivertochoosetheactionmostfavourablefor
the sender. Now, Tab l e presents the minimum probability
threshold that can make the strategic move successful. at
is, if credibility is suciently high the message works and
achieves its purpose, in the conict game with nonbinding
communication illustrated in Figure .
In Section we assumed that the sender could communi
cate a completely credible message in order to inuence her
counterpart. e question is, how robust is this equilibrium
if we reduce the level of commitment? Proposition sum
marises the condition for the receiver to choose dove as the
optimal strategy. It is the way for calculating the beliefs that
are shown in Table .
Proposition 11 (incentives to cooperate). ∗
1(2)=if and
only if (1−)(2)+(0.75)(2)+(−)(2)+(−
0.25)(2)≥0.
Based on Proposition ,thesecondresultisthatcheap
talk always has meaning in equilibrium. We consider that
this equilibrium selection is relevant because the sender
focuses on the communication in the literal meanings of the
statements but understands that some level of credibility is
necessary to improve her bargaining power. Table sum
marises the true enough property of the statements. Here, the
receiverupdateshisbeliefsinarationalwayandhechoosesto
play dove if and only if it is his expected best response. We can
interpret the beliefs in Table as a threshold, because if this
condition is satised, the sender is successful in her intention
of manipulating the receiver’s behaviour. us, some level of
credibility is necessary, but not at a % level.
Itisclearthatiftheconictishigh,thecommitment
threshold is also higher. In C and C cases the sender must
commit herself to implement the warning strategy, which
is a weakly dominant strategy. In the C case the strategic
movement implies a threat or a promise,formulatingan
aggressive statement in order to deter the receiver from
behaving aggressively. e worst situation is the C case,
where there is only one way to avoid the disagreement
point, to implement a promise. e promise in this game is
a commitment that avoids the possibility of exploiting the
opponent, because fear can destroy the agreement of mutual
cooperation.
In the scope of this paper, threats are not only pun
ishments and promises are not only rewards. ere is a
credibility problem because these strategic moves imply a
lack of freedom in order to avoid the rational selfserving
behaviourinasimpleonestepofthinking.eparadox
is that this decision is rational if the sender understands
that her move can inuence other players’ choices, because
communication is the way to increase her bargaining power.
is implies a second level of thinking, such as a forward
induction reasoning.
5. Conclusions
In this paper we propose a behavioural model following
Schelling’s tactical approach for the analysis of bargaining.
In his Essay on Bargaining , Schelling analyses situations
where subjects watch and interpret each other’s behaviour,
each one better acting taking into account the expectations
that he creates. is analysis shows that an opponent with
rational beliefs expects the other to try to disorient him
and he will ignore the movements he perceives as stagings
especially played to win the game.
e model presented here captures dierent levels
of conict by means of a simple parameterisation. In
Journal of Applied Mathematics
a bilateral bargaining environment it analyses the strategic
use of binding and nonbinding communication. Our ndings
show that when messages are binding, there is a rst mover
advantage. is situation can be changed in favour of the
second mover, if the latter sends threats or promises in a
preplay move. On the other hand, when players have the
possibility to send nonbinding messages, their incentives to
lie depend on the level of conict. When conict is low, the
sender has strong incentives to tell the truth and cheap talk
will almost fully transmit private information. When conict
is high, the sender has strong incentives to blu and lie.
erefore, in order to persuade the receiver to cooperate with
her nonbinding messages, the sender is required to provide a
minimumlevelofcredibility(notnecessarilya%).
In summary, the equilibrium that satises truthtelling
beliefs and sender’s bargaining power allows us to show that
thelessconictthegamehas,themoreinformativethe
equilibrium signal is, and the less stronger the commitment
needed to implement it is. Our equilibrium selection is based
on the assumption that in reality people do not seem to lie
as much, or question each other’s statements as much, as
rational choice theory predicts. For this reason, the conict
game with nonbinding messages is a good environment to test
dierent game theoretical hypotheses, because it is simple
enough to be implemented in the lab.
With this in mind, the strategic use of communication
in a conict game, as illustrated in our model, is the right
way to build a bridge between two research programs:
the theory on bargaining and that on social dilemmas. As
Bolton [] suggested, bargaining and dilemma games have
been developed in experimental research as fairly separate
literatures. For bargaining, the debate has been centred on
the role of fairness and the nature of strategic reasoning.
For dilemma games, the debate has involved the relative
weights that should be given to strategic reputation building,
altruism, and reciprocity. e benet of the structure and
payo scheme we propose is to study all these elements at
the same time. Our model provides a simple framework
to gather and interpret empirical information. In this way,
experiments could indicate which parts of the theory are most
usefultopredictsubjects’behaviourandatthesametimewe
can identify behavioural parameters that the theory does not
reliably determine.
Moreover, the game presented here can be a very useful
tool to design economic experiments that can lead to new
evidence about bilateral bargaining and, furthermore, about
human behaviour in a wider sense. On the one hand, it
can contribute to a better understanding of altruism, self
ishness, and positive and negative reciprocity. A model that
only captures one of these elements will necessarily portray
an incomplete image. On the other hand, bargaining and
communication are fundamental elements to understand the
power that one of the parts can have.
In further research, we are interested in exploring the
emotional eects of cheating or being cheated on, particularly
by considering the dilemma that takes place when these
emotional eects are compared to the possibility of obtaining
material advantages. To do so, it is possible to even consider
a simpler version of our model using a coarser type space
(e.g., only hawk and dove). is could illustrate the existing
relationship between the level of conict and the incentives
to lie. As the model predicts, the higher the level of conict
the more incentives players have to not cooperate, but they are
better o if the counterpart does cooperate. erefore, players
with type hawk wouldbemoreinclinedtolieanddisguise
themselves as cooperators. By measuring the emotional
componentoflyingandbeingliedto,wewillbeableto
show that people do not only value the material outcomes of
bargainingbutthatthemeansusedtoachievethoseendsare
also important to them.
Appendix
Proof of Proposition 2.Suppose that 1(∗
1,∗
2)=and
2(∗
1,∗
2)=;then≥1.If∗
2=,then1(,) ≥
1(,) and ≥, but by assumption >.If∗
2=,
then 1(,) ≥ 1(,)and ≥.,andatthesametime
2(,) ≥ 2(,).eonlycompatiblecaseis0.25 = ,
but by assumption . =. erefore, 1(∗
1,∗
2) =and
2(∗
1,∗
2) =.
Proof of Proposition 5.Let
2be a threat or a promise.
Following Denitions and ,2(∗
1
,
∗
2)≥
2(∗
1,∗
2).
Suppose that ∗
1
=; then there are two possibilities,
∗
2=∗
2
or 2(,
∗
2)≥2(∗
1,∗
2).If
∗
2=∗
2, then by denition
∗
2is
neither a threat nor a promise. If 2(,
∗
2)≥2(∗
1,∗
2),then
∗
1=or ∗
1=.If∗
1=, by assumption 2(,
∗
2)<
2(,∗
2).If∗
1=and
2is a threat, then 2(,
∗
2)<
2(∗
1,∗
2).If∗
1=and
2isapromise,itmustfulll
2(,
∗
2)≥
2(,∗
2)and 2(,
∗
2)<
2(,∗
2).eC
and C games are not under consideration because ∗
1=
and for C y C cases there are no messages for which these
conditions are true at the same time. erefore, ∗
1
=.
Proof of Proposition 6.Let us consider the message 2=
.ByProposition we know that 2(∗
1,∗
2) =,and
by assumption 1(,) > 1(,),then2=is a
commitment message, because 2(,)=1≥
2(∗
1,∗
2).
If 2(,) > 2(∗
1,∗
2),then1>
2(∗
1,∗
2), to satisfy this
condition and using Proposition again; we conclude that
(∗
1,∗
2) = (,∗
2).As∗
2=
∗
2 and it is part of the SPNE,
then 2(,∗
2)>2(,∗
2), and therefore >1.
e proof in the other direction is as follows. Let >1;
then ∗
2=∗
2.UsingProposition we know that 1(∗
1,∗
2) =
; therefore ∗
1=.Now2(∗
1,∗
2)<1.Asweshowin
the rst part, 2=is a commitment message such
that 2(∗
1,) = 1. erefore, there exists a commitment
message such that 2(∗
1,∗
2)>2(∗
1,∗
2).
Proof of Proposition 8.Consider the senders’ types =
and =.If∗
2is a completely truthtelling
message, then ∗
 =1and ∗
 =1. By assumptions
1(,1(−),,−)=1and 1(,1(−),,−)=
0.25,then∗
1 =.Inthesameway,1(,1(−),,−)=
and 1(,1(−),,−)=;then∗
1 =. erefore,
the utility for the sender is (,1(−),,−)=1
and (,1(−),,−)=. ese conditions imply
that the sender type has incentives to deviate and 2=
(,,,)cannot be part of any PBE.
Journal of Applied Mathematics
Proof of Proposition 10.Let
2=
∗
2be a commitment
message in the conict game with perfect information and
∗
1∗(
2)=.If
∗
2=
2is not a selfcommitting mes
sage, then another message
2must exist such that
2(,1(−∗(
2)),
∗
2,∗
−
2)<
2(∗
1,
2,∗
−
2).Giventhe
payo assumptions,
2(,1(−∗(
2)),
∗
2,∗
−
2)≥
2(∗
1,
∗
2,∗
−
2)for every ∗
2∈Θ
2. erefore,
∗
2=
2is a self
committing message.
ProofofCorollarytoProposition 10.e proof to the corollary
follows from Propositions and , and thus it is omitted.
Proof of Proposition 11.e expected utility for each receiver’s
strategy is as follows:
1(,2)=1
(2)+1
(2)+
(2)+
(2),
1(,2)=
(2)+0.25(2)+(2)+
0.25(2),
therefore, 1(,2)≥
1(,2)if and only if (1−
)(2)+(0.75)(2)+(−)(2)+(−
0.25)(2)≥0.
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
Acknowledgments
is paper was elaborated during the authors’ stay at the
University of Granada, Spain. e authors are grateful for
the help and valuable comments of Juan Lacomba, Francisco
Lagos, Fernanda Rivas, Aurora Garc´
ıa, Erik Kimbrough,
Sharlane Scheepers, and the seminar participants at the
VI International Meeting of Experimental and Behavioural
Economics (IMEBE) and the Economic Science Association
World Meeting (ESA). Financial support from the
Spanish Ministry of Education and Science (Grant code
SEJ/ECON), the Proyecto de Excelencia (Junta de
Andaluc´
ıa, PSEJ), and the Project VIE from the
Universidad Industrial de Santander (UIS) in Bucaramanga,
Colombia, is also gratefully acknowledged.
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