costly signalling 1

Costly Signaling Meshe Hoffman and Erez Yocli 1f you are working with a pastner, you and your partner may tum in a single copy of the problem sek. Please shonw your work and acknowledge any additional resources consulted. 1. Simplest Model of Costly Signaling Why do poople buy BMWs even though they can't get them to work any faster in morsing traffic? O Guce t-shirts that are o warmes, softer, or longer-lsting than $10 ones from Fraitof-the-Loom? Why do axtists or fashion designers revel in making ast within seemingly pointless constraints, like when Emincm rhymes, or Shakespeare uses lambic Pentometer? Why is it fashionable to wear clothes that are particularly dificult to care for, like white sncakers? Why, as we discussed in class, did our preforences with respect to BMI change with time, and why do they vary across culture? Similarly for skin tono? Why is it attractive to grow long “pinky-nails in some cultures? We'll use the costly signaling model to explain such puszing features of our sense of assthetics. “This i the very simple model of costly signaling that was presented in class « There aze two players, ¥ = {S. 1), a sender (S) and a receiver (R). « The states of the world are © = (L, 1) These represent the sender's type, low (L) or high (H). The lkelibood the sender ‘Sendors obscrve their type, then choose a signal to send from the set As = {sp,51). Technically, sender's strategics are functions that specify which signal they send given their type, s £ ©2 - As. Valid strategics for the sender are thus: always send so, send 1 only if H, send so only if H, and always send s1. We'll discuss this further when wo o o payoffs, but there is a cast associated with sending ;. 1t i ¢, for low types and i for high types, whero eyr < ey Tho cost of sonding sq is zer0 for both types. « Reccivers do not observe the seader’s type, but they do observe the sender's signal. Then, they choose whether to accept or rejeet the sender, Ag = (A, K). Technically, receiver's strategies are functions that specify whether they accept given the sender’s signal, 7 : As -+ Ap. The strategies available to the reciver are: never accept, only accopt the sender f ho sends 1, only accept if ho sends 5o, oF always accopt. « High senders roccive a payoff o ay from being accopted. Low senders roceive a payoff . from being accepted. Both receive a payofl of 0 if they are not accepted. From these payoffs, we must sublract the cost of sonding s, when it was sent. Receivers roceive a payoff of by from accopting a high type, by from accepting low type, and 0 from accepting no one. We assume by > by Technically both players’ payofls are Functions that depend on the sender’ type, the signal sent, and whoter the recciver ccepted us : 2 x As X Ar -+ R, un : ©x As x Ag - R. Notice, however, that the rocelver's payofis do not dircetly depend on the sender’s signal, only on the sender’s type, and wheter the recciver accepted tho sender. An example: when the sender is type L, sends signal s, and the receiver accepts, the sender and receiver payoffs are us(Lys1,4) = ai ~ et and un(L, 51, 4) = b respectively. (8) The following strategy profile which we call 5%, is the one of greatest interest to s the sender sends 51 only if H, and reciver only acoepts i the sender sent s, & When players play according to 6°%, what are the sender’s payoffs when his type is L7 When it is H? 6. What are the roceiver’s payoffs in cach case? . Under what conditions is this strategy profle an eq rating equilibrium)? To answer this, check the following. © What deviations are available to the sender whea bis type is L? H? How do bis payolfs compare to those you found in question 1(a)i? © What deviations are available to the receiver when she observes 7 517 How do these compare 1o those you found in question 1(a)i? v, A key, counter-intuitive result i that signals must be sufficiently costly, or we will not observe costly signaling. Point to the condition that yields this result. Which deviation s this condition proventing? Can you clucidate how the cxamples from. class like skin-tone, BMI, and fingernails relate to this condition? (1) Now consider the following two strategy profies. Pooling with refection: the sender always sends s, and the receiver never accepts. + Pooling with acceptance: the sender always sends s and the receiver always accepts. Under what conditions, if any, are these strategy profies oquilibria? How do theso conditions depend on py and b,