# The Dynamics of Costly Signaling

## Abstract

**:**

## 1. Introduction

## 2. Job Market Signaling

## 3. Pruning Spence’s Game

**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).

$Low$ | $1+{p}_{H},-{p}_{L}{p}_{H}$ | $1,-{p}_{H}$ |

$Sep$ | $1+{p}_{H}-\frac{{p}_{H}{e}^{*}}{2},-{p}_{L}{p}_{H}$ | $1+{p}_{H}-\frac{{p}_{H}{e}^{*}}{2},0$ |

$1+{p}_{H}-{p}_{L}{e}^{*}-\frac{{p}_{H}{e}^{*}}{2},-{p}_{L}{p}_{H}$ | $2-{p}_{L}{e}^{*}-\frac{{p}_{H}{e}^{*}}{2},-{p}_{L}$ |

**Figure 2.**Best response correspondences for the pruned Spence signaling game with (a) $1<{e}^{*}<2$ and (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. ${x}_{2}$ signifies the probability that the sender plays $Sep$, ${x}_{3}$ the probability that the sender plays $High$, and ${y}_{2}$ the probability that the receiver plays $Sep$. Nash equilibria are highlighted by black dots.

## 4. Dynamics

#### 4.1. Separating Equilibria

**Figure 3.**Phase portraits showing the dynamics of the pruned Spence signaling game with $1<{e}^{*}<2$ for (a) the replicator dynamic and (b) the best response dynamic. Black and grey dots indicate stable and unstable rest points respectively.

**Figure 4.**The proportion of phase space that converges to separating under both dynamics with ${p}_{H}=.5$. Each data point for the replicator dynamic is the average of 1000 randomly chosen initial conditions.

#### 4.2. Hybrid Equilibria

**Theorem 1.**The hybrid equilibrium $H=(1-{p}_{H},{p}_{H},{e}^{*})$ is neutrally stable under the replicator dynamic when $0<{e}^{*}<1$.

**Figure 5.**Phase portraits showing the dynamics of the pruned Spence signaling game with $0<{e}^{*}<1$ for (a) the replicator dynamic and (b) the best response dynamic. Black and grey dots indicate stable and unstable rest points respectively.

**Theorem 2.**The hybrid equilibrium $H=(1-{p}_{H},{p}_{H},{e}^{*})$ is asymptotically stable under the best response dynamic.

**Figure 6.**The proportion of phase space that converges to the ${x}_{1}=0$ boundary face under both dynamics with ${p}_{H}=.5$. Each data point for the replicator dynamic is the average of 1000 randomly chosen initial conditions.

#### 4.3. Separating vs. Hybrid Equilibria

$Low$ | $\frac{3}{2},-\frac{1}{4}$ | $1,-\frac{1}{2}$ | $1,-\frac{1}{2}$ |

$Se{p}_{{e}_{1}^{*}}$ | $\frac{3}{2}-\frac{{e}_{1}^{*}}{4},-\frac{1}{4}$ | $\frac{3}{2}-\frac{{e}_{1}^{*}}{4},0$ | $1-\frac{{e}_{1}^{*}}{4},-\frac{1}{2}$ |

$Hig{h}_{{e}_{1}^{*}}$ | $\frac{3}{2}-\frac{3{e}_{1}^{*}}{4},-\frac{1}{4}$ | $2-\frac{3{e}_{1}^{*}}{4},-\frac{1}{2}$ | $1-\frac{3{e}_{1}^{*}}{4},-\frac{1}{2}$ |

$\frac{3}{2}-\frac{{e}_{2}^{*}}{4},-\frac{1}{4}$ | $\frac{3}{2}-\frac{{e}_{2}^{*}}{4},-\frac{1}{2}$ | $\frac{3}{2}-\frac{{e}_{2}^{*}}{4},0$ |

**Figure 7.**The number of randomly chosen initial conditions that converged to pooling, separating, and the hybrid face under both the replicator dynamic and the logit dynamic with $\eta =.01$. ${e}_{2}^{*}$ is held fixed at $1.025$ while ${e}_{1}^{*}$ is varied.

#### 4.4. Riley Equilibria

**Figure 8.**Phase portraits for the game in Table 2 limited to the sending and receiving strategies $Se{p}_{{e}_{1}^{*}}$ and $Se{p}_{{e}_{2}^{*}}$ for (a) the replicator dynamic and (b) the best response dynamic. The horizontal direction shows the sending population and the vertical direction shows the receiving. Black, grey, and white dots indicate asymptotically stable, neutrally stable, and unstable rest points respectively.

**Figure 9.**The number of randomly chosen initial conditions that converged to each stable set under the replicator dynamic. For all trials, ${e}_{2}^{*}=1.9$ and ${p}_{H}=.5$.

## 5. Discussion

## Acknowledgments

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## A. Proofs

**Figure 10.**The surfaces ${\Sigma}_{1},{\Sigma}_{2},{\Sigma}_{3},$ and ${\Sigma}_{4}$ are illustrated in (a) along with a solution orbit starting from ${\Sigma}_{1}$. The iteration of the first return map and convergence to H is demonstrated in (b).

^{2}They 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],^{3.}Furthermore, 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.^{4}Note 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.^{5}It is convenient here to work directly with ${x}_{2},{x}_{3}$, and ${y}_{2}$ instead of ${x}_{1}$ or ${y}_{1}$.^{6}Since $BR$ is a set-valued function, the best response dynamic is not technically a dynamical system. Instead it is a differential inclusion.

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Wagner, E.O. The Dynamics of Costly Signaling. *Games* **2013**, *4*, 163-181.
https://doi.org/10.3390/g4020163

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Wagner EO. The Dynamics of Costly Signaling. *Games*. 2013; 4(2):163-181.
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Wagner, Elliott O. 2013. "The Dynamics of Costly Signaling" *Games* 4, no. 2: 163-181.
https://doi.org/10.3390/g4020163