The Stock Market Game and the Kelly-Nash Equilibrium

The Stock Market Game and the Kelly-Nash Equlbrum Carlos Alós-Ferrer, Ana B. Ana Department of Economcs, Unversty of Venna. Hohenstaufengasse 9, A-1010 Venna, Austra. July 2003 Abstract We formulate the stock market of Blume and Easley 1992) and Hens and Schenk- Hoppé 2001) as a non-cooperatve game. We fnd that, regardless of the asset market structure e.g. ncomplete markets are allowed), there exsts a unque pure-strategy Nash equlbrum, whch s strct and symmetrc. Ths equlbrum s also evolutonarly stable n the sense of Schaffer 1988). In ths equlbrum, traders allocate ther wealth proportonally to the expected returns of the assets. Ths rule concdes wth the Kelly Crteron bettng your belefs ) n the partcular case of Arrow securtes. Allowng for rsk averson, we stll fnd that the same profle s a Nash equlbrum, provded there s no aggregate rsk n the asset market. Under rsk averson e.g. logarthmc utlty) and aggregate rsk, the symmetrc Nash equlbrum of the game dffers. Keywords: stock market game, Kelly crteron, ncomplete markets, evolutonary stablty. JEL classfcaton: C72, G11, D83. 1 Introducton Consder a smple world wth a fnte number of states and an equal number of assocated assets Arrow securtes), where each asset delvers one monetary unt f and only f the correspondng state happens. In ths smple world, suppose states follow an..d. process. In an equvalent context, t was shown n Breman 1961) that dvdng wealth among assets proportonally to the probablty of ther correspondng states maxmzes the growth rate of wealth. Such a dvson can be generated by maxmzng the expected logarthm of relatve returns, a procedure known as the Kelly crteron see Kelly 1956)). More recently, Blume and Easley 1992) found that n a framework wth dagonal securtes), f some nvestors were to use the rule above n a closed equlbrum model, they would asymptotcally come to domnate the market, where domnance s to be expressed n terms of ther share of the aggregate wealth. In the framework of Bayesan learnng, where agents are tryng to learn the true probabltes of the states, Blume and Easley 1992) descrbe ther behavor as bettng your belefs. An specally attractve feature of Blume and Easley 1992) s that t actually models a market, where the prces of the assets are determned by supply and demand through a We are ndebted to Larry Blume, Davd Easley, Pero Gottard, Thorsten Hens, Manfred Nermuth, and two anonymous referees for valuable suggestons whch greatly mproved the paper. All remanng errors are ours. We gratefully acknowledge fnancal support from the Austran Scence Fund FWF) under Project P15281, and fromthe Austran Exchange Servce ÖAD) and the Spansh Mnstry of Educaton and Culture under the Span-Austra Accones Integradas respectve projects 18/2003 and HU02-4. 1

market clearng condton. However, Blume and Easley 1992) s a model of compettve equlbrum, n the sense that agents gnore all nformaton not contaned n the prces, neglectng any strategc consderaton as they would arse n a game. Moreover, Blume and Easley 1992), as Breman 1961), focus on asymptotc optmalty propertes. Ths model can be generalzed beyond the Arrow securtes case. Hens and Schenk- Hoppé 2001) consder a generalzaton of the model n Blume and Easley 1992), allowng for ncomplete markets, wth a general not necessarly fnte) state space, and sngle out a generalzaton of the Kelly crteron as an attractor of a random dynamcal system of evolutonary type. The generalzaton of the Kelly crteron dentfed by Hens and Schenk- Hoppé 2001) prescrbes that traders should allocate ther wealth proportonally to the expected returns of the assets. Bell and Cover 1980, 1988) note that, n addton to these asymptotc, long-run propertes, nvestors should be nterested n good short term compettve performance. They postulate a one-shot, two-player, constant-sum game where the goal of a player s to obtan more captal than the opponent payoffs beng, e. g., the probablty that the own captal wll exceed the opponent s). Ths approach presents obvous problems from an economc pont of vew. Apart from the lmtaton to the two player framework, the returns process s completely exogenous n that no market clearng s taken nto account, and, moreover, payoffs are not monetary, but qualtatve outperform the opponent). However, the quoted papers show that the Nash equlbrum of the two-player game so defned nvolves, frst, a randomzaton of captal, and second, an nvestment accordng to the Kelly crteron. Our am here s to model a general stock market not restrcted to the Arrow securtes case) as a game, to allow for strategc behavor, where payoffs are the monetary ones and any fnte) number of nvestors s allowed. We wll then analyze ths game to establsh short-term optmalty propertes, as opposed to long-run asymptotc) ones. Specfcally, we wll show that the generalzaton of the) Kelly crteron s both a strct) Nash equlbrum and an Evolutonarly Stable Strategy n the approprate sense). The motvaton s as follows. Suppose the Kelly crteron or ts generalzaton for arbtrary asset markets) s Evolutonarly Stable n a well-defned sense yet to be clarfed). Then, we can ntutvely expect t to be a long-run outcome of a dynamcal system of evolutonary type, where nvestors are assumed to be boundedly ratonal. Suppose, further, that the Kelly crteron also nduces a Nash equlbrum once the stock market s modelled as a game. Then, also strategc ratonal nvestors may adopt and contnue to use t, because they wll have no ncentves to try other nvestment crtera once the populaton has adopted t. Both ratonal and boundedly ratonal nvestors wll, hence, behave as f they were nterested n wealth accumulaton snce the Kelly crteron s a maxmzer n that respect accordng to Breman 1961)). 2 The Stock Market Game 2.1 The Asset Market There are S 2 possble states of the world, s = 1,..., S. State s occurs wth probablty q s > 0, wth S s=1 q s = 1. We consder a general asset market wth K 2 assets, k = 1,..., K. In state s asset k yelds postve payoff A k s) 0. We assume that total payoff of all assets s strctly postve n each state,. e., K A k s) > 0 k=1 2

and that each asset has a strctly postve payoff for some state,. e., for each asset k, there exsts at least some state s such that A k s) > 0. We explctly allow for ncomplete asset markets e. g. f S > K). There are n nvestors players), = 1,..., n. Each nvestor owns an ntal wealth r0 > 0, and we normalze total wealth so that n =1 r 0 = 1. Let K = {z = z 1,..., z k ) R K K + k=1 z k = 1} denote the K 1)-dmensonal smplex. Investor has to choose a vector α = α1,..., αk ) K, where αk denotes the fracton of s wealth allocated to asset k. Gven a profle α = α 1,..., α n ), we denote α k = αk 1,..., αn k ) for convenence. 2.2 Market Clearng and Prces The Stock Market Game ntroduced n ths secton has a structure smlar to that of a Strategc Market Game as n Shapley and Shubk 1977). There s one unt of each asset k avalable. Gven a strategy profle α, the market-clearng prces are determned as p k α k ) = αkr 0 1) provded at least some αk > 0. Let x k α k) denote the number of unts of asset k purchased by nvestor. If nvestor offers a proporton of her wealth αk > 0, she receves a number of unts of asset k whch s computed as the rato of the amount of wealth offered to the prce of that asset. If αk = 0, nvestor does not purchase asset k. I. e., { α x k r 0 kα k ) = p k α k ) f αk > 0 0 f αk = 0 2) Provded at least some αk > 0, Equatons 1) and 2) mply x kα k ) = 1 3) =1 Formally, the market prce for any asset k remans undetermned when there s no trade of that asset, that s when αk = 0 for all. In that case, though, the prce s not relevant for the assgnment of assets to nvestors, snce an nvestor who does not nvest n asset k receves 0 unts of that asset. For convenence, we can set p k α k ) = 0 f asset k s not traded ths s nconsequental for the analyss). Note that p 1 α 1 ),..., p K α K )) K, snce p k α k ) 0 and K K p k α k ) = αkr 0 = k=1 k=1 =1 =1 r0 =1 k=1 K αk = r0 = 1 4) Note also that, whenever all nvestors allocate the same numercal share z > 0 to a gven asset, ts market prce wll be numercally equal to that share: α k = z,..., z), z > 0 = p k α k ) = zr0 = z r0 = z 5) Moreover, the number of unts of asset k acqured by each nvestor n that case s numercally equal to her ntal wealth. To see ths, we use 5): =1 α k = z,..., z), z > 0 = x kα k ) = zr 0 p k α k ) = zr 0 z = r 0 6) =1 =1 3

The x k are dscontnuous functons at the ponts where α k = 0 for all note that x k z,..., z) = r 0 for all z > 0, but x k 0,..., 0) = 0). Indeed, f no player s nvestng n a gven asset k, a sngle player devatng to nvest ε > 0, no matter how small, on that asset wll be able to acqure ts entre supply: αk = ε > 0, α j k = 0 j = x kα k ) = εr 0 p k α k ) = εr 0 εr0 = 1 7) On the other hand, at all profles such that α j k > 0 for some j n partcular, n the nteror of the own strategy space K ), the functon x k s twce dfferentable. We compute, for reference, x k α k) αk = r 0p k α k ) αk r 0) 2 [p k α k )] 2 = r 0 p k α k ) k α k ) ) 8) and 2 x k α k) r [ 0 x αk = k α k ) )2 [p k α k )] 2 αk p k α k ) + kα k ) ) ] r0 = ) r 2 = 2 0 p k α k ) kα k ) ) 9) Provded some α j k > 0, j, the rght hand sde of Equaton 8) s strctly postve and that of Equaton 9) s strctly negatve and hence x k s a strctly ncreasng and strctly concave functon of αk. That s, the number of unts acqured of a gven asset s ncreasng and concave n the share of wealth nvested n that asset holdng other traders nvestments constant). Intutvely, the number of unts that a trader receves ncreases when she allocates more wealth to t, but snce an ncrease n demand mples an ncrease n prce, the nvestor wll be able to acqure less unts per monetary unt nvested. The fact that x k s ncreasng shows that the frst, drect effect domnates the second. The latter, though, becomes stronger as the allocated share of wealth ncreases,. e. there are decreasng returns to nvestment x k s concave). 2.3 Payoffs Assume that nvestors are rsk neutral. Ths assumpton wll be relaxed n Secton 5. Gven a profle α = α 1,..., α n ), we wrte α = α, α ) followng the standard game-theoretc conventon. The expected) monetary payoff of player s then S K ) π α, α ) = q s A k s)x kα k ) 10) Defnng the expected payoff of asset k as Equaton 10) can be rewrtten as E k = s=1 k=1 S q s A k s) > 0 11) s=1 π α, α ) = K E k x kα k ) 12) k=1 The tuple K, r0, π ) n =1 defnes the Stock Market Game. 4

3 The Kelly-Nash Equlbrum We am to dentfy all Nash Equlbra n pure strateges) of the Stock Market Game. Due to the dscontnuty of the functon x k when asset k s not traded, the analyss must avod dfferental calculus at ths pont. Therefore, we show frst that at any Nash equlbrum all assets must be traded. For, f any gven asset s not traded, an nvestor devatng any fracton of her wealth to nvest n that asset wll completely determne ts prce. Snce all assets have postve expected return, the devaton wll be proftable for a small enough nvestment and hence prce). Lemma 1. If α s a Nash equlbrum, then all assets are traded and hence p k α k ) > 0 for all k = 1,..., K. Proof. Suppose αk = 0 for all = 1,..., n. Choose any player and any other ˆk such that α ˆk > 0 hence pˆkαˆk) > 0). Let 0 < ε < α ˆk. If player were to devate and nvest ε n asset k, she would receve the entre supply of asset k by 7),. e. x k 0,..., εr 0,..., 0) = 1. Defne, then α ε) K such that αk ε) = ε, αˆkε) = α ˆk ε, and αk ε) = α k for any k k, ˆk. The payoff of player when devatng to strategy α ε) s ) π α ε), α ) = α ˆk ε r E k x 0 kα k ) + E k + Eˆk pˆkαˆk) εr0 13) k k,ˆk and hence the devaton wll be proftable f and only f ) α ˆk ε r π α ε), α ) > π α, α 0 ) E k + Eˆk pˆkαˆk) εr0 α ˆkr 0 > Eˆk pˆkαˆk) 14) whch holds by contnuty for ε small enough, snce E k > 0. In order to study the set of Nash equlbra, we perform a prelmnary computaton. Consder a stuaton n whch all assets are traded, mplyng p k α k ) > 0 for all k. Suppose that a player, who s nvestng a strctly postve fracton of her wealth n asset k 2, were to transfer a fracton 0 ε < αk 2 to asset k 1. Analogously to the prevous proof, we can then defne α ε) K such that α k 1 ε) = α k 1 + ε α k 2 ε) = α k 2 ε α kε) = α k for any k k 1, k 2 15) Gven the strateges of all other players, we compute the payoffs to nvestor of ths devaton: π α ε), α ) = α E k x k α k ) + E k1 + ε ) r0 α k1 p k1 α k1 ) + εr + E ε ) r0 k k 1,k 2 0 p α ) εr0 16) Ths expresson s a dfferentable) functon of ε such that π α 0), α ) = π α, α ). Hence, whenever d dε π α ε), α ) > 0 17) ε=0 5

we can conclude that player has an ncentve to devate by transferrng at least a small fracton of wealth from asset k 2 to k 1. Thus, we compute: d dε π α ε), α ) = [ ε=0 ) r0 pk1 α k1 ) + εr0 α E k1 + ε ) ) r0) 2 r0 p α ) εr0 α k1 ) pk1 α k1 ) + εr0 2 E ε ) ] r0) 2 p α ) εr0) 2 = [ ) )] ε=0 r 0 E k1 p k1 α k1 ) [ r 0 E k1 p k1 α k1 ) 1 α k 1 r 0 p k1 α k1 ) E k 2 p α ) 1 α k 2 r 0 p α ) k1 α k1 ) ) E k 2 p α ) α ) )] 18) Let us try to nterpret Equaton 18). Thnk of an nvestor consderng to reallocate wealth from asset k 2 to asset k 1 at state α. Gven the prces of the assets at state α, = E k1 p k1 α k1 ) E k 2 p α ) 19) s the expected proft of movng one monetary unt from asset k 2 to asset k 1. If ths dfference s postve, ths reallocaton of wealth has a postve drect effect on expected returns. On the other hand, ths has the effect of rasng the prce of asset k 1 and lowerng the prce of asset k 2. In turn, the number of affordable unts of asset k 1 resp. k 2 ) per unt of wealth nvested decreases resp. ncreases). At the margn, the dfference n expected returns due to ths change n the holdngs of each asset s gven by 1 x k E 2 α ) p α ) E x k 1 α k1 ) k 1 p k1 α k1 ) 20) For each asset ths ndrect effect s larger the larger the number of unts bought so far, whch equals x k j α kj ). Equaton 18) says that there s an ncentve to move resources from k 2 to k 1 whenever the gans from the drect effect are larger than the losses from the ndrect effect through prces. Equaton 18) s the key to show that, n any equlbrum, asset prces must not only be postve, but they must be numercally equal to the normalzed or relatve) expected returns gven by R k = E k E where E = K E k. 21) k =1 Lemma 2. If α s a Nash equlbrum, then p k α k ) = R k for all k = 1,..., K. Proof. Let α be a Nash equlbrum. We proceed by contradcton. Note that the vector R 1,..., R K ) s an element of the smplex K. Recall also that p 1 α 1 ),..., p K α K )) K by 4). Hence, f these two vectors of the smplex are not dentcal, t follows that there exst 1 The margnal change n the number of affordable shares per monetary unt transferred s gven by ) 1 lm ε 0 εr0 p kj α kj )x 1 1 x k α kj ) j k j α kj ) p kj α kj ) εr0 = p kj α kj ) p kj α kj ). 6

assets k 1, k 2 such that R k1 > p k1 α k1 ) and R < p α ). Snce, by Lemma 1, all prces are strctly postve, t follows that R k1 p k1 α k1 ) > 1 and R p α ) < 1 22) Let be a player wth αk 2 > 0 and defne α ε) as n 15), wth 0 ε < αk 2. Dvdng Equaton 18) by the constant E, and takng 22) nto account, we obtan that ) 1 d E dε π α ε), α ) = ε=0 [ r0 R k1 p k1 α k1 k1 α k1 ) ) R k 2 p α ) α ) )] > r0 [ k1 α k1 ) ) k 2 α ) )] = r0 [ x α ) x k 1 α k1 ) ] 23) We clam that there exsts a player wth αk 2 > 0 such that x k 2 α ) x k 1 α k1 ) 0. If we can establsh ths, then t follows from 23) that ths partcular player has an ncentve to devate, a contradcton whch completes the proof. Suppose, then, that x k 2 α ) x k 1 α k1 ) < 0 for all players such that αk 2 > 0. Then, snce x j k 2 α ) = 0 for all players wth α j k 2 = 0, and recallng 3), 1 = {x k 2 α ) αk 2 > 0} < =1 {x k 1 α k1 ) αk 2 > 0} =1 x k 1 α k1 ) = 1 24) =1 a contradcton. The ntuton behnd Lemma 2 s the followng. By Equaton 22), whenever an asset s undervalued n the sense that the expected return per monetary unt nvested n that asset exceeds one, there must be another asset that s overvalued. In that case, movng wealth from the latter to the former yelds postve drect gans as measured by Equaton 19). Moreover, an nvestor can be found who owns a larger number of shares of the overvalued asset than of the undervalued one. For ths nvestor the ndrect effect through prces gven by Equaton 20) cannot offset the drect gans of transferrng wealth from the overvalued to the undervalued asset. Lemma 2 shows that, n any Nash equlbrum, prces have to be far, n the sense that every asset yelds the same expected returns per nvested monetary unt. That s, prces are numercally equal to the normalzed) expected returns. A pror, ths s not enough to dentfy the equlbra, because far prces wll obtan whenever aggregate nvestment s n accordance wth expected returns n the market-clearng equatons 1). That s, dfferent strategy profles can lead to far prces. We wll show, though, that n fact there exsts a unque equlbrum n pure strateges, correspondng to the followng defnton. Defnton 1. The Kelly-Nash equlbrum s the strategy profle α gven by α k = R k for all = 1,..., n and all k = 1,..., K. In the case of Arrow securtes S = K and A k s) = 1 f s = k, A k s) = 0 otherwse), the Kelly-Nash equlbrum reduces to the rule n Breman 1961) and the nvestment rule called bettng your belefs by Blume and Easley 1992), also known as the Kelly Crteron, whch s gven smply by α s = q s. For general assets t concdes wth the nvestment rule 7

dentfed by Hens and Schenk-Hoppé 2001), whch can be seen as the long-run outcome of an approprate dynamcal system of evolutonary type. 2 We remark that n the case of dagonal securtes S = K and A k s) > 0 f s = k, A k s) = 0 otherwse), the Kelly-Nash equlbrum and the bettng your belefs rule α s = q s would dffer unless we are n the Arrow securtes case. For example, consder two states of the world s = 1, 2 whch occur each wth probablty 1/2, and two assets k = 1, 2. Asset 1 pays 2 unts f state 1 occurs, and 0 otherwse. Asset 2 pays 3 unts f state 2 occurs, and 0 otherwse. Whle a bettng your belefs rule would prescrbe α 1 = α 2 = 1/2, the Kelly-Nash equlbrum prescrbes α 1 = 2/5 and α 2 = 3/5. Theorem 1. The only pure-strategy Nash equlbrum of the Stock Market Game s the Kelly-Nash equlbrum. Moreover, t s a strct equlbrum. Proof. We already know from Lemmata 1 and 2 that n any Nash equlbrum all assets are traded and p k α k ) = R k > 0 for all k. Consder any trader. We clam that, n equlbrum, α k = R k for all assets k. Suppose otherwse. Snce both α and R 1,..., R K ) are n the smplex K, t follows that there exst two assets k 1, k 2 such that 0 α k 1 < R > 0 and α k 2 > R > 0. In partcular, ths mples that trader has more unts of asset k 2 than of asset k 1, because snce the prces are p k α k ) = R k, x α ) = α k 2 r 0 R > r 0 > α k 1 r 0 R k1 = x k1 α k1 ). Defne α ε) as n 15), wth 0 ε < α k 2. From Equaton 18), and snce by Lemma 2 the equlbrum prces are p k = R k, we have that ) 1 d E dε π α ε), α ) = ε=0 r 0 [ R k1 p k1 α k1 ) k1 α k1 ) ) R k 2 p α ) α ) )] = r0 [ x α ) x k 1 α k1 ) ] > 0 25) Hence, player could proftably devate by transferrng some small fracton of wealth to asset k 1 and we conclude that the only possble Nash equlbrum s gven by α k = R k for all k and. It remans to show that ths profle s ndeed a Nash equlbrum so far we have only proven that no other profle can be a Nash equlbrum). We wll do ths by analyzng trader s maxmzaton problem. 3 Suppose, then, that α j k = R k for all k and all j. We have to show that α k = R k for all k s a best response for player. Player s maxmzaton problem s equvalent to 1 max α K E π α, α ) 26) 2 We do not clam, though, that the Kelly-Nash equlbrum s the approprate generalzaton of the Kelly Crteron to general asset markets. We study here short-run, strategc propertes, whereas such a generalzaton should be based on the long-run propertes. For example, Blume and Easley 1992) show that, for Arrow securtes, nvestors followng the Kelly Crteron wll eventually domnate the market; Sandron 2000, Sec. 3), though, llustrates the dffcultes of a generalzaton of such a long-run property to more general asset markets. 3 It s not dffcult to show unqueness also from trader s optmzaton problem and Lemma 2. However, we fnd the argument above more ntutve. 8

whose soluton must fulfll the frst order condtons R k x k α k) α k λ = 0 27) for k = 1,..., K, where λ s the Lagrange multpler assocated to the constrant k α k = 1. For αk = R k, we have that p k α k ) = R k and x k α k) = r0, and hence recallng 8)) the frst order condtons are fulflled wth λ = r0 ) 1 r 0 28) Thus, the Kelly-Nash equlbrum fulflls the frst order condtons for each trader s optmzaton problem wth the Lagrange multpler gven by 28). Snce x k s strctly concave for all k whenever asset k s traded, the payoff functon gven by 12) s also strctly concave. Therefore, the Kelly-Nash equlbrum s a strct Nash equlbrum. We fnd Theorem 1 rather strong, specally the unqueness part. To understand why, consder the followng. It s straghtforward to show that the Kelly-Nash equlbrum s also a compettve equlbrum, because f traders take the prces p k = R k as gven, ther profts are constant n own strategy: π = k E k α k r 0 R k = r 0 E. For the same reason, any strategy profle nducng far prces wll consttute a compettve equlbrum,.e. we have a multplcty of compettve equlbra. Ths s n sharp contrast wth our result: once strategc effects are taken nto account, the Kelly-Nash equlbrum s the only Nash equlbrum. Intutvely, the reason for ths s the fact that, f prces are far, transferrng wealth among assets wll have no drect effects as measured by expresson 19). Snce ths expresson s ndependent of the traders strateges, any profle wll be a compettve equlbrum f t nduces far prces. Indrect, strategc effects through changes n prces, though, as measured by expresson 20), are stll present. As mentoned above, ths expresson does depend on the traders strateges through the number of unts of each asset acqured. These effects only dsappear f the number of unts hold s the same for all assets. Ths s precsely the force at work n the prevous proof. Inequalty 25) above amounts to the clam that, provded prces are far, f a trader holds more unts from one asset than from another asset, she can proftably devate by movng to a more balanced allocaton. Ths s the key property of the Kelly-Nash equlbrum: for each trader the number of unts hold from each asset s constant and numercally equal to the ntal wealth. 4 Hence, at the Kelly-Nash equlbrum there are no ncentves to devate. Intutvely, drect effects of transferrng wealth among assets do not exst because prces are far recall equaton 19)), and ndrect effects also fal to exst because the number of unts of each asset hold by a gven trader s also constant recall equaton 20)). Of course, the number of unts hold of each asset s a pror economcally rrelevant, but when prces are far, holdng the same number of unts of each asset means that the trader s holdng an equally strong poston n each asset, not only n the number of unts, but also n terms of nvested wealth relatve to prces: a small nvestment n cheap assets, a large 4 The fact that share holdngs for a gven trader are constant across assets, n turn, comes from symmetry across traders nvestments recall 6)). It s obvous that the reverse mplcaton also holds, and thus the Kelly-Nash equlbrum can be smply defned as the only symmetrc profle nducng far prces. 9

nvestment n expensve assets. If ths were not the case, the trader could dvert rent away from assets were she has a stronger poston, manpulatng ts prce down, at the expense of ncreasng the prce of another asset where she had a weaker poston. Snce the prce change s symmetrc prces add up to one), the gans n the former asset would offset the loss n the latter. Ths s llustrated n the followng example. Example 1. There are two equally lkely states of nature, s = 1, 2, and two assets k = 1, 2. Asset 1 pays 2 unts n state 1 and zero otherwse whle asset 2 pays 3 unts n state 2 and zero otherwse. Hence E 1 = 1 and E 2 = 3 2. These are dagonal securtes but not Arrow securtes. There are two nvestors, Adam and Eve, endowed wth ntal wealth r0 A = r0 E = 1 2. The Kelly-Nash equlbrum prescrbes both of them to nvest 1/ 5 2 )) = 0.4 of ther wealth n asset 1 and the remanng 0.6 n asset 2, whch leads to prces p 1 = 0.4 and p 2 = 0.6. Suppose nstead Adam 1 nvests 0.7 of hs wealth n asset 1, whle Eve only nvests 0.1 of her wealth n that asset. The prce s stll gven by p 1 = 1 2 0.7 + 0.1) = 0.4 and thus p 2 = 0.6 and the descrbed profle s a compettve equlbrum snce the prces are far). Eve s recevng 0.125 unts of asset 1 and 0.75 unts of asset 2. Every nvested monetary unt yelds an expected return of 2.5 and hence the total expected payoffs for both Adam and Eve are equal to the ntal wealth tmes the expected return, or 1.25. Suppose, though, that Eve dverts 0.1 of her wealth from asset 2 to asset 1, so that now she s nvestng 0.2 of her wealth n asset 1 and 0.8 n asset 2. The prce of asset 1 ncreases to p 1 ε) = 1 2 0.7 + 0.2) = 0.45 and the prce of asset 2 decreases to 0.55 the change n prces s of course symmetrc, snce prce vectors always add up to one). Consequently, Eve now receves 1 2 0.2) /0.45 0.222 unts of asset 1 and 1 2 0.8) /0.55 0.727 unts of asset 2. Her new expected payoffs are π E 0.222 1 + 0.727 1.5 1.313 > 1.25 and thus she s better off than wth her prevous nvestment. The explanaton s the followng. Wth her new nvestment plan, Eve has acqured addtonal unts of asset 1, but has ncreased ts prce from 0.4 to 0.45. The return per monetary unt nvested n that asset, though, s now 1/0.45 2.222 rather than 2.5. Wth ths operaton, Eve s losng 2.5 2.222 0.277 wth every monetary unt nvested n asset 1,. e. wth 0.2 of her wealth), relatve to the prevously acheved returns. The prce of asset 2, though, has dropped from 0.6 to 0.55, and even though she s now recevng less unts of t, the return per monetary unt nvested n asset 2 s now 3 2 )/0.55 2.727 rather than 2.5. She s now earnng 0.227 more than before per each monetary unt nvested n asset 2,. e. wth 0.8 of her ntal wealth. Even though her addtonal per-monetary-unt earnngs n asset 2 are smaller than her per-monetary-unt losses n asset 1, she has a much larger nvestment n asset 2. Total payoff varaton s 0.2 0.272) + 0.8 +0.227) 0.127 > 0. By transferrng wealth between the two assets, Eve has successfully manpulated prces to her advantage, strategcally ncreasng her earnngs n an asset where her share holdngs are large at the expense of an asset where her share holdngs are small. 4 Evolutonary Stablty So far we have adopted the classcal approach n game theory, assumng that all nvestors know the assets dstrbutons of payoffs and maxmze ther expected utlty gven the nvestment strategy of the opponents. In that settng we have shown that the profle gven by 10

Defnton 1 s a symmetrc Nash equlbrum f all nvestors use the prescrbed nvestment plan, none of them can beneft from devatng to a dfferent plan. In the present secton we adopt an evolutonary approach wth nvestors that have hardwred strateges. In partcular, they need not be nformed about the assets dstrbutons of payoffs and they do not act strategcally. An evolutonarly stable strategy ESS) s then an nvestment strategy such that, once adopted by all nvestors, cannot be outperformed by any dfferent nvestment strategy. The dea s that no nvestor who would experment wth a dfferent strategy would obtan a larger return per monetary unt nvested than the other nvestors stll usng the status quo strategy. If ths were possble, other nvestors would follow the successful expermenter and the orgnal strategy would not be stable. The defnton of evolutonary stablty we use here s due to Schaffer 1988) and apples to a any fnte number of players. Although based on the same prncple of non-nvadablty, t dffers from the standard defnton of Maynard Smth 1982). The latter, whch s a refnement of Nash equlbrum, apples to a contnuum populaton of players who are randomly drawn n pars to play a two-person game. Schaffer s 1988) defnton s better suted to model market settngs, where the payoffs to any player depend on the actons of all market partcpants, and not only on the acton chosen by a randomly pcked opponent. It s mportant to note, however, that ths concept of evolutonary stablty s not related to Nash equlbrum n general. 5 The reason for ths mportant dfference s due to fnte populaton effects see Vega-Redondo 1996)) for a dscusson). In partcular, as we wll see n more detal below, a Schaffer ESS ams at maxmzng the dfference between own and opponents payoffs, a feature that s known as spteful behavor. A devaton from a Nash equlbrum strategy to an ESS may be worth undertakng f, even when t reduces own payoffs, t weakens the opponent n relatve terms. Ths responds to an dea of selecton of strateges based on relatve performance. Takng returns per monetary unt as the relevant payoffs, the game s symmetrc. We now proceed to adapt Schaffer s 1988) concept to ths framework. Defnton 2. We say that ˆα K s an evolutonarly stable strategy n the stock market game abbrevated ESS) f, for any and for any α K, α ˆα, 1 π α, ˆα ) 1 r j π j α, ˆα ) 29) 0 r 0 for any j, where ˆα = ˆα, n 1..., ˆα). The ESS s strct f the nequalty n 29) holds strctly. Note that the returns per monetary unt for any two nvestors choosng ˆα s the same. Hence only one comparson between the expermenter and any status quo nvestor s necessary n 29). That s, for any j, j, 1 π j α, ˆα ) = 1 π j α, ˆα ). r j 0 It follows drectly from Equaton 29) that an ESS ˆα solves the followng problem ) 1 max π α, ˆα ) 1 α K r j π j α, ˆα ) 0 r 0 r j 0 30) 5 As wll be seen later n ths secton, ths paper provdes one example of a Schaffer ESS that s also a Nash equlbrum. To our knowledge, ths s the frst non-trval example n the evolutonary lterature where a Schaffer ESS s also a Nash equlbrum. 11

for any j. It s n ths sense that an ESS maxmzes relatve performance. We remark that our concept of ESS s exactly that n Schaffer 1988) f we redefne the payoffs to be per-unt returns. Such normalzaton of payoffs would not affect the set of Nash equlbra. Thus the Kelly-Nash equlbrum s stll the only equlbrum of the game. In the stock market game ths equlbrum also turns out to be evolutonarly stable. Proposton 1. The nvestment strategy prescrbed by the Kelly-Nash equlbrum s a strct ESS n the stock market game. Proof. An ESS ˆα K solves problem 30), whch can be rewrtten as follows for α j k = ˆα k for all j K αk max E ˆα ) k k α K αk r 0 + ˆα k1 r0 ) 31) k=1 snce p k α k ) = αk r 0 + ˆα k 1 r0) f the expermental nvestment strategy s α. The frst order condtons for an nteror soluton are then α E k r0 + ˆα k 1 r0) αk ˆα ) ) k r 0 k α k r0 + ˆα k1 r0 )) 2 λ = 0 32) for every k = 1,..., K, where λ s the Lagrange multpler assocated to the constrant = 1. These condtons smplfy to k α k E k ˆα k = λ α kr 0 + ˆα k 1 r 0) ) 2 33) Note that f ˆα k = R k for all k, then α k = R k for all k fulflls the frst order condton wth λ = E. Snce the matrx of second dervatves of the lagrangean functon s dagonal see 32)), the second order condton amounts to E k ˆα k r0 2 α k r0 + ˆα k1 r0 )) 3 < 0 34) whch s obvously fulflled and completes the proof. A word of cauton s necessary here. The concept of ESS both Maynard Smth s and Schaffer s) s a statc one, based on one-shot comparsons. The evolutonary lterature usually proceeds from such statc defntons to later establsh ther dynamc propertes. For the case of Schaffer ESS, dynamc stablty propertes are establshed n Alós-Ferrer and Ana 2002). Hens and Schenk-Hoppé 2001), though, defne a concept of evolutonary stablty whch s drectly based on the stochastc dynamcs and s hence unrelated to those classc, statc defntons. No confuson should arse wth ths approach. 5 General utlty functons The payoff specfcaton gven by 10) assumes a lnear utlty of wealth,. e. rsk-neutralty. A more general specfcaton allowng e. g. for rsk averson would be ) S π α, α ) = q s u x kα k )A k s) 35) s=1 k 12

wth u : R + R a dfferentable) strctly ncreasng and concave functon. The frst-order condtons for an nteror Nash equlbrum are then ) S q s u x k α k )A k s) x k α k) α A k s) λ = 0 36) k k s=1 for k = 1,..., K, where λ s the lagrange multpler of the constrant k α k = 1. Usng 8), these condtons can be wrtten as ) S r0 q s u x k α k )A k s) k α k) A k s) = λ 37) p k α k ) k s=1 We check for symmetrc solutons. If αk = α k > 0 for all, then from 5) p kα k ) = αk and from 6) x k α k) = r0. Then, 37) turns nto r01 r0) ) q s u r 0 A k s) A k s) = λ αk 38) s Addng up n k, ths yelds λ = r 01 r 0) s k ) q s u r 0 A k s) A k s) 39) and replacng back nto 38), we obtan ) ) ) q s u r 0 A k s) A k s) = αk q s u r 0 A k s) A k s) s k We say that the asset market presents no aggregate rsk f As) = k A k s) s ndependent of s. Under ths addtonal assumpton, u r0 k A k s)) > 0 s also ndependent of the state and condton 40) yelds E k = αk s k k k k 40) k E k 41).e. αk = R k. In summary, the only canddate for a symmetrc) Nash equlbrum s also the Kelly-Nash equlbrum. Take, hence, α j k = R k > 0 for all j. Recall that x k α k) s a strctly concave functon of αk and that u ) s concave and strctly ncreasng. Then, t s straghtforward to show that the payoff functon π α, α ) gven by 35) s strctly concave n α. Thus, the frst order condton s suffcent for a strct, global maxmum and we have proven: Proposton 2. Consder the stock market game wth utlty of wealth gven by u : R + R, concave rsk averson) and strctly ncreasng. If there s no aggregate rsk, the Kelly-Nash equlbrum s a strct Nash Equlbrum. Interestngly enough, the assumpton of no aggregate rsk s not necessary n the case of rsk neutralty Theorem 1), but t s necessary to obtan the result under rsk averson. Wthout ths assumpton, n general condton 40) would yeld asymmetrc Nash equlbra unless r0 = r j 0 for all, j. An example of partcular nterest where the Nash equlbrum s stll symmetrc s posed by the case of logarthmc utlty, uz) = lnz. Drect computaton from equaton 40 yelds: 13

Proposton 3. Consder the stock market game wth utlty of wealth gven by uz) = lnz. Independently of whether there s aggregate rsk or not, the strct) symmetrc Nash equlbrum s gven by αk = A k s) q s As) s where As) = k A k s). Ths equlbrum concdes wth the Kelly-Nash equlbrum only f there s no aggregate rsk. Moreover, n the case of dagonal securtes K = S and A k s) = 0 f k s), n equlbrum αs = q s,. e. traders bet ther belefs. Whereas the Kelly-Nash equlbrum prescrbes to nvest accordng to the normalzed) expected returns of the assets R k = E k E ), the Nash equlbrum n the case of logarthmc utlty prescrbes to nvest accordng to the expected normalzed returns. As an llustraton of t he dfference, consder agan the case of dagonal securtes and aggregate rsk descrbed n Example 1. Under rsk neutralty, n the symmetrc Nash equlbrum traders would nvest accordng to α1 E 1 = = 2 α E 2 2 = = 3 E 1 + E 2 5 E 1 + E 2 5. Under logarthmc utltes, though, they would nvest accordng to α1 = 1 2 2 2 + 1 2 0 3 = 0.5 α 2 = 1 0 2 2 + 1 3 2 3 = 0.5. e. they would ndeed bet ther belefs Blume and Easley 1992)). 6 Fnal comments The Kelly crteron exhbts well-known asymptotc optmalty propertes as shown by Breman 1961), and long-run evolutonary stablty propertes as llustrated by Blume and Easley 1992). We have shown that t also exhbts good short-run propertes n a strategcally compettve market envronment. In summary, regardless of the asset market structure, we have found a generalzaton of the Kelly crteron, whch we call Kelly-Nash equlbrum, wth the followng propertes. Frst, t corresponds to the unque Nash equlbrum n a stock market game wth market clearng. Second, t s also evolutonarly stable n the sense of Schaffer 1988). Thrd, t s stll a Nash equlbrum f we allow for rsk averson on the sde of nvestors, provded n ths case) there s no aggregate rsk. Moreover, the Kelly- Nash equlbrum concdes wth the rule dentfed by Hens and Schenk-Hoppé 2001) as a long-run predcton for certan stochastc dynamcal systems. Such an overabundance of optmalty propertes for the rule at hand should make ntutvely clear that, e. g., any reasonable dynamc process should sngle t out as a long run predcton. Ths casts lght on the results by Blume and Easley 1992) and Hens and Schenk-Hoppé 2001). Followng Bell and Cover 1980, 1988), we have gone to the extreme n our analyss of the short-term and allowed the stock market to collapse nto a one-shot game. It would be more nterestng to consder a fntely repeated and explctly stochastc game, whch would brng us closer to the typcal termnal wealth problem analyzed n fnance. However, the man lesson learned from the present exercse would reman unchanged by such generalzatons: regardless of ts long-run, asymptotc mplcatons, the rule at hand exhbts strong shortterm optmalty propertes. 14

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