Costly signalling 1

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School

Stanford University *

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Course

167G

Subject

Chemistry

Date

Oct 30, 2023

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Uploaded by MajorLobsterMaster909

1. Simplest Model of Costly Signaling Why do people buy BMWSs even though they can't get them to work any faster in morning traffic? Or Gueci t-shirts that are no warmer, softer, or longer-lasting than $10 ones from Fruit-of-the-Loom? Why do artists or fashion designers revel in making art within seemingly pointless constraints, like when Eminem rhymes, or Shakespeare uses lambic Pentameter? Why is it fashionable to wear clothes that are particularly difficult to care for, like white sneakers? Why, as we discussed in class, did our preferences with respect to BMI change with time, and why do they vary across culture? Similarly for skin tone? Why is it attractive to grow long ‘pinky’-nails in some cultures? We'll use the costly signaling model to explain such puzzling features of our sense of aesthetics. This is the very simple model of costly signaling that was presented in class: (a) (b) There are two players, N = {S, R}, a sender (S) and a receiver (R). The states of the world are 2 = {L, H}. These represent the sender’s type, low (L) or high (H). The likelihood the sender is a high type is py. Senders observe their type, then choose a signal to send from the set Ag = {sg,5,}. Technically, sender’s strategies are functions that specify which signal they send given their type, og : 2 <> Ag. Valid strategies for the sender are thus: always send sy, send 51 only if H, send sp only if H, and always send s). We'll discuss this further when we get to payoffs, but there is a cost associated with sending s,. It is ¢, for low types and ¢y for high types, where ¢y < ¢;. The cost of sending s is zero for both types. Receivers do not observe the sender’s type, but they do observe the sender's signal. Then, they choose whether to accept or reject the sender, Ay = {A, R}. Technically, receiver's strategies are functions that specify whether they accept given the sender’s signal, o : As <+ Ap. The strategies available to the receiver are: never accept, only accept the sender if he sends 51, only accept if he sends so, or always accept. High senders receive a payoff of ay from being accepted. Low senders receive a payoff ar from being accepted. Both receive a payoff of 0 if they are not accepted. From these payoffs, we must subtract the cost of sending s, when it was sent. Receivers receive a payoff of by from accepting a high type, by from accepting a low type, and 0 from accepting no one. We assume by > by. Technically both players’ payoffs are functions that depend on the sender’s type, the signal sent, and whether the receiver accepted ug : 2 x Ag x Ag = R, up : 2 x Ag x Ap - R. Notice, however, that the receiver's payoffs do not directly depend on the sender’s signal, only on the sender’s type, and whether the receiver accepted the sender. An example: when the sender is type L, sends signal s), and the receiver accepts, the sender and receiver payoffs are us(L,s1, A) = ar — e and ur(L, s1, A) = br, respectively. The following strategy profile, which we call 0©%, is the one of greatest interest to us: the sender sends s; only if H, and receiver only accepts if the sender sent s,. i. When players play according to oS, what are the sender’s payoffs when his type is L? When it is H? ii. What are the receiver’s payoffs in each case? iii. Under what conditions is this strategy profile an equilibrium (it is called the sepa- rating equilibrium)? To answer this, check the following. e What deviations are available to the sender when his type is L7 H? How do his payoffs compare to those you found in question 1(a)i? e What deviations are available to the receiver when she observes ;7 5,7 How do these compare to those you found in question 1(a)ii? iv. A key, counter-intuitive result is that signals must be sufficiently costly, or we will not observe costly signaling. Point to the condition that yields this result. Which deviation is this condition preventing? Can you elucidate how the examples from class like skin-tone, BMI, and fingernails relate to this condition? Now consider the following two strategy profiles. e Pooling with rejection: the sender always sends sg, and the receiver never accepts. e Pooling with acceptance: the sender always sends sy and the receiver always accepts. Under what conditions, if any, are these strategy profiles equilibria? How do these conditions depend on py and by”

Costly signalling 1

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