Single-Event Multinomial Full Kelly via Implicit State Positions

Single-Ev en t Multinomial F ull Kelly via Implicit State P ositions Christopher D. Long Abstract F or a single ev ent with finitely man y mutually exclusiv e outcomes, the full Kelly problem is to maximize exp ected log w ealth o ver nonnegative stak es together with an optional cash p osition. The optimal form ula is classical, but the supp ort-selection step is often presen ted via Lagrange m ultipliers. This note giv es a shorter state-price deriv ation. A cash fraction c acts as an implicit p osition in ev ery outcome: in terminal-w ealth terms, it is equiv alen t to a baseline stake cq i on outcome i , where q i is the state price. On an y activ e supp ort, explicit b ets therefore only top up fa vorable outcomes from this baseline cq i to the optimal total stake p i . This yields the form ula x i = ( p i − cq i ) + , the threshold rule p i /q i > c , and, after sorting outcomes by p i /q i , a one-pass greedy algorithm for supp ort selection. The result is standard in substance, but the implicit-position viewp oint giv es a compact proof and a conv enien t wa y to remem b er the solution. Keyw ords. Kelly criterion; m ultinomial betting; state prices; log-optimal growth; horse-race w agering. 1 In tro duction The Kelly criterion maximizes exp ected logarithmic gro wth and occupies a cen tral place in gam bling, p ortfolio theory , and information theory [ 1 , 2 , 3 , 4 ]. In the horse-race mo del, or more generally in a finite state market with state-contingen t claims, the optimization problem is conca ve and completely explicit [ 5 ]. F or a single multinomial ev ent with mutually exclusive outcomes, explicit treatmen ts also app ear in the pari-m utuel literature, notably in Rosner’s early analysis [ 6 ] and in the closed-form solution of Smo czynski and T omkins [ 7 ]. The natural question is therefore not whether the problem can b e solv ed, but how to present the solution so that the supp ort structure and the form ula are b oth immediate. This note giv es a short answer. The k ey observ ation is that cash itself is a state-contingen t claim. If the b ettor k eeps cash fraction c , then in state i this contributes wealth c ; equiv alen tly , relativ e to outcome i , it is as if the b ettor were already holding an implicit stak e cq i . Once that is recognized, the fixed-supp ort problem collapses to the elementary full-inv estmen t Kelly rule, and the global problem b ecomes a monotone threshold search. The resulting algorithm is greedy: sort b y the edge ratio p i /q i and k eep adding outcomes while that ratio exceeds the current cash level. Nothing here changes the classical theory . The con tribution is exp ository: a short state-price pro of in which cash is an all-state claim, so active b ets merely top up fa vorable outcomes from cq i to p i . This isolates the idea that makes the formulas memorable and keeps the supp ort-selection step nearly algebra-free. 1 2 Mo del Consider a single ev ent with outcomes i = 1 , . . . , n . Let p i > 0 , n X i =1 p i = 1 , b e the b ettor’s sub jective probabilities. Outcome i is a v ailable at decimal o dds O i > 1 , and we write q i := 1 O i > 0 for the corresp onding state price or implied probabilit y . The ov erround is P i q i , whic h may exceed 1 . A strategy consists of nonnegativ e stakes x i ≥ 0 and a cash holding c ≥ 0 satisfying c + n X i =1 x i = 1 . If outcome i o ccurs, terminal wealth is W i ( c, x ) = c + x i q i . The single-p erio d full Kelly problem is max c ≥ 0 , x i ≥ 0 G ( c, x ) := n X i =1 p i log  c + x i q i  sub ject to c + n X i =1 x i = 1 . (1) It is con venien t to define the e dge r atios r i := p i q i . These compare the b ettor’s probabilities to the mark et-implied probabilities. They will also b e the optimal terminal w ealth levels on active outcomes. Although the terminal wealth vector is unique, the pair ( c, x ) need not b e unique in degenerate tie cases. The canonical strategy constructed b elo w is unique whenev er no edge ratio equals the final cash cutoff. 3 A w eigh ted full-in v estmen t lemma The only optimization fact needed is the following standard weigh ted AM-GM/Jensen lemma. Lemma 1. L et a i > 0 with A := P m i =1 a i , and let S > 0 . T hen m X i =1 a i log z i is uniquely maximize d over z i > 0 subje ct to P m i =1 z i = S at z i = S A a i ( i = 1 , . . . , m ) . 2 Pr o of. W rite m X i =1 a i log z i − m X i =1 a i log a i = A m X i =1 a i A log  z i a i  . By conca vity of log , m X i =1 a i A log  z i a i  ≤ log m X i =1 a i A z i a i ! = log 1 A m X i =1 z i ! = log  S A  . Equalit y holds only when z i /a i is constan t in i , and the constraint P i z i = S forces that constant to b e S/ A . 4 Optimization on a fixed supp ort Let A ⊆ { 1 , . . . , n } b e nonempt y and prop er. W rite P A := X i ∈ A p i , Q A := X i ∈ A q i . W e ask for the b est strategy whose explicit b ets are confined to A , so that x j = 0 for j / ∈ A . Prop osition 2 (Fixed-supp ort optimizer) . L et A ⊊ { 1 , . . . , n } b e nonempty and assume Q A < 1 . Define c A := 1 − P A 1 − Q A . (2) If p i > c A q i ( i ∈ A ) , (3) then among al l str ate gies with x j = 0 for j / ∈ A , the unique maximizer of (1) is c = c A , x i = p i − c A q i ( i ∈ A ) , x j = 0 ( j / ∈ A ) . (4) Its terminal we alth is W i = p i q i = r i ( i ∈ A ) , W j = c A ( j / ∈ A ) . (5) Pr o of. Fix c ∈ (0 , 1) . F or i ∈ A , define the effe ctive total stake y i := x i + cq i . This is natural b ecause cq i stak ed on outcome i w ould pay exactly c in state i ; cash is therefore an implicit p osition in every outcome. F or i ∈ A , W i = c + x i q i = y i q i , while for j / ∈ A we hav e W j = c . Since x j = 0 for j / ∈ A and c + P i x i = 1 , X i ∈ A y i = X i ∈ A x i + cQ A = 1 − c + cQ A = 1 − c (1 − Q A ) . 3 Therefore, for fixed c , maximizing exp ected log wealth is equiv alen t to maximizing X i ∈ A p i log y i o ver y i > 0 with X i ∈ A y i = 1 − c (1 − Q A ) , since the terms − P i ∈ A p i log q i + (1 − P A ) log c are constan t. By Lemma 1, the unique maximizer for this fixed c is y i = 1 − c (1 − Q A ) P A p i ( i ∈ A ) . Substituting bac k yields the v alue function Φ A ( c ) = C A + P A log  1 − c (1 − Q A )  + (1 − P A ) log c, where C A is constan t in c . Since Φ A is strictly conca ve on (0 , 1) , its unique maximizer is characterized b y Φ ′ A ( c ) = − P A (1 − Q A ) 1 − c (1 − Q A ) + 1 − P A c = 0 , whic h gives (2). At c = c A one has 1 − c A (1 − Q A ) = P A , so the optimal effective stakes are simply y i = p i , hence x i = p i − c A q i . Condition (3) guaran tees that these explicit stak es are strictly p ositive on A . F ormula (5) follows immediately . The prop osition explains the formula: if cash level c A is held bac k, then the b ettor already has an implicit stake c A q i in every outcome. On the active supp ort, the total stake should therefore equal p i , so the additional explicit stak e is p i − c A q i . If A is prop er but Q A ≥ 1 , then the same v alue function Φ A ( c ) is strictly increasing on (0 , 1) , so no optimizer confined to A can keep every state in A activ e; the optim um necessarily collapses to a smaller supp ort. Thus any genuinely active prop er supp ort m ust satisfy Q A < 1 . 5 Greedy supp ort selection Sort the outcomes so that r 1 ≥ r 2 ≥ · · · ≥ r n . F or k = 0 , 1 , . . . , n , let A k := { 1 , . . . , k } , P k := k X i =1 p i , Q k := k X i =1 q i , with the con ven tion P 0 = Q 0 = 0 and c 0 = 1 . Whenever k < n and Q k < 1 , define c k := 1 − P k 1 − Q k . F or notational conv enience set c n := 0 . 4 Theorem 3 (Greedy characterization of the canonical Kelly strategy) . Start fr om A 0 = ∅ and c 0 = 1 . A t step k < n , c omp ar e r k +1 with c k . • If r k +1 > c k , add outc ome k + 1 to the supp ort and up date c ash to c k +1 = 1 − P k +1 1 − Q k +1 . • If r k +1 ≤ c k , stop. L et k ∗ b e the first index at which the algorithm stops, with the c onvention k ∗ = n if no stop o c curs e arlier. Then a c anonic al optimal str ate gy is c ∗ = c k ∗ , x ∗ i =  p i − c ∗ q i  + ( i = 1 , . . . , n ) , (6) and the unique optimal terminal we alth ve ctor is W ∗ i = max  c ∗ , p i q i  . (7) In p articular, the active outc omes form a pr efix after sorting by p i /q i . If r i  = c ∗ for every i , then the optimal str ate gy itself is unique. Pr o of. Supp ose first that k < n and r k +1 > c k . Then q k +1 < p k +1 c k ≤ 1 − P k c k = 1 − Q k , so Q k +1 < 1 and c k +1 is w ell defined. Moreov er, c k +1 < c k ⇐ ⇒ 1 − P k +1 1 − Q k +1 < 1 − P k 1 − Q k ⇐ ⇒ p k +1 > c k q k +1 ⇐ ⇒ r k +1 > c k . Th us every accepted addition strictly low ers cash. Since the r i are decreasing, r i ≥ r k +1 > c k > c k +1 (1 ≤ i ≤ k + 1) , so Prop osition 2 applies to A k +1 . In particular, once an outcome b ecomes activ e, it nev er drops out. No w supp ose the algorithm stops at k = k ∗ < n , so r k +1 ≤ c k . Consider an y strict enlargement B = A k ∪ S , where S ⊆ { k + 1 , . . . , n } is nonempty . W rite P S := X j ∈ S p j , Q S := X j ∈ S q j . Because r j ≤ r k +1 ≤ c k for ev ery j ∈ S , P S ≤ c k Q S . (8) If Q B < 1 , the cash level attached to supp ort B is c B := 1 − P k − P S 1 − Q k − Q S , and a one-line calculation giv es c B < c k ⇐ ⇒ P S > c k Q S . 5 By (8) this cannot happ en, so c B ≥ c k . Hence every j ∈ S satisfies r j ≤ c k ≤ c B , whic h contradicts the strict p ositivity condition r j > c B required by Prop osition 2 for an optimizer with supp ort confined to B . Therefore no strict enlargement of A k with Q B < 1 can b e optimal. If Q B ≥ 1 , the preceding remark after Prop osition 2 shows that an optimizer confined to B cannot k eep all states in B active, so its actual active supp ort is smaller and has already b een ruled out. An y smaller prefix A m with m < k is also sub optimal. Indeed, the algorithm did not stop at m , so r m +1 > c m . By the first part of the pro of, Prop osition 2 applies to A m +1 and pro duces a distinct optimizer among strategies confined to A m +1 . Since the optimizer on A m is feasible for that larger class but not optimal there, A m cannot b e globally optimal. Th us the canonical optimizer is attained at the stopping prefix A k . F ormula (6) is exactly the fixed-supp ort form ula there, extended by zeros off the activ e set, and (7) is just a restatement of the activ e and inactive wealth levels. If the algorithm never stops b efore n , then every prefix extension is accepted and the canonical c hoice is c ∗ = 0 , x ∗ i = p i for all i , whic h again yields (7) . Finally , strict concavit y in the terminal w ealth vector gives uniqueness of W ∗ , and the strategy is unique whenever no ratio r i lies exactly at the cutoff c ∗ . Remark 4 (Binary reduction) . F or a binary event, if only outc ome 1 is active then c ∗ = 1 − p 1 1 − q 1 , x ∗ 1 = p 1 − c ∗ q 1 = p 1 − q 1 1 − q 1 , which is the classic al one-b et K el ly fr action written in state-pric e form. Remark 5 (In terpretation) . The optimal terminal we alth has the clipp e d form W ∗ i = max { c ∗ , r i } . On outc omes with sufficiently lar ge e dge r atio p i /q i , the b ettor r aises we alth fr om the c ash flo or c ∗ up to p i /q i ; al l other outc omes ar e left at the flo or. The cutoff c ∗ is ther efor e the unique level at which the set of favor able states is sep ar ate d fr om the r est. 6 Concluding commen t F or a single multinomial ev ent, the full Kelly solution is already known and completely explicit. What is useful in practice is a deriv ation that keeps the intuition visible. The implicit-p osition viewp oin t do es exactly that: cash is an all-state claim, so active b ets only need to top up eac h fa vorable outcome from cq i to p i . Once written this wa y , the threshold rule and greedy supp ort selection b ecome immediate. References [1] J. L. 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