The evolution of Horn's rule

Abstract

Horn's rule says that messages can be kept ambiguous if only a single interpretation is plausible. Speakers only perform costly disambiguation to convey surprising information. This paper shows that, while non‐cooperative game theory cannot justify Horn's rule, evolutionary game theory can. In order to model the evolution of signalling, the pooling equilibrium needs to be one's starting point. But in such an equilibrium, the plausible interpretation is made, and the receiver is therefore already predisposed to interpret absence of a signal as referring to a plausible event. From there on, a marked signal referring to an implausible event can evolve. At the same time, the paper identifies an exception to Horn's rule. If giving a plausible interpretation for an implausible event is very costly, then in the pooling equilibrium the implausible interpretation is always made. In this exceptional case, only an inefficient separating equilibrium disobeying Horn's rule can evolve.

Keywords:

• Horn's rule
• signalling
• evolutionary game theory
• evolutionary drift
• pragmatics

Acknowledgements

I would like to thank one anonymous referee, Robert van Rooij, Christina Pawlowitsch, and participants of the 6th workshop on Economics and Philosophy (Economics and Language), Fundación Urrutia Elejalde, Madrid, 15–17 June, for helpful comments. Financial support of the Tjalling C. Koopmans Institute is gratefully acknowledged. Any remaining mistakes are my own.

Notes

1. As pointed out by a referee, in the standard statement of Horn's rule, the choice is between a cheap message and an expensive message, and not between sending a costly message and sending no message at all, as is the case in the model. However, it suffices to interpret the sender's action of ‘doing nothing’ as the sending of a costless signal to obtain a model that fits the standard statement of Horn's rule. Moreover, the argument extends to the case where the choice is between a cheap signal (which still has a positive cost attached to it) and a costly signal. It should be noted that in these alternative models, in the pooling Nash equilibrium, the cheap signal will always be sent.

2. The argument extends to a more realistic environment where players sometimes play the role of signallers, and sometimes play the role of receivers. An asymmetric population game is considered here for simplicity.

3. Why take the Nash pooling equilibrium as a starting point, and not for instance a population state where both sender types send the signal, and where receivers take action F? From the latter population state, by the same argument as used in the paper, the inefficient separating equilibrium can evolve. However, this neglects the question of how such a population state could ever arise. Surely, if we are to model the evolution of signalling, our starting point should be a situation without signalling. From such a pooling equilibrium, a situation where all sender types send the signal can never evolve.

4. As pointed out by an anonymous referee, one could still argue that the separating equilibrium that can evolve under equation (10) is Pareto‐efficient for the following reason. Suppose that the cost of sending a message is negligible, and that the message gets lost with probability ε. Then sending a message in state F is Pareto‐superior to sending a message in state I if . It is easy to see that this boils down to equation (10). Simply, if the cost of taking action F in state I is very high, and if messages sometimes get lost, then it is better to send a message in state F than to send a message in state I. It should be stressed that this argument is quite different from Horn's rule, which focuses on the cost of sending a message, and does not consider noise.