Signaling Among Relatives. III. Talk Is Cheap (Costly Signaling͞cost-Free Signaling͞handicap Principle͞sir Philip Sidney Game͞parent–Offspring Conflict)
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Proc. Natl. Acad. Sci. USA Vol. 95, pp. 5100–5105, April 1998 Evolution Signaling among relatives. III. Talk is cheap (costly signalingycost-free signalingyhandicap principleySir Philip Sidney gameyparent–offspring conflict) CARL T. BERGSTROM* AND MICHAEL LACHMANN Department of Biological Sciences, Stanford University, Stanford, CA 94305 Edited by John Maynard Smith, University of Sussex, Brighton, United Kingdom, and approved February 16, 1998 (received for review September 8, 1997) ABSTRACT The Sir Philip Sidney game has been used by hunger but send a continuously intensifying signal as hunger numerous authors to show how signal cost can facilitate increases above this threshold. The mother then would be able honest signaling among relatives. Here, we demonstrate that, to distinguish among the hunger levels of each nestling above in this game, honest cost-free signals are possible as well, the threshold, while the nestlings below the threshold would all under very general conditions. Moreover, these cost-free share a common signal despite some differences in hunger signals are better for all participants than the previously level. explored alternatives. Recent empirical evidence suggests that Previous biological models of signaling among relatives have begging is energetically inexpensive for nestling birds; this focused on only the separating equilibria; this omission is a finding led some researchers to question the applicability of consequence of the differential equation-based approach used the costly signaling framework to nestling begging. Our in these models, as discussed in Signaling II. Maynard Smith’s results show that cost-free or inexpensive signals, as observed model (8), treated in Section 1, is an exception—it is con- empirically, fall within the framework of signaling theory. strained to allow fewer available signals than there are differ- ent signaler states. How can relatives communicate honestly despite interests that At these separating equilibria, communication is very costly. sometimes conflict? Several authors have addressed this ques- Bergstrom and Lachmann (15) (referred to from here on as tion by extending the costly signaling framework of Zahavi (1, Signaling I) examined the discrete and continuous Sir Philip 2) and Grafen (3) to treat signaling of need among relatives Sidney (SPS) games of Maynard Smith and Johnstone and (4–9). Broadly speaking, their results demonstrate that signal Grafen (5, 7) and prove that, although costly signaling among cost plays much the same role in signaling of need among relatives can indeed be a stable equilibrium, it often leaves relatives as it does in the examples of sexual selection signaling both participants worse off than if no signaling were to have originally treated by Grafen (3, 10)—namely, cost serves to occurred at all. Rodriguez-Girones et al. (16) demonstrate that ensure honest signaling. These signaling games often allow similar results hold for Godfray’s (6) model of nestling begging. multiple signaling equilibria [Lachmann and Bergstrom (11), In these models, the separating signaling equilibrium takes the referred to from here on as Signaling II]. A signaling equilib- form of an inferior equilibrium of a coordination game. That rium is defined as a pair of signaler and signal-receiver is, the sender and receiver of the signal aim to coordinate strategies that is an equilibrium in the game-theoretic Nash either on the mutual assumption of signaling or on the mutual equilibrium sense, in which neither player will gain an advan- assumption that there will be no signaling. Signaling is too tage from unilaterally altering strategy. The equilibria may be costly in these cases because when signaling equilibria take this separating equilibria, in which signalers in different states each separating form, the cost of signaling is independent of the send distinct signals, or they may be pooling or partially pooling distribution of signaler states, whereas the average value of equilibria, in which some signalers in different states share a signaling—the fitness gain made possible by the signal— common signal. (We borrow these definitions from economics; depends on this distribution (see Signaling I). Consequently, see, e.g., refs. 12–14.) Here, signals are taken to be distinct if rare signaler types can inflate greatly the signal cost without they are distinguished by some signal receiver. appreciably altering signal value, and thus signals can cost Both kinds of equilibria are plausible in biological signaling more than they are ‘‘worth’’ to either sender or receiver. systems. Consider a situation akin to that described by Godfray These results raise problems for the application of signaling (6, 9), in which a mother bird arrives at her nest and must theory to empirical systems. First, theoretical models predict decide how much to feed each of her nestlings. The nestlings relatively large signal costs, whereas empirical data suggest signal their hunger level by begging. If the mother is able to costs are generally small (see Section 3). Second, if separating distinguish the hunger levels of each nestling by observing the signaling equilibria are too costly to be worthwhile for sender (continuously distributed) volume of each nestling’s vocaliza- or receiver, then signaling systems only would arise among tion, we may have a separating equilibrium. If instead she is relatives when evolution became stuck at a ‘‘bad’’ equilibrium. unable or unwilling to distinguish among different degrees of This seems unlikely, given that no-signaling is most likely the begging and only observes whether or not each bird is begging, ancestral condition (15, 16). As mentioned above, however, we will have a pooling equilibrium. From the perspective of the separating equilibria are not the only possible signaling equi- signal receiver (i.e., the mother), birds with different levels of libria. In Signaling II, we show that a large number of pooling hunger will all be sharing a common signal—‘‘squawk,’’ or, and partially pooling equilibria can be stable in signaling games alternatively, ‘‘don’t squawk.’’ We also could imagine a par- played among relatives. With the possibility of pooling equi- tially pooling equilibrium, in which nestlings send the same libria comes the opportunity to use inexpensive or even signal—not begging at all—below some threshold level of zero-cost signals, with only a modest decline in signal value. In this paper, we explore these cheap signaling equilibria in detail. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked ‘‘advertisement’’ in This paper was submitted directly (Track II) to the Proceedings office. accordance with 18 U.S.C. §1734 solely to indicate this fact. Abbreviation: SPS, Sir Philip Sidney. © 1998 by The National Academy of Sciences 0027-8424y98y955100-6$2.00y0 *Towhomreprintrequestsshouldbeaddressed:[email protected]. PNAS is available online at http:yywww.pnas.org. edu. 5100 Downloaded by guest on September 29, 2021 Evolution: Bergstrom and Lachmann Proc. Natl. Acad. Sci. USA 95 (1998) 5101 Because multiple signaling equilibria exist, models of signal will allow us to address these issues. For discussion of the evolution are required to predict which equilibrium will be additional complexities that arise in defining signal cost, see observed in a given signaling game (see Signaling II). More- Hasson (17) and Maynard Smith and Harper (18). over, because signaler and receiver interests do not entirely First of all, we will treat signal cost as a relative quantity. We coincide, the equilibrium that is best for the signaler may not always will look at the difference between the cost of sending also be the best for the signal receiver. As a result, there is no a signal and the cost of taking some other action—for example, obvious way to define which signaling equilibrium is optimal in sending a different signal or even doing nothing. (Notice that a particular signaling system. For this reason, we do not doing nothing can itself be a signal and therefore usually will concern ourselves here with finding the optimal equilibria. be considered as one of the possible alternative signals, as Instead, in Section 1, we demonstrate mathematically that discussed in Signaling I). Computing signal costs as a relative cost-free signaling equilibria are far more common than measure makes intuitive sense—costs generally are computed previously believed in systems of signaling among relatives. We relative only to the set of feasible alternatives. Moreover, this show that, in the SPS game, these cost-free equilibria are better treatment of cost is in accordance with the theoretical basis for for both sender and receiver than either the no-signaling stable signaling. In Signaling II, we showed (Section 3, expres- equilibrium, for which all signalers share the cheapest signal, sions 4, 6, and 8) that the conditions for signal stability involve or the separating equilibrium considered in previous analyses. only the differences in signal costs and not the absolute signal We consider two different reasons why signaling may be costs. cost-free. First, as depicted in Fig. 1 and treated in detail by This said, we define the difference in signal cost between Maynard Smith (8), signaling can be cost-free when there is no signal A and signal B as the change in fitness that results from gain in misrepresenting one’s condition to anyone. Second, as sending signal B instead of signal A, with the response of the depicted in Fig. 4, we demonstrate that signaling can be signal receiver held constant. (In the present model, signal cost cost-free when there is a gain to misrepresenting one’s con- is independent of the receiver response; in other models the dition to some, as long as there is on average no gain in two may co-vary. In that case, signal cost can be defined only misrepresenting one’s condition to everyone. with respect to a particular receiver response.) Notice that, In Section 2, we show that, in some cases, no cost-free because receiver response is held constant under this defini- signaling equilibria exist.