answer correctly.

QUESTION 4 Two generals need to coordinate their attacks on an enemy position. The position is either lighy fortified {L} or heavily fortied {H}. Each general has two choices: Attack {A} or Not Attack {N} and the decisions will be made \"sitnultaneously" by the two generals {more precisely. each general makes her decision without krmving the decision of the other general]. Today is Dayalttltltettvcgeneralaarcintltesamelccationandeandiscuasmstters.bmtomorrowthey will take up positions in t'ermit locations. It is common knowledge between the two generals that from her location Gerteral 2 [= Player 2} will not be able to determine if\" the enemy position is I. or H, while General ] I:- Player 1] will be able to con-scrip determine if the enemy position is I. or H. They will be able to commtu'ticate by e-mail via a satellite link that is not completely reliable, inthe sense that every message that issenthasaprobability a {Ilsa-ct } efnor getting delivered. (in Day ll the players agree that Player 1 will send an email to Player 2 at precisely 15:00am grand only if the enemy position is L. Cototnunicarion is done using computers: each computer is programmed to automatically and instantaneously send an acknowledgment whenever it receives a message and at each moment the computer scrmn displays the nrunber or messages received item the other player. Communication is almost instantaneous so that if\" one minute has elapsed since the last. message was received by Computerr' [i = 1,2} than it means that the lad: acknowledgment sent by lIZomputer l was either not received critwas received but the automatically generated acknowledgment by Cornputerj {f at r] was lost; thus, if no message is received within one minute of the remption of the last message, the computer's screen ashes the messsg \"End of communication" and, on the next line, \"total number ofmessages received by this computer: a\". if Computer 2 {Player 2's computer} has not received any messages by dllam the screen displays the message "No messages received - End nfcommunication\". Eta Day it both players agree that the probability that the enemy position is L {and thus that Player 1 will send the nite-mail] isp. with Us: pet]. {a} Draw atree where atthe rstnode there aretwo edges, onerepresenng the possibility that the rst message is sent and the other the possibility that it is not sent. and'at every successive node preceded by I sent message. there are m edges, one representing the possibility that. the automatically generated acknowledgmem is not delivered and the other the possibility that the automatically generated acknowledgment is delivered. Label each terminal node with the total monber of messages sent over the channel and the prior probability [that is, the probability as assessed on Day {I} that tltat node is reached. {bl Using as states the terminal nodes of the tree of part {a}, draw the information partition of Player 1 (that is, the possible titans states of lmowledge of Player I as assessed on Day {I}. it} Using as states the tannins] nodes of the tree of part is}. draw the interaction panirion of Player 2 [that is, the possible future states of knowledge of Player 2 as assessed on Day D]. (d) Suppose that, as a matter of fact, the enemy position is & so that Player 1 at 6:00am of Day 1 sends the notification to Player 2. How many messages need to be successfully exchanged between the two players for it to become common knowledge that the enemy position is L? [continues on the next page] Page 6 of 7 Let the von Neumann-Morgenstern payoffs be as follows (4 means "Attack" and / means "Do Not attack"), where c > 0. If enemy position is _ If enemy position is H Player 2 Player 2 A N A N 1+c 0 C I+c C Player 1 Player 1 N N For the following questions, suppose that on Day 0 the players agree to follow this strategy, call it strategy s, : if a player knows that a total of at least & messages were sent (with k > 0 ), then that player will attack, otherwise she/he will not attack. (e) Suppose that c =2. Are there values of & such that it is in the interest of each player to follow strategy s, if she trusts that the other player will follow strategy s, ? Explain [Hints: (1) you need to use Bayes' rule to obtain posterior beliefs; (2) consider first the case where k is odd and then the case where * is even.](f) Suppose now that 0 0. You find yourself in a situation of asymmetric information: each Davis resident knows his/her own (r, x), but you (the owner of Unter) only know the distribution P over the set of possible pairs (r,x) [ P(r, K) is the fraction of Davis residents who are characterized by the pair (r, x)]. You are risk neutral. Let / be the number of Davis residents. Because of tax reasons, you need to run a small company by enrolling as members not more than K residents. From now on by "price" we will mean the membership fee. Let us start with a simple example, where /= 24,000 and K = 50; furthermore, the possible pairs (r, A) and, for each pair, the number of residents who are characterized by that pair, are given in the following table. (r,K): (100,8) (120, 12) (100,16) (100, 24) (140,24) (120,36) (140,36) (120,40) (140,72) sidents: 2,000 1, 000 3,000 . 2, 000 1, 000 2, 000 4, 000 4,000 5,000 What price will you charge? [You preferences reflect the capitalist society in which you grew up: you want to make as much money as possible! Clearly, your answer will have to be conditional on the value of c, which is the cost of providing each ride.] (b) For this question assume that K > n, = N). Assume a uniform distribution, that is, 1=1 j=1 ny = ny, for every i, j, s, te (1, 2,...,10}