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# Game Theory ECON 617

UM

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This 6 page Class Notes was uploaded by Leopoldo Ritchie DDS on Thursday October 29, 2015. The Class Notes belongs to ECON 617 at University of Michigan taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/231657/econ-617-university-of-michigan in Economcs at University of Michigan.

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Date Created: 10/29/15

Economics of Rumours l Motivation Motivation Economics of Rumours Economic Reasons Abhijit V Banerjee a Volatility in asset markets I I Strong co movement among seeming unrelated financial assets The Review Of Economic Studies April 1993 I Herd like or imitative behaviour of participants in asset markets Presented by Aditya Mahajan SOCIaI Reasons ECON 617 Game Theory 3 Brand choice by customers I Spreading of fads fashions and ideas within society I Should we trust rumours Economics of Rumours Economics of Rumours l General Model l General Model The Model Information Transmission 3 Unknown Investment Opportunity Returns Probability Low b 1 p a 1 0 b C a a and its state ie its returns And decide whether to invest of not At Start of the Process Some randomly chosen investors 3 Learn of the opportunity a Two kinds of agents Cost Probability LOW 0 q 3 Rest of the population does not yet know of the opportunity High c 1 q I A portion of the remaining population hear that someone has Basic Assumptions aneSted I Investment opportunity is unknown HoweVequot they do quot0t know gt agents do not know whether the opportunity exists 39 the I GtUI n Of the opportunity gt agents do not know the returns ie the state of the opportunity the costs of Others who had invested I Cost of investment is private information Have to decide Whether to invest or not I agt cgt bgt0 High cost investors invest when returns are high Economics of Rumours I General Model Rumour Process I Investor has to decide whether to invest or not I If he opts to invest he becomes a source of information to others Decision to believe in the rumour and to pass it on is based on optimizing behaviour Solution Concept Nash Equilibrium of a Bayesian Extensive Game I On hearing the rumour the investors update their beliefs about the state of the opportunity Let u be the updated Economics of Rumours I Information Transmission Different Models of Information Transmission Information Transmission Exogenous or Endogenous 1 Probability of hearing the rumour does not depend on the number of agents who have invested in the past 2 Probability of hearing the rumour is an increasing function of the number of people who have already invested a contagion rocess belief of the opportunity being in high return state a p I The high cost investor invests only if c b ua1 ub2c E ltu a b Economics of Rumours Economics of Rumours I Informalion Transmission I Informalion Transmission Two Simple Models Exogenous Information Tra nsmission I Unknown Investment Opportunity Returns Probability Low b 1 p High 3 a Two kinds of agents Cost Probability Low 0 q High c 1 q I Population is the set of positive integers indexed by i I Discrete time process at each time only one investor gets a chance to invest o Investor i gets his chance in period i I Only the first investor is told the state of the world I All other investors only observe the decision taken by the previous investor 0 b C Existence of Investment Opportunity is Unknown The First Model I The first investor is told the opportunity exists and what the returns are 0 Low Cost ie 0 Always Invests 0 High Cost ie c Invests when the returns are high Le a but not when the returns are low Le b I Investor 2 observes or hears about whether or not investor 1 had invested I If he did not invest the second investor does not even find out that the opportunity exists and the rumour stops o If the first investor invested investor 2 updates his beliefs using Baye39s rule Pra investor 1 invests L p1 pq Economics of Rumours Economics of Rumours I Informalion Transmission Two Simple Models I Informalion Transmission Two Simple Models Two Assumptions The Game Tree Assumption P gt c b p1 pq a b The second investor even if he has high costs will invest if he observes that the first investor has invested Assumption c b gt a b p A high cost investor who only knows the ex ante distribution will not invest Economics of Rumours Economics of Rumours I Informalion Transmission Two Simple Models I Informalion Transmission Two Simple Models Conclusion of the First Model Existence of Investment Opportunity is Known The Second Model I The first investor is told the state of the investment Proposition 21 opportunity The informativeness of the rumour does not change over time in the sense that the ex post conditional on observing that the previous person invested probability that the return is 3 remains the same Everyone invests if and only if the first person invests 0 Low Cost ie 0 Always Invests 0 High Cost ie c Invests when the returns are high Le a but not when the returns are low ie b 3 Everybody knows that the Opportunity exists Key assumptions for the proof everyone who has low costs will invest in their respective periods a The high cost investor however will invest only if state a is sufficiently likely c b mltut1 3 Information transmission process is unaffected by the state of the world 3 Agents do not know that the opportunity exists L P 1 PZf Prinvestor t invested b where zt Prlnvestor t Invested a Economics of Rumours I Information Transmission Two Simple Models Conclusion of the Second Model Preposition 22 The informativeness of a rumour dies away over time in the sense that the ex post conditional on observing that the previous person invested probability that the return is a converges over time to the ex ante probability I If assumption holds there is a time t after which high cost investors do not invest even if the previous person had invested I If assumption does not hold once anyone invests all subsequent investors of either cost type will invest Economics of Rumours I Information Transmission Two Simple Models Conclusion of the Second Model Key steps for the proof I Initially all investors invest if their predecessor had invested but only the low cost investor invest if their predecessor did not invest zt1 1 qt gt1 2 mt ep Informativeness of the rumour decays with time 3 There exists some tquot Z 2 such that high cost investor tquot 1 will not invest even if investor t invests Reason If it were known that a particular investor had low costs his decision to invest will provide no information to the next investor and from then on all subsequent investors will learn nothing from hearing the rumour Economics of Rumours Model with Endogenous Information Transmission Endogenous Information Transmission Drawback with the Simple Models I Fail to capture that speed of transmission should depend on the value of the information a If the information is such that it only affects relatively few people it is unlikely to spread very fast Endogenous Information Transmission Model I Each person has some probability of getting the information at any instant a This probability depends on how many people in the past have used this information to invest a The probability is proportional to the total number of people who have invested at any time in the past I There is no informational delay Each instance of an investment remains a perpetual source of the rumour simple epidemic model Economics of Rumours Model with Endogenous Information Transmi ion Endogenous Information Transmission The Rumour Process 3 The rumour takes form of information that someone has invested I This also tells the potential investor that the investment opportunity is available a This is all the information he gets to know I He does not know the number of people who have already invested a He does not find out what information anyone else in the population had a He gets no negative information I He gets to hear the rumour only once Economics of Rumours Economics of Rumours l Model with Endogenous Information Transmission l Model with Endogenous Information Transmission The Model Some Definitions I Unknown Investment opportunity I For i ab and any time instant 5 define RetUrns Probability Ni5 The proportion of population that has invested Low b 1 p until instant 5 given state i High 3 Pi5 The proportion of the population that has not Two kinds of agents 0 b C a heard the rumour until instant 5 given state i Prri5 The probability that in state i agent hears the Cost Probabilit y rumour for the first time between 5 and 5 d5 Low 0 q High C 1q I y Is the rate of spread of Information At any Instant 5 the probability that a person who ha not yet heard the rumour will I Population given by the 01 interval hear it between 5 and 5d5 is given by I Continuous time process I At time 0 each person has X probability of begin told that the IHItial COHditions opportunity exists as well as whether returns are a or b y x Ni5 gtlt d5 Pa0l X Pb01 X Na0x Nb0xq I After that no one is told the true state of the world any more the only source of information is the rumour Economics of Rumours Economics of Rumours Model with Endogenous Information Transmission Model with Endogenous Information Transmi ion The Decision Problem Dynamics of the Decision Rule Suppose an agent hears the rumour in some time interval 55 d5 I If the agent has low costs she will always invest Prri5 ytNi5tPi5 d5 I If the agent has high costs she will invest only if c b yNbsPbs M5 Pra hears rumour In 55 15 2 3 b 25 yNa39s Ma39s 33 Updated Beliefs Dynamics in Regime 1 By Baye s rule p Prr b5 I dP dN 5 p 1 p25 W ere 25 Mr I 35 yNIsPIs yNIsPIs 34 Decision Rule Dynamics in Regime 2 gs zf 32 1 PCb dP dim 5 5 25 5 2 Regime 1 otherwise Regime 2 T yNIr5PIr5r 5 YQNIISPIr5 35 Both types invest Only low cost invest Economics of Rum Economics of Ru Modei with Endogenous Information Transmission Modei with Endogenous Information Transmission Two Lemmas The Main Result Lemma 31 Behaviour in Regime 1 If the system has been in Regime 1 at all dates before t Pa5 lt Pb5 and Na5 gt Nb5 for 0 lt5 lt 1 Proposition 31 25 increases monotonically over time and is unbounded Therefore for any value of 2 there will be an instant tquot at which there will be a transition from Regime 1 to Regime 2 After iquot the Lemma 32 Behaviour in Regime 2 If the system has been in Regime 1 at all dates before and including t and has been in Regime 2 at all dates after iquot but before 1 system Wlll remain In Regime 2 Pa5ltPb5 and Na5gtNb5 for0lt5ltt Implications Rumour travels faster in high return state Economics of Rumours Economis of Rumours Modei with Endogenous Information Transmission Disoussion Implication of the Main Result Comparison of the Three Models 39 ex P0St after hearing the rumour Pmbablllty Of State 3 goes First Model lnformativeness of the rumour does not change over to 0 over time time a If the rumour is sufficiently old when one first hears it the returns are almost certainly low a Rumour gives us very precise information Second Model lnformativeness of the rumour decays over time till the information becomes essentially uninformative Main Model A rumour that has been around a long time becomes Reason extremely precise Nbs Pbs Mars Pas Nbs klt1 Pbs Nas m unbounded 25 3 Observation of the rumour becomes less and less informative 3 Information from the time it takes for the rumour to first reach us becomes very precise over time

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