MA Game Theory 2005, LSE

Transcription

1 MA. Game Theor, LSE Problem Set The problems in our third and final homework will serve two purposes. First, the will give ou practice in studing games with incomplete information. Second (unless indicated below) the are formatted like our eam questions. M part of our eam will consist of four such questions, and ou will have to answer three of them. Notice that the questions ma look long, but that is onl to introduce the problem. The answers that I ask for are generall quite short. I do not epect, nor encourage, long answers, just answers that are to the point. However, ou must eplain wh ou get what ou do, otherwise ou will not get full credit. Some of the problems below are adapted from Martin Osborne s tetbook. [] Two persons are bidding on an object for auction. Each person has independent, private valuations for the object that are drawn from a uniform distribution over the interval [, ]. The auction is sealed-bid, first price. (a) Formall describe this situation as a Baesian game, taking care to define states and strategies in particular. (b) Find the smmetric equilibrium of this game, in which each plaer uses the same bid function. (c) Using the equilibrium bid function, prove that the outcome of the auction ields the same epected revenue as a second price auction in which the equilibrium in weakl dominant strategies is plaed. [In answering this question, ou can simpl state what the equilibrium in weakl dominant strategies is for the second-price auction, without an proof.] [] In this question, we stud a problem of public goods provision with incomplete information. Suppose that a well is to be dug in a village with n people. It costs c to get the well running. Each person has a valuation for the well. Each person i knows her true valuation v i. But when i j, i onl knows that j s valuation is drawn independentl from a uniform distribution (the same one for everbod) over the range [v, v]. Assume that v < c < v. The following mechanism is to determine whether the well will be dug. All the individuals simultaneousl submit sealed envelopes to an arbitrator. The envelopes can either contain a contribution of c or contain nothing (no intermediate contributions are allowed). If everone submits nothing, the well is not dug and everone gets. If at least one person submits c, then each person i who did contribute c gets v i c (no ecess contributions are returned), and each person i who contributed nothing gets free use of the well, with paoff v i. (a) B comparing the costs and benefits of contributing c, find a smmetric Baesian Nash equilibrium in which each person contributes if and onl if her valuation is above some

2 threshold v. Prove that v is the unique solution to the equation ( v () v ) v n = c. (b) The probabilit that the good is provided is obviousl equal to ( v ) v n (because this is the chance that at least one person has a valuation eceeding v ). Using equation (), prove that ( v ) v n = c F (v ) v, and using this equation, what can ou sa about the probabilit of the public good being provided as the number of individuals n increases? Does it rise? Fall? Sta the same? Eplain our answer. [] A defendant in a case is either Guilt G or Innocent I. The judge doesn t knows which, but has a prior that the defendant is G with probabilit π. We summarize the progress of the case b saing that the judge receives one of two signals. A i-signal indicates innocence, a g-signal guilt. If the defendant is trul guilt, the g-signal is received with probabilit p > / (but the i-signal can also be received, with probabilit p). Similarl, if the defendant is trul innocent, the i-signal is received with probabilit q > / (but the g-signal can also be received, with probabilit q). That is, the signals are nois but indicative of true guilt or innocence. The judge s paoffs are: = if a trul guilt person is convicted or a trul blameless person is acquitted. = z if an innocent person is convicted. = ( z) if a guilt person is acquitted. (a) Show that z can be viewed as a measure of resistance to conviction in the following sense: if the judge assesses the defendant to be guilt with probabilit r, then acquital is weakl better than conviction iff r z. (b) Prove that the judge makes different decisions depending on her received signal (i.e., convict if she sees g, acquit if she sees i) whenever () [You will have to use Baes Rule.] ( p)π ( p)π + q( π) < z < pπ pπ + ( q)( π). [] [Too long to be an eam question but good practice.] Now we replace the judge b n identical jurors, all with the same signal structure and preferences as the judge in the previous question. But the receive their signals independentl (in particular, two jurors could get different signals). Each juror looks onl at her own signal no one shares signals in this problem and votes to acquit or convict.

3 Conviction occurs if and onl everone votes to convict, otherwise the defendant is acquitted. Assume that the condition in equation () is satisfied, so that if each juror were to be full in charge of the case, she would vote to convict on seeing a g-signal and to acquit on seeing an i-signal. In this question, we want to see whether this voting according to signal can continue to be an equilibrium when there are several jurors, instead of just one. (a) Assuming that each juror votes according to her signal. Prove that an individual s vote onl makes a difference when all other jurors have received a g-signal. (b) B part (a), it suffices to look at an individual juror s behavior in the event that everone else gets a g signal. Now suppose the juror does receive an i-signal. Show that she will use it i.e., vote to acquit in this case onl if z + q π p π ( ) n q p (c) Notice there is something odd about the condition in (b). Show that its right hand side converges to as n becomes large. This means that for a given resistance to conviction (see the interpretation of z in the previous question) each juror cannot vote according to signal as n becomes large! (d) Interpret this result. Recall part (a) to do so. (e) Just for fun, and to see how quickl signals ma start getting ignored under these presumptions, get a calculator and calculate the value of the RHS when n =, π =., and p = q =.8 (prett reasonable parameters). (f) So for n large, there is no Nash equilibrium in which ever juror votes strictl according to her signal. What about miing? Prove that there is no smmetric Nash equilibrium in which jurors mi if the get a g-signal. [Hint: if the do so, the must be indifferent when the get a g-signal, so must strictl prefer to acquit when the get a b-signal. Now show that same argument as in part (c) applies.] (g) [Etra credit; consult Osborne s tet for help if needed.] So the onl kind of smmetric Nash equilibrium involves a juror voting to convict with some probabilit β when the signal is i and voting to convict for sure when the signal is g. Compute this equilibrium and show that it must have the propert that ( ) q + qβ n = p + ( p)β ( p)π( z). q( π)z [To show this, note that we must have indifference when a b-signal is received. But use part (a): this means that we must have indifference in the pivotal case when everone else is voting to convict. Pr(G signal is b and n votes for C ) = z. Now epand this epression. You will have to use Baes Rule again.] [] Consider the following three-stage version of the centipede game:

4 (a) Carefull describe the subgame perfect equilibrium of this game. (b) Now amend this game as follows. With probabilit ɛ plaer s preferences are as shown in the diagram (call this the rational tpe of ). But with probabilit ɛ, he is the sort who alwas likes to pass (move horizontall in the diagram) without regard to his paoff. Plaer is unchanged. Prove that in an Baesian Nash equilibrium, it cannot be the case that the rational tpe of plaer ends the game for sure in the first stage. (c) On the other hand, prove that if ɛ < /, it also cannot be an equilibrium strateg for the rational tpe of plaer to pass for sure. This part, combined with part (b), means that the rational tpe of plaer must mi between grabbing and passing at the ver first date. (d) [etra credit, great practice] Tr and solve the mied strateg Baesian Nash equilibrium of this game. [6] This is a formalization of the idea of conspicuous consumption. Suppose that Jim s wealth is either high H or low L, with H > L >. Jim knows which one it is; the world at large doesn t. But Jim would like people to think that he has high wealth; this makes him happier. Assume that if people think he has high wealth with probabilit q, his paoff equals q. Initiall the world has a prior on Jim: it thinks Jim is rich with probabilit p. Now, suppose that Jim has the option to bu flash things to signal his wealth. Let c be the conspicuous consumption of these flash items and suppose that the bring no intrinsic pleasure to Jim. There is a cost, though. Let the cost of c units be c/w, where w is Jim s wealth, which can take on one of two values. The world observes c and updates its priors on Jim. Jim s final net paoff is q c/w, where q is the updated posterior and w is Jim s private wealth. (a) Are there separating sequential equilibria in this model? (b) Are there pooling sequential equilibria in this model? (c) Are there hbrid equilibria? (d) Describe what happens if the intuitive criterion is applied to these equilibria. [7] Finall, a little puzzle with the sequential equilibrium definition. Look at this game:

5 u a b (a) Find all the sequential equilibria of this game. Now look at this game: u r a b (b) Find all the sequential equilibria of this game. (c) Notice that plaing u can be part of a sequential equilibrium in (a) but not in (b). Wh are these two games different? [Think of our discussion in class regarding off-equilibrium moves as errors.]