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.

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

1.1 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 $\left( z_{1},z_{2}\right)$. $\square$

1.2 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. $\square$

1.3 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. $\square$