The Market Organism: Long Run Survival in Markets with Heterogeneous Traders

Transcription

1 The Market Organsm: Long Run Survval n Markets wth Heterogeneous Traders Lawrence E. Blume Davd Easley SFI WORKING PAPER: SFI Workng Papers contan accounts of scentfc work of the authors) and do not necessarly represent the vews of the Santa Fe Insttute. We accept papers ntended for publcaton n peer-revewed journals or proceedngs volumes, but not papers that have already appeared n prnt. Except for papers by our external faculty, papers must be based on work done at SFI, nspred by an nvted vst to or collaboraton at SFI, or funded by an SFI grant. NOTICE: Ths workng paper s ncluded by permsson of the contrbutng authors) as a means to ensure tmely dstrbuton of the scholarly and techncal work on a non-commercal bass. Copyrght and all rghts theren are mantaned by the authors). It s understood that all persons copyng ths nformaton wll adhere to the terms and constrants nvoked by each author's copyrght. These works may be reposted only wth the explct permsson of the copyrght holder. SANTA FE INSTITUTE

2 The Market Organsm: Long Run Survval n Markets wth Heterogeneous Traders Lawrence Blume and Davd Easley March 2008 Frst verson, November Research support from the Natonal Scence Foundaton for research support under grant SES and the Cowles Foundaton for Research n Economcs at Yale Unversty s gratefully acknowledged. Cornell Unversty Santa Fe Insttute

3 Abstract: The nformaton content of prces s a central problem n the general equlbrum analyss of compettve markets. Ratonal expectatons equlbrum dentfes condtonng smultaneously on contemporaneous prces and prvate nformaton as the mechansm by whch nformaton enters prces. Here we look to the ecology of markets for an explanaton of the nformaton content of prces. Markets could select across traders wth dfferent belefs, or, remnscent of the wsdom of crowds, markets could balance the dverse nformaton of many partcpants. We provde theoretcal support n favor of the frst mechansm, and aganst the second. Along the way we demonstrate that the necessary condton for long-run survval n complete markets found n Sandron 2000) and Blume and Easley 2006) s not suffcent for long run survval. We also demonstrate some surprsng behavor of market prces when several trader types wth dfferent belefs survve. Ths paper contnues the research program of Blume and Easley 1992) and 2006). Correspondng Author: Lawrence Blume Department of Economcs Urs Hall Cornell Unversty Ithaca NY USA lb19@cornell.edu

4 1 Amazngly, the stock market knows better than the analysts do. Henry Herrmann, Chef Investment Offcer, Waddell & Reed The stock market has predcted nne out of the last fve recessons. Paul Samuelson, Newsweek, Introducton Economsts, busnessmen and the laety all talk about the knowledge of the market: The market learns... and the market knows... are all accepted explanatons for observed prcng phenomena. The ultmate expresson of ths dea s the wde and recent nterest n predcton markets. Despte the embarrassment of the Unted States Defense Department s FutureMAP program, companes lke Mcrosoft, El Llly and Hewlett- Packard use nformaton markets as a way of elctng nformaton from workers and managers n order to gude decsonmakng. 1 Even very early studes of market effcency compared market performance to the performance of experts, and most found that the market dd at least as well Fglewsk 1979, Snyder 1978). There are three ways n whch a market could be sad to predct: The market can balance the dfferent belefs of traders. On ths account the market could be more accurate than any sngle trader s nformaton. Ths seems to be the dea behnd arguments for the wsdom of crowds. The market can select belefs; that s, markets favor traders wth more accurate nformaton, and as these traders grow n wealth, market prces come to reflect ther vews. Ths s an old Chcago School argument often attrbuted ncorrectly) to Mlton Fredman. Its mplcatons for asset markets were drawn out by Fama 1965) and Cootner 1964). Fnally, the market can exchange nformaton among traders; that s traders can learn what others know from market prces. Ths s the dea behnd ratonal expectatons and the lterature on learnng from prces. The lterature on nformatonal exchange n markets s huge. Market balancng and market selecton, on the other hand, are much less studed. Here we wll buld some smple dynamc equlbrum models to nvestgate the role of balancng and selecton n the long run behavor of asset prces n markets wth heterogeneous belefs. We buld upon the market selecton results of Blume and Easley 1992), Sandron 2000) and Blume and Easley 2006). Along the way we wll extend the analyss of these papers.

5 2 In partcular, we wll show that the necessary condtons for traders long-run survval developed n these three papers are not suffcent. We study complete markets. The assets we prce are Arrow securtes. More complex assets can be prced by arbtrage from these assets. We do not allow traders to learn. Blume and Easley 2006) conduct a detaled examnaton of the market selecton hypothess when traders learn, and the mplcatons of that analyss for asset prces could be traced out. The nteracton of selecton and asymmetrc nformaton s addressed n Malath and Sandron 2003). Here we prefer to study the effects of balancng and selecton n solaton, wthout the nterestng but confoundng effects of nformaton sharng. 2 The Model Our model s an nfnte horzon exchange economy whch allocates a sngle commodty. Our method s to examne Pareto optmal consumpton paths and the prces whch support them. The frst welfare theorem apples to the economes we study, so every compettve path s Pareto optmal. Thus any property of all optmal paths s a property of any compettve path. In ths secton we establsh basc notaton, lst the key assumptons and characterze Pareto optmal allocatons. We assume that tme s dscrete and begns at date 0. The possble states at each date form a fnte set S = {1,..., s}, wth cardnalty s = S. The set of all nfnte sequences of states s Σ wth representatve sequence σ = σ 0,...), also called a path. σ t denotes the value of σ at date t, and σ t = σ 0,..., σ t ) denotes the partal hstory through date t of the path σ. Let H t denote the set of all partal hstores through date t, let H 0 = {σ 0 }, the set contanng the null hstory, and let H = t=0,1,... H t denote the set of all partal hstores. The set Σ together wth ts product sgma-feld s the measurable space on whch everythng wll be bult. Let p denote the true probablty measure on Σ. It s the dstrbuton on sequences consstent wth d draws from probablty dstrbuton ρ on S. The true probablty of state s s ρs). Snce the processes and belefs are d, counts wll be mportant. Let n s tσ) = {τ t : σ τ = s}.

6 3 Expectaton operators wthout subscrpts ntend the expectaton to be taken wth respect to the measure p. For any probablty measure p on Σ, p tσ) s the margnal) probablty of the partal hstory σ t = σ 0,..., σ t ). That s, p tσ) = p {σ 0 } {σ t } S S ). In the next few paragraphs we ntroduce a number of random varables of the form x t σ). All such random varables are assumed to be date-t measurable; that s, ther value depends only on the realzaton of states through date t. Formally, F t s the σ-feld of events measurable through date t, and each x t σ) s assumed to be F t -measurable. 2.1 Traders An economy contans I traders, each wth consumpton set R ++. A consumpton plan c : Σ t=0 R ++ s a sequence of R ++ -valued functons {c t σ)} t=0 n whch each c t s F t -measurable; that s, c t : H t R ++. Each trader s endowed wth a partcular consumpton plan e, called the endowment stream. Trader has a utlty functon U c) whch assgns to each consumpton plan the expected presented dscounted value of the plan s payoff stream wth respect to some belefs. Specfcally, trader has belefs about the evoluton of states, whch are represented by a probablty dstrbuton p on Σ. She n fact beleves that states are d draws from probablty dstrbuton ρ on S. She also has a payoff functon u : R ++ R on consumptons and a dscount factor β strctly between 0 and 1. The utlty of a consumpton plan s { U c) = E p βu t ct σ) )}. t=0 We wll assume throughout the followng propertes of payoff functons: A. 1. The payoff functons u are C 1, strctly concave, strctly monotonc, and satsfy an Inada condton at 0.

7 4 Each trader s endowment s a consumpton plan. We assume that endowments are strctly postve and that the aggregate endowment s unformly bounded. Let e = e ; then e t σ) = e tσ) denotes the aggregate endowment at date t on path σ. A. 2. There are numbers > F f > 0 such that for each trader, all dates t and all paths σ, f nf t,σ e t σ) sup t,σ e t σ) < F. The upper bound n partcular s mportant to the dervaton of our results. The conclusons hold when F grows slowly enough, but may fal when F grows too quckly. The followng assumpton about belefs wll be convenent, and entals no essental loss of generalty. A. 3. For each trader and s S, f ρs) > 0 then ρ s) > 0. If there s a possble state s whch trader beleves to be mpossble, the trader would trade away all clams to the commodty n any partal hstory n whch s s ever reached. Every possble state wll almost surely be realzed at some date, so the trader wll almost surely not survve. Hence there s no cost to dscardng such traders at the outset. 2.2 Pareto Optmalty Standard arguments show that n ths economy, Pareto optmal consumpton allocatons can be characterzed as maxma of weghted-average socal welfare functons. If c = c 1,..., c I ) s a Pareto optmal allocaton of resources, then there s a non-negatve vector of welfare weghts λ 1,..., λ I ) 0 such that c solves the problem max c 1,...,c I ) such that λ U c) c e 0 t, σ c tσ) 0 1)

8 5 where e t = e t. The frst order condtons for problem 1 are: For all t there s a postve F t 1 -measurable random varable η t such that λ βu t c t σ) ) ρ s) ns t σ) η t σ) = 0 2) almost surely, and s c tσ) = e t σ) 3) These equatons wll be used to characterze the long-run behavor of consumpton plans for ndvduals wth dfferent preferences, dscount factors and belefs. 2.3 Compettve Equlbrum A prce system s a prce for consumpton n each state at each date such that the value of each trader s endowment s fnte. Defnton 1. A functon p : Σ t=0 R ++ such that each p t s F t 1 -measurable s a present value prce system f, for all traders, σ t H p tσ) e tσ) <. As s usual, a compettve equlbrum s a prce system and, for each trader, a consumpton plan whch s affordable and preference maxmal on the budget set such that all the plans are mutually feasble. The exstence of compettve equlbrum prce systems and consumpton plans s straghtforward to prove. See Peleg and Yaar 1970). At each partal hstory σ t and for each state s there s an Arrow securty whch trades at partal hstory σ t and whch pays off one unt of account n partal hstory σ t, s) and zero otherwse. The prce of the state s Arrow securty n unts of consumpton at partal hstory σ t, the securty s current value prce, s the prce of consumpton at partal hstory σ t, s) n terms of consumpton at partal hstory σ t, whch s q s t σ) p t+1 σ t, s)/p t σ). Under our assumptons, every equlbrum present-value prce system wll be strctly postve because every partal hstory s beleved to have

9 6 postve probablty, and because condtonal preferences for consumpton n each possble state are non-satated), and so all current value prces are well defned. We wll be partcularly nterested n normalzed current-value prces: q s t σ) = q s t σ)/ ν qν t σ). It s not obvous what t means to prce an Arrow securty or any other asset) correctly. The lterature contans notons such as for long-lved assets): Prces should equal the present dscounted value of the dvdend stream. But n a world n whch traders dscount factors are not all dentcal, t s not ntutvely obvous what the dscount rate should be; and to say that the correct dscount rate s the market dscount rate s to beg the queston. Is the market dscount rate, after all, correct? Wth Arrow securtes, t seems that prces should be related to the lkelhood of the states. But n a market wth endowment rsk n whch atttudes to rsk are not all dentcal, rsk prema should matter too, and agan n a market n whch not all traders have the same atttude to rsk, t s not obvous what the correct rsk premum s. Only so that we can meanngfully talk about correct prces, we make the followng assumpton: A. 4. There s an e > 0 such that for all paths σ and dates t, e t σ) e. That s, there s no aggregate rsk. The only rsk n ths economy s who gets what, not how much s to be gotten. The reason for ths assumpton s the followng result: Theorem 1. Assume A If all traders have dentcal belefs ρ, then for all dates t and paths σ, q s t σ) = ρ. 2. On each path σ at each date t and for all ɛ > 0 there s a δ > 0 such that f c tσ) e < δ, then q t σ) ρ < ɛ. Ths s to say that prces reflect belefs. A consequence of the frst pont s that n a ratonal expectatons equlbrum, the Arrow securtes spot prces wll be ρ, the true probabltes of the state realzatons. Thus we now know what t means for assets to be correctly prced. The second pont asserts that when one trader s domnant n the sense that hs demand s very large relatve to that of the other traders, the equlbrum wll prmarly reflect her belefs. The proof of both ponts s elementary, n the frst case from a calculaton and n the second from a calculaton and the upper hem-contnuty of the equlbrum correspondence.

10 7 3 Selecton By selecton we mean the dea that markets dentfy those traders wth the most accurate nformaton, and the market prces come to reflect ther belefs. We llustrate ths dea wth an example. 3.1 A Leadng Example Consder an economy wth two states of the world, S = {A, B}. In a small abuse of our notaton, take the probablty of state A at any date t to be ρ. Traders have logarthmc utlty, and have dentcal dscount factors, 0 < β < 1. Trader beleves that A wll occur n any gven perod wth probablty ρ. Ths s bascally just a bg Cobb-Douglas economy, and equlbrum s easy to compute. Let w0 denote the present dscounted value of trader s endowment stream, and let wtσ) denote the amount of wealth whch s transferred to partal hstory σ t, measured n current unts. The optmal consumpton plan for trader s to spend fracton 1 β)β t ρ ) na t σ) 1 ρ ) nb t σ) of w0 on consumpton at date-event σ t. Ths can be descrbed recursvely as follows: In each perod, eat fracton 1 β of begnnng wealth, wt, and nvest the resdual, βwt, n such a manner that the fracton αt of date-t nvestment whch s allocated to the asset whch pays off n state A s ρ. Let qt A denote the prces of the securty whch pays out 1 n state A at date t and 0 otherwse; let qt B denote the correspondng prce for the other date-t Arrow securty. Gven the begnnng-of-perod wealth and the market prce, trader s end-of-perod wealth s determned only by that perod s state: w t+1a) = βρ w t q A t w t+1b) = β1 ρ )w t q B t Each unt of Arrow securty pays off 1 n ts state, and the total payoff n that state must be the total wealth nvested n that asset. Thus n equlbrum, βρ w t q A t = j βw j t,

11 8 and so the asset prces at date t are q A t = = ρ wt j wj t ρ r t q B t = 1 ρ )r t where r t s the share of date t wealth belongng to trader. That s, the prce of asset s at date t s the wealth share weghted average of belefs. So at any date, the market prces states by averagng traders belefs. Of course there s no reason for ths average to be correct snce the ntal dstrbuton of wealth was arbtrary. But the process of allocatng the assets and then payng them off reallocates wealth. The dstrbuton of wealth evolves through tme, and the lmt dstrbuton of wealth determnes prces n the long run. We can work ths out to see how the market learns. In ths model t should be clear what correct asset prcng means. If all traders had ratonal expectatons, then the prce of the A Arrow securty at any pont n the date-event tree would be ρ, and the prce of the B Arrow securty would be 1 ρ. Let 1 A s) and 1 B s) denote the ndcator functons for states A and B, respectvely. Along any path σ of states, ρ wt+1σ) ) 1A σ t+1 ) 1 ρ ) 1B σ t+1 ) = β w qt A σ) qt B σ t ) tσ t ), and so the rato of s wealth share to js evolves as follows: r t+1σ) r j t+1σ) = ρ ρ j ) 1Aσt+1) 1 ρ 1 ρ j ) 1Bσt+1) r tσ) r j t σ). Ths evoluton s more readly analyzed n ts log form: log r t+1σ) ρ ) rt+1σ) j = 1 Aσ t+1 ) log + ρ j 1 B σ t+1 ) log 1 ρ 1 ρ j ) + log r tσ) r j t σ). 4)

12 9 To understand how the market can learn, consder a Bayesan whose pror belefs about state evoluton contan I d models n her support, {ρ 1,..., ρ I }, and let rt denote the probablty she assgns to model posteror to the frst t observatons. The Bayesan rule for posteror revson s exactly that of equaton 4). The market s a Bayesan learner. The evoluton of the dstrbuton of wealth parallels the evoluton of posteror belefs. Market prces are wealth share-weghted averages of the traders models, and so the prcng functon for assets s dentcal to the rule whch assgns a predctve dstrbuton on outcomes to any pror belefs on states. In other words, the prce of asset A n ths example s the probablty the Bayesan learner would assgn to the event that the next state realzaton wll be A. In ths sense, the market s a Bayesan. We are not commttng economc anthropomorphsm; we smply note the dentty of the equaton descrbng the evoluton of posteror belefs for a hypothetcal Bayesan learner and that descrbng the evoluton of the wealth share dstrbuton. From these observaton we can draw several conclusons. If some trader holds correct belefs, then n the long run hs wealth share wll converge 1, and the market prce wll converge to ρ. The assets are prced correctly n the long run. Second, f no model s correct, the posteror probablty of any model whose Kullback-Lebler dstance from the true dstrbuton s not mnmal converges a.s. to 0. In ths example, selecton cannot make the market do better than the best-nformed trader. In partcular, f there s a unque trader whose belefs ρ are closest to the truth, but are not correct, then prces converge n the long run to ρ almost surely, and so assets are msprced. 3.2 Selecton n Complete IID Markets Traders are characterzed by three objects: A payoff functon u, a dscount factor β and a belef ρ. We wll see that payoff functons are rrelevant to survval. Only belefs and dscount factors matter. We would expect that dscount factors matter n a straghtforward way: Hgher dscount factors reflect a greater wllngness to trade present for future consumpton, and so they should favor survval. Smlarly, traders wll be wllng to trade consumpton on unlkely paths for consumpton on those they thnk more lkely. Those traders who allocate the most to the hghest-probablty paths have a survval advantage. Ths advantage, as we wll see, can be measured by the Kullback-Lebler dstance of belefs from the truth, the relatve entropy of ρ wth respect

13 10 to ρ : I ρ ρ ) = ρ s log ρ s ρ s s The Kullback-Lebler dstance s not a true metrc. But t s non-negatve, and 0 ff ρ = ρ. 2 Assumpton A.3. ensures that I ρ ρ ) <. Our results wll demonstrate several varetes of asymptotc experence for traders n d economes. Traders can vansh, they can survve, and the survvors can be dvded nto those who are neglgble and those who are not. Defntons are as follows: Defnton 2. Trader vanshes on path σ f lm t c tσ) = 0. She survves on path σ f lm sup t c tσ) > 0. A survvor s neglgble on path σ f for all 0 < r < 1, lm T 1/T ) {t T : c tσ) > re t σ)} = 0. Otherwse she s non-neglgble. In the long run, traders can ether vansh or not, n whch case they survve. There are two dstnct modes of survval. A neglgble trader s someone who consumes a gven postve share of resources nfntely often, but so nfrequently that the long-run fracton of tme n whch ths happens s 0. The defntons of vanshng, survvng and beng neglgble are remnscent of transence, recurrence and null-recurrence n the theory of Markov chans. 3.3 The Basc Equatons Our method uses the frst order condtons to solve for the optmal consumpton of each trader n terms of the consumpton of some partcular trader, say trader 1. We then use the feasblty constrant to solve for trader 1 s consumpton. The fact that we can do ths only mplctly s not too much of a bother. Let κ = λ 1 /λ. From equaton 2) we have that u c t σ) ) ) t c 1 t σ) ) = κ β1 ρ 1 s u 1 β s S ρ s ) n s t σ) Note that f two traders and j have dentcal belefs and dscount factors, then condtonal on the total amount the two traders consume, the dvson between the two 5)

14 11 traders s non-stochastc. If the total amount they consume s c j t σ) on path σ at date t, then c t, c j t) solves the problem u j c j t σ) ) = max c,c j ) such that λ u c ) + λ j u j c j ) λ + λ j c + c j c j t σ t ) 0 t, σ, and c, c j 0 because the common dscount factors and belefs cancel out. Thus f n problem 1) traders and j wth weghts λ and λ j are replaced by the collectve trader wth weght λ + λ j and payoff functon u j, a consumpton allocaton solves the dsaggregated problem only f the consumpton allocaton formed by summng s and j s consumpton and leavng everythng else unchanged solves the aggregated problem. Conversely, f an allocaton solves the aggregated problem, then the consumpton allocaton constructng by dsaggregatng the j-trader s consumpton at each date-event par by solvng problem 6), and leavng all other consumpton unchanged, solves the dsaggregated problem. In summary, there s no loss of generalty n assumng that the traders are representatve traders, each representng a class of traders wth dentcal dscount factors and belefs but perhaps dfferent payoff functons. In patcular, wthout loss of generalty we can assume that each trader has a unque dscount factor-belef par β, ρ ). It wll sometmes be convenent to have equaton 5) n ts log form: log u c t σ) ) u 1 c 1 t σ) ) = log κ + t log β 1 ) n s β tσ) log ρ s log ρ1 s. ρ s s ρ s We can decompose the evoluton of the rato of margnal utltes nto two peces: The mean drecton of moton, and a mean-0 stochastc component. log u c t σ) ) u 1 c 1 t σ) ) = log κ + t log β 1 t ) ρ s log ρ s log ρ1 s + β ρ s s ρ s ) ) n s t σ) tρ s log ρ s log ρ1 s ρ s ρ s s = log κ + t log β 1 I ρ ρ 1 ) ) t log β I ρ ρ ) ) ) ) n s t σ) tρ s log ρ s log ρ1 s ρ s ρ s s 6)

15 12 The mean term n the precedng equaton gves a frst order characterzaton of traders long fun fates. Defnton 3. Trader s survval ndex s s = log β I ρ ρ ). Then log u u 1 c t σ) ) c 1 t σ) ) = log κ + ts 1 s ) s n s t σ) tρ s ) log ρ s log ρ 1 s ) 7) 3.4 Who Survves? Necessty Necessary condtons for survval have been studed before, notably by Blume and Easley 2006) and Sandron 2000). In ths economy, a suffcent condton guaranteeng that trader vanshes s that trader s survval ndex s not maxmal among the survval ndex of all traders. Consequently, a necessary condton for survval s that the survval ndex be maxmal. Theorem 2. Assume A.1 3. If s < max j s j, then trader vanshes. The analyss compares one trader, say trader 1, to other traders n the economy. We use equaton 7) to show that f trader has a larger survval ndex, than trader 1, trader 1 must vansh. The frst step s to relate long-run survval outcomes to the ratos of traders margnal utltes, the lhs of 7). Lemma 1. If on a sample path σ, log u c t σ) ) /u 1 c 1 t σ) ) for some trader, then lm t c 1 t σ) = 0. On the other hand, f lm sup mn log u c t σ) ) /u 1 c 1 t σ) ) >, then lm sup t c 1 t σ) > 0. Proof: Suppose frst that the lmt of the log of the rato of margnal utltes converges to along a path σ. Ths can happen n one of two ways: Ether the denomnator converges to 0 or the numerator dverges to nfnty. It must be the latter, because the denomnator s bounded below by u F ) > 0. Consequently, on any such path, c tσ) 0.

16 13 In every perod t, there s a trader t) who consumes at least c t) σ) f/i. If trader 1 were to vansh, then lm t log u t) t) c t σ) ) /u 1 c 1 t σ) ) converges to snce the number of traders s fnte). But f the lmsup condton s satsfed, then there s an ɛ such that log u t) t) c t σ) ) /u 1 c 1 t σ) ) > ɛ nfntely often. Proof of Theorem 2. We prove ths theorem by examnng equaton 7). Take tme averages of both sdes and observe that for each s, t 1 n s t ρ s converges p-almost surely to 0, to conclude that for almost all paths σ t, 1 lm t t log u c t σ) ) c 1 t σ) ) = s 1 s. u 1 If s 1 s not maxmal, there s an for whch s 1 s < ɛ < 0. For almost all paths σ t there s a T such that f t > T, then u c t σ) ) /u 1 c 1 t σ) ) < ɛt. Accordng to lemma 1, trader 1 vanshes. Ths result s a consequence of the SLLN. Blume and Easley 2006) extend ths result to dentfy necessary condtons for survval n many dfferent, non-iid settngs. Lemma 1 also gves a lower bound rate at whch traders vansh. Let s be the maxmal survval ndex n the trader populaton. Corollary 1. If s 1 s not maxmal, then u 1 c 1 t σ t ) ) exp s 1 s )t > 0 a.s. Proof. Replace u c t σ t ) ) wth the smaller u F ) and calculate. If trader 1 s of the CRRA class, for nstance, t follows that her consumpton declnes at an exponental rate, whch depends upon the coeffcent of relatve rsk averson. Ths shows that although whether a trader vanshes or not s ndependent of the payoff functon, the rate at whch she vanshes s payoff functon-dependent. More rsk-averse traders vansh at a slower rate, and n fact, n the rate there s a trade-off between the survval ndex and the coeffcent of relatve rsk averson γ, snce c t s Oexp ts 1 s )/γ).

17 Market Equlbrum Selecton The mplcatons for long-run asset prcng are already llustrated n the example whch began ths secton. Corollary 2. If there s a unque trader wth mnmal survval ndex among the trader populaton, then market prces converge to ρ almost surely. The lmt market prce ρ s not necessarly the prce representng the most accurate belefs n the market due to the tradeoff n the survval ndex between belef accuracy and patence. The lmt prce wll be the best belefs n the market f all traders have dentcal dscount factors. Proof. Ths Corollary s an mmedate consequence of Theorems 1 and 2. If only trader has maxmal survval ndex, then almost surely all other traders vansh and q t converges to ρ. Usng the tools of Blume and Easley 2006) we can extend ths result n varous ways. For nstance, we can provde a survval ndex analyss of fnte state) Markov economes wth traders who hold Markov models of the economy or even traders who hold msspecfed) d models. If all traders are Bayesan learners satsfyng certan regularty condtons, and the truth s n the support of ther belefs, then all wll eventually learn the true state dstrbuton and so prces wll ultmately be correct. But those traders wth low-dmensonal belef supports wll learn faster than those wth hgher-dmensonal belef supports, and prces wll converge to the true prces at the faster rate. 4 Balancng When a sngle trader type) has the hghest survval ndex, market prces converge to hs vew of the world. There s no room for balancng of dfferent belefs because, n the long run, there s only one belef and dscount factor present n the market. But f the

18 15 market process s more complcated than the world vew of any sngle trader so that no trader has correct belefs, or f traders are asymmetrcally nformed, t s possble that multple traders could have maxmal survval ndex. Wll all such traders survve, and what are the mplcatons for suffcency? 4.1 Who Survves? Suffcency Theorem 2 shows that traders wth survval ndces that are less than maxmal n the populaton vansh. Ths does not mply that all those wth maxmal survval ndces survve. The rhs of equaton 7) s a random walk, and the analyss of the prevous secton s based on an analyss of the mean drft of the rhs of equaton 7). Theorem 2 shows that a non-zero drft has mplcatons for the survval of some trader. When two traders wth maxmal survval ndces are compared, the drft of the walk s 0, and further analyss of equatons 5) and 7) s requred. Snce the long run behavor of a random walk depends upon the dmenson of the space beng walked through, our results wll depend upon the number of states s. More defntons are requred. For a probablty dstrbuton θ on S, defne the vector of log-probabltes: loθ) = logθs)/θs) ) ) s 1 s=1. Let Sur denote the set of traders wth maxmal survval ndex. Theorem 1 ndcates that these are the only potental survvors. The fate of a trader n Sur s determned by how her belefs, as represented by loρ ), are postoned relatve to the belefs of the other traders n Sur. Denote by C{loρ j )} j Sur the closed convex cone generated by the log-probablty vectors of the traders. Defnton 4. Trader s nteror f loρ ) s n the relatve nteror of C{loρ j )} j Sur. She s extremal f loρ ) s an extreme pont, that s, not a non-negatve lnear combnaton of the other loρ j ). We are nterested n markets wth heterogeneous potental survvors. Theorem 3. Assume A.1 3, and suppose s 3 and 0 < r < If Sur, then trader survves. 2. Extremal traders are non-neglgble and nteror traders are neglgble

19 16 3. If trader s extremal, then lm T 1/T ) {t T : c t > re t } > 0 a.s. When s 3, a maxmal survval ndex s suffcent as well as necessary) for survval. But how one survves depends upon ones poston n the group of survvors. Interor survvors are neglgble. The fracton of tme they consume a postve share of aggregate endowment s 0. Extremal traders, on the other hand, have hghly volatle consumpton. The fracton of tme each consumes an arbtrarly small share of aggregate endowment n postve, as s the fracton of tme each consumes nearly all of the aggregate endowment. When s > 3, the pcture s even more stark. Interor traders vansh. Maxmalty of a trader s survval ndex s no longer a suffcent condton for survval. Theorem 4. Assume A.1 3, and suppose s > 3 and 0 < r < Interor traders vansh. 2. Extremal traders survve and are non-neglgable. 3. If trader s extremal, then lm T 1/T ) {t T : c t > re t } > 0 a.s. The dscusson of survval possbltes s complcated by the possblty of heterogeneous dscount factors. If dscount factors are homogeneous, all traders wth maxmal survval ndex survve. Corollary 3. If for all and j, β = β j then survves ff Sur. Proofs of Theorems 3 4 and Corollary 3. Wthout loss of generalty we can take the welfare weghts to be 1 by multplyng each payoff functon by an approprate postve weght). We already know that f a trader s not n Sur, she vanshes, so suppose that trader 1 s n Sur, and consder trader 1 who s also n Sur. Nothng addtonal s learned by studyng traders who are not n Sur). Snce 1 and are both n Sur, they have dentcal survval ndces, and so the rght hand sde of equaton 7) becomes ẑ tσ) = s n s t σ) tp s ) log ρ 1 s ρ s

20 17 We want to nvestgate f, nfntely often, for all such traders 1 n Sur, ẑ s arbtrarly large, arbtrarly small, or both. To do ths we study the evoluton of the system zt) Sur, 1 = A w 1t. w s 1,t where a s = log ρ 1 s/ρ 1 s log ρ s/ρ s for s = 1,..., s 1 and Sur, 1, and w t s the mean-0 random walk such that w st σ) = n s tσ) tρ s for s = 1,..., s 1. We can neglect the traders not n The random walk n s tσ) tρ s ) s s=1 s an s 1 dmensonal random walk snce the sum of all elements of the vector s 0, so we have chosen a representaton whch drops the s coordnate. Thus the vector z t σ) = ) z t σ) Sur, 1 s a random walk n R #Sur 1. For ease of reference, defne loρ ) = log ρ s/ρ s) s 1 s=1. We consder outcomes for trader 1 correspondng to two types of A matrx. 1) There s a vector drecton) x such that Ax 0. 2) for each vector x 0 there s a row vector a of A such that a x < 0. The remanng possblty s that Ax 0, but s not strctly greater than 0. As we wll see, cases 1) and 2) correspond to extremal and nteror traders. The remanng possblty s that a trader s belefs are on the boundary of the polytope C{loρ j )} j Sur but are not extreme ponts. That s, they le n the relatve nteror of some facet of the polytope. We call such traders boundary traders. We have no results for them, for reasons that wll be dscussed below. If case 1) holds, then there s an open cone C such that for all x C, Ax 0. Whenever w t C, log u c t σ t ) ) /u 1 c 1 t σ t ) ) s postve for all Sur, 1. Furthermore, for all consumpton shares r < 1 there s a bound br) such that f, for all Sur, 1, a w t σ) > br), then c 1 t σ) > re t σ). The set of such w values s the open cone less a compact set contanng the orgn. Such sets are recurrent, so Pr{c 1 t σ) > re t σ).o.} = 1. If case 2) holds, then for any drecton of the walk, there s a trader Sur, 1 such that log u c t)/u 1c 1 t ) s arbtrarly negatve when the walk s far enough out n that drecton. Consumpton for trader 1 s bounded away from 0 only on compact sets contanng the orgn. Such sets are null-recurrent for two dmensonal walks, and transent for walks n dmenson three or hgher. So when s 3 so that the dmenson of the walk does not exceed 2), trader 1 s neglgble; otherwse trader 1 vanshes.

21 18 Cases 1 and 2, respectvely, correspond to extremal and nteror traders. The next lemma establshes ths, whch completes the proof. Lemma 2. The nequalty system Ax 0 has a soluton ff loρ 1 ) s an extreme pont of the convex hull of the lo ρ ) Sur, that s, ff trader 1 s extremal. If trader 1 s nteror, then for all drectons x there s an a such that a x < 0. Proof of Lemma 2. A theorem of the alternatve see the Appendx) states that there s an x such that Ax 0 ff no non-trval, non-negatve lnear combnaton of the rows s 0. That s, there s no non-trval and non-negatve set of weghts {λ } 2 such that 2 λ loρ ) = loρ 1 ). In other words, a strctly postve drecton x exsts ff loρ 1 ) s extremal n C{loρ j )} I j=1. A Theorem of the Alternatve Gale 1960) also shows that ether Ax 0 and not equal to 0 n every component) has a soluton, or the a are lnearly dependent wth strctly postve weghts. Thus f trader 1 s nteror, there s no non-negatve drecton. Ths proves both theorems. To prove the corollary, suppose that dscount factors are dentcal. Then s = I ρ ρ ) and for some s and Sur, I ρ ρ ) = s. Ths functon s strctlyconcave n ρ, so the set of all µ such that I ρ ν) s s strctly convex, and all traders wth maxmal survval ndex are extreme ponts of ths convex set. The followng fgures demonstrate the geometry of Theorem 3 when s = 3. The left llustraton n Fgure 1 plots the log-odds ratos of fve survvng belefs. The dscount factors for all traders cannot be all the same. The log-odds vector of the true dstrbuton can be anywhere n the plane, but t s most entertanng to thnk of t as beng nsde the trangle, and perhaps even concdent wth E. Fgure 1 plots the logodds ratos of fve belefs n Sur when s = 3. The dscount factors for all traders cannot be all the same.) The upper left fgure dsplays the polytope C{loρ j )} j Sur. Traders A, B and C are extremal, E s nteror and D s boundary. The fgure ndcates there are many drectons of ncrease for extremal trader A, only 1 drecton for boundary trader D, and no drectons of ncrease for nteror trader E.

22 19 B A A D A C E A E C D A B D C E D E C D E D A E A D B E B Fgure 1: Fve Belefs n Sur. The three remanng fgures ndcate the drectons of ncrease. The cone on the upper rght ndcates the drectons n whch the random walk can move so as to ncrease the wealth share of extremal trader A relatve to all other traders. Far enough out n the cone, trader A consumes an arbtrarly large share of aggregate endowment. On the bottom left, any drecton n the shaded half-space mproves D relatve to B and E, but these drectons mostly mprove ether A or C relatve to B and E as well, and n fact mprove one or the other of them more, so far enough out ether A or C s consumng most of the endowment. When the random walk crosses the lne orthogonal to the boundary of the half space, A, C and D have equal margnal utltes, and far out on ths lne thus mnmzng the consumpton of B and E) s the best D can do. Thus D s share s bounded from above. Whether D vanshes or not depends upon the recurrence propertes of lower dmensonal cones, about whch we have lttle to say at ths level of generalty. Fnally on the rght, any drecton s better for some trader relatve to E. The best E does s when the walk s near the orgn. But when the walk s n a neghborhood of the orgn, E s share s stll bounded, and when s > 3 such neghborhoods are n any event transent.

23 Market Equlbrum Balancng The mplcaton for market equlbrum from the exstence of multple survvers s perhaps surprsng: Corollary 4. If multple traders have maxmmal survval ndex, then for all extremal traders and all ɛ > 0, q t ρ < ɛ nfntely often. If s > 3 t s possble that for ɛ > 0 suffcently small, the event q t ρ s transent, even f some survvor has ratonal expectatons. Wth multple survvors, asset prces are volatle. Furthermore, asset prces need be approxmately rght; specfcally, approxmately rght prces may be transent. One mght hope that, nonetheless, the tme average of prces s approxmately correct. We beleve that ths weaker noton of correct asset prcng may fal, and we hope to have a proof shortly. Fgure 4.2 llustrates some of the possbltes for prces wth multple survors n the leadng example of the prevous secton wth log utlty. In ths fgure the true dstrbuton s ρ. The closed curve connectng ponts P, Q and R s a curve of constant relatve entropy, n ths case Suppose all traders have dentcal factors, and all have belefs whch are on or outsde the curve. Those traders wth belefs outsde the curve wll vansh. Suppose now that Sur contans three traders wth belefs P, Q and R. All three wll survve. The equlbrum prce wll wander around nsde the convex hull of these three ponts. As the three ponts are drawn, ρ s n ther convex hull, and t s at least possble that the average behavor of prces over tme could be approxmately correct. On the other hand, suppose Q and R were hgher up on the so-relatve entropy curve, nearer to P. It s possble to arrange them so that ρ s no longer n the convex hull, and so the long-run tme average of prces would be nowhere near ρ. Fnally, consder movng pont Q off the curve. If t moves n, ths trader s the unque survver, and selecton dctates that prces converge to Q. On the other hand, f Q moves out, ths trader s no longer a survvor. The two survvors are P and R, and n the long run prces wll move up and down on the lne segment connectng these two ponts. Agan there s no connecton between the long-run behavor of prces and ρ.

24 P Ρ R Q Fgure 2: Multple survvors, s = 3. 5 Concluson Ths analyss suggests that, contrary to Henry Herrman s vew n the epgram whch begns ths paper, the market knows not better but only as well as the best) analysts do. The market can be no better nformed than the most ft trader accordng to the ftness ndex metrc, and f there are several most-ft traders wth dstnct belefs, then the market belefs as expressed n lmt equlbrum prces may fal to converge. The necessty of a maxmal survval ndex for long-run selecton has frequently been notced, ncludng Sandron 2000) and Blume and Easley 2006). The observaton that t s not suffcent, and the suffcent condton derved here for the d economy, are both new. It s not surprsng that only belefs matter for the suffcent condton.

25 22 Snce dscount factors are non-stochastc, they are part of the mean term whch gves the necessary condton. Were dscount factors themselves stochastc, the analyss of necessary condtons for survval would reman essentally unchanged, but devatons from the mean log-dscount factor would appear n the suffcent condton. The utlty of the survval ndex s that t separates the stochastc evoluton of margnal utlty ratos nto a determnstc drft and and a mean-0 stochastc component. One contrbuton of ths paper s to show that the stochastc component matters for survval and long run asset prces. Ths analyss extends to Markov envronments, where a survval ndex, the same knd of decomposton, also exsts. In more complcated envronments, such as that created by Bayesan traders, we already know that the smple decomposton nto a determnstc drft and a stochastc resdue n nsuffcent. We showed n Blume and Easley 2006) that when dscount factors among Bayesan traders are dentcal, all survval ndces converge to the log of the common dscount factor, and that a necessary condton for survval s that the rate of convergence be fastest. The rate turns out to be related to the dmenson of the support of pror belefs; more parameters to learn slow down learnng. We have not yet gone beyond ths. Among traders who learn the fastest, whose parameter spaces are of mnmal dmenson, there are stll dfferences due to dfferent pror belefs whch appear n a stochastc term whch s convergng to 0. If the convergence s slow enough, there s stll room for the knd of effects we descrbe here to work, but we have not even attempted ths analyss. In stll more complex envronments, the analog of the survval ndex s a seres of condtonal means whch are compared by dvergence rates. We can construct example envroments where exactly the effects we descrbe here work, and so the analyss of Blume and Easley 2006) s agan) necessary but not suffcent. These examples, however, are very artfcal, and ths level of analyss s probably too abstract to say anythng useful. Perhaps an even more compellng ssue s an asymptotc analyss of wealth shares and prces when markets are ncomplete. Blume and Easley 2006) have some smple examples of how ncomplete markets can select for the wrong trader, whch makes the market, n the lmt, less smart than ts smartest trader. In the most compellng example, an excessvely optmstc trader oversaves, and thus comes to domnate n the lmt. Becker et. al. 2006) analyse a market n whch the only assets are money and one rsky asset, so that wth enough states) the market s ncomplete. They too fnd that long-run prce volatlty wth multple survvors. In partcular, f two or more traders

26 23 survve n the long run, then the each trader consumes arbtrarly lttle nfntely often. In studyng predcton markets lke those contracts traded on Iowa Electronc Market whch make book on poltcal races, t s mportant to take account of learnng through prces, and to entertan the possblty that the accurate performance of these markets s due at least as much to trader learnng from prces as opposed to more outsde nformaton) as t s to market selecton. In our vew, ths s less mportant when t comes to large markets for securtes and other fnancal assets. Ths s not to say that learnng s not mportant; surely t s. But these markets are suffcently complcated, and tradng occurs for so many dverse motves, that the possblty of consstent learnng rules seems to us remote. Ths leaves room for the market to be smarter n the long run than ts traders; and so we are led to ask, how s the market s learnng experence dfferent than that of ts traders? The leadng example of secton 3.1 s a frst step towards answerng ths queston. Appendx: Lnear Algebra Let A be an n m matrx. Theorem 1. One and only one of the followng equaton systems has a soluton: Ax 0 8) ya = 0, y 0, y 0 9) Ths Theorem s an mmedate consequence of the followng theorem, due to Fan, Glcksberg, and Hoffman 1957), concernng m convex functons, each mappng the non-empty convex set K to R. Theorem 2 Fan et. al.). One and only one of the followng alternatves holds: 1. The system of nequaltes f x) < 0, = 1,..., m, x K has a soluton; 2. There are non-negatve scalars λ, not all 0, such that λ f x) 0 for all x K.

27 24 Proof of Theorem 1. Take f x) = a x, where a s the th row of the matrx A. The f are convex functons and W s a convex set. If 8) has no soluton, then accordng to Fan et. al., there are non-negatve scalars y not all 0 such that for all x W, y a x) 0. 10) In partcular, y a x = 0 for all x, because f not t wll be possble to make ths term arbtrarly negatve by sutable choce of x, and so the nequalty wll be volated for some x. Ths wll be true f and only f y a = 0.

28 25 Notes 1 The End of Management, Tme Magazne Bonus Secton, August 2004, 1,00.html. For a popular account of predcton markets, see Suroweck 2003). 2 In fact, t s jontly convex n ρ, ρ ), but we wll not need to make use of ths fact.

29 26 References Blume, L., and D. Easley 1992): Evoluton and Market Behavor, Journal of Economc Theory, 581), Blume, L. E., and D. Easley 2006): If You re so Smart, why Aren t You Rch? Belef Selecton n Complete and Incomplete Markets, Econometrca, 744), Cootner, P. 1964): The Random Character of Stock Market Prces. MIT Press, Cambrdge, MA. Fama, E. 1965): The Behavor of Stock Market Prces, Journal of Busness, 381), Fan, K., I. Glcksberg, and A. J. Hoffman 1957): Systems of Inequaltes Involvng Convex Functons, Proceedngs of the Amercan Mathematcal Socety, 83), Fglewsk, S. 1979): Subjectve Informaton and market Effcency n a Bettng Model, Journal of Poltcal Economy, 871), Gale, D. 1960): Lnear Economc Models. McGraw-Hll, New York. Malath, G., and A. Sandron 2003): Market Selecton and Asymmetrc Informaton, Revew of Economc Studes, 702), Peleg, B., and M. E. Yaar 1970): Markets wth Countably Many Commodtes, Internatonal Economc Revew, 113), Sandron, A. 2000): Do Markets Favor Agents Able to Make Accurate Predctons?, Econometrca, 686), Snyder, W. W. 1978): Horse Racng: Testng the Effcent Markets Model, Journal of Fnance, 334), Suroweck, J. 2003): Decsons, Decsons, The New Yorker, March 24, p. 29.