Calculated Boldness: Optimizing Financial Decisions with Illiquid Assets
Stanislav Shalunov, Alexei Kitaev, Yakov Shalunov, Arseniy Akopyan
CCalculated Boldness
Optimizing Financial Decisions with Illiquid AssetsStanislav Shalunov , Alexei Kitaev , Yakov Shalunov , and Arseniy Akopyan FORA Capital California Institute of Technology, Pasadena, CA 91125, USA IITP RAS (Kharkevich Institute), Moscow, Russia
December 23, 2020
Abstract
We consider games of chance played by someone with external capital that cannot beapplied to the game and determine how this affects risk-adjusted optimal betting. Specifically,we focus on Kelly optimization as a metric, optimizing the expected logarithm of total capitalincluding both capital in play and the external capital. For games with multiple rounds,we determine the optimal strategy through dynamic programming and construct a closeapproximation through the WKB method. The strategy can be described in terms of short-term utility functions, with risk aversion depending on the ratio of the amount in the gameto the external money. Thus, a rational player’s behavior varies between conservative playthat approaches Kelly strategy as they are able to invest a larger fraction of total wealth andextremely aggressive play that maximizes linear expectation when a larger portion of theircapital is locked away. Because you always have expected future productivity to account foras external resources, this goes counter to the conventional wisdom that super-Kelly bettingis a ruinous proposition.
Suppose you—the reader—have arrived at a conference in Los Vegas and are offered an opportu-nity. You are given $1 ,
000 and the chance to play a special game. The game is simple: you canstake as much of the money on a coin flip as you want, but no other money you may have. If youwin, you earn +0 . − .
25 of the stake. Then you doit 999 more times.If you wanted to, you could just walk away with the $1 ,
000 and go about your day. But thisis a positive expectation game. On average, you win +0 . x every game. With 1 ,
000 games, theexpected returns of putting down all your available money on every flip would be approximately$53 , , , , ¢ .
3. If you wish to walk away with more than your starting$1 , a r X i v : . [ q -f i n . P M ] D ec hus, if k is the number of winning flips, k <
524 gives less than $1 ,
000 at the end, and a simplebinomial distribution lets you calculate the probability). All your winnings are concentrated in asmall number of unlikely, massive outcomes. Not playing at all is wasteful, but betting everythingis too risky. This raises an obvious question: how should you play?This is not a meaningless hypothetical: we are not considering this game arbitrarily but ratherbecause it is a simplified model of life. We can view career progression as a series of decisions abouthow much effort to invest in opportunities as they come up. Furthermore, since overall wealth isgrowing, we can assume that these opportunities average to a positive expectation. However, sincejust investing everything blindly in each one obviously leads to ruin, we know the median result ofopportunities is negative. So we come to the rough conclusion that life is a series of rounds, eachwith a positive expectation but negative median result, making this game a reasonable proxy andan interesting problem to solve.To account for the cost of risk and variance, the best option is not to optimize expected returnsbut to optimize the value of some utility function. We go into more detail on utility functions inthe next section, but the one which we will be studying is the logarithm. It was first proposedin 1738 by Bernoulli [1], and its modern use in repeated gambling and investment is credited toKelly [2]. If you simply optimize the logarithm of returns, you get the far more reasonable strategyof placing of your capital on every flip. Your probability of walking away with more than $1 , , ,
000 you’ve been given to play with. Even if you are completely broke, you have all your futureearnings and prospects to factor in to play optimally. In other words, you need to optimize ln(1+ x )(normalizing external capital to 1), a problem far more complicated than simply optimizing ln( x )(simply betting ) and the one we will be solving in this paper.Part of conventional wisdom is that reasonable behavior is at least as risk-averse as a logarithmand that less risk aversion leads to ruin. Using the family of isoelastic utility functions (seesection 2.1) we can see this intuitively by noting that logarithm and all more risk-averse functionsgive a value of −∞ at 0 while anything less risk-averse than logarithm gives it a value of 0. It’seasy to see that no bet that leads to a possibility of ruin will ever be taken in the former case, whilein the latter case bets with a chance of ruin can be taken, meaning the probability of bankruptcyapproaches 1 with time.However, one of the key results we see here is that cases where it is rational to bet more thanthe “Kelly bet” not only exist in life but are actually perfectly normal, even for individuals whoseoverall utility function places true ruin at −∞ . We have already mentioned Bernoulli’s thesis: for a person with total wealth w , the utility of thissum is given by ln w . In any gamble, a rational player should maximize the expectation valueof this quantity (but not, for example, the expectation value of w or √ w ). However, it appearsthat such a sweeping statement cannot be true because different people have different preferencesabout acceptable risks and how much money they need. For example, − w appears to representbehavior of some risk-averse individuals well. An objective analysis of different utility functions2 . − . . . . . . . α . . . . . . λ ∗ ( α ) Figure 1: Optimal betting fraction λ ∗ as a function of the risk parameter α for the gamble withgain factors a = 1 . a = 0 .
75 and probabilities p = p = 0 .
5. In this example, the Kellyfraction is λ ∗ (0) = 1 /
3, and the gamble is attractive (i.e. λ ∗ ( α ) = 1) for α ≥ α ≈ . In a deterministic situation, the utility can be defined by any monotonically increasing function u ( w ), and maximizing it is equivalent to maximizing w . When chance is involved, it is natural toconsider concave functions because attaining a certain wealth w is usually better than expecting arandom gain with the same average. (In any case, the first option can be converted to the secondby going to a casino.) Among monotone, concave functions, we focus on functions of the form u ( x ) = cx α + const. The constant term and the coefficient c do not matter as long as c has thecorrect sign, that is, positive if 0 < α < α <
0. Let us also include the α = 0case: u α ( x ) = (cid:40) α − ( x α −
1) if α ∈ ( −∞ , ∪ (0 , , ln x = lim s → s − ( x s −
1) if α = 0 . (1)This is termed the “isoelastic utility function with risk aversion 1 − α ”. We prefer to work with α and call it the risk parameter . It will be shown that using α > α ≤ E [ u α ( w )] in a one-time gamble. The player’s decision isrepresented by a single parameter λ ∈ [0 , w in he is willing tobet. The game is defined by some gain factors a j and probabilities p j . (For example, each roundof the game described in the introduction has a = 1 . a = 0 .
75, and p = p = 0 . w j ( λ ) = (1 + λ ( a j − w in (2)3ith probability p j . Adopting the utility function u α , we are interested in maximizing its expec-tation value, U α ( λ ) = E j (cid:2) u α ( w j ( λ )) (cid:3) = (cid:88) j p j u α ( w j ( λ )) . (3)This is a concave function of λ , and its derivative is monotonically decreasing: U (cid:48) α ( λ ) = w α in (cid:88) j p j a j − λ ( a j − − α . (4)Therefore, there are three possibilities for the value of λ = λ ∗ at which U α ( λ ) attains its maximum: λ ∗ = 0 if U (cid:48) α (0) = w α in (cid:80) j p j ( a j − ≤ λ ∗ = 1 if U (cid:48) α (1) = w α in (cid:80) j p j ( a αj − a α − j ) ≥ < λ ∗ < U (cid:48) α ( λ ∗ ) = 0 (intermediate case). (5)Note that “favorability” is a property of the gamble itself; it simply means that the average gainfactor ¯ a = (cid:80) j p j a j is greater than 1. “Attractiveness”, on the other hand, depends on the riskparameter α . See figure 1 for illustration.Now, we will examine the risks the utility function u α entails, particularly, for α >
0. It issufficient to consider only attractive gambles because a λ ∗ bet is equivalent to betting all of one’smoney in a modified game, where ˜ a j = 1 + λ ∗ ( a j − p ( a α − a α − ) + p ( a α − a α − ) ≥ , p + p = 1 . (6)They can be satisfied for arbitrarily small positive numbers p , a by choosing a sufficiently large a . (Here we have assumed that α > u α , α = 1 / a = 1000, a = 0 . p = 0 . p = 0 .
9, whichimplies a tenfold loss with 0 . α = 0) helps avoid such reckless behavior. In this case, theattractiveness condition becomes (cid:80) j p j a − j ≤
1, or simply p ≤ a for two-outcome games with a → ∞ . Thus, a Bernoulli player will risk a loss by factor of A if it occurs with probability 1 /A or less. That is what we called “relatively safe”. However, the risk increases if fractional betsare not allowed. Here we come to an implicit assumption on which Bernoulli’s theory rests—thepossibility to manage risks by splitting capital (e.g. into a bet and a safe portion). This conceptis formalized below as a convex game. A general game of chance is described by a collection of positive real numbers w J (Λ), where Λrepresents a playing strategy and J some random event, occurring with probability p J . Thisdefinition is suitable as a description of each gambling round as well as the game as a whole.In the latter case, Λ is some function that prescribes player’s actions throughout the game. We4ssume that the randomness is external to the player, who makes deterministic decisions basedon available information. We are interested in optimizing the average Bernoulli utility, U (Λ) = E J (cid:2) ln( w J (Λ)) (cid:3) = (cid:88) J p J ln( w J (Λ)) . (7)The strategy Λ maximizing U (Λ) is called the Kelly strategy, though in his paper, Kelly focusedon repeated gambling.A convex game enjoys the property that for any strategies Λ (0) , Λ (1) and any number 0 ≤ t ≤ interpolating strategy Λ ( t ) , namely, one satisfying the condition w J (cid:0) Λ ( t ) (cid:1) = (1 − t ) w J (cid:0) Λ (0) (cid:1) + t w J (cid:0) Λ (1) (cid:1) for all J. (8)Logarithmic optimization for convex games, and, in particular, games based on investment port-folios, was studied in [3, 4]. These papers, among several others, demonstrate that the Kellystrategy is superior not only in terms of maximizing a particular function, but in a more objectivesense. We now discuss some simple properties of this kind. The following result (in the contextof portfolios) appears as Corollary 2 in Ref. [3]. Proposition 1.
Let Λ ∗ and Λ be, respectively, the Kelly-optimal strategy and an arbitrary strategyin a convex game, and let A ≥ . Then the probability of the event w J (Λ) /w J (Λ ∗ ) ≥ A is at most /A .Proof. Consider the strategy Λ ( t ) interpolating between Λ ∗ and Λ. Since Λ ∗ is optimal, we have0 ≥ dU (cid:0) Λ ( t ) (cid:1) dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 = (cid:88) J p J w J (Λ) − w J (Λ ∗ ) w J (Λ ∗ ) = E J (cid:20) w J (Λ) w J (Λ ∗ ) (cid:21) − . (9)The statement in question follows immediately by the Markov inequality.Essentially, strategy Λ ∗ is hard to beat by a large factor. This can be compared intuitivelyto the idea that if you have some amount of money, say $1, and any form of non-advantageouscasino, you will be able to obtain $ A with probability no greater than 1 /A . Obviously, playing ina non-advantageous casino is hardly a good idea.A key observation in Kelly’s original paper [2] is that for repeated gambling or investments,maximizing the logarithmic utility is the same as maximizing growth rate. This can be shownrather simply. If w n is the wealth after round n (for n = 0 , , , . . . ), then the growth rateis defined as lim n →∞ n ln w n w . For any myopic strategy, this limit is equal to E (cid:2) ln (cid:0) w n +1 w n (cid:1)(cid:3) = E [ln w n +1 ] − E [ln w n ] with probability 1. This follows from the strong law of large numbers.Note that maximizing E (cid:2) ln (cid:0) w n +1 w n (cid:1)(cid:3) (and thus, the growth rate) is different from maximizing theexpectation value of w n +1 w n .The Kelly strategy dominates any other strategy in the long run. A naive statement of thisproperty would be that if w ∗ n and w n are the results of playing by the Kelly strategy and anyother strategy, respectively, then Pr[ w ∗ n ≥ w n ] → n → ∞ . But this is true only for myopicstrategies; in the general case, Thorp gives a counterexample [5]. There are different ways toformulate asymptotic optimality for general strategies, see e.g. [6].Because of these advantages, we consider specifically Kelly optimization with external capital,rather than optimization of some other utility function.5 Repeated gambling and dynamic programming
Now we define the main problem formally. The game consists of n identical rounds. We countthem from the end, since the important factors in making a decision is the remaining number ofrounds k and the current amount x k . (Our notation is a bit complex as we keep record of thegame history.) The initial capital x n = x is fixed. Each round is a favorable gamble with gainfactors a j and probabilities p j . The player can bet any fraction λ k ∈ [0 ,
1] of the current amount.Depending on the chance event j = j k occurring with probability p j k , the new amount is x k − = (1 + λ k ( a j k − x k . (10)A playing strategy is a sequence of functions Λ = (Λ n , . . . , Λ ), where Λ k prescribes the bettingfraction in the k -th round using all available information: λ k = Λ k ( x ; j n , . . . , j k +1 ). By composingindividual steps like (10), we get the functional dependence x k − = X k − ( x ; Λ n , . . . , Λ k ; j n , . . . , j k ).Thus, the final amount may be written as x = X ( x, Λ , J ), where J = ( j n , . . . , j ) is the wholesequence of chance events.This game (or more exactly, the set of functions ( x, J ) (cid:55)→ X ( x, Λ , J ) corresponding to dif-ferent strategies) is convex. Indeed, an interpolation between given strategies Λ (0) , Λ (1) can beconstructed by successively defining λ ( t ) k = Λ ( t ) k ( x ; j n , . . . , j k +1 ) as follows: λ ( t ) k = (1 − t ) x (0) k λ (0) k + tx (1) k λ (1) k x ( t ) k , (11)where x ( t ) n = x , and x ( t ) k for k = n − , . . . , x ( t ) k = (1 − t ) x (0) k + tx (1) k for all x , J , and k. (12)The k = 0 case of the last equation is exactly the desired convexity property.We actually consider the n -round game in a bigger setting. In addition to the invested capital,the player has some side money. Without loss of generality, we take this amount to be 1. Noadditional investments are allowed during the game. Thus, the total attained wealth is w = 1 + x .We are interested in maximizing its expected Bernoulli utility, E [ln(1 + x )]. More exactly, thegoal is to calculate f n ( x ) = max Λ E J (cid:2) ln(1 + X ( x, Λ , J )) (cid:3) (13)and to find the corresponding optimal strategy. The latter depends only on the current amountrather than the full history, i.e. λ k = Λ k ( x k ).A general solution is obtained using dynamic programming. Obviously, f ( x ) = ln(1 + x ) . (14)For k >
0, the situation in the k -th round is equivalent to a one-time gamble with utility function f k − . Hence, f k ( x ) = max λ ∈ [0 , (cid:88) j p j f k − (cid:0) (1 + λ ( a j − x (cid:1) , (15)whereas Λ k ( x ) is given by the value of λ at which the maximum is attained.6 − − − − x − − − − − − f n ( x ) n = 0 n = 250 n = 500 n = 750 n = 1000 10 − − − x . . . . . . . . Λ n ( x ) n = 1 n = 250 n = 500 n = 750 n = 1000 Figure 2: Numerical solution of equation (15) with the initial condition f ( x ) = ln(1 + x ). Thesecond plot shows the optimal betting fractions, which vary from 1 to 1 / x →
0, we may use the approximation f ( x ) ≈ x . To maximize E [ x ], one should bet all available money (because we have assumed thatthe gamble is favorable). We can now see, using induction in k , that f k is a linear function and thatbetting all the money is always good. (This is an example of a myopic strategy.) Indeed, if f k − is linear, then the maximum in eq. (15) is attained at λ = 1, and hence, f k ( x ) = (cid:80) j p j f k − ( a j x )is also linear. More specifically, the solution is: f k ( x ) ≈ ¯ a k x for x → , where ¯ a = (cid:88) j p j a j > . (16)In the x → ∞ case, the utility function f ( x ) may be approximated by ln x so that the optimalbetting fraction is λ Kelly = λ ∗ (0) (see section 2.1, particularly, figure 1). As in the previous case,the same myopic strategy can be used throughout the game. Thus, the expected utility grows ata constant rate v : f k ( x ) ≈ ln x + kv for x → ∞ , where v = (cid:88) j p j ln(1 + λ Kelly ( a j − . (17)A numerical solution of equation (15) for our exemplary gamble (with a = 1 . a = 0 . p = p = 0 .
5) is shown in figure 2. Note that the plots cover a broad range of parameters,including some ridiculously small values of x . Furthermore, the expected utility f n ( x ) can beso small that it would not be worth the optimization effort. (In general, a tiny profit is worthpursuing only if it can be combined with other tiny profits.) However, the study of the problemfor such extreme parameters helps one understand the large n behavior, which is the subject of thenext section. The example shown in figure 3 is less extreme. It illustrates the difference betweenthe optimal and Kelly strategies for sufficiently large n but intermediate x . The Kelly strategyguarantees a higher median value of the final capital x . The optimal strategy, on the other hand,achieves a higher probability of gaining an amount x of the order of 1. This probability can besmall (in fact, it falls exponentially with n ), but in this case, it is better to take the risk thanhaving an assured but tiny gain. This will be explained in terms of an effective utility functionwith a suitably chosen risk parameter 0 ≤ α ≤ − − − x p r o b a b ili t y d e n s i t y Kellyoptimal − − − x . . . . . . t a il p r o b a b ili t y Kellyoptimal
Figure 3: Probability distributions of the final capital x when using the optimal strategy vs.Kelly’s. The game consists of n = 1000 rounds, and the initial capital is x n = 10 − . (this choiceof initial capital is discussed in the conclusion). The second plot shows the tail distribution, i.e.the probability that the final capital exceeds a given amount.We conclude this section with a rather technical remark. To solve equation (15) in a finiteinterval while avoiding an unreasonable computational cost, one has to impose some boundaryconditions, for example, f n (0) = 0 and f n ( x ) = ln x + nv for x > x max . A naive implementationof the second condition may cause an instability. To avoid this and other possible instabilities,we maintained an invariant of the exact problem—that the function f n is concave for all n . Moreexactly, if f n − is concave, then f n is concave; this property is closely related to the convexity ofthe game itself. To reconstruct the function from its grid values, we used a concavity-preservingquadratic interpolation. When gluing the numerical solution for x < x max and the Kelly asymp-totics for x > x max , we cut the “tooth” at the boundary so that the resulting function is concave. Let us now find the asymptotic form of the functions f n at large n . The key insight, which can begleaned from the numerics, is that d ln f n ( x ) d ln x (i.e. the slope of the curves in the left plot in figure 2)varies slowly over a broad range of x . For the purpose of optimizing the betting fraction at a givenpoint x = x n , we may assume the slope α to be constant. That is, we may use the approximation f n − ( x ) ≈ c n − x α , ≤ α ≤ x n . Power-law functions are invariant under the iteration byequation (15), with the overall factor growing exponentially. More exactly, c n = e κ ( α ) c n − , where κ ( α ) = max λ ∈ [0 , r ( α, λ ) , r ( α, λ ) = ln (cid:18) (cid:88) j p j (1 + λ ( a j − α (cid:19) . (19)The function κ ( α ) and its derivative κ (cid:48) ( α ) for the specific gamble are plotted in figure 4a. Wewill show that κ is convex, and hence, κ (cid:48) is monotone.8 . . . . . . α . . . . . . . . κ ( α ) κ ( α ) time t = in spatial coordinate q = ln x momentum p = − iα Hamiltonian H = κ ( α )velocity dqdt = ∂H∂p = i κ (cid:48) ( α )a) b)Figure 4: a) Growth rate κ and its derivative κ (cid:48) as functions of the risk parameter α . b) Mappingbetween quantum mechanics of a 1d particle and the gambling problem.When using the approximation (18) locally, the equation c n = e κ ( α ) c n − should be replacedwith this one: ∂ ln f n ( x ) ∂n = κ ( α n ( x )) with α n ( x ) = ∂ ln f n ( x ) ∂ ln x . (20)Here n is treated as a continuous variable, the second variable being q = ln x . This is, essentially,the WKB approximation used in a slightly unusual situation. Let us digress a bit and elaborateon this analogy.The WKB approximation is commonly used in quantum mechanics. It amounts to writingthe wavefunction of a particle moving along the q axis as ψ ( t, q ) = e iS ( t,q ) , up to some factor thatvaries slowly in space and time. Then the Schr¨odinger equation for ψ is reduced to the Hamilton-Jacobi equation, ∂Sdt = − H (cid:0) q, p ) with p = ∂S∂q . One can readily see the analogy with equation (20);the variable mapping between the two problems is shown in figure 4b. Note that the momentum p = − iα is imaginary, like in the quantum tunneling problem. Studying quantum evolution inimaginary time is also a common trick, used to calculate thermodynamic properties.Returning to the main subject, equation (20) is first-order, and thus, can be solved by themethod of characteristics. Characteristics are lines of constant α in the (ln x, n ) plane. They aregiven by the equation d ln xdn = − κ (cid:48) ( α ) . (21)Indeed, let us differentiate both sides of the first equation in (20) with respect to ln x and express ∂ ln f n ( x ) ∂ ln x on the left-hand side as α = α n ( x ). The result is ∂α∂n = κ (cid:48) ( α ) ∂α∂ ln x , implying that α is constant on lines (21). These lines are projected from points x on the n = 0 line, where α = α ( x ) is as follows: α ( x ) = d ln f ( x ) d ln x = x (1 + x ) ln(1 + x ) → (cid:40) x → , x → ∞ . (22)On each characteristic, equation (20) is reduced to an ordinary differential equation, which has a9 − − − x n Figure 5: WKB characteristics (left) and asymptotic regions in the n → ∞ limit (right). Theblue area in the left figure will be studied in section 5.simple solution: f WKB n (cid:0) e − n κ (cid:48) ( α ) x (cid:1) = e − n ( α κ (cid:48) ( α ) − κ ( α )) f ( x ) , where α = α ( x ) . (23)From the algorithmic point of view, f WKB n ( x ) is computed by shooting a characteristic witha suitable slope − κ (cid:48) ( α ) from the point (ln x, n ) so as to satisfy the equation α = α ( x ) at theendpoint, see figure 5. This is an explicit expression for κ (cid:48) ( α ): κ (cid:48) ( α ) = ∂r ( α, λ ) ∂α (cid:12)(cid:12)(cid:12)(cid:12) λ = λ ∗ ( α ) = (cid:80) j p j ˜ a αj ln ˜ a j (cid:80) j p j ˜ a αj , where ˜ a j = 1 + λ ∗ ( α ) ( a j − . (24)In particular, κ (cid:48) (0) is equal to the Kelly rate v defined in equation (17), and we will denote κ (cid:48) (1)by v . Let us use the following approximation: α ( x ) ≈ x < , x > , any number between 0 and 1 if x = 0 . (25)It incurs error in a relatively small interval, x ∼
1, and the overall precision loss is comparableto that due to the WKB approximation itself. Thus, we arrive at the asymptotic region pictureshown in figure 5. In the intermediate region, located between the lines with slopes − v and − v ,the approximate solution is f n ( x ) ∼ e − nh ( − ln xn ) , (26)where h the Legendre transform of the function κ : h ( v ) = max α ∈ [0 , ( αv − κ ( α )) for v ≤ v ≤ v . (27)That is, h ( v ) = α κ (cid:48) ( α ) − κ ( α ) for the unique α ∈ [0 ,
1] satisfying the equation κ (cid:48) ( α ) = v .Let us now establish some useful properties of the functions κ ( α ), r ( α, λ ) (see equation (19)),and h ( v ). Some of the subsequent arguments are borrowed from statistical mechanics and thederivation of Chernoff’s bound. 10 roposition 2. The functions κ ( α ) and r ( α, λ ) are convex in α .Proof. Let ∆ be the set of probability distributions on the outcomes of a single gamble. Then r ( α, λ ) = max q ∈ ∆ (cid:18)(cid:88) j q j ln p j ˜ a αj q j (cid:19) , where ˜ a j = 1 + λ ( a j − . (28)Indeed, the maximum is attained at an interior point of ∆, which can be determined by takingpartial derivatives with respect to q j and using a Lagrange multiplier. Specifically, q j = p j ˜ a αj /A with A = (cid:80) j p j ˜ a αj , and hence, (cid:80) j q j ln p j ˜ a αj q j = ln A = r ( α, λ ).Next, we interpret the maximum in equation (28) as a Legendre transform of some functionin α : r ( α, λ ) = max v ( αv − s ( v, λ )) , (29)where s ( v, λ ) = min q ∈ Q ( v ) (cid:18)(cid:88) j q j ln q j p j (cid:19) , Q ( v ) = (cid:26) q ∈ ∆ : (cid:88) j q j ln ˜ a j = v (cid:27) . (30)(The minimum over an empty set is defined as + ∞ .) It follows that r ( α, λ ) is convex in α , andhence, κ ( α ) = max λ ∈ [0 , r ( α, λ ) is also convex. Proposition 3.
The function h defined by equation (27) admits the following representation: h ( v ) = min λ ∈ [0 , s ( v, λ ) for v ≤ v ≤ v , (31) where s ( v, λ ) is the minimum relative entropy, see equation (30) .Proof. It follows from the convexity of the function q (cid:55)→ (cid:80) j q j ln q j p j that s ( v, λ ) is convex in v .Applying the Legendre transform to a convex function twice gives the same function; therefore, s ( v, λ ) = sup α ∈ R ( αv − r ( α, λ )) . (32)Let us temporarily restrict α to the interval [0 , α ∈ [0 , r ( α, λ ) is concave in λ ,and thus, we can apply von Neumann’s minimax theorem [7]: h ( v ) = max α ∈ [0 , min λ ∈ [0 , ( αv − r ( α, λ )) = min λ ∈ [0 , max α ∈ [0 , ( αv − r ( α, λ )) . (33)We claim that the constraint α ∈ [0 ,
1] on the last maximum can be dropped, provided v ≤ v ≤ v .To see this, let us assume that both inequalities are strict; the general case follows by continuity.If v < v < v , the maximin is attained at some point ( α ( v ) , λ ( v )) with 0 < α ( v ) <
1. This pointis a saddle of the function ϕ ( α, λ ) = αv − r ( α, λ ), meaning thatmin λ ∈ [0 , ϕ ( α ( v ) , λ ) = ϕ ( α ( v ) , λ ( v )) = max α ∈ [0 , ϕ ( α, λ ( v )) . (34)Since 0 < α ( v ) < ϕ ( α, λ ( v )) is concave in α for α ∈ R , the last maximum is, actually,global. Thus, ( α ( v ) , λ ( v )) satisfies the saddle condition for λ ∈ [0 , α ∈ R , and hence, is alsoa minimax point in this domain. We conclude that h ( v ) = min λ ∈ [0 , sup α ∈ R ( αv − r ( α, λ )), wherethe supremum is equal to s ( v, λ ) due to equation (32).11 − − − − x − − − − − − f ( x ) exactWKBdiffusion − − − − x f ( x ) exactWKBdiffusion Figure 6: Comparison of the exact solution f n with those obtained using the WKB and diffusionapproximations for n = 1000. The dashed vertical line is positioned at e − nv , which correspondsto the boundary of the α = 0 region.WKB characteristics have an interesting probabilistic interpretation. Its exact statement wouldhave to be elaborate and the proof would involve a Chernoff bound, with the relative entropyplaying its usual role. We will not attempt that but notice that this qualitative picture explainsthe numerics well. Let x n be the initial capital, and let p ( x ) denote the tail distribution ofthe final capital, assuming that the player uses the optimal strategy. Then the expected utility f n ( x n ) = E [ f ( x )] can be expressed as follows: f n ( x n ) = (cid:90) f ( x ) (cid:18) − dp ( x ) dx (cid:19) dx . (35)We claim that this integral is dominated by a small neighborhood of the endpoint of the character-istic passing through (ln x n , n ). Given that, the characteristic represents an “optimistic trajectory”the player hopes to follow. The expected utility along this trajectory increases exponentially at therate − d ln f n ( x ) dn = α κ (cid:48) ( α ) − κ ( α ). Here, we refer to the conditional expectation value, assuming thatthe player stays on track. However, the probability of this event decreases at the same rate, suchthat the total expectation value is conserved. (The probability could drop even faster if there wereother paths to the destination. However, if the player ever falls behind, the chances to recover arevery slim.) Thus, we may call v = κ (cid:48) ( α ) the “optimistic growth rate” and h ( v ) = α κ (cid:48) ( α ) − κ ( α )the “failure rate”. Our asymptotic analysis is all about the balance between these quantities, asis evident from this expression for the WKB solution (23): f WKB n ( x ) = max v e − nh ( v ) f (cid:0) e nv x (cid:1) . (36) As already mentioned, the expected gains f n ( x ) in the α = 1 region and the intermediate 0 < α < n . Let us now study the boundary between the intermediateregion and the Kelly region (with α = 0), where the gains can be significant. The Kelly region12ill also be covered by our analysis. The plan is to expand κ ( α ) to the second order in α andwrite the corresponding differential equation, which is, essentially, the diffusion equation.We begin with a quadratic expansion of r ( α, λ ) in α : r ( α, λ ) = α (cid:0) v ( λ ) + D ( λ ) α + O ( α ) (cid:1) , (37)where v ( λ ) = (cid:88) j p j ln ˜ a j , D ( λ ) = 12 (cid:18)(cid:88) j p j (ln ˜ a j ) − (cid:16)(cid:88) j p j ln ˜ a j (cid:17) (cid:19) , ˜ a j = 1 + λ ( a j − . (38)The following intermediate calculations use the assumption λ Kelly = λ ∗ (0) <
1, but the result isalso valid for λ Kelly = 1. Expanding v ( λ ) and D ( λ ) in λ − λ Kelly , we go one order higher thannecessary: v ( λ ) = v ( λ Kelly ) + v (cid:48) ( λ Kelly ) ( λ − λ Kelly ) + v (cid:48)(cid:48) ( λ Kelly )2 ( λ − λ Kelly ) + · · · ,D ( λ ) = D ( λ Kelly ) + D (cid:48) ( λ Kelly ) ( λ − λ Kelly ) + · · · . (39)The higher accuracy helps to illustrate how the λ optimization works. Note that v (cid:48) ( λ Kelly ) = 0because by definition, λ Kelly is the point at which v ( λ ) = (cid:80) j p j ln(1+ λ ( a j − λ ∗ ( α ) = λ Kelly − D (cid:48) ( λ Kelly ) v (cid:48)(cid:48) ( λ Kelly ) α + O ( α ) . (40)However, the approximation λ ∗ ( α ) ≈ λ Kelly will be sufficient for our purposes, and it is alsoapplicable if λ Kelly = 1. Using either this approximation or equation (40), we get: κ ( α ) = v α + Dα + O ( α ) , where v = v ( λ Kelly ) , D = D ( λ Kelly ) . (41)Independently of the concrete expression for κ ( α ), the approximation λ ≈ λ Kelly makes theproblem linear: f n ( x ) ≈ (cid:88) j p j f n − (˜ a j x ) , ˜ a j = 1 + λ Kelly ( a j − . (42)It is convenient to write it in lin-log coordinates, i.e. in terms of the variable y = ln x and thefunction g n ( y ) = f n ( e y ) . (43)Thus, the recurrence relation (42) becomes g n ( y ) ≈ (cid:88) j p j g n − ( y + ln ˜ a j ) . (44)To simplify it further, we will fit the set of standard solutions g n ( y ) = e κ ( α ) n + αy with a linearsecond-order differential equation using the expression (41) for κ ( α ): ∂g∂n = v ∂g∂y + D ∂ g∂y . (45)13ts solution can be written explicitly: g n ( y ) = (cid:90) K n ( y, z ) g ( z ) dz, where K n ( y, z ) = 12 √ πDn exp (cid:18) − ( y − z + v n ) Dn (cid:19) . (46)Replacing g ( y ) = ln(1 + e y ) with a crude approximation, g ( y ) = y θ ( y ) , (47)we get: g n ( y ) = √ Dn g (cid:18) y + v n √ Dn (cid:19) , g ( t ) = t t )2 + e − t √ π , (48)where θ is the step function and erf is the error function. We are now equipped with the tools to return to the original question and determine how oneshould realistically play this game, specifically considering our youngest likely readers. We firstestimate the external capital for this reader, who we place as a grad student. There are approacheswe can take to estimate this. The EPA puts the value of a statistical life at $7 . . . . ,
000 per month for a 25-year-old with $2 , ,
000 initial investment. This is substantially less than the expected income, butwe must consider that this already factors in retirement savings (since the annuity makes nodistinction), which brings it up to the equivalent of roughly $70 ,
000 annually (assuming a fairlytypical 15% retirement savings rate). Considering this is income one gets never working again (i.e.an extremely early retirement), and that this game is best viewed on a logarithmic scale (so thedifference between $2 and $3 million is minor), we will use $2 , ,
000 as our estimate for externalcapital.Since rather than solving ln( x ext + x int ), we are normalizing to ln(1 + x ), our initial x n is $1 , , , = 5 · − ≈ − . , n = 1 , .
494 for the first move. If that coin-flip is good, our capitalincreases and our next bet drops slightly to 0 .
492 of our total capital. If it’s bad, our bet increasessimilarly. In general, the optimal bet will increase to infinity (but is capped at 1 by the rules) as14
10 20 30 40 50 x ($1 , , . . . . . . t a il p r o b a b ili t y Kellyoptimal
Figure 7: Linear scale tail distribution of the final capital x (in millions of dollars). The gameconsists of n = 1000 rounds and the external capital is $2 , , (the ln( x ) optimization) as your internal capitaloverwhelms external capital. As a consequence, bad luck in this game reinforces itself but goodluck will encourage you to be more cautious, increasing the chances for further success.We can summarize from figure 3 that optimizing ln(1+ x ) leads to “double or nothing” behavior:it’s a waste of effort to go for a mediocre result and instead optimal to risk the small amount inplay for results on order with external capital. We can see in figure 7 that the probabilities oflarge sums are substantially higher under optimal play.These results stand in stark contrast to the conventional wisdom that super-Kelly betting isa path to ruin. We see that, in a situation where there is capital that cannot be applied, bettingmore aggressively than the Kelly strategy is a sound strategy. Because everyone has capital theycannot apply, whether it be explicit non-liquid investments or simply future productivity, weconclude that super-Kelly betting can be a reasonable option. Acknowledgments
We thank Gregory Falkovich for useful comments. A.K. is supported by the Simons Foundationunder grant 376205 and by the Institute of Quantum Information and Matter, a NSF Frontiercenter funded in part by the Gordon and Betty Moore Foundation.
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