The dynamics of costly signaling

Wagner, Elliott O. Article The dynamics of costly signaling Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Wagner, Elliott O. (2013) : The dynamics of costly signaling, Games, ISSN 2073-4336, MDPI, Basel, Vol. 4, Iss. 2, pp. 163-181, https://doi.org/10.3390/g4020163 This Version is available at: https://hdl.handle.net/10419/98502 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/3.0/ Games 2013,4, 163-181; doi:10.3390/g4020163 OPEN ACCESS games ISSN 2073-4336 www.mdpi.com/journal/games Article The Dynamics of Costly Signaling Elliott O. Wagner Universiteit van Amsterdam, Institute for Logic, Language, and Computation, Science Park 904, 1098XH Amsterdam, Netherlands; Tel.: +31 (0)20 525 8253; E-Mail: [email protected] Received: 11 February 2013; in revised form: 1 April 2013 / Accepted: 11 April 2013 / Published: 26 April 2013 Abstract: Costly signaling is a mechanism through which the honesty of signals can be secured in equilibrium, even in interactions where communicators have conflicting interests. This paper explores the dynamics of one such signaling game: Spence’s model of education. It is found that separating equilibria are unlikely to emerge under either the replicator or best response dynamics, but that partially communicative mixed equilibria are quite important dynamically. These mixtures are Lyapunov stable in the replicator dynamic and asymptotically stable in the best response dynamic. Moreover, they have large basins of attraction, in fact larger than those of either pooling or separating equilibria. This suggests that these mixtures may play significant, and underappreciated, roles in the explanation of the emergence and stability of information transfer. Keywords: game theory; evolutionary dynamics; signaling; costly signaling 1. Introduction How can the honesty of communication between two agents be ensured when their interests do not coincide? This is one way of framing the question Spence [1] posed in his seminal paper about job market signaling. His famous answer was that a particular structure of costly signaling can guarantee honesty in equilibrium. However, a complication typical of signaling models in this tradition is that they often have an infinite number of equilibria. Such an abundance of equilibria makes equilibrium selection a daunting task. Those seeking to address this issue have often posited equilibrium refinements with the aim of identifying only a small number of plausible equilibrium outcomes.1 1N¨ oldeke and Samuelson [2] and Jacobsen et al. [3] are exceptions. Their findings are outlined in Section 2and related to the results of this study in Section 5. Games 2013,4164 This paper takes a different approach to equilibrium selection in Spence’s original model of education. Instead of applying an equilibrium refinement, Spence’s game will be embedded into two common game dynamics: the replicator and best response dynamics. These dynamics arise from very different modeling assumptions. The replicator dynamic is a paradigm example of an unsophisticated process of imitation, whereas the best response dynamic is an archetype of myopic rational behavior. Nonetheless, both have been proposed as models of learning in games [4,5]. In order to study equilibrium selection and maintenance, the dynamic stability of the equilibria in Spence’s model and the sizes of the basin of attraction of each attractor will be investigated under both these dynamics. Section 2reviews Spence’s game and Section 3carries out this study into the dynamics. It is found that mixed equilibria, which are largely ignored in the signaling literature, play a very important role in the emergence and stability of information transfer. It is proven that these mixtures are Lyapunov stable in the replicator dynamic and asymptotically stable in the best response dynamic. It is also shown that these mixtures have large basins of attraction; so large, in fact, that randomly chosen initial conditions are more likely to lead to a mixture than to one of the pure equilibria, which are those generally selected by standard refinements. Since both dynamics are most easily interpreted as models of large populations, these mixtures are naturally interpreted as polymorphic market states. Additionally, when multiple separating equilibria are present in the model, only the separating equilibrium in which high quality senders use the cheapest signal that is too costly for low quality types to profit by duplicating—called the Riley equilibrium—is stable under the best response dynamic. The Riley equilibrium also has the largest basin of attraction, at least when compared with other separating possibilities, under the replicator dynamic. These results are discussed and connected to the literature in Section 5. It is suggested that this class of mixed equilibria may play a significant, and under-appreciated, role in the explanation of information transfer in competitive markets. Due to the fact that costly signaling games arise also in models of various biological [6,7] and linguistic phenomena [8–11], these results indicate that attention to mixed equilibria may be fundamental for an understanding of the evolution of communication more generally. 2. Job Market Signaling The standard presentation of Spence’s [1] model of education takes the form of a signaling game with two players: a worker (the sender) and an employer (the receiver); see, e.g., [12], [13], or [14] for textbook accounts. The worker knows her ability level θ, but her prospective employer does not. After observing θ, the worker chooses some level of education e∈R+to purchase. The worker incurs a cost c(θ, e)obtaining this education. It is assumed that the value of the worker to the employer is θ, and that, after observing the worker’s message (choice of e), the employer pays the worker a wage wthat is equal to the employer’s expectation of θ. To model this assumption the employer is frequently presumed to seek to minimize the quadratic difference between the wage and θ, hence the payoff to the employer is given as −(w−θ)2. The payoff to the employee is w−c(θ, e). To simplify analysis, it is typically assumed that employees are one of two types; i.e.,θ∈ {θL, θH}. Denote the probability of a worker being of these types pLand pH(with pH= 1 −pL). Following Spence’s original article, let c(θ, e) = e θ,θL= 1, and θH= 2, so that c(θL, e) = eand c(θH, e) = e 2. Games 2013,4165 These assumptions are not necessary for the equilibrium analysis of Spence’s model, but they make the dynamical analysis of the next section tractable, so for ease of exposition they are made now. It is well known that this game has an infinite number of perfect Bayesian equilibria. Coupled with the appropriate beliefs, they come in three flavors: pooling, separating, and hybrid. In a pooling equilibrium both types of worker send the same message e∗and the employer offers the wage w∗=pL+2pH= 1+pH upon the receipt of this message. In this sort of equilibrium the worker does not reveal any information about her type. In a separating equilibrium, type θHworkers send message e∗∈[1,2] and type θL workers send eL= 0. The employer then offers the wage 2upon receipt of message e∗and 1 upon receipt of message eL. Education level functions as a perfect indicator of worker productivity in a separating equilibrium. Hybrid equilibria are mixtures in which one type of worker chooses one level of education with certainty and the other type randomizes between pooling and separating. In these mixed equilibria, education level carries some information, but information transfer is imperfect. An example of a particular hybrid equilibrium is given in the following section. Hybrid equilibria are often ignored in discussions of Spence’s signaling game 2, but the dynamic analysis presented in the rest of this paper suggests that they may play an important role in the informational structure of markets. So there are three types of equilibria and a continuum of each type. Many refinements have been proposed to limit this variety to just the reasonable equilibria. Indeed, the signaling and screening refinement literature is so immense that it is impossible to summarize even a small fraction of it here. Here, however, are two influential refinements. Riley [17] showed (in a more general model) that there is a unique equilibrium that Pareto dominates the family of separating equilibria, and this equilibrium (sometimes called the Riley equilibrium) is one in which high quality senders use the cheapest signal that is too costly for low quality senders to duplicate. In light of this fact, it is natural to think that if a market finds its way to a separating equilibrium, we should expect it to be the Riley equilibrium. After all, why would high quality senders invest in a more expensive signal than is necessary? In terms of Spence’s model as described above, the Riley equilibrium is the separating equilibrium with e∗= 1. In a different approach to equilibrium selection, Cho and Kreps [18] suggested refining beliefs off the equilibrium path. Their Intuitive Criterion, as applied to Spence’s model, has a great deal of bite. It eliminates all equilibrium outcomes except the Riley equilibrium. In a related paper Cho and Sobel [16] analyze another refinement of off equilibrium path beliefs that selects the Riley equilibrium in this game.3 N¨ oldeke and Samuelson [2] pioneered an alternative approach to equilibrium selection in Spence’s model. Finding inspiration in Spence’s remark that certain equilibrium outcomes would not survive a dynamic process of belief and strategy revision coupled with arbitrarily small perturbations, they investigated such a dynamic model explicitly. Their “Spencian” dynamic leads to one of three types of recurrent sets. One such set is the Riley equilibria. Another contains pooling equilibria. The third contains a two period cycle. The cycles consists of type θHsenders always sending the message e∗ 2They are not mentioned in either Fudenberg and Tirole [12] or Osborne and Rubinstein [14], and are not discussed in Spence [1]. Similarly, in his expansive survey of screening and signaling research, Riley [15] does not cover these mixtures. They are, however, discussed in Cho and Sobel [16] and Gibbons [13], 3Furthermore, in the pruned Spence game that is introduced in the next section, their refinement selects the hybrid equilibrium when a separating equilibrium does not exist. Games 2013,4166 and type θLsenders switching between sending e∗and eLon each iteration of the dynamic. These cycles roughly correspond to the hybrid equilibria described above, and N¨ oldeke and Samuelson show that this cycle exists if an only if there is a mixed sequential equilibrium of the variant of Spence’s model developed in [18]. This approach has been further pursued by Jacobsen et al. [3], who extended Young’s [19] stochastic dynamic for agents with limited memories to games of incomplete information. They found that this stochastic process yields a strong prediction: whenever the signaling game has separating equilibria, in the long-run this system’s limiting distribution will put almost all its weigh on the Riley equilibrium. Here a complementary approach is pursued. Instead of investigating a dynamic tailor-made for Spence’s game (like N¨ oldeke and Samuelson [2]) or the limiting distribution of a Markov process (like Jacobsen et al. [3]), I will simply embed this game into the two most vanilla learning dynamics around (the replicator and best response dynamics). Section 5compares the results yielded by these different approaches. 3. Pruning Spence’s Game The previous section reviewed how it is that message cost structure can allow honest signaling in equilibrium. However, a purely static analysis limited to a game’s equilibrium structure leaves some important questions unanswered. For example, how likely is it that a population of senders and receivers ends up at a separating equilibrium instead of at pooling or at a mixture? Another unaddressed question is whether or not such a population is guaranteed to converge to an equilibrium. N¨ oldeke and Samuelson’s discrete-time revision protocol did not always converge to an equilibrium, but what about more run-of-the-mill dynamical systems? Such dynamics are known to exhibit complex behavior in some circumstances. For example, neither the replicator dynamic nor the best response dynamic is guaranteed to converge to the mixed Nash equilibrium of generalized rock-paper-scissors. Instead, the population may cycle around the equilibrium or spiral outward toward the boundary of phase space under the replicator dynamic, or end up oscillating endlessly in a stable limit cycle under the best response dynamic [20]. In any case, results of this sort demonstrate the importance of understanding out-of-equilibrium play and how it can lead to—or deviate from—equilibrium. However, in order to study the dynamics here in detail, it is necessary to prune the strategy space of Spence’s game because there is not a thoroughly developed theory of adaptive dynamics for games, and in particular asymmetric Bayesian games, with infinite strategy spaces. If the important dynamical question is understanding how a system evolves to either pooling, separating, partial communication, or non-convergence, then an obvious way to shrink the strategy space is to limit workers to just two messages: the costless message eL= 0 and a more expensive message e∗. Then it is natural to focus on just three pure strategies for the worker: pool by always sending eL, separate by sending eLwhen type θLand e∗when type θH, and pool by always sending e∗. Call these strategies Low,Sep, and High. This restriction enables us to focus on just two pure employer strategies: act as though the signal carries no information about type by offering the pooling wage 1 + pH, and act as though the signal perfectly Games 2013,4167 identifies the type (i.e., offer 1if eLis received and offer 2if e∗is received). Call the former strategy Pool and the latter Sep. This pruned extensive form game is shown in Figure 1.4 Figure 1. The extensive form representation of Spence’s game in which workers choose from only two levels of education and employers either act as though senders are pooling (P) or act as though the message indicates low quality (L) or act as though the message indicates high quality (H). aN θL θH q 1 e= 0 e=e∗ q 1 e= 0 e=e∗ p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 2 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 2 q@ @ @ @     P L H q 1 + pH,−p2 H q 1,0 q 2,−1 q@ @ @ @     P L H q 1 + pH,−p2 L q 1,−1 q 2,0 q  @@@ @ P L H q1 + pH−e∗,−p2 H q1−e∗,0 q2−e∗,−1 q  @@@ @ P L H q1 + pH−e∗ 2,−p2 L q1−e∗ 2,−1 q2−e∗ 2,0 Both the replicator dynamic and the best response dynamic are infinite population models, and payoffs to strategy types are given by the type’s expected payoff when matched with a random member of the population. Therefore, we can now focus analysis on the 3×2normal game shown in Table 1in which the payoffs are the expectations of payoffs from the extensive form game. Notice that if the receiver plays Pool, the sender’s unique best response is to play Low. Likewise, the receiver’s best response to Low is to play Pool. Thus, the profile (Low, P ool)is a strict Nash equilibrium. It corresponds to a pooling equilibrium in Spence’s original game; workers do not purchase education and employers do not listen to signals. Similarly, the receiver’s unique best response to High is to play P ool. Likewise, the receiver’s unique best response to a sender who separates is to play Sep herself. These best response relationships are independent of the cost of the more expensive message, e∗. To determine the equilibrium structure of this game, It only remains to determine sender’s best response to a receiver playing the pure strategy Sep. Sep will be the sender’s unique best response if and only if 1< e∗<2. Accordingly, when 1< e∗<2, the profile (Sep, Sep)corresponds to a separating equilibrium in the full game. On the other hand, if 0< e∗<1then this separating profile is not an equilibrium. Nonetheless, an important mixed equilibrium exists for these values of e∗. The profile in which the sender randomizes 4Note that the interaction here is modeled as a two-player game. One player is the worker, and nature chooses her type. As was pointed out by a referee, in some economic applications it may be more natural to treat different worker types as different players. Although this is true, such a model would not be amenable to straightforward analysis in the dynamics considered here. Games 2013,4168 between Sep and High with probabilities pLand pH, and the receiver randomizes between Pool and Sep with probabilities 1−e∗and e∗respectively is a Nash equilibrium when 0< e∗<1. It corresponds to a hybrid equilibrium in the original game in which high productivity workers send message e∗with certainty and low productivity workers randomize between separating from and pooling with the high type. The mixed strategy space and best response correspondences for both cases are drawn in Figure 2. These correspondences are crucial for analyzing the best response dynamic below. Table 1. The pruned Spence signaling game. Pool Sep Low 1 + pH,−pLpH1,−pH Sep 1 + pH−pHe∗ 2,−pLpH1 + pH−pHe∗ 2,0 High 1 + pH−pLe∗−pHe∗ 2,−pLpH2−pLe∗−pHe∗ 2,−pL Figure 2. Best response correspondences for the pruned Spence signaling game with (a) 1< e∗<2and (b) 0< e∗<1. The sender’s best reply is shown by the thick line. The receiver’s best reply is shown by the translucent surface. x2signifies the probability that the sender plays Sep,x3the probability that the sender plays High, and y2the probability that the receiver plays Sep. Nash equilibria are highlighted by black dots. y2=e* 2 x2=pH x3=pH Low Sep High Sep Pool y2=e* 2 x2=pH y2=e* x3=pH Low Sep High Sep Pool (a) (b) Although this 3×2game has eliminated an infinite number of sending and receiving strategies, it still captures the spirit of Spence’s model. Pooling is always an equilibrium outcome, and separating can be an equilibrium if the high quality types send a sufficiently costly message. Thus, just like in Spence’s model, a costly education can signal high quality and secure high wages even though education itself may not increase productivity. Now that we have a two player normal form game that retains some Games 2013,4169 of the structure of Spence’s original model, it is possible to proceed in analyzing the dynamics of job market signaling. 4. Dynamics The adaptive dynamics considered here are the two-population replicator and best response dynamics. The first population is the population of workers. Denote the proportion that chooses each strategy Low, Sep, and High as x1,x2, and x3. The second population consists of employers. Let y1and y2be the proportions of the population that play Pool and Sep. Because x1+x2+x3= 1 and y1+y2= 1, the dynamics for this system live in the three dimensional space ∆3×∆2where ∆nis the n−1dimensional simplex {(p1, . . . , pn)|pi≥0,Ppi= 1}. Coordinates in phase space will be written (x2, x3, y2).5 The replicator dynamic for the pruned game is given by the three differential equations ˙x2=x2[(Ay)2−x·Ay] ˙x3=x3[(Ay)3−x·Ay] ˙y2=y2[(Bx)2−y·Bx] (RE) where Ais the sender’s 3×2payoff matrix and Bis the receiver’s 2×3payoff matrix. Although this dynamic was originally formulated by [21] to model natural selection in an asexually reproducing population, it also provides a model of cultural learning in economic situations. In this context, the equations give the fluctuations in strategy distributions as agents imitate successful members of their population. In other words, these equations describe large populations of employers and workers in which individual agents, when called on to revise their strategy choice, choose to imitate a more prosperous player [22]. An advantage of focusing on the replicator dynamic is that stability analysis of its rest points can provide information about the stability of these points under all two population uniformly monotone selection dynamics. This family of game dynamics is characterized by a positive linear correlation between relative growth rates and payoff differences. Since a rest point is asymptotically stable for the replicator dynamic if and only if it is asymptotically stable under every two population uniformly monotone selection dynamic ([23], Theorem 3.5.3), studying this process provides an understanding of the behavior of this larger class of selection dynamics. The best response dynamic for the pruned game is written as ˙x2=BR(y)−x2 ˙x3=BR(y)−x3 ˙y2=BR(x)−y2 (BR) where BR(y) = {ˆx∈∆3|ˆx·Ay ≥x·Ay for all x∈∆3}and BR(x)is defined similarly.6The usual interpretation of this dynamic is that a small fraction of each large population revises their strategy at each time interval. Upon revision, they choose a best reply to the current state. A complication in 5It is convenient here to work directly with x2, x3, and y2instead of x1or y1. 6Since BR is a set-valued function, the best response dynamic is not technically a dynamical system. Instead it is a differential inclusion. Games 2013,4170 analyzing this system is that it is not in general differentiable at rest points (Nash equilibria) because it is at these points where BR abruptly changes and is often many-valued. However, an analytic virtue of this dynamic is that piecewise linear solutions can be constructed from any initial condition. This is due to the fact that, at every state, the best response dynamic moves the population in a straight line toward the current best reply profile; see, e.g. [24] or [23]. This fact will be used in the constructions below. 4.1. Separating Equilibria When 1< e∗<2the dynamics of the pruned game shown in Table 1are straightforward. There are two strict Nash equilibria: (Low, P ool)and (Sep, Sep). Therefore, the corresponding states (0,0,0) and (1,0,0) are asymptotically stable under both dynamics. Furthermore, the pure sending strategy High is strictly dominated by the pure strategy Sep. Pure strategies that are strictly dominated by other pure strategies are driven to extinction by both dynamics. Thus every initial condition in the interior of phase space is brought to the x3= 0 boundary face. On this boundary face there is a mixed Nash equilibrium at (pH,0,e∗ 2). Through linearization (for the replicator dynamic) or inspection of the best response correspondences (for the best response dynamic), it is easy to see that this mixed equilibrium is unstable. Phase portraits for both systems are illustrated in Figure 3. Additionally, since asymptotically stable rest points in the replicator dynamic are asymptotically stable for every two population uniformly monotone selection dynamic, we can immediately conclude that the pooling and separating equilibria are asymptotically stable under all such dynamics. Figure 3. Phase portraits showing the dynamics of the pruned Spence signaling game with 1< e∗<2for (a) the replicator dynamic and (b) the best response dynamic. Black and grey dots indicate stable and unstable rest points respectively. y2=e* 2 x2=pH Low Sep High Sep Pool y2=e* 2 x2=pH Low Sep High Sep Pool (a) (b) Games 2013,4177 what has been shown above: although separating may be a Nash equilibrium, it may be a very unlikely outcome of evolutionary and learning dynamics. Thus, if we want to understand costly signals seen in nature, it might be prudent to look beyond separating equilibria to the game’s hybrid equilibria. This conclusion can also be adapted for costly signaling models found in linguistics. Spence’s game has been co-opted to provide a first shot at a theory of politeness [8], the assumption being that polite messages are more costly to utter. However, although politeness may be a strict Nash equilibrium, the results seen here indicate that it may be an infrequent outcome of social learning. Similarly, linguists often use common-interest signaling games with costly signals to model a range of phenomena [9–11]. The conditions under which these games have evolutionarily stable and neutrally stable strategies are known [11], but the results seen here reveal that this static picture only provides part of the story. In addition to understanding the equilibrium structure (and even in addition to knowing which states have basins of attraction), we would like to know which are likely outcomes of the evolution or learning. The case study here shows that partially communicative mixed strategies may have large basins, and such information cannot be obtained through an analysis of equilibrium structure alone. On the other hand, at least one prediction of more traditional game-theoretic analysis is recaptured through the dynamic approach above. Namely, this paper provides further evidence for the claim that the Riley equilibrium is the most likely separating outcome. Less efficient separating equilibria have much smaller basins under the replicator dynamic and are not even stable under the best response dynamic. These results buttress those from both N¨ oldeke and Samuelson [2] and Jacobsen et al. [3]. N¨ oldeke and Samuelson showed that under their Spencian dynamic the Riley equilibrium is the only stable separating equilibrium. Jacobsen et al. showed that in Young’s [19] stochastic dynamic, the Riley equilibrium is the only separating equilibrium with any weight in the system’s limiting distribution. Thus the observation that the Riley equilibrium is the most likely separating outcome is robust across a variety of models. Acknowledgments I would like to thank Brian Skyrms, Simon Huttegger, Michael McBride, Rory Smead, and Kevin Zollman for helpful comments on earlier drafts of this paper. I would also like to thank two anonymous referees for their very thoughtful comments. References 1. Spence, M. Job market signaling. Q. J. Econ. 1973,87, 355-374. 2. N¨ oldeke, G.; Samuelson, L. A dynamic model of equilibirum selection in signaling markets. J. Econ. Theor. 1997,73, 118-156. 3. Jacobsen, H.J.; Jensen, M.; Sloth, B. Evolutionary learning in signalling games. Game. Econ. Behav. 2001,34, 34-63. 4. Schlag, K.H. Why imitate, and if so, how? A boundedly rational approach to multi-armed bandits. J. Econ. Theor. 1998,78, 130-156. 5. Gilboa, I.; Matsui, A. Social stability and equilibrium. Econometrica 1991,59, 859-869. 6. Zahavi, A. Mate selection: A selection for a handicap. J. Theor. Biol. 1975,53, 205-214. Games 2013,4178 7. Grafen, A. Biological signals as handicaps. J. Theor. Biol. 1990,144, 517-546. 8. Van Rooij, R. Being polite is a handicap: Towards a game theoretical analysis of polite linguistic behavior. Proceedings of the 9th Conference on Theoretical Aspects of Rationality and Knowledge, Bloomington, IN, USA, 2003, pp. 45-58. 9. Van Rooij, R. Signalling games select Horn strategies. Linguis. Philos. 2004,27, 493-527. 10. J¨ ager, G. Evolutionary game theory and typology: A case study. Language 2007,83, 74-109. 11. J¨ ager, G. Evolutionary stability conditions for signaling game with costly signals. J. Theor. Biol. 2008,253, 131-141. 12. Fudenberg, D.; Tirole, J. Game Theory; MIT Press: Cambridge, UK, 1991. 13. Gibbons, R. Game Theory for Applied Economists; Princeton University Press: Princeton, NJ, USA, 1992. 14. Osborne, M.J.; Rubinstein, A. A Course in Game Theory; MIT Press: Cambridge, UK, 1994. 15. Riley, J.G. Silver signals: Twenty-five years of screening and signaling. J. Econ. Lit. 2001, 39, 432-478. 16. Cho, I.; Sobel, J. Strategic stability and uniqueness in signaling games. J. Econ. Theor. 1990, 50, 381-413. 17. Riley, J. Informational equilibrium. Econometrica 1979,47, 331-359. 18. Cho, I.; Kreps, D.M. Signaling games and stable equilibria. Q. J. Econ. 1987,102, 179-221. 19. Young, P.H. The evolution of conventions. Econometrica 1993,61, 57-84. 20. Gaunersdorfer, A.; Hofbauer, J. Fictitious Play, Shapley Polygons, and the Replicator Equation. Game. Econ. Behav. 1995,11, 279-303. 21. Taylor, P.D.; Jonker, L. Evolutionarily stable strategies and game dynamics. Math. Biosci. 1978, 40, 145-156. 22. Weibull, J.W. Evolutionary Game Theory; MIT Press: Cambridge, UK, 1997. 23. Cressman, R. Evolutionary Dynamics and Extensive Form Games; MIT Press: Cambridge, UK, 2003. 24. Hofbauer, J.; Sigmund, K. Evolutionary Games and Population Dynamics; Cambridge University Press: Cambridge, UK, 1998. 25. Hirsch, M.W.; Smale, S.; Devaney, R.L. Differential Equations, Dynamical Systems, & an Introduction to Chaos; Elsevier Academic Press: London, UK, 2004. 26. Berger, U. Fictitious play in 2×ngames. J. Econ. Theor. 2005,120, 139-154. 27. Fudenberg, D.; Levine, D.K. The Theory of Learning in Games; MIT Press: Cambdrige, UK, 1998. 28. Bergstrom, C.T.; Lachmann, M. Signalling among relatives. I. Is costly signalling too costly. Philosophical Transactions of the Royal Society, London 1997,352, 609-617. 29. Lachmann, M.; Sz´ amad´ o, S.; Bergstrom, C.T. Cost and conflict in animal signals and human language. Proceedings of the National Academy of Science 2001,98, 13189-13194. 30. Huttegger, S.M.; Zollman, K.J.S. Dynamic stability and basins of attraction in the Sir Philip Sidney game. Proceedings of the Royal Society B: Biological Sciences 2010,277, 1915-1922. 31. Maynard Smith, J. Honest signalling: The Philip Sidney game. Anim. Behav. 1991,42, 1034- 1035. Games 2013,4179 32. Carr, J. Applications of Center Manifold Theory; Springer: New York, NY, USA, 1981. 33. Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields; Springer: New York, NY, USA, 1983. Appendix A. Proofs Proof of Theorem 1.At H, the Jacobian matrix of the replicator dynamic for the pruned Spence signaling game reduces to J=   1 2(p2 He∗−pHe∗)1 2(p2 He∗−pHe∗)−p3 H+ 2p2 H−pH −1 2e∗p2 H−1 2e∗p2 Hp3 H−2p2 H+pH e∗pH−e∗2pH−e∗+e∗2+ 2e∗pH−2e∗2pH0    The characteristic equation det(J−λI) = 0 has solutions λ1=−1 2pHe∗and λ2,3=±q−pHe∗(e∗−1) (pH−1)3 These solutions are the eigenvalues of J.0< pH<1and, by hypothesis, 0< e∗<1. Therefore (e∗−1) <0and (pH−1) <0, and consequently (e∗−1)(pH−1)3>0. Therefore, λ1is real and negative, and both λ2and λ3are purely imaginary. From the Center Manifold Theorem [32,33] it follows that there exists a stable manifold tangent to λ1’s eigenspace and a center manifold tangent to the eigenspace corresponding to λ2and λ3. According to the Center Manifold Theorem, since Jhas no eigenvalues with positive real part, the stability of rest point Hdepends solely on the dynamics upon the center manifold. The center eigenspace here is simply the x1= 0 boundary face of phase space (indeed, the center manifold is identical to this boundary face). From inspecting the pruned game, it is easy to see that this face is characterized by a best response cycle. Indeed, the dynamics on this face are the dynamics of matching pennies. Under the two population replicator dynamic, it is well known that matching pennies yields closed periodic orbits centered on the mixed Nash equilibrium [22–24]. Thus, this is also the behavior of the system on the center manifold. Since the mixed Nash is neutrally stable in matching pennies, the hybrid equilibrium is neutrally stable in the pruned Spence signaling game. Proof of Theorem 2.This result will be shown by the construction of a first return map from Σ1to Σ1 such that iteration of the map leads to H. Let Σ1=(x2, x3, y2)∈R3|x2=pH+αx3,0< x3< pH, e∗< y2<1 if pH>e−1 e−2and Σ1=(x2, x3, y2)∈R3|x2=pH+αx3,0< x3< pH, e∗< y2<e(pH−1) 2pH−1 Games 2013,4180 otherwise. This adjustment when pH<e−1 e−2is necessary to guarantee that a point on Σ1returns to Σ1 instead of converging to the pooling equilibria. Likewise let Σ2=(x2, x3, y2)∈R3|0< x2<1−pH,max 0,x2−pH α< x3<1−x2, y2=e∗ Σ3=(x2, x3, y2)∈R3|x2=pH+αx3,0< x3< pH,e∗ 2< y2< e∗ Σ4=(x2, x3, y2)∈R3|pH< x2<1,0< x3<min 1−x2,x2−pH α, y2=e∗ with α=1−2pH pH. These surfaces (shown in Figure 10) are the locations at which the solution trajectories to the best response dynamic abruptly change direction. Figure 10. The surfaces Σ1,Σ2,Σ3,and Σ4are illustrated in (a) along with a solution orbit starting from Σ1. The iteration of the first return map and convergence to His demonstrated in (b). y2=e* 2 x2=pH y2=e* x3=pH Low Sep High Sep Pool S1 S2 S3 S4 S1 Hx2 H0L,x3 H0L,y2 H0LLÎS1 Hx2 H1L,x3 H1L,y2 H1LL Hx2 H2L,x3 H2L,y2 H2LL Hx2 H3L,x3 H3L,y2 H3LL H (a) (b) Due to the piecewise linear nature of the solution orbits, every orbit leaving each of these surfaces travels in a straight line toward the current best response profile. For instance, a solution from an initial condition on Σ1will travel in a line toward (0,1,0) until it intersects Σ2at which point the solution changes direction to point to (1,0,0) until intersecting Σ3, and so on. Working all this out, one can see that an initial condition on Σ1at coordinates (x2, x3, y2)will intersect Σ2at coordinates (x0 2, x0 3, e∗) given by the linear equation (x0 2, x0 3, e∗)T= (−pH−αx3,1−x3,−y2)Ty2−e∗ y2+ (pH+αx3, x3, y2)T Then the solution will be carried to Σ3with an intersection at (x00 2, x00 3, y00 2)T= (1 −x0 2,−x0 3,−e∗)TpH−x0 2+αx0 3 1−x0 2+αx0 3+ (x0 2, x0 3, e∗)T Next, the solution aims to Σ4with an intersection at (x000 2, x000 3, e∗)T= (1 −x00 2,−x00 3,1−y00 2)Te∗−y00 2 1−y00 2+ (x00 2, x00 3, y00 2)T Games 2013,4181 Finally, the solution returns to Σ1at the location given by (x0000 2, x0000 3, y0000 2)T= (−x000 2,1−x000 3,1−e∗)TpH−x000 2+αx000 3 α(x000 3−1) −x000 2+ (x000 2, x000 3, e∗)T Putting the four linear components together, the first return of a state on Σ1with coordinates x(n) 3and y(n) 2will be at the location given by the two dimensional map x(n+1) 3=pH(e∗−e∗x(n) 3+e∗2x(n) 3−y(n) 2) e∗−e∗pH+e∗2pH−y(n) 2 y(n+1) 2=e∗(e∗+e∗pHy(n) 2−pHy(n) 2−y(n) 2) e∗−e∗pH+e∗2pH−y(n) 2 As this fractional linear recurrence is iterated, the coordinates of the nth return to Σ1are x(n) 3=npH(e∗−y(0) 2)−pHe∗x(0) 3+e∗2x(0) 3 n(e∗−y(0) 2)−e∗pH+e∗2pH y(n) 2=ne∗(e∗−y(0) 2)−e∗pHy(0) 2+e∗2pHy(0) 2 n(e∗−y(0) 2)−e∗pH+e∗2pHy(0) 2 . This solution (as well as all of the above linear algebra) can be easily verified in Mathematica. Looking at asymptotic behavior one sees that, limn→∞ x(n) 3=pHand limn→∞ y(n) 2=e∗, so any initial condition on Σ1will be taken to Hunder the best response dynamic. Therefore, the point His asymptotically stable. c 2013 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).