Game-Theoretic Pragmatics Under Conflicting and Common Interests

Abstract

This paper combines a survey of existing literature in game-theoretic pragmatics with new models that fill some voids in that literature. We start with an overview of signaling games with a conflict of interest between sender and receiver, and show that the literature on such games can be classified into models with direct, costly, noisy and imprecise signals. We then argue that this same subdivision can be used to classify signaling games with common interests, where we fill some voids in the literature. For each of the signaling games treated, we show how equilibrium-refinement arguments and evolutionary arguments can be interpreted in the light of pragmatic inference.

Appendix: Proofs

Proof of Proposition 1

(i)–(ii) In a pooling PBE where R always does a ₁ (a ₂), S in state t ₂ (t ₁) can only send m ₀. Given the condition on R’s payoffs, if S in state t ₁ (t ₂) sends m ₁ (m ₂) with sufficiently low probability, R will continue to always do a ₁ (a ₂). A pooling PBE where R always does a ₂ can only be maintained if R believes that m ₁ comes from t ₂. Yet, as soon as m ₁ is sent with small probability, such beliefs cannot be maintained. It follows that such a pooling PBE is not a sequential equilibrium. A pooling PBE where R always does a ₁ can be maintained if R believes that m ₁ (m ₂) comes from t ₁ (t ₂). This is therefore a sequential equilibrium.

(iii) Let R do a ₁ when receiving m ₀. Then S sends m ₀ in t ₂, and we cannot have a separating PBE. It follows that in any separating PBE, if m ₀ is sent, it can only be followed by a ₂. In turn, it follows that S in t ₁ always sends m ₁ in a separating PBE. S in t ₂ may mix in any sort of way between sending m ₀ and m ₂. Further, in a pooling PBE as under (ii), an alternative best response for R to S’s strategy is to do a ₁ (a ₂) upon m ₁ (m ₂), and a ₂ upon m ₀, which is in line with R’s best response in the separating PBE. It follows that separating PBE are attainable from the pooling PBE under (ii). Finally, in a pooling PBE as under (i), given that a ₁ is done upon m ₀, no separating PBE is attainable from this pooling PBE. □

Proof of Proposition 2

Let S send a message $m_{i}^{1}$ for whatever cue i takes on a value of 1. Then R prefers to do a ₂ given that

$$ \begin{aligned} & \frac{{2x(1 - x)u_{R} \left( {t_{2} ,a_{2} } \right) + (1 - x)^{2} u_{R} \left( {t_{1} ,a_{2} } \right)}}{{2x(1 - x) + (1 - x)^{2} }} > \frac{{2x(1 - x)u_{R} \left( {t_{2} ,a_{1} } \right) + (1 - x)^{2} u_{R} \left( {t_{1} ,a_{1} } \right)}}{{2x(1 - x) + (1 - x)^{2} }} \\ & \quad \Leftrightarrow \frac{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}} > \frac{(1 - x)}{2x} \\ \end{aligned} $$

It follows that there is no separating PBE where S sends a message $m_{i}^{1}$ for whatever cue takes on a value of 1.

S can instead employ a strategy where she only sends $m_{i}^{1}$ if a particular cue i takes on a value of 1, randomizes in any sort of way about sending $m_{i}^{2}$ or m ₀ when i takes on a value of 2, and never sends messages $m_{j}^{1}$ or $m_{j}^{2}$. If R now observes $m_{i}^{1}$, it is a best response for him to do a ₁ iff

$$ \begin{aligned} & \frac{{x(1 - x)u_{R} \left( {t_{2} ,a_{1} } \right) + (1 - x)^{2} u_{R} \left( {t_{1} ,a_{1} } \right)}}{{x(1 - x) + (1 - x)^{2} }} > \frac{{x(1 - x)u_{R} \left( {t_{2} ,a_{2} } \right) + (1 - x)^{2} u_{R} \left( {t_{1} ,a_{2} } \right)}}{{x(1 - x) + (1 - x)^{2} }} \\ & \quad \Leftrightarrow \frac{(1 - x)}{x} > \frac{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}} \\ \end{aligned} $$

When receiving $m_{i}^{2}$ or m ₀, it is a best response for R to do a ₂, as given S’s equilibrium strategy, this automatically means that event t ₂ occurs. Further, given that S does not send $m_{j}^{1}$ or $m_{j}^{2}$, it is a weak best response for R to do a ₂ whenever one of these messages is received.

In turn, given that R does a ₁ when receiving $m_{i}^{1}$ and a ₂ otherwise, it is a weak best response for S never to send $m_{j}^{1}$ or $m_{j}^{2}$ and m ₀ or $m_{i}^{2}$ when cue i has a value of 2, and a strict best response to send $m_{i}^{1}$ when cue i has a value of 1.

In any pooling PBE, response a ₂ is adopted as $\frac{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}} > \frac{{\pi_{1} }}{{\pi_{2} }}$, where $\frac{{\pi_{1} }}{{\pi_{2} }} = \frac{{(1 - x)^{2} }}{{2x(1 - x) + x^{2} }}$. This is because we have assumed that $\frac{(1 - x)}{2x} < \frac{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}}$, and because $\frac{(1 - x)}{2x} > \frac{{(1 - x)^{2} }}{{2x(1 - x) + x^{2} }}$. The pooling PBE is Pareto-inefficient as each player is better off in the specified partial separating PBE. A pooling PBE where S always sends m ₀ is a sequential equilibrium, as a message $m_{i}^{1}$ can always have come from S who observed value 2 for cue j. Further, such a pooling PBE meets the intuitive criterion, as it is not the case that only S who observes event t ₁ could benefit from sending $m_{i}^{1}$. The specified partial separating PBE is attainable from the pooling PBE where S always sends m ₀, because it is a best response for R to S’s strategy in this pooling PBE to plan to do a ₂ when receiving $m_{j}^{1}$, $m_{j}^{2}$, $m_{i}^{2}$ or m ₀, and to plan to do a ₁ when receiving $m_{i}^{1}$, as in the partial separating PBE. □

Proof of Proposition 3

If S plays the strategy of the specified candidate separating PBE, it is a best response for R to do a ₁ when receiving message m, and to do a ₂ otherwise. If R plays the strategy of the specified candidate separating PBE, it is a best response for S to send message m in state t ₁ and not to send m ₀ in state t ₂ iff $u_{S} \left( {t_{2} ,a_{2} } \right) \ge u_{S} \left( {t_{2} ,a_{1} } \right) - C\left( {t_{2} ,m} \right)$ and $u_{S} \left( {t_{1} ,a_{1} } \right) - C\left( {t_{1} ,m} \right) \ge u_{S} \left( {t_{1} ,a_{2} } \right)$. The conditions follow.

Under (i), the pooling PBE is neologism proof and meets the intuitive criterion because S in t ₁ already obtains her most preferred response a ₁ without sending a message. For the same reason, while sending a message is dominated for S in t ₂, the pooling PBE survives dominance.

Under (ii), the pooling PBE is not neologism proof, does not meet the intuitive criterion, and is eliminated by dominance. Only S in t ₁ benefits from deviating from the pooling PBE by sending m; moreover, sending m rather than m ₀ is dominated for S in t ₂. The results on attainability and on pooling-based neologism proofness and pooling-based intuitive criterion now follow straightforwardly. □

Proof of Proposition 4

We first show that under the given condition, the pooling PBE takes on the form given under (i).

$$ \begin{aligned} & \pi_{1} u_{R} \left( {t_{1} ,a_{3} } \right) + \pi_{2} u_{R} \left( {t_{2} ,a_{3} } \right) \ge \pi_{1} u_{R} \left( {t_{1} ,a_{1} } \right) + \pi_{2} u_{R} \left( {t_{2} ,a_{1} } \right) \\ & \quad \Leftrightarrow \pi_{2} \left[ {u_{R} \left( {t_{2} ,a_{3} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right] \ge \pi_{1} \left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{3} } \right)} \right] \\ & \quad \Leftrightarrow \frac{{\pi_{2} }}{{\pi_{1} }} \ge \frac{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{3} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{2} ,a_{3} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}} \\ \end{aligned} $$

(6)

$$ \begin{aligned} & \pi_{1} u_{R} \left( {t_{1} ,a_{3} } \right) + \pi_{2} u_{R} \left( {t_{2} ,a_{3} } \right) \ge \pi_{1} u_{R} \left( {t_{1} ,a_{2} } \right) + \pi_{2} u_{R} \left( {t_{2} ,a_{2} } \right) \\ & \quad \Leftrightarrow \pi_{1} \left[ {u_{R} \left( {t_{1} ,a_{3} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right] \ge \pi_{2} \left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{3} } \right)} \right] \\ & \quad \Leftrightarrow \frac{{\left[ {u_{R} \left( {t_{1} ,a_{3} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{3} } \right)} \right]}} \ge \frac{{\pi_{2} }}{{\pi_{1} }} \\ \end{aligned} $$

(7)

Combining (6) and (7), the condition for R’s utility function stated in the proposition is obtained. The pooling PBE where R does a ₃ in any event is neologism proof: S in both states of the world prefers to send any message m ₁ that leads R to do a ₁ with positive probability, rather than to send message m ₀.

We now show the existence of the separating PBE in (ii). When not receiving any signal, R should prefer to do a ₃ to a ₁:

$$ \begin{aligned} & \frac{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right)}}{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right) + \pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}u_{R} \left( {t_{1} ,a_{3} } \right) + \frac{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right) + \pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}u_{R} \left( {t_{2} ,a_{3} } \right) \\ & \quad \ge \frac{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right)}}{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right) + \pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}u_{R} \left( {t_{1} ,a_{1} } \right) + \frac{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right) + \pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}u_{R} \left( {t_{2} ,a_{1} } \right) \\ & \quad \Leftrightarrow \pi_{2} \left[ {u_{R} \left( {t_{2} ,a_{3} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]\mu \left( {\left. {m_{0} } \right|m_{2} } \right) \ge \pi_{1} \left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{3} } \right)} \right]\mu \left( {\left. {m_{0} } \right|m_{1} } \right) \\ \end{aligned} $$

(8)

$$ \Leftrightarrow \frac{{\mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}{{\mu \left( {\left. {m_{0} } \right|m_{1} } \right)}} \ge \frac{{\pi_{1} \left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{3} } \right)} \right]}}{{\pi_{2} \left[ {u_{R} \left( {t_{2} ,a_{3} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}, $$

(9)

where the right-hand side in (9) is smaller than 1. It follows from (9) that a separating PBE can only exist if $\mu \left( {\left. {m_{0} } \right|m_{2} } \right) > 0$. Also, when not receiving any signal, R should prefer to do a ₃ to a ₂:

$$ \begin{aligned} & \frac{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right)}}{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right) + \pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}u_{R} \left( {t_{1} ,a_{3} } \right) + \frac{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}{{\pi_{1} \mu \left( {\left. {m_{0} } \right|S_{1} } \right) + \pi_{2} \mu \left( {\left. {m_{0} } \right|S_{2} } \right)}}u_{R} \left( {t_{2} ,a_{3} } \right) \\ & \quad \ge \frac{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right)}}{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right) + \pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}u_{R} \left( {t_{1} ,a_{2} } \right) + \frac{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m_{1} } \right) + \pi_{2} \mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}u_{R} \left( {t_{2} ,a_{2} } \right) \\ & \quad \Leftrightarrow \mu \left( {\left. {m_{0} } \right|m_{1} } \right)\pi_{1} \left[ {u_{R} \left( {t_{1} ,a_{3} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right] \ge \mu \left( {\left. {m_{0} } \right|m_{2} } \right)\pi_{2} \left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{3} } \right)} \right] \\ \end{aligned} $$

(10)

$$ \Leftrightarrow \frac{{\pi_{1} \left[ {u_{R} \left( {t_{1} ,a_{3} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}}{{\pi_{2} \left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{3} } \right)} \right]}} \ge \frac{{\mu \left( {\left. {m_{0} } \right|m_{2} } \right)}}{{\mu \left( {\left. {m_{0} } \right|m_{1} } \right)}}, $$

(11)

where the left-hand side in (11) is larger than 1. By (8) and (10), it should be the case that $\left[ {u_{R} \left( {t_{1} ,a_{3} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right] > 0$ and $\left[ {u_{R} \left( {t_{2} ,a_{3} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right] > 0$. Combined with (9) and (11), it follows that a necessary condition for the existence of a separating PBE is that $\left[ {u_{R} \left( {t_{1} ,a_{3} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]*\left[ {u_{R} \left( {t_{2} ,a_{3} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right] \ge \left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{3} } \right)} \right]*\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{3} } \right)} \right].$

At the same time, S should prefer to send the right message in each state, and prefer this to not sending any message at all. That is, for S who in state t ₁, we must have:

$$ \mu \left( {\left. {m_{1} } \right|m_{1} } \right)u_{S} \left( {t_{1} ,a_{1} } \right) + \mu \left( {\left. {m_{0} } \right|m_{1} } \right)u_{S} \left( {t_{1} ,a_{3} } \right) \ge \mu \left( {\left. {m_{2} } \right|m_{2} } \right)u_{S} \left( {t_{1} ,a_{2} } \right) + \mu \left( {\left. {m_{0} } \right|m_{2} } \right)u_{S} \left( {t_{1} ,a_{3} } \right) $$

(12)

and

$$ \mu \left( {\left. {m_{1} } \right|m_{1} } \right)u_{S} \left( {t_{1} ,a_{1} } \right) + \mu \left( {\left. {m_{0} } \right|m_{1} } \right)u_{S} \left( {t_{1} ,a_{3} } \right) \ge u_{S} \left( {t_{1} ,a_{3} } \right) $$

(13)

For S who observes state t ₂, we must have:

$$ \mu \left( {\left. {m_{2} } \right|m_{2} } \right)u_{S} \left( {t_{2} ,a_{2} } \right) + \mu \left( {\left. {m_{0} } \right|m_{2} } \right)u_{S} \left( {t_{2} ,a_{3} } \right) \ge \mu \left( {\left. {m_{1} } \right|m_{1} } \right)u_{S} \left( {t_{2} ,a_{1} } \right) + \mu \left( {\left. {m_{0} } \right|m_{1} } \right)u_{S} \left( {t_{2} ,a_{3} } \right) $$

(14)

and

$$ \mu \left( {\left. {m_{2} } \right|m_{2} } \right)u_{S} \left( {t_{2} ,a_{2} } \right) + \mu \left( {\left. {m_{0} } \right|m_{2} } \right)u_{S} \left( {t_{2} ,a_{3} } \right) \ge u_{S} \left( {t_{2} ,a_{3} } \right) $$

(15)

By the fact that $u_{S} \left( {t_{1} ,a_{2} } \right) > u_{S} \left( {t_{1} ,a_{3} } \right)$, $u_{S} \left( {t_{2} ,a_{1} } \right) > u_{S} \left( {t_{2} ,a_{3} } \right)$, constraints (13) and (15) are slack. Constraints (12) and (14) can respectively be rewritten as:

$$ \left[ {1 - \mu \left( {\left. {m_{0} } \right|m_{1} } \right)} \right]\,\left[ {1 - \mu \left( {\left. {m_{0} } \right|m_{2} } \right)} \right]^{ - 1} \ge \left[ {u_{S} \left( {t_{1} ,a_{2} } \right) - u_{S} \left( {t_{1} ,a_{3} } \right)} \right]\,\left[ {u_{S} \left( {t_{1} ,a_{1} } \right) - u_{S} \left( {t_{1} ,a_{3} } \right)} \right]^{ - 1} $$

(16)

and

$$ \left[ {u_{S} \left( {t_{2} ,a_{2} } \right) - u_{S} \left( {t_{2} ,a_{3} } \right)} \right]\,\left[ {u_{S} \left( {t_{2} ,a_{1} } \right) - u_{S} \left( {t_{2} ,a_{3} } \right)} \right]^{ - 1} \ge \left[ {1 - \mu \left( {\left. {m_{0} } \right|m_{1} } \right)} \right]\,\left[ {1 - \mu \left( {\left. {m_{0} } \right|m_{2} } \right)} \right]^{ - 1} $$

(17)

In (16), the right-hand side is smaller than 1; in (17), the left-hand side is also smaller than 1. From the latter, it follows that in any separating PBE, it must be the case that $\mu \left( {\left. {m_{0} } \right|m_{1} } \right) > \mu \left( {\left. {m_{0} } \right|m_{2} } \right)$. Further, it follows from (16) and (17) that a separating PBE can only exist if $\left[ {u_{S} \left( {t_{2} ,a_{2} } \right) - u_{S} \left( {t_{2} ,a_{3} } \right)} \right] *$ $\,\left[ {u_{S} \left( {t_{1} ,a_{1} } \right) - u_{S} \left( {t_{1} ,a_{3} } \right)} \right]$ ≥ $\left[ {u_{S} \left( {t_{1} ,a_{2} } \right) - u_{S} \left( {t_{1} ,a_{3} } \right)} \right]\,*$ $\,\left[ {u_{S} \left( {t_{2} ,a_{1} } \right) - u_{S} \left( {t_{2} ,a_{3} } \right)} \right]$. Therefore, under the conditions specified in the proposition, levels of $\mu \left( {\left. {m_{0} } \right|m_{1} } \right)$ and $\mu \left( {\left. {m_{0} } \right|m_{2} } \right)$ exist such that constraints (9), (11), (16) and (17) are valid, so that a separating PBE exists.

The fact that there is a unique Pareto-efficient separating PBE with minimal noise can be seen by plotting (9), (11), (16) and (17) in graph with $\mu \left( {\left. {m_{0} } \right|m_{1} } \right)$ on one axis and $\mu \left( {\left. {m_{0} } \right|m_{2} } \right)$ on the other (see Fig. 1). From Fig. 1, it can be seen that, starting from any Pareto-inefficient separating PBE, a single neologism that is less noisy, but keeps $\mu \left( {\left. {m_{0} } \right|m_{1} } \right)$ and $\mu \left( {\left. {m_{0} } \right|m_{2} } \right)$ in the range of PBE, is credible to R. Still, each separating PBE is attainable from the pooling PBE where R does a ₃ in each circumstance; simply, by evolutionary drift, R can become predisposed to respond in the given way to messages with particular levels of noise. □

Fig. 1

Noisy signaling conflict-of-interest model

Proof of Proposition 5

• (i) There is no separating PBE where S sends a different message in each state. This is because if there are three different messages which induce R to do respectively a ₁, a ₂ and a ₃, then S in state t ₃ would send the message inducing action a ₂, and S in state t ₂ sends the message inducing action a ₁.

  There is no partial separating PBE where S pools states t ₂ and t ₃. This is because S in state t ₂ would then send the message inducing a ₁. As $\frac{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}} > \frac{{\pi_{2} }}{{\pi_{1} }}$, when S in a candidate partial separating PBE pools t ₂ and t ₁, R does a ₁. It follows that S in state t ₁ and t ₂ does not want to send the message inducing a ₃. At the same time, S in t ₃ does not want to send the message that is sent in both states t ₂ and t ₁, as S in t ₃ prefers action a ₃ to action a ₁.

• (ii) Pooling PBE are not neologism proof, whatever the form they take. If a ₃ is R’s best response in the pooling PBE, consider a neologism “t ₃ does not occur”. Following Eq. (1), by $u_{S} \left( {t_{3} ,a_{3} } \right) > u_{S} \left( {t_{3} ,a_{1} } \right)$, $\frac{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}} > \frac{{\pi_{2} }}{{\pi_{1} }}$, S in state t ₃ does not have any incentive to send such a message if R considers the message to be true. By $u_{S} \left( {t_{2} ,a_{3} } \right) < u_{S} \left( {t_{2} ,a_{1} } \right)$, $u_{S} \left( {t_{1} ,a_{3} } \right) < u_{S} \left( {t_{1} ,a_{1} } \right)$, S in state t ₁ and t ₂ has an incentive to send such a message if it is considered to be truthful. If a ₁ is the best response in the pooling PBE, consider a neologism of the form “state t ₃ occurs”. By $u_{S} \left( {t_{2} ,a_{3} } \right) < u_{S} \left( {t_{2} ,a_{1} } \right)$, $u_{S} \left( {t_{1} ,a_{3} } \right) < u_{S} \left( {t_{1} ,a_{1} } \right)$, and $u_{R} \left( {t_{3} ,a_{3} } \right) > u_{R} \left( {t_{3} ,a_{1} } \right)$, S in states t ₁ and t ₂ does not have an incentive to send such a message if it is considered truthful. By $u_{S} \left( {t_{3} ,a_{3} } \right) > u_{S} \left( {t_{3} ,a_{1} } \right)$, S in state t ₃ does have an incentive to send such a message if it is considered truthful. If a ₂ is the best response in the pooling PBE, consider a neologism of the form “state t ₂ or t ₁ occurs”. By $u_{S} \left( {t_{3} ,a_{2} } \right) > u_{S} \left( {t_{3} ,a_{1} } \right)$ and $\frac{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}} > \frac{{\pi_{2} }}{{\pi_{1} }}$, S in state t ₃ does not have any incentive to send a message if it is considered truthful, as it is then met with response a ₁. By $u_{S} \left( {t_{2} ,a_{2} } \right) < u_{S} \left( {t_{2} ,a_{1} } \right)$, $u_{S} \left( {t_{1} ,a_{2} } \right) < u_{S} \left( {t_{1} ,a_{1} } \right)$, S in states t ₂ and t ₁ does prefer to send such a message if it is considered truthful.

A partial separating PBE is attainable from each of the given pooling PBE, as evolutionary drift can always lead R to plan to do a ₃ when receiving one out-of-equilibrium message, and a ₁ when receiving another out-of-equilibrium message.□

Proof of Proposition 6

For proving the existence of the pooling PBE under (i) and (ii), see the proof of Proposition 2. We next prove the existence of the separating PBE in (iii). If R believes that a single revealed “1” means that both cues have a value of 1, then given common interests S follows the candidate equilibrium strategy. This in turn makes it a best response for R to follow the candidate equilibrium strategy. The same applies to the other separating PBE.

The pooling PBE in (i) is a sequential equilibrium, because a single revealed cue “1” can also be revealed by S in state (1, 2) or (2, 1). The pooling PBE in (ii) is not a sequential equilibrium, because a single revealed “2” can only be revealed in states (1, 2), (2, 1) or (2, 2).

The separating PBE where S reveals a “1” if both of the cues have a value of 1, and does not reveal anything otherwise, is attainable from the pooling PBE under (i) (conditional on $\frac{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}} > \frac{{\pi_{1} }}{{\pi_{2} }}$), as in this pooling PBE R already does a ₂ when nothing is revealed. The separating PBE where S reveals a “2” if at least one of the cues has a value of 2, and does not reveal anything otherwise, is attainable from the pooling PBE under (ii) (conditional on $\frac{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}} < \frac{{\pi_{1} }}{{\pi_{2} }}$), as in this pooling PBE R already does a ₁ when nothing is revealed. The separating PBE where S reveals a “1” if both of the cues have a value of 1, and reveals a “2” if at least one of the cues has a value of 2, is attainable from any of the pooling PBE, since starting from any of the pooling PBE evolutionary drift can cause R to respond in the appropriate way. □

Proof of Proposition 7

If $\pi_{2} u_{R} \left( {t_{2} ,a_{2} } \right) + \pi_{1} u_{R} \left( {t_{1} ,a_{2} } \right) > \pi_{2} u_{R} \left( {t_{2} ,a_{1} } \right) + \pi_{1} u_{R} \left( {t_{1} ,a_{1} } \right)$, then in pooling PBE a ₂ is always done, if $\pi_{2} u_{R} \left( {t_{2} ,a_{2} } \right) + \pi_{1} u_{R} \left( {t_{1} ,a_{2} } \right) < \pi_{2} u_{R} \left( {t_{2} ,a_{1} } \right) + \pi_{1} u_{R} \left( {t_{1} ,a_{1} } \right)$, then in pooling PBE a ₁ is always done. If S sends a message in state t ₁ (t ₂), then it is a best response for R to take action a ₁ (a ₂) when a message is received, and to take action a ₂ (a ₁) when no message is not received. This response by R in turn makes the specified strategy by S a best response. For R, it does not matter which separating PBE is played. S is better off if the message is only sent infrequently.

Consider the case where in the pooling PBE, a ₂ (respectively a ₁) is always done. If the messages are from a common language, given that players have common interests, the PBE is not neologism proof, since only R in state t ₁ (respectively t ₂) has an incentive to send a message with common meaning “do a ₁” (respectively “do a ₂”), if

S expects that R will act on the literal meaning of this message. For this reason, only the efficient separating PBE (respectively inefficient separating PBE) is pooling-based neologism proof. If the messages are not from a common language, in the pooling PBE, it is equilibrium dominated for S to send a costly message in state t ₂ (respectively t ₁). The pooling PBE therefore does not meet the intuitive criterion, as R should always interpret that a costly message came from S in state t ₂ (respectively t ₁). For this reason, only the efficient separating PBE (respectively inefficient separating PBE) meets the pooling-based intuitive criterion. Finally, in the pooling PBE, it is an alternative best response for R to plan to do a ₂ (respectively a ₁) when not receiving a message, and plan to do a ₁ (respectively a ₂) when receiving a message. It follows that only the efficient separating PBE (respectively inefficient separating PBE) is attainable from the pooling PBE. □

Proof of Proposition 8

As by assumption a message can only be received if it was sent, the constraint determining the existence of a separating PBE is what R does when not receiving a message. In the separating PBE where a message is sent in state t ₁, R strictly prefers to do a ₂ when not receiving a message iff

$$ \begin{aligned} & \frac{{\pi_{2} }}{{\pi_{2} + \pi_{1} \mu \left( {\left. {m_{0} } \right|m} \right)}}u_{R} \left( {t_{2} ,a_{2} } \right) + \frac{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m} \right)}}{{\pi_{2} + \pi_{1} \mu \left( {\left. {m_{0} } \right|m} \right)}}u_{R} \left( {t_{1} ,a_{2} } \right) \\ & \quad > \frac{{\pi_{2} }}{{\pi_{2} + \pi_{1} \mu \left( {\left. {m_{0} } \right|m} \right)}}u_{R} \left( {t_{2} ,a_{1} } \right) + \frac{{\pi_{1} \mu \left( {\left. {m_{0} } \right|m} \right)}}{{\pi_{2} + \pi_{1} \mu \left( {\left. {m_{0} } \right|m} \right)}}u_{R} \left( {t_{1} ,a_{1} } \right) \\ & \quad \Leftrightarrow \frac{{\pi_{2} \left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}{{\pi_{1} \left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}} > \mu \left( {\left. {m_{0} } \right|m} \right) \\ \end{aligned} $$

(18)

A pooling PBE where R does a ₂ only exists if $\frac{{\pi_{2} \left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}}{{\pi_{1} \left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}} > 1$, so that this constraint is not binding. In the separating PBE where a message is sent in state t ₂, R strictly prefers to do a ₁ when not receiving a message iff

$$ \begin{aligned} & \frac{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m} \right)}}{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m} \right) + \pi_{1} }}u_{R} \left( {t_{2} ,a_{1} } \right) + \frac{{\pi_{1} }}{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m} \right) + \pi_{1} }}u_{R} \left( {t_{1} ,a_{1} } \right) \\ & \quad > \frac{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m} \right)}}{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m} \right) + \pi_{1} }}u_{R} \left( {t_{2} ,a_{2} } \right) + \frac{{\pi_{1} }}{{\pi_{2} \mu \left( {\left. {m_{0} } \right|m} \right) + \pi_{1} }}u_{R} \left( {t_{1} ,a_{2} } \right) \\ & \quad \Leftrightarrow \frac{{\pi_{1} }}{{\pi_{2} }}\frac{{u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)}}{{u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)}} > \mu \left( {\left. {m_{0} } \right|m} \right) \\ \end{aligned} $$

(19)

This is the condition stated in the proposition.

R (and S) prefers the separating PBE where a message is sent in t ₁ rather than in t ₂ if

$$ \begin{aligned} & \pi_{2} u_{R} \left( {t_{2} ,a_{2} } \right) + \pi_{1} \left[ {\mu \left( {\left. m \right|m} \right)u_{R} \left( {t_{1} ,a_{1} } \right) + \mu \left( {\left. {m_{0} } \right|m} \right)u_{R} \left( {t_{1} ,a_{2} } \right)} \right] \\ & \quad > \pi_{2} \left[ {\mu \left( {\left. m \right|m} \right)u_{R} \left( {t_{2} ,a_{2} } \right) + \mu \left( {\left. {m_{0} } \right|m} \right)u_{R} \left( {t_{2} ,a_{1} } \right)} \right] + \pi_{1} u_{R} \left( {t_{1} ,a_{1} } \right) \\ & \quad \Leftrightarrow \pi_{2} u_{R} \left( {t_{2} ,a_{2} } \right) + \pi_{1} u_{R} \left( {t_{1} ,a_{2} } \right) > \pi_{2} u_{R} \left( {t_{2} ,a_{1} } \right) + \pi_{1} u_{R} \left( {t_{1} ,a_{1} } \right) \\ \end{aligned} $$

(20)

The latter condition is identical to the condition for a ₂ to be the best response in the pooling PBE.

The pooling PBE where R does a ₂ in any event is not neologism proof, because when R receives a noisy message m with a common meaning “Do a ₁”, R realizes that only S in state t ₁ has an incentive to send it, if S acts upon the belief that R will follow the advice contained in the noisy message. For the same reason, the efficient separating PBE is pooling-based neologism proof, whereas the inefficient separating PBE is not. Only the efficient separating PBE is attainable from the pooling PBE where R does a ₂ in any event, because in this PBE it is an alternative best response for R to plan to do a ₁ when receiving a message m. □

Proof of Proposition 9

• (i) The pooling PBE takes this form by the fact that $\pi_{1} \left[ {u_{R} \left( {t_{1} ,x} \right) - u_{R} \left( {t_{1} ,a_{3} } \right)} \right] + \pi_{2} \left[ {u_{R} \left( {t_{2} ,x} \right) - u_{R} \left( {t_{2} ,a_{3} } \right)} \right] < \pi_{3} \left[ {u_{R} \left( {t_{3} ,a_{3} } \right) - u_{R} \left( {t_{3} ,x} \right)} \right]$ for $x = a_{1} ,a_{2}$. In any pooling PBE where S never sends any messages, R should conclude that only S in state t ₁ or t ₂ has an incentive to send a message with commonly known meaning “t ₁ or t ₂ occurs”, if S believes that R will take the literal meaning of this message to be true. At the same time, R should conclude that only S in state t ₁ has an incentive to send a message with commonly known meaning “t ₁ occurs”, with the same restrictions on the beliefs of S. This illustrates that the given pooling PBE are not neologism proof.

• (ii) In the first type of partial separating PBE, t ₁ and t ₂ get pooled. By $\frac{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{1} } \right)} \right]}} < \frac{{\pi_{2} }}{{\pi_{1} }}$, R does a ₂ when getting a pooled signal. Given that $u_{S} \left( {t_{1} ,a_{2} } \right) > u_{S} \left( {t_{1} ,a_{3} } \right)$, $u_{S} \left( {t_{3} ,a_{3} } \right) > u_{S} \left( {t_{3} ,a_{2} } \right)$, and $u_{S} \left( {t_{2} ,a_{2} } \right) > u_{S} \left( {t_{2} ,a_{3} } \right)$, S prefers to tell the truth. Part (i) of the proof already showed that this PBE is not pooling-based neologism proof. Also, in the pooling PBE where S never sends messages and R does a ₃ in any event, an alternative best response for R is to plan to do a ₂ when receiving a message, so that the specified separating PBE is attainable from this pooling PBE.

• (iii) In the second type of partial separating PBE, t ₂ and t ₃ get pooled. By $\frac{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{3} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{3} ,a_{3} } \right) - u_{R} \left( {t_{3} ,a_{2} } \right)} \right]}} < \frac{{\pi_{3} }}{{\pi_{2} }}$, R does a ₃ when getting a pooled signal. Given that $u_{S} \left( {t_{1} ,a_{1} } \right) > u_{S} \left( {t_{1} ,a_{3} } \right)$, $u_{S} \left( {t_{3} ,a_{3} } \right) > u_{S} \left( {t_{3} ,a_{1} } \right)$, and $u_{S} \left( {t_{2} ,a_{3} } \right) > u_{S} \left( {t_{2} ,a_{1} } \right)$, S prefers to tell the truth. Part (i) of the proof shows that this PBE is not pooling-based neologism proof. Also, in the pooling PBE where S never sends messages and R does a ₃ in any event, an alternative best response for R is to plan to do a ₁ when receiving a message, so that the specified partial separating PBE is attainable from this pooling PBE.

We end by deriving the condition under which the partial separating PBE under (iii) is Pareto superior to the partial separating PBE under (ii). This is the case iff

$$ \begin{array}{*{20}c} \begin{gathered} \pi_{1} u_{R} \left( {t_{1} ,a_{2} } \right) + \pi_{2} u_{R} \left( {t_{2} ,a_{2} } \right) + \pi_{3} u_{R} \left( {t_{3} ,a_{3} } \right) \hfill \\ \quad < \pi_{1} u_{R} \left( {t_{1} ,a_{1} } \right) + \pi_{2} u_{R} \left( {t_{2} ,a_{3} } \right) + \pi_{3} u_{R} \left( {t_{3} ,a_{3} } \right) \hfill \\ \end{gathered} & {\text{iff}} & {\frac{{\left[ {u_{R} \left( {t_{2} ,a_{2} } \right) - u_{R} \left( {t_{2} ,a_{3} } \right)} \right]}}{{\left[ {u_{R} \left( {t_{1} ,a_{1} } \right) - u_{R} \left( {t_{1} ,a_{2} } \right)} \right]}} < \frac{{\pi_{1} }}{{\pi_{2} }}.} \\ \end{array} $$

□