Buried Signals Game Theory and Social Behavior If you are working with a partner, you and your partner may turn in a single copy of the problem set. Please show your work and acknowledge any additional resources consulted. 1. Setup The signal-burying game is a more complicated signaling game than the previous costly signaling model we saw. Now there will be three types of senders and two types of receivers, Moreover, the sender will have the opportunity to "bury\" her signal, in which case there is a chance the signal will go undetected. Our key question will be: under what conditions might a sender choose to do this. At the beginning of the game, each player's type is randomly determined. The sender's type will be High (H), Medium (M), or Low (L), with probability pur, par, pr respectively. The receiver's type will be discerning (D) and undiscerning (U}, with probabilities gp. g respectively. Senders choose between three actions, no signal, (n), elear signal, (c) and buried signal(h). Clear signals and buried signals each cost ; = 0 to send, where this cost depends on the sender's type i {L, M, H}). Buried signals will remain buried, and look indistinguishable from n to the Receiver, with probability 1 - i, which again may depend on the Sender's type, ie {L M H} The sender learns her type before choosing among her available actions. Then the Receiver observes either . or b, depending on the Sender's choice and whether the Buried signal remained buried. The Heceiver then decides whether to reject or accept the Sender. This decision can depend on the Receiver's type as well as what she observes, but not directly on the Sender's type, which the Receiver never observes. Senders receive a payoll a;; > 0 if they are type i and accepted by receivers of type j. Receivers obtain payoff b, if they are type j and accepted a sender of type i. When the receiver chooses to reject, both the sender and receiver get 0, regardless of their type. We are interested in the strategy profile in which: o Low senders send no signal n o Medium senders send the clear signal s High type senders bury b s Undiscerning receivers accept either the clear signal or the buried signal b s Discerning receivers only sccept buried signals b 2. Questions (a) Finding the Equilibrium Conditions i. What payoffs does each type of each player expect to get, if everyone plays according to this strategy profile? i, List out the possible deviations for each type of sender. What payolfs would they get for those deviations, holding constant what the receivers are deing\" Hint: each type of sender can deviate in more than one way. Make sure to include all of them, iil. List out the possible deviations for each type of receiver. What payoffs would they get for each of those deviations, holdiing constant what the senders are doing? Hint: each type of recelver can deviate in more than one way., Make sure to include all of thermn. iv. Using your answers to the above questions, list oll the inequalities that need to be satisfied to ensure that the above strategy profile is an equilibrium. Hint: recall that for a strategy profile to be an equilibrivin no type of any player can benefit from unilaferally deviating, These inequalities, together, are called the \"equilibrium conditions. Hint: you should end wp with tnequalitics like the follouring: =prbrn = (1 = rg)pubnp (1) 2) 3) ) prbey = (1 ra)pubau e e 2 quagy e Z rilgpaLp + qray) (4 The first two inequalities come from ensuring that neither type of receiver wants to accept the "no signal,\" taking into account what the varions senders are doing. The second two inequalities come from ensuring low type of sender doesn't want to send either the buried or clear signal, taking into account how the receivers react. (b) Bonus Questions: Interpretation i. ii. See if you can interpret each of the 'equilibrium conditions' you obtained above. Hint: For instance, the first two conditions are interpreted as \"both types of receivers sufficiently dislike the low types, that they prefer not to accept the \"no signal\