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# Economics of Uncertainty, Information, and Games ECON 211A

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This 68 page Class Notes was uploaded by Mrs. Carmelo Deckow on Friday September 4, 2015. The Class Notes belongs to ECON 211A at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 89 views. For similar materials see /class/177945/econ-211a-university-of-california-los-angeles in Economcs at University of California - Los Angeles.

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Date Created: 09/04/15

Carlsson and van Damme Invest Notlnvest Invest 66 6 10 Notlnvest 06 1 00 Three cases 6gt1 dominant strategy to invest 66 01 two pure equilibria coordination problem 6lt1 dominant strategy not to invest incomplete information about 6 each player observes a noisy signal 3326 osi where 82 are independent normal random variables with zero mean and unit variance improper uniform prior over 6 each player sees 6 as normal with mean 32 and variance 02 each sees their opponents signal as the sum of this normal and an independent normal with mean zero and variance 02 that is a normal with mean 32 and variance 202 expected utility gain from investing if probability of opponent notlnvesting is IQ12 is Elelxi qi so best response is invest if this is nonnegative since Elelxi xi this can be written as 32 q12 Suppose you believe your opponent invests for 1Zgt b Then qaiSltIgt b i 2120 hence you must invest if xigtltIDlt ltbxigt lt212ogtgt Suppose you believe your opponent notlnvests for 1zlt b Then qi2 fly32 212o hence you notlnvest if xiltltIgtlt ltb xigt lt212ogtgt Implicitly define the function bkltIgtbk k 212 0 this has a unique solution because lhs strictly increasing in b and rhs strictly decreasing in b since rhs strictly increasing in k bk is strictly increasing Mk has a unique fixed point at 12 why substitute blltk and the RHS becomes CIgt0 1 2 Figure 21 Emction 21c any strategy that is not dominated must satisfy Invest 1gt1 s at Notlnvest lt0 suppose you know your opponent will choose Notlnvest for ltk dominance implies you should choose Notlnvest for azlt bk suppose you know your opponent will choose Invest for gtk dominance implies you should choose Invest for azgtbk so after H round of iterated dominance Invest 13gt b 1 333 Notlnvest 1lt b O Since bk strictly increasing and has a unique fixed point at1 2 lim b 0b 112 see the diagram n gtoo so the only thing to survive iterated weak dominance is the cutpoint strategy Invest 13gt1 2 s at NotInvest 1131 2 and this is a best response to itself since b1 21 2 so it is and equilibrium as well as the only thing to survive iterated dominance weak or strong dominance conditional on 6 the choice of the two players is independent and the probability of investment is ltIgtltlt egt ogt also a continuum of players result payoff to investing 6 1 l where l is fraction of players investing iterated deletion of dominated strategies leaves only lnvest when you get a signal greater than 1 2 Learning in the Worst Case David K Levine September 1 2005 Global Convergence grail of learning research global convergence theorem for convincing learning processes easy to construct examples of learning processes that don t converge nonconvergence looks like cobweb people repeat the same mistakes over and over not terrifically plausible 0 we seem to see much equilibriumness around us traffic example 0 possible and difficult to construct learning processes with global convergence properties more or less must be stochastic to Nash equilibrium but the processes don t make much sense fishing for Nash equilibrium 0 I ll try to convince you that all sensible learning procedures lead in the longrun to correlated equilibrium 0 I ll start by motivating learning processes from an individual perspective ie processes that work 0 I m only going to talk about pure forecasting no causality Worstcase or Universal analysis vs Bayesian analysis 0 opponents may be smarter than you 0 their process of optimization may result in play not in the support of yourp or o probability 1 with respect to your own beliefs is not meaningful in the setting of a game 0 example everyone believing that they face a stationary process a common statistical assumption implies that no one will actually behave in a stationary way 0 these deficiencies in the robustness of Bayes learning are why there is no satisfactory global convergence theorem for learning procedures Classical Case of Fictitious Play keep track of frequencies of opponents play begin with an initial or prior sample play a bestresponse to historical frequencies not well defined if there are ties but for generic payoffprior there will be no ties optimal procedure against iid opponents how well does fictitious play do if the iid assumption is wrong How well can ctitious play do in the longrun 0 notice that fictitious play only keeps track of frequencies can fictitious play do as well in the longrun as if those frequencies but not the order of the sample was known in advance 0 alternatively suppose that a player is constrained to play the same action in every period so that he does not care about the order of observations Universal Consistency let u be actual utility at time 2 let j be frequency of opponents play joint distribution over Squot suppose that for all note that this does not say for almost all sequences of opponent play liminfTgtW1 TZ1u maxsi my 132 0 then the learning procedure is universally consistent ls ctitious play universally consistent Fudenberg and Kreps example 00 11 11 00 this coordination game is played by two identical players suppose they use identical deterministic learning procedures then they play UL or DR and get 0 in every period this is not individually rational let alone universally consistent Theorem Monderen Samez Sela Fudenberg Levine fictitious play is consistent provided the frequency with which the player switches strategies goes to zero Smooth Fictitious Play instead of maximizing uisquot j1 maximize ui0i 1 vi0i where vquot is smooth concave and has derivatives that are unbounded at the boundary of the unit simplex example the entropy vimquot Si ailtsilogaquotltsquot as xi 0 this results in an approximate optimum to the original problem however the solution to ui0i 1 vi0i is smooth and interior always puts positive weight on all pure strategies Theorem Blackwell Hannah Fudenberg and Levine and others smooth fictitious play is a universally consistent with g a 0 as a 0 9 Conditional Probability Models Experts allow time dependent games liminfTgtW1TZ1uj 1 TZnaxsiusisti 2 0 same theorem holds without change in proof a model makes conditional probability forecasts an expert makes recommendations about how to play s eih1 set veisti uieih1sti conclusion can do as well as if you knew who the best expert was in advance 1O Conditional Probability Models Direct classify observations into subsamples countable collection of categories P classi cation rule VEH x S a 1 Vi hi 1 a 5 igy empirical distribution of opponent s play conditional on the category w may is number of time category has occured effective categories minimal finite subset Pt 6 Pwith all observations through time t m denotes the number of effective categories 11 Assumption 1 limHW mt l 0 This is essentially the method of sieves 12 Universal Conditional Consistency total utility actually received in the subsamplew is um MW uiSi t i lm MW gt 0 CtW 0 universal conditional consistency limsup1TZWEP cmy s 0 13 Non Calibrated Case categorization rule depends only on history not on own plans 1 given hj1 yhj1 chooses the category 2 play a smooth fictitious play against the sample in the chosen category 9110 3 add the new observation sf to the category whj1 Works like smooth fictitious play within each category so universally conditionally consistent 14 Calibrated Case try to use a rule gyhj1sj focus on special case ySf P S each category 1 has a corresponding smooth fictitious play 0quot quot1gu suppose we choose category 1 with probability My then overall play is pmquot ZWMWMMIQWMSU 15 but categories correspond to own strategies fixed point property Ms prs Msquot ZVlw0quot 1wsi unique fixed point solvable by linear algebra 16 Interpretation of Calibration weather forecasting example calibrated beliefs versus calibrated ac ons consequence of universal calibration global convergence to the set of correlated equilibria 17 Shapley Example A M B 00 01 10 10 00 01 01 10 00 18 smooth fictitious play time in logs Exponential Fictitious Play 5300 ioou 06 0 4 time average frequency 53 o 0 O1 0 N 01 W 0 i i i quot 39 i quot N LO V O N LO V O N LO V O N V Lo N LO N V 07 07 00 N LO O O O 0 N V 00 L0 num ber of periods condition on opponents last period play time in logs 19 time average frequency Learning Conditional on Opponent39s Play 0 a num ber of periods 16382 20 Discounted Learning A learning procedure 3 is gas good as a procedure p if for all sequences of discount factors t and all histories h ZZl tMMht D St i S 2l tult ht 1gtast igt 5 Proposition 2 For any learning procedure p and any 8 there exists a categorical smooth fictitious play bthat is gas good as p exploits the fact that the time average result must be true for at every time 21 The Folk Theorem Review of Long Run vs Short run max u1a mixed precommitmentStackelberg V1 best dynamic equilibrium best dynamic equilibrium w moral hazard pure precommitmentStackelberg static Nash worst dynamic equilibrium w moral hazard 21 worst dynamic equilibrium minmax min u1a 0 structure of an equilibrium 0 role of reputation can do strictly better when there is moral hazard Simple Folk Theorems 0 socially feasible 0 individually rational Statement of Folk Theorem Prisoner s Dilemma Game R L U 22 03 D 30 11 o Nash with time averaging o perfect Nash threats with discounting public randomization vs discount factors near one Vt 1 ut vt1 vt1 6 1vt 1 661ut note that coefficient add up to one Fudenberg Maskin Theorem issue perfection and minmaxing minmax followed by reversion to another equilibrium note simultaneous determination of equilibria Moral Hazard Folk Theorem incentive constraints convexity of space halfspaces and necessity smooth approximations halfspaces and sufficiency Information Conditions At a Point 0 enforcible o pairwise identifiability o br for playeri o coordinate vs regular hyperplanes enforcible br gt coordinate o enforcible pairwise identifiability gt regular full rank gt enforcible pairwise full rank gt pairwise identifiability Modeling Altruism in Experiments David K Levine September 1 2005 Ultimatum Roth et al 1991 ultimatum bargaining in four countries extensive form x10x A R OD usual selfish case with a 0 player 2 accepts any demand less than 10 subgame perfection requires player 1 demand at least 995 Table 1 below pools results of the final of 10 periods of play in the 5 experiments with payoffs normalized to 10 Demands of Acceptance Table 1 Altruistic Preferences players 239 1n at terminal nodes direct utility of ul coefficient of altruism 1 lt a lt I adjusted utility V1 u Zj aiuj d i Vu u ftquot 11 03131 objective is to maximize adjusted utility since the stakes are small ignore risk aversion and identify direct utility with monetary payoffs prior to start of play players drawn independently from population with a distribution of altruism coefficients represented by a common cumulative distribution function Fa each player s altruism coefficient a is privately known the distribution F is common knowledge we model a particular game as a Bayesian game augmented by the private information about types marginal utility of money returned to experimenter is assumed zero Related Work VI u Zm ijuj 6 determined from players types or other details about the game 0 Ledyard 1995 y 2 7uf u is undefined fair amount 0 Rabin 1993 y 2 7ul ulf player cares about fairfor himself rather than fair for the other player fair amount is a fixed weighted average of the maximum and minimum Pareto efficient payoff given player i s own choice of strategy coefficient 7 endogenous in complicated way Andreoni and Miller 1996 Palfrey and Prisbrey 1997 warm glow effect value of contributions to other players not so important as the cost of the donation there is a warm glow players wish to incur a particular cost of contribution regardless of the benefit 4person public goods contribution game players must decide whether or not to contribute a single token each period each player randomly draws value 5 for token uniformly distributed on 1 to 20 token kept the value of token is paid token contributed fixed amount 7 paid to each player 1 5139 gimi 7221771139 39 each player 20 rounds with fixed value of 7 four times with different values of 7 each round players shuffled results from the second 10 rounds with each value of 7 so players relatively experienced 73 715 i y Gain m Gain m ratio ratio 5 18 000 90 060 34 27 018 131 067 12 68 027 337 079 0 088 086 Table 2 data is pooled as indicated in the table Ultimatum Roth et al 1991 ultimatum bargaining in four countries extensive form x10x A R OD usual selfish case with a 0 player 2 accepts any demand less than 10 subgame perfection requires player 1 demand at least 995 Table 2 below pools results of the final of 10 periods of play in the 5 experiments with payoffs normalized to 10 Demand Obs Frequencyof Probabilityof Adjusted Observations Acceptance Acceptance 500 37 28 100 100 600 67 52 082 080 700 26 20 065 065 Table3 10 Proposition 1 No demand will be made for less than 500 and any demand of 500 or less will be accepted In fact in the data only was offer of less than 500 was ever made and it was for 475 and was accepted so the data are consistent with Proposition 1 11 assume that the distribution F places weight on three points c7 gt do gt g altruistic normal and spiteful types since there are three demands made in equilibrium and more altruistic types will prefer to make lower demands we look for an equilibrium in which the altruistic types demand 500 the normal type 600 and the spiteful type 700 also require that no type wants to demand more than 700 so probabilities of the three types are 028 052 and 020 respectively as this is the frequency of demands in the sample 12 500 demand is accepted by all three types 600 demand is accepted by 82 of the population but attribute the difference between 80 and 82 to sampling error can t reject at 28 level so assume exactly spiteful types reject 700 demand accepted by 65 of the population corresponding to all the altruistic types 28 and 71 071gtlt052 z 037 of the normal types so normal types must be indifferent between accepting and rejecting a 700 demand 13 consider the 500 demand all types will accept this demand the adjusted utility received by a player demanding this amount is a x128c752a020g 5 1 if the spiteful type accepts all types will accept the demand 5 since offer is known to be made by the altruistic type for spiteful type to accept we must have 11 this inequality is always satisfied for M gt 1 14 16a0 1356765a0 485 a0 1280752a020Q 5 2 0 11 11 26J135c765a0 48 5CT128CT52a020Q 5 S 0 11 11 3 4 LM06 s 0 1 1 7 a 143c757a0 65 11 4x7 g 143J57a0 3065 6 g 35676561048 2 0 11 11 57 a0 143c757a0 365 6 a0 135c765a0 108 S 0 11 11 a 161 6 3 0 7 11 15 a sequential equilibrium matching the data will be given by parameters 1gt 67 gt do gt g gt 10 3 xi 3 1 such that the inequalities 1 through 5 and the equality 6 above are satisfied Proposition 2 There is no equilibrium with xi 0 Proposition 3 In equilibrium 3013 a0 3 O95 1 lt g lt 23 12 xi 2 0222 16 Parameter s consistent with sequential equilibrium a 01 0 0 0 0 0 0 0 0 O 08 O O O O O 04 Table 4 it appears to be difficult to get g larger than 087 versus the known lower bound of 23 values of are difficult to find lower than 035 against the known lower bound of 022 values of are difficult to get higher than 049 although I have not been able to get an analytic upper bound on xi other than 1 couldn t find equilibria with values of a0 below 02 although the known lower bound is only 301 17 Competitive Auction Sanity Check Roth et al report a market game experiment under similar experimental conditions Nine identical buyers submit an offer to a single seller to buy an indivisible object worth nothing to the seller and 1000 to the buyer If the seller accepts he earns the highest price offered and a buyer selected from the winning bids by lottery earns the difference between the object s value and the bid Each player participates in 10 different market rounds with a changing population of buyers game has two subgame perfect equilibrium outcomes with selfish players either the prices is 1000 or everyone bids 995 in the experiment by round 7 the price rose to 995 or 1000 in every experiment and typically this occurred much earlier 18 let a be the coefficient of altruism adjusted for the opponent s altruism seller accepts X if x a1 x 2 0 a gt 1 so true provided that x 2 500 buyers if there are multiple offers at 1000 then no seller can have any effect on their own utility since the seller always gets 1000 and the buyers 000 regardless of how any individual seller deviates more generally suppose that seller offers are independent of how altruistic they are an offer X accepted with probability p gives utility p1 x00c1 pa a1 ap1 x which regardless of a are the same preferences as l x since preferences are independent of altruism players are willing to use strategies that are independent of how altruistic they are so every equilibrium without altruism is an equilibrium with altruism 19 Centipede McKelvey and Palfrey 1992 29 experiments over the last 5 of 10 rounds of play P P P P 062 0521 0235 014s64o160 T1 008 T2 049 T3075 T4082 04oo1 0 020080 1 60040 080320 Figure 1 does not make much sense with selfish players 18 of player 2 s who reach the final move choose to throw away money with selfish preferences the unique Nash equilibrium is for all player 1 s to drop out immediately 20 model the same model of three types we used to analyze ultimatum assume 2 045 g 09 and a0 022 which are parameters that have been narrowed down by the data on ultimatum probabilities of the spiteful normal and altruistic groups are 020052028 respectively 21 virtually no player 1 s drop out in the first move so that the distribution of types the second time player 1 moves should be essentially the prior distribution second move by player 1 25 of the players choose to continue which within the margin of sampling error is quite close to the 28 of player 1 s that are altruistic So we will assume that in player 1 s final move all the altruistic types pass and all the other types take and we will analyze the following modified data 160 0521 0238 0148640160 T1 000 T2 049 T3072 T4082 040010020080160040080320 Figure 2 22 player 2 s at the final node first spiteful and selfish types drop out before altruists and fewer players pass than the 28 of the population that are altruists we conclude that the altruistic types must be indifferent between passing and taking all player 1 s are known to player 2 to be altruists at this point it follows that c7xic7 c7xic7 080 160 11 11 From this we may calculate c7 27 z 029 This is one of the wide range of values consistent with the ultimatum data 320 640 23 consider player 1 s final decision to pass or take 51 of the player 2 s previously passed including all the altruistic player 2 s so 028051 055 of the player 2 s are altruists and the remaining 045 are selfish types player 1 takes he then places a weight on his opponents utility of a a0zlt055m045xaogt 013 T 11 I utility if he takes is 160 aT 040 155 pass has a 018 chance of an altruistic opponent gets 640 for himself and 1 60 for the opponent or 631 faces a 082 chance of an opponent who is 045082 055 likely to be selfish and 045 likely to be altruistic 24 yields a utility of 033 averaging over his opponent passing and taking in the final round yields the expected utility to passing of 1 40 less than the utility of taking selfish type should take so should spiteful type since normal type nearly indifferent altruistic type passes 25 utility from taking and passing Node Type Take Pass Difference Utility Utility 1 slast a0 155 140 014 move 2 sfirst a0 076 085 009 move 1 sfirst Q 033 049 016 move Table 5 spiteful type 1 player willing to pass in the first period only inconsistency selfish type of player 2 first move should be indifferent between passing and taking and in fact prefers to pass 27 Public Goods Contribution Game public goods contribution game studied by Isaac and Walker 1988 simultaneous move n person game each individual may contribute a number of tokens to a common pool or consume them privately ml is the number of tokens contributed normalize so that the total number of available tokens per player is 1 the direct utility is given by n u 2 ml yZFlmj 28 four treatments were used with different numbers of players and different values for the marginal per capital return 7 consider the final round of play only each treatment was repeated three times The data from the experiments n migt0 gt1 0 01 O O Table 6 29 vs 28 altruists w average coefficient of 029 as above assume 1 045 g 09 do 022 c7 029 w probabilities 020 052 028 mean population altruism d 021 30 adjusted utility of contributing az i Ad 1 A where LZ is the mean contribution by players otherthan player i Ur m 7mz39 m4 n 1 77AL 7mz39 n DWALz39 differentiating with respect to own contribution GZ39 1 gt0 1A01 h 17 And cacuate cutoff agtlltlt171 gt amp W DV A39 31

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