Utility-Invariant Support Selection and Eventwise Decoupling for Simultaneous Independent Multi-Outcome Bets

Utilit y-In v arian t Supp ort Selection and Ev en t wise Decoupling for Sim ultaneous Indep enden t Multi-Outcome Bets Christopher D. Long Abstract F or sim ultaneous indep enden t ev ents with finitely many outcomes, consider the exp ected- utilit y problem with nonnegative wagers and an endogenous cash p osition. W e prov e a short supp ort theorem for a broad class of strictly increasing strictly concav e utilities. On any fixed supp ort family and at an y optimal p ortfolio with p ositiv e cash, summing the activ e first-order conditions and comparing that sum with cash stationarity yields the exact iden tity λ K ( U ) ℓ = 1 − P ℓ,A 1 − Q ℓ,A , where P ℓ,A and Q ℓ,A are the activ e probability and price masses of ev ent ℓ , λ is the budget m ultiplier, and K ( U ) ℓ is the con tinuation factor seen by inactiv e outcomes of that even t. Con- sequen tly , after sorting eac h even t b y the edge ratio p ℓi /π ℓi , the exact active supp ort is the ev ent wise union of the single-even t supp orts, and this supp ort is indep enden t of the utility function. The single-even t utilit y-inv ariant supp ort theorem is already explicit in the free- exp osure pari-m utuel setting in Smo czynski and Miles; the p oint of the presen t note is that the sim ultaneous indep enden t-ev ents analogue follows from the same state-price geometry once the righ t con tinuation factor is iden tified. 1 In tro duction F or a single multi-outcome ev ent, the active-set structure of the full Kelly problem is classical; see, among others, Kelly [ 1 ], Rosner [ 3 ], Smo czynski and T omkins [ 5 ], and Whelan [ 6 ]. In state-price language, cash is an implicit all-state p osition, so active wagers merely top up fa v orable outcomes b ey ond the cash flo or; see Long [ 2 ]. F or general increasing concav e utilit y in the free-exp osure pari-m utuel setting, Smo czynski and Miles [ 4 ] prov e that the optimal set of horses is independent of utilit y . The purp ose of this note is to prov e the simultaneous indep enden t-ev ents analogue in a short self-con tained form. The decisive structural observ ation is that, for a fixed even t, ev ery inactiv e outcome sees the same background wealth from the rest of the p ortfolio. This common contin uation factor is the m ulti-ev ent analogue of the single-even t cash flo or. Once that ob ject is named, the pro of is a short stationarity-decomposition argumen t: sum the active first-order conditions, condition the cash stationarity equation on one ev ent, and subtract. The result isolates a clean division of labor. Utilit y curv ature affects the active weights , but not the supp ort . In the sim ultaneous indep enden t problem, supp ort selection decouples even twise and is gov erned by exactly the same threshold (1 − P ) / (1 − Q ) as in the single-ev ent problem. This sharp ens the structural in terpretation of sim ultaneous Kelly-type optimization b ey ond the algorithmic literature on man y sim ultaneous b ets; see Whitro w [ 7 ]. 1 2 Mo del and admissible utilities Consider m ≥ 1 indep enden t ev ents. Ev ent ℓ has outcomes i ∈ { 1 , . . . , n ℓ } , sub jectiv e probabilities p ℓi > 0 , n ℓ X i =1 p ℓi = 1 , and state prices π ℓi > 0 . A portfolio consists of a cash p osition c ≥ 0 and nonnegativ e w ager sizes g ℓi ≥ 0 , sub ject to the bankroll constraint c + m X ℓ =1 n ℓ X i =1 π ℓi g ℓi = 1 . (1) If the realized pro duct-state is X = ( X 1 , . . . , X m ) , terminal w ealth is W ( X ) = c + m X ℓ =1 g ℓ,X ℓ . (2) W e restrict atten tion to portfolios satisfying W ( x ) > 0 for every pro duct-state x. Definition 1. A n admissible utility is a function U : (0 , ∞ ) → R such that: (U1) U ∈ C 1 ((0 , ∞ )) , (U2) U ′ ( w ) > 0 for every w > 0 , (U3) U is strictly c onc ave on (0 , ∞ ) . The optimization problem is max Φ( c, g ) := E  U ( W ( X ))  sub ject to ( 1 ), c ≥ 0 , g ℓi ≥ 0 , W ( x ) > 0 ∀ x. (3) F or each outcome define the edge ratio r ℓi := p ℓi π ℓi . (4) A supp ort family is a tuple A = ( A 1 , . . . , A m ) with A ℓ ⊆ { 1 , . . . , n ℓ } . W e say that ( c, g ) is supp orte d on A if g ℓi > 0 for i ∈ A ℓ , g ℓi = 0 for i / ∈ A ℓ . F or fixed ℓ , define the bac kground wealth from all other even ts b y R ℓ ( X − ℓ ) := c + X r  = ℓ g r,X r . (5) Then on the slice { X ℓ = i } one has W ( X ) = g ℓi + R ℓ ( X − ℓ ) . In particular, if i / ∈ A ℓ then g ℓi = 0 and the slice wealth is exactly R ℓ ( X − ℓ ) , indep enden t of whic h inactiv e outcome is b eing tested. 2 3 Fixed-supp ort stationarit y and reduced costs The next proposition records the only fixed-supp ort facts needed for the theorem. Prop osition 2 (Fixed-supp ort first-order and reduced-cost conditions) . Fix a supp ort family A = ( A 1 , . . . , A m ) , and let ( c, g ) b e an optimal p ortfolio for the r estriction of ( 3 ) to p ortfolios supp orte d on A . A ssume c > 0 . Then ther e exists a multiplier λ > 0 such that λ = E  U ′ ( W ( X ))  , (6) and, for every event ℓ and every active outc ome i ∈ A ℓ , p ℓi E − ℓ  U ′ ( g ℓi + R ℓ ( X − ℓ ))  = λπ ℓi . (7) If j / ∈ A ℓ , define K ( U ) ℓ := E − ℓ  U ′ ( R ℓ ( X − ℓ ))  . (8) Then the one-side d dir e ctional derivative in the fe asible dir e ction g ℓj 7→ g ℓj + ε at ε = 0 + is p ℓj K ( U ) ℓ − λπ ℓj , and optimality ther efor e for c es the r e duc e d-c ost ine quality p ℓj K ( U ) ℓ ≤ λπ ℓj . (9) In p articular, for fixe d ℓ , every inactive outc ome shar es the same c ontinuation factor K ( U ) ℓ . Pr o of. Restricting to supp ort family A means that the v ariables g ℓi with i / ∈ A ℓ are fixed at zero, while c and the active v ariables g ℓi with i ∈ A ℓ v ary su bject to ( 1 ) . Because U is strictly concav e and wealth is affine in the decision v ariables, the restricted ob jectiv e is strictly concav e. Standard Kuhn-T uc ker conditions therefore apply . W rite the Lagrangian L ( c, g ; λ ) = Φ( c, g ) − λ   c + X ℓ,i π ℓi g ℓi − 1   . Because c > 0 , complementary slackness yields stationarity with respect to c . Since ∂ W ( X ) /∂ c = 1 in every state, ∂ Φ ∂ c = E  U ′ ( W ( X ))  , whic h giv es ( 6 ). If i ∈ A ℓ , only the slice { X ℓ = i } contributes to the deriv ative with resp ect to g ℓi , so ∂ Φ ∂ g ℓi = p ℓi E − ℓ  U ′ ( g ℓi + R ℓ ( X − ℓ ))  . Stationarit y in the active v ariable g ℓi yields ( 7 ). If j / ∈ A ℓ , then g ℓj = 0 . The directional deriv ativ e of the ob jectiv e in the feasible direction g ℓj 7→ g ℓj + ε at ε = 0 + equals p ℓj E − ℓ  U ′ ( R ℓ ( X − ℓ ))  = p ℓj K ( U ) ℓ . The corresponding directional deriv ativ e of the Lagrangian is therefore p ℓj K ( U ) ℓ − λπ ℓj , and optimalit y implies ( 9 ). The common-factor claim follows directly from ( 8 ). 3 4 The threshold iden tit y F or a support family A , define the active probability and price masses of ev en t ℓ b y P ℓ,A := X i ∈ A ℓ p ℓi , Q ℓ,A := X i ∈ A ℓ π ℓi . (10) If the outcomes of even t ℓ are sorted so that r ℓ 1 ≥ r ℓ 2 ≥ · · · ≥ r ℓn ℓ , and A ℓ = { 1 , . . . , k ℓ } is a prefix, we abbreviate P ℓ,k ℓ := X i ≤ k ℓ p ℓi , Q ℓ,k ℓ := X i ≤ k ℓ π ℓi . Theorem 3 (Utility-in v arian t threshold iden tity) . Fix a supp ort family A = ( A 1 , . . . , A m ) , and let ( c, g ) b e optimal for the r estriction of ( 3 ) to p ortfolios supp orte d on A . A ssume c > 0 . Then for every event ℓ , λ K ( U ) ℓ = 1 − P ℓ,A 1 − Q ℓ,A , (11) wher e λ is the multiplier fr om Pr op osition 2 and K ( U ) ℓ is define d by ( 8 ) . Pr o of. Fix an ev ent ℓ . Summing the active stationarit y equations ( 7 ) ov er i ∈ A ℓ giv es λQ ℓ,A = X i ∈ A ℓ p ℓi E − ℓ  U ′ ( g ℓi + R ℓ ( X − ℓ ))  . (12) On the other hand, conditioning the cash stationarity equation ( 6 ) on X ℓ yields λ = X i ∈ A ℓ p ℓi E − ℓ  U ′ ( g ℓi + R ℓ ( X − ℓ ))  + X j / ∈ A ℓ p ℓj E − ℓ  U ′ ( R ℓ ( X − ℓ ))  = X i ∈ A ℓ p ℓi E − ℓ  U ′ ( g ℓi + R ℓ ( X − ℓ ))  + (1 − P ℓ,A ) K ( U ) ℓ . (13) Substituting ( 12 ) in to ( 13 ) gives λ = λQ ℓ,A + (1 − P ℓ,A ) K ( U ) ℓ , whic h rearranges to λ (1 − Q ℓ,A ) = (1 − P ℓ,A ) K ( U ) ℓ . Dividing by K ( U ) ℓ > 0 pro v es ( 11 ). Corollary 4 (Ev ent wise prefix supp ort and exact decoupling) . A ssume, in addition, that e ach event is sorte d so that r ℓ 1 ≥ r ℓ 2 ≥ · · · ≥ r ℓn ℓ . L et ( c, g ) solve the simultane ous pr oblem ( 3 ) with c > 0 . Then for e ach event ℓ ther e exists an inte ger k ℓ ∈ { 0 , 1 , . . . , n ℓ } such that g ℓi > 0 for i ≤ k ℓ , g ℓi = 0 for i > k ℓ . 4 Mor e over, r ℓ,k ℓ +1 ≤ 1 − P ℓ,k ℓ 1 − Q ℓ,k ℓ ( k ℓ < n ℓ ) , (14) and the exact supp ort of event ℓ is the same supp ort obtaine d by solving the c orr esp onding single-event pr oblem with the same pr ob abilities and pric es. Henc e the simultane ous exact supp ort is the eventwise union of the single-event supp orts. Pr o of. Let A ℓ := { i : g ℓi > 0 } . If i ∈ A ℓ , then g ℓi > 0 , so strict concavit y implies U ′ ( g ℓi + R ℓ ( X − ℓ )) < U ′ ( R ℓ ( X − ℓ )) for every X − ℓ . T aking exp ectations and using ( 7 ) and ( 8 ), λπ ℓi p ℓi = E − ℓ  U ′ ( g ℓi + R ℓ ( X − ℓ ))  < K ( U ) ℓ . Th us ev ery active outcome satisfies r ℓi > λ K ( U ) ℓ . (15) F or every inactiv e outcome j / ∈ A ℓ , the reduced-cost inequalit y ( 9 ) gives r ℓj ≤ λ K ( U ) ℓ . (16) Hence, after sorting b y decreasing r ℓi , the activ e set of ev ent ℓ m ust b e a prefix { 1 , . . . , k ℓ } . Applying Theorem 3 to that prefix support yields λ K ( U ) ℓ = 1 − P ℓ,k ℓ 1 − Q ℓ,k ℓ , so ( 16 ) for the next inactive outcome b ecomes exactly ( 14 ) . This is the same stopping rule as in the single-ev ent problem, so support disco very decouples ev ent wise. Corollary 5 (Single-ev ent utilit y-inv ariant supp ort) . F or m = 1 , the optimal active supp ort is the gr e e dy pr efix determine d by r k +1 ≤ 1 − P k 1 − Q k , indep endently of the admissible utility U . Pr o of. When m = 1 , one has R 1 ≡ c and therefore K ( U ) 1 = U ′ ( c ) . Theorem 3 and Corollary 4 then yield the stated cutoff immediately . Remark 6 (One-dimensional reduction in the single-even t problem) . Onc e the single-event supp ort A is known, the active first-or der c onditions imply U ′ ( W i ) = λ r i ( i ∈ A ) , so W i = ( U ′ ) − 1  λ r i  ( i ∈ A ) . 5 θ A := 1 − P A 1 − Q A , The or em 3 gives λ/U ′ ( c ) = θ A , henc e c = ( U ′ ) − 1  λ 1 − Q A 1 − P A  . The budget c onstr aint b e c omes a single sc alar e quation for λ : (1 − Q A )( U ′ ) − 1  λ 1 − Q A 1 − P A  + X i ∈ A π i ( U ′ ) − 1  λ r i  = 1 . F or CRRA utility U ( w ) = w 1 − γ / (1 − γ ) , this e quation is explicit. Remark 7 (Log utilit y and CRRA as special cases) . F or U ( w ) = log w , The or em 3 r e c overs the supp ort thr eshold in L ong [ 2 ]. F or CRRA utility U ( w ) = w 1 − γ / (1 − γ ) with γ > 0 , the the or em shows that the active supp ort is unchange d acr oss the entir e CRRA family: risk aversion changes the active weights, but not the active set. Remark 8 (Boundary case c = 0 ) . When c = 0 , c ash stationarity b e c omes one-side d. Mor e pr e cisely, ther e is a slack term ν := λ − E  U ′ ( W ( X ))  ≥ 0 . R ep e ating the pr o of of The or em 3 gives λ (1 − Q ℓ,A ) = (1 − P ℓ,A ) K ( U ) ℓ + ν , so λ K ( U ) ℓ = 1 − P ℓ,A 1 − Q ℓ,A + ν (1 − Q ℓ,A ) K ( U ) ℓ . Thus the exact utility-invariant thr eshold identity is an interior phenomenon. This clarifies why exo genous exp osur e c onstr aints or fr actional-b etting c onstr aints gener al ly destr oy supp ort invarianc e. Remark 9 (Indep endence as the structural b oundary) . The pr o of uses indep endenc e at exactly one p oint: for an inactive outc ome j / ∈ A ℓ , the c ontinuation factor K ( U ) ℓ = E − ℓ [ U ′ ( R ℓ ( X − ℓ ))] must b e the same for every such j . Under dep endenc e, the c onditional law of X − ℓ given X ℓ = j varies with j , so differ ent inactive outc omes se e differ ent c ontinuation factors. The c onditioning identity ( 13 ) then no longer c ol lapses to a single sc alar thr eshold. Indep endenc e is ther efor e not a mer e c onvenienc e; it is the exact structur al b oundary of the the or em. Remark 10 (Strict vigorish and canonical uniqueness) . Supp ose every event has strict overr ound, n ℓ X i =1 π ℓi > 1 for every ℓ. Then the fair-event c ash-shift de gener acy is absent: one c annot move a c onstant amount fr om c ash into every outc ome of a single event without changing the budget. If, in addition, ther e ar e no exact thr eshold ties, then the eventwise pr efix supp ort in Cor ol lary 4 is unique. If e quality o c curs at a thr eshold, the natur al c anonic al tie-br e ak is to pr efer c ash, i.e. to tr e at e quality outc omes as inactive. 6 References [1] J. L. Kelly , Jr., A new in terpretation of information rate, Bel l System T e chnic al Journal 35 (1956), 917–926. [2] C. D. Long, Single-Ev en t Multinomial F ull Kelly via Implicit State Positions, 2026. [3] B. Rosner, Optimal allo cation of resources in a pari-mutuel setting, Management Scienc e 21 (1975), 997–1006. [4] P . Smo czynski and R. E. Miles, Utilit y-optimal infra-marginal pari-m utuel bets: Monotone utilities, 2025. [5] P . Smo czynski and D. T omkins, An explicit solution to the problem of optimizing the allo cations of a b ettor’s wealth when wagering on horse races, The Mathematic al Scientist 35 (2010), 10–17. [6] K. Whelan, F ortune’s F ormula or the Road to R uin? The Generalized Kelly Criterion With Multiple Outcomes, working pap er, 2023. [7] C. Whitrow, Algorithms for optimal allo cation of b ets on many sim ultaneous ev ents, Journal of the R oyal Statistic al So ciety: Series C (Applie d Statistics) 56 (2007), 607–623. 7