This paper studies the conditions that improve bargaining power using threats and promises. We develop a model of strategic communication, based on the
Bargaining power refers to the relative ability that a player has in order to exert influence upon others to improve her own wellbeing. It is related also to idiosyncratic characteristics such as patience, so that a player turns the final outcome into her favour if she has better outside options or if she is more patient [
In bargaining theory,
Commitment theory was proposed by Schelling [
There are three principal reasons for modelling preplay communication: information disclosure (signalling), coordination goals (cheap-talk), and strategic influence (in Schelling’s sense). Following Farrell [
Our paper contributes to the strategic communication literature in three ways. First, we propose a particular characterisation of
Second, we model strategic moves with nonbinding messages, showing that choosing a particular message and its credibility are related to the level of conflict. In this way, the
Third, 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 environment, information and communication influence 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 is the best he could do. Players, in a first stance, cannot influence others’ behaviour. On the contrary, Schelling [
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:
The next step is to show that credibility is related to the probability that the sender fulfills the action specified in the nonbinding message. For this, we highlight that players share a common language, and the literal meaning must be used to evaluate whether a message is credible or not. Hence, the receiver has to believe in the literal meaning of announcements if and only if it is highly probable to face the truth. Technically, we capture this intuition in two axioms:
The paper is organised as follows. In Section
The
The
Player 2 | |||
|
| ||
|
|||
Player 1 |
|
1, 1 |
|
|
|
0.25, 0.25 |
Note that both mutual cooperation and mutual defection lead to equal payoffs, and the combination of strategies
Under these assumptions,
Based on the system of incentives, it is possible to explain why these games are ordered according to their level of conflict, from lowest to highest (see Table
Nash equilibria in the conflict game.
|
|
Pareto optimal | |
|
|||
C1 |
|
(1, 1) | Yes |
|
|||
C2 |
|
(1, 1) | Yes |
|
(0.25, 0.25) | No | |
|
|||
C3 |
|
|
Yes |
|
|
Yes | |
|
|||
C4 |
|
(0.25, 0.25) | No |
The C3 game portrays an environment with higher levels of conflict, since there are two equilibria with unequal payoffs. 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
Until this moment we have used equilibrium unicity and its optimality to argue that the games are ordered by their level of conflict. However, it is possible to understand the difference in payoffs
The conflict game: illustrative cases.
C1:
2 | |||
|
| ||
|
|||
1 |
|
1, 1 | 0.5, 0.5 |
|
0.5, 0.5 | 0.25, 0.25 |
C2:
2 | |||
|
| ||
|
|||
1 |
|
1, 1 | 0, 0.5 |
|
0.5, 0 | 0.25, 0.25 |
C3:
2 | |||
|
| ||
|
|||
1 |
|
1, 1 | 0.5, 1.5 |
|
1.5, 0.5 | 0.25, 0.25 |
C4:
2 | |||
|
| ||
|
|||
1 |
|
1, 1 | 0, 1.5 |
|
1.5, 0 | 0.25, 0.25 |
We consider now the conflict game with a sequential decision making protocol. The idea is to capture a richer set of strategies that allows us to model threats and promises as self-serving 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 [
Suppose that player
The conflict game with perfect information.
In this sequential game with perfect information a strategy profile is
The strategy profile
The strategy
If
The intuition behind Proposition
SPNE in the conflict game with perfect information.
|
|
Pareto optimal | |
|
|||
C1 |
|
(1, 1) | Yes |
C2 |
|
(1, 1) | Yes |
C3 |
|
|
Yes |
C4 |
|
(0.25, 0.25) | No |
We can see that the possibility to play a response rule is not enough to increase player 2’s bargaining power. For this reason, we now consider the case where player
Following Schelling [
The idea behind
(1) The commitment message
(2) The commitment message
(3) The commitment message
The purpose of a
The strategic goal in the conflict game is to deter the opponent of choosing hawk, because by assumption
If
The intuition behind Proposition
Proposition
There exists a commitment message
Therefore, threats and promises provide a material advantage upon the adversary only in cases with high conflict (e.g., C3 and C4). Thus, the condition
An essential element of commitments is to determine under what conditions the receiver must take into account the content of a message, given that the communication purpose is to change the rival’s expectations. The characteristic of a
Commitment messages.
Warning |
|
Threat |
|
Promise |
|
|
|
||||||
C1 |
|
(1, 1) |
|
(1, 1) | ||
C2 |
|
(1, 1) |
|
(1, 1) | ||
C3 |
|
|
|
|
|
(1, 1) |
C4 |
|
(0.25, 0.25) |
|
(1, 1) |
Up to this point we have considered the first 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 effective (i.e., self-serving announcements). Now we inquire about the credibility of these strategic moves, because if the 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. The message is not enough to increase the bargaining power. It is necessary that the specified action is actually the one that will be played, or at least that the sender believes it. The objective in the next section is to stress the credibility condition. It is clear that binding messages imply a degree of commitment at a 100% 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 1.
The credibility problem is related to how likely it is that the message sent coincides with the actions chosen. The sender announces her way of playing, but it could be a bluff. 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 first stage
Following the intuition behind credible message profile in Rabin [
The objective of this section is to formalise these ideas using our benchmark
Consider a setup in which player
In order to characterise the utility function we need some notation. A message profile
There is imperfect information because the receiver can observe the message, but the sender’s type is not observable. Thus, the receiver has four different information sets, depending on the message he faces. A receiver’s strategy
In this specification, messages are payoff irrelevant and what matters is the sender’s type. For this reason, it is necessary to define the receiver’s beliefs about who is the sender when he observes a specific message. The receiver’s belief
All the elements of the
Conflict game with nonbinding messages.
The signalling conflict 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 characterisation of the communication equilibrium. For this reason the appropriate concept in this case is the perfect Bayesian equilibrium.
A perfect Bayesian equilibrium is a sender’s message profile
The conditions in this definition are incentive compatibility for each player and Bayesian updating. The first condition requires message
There are, in general, several different equilibria in the
If the receiver faces a message
Following Farrell and Rabin [
More precisely,
Perfect Bayesian equilibria that satisfy Axioms
Message by type | Player 1’s best resp. | Belief of truth-telling | |
|
|
| |
|
|||
C1 |
|
|
|
C2 |
|
|
|
C3 |
|
|
|
C4 |
|
|
|
If
Axiom
With this in mind, it is possible to stress that a contribution of our behavioural model is to develop experimental designs that aim to unravel the strategic use of communication to influence (i.e., manipulate) others’ behaviour. That 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 different techniques for turning the other’s behaviour into their favour. In our case, the sender asks herself, “This is the outcome I would like from this game; is there anything I can do to bring it about?”
The completely truth-telling messages profile
Proposition
Let
We introduce the
Let
Corollary to Proposition
The intuition behind Proposition
As we can see in the
The required elements for a perfect Bayesian equilibrium at each game are shown in Tables
Beliefs that support the perfect Bayesian equilibrium.
Warning | Threat | Promise | |
|
|||
C1 |
|
Truth | |
C2 |
|
|
|
C3 | Lie |
|
|
C4 | Lie |
|
The problem of which message must be chosen is as simple as follows in the next algorithm: first, the sender tells the truth. If the truth-telling message leads the receiver to play dove, then she does not have any incentive to lie. In the other case, she must find 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 indifferent between them.
Table
In addition, notice that Table
In Section
Based on Proposition
It is clear that if the conflict is high, the commitment threshold is also higher. In C1 and C2 cases the sender must commit herself to implement the
In the scope of this paper, threats are not only punishments and promises are not only rewards. There is a credibility problem because these strategic moves imply a lack of freedom in order to avoid the rational self-serving behaviour in a simple one step of thinking. The paradox is that this decision is rational if the sender understands that her move can influence other players’ choices, because communication is the way to increase her bargaining power. This implies a second level of thinking, such as a forward induction reasoning.
In this paper we propose a behavioural model following Schelling’s tactical approach for the analysis of bargaining. In his Essay on Bargaining 1956, Schelling analyses situations where subjects watch and interpret each other’s behaviour, each one better acting taking into account the expectations that he creates. This 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.
The model presented here captures different levels of conflict by means of a simple parameterisation. In a bilateral bargaining environment it analyses the strategic use of binding and nonbinding communication. Our findings show that when messages are binding, there is a first mover advantage. This 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 conflict. When conflict is low, the sender has strong incentives to tell the truth and cheap talk will almost fully transmit private information. When conflict is high, the sender has strong incentives to bluff and lie. Therefore, in order to persuade the receiver to cooperate with her nonbinding messages, the sender is required to provide a minimum level of credibility (not necessarily a 100%).
In summary, the equilibrium that satisfies
With this in mind, the strategic use of communication in a conflict 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 [
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, selfishness, 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 effects of cheating or being cheated on, particularly by considering the dilemma that takes place when these emotional effects 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). This could illustrate the existing relationship between the level of conflict and the incentives to lie. As the model predicts, the higher the level of conflict the more incentives players have to not cooperate, but they are better off if the counterpart does cooperate. Therefore, players with type
Suppose that
Let
Let us consider the message
The proof in the other direction is as follows. Let
Consider the senders’ types
Let
The proof to the corollary follows from Propositions
The expected utility for each receiver’s strategy is as follows:
therefore,
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was elaborated during the authors’ stay at the University of Granada, Spain. The authors are grateful for the help and valuable comments of Juan Lacomba, Francisco Lagos, Fernanda Rivas, Aurora García, 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 2010 (ESA). Financial support from the Spanish Ministry of Education and Science (Grant code SEJ2009-11117/ECON), the Proyecto de Excelencia (Junta de Andalucía, P07-SEJ-3261), and the Project VIE-1375 from the Universidad Industrial de Santander (UIS) in Bucaramanga, Colombia, is also gratefully acknowledged.