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Believing when credible: talking about future intentions and past actions

Abstract

In an equilibrium framework, we explore how players communicate in games with multiple Nash equilibria when messages that make sense are not ignored. Communication is about strategies and not about private information. It begins with the choice of a language, followed by a message that is allowed to be vague. We focus on equilibria where the sender is believed whenever possible, and develop a theory of credible communication. We show that credible communication is sensitive to changes in the timing of communication. Sufficient conditions for communication leading to efficient play are provided.

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Notes

  1. Note that we do not allow for the explicit communication of arbitrary mixtures of actions. The reasons are as follows. First, our aim is to formulate a model that is close to what we think happens in reality. We do not find that people communicate mixtures. Second, we wanted to formulate messages that could be verified ex post which we do not think is possible for mixtures.

  2. Andras Kornai helped us to clarify the justification of the use of partitions (via private communication).

  3. Assume that there are two messages \(m_{1}\) and \(m_{2}.\) The following constitutes a communicative sequential equilibrium. Player one sends \(m_{1},\) both play the mixed equilibrium after \(m_{1}\) and \( \left( R,R\right) \) after \(m_{2}.\) One can also find inefficient pure strategy communicative sequential equilibria in larger games. For instance, consider the symmetric pure coordination game with payoffs 10,  2 and 1 on the diagonal and 0 on the off-diagonal. Then \(\left( 2,2\right) \) can be supported by this concept.

  4. \(\left\{ m_{1},\ldots ,m_{k}\right\} \) is a partition of \(S_{1}\) if \(\cup _{i=1}^{k}m_{i}=S_{1}\) and for all ij in \( \left\{ 1,\ldots ,k\right\} \) with \(i\ne j\) we have that \(m_{i}\ne \emptyset ,\) \(m_{i}\subseteq S_{1}\) and \(m_{i}\cap m_{j}=\emptyset \).

  5. Along the lines of this result, an alternative and equivalent approach would be to drop the explicit choice of a language, and instead define the notion of a communication-proof equilibrium in the spirit of Zapater (1997).

  6. Generically, in any common interest game there is a unique efficient outcome—a Nash equilibrium outcome—which can be attained by a pure strategy profile. In particular, this outcome is the favorite Nash equilibrium outcome of both players.

  7. In our notation \(\left( \left( p_{1},p_{2},p_{3}\right) ,\left( q_{1},q_{2},q_{3}\right) \right) ,\) \(p_{1},\) \(p_{2}\) and \(p_{3}\) denote the probability of choosing T, M, and B, respectively. Similarly, \(q_{1},\) \(q_{2}\) and \(q_{3}\) denote the probability of choosing Z, N, and R, respectively.

  8. In our notation \((\left( p_{1},p_{2},p_{3}\right) ,\left( q_{1},q_{2},q_{3}\right) ) ,\) \(p_{1},\) \(p_{2}\) and \(p_{3}\) denote the probability of player one bidding 0,1, and 2, respectively. Similarly, \( q_{1},\) \(q_{2}\) and \(q_{3}\) denote the probability of player two bidding 0,  1,  and 2,  respectively.

  9. For the definition of supermodularity, see the Appendix.

  10. In fact, credible communication can easily be defined also for infinite action sets. However, one then needs to add some technical qualifications to be able to properly model mixed strategies and beliefs.

  11. Other variants of the model such as for example extending credible communication to more general languages, letting player two choose the language or when player one can talk, publicly or privately, to multiple audiences is investigated in the working paper version (Schlag and Vida 2019).

  12. In our model, the language explains the context of the message that is being sent. Hence we assume that the language is chosen before sending the message and after the action has been chosen. However, in some settings, the alternative timing in which the language is chosen before the action is chosen may be more appropriate.

  13. The rest of Remark 1 does not remain true, we provide a counter example for point 1-(c) in the online Appendix, while the rest is obvious. Obviously, point 3 of Proposition 2 fails to be equivalent with points 1 and 2.

  14. For the definitions of strict supermodularity and positive spill-overs, see the Appendix.

  15. It is easy to see that in our setup, player one gets his favorite Nash equilibrium outcome under talk about intentions, if (a) each action is part of a unique pure strategy Nash equilibrium and (b) his favorite Nash equilbrium outcome is attained by one of them. Lo states that self-committing (which implies (a)), together with self-signaling (which implies (b)), implies that the sender gets his favorite equilibrium.

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Correspondence to Péter Vida.

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We would like to thank Attila Ambrus, Stefano Demichelis, Olivier Gossner, Takakazu Honryo, Andras Kornai, Tymofiy Mylovanov, Martin Peitz, Larry Samuelson, Joel Sobel, Thomas Troeger, two anonymous referees and the participants of the game theory class at University of Vienna for useful comments. Péter Vida received financial support from SFB/TR 15, from the Cowles Foundation and from Labex MME-DII which is gratefully acknowledged. An earlier version of this paper had the title “Commitments, Intentions, Truth and Nash Equilibria”.

Supplementary Information

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Supplementary material 1 (pdf 167 KB)

Appendices

Appendices

Some definitions

Consider G with \(S_{1}=\{1,2,\ldots ,p\}\) and \(S_{2}=\{1,2,\ldots ,q\}\).

Definition 5

We say that G is supermodular if for all \(k,k^{\prime }\in S_{1}\) with \( k\le k^{\prime }\) and for all \(l,l^{\prime }\in S_{2}\) with \(l\le l^{\prime }\) we have that:

$$\begin{aligned} u_{1}(k^{\prime },l^{\prime })-u_{1}(k,l^{\prime })\ge u_{1}(k^{\prime },l)-u_{1}(k,l)\;\text {and}\;u_{2}(k^{\prime },l^{\prime })-u_{2}(k^{\prime },l)\ge u_{2}(k,l^{\prime })-u_{2}(k,l). \end{aligned}$$

Definition 6

We say that the game G exhibits positive spill-overs if for all \( l,l^{\prime }\in S_{2}\) with \(l<l^{\prime }\) and for all \(k\in S_{1}\) we have that \(u_{1}(k,l)<u_{1}(k,l^{\prime }),\) and for all \(k,k^{\prime }\in S_{2}\) with \(k<k^{\prime }\) and for all \(l\in S_{2}\) we have that \( u_{2}(k,l)<u_{2}(k^{\prime },l).\)

Let us denote by \(b_{i}(s_j)=argmax_{s_i\in S_i} u_i(s_i,s_j)\).

Definition 7

We say that the game G has (strictly) increasing best responses if \(b_1(\cdot ),b_2(\cdot )\) are functions such that for all \(l,l^{\prime }\in S_{2}\) with \(l<l^{\prime }\) we have that \( b_{1}(l)\le (<)b_{1}(l^{\prime })\) and for all \(k,k^{\prime }\in S_{1}:k<k^{\prime }\) we have that \(b_{2}(k)\le (<)b_{2}(k^{\prime }).\)

It is easy to see that supermodular games have increasing best responses.

Definition 8

We say that the game G is strictly supermodular if it is supermodular, and in addition, has strictly increasing best responses.

Omitted proofs of propositions

Proof of Proposition 1

Proof

Consider a CCE \((L_{1},m_{1},\sigma _{1},\sigma _{2},\mu _{2})\) with outcome \(\left( z_{1},z_{2}\right) \). Consider any \(L\in C(L_1)\) and any \(m\in C(m_1^L)\) then \((\sigma _1^L(m),\sigma _2^L(m))\) is a Nash equilibrium of G. Player one must be indifferent between the different Nash equilibria played with positive probability on the equilibrium path. By distinctness, player two will also be indifferent between these Nash equilibria. It follows that the convex combination of such Nash equilibrium outcomes is a Nash equilibrium outcome, that of \(\sigma ^L(m)\). Finally, we show that no communication can be chosen to be the equilibrium language resulting in the outcome \(\left( z_{1},z_{2}\right) \). First, keeping everything else unchanged as in the original equilibrium, set the beliefs after the language no communication such that play results in the Nash equilibrium with outcome \(\left( z_{1},z_{2}\right) \), namely set \(\mu _2^{\{S_1\}}(S_1)=\mu _2^L(m)=\sigma _1^{\{S_1\}}(S_1)=\sigma _1^L(m)\) and \(\sigma _2^{\{S_1\}}(S_1)=\sigma _2^L(m)\). This is still a CCE. Next, change the equilibrium language choice to no communication, namely set \(L_1=\{S_1\}\). Again, this is a CCE with no communication on the equilibrium path resulting in the Nash outcome \(\left( z_{1},z_{2}\right) \). \(\square \)

Proof of Proposition 2

Proof

Point 2 follows from point 1, as otherwise for all \(L\in {\mathcal {L}}\) we have \((m_{1}^{L},\sigma _{1}^{L},\sigma _{2}^{L},\mu _{2}^{L})\) which is a CCE under L resulting in \(z_1^L<z_1^*\) for player (for \(L\notin {\mathcal {L}}\) one can choose \(m_{1}^{L},\sigma _{1}^{L}=\mu _{2}^{L}, \sigma _{2}^{L}\) in a way that \(\sigma ^L(m)\) is a Nash equilibrium for all \(m\in M\) resulting in the worst Nash outcome for player one). One can then choose \(L_1\in argmax_{L\in {\mathcal {L}}}z_1^L\) to be the equilibrium language. Completing the strategy profile with \((m_{1},\sigma _{1},\sigma _{2},\mu _{2})\) one obtains a CCE resulting in \(z_1^{L_1}< z_1^*\) (this is basically an implication of point 6 in Remark 2). Point 1 follows from point 2 because player one can choose the language which gives him his favorite outcome in any CCE under this language. Point 2 follows from point 3, as L can be chosen to be \(\{C(\sigma _{1}), S_1{\setminus } C(\sigma _{1})\}\) in case (a),(b) and (c) holds or to be \(\{S_1\}\) when there is a unique equilibrium outcome for player one of G. Finally point 3 follows from point 2. Suppose the language satisfying point 2 can be chosen to be \(\{S_1\}\). Then G must have a unique equilibrium outcome for player one, as any Nash equilibrium outcome is a CCE outcome under \(\{S_1\}\). Suppose now the language satisfying point 2 can be chosen to be \(L\ne \{S_1\}\). It is clear that (a) and (b) must hold because there are at least two different (disjoint) messages in L after which Nash equilibria of G must be played in any CCE under L (see points 1 and 2 in Remark 1). Suppose (c) is not satisfied. We can assume that for all \(m\in L\) there is a \(\sigma ^m\) Nash equilibrium such that \(C(\sigma _1^m)\subseteq m\) yielding an outcome different from \(z_1^*\) for player one, as otherwise there is \(m\in L\) such that (a), (b) and (c) holds choosing \(\sigma \) to be any Nash equilibrium for which \(C(\sigma _1)\subseteq m\). Now, it is easy to construct a CCE under L with an outcome different from \(z_1^*\) for player one by setting \(\sigma ^{L}(m)=\sigma ^m\) and choosing \(m_1^{L}\in argmax_{m\in L}u_1(\sigma ^m), \mu _2^{L}(m)=\sigma _1^m\) which is a contradiction. \(\square \)

Proof of Proposition 4

Proof

Consider G with \(S_{1}=\{1,2,\ldots ,p\}\) and \(S_{2}=\{1,2,\ldots ,q\}\). Given that strict supermodularity implies strictly increasing best responses (see the definition in the Appendix) it must be that \(p=q\) and the pure strategy Nash equilibria are \((k,l)\in S_1\times S_2\), where \(k=l\). Given supermodularity and positive spill-overs, \((k,l)=(p,p)\) results in the unique efficient Nash equilibrium outcome \(z^*=(z_1^*,z_2^*)\) which also Pareto dominates all the other, possibly non-Nash equilibrium outcomes. Players get their overall best payoffs.

To prove the “if” statement it is easy to check that \(\sigma _1=p\), for all \(k\in S_1\), \(L_1(k)=\{S_1\},m^{\{S_1\}}_1(k)=S_1\) and \(\sigma _2^{\{S_1\}}(S_1)=\mu _2^{\{S_1\}}(S_1)=p\) is part of a CCE where the rest of the strategy profile can be chosen arbitrarily satisfying the conditions for CCE. After the choice of any other credible language player one can be just worse off no matter the choice of k. After non-credible languages play and beliefs can be set arbitrarily respecting sequential rationality.

The “only if” statement is proven by contradiction. It is enough to prove that \(L=\{S_1\}\) is the unique credible language for which \(z^*\) is a CCE outcome under L. It then follows that for all \(k\in S_1\) player 1 must choose \(L_1(k)=\{S_1\}\) in any CCE resulting in \(z^*\). Hence, suppose to the contrary that there is a credible language L, different from no communication, such that \(z^*\) is a credible outcome under L. Suppose that the equilibrium message is \(m^L_1(p)=m\ne S_1\), which follows from the fact that \(L\ne \{S_1\}\) is a partition, \(p\in m\) and it induces player two to play p, i.e. \(\sigma _2^L(m)=p, \mu _2^L(m)=p\). We show now that for any other \(m'\in L\) for which \(p\notin m'\), necessarily by the partition structure of L, we have that for all \(\mu _2^L(m')\) supported within \(m'\), \(p-1\) is a better response of player two than p. Hence, whenever \(k\in m'\), player one will deviate and send the message m resulting in the highest action of player 2, in p, and by positive spill-overs, in a higher utility for player one compared to sending \(m'\). We prove that:

$$\begin{aligned} p\notin argmax_{l\in S_2}\sum _{k\in S_1{\setminus } \{p\}}\mu _2^L(m')(k)u_2(k,l), \end{aligned}$$

where \(C(\mu _2^L(m'))\subseteq m'\). It is enough to prove that \(u_2(k,p-1)>u_2(k,p)\) for all \(k\ne p\) in \(S_1\). This follows from the fact that

$$\begin{aligned} 0>u_2(p-1,p)-u_2(p-1,p-1)\ge u_2(k,p)-u_2(k,p-1) \end{aligned}$$

for every \(k\le p-1\), where the first inequality follows from the fact that \((p-1,p-1)\) is a Nash equilibrium (and that the best responses are unique) and the second inequality follows from supermodularity. \(\square \)

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Schlag, K.H., Vida, P. Believing when credible: talking about future intentions and past actions. Int J Game Theory 50, 867–889 (2021). https://doi.org/10.1007/s00182-021-00772-2

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Keywords

  • Pre-play communication
  • Credibility
  • Coordination
  • Language
  • Multiple equilibria
  • Virtual communication

JEL Classification

  • C72
  • D83