RESEARCH ARTICLE MOELING FIXE OS BETTING FOR FUTURE EVENT PREICTION Weyun Chen eartment of Educatona Informaton Technoogy, Facuty of Educaton, East Chna Norma Unversty, Shangha, CHINA {weyun.chen@qq.com} Xn L eartment of Informaton Systems, Coege of Busness, Cty Unversty of Hong Kong, Hong Kong, CHINA {xn..hd@gma.com} ane Zeng eartment of Management Informaton Systems, Unversty of Arzona, Tucson, AZ 8572 U.S.A., and State Key Laboratory of Management and Contro for Comex Systems, Insttute of Automaton, Chnese Academy of Scences, Beng, CHINA {zeng@ema.arzona.edu} Aendx Theorem : If α ()= β (-), then f(x;,θ) = f(-x;-,θ). Proof: λ f( x;, θ ) = Beta x;, = λ = = ( α β ) λ = Beta x = λ ( - ; β, α ) λ = Beta ( - x; α, ) ( ;, ) β = f x θ = λ = # Theorem 2: E[Beta(x; α (),β ())] s strcty ncreasng wth. Proof: α E[Beta(x; α (),β ())] = = α + β β + α Snce s wthn (0, ), both α () and β () are ostve. MIS Quartery Vo. 4 No. 2 Aendx/June 207 A
Chen et a./modeng Fxed Odds Bettng β β α α Z Z h h α ( A) β α u uh β u + uh A = h= 2 h= 2 = α α β Ceary, α and the equa sgn hods ony f a u h =0 when s wthn range (0, ). 0 If a u h =0, both α () and β () equa 0, whch s n confct wth our assumtons. β () α Thus s strcty decreasng n, and s strcty ncreasng wth. # α() α + β Theorem 3: If arameters q,, ρ, γ, τ, o A, o B are a ostve, there exsts and ony exsts one beef vaue c 0 (0,), caed baance beef hereafter, satsfyng U(c, o A ) = U( c, o B ). Proof: When q,, ρ, γ, τ are ostve, both w + ( ) and w - ( ) are strcty ncreasng functons. Accordngy, the utty functon U(x, o) s a strcty ncreasng functon and U( x,o) s a strcty decreasng functon n beef x. Gven o A > 0 and o B > 0, [U(x, o A ) U(-x, o B )] s strcty ncreasng. It s easy to verfy that U(x = 0, o A ) < 0 < U(x =, o B ) and U(x =, o A ) > 0 > U(x = 0, o B ). Thus, [U(x, o A ) U(-x, o B )] < 0 for x = 0 and [U(x, o A ) U( x, o B )] > 0 for x =. As such, there must exst one and ony one baance beef x = c, satsfyng U(c, o A ) = U( c, o B ). # Theorem 4: For suffcenty arge m, maxmzng equaton 7 reduces to sovng PA(,θ) = s A. Proof: Lc(,θ) n equaton (7) s contnuous and dfferentabe. Snce 0 < <, the vaue of that maxmzes Lc(,θ), f t exsts, must satsfy the Lc(, θ) frst-order condton = 0. Lc(, θ ) PA(, θ) PA(, θ) = ms A + m( sa) + R + ( R) PA(, θ) PA(, θ) ( s ) ( R ) PA(, θ ) s A A R = m - + - PA(, θ) PA(, θ) ( θ ) θ ( θ ) PA(, ) (, ) - (, ) θ sa PA sa PA θ R R = m + PA(, ) PA(, ) PA(, θ) sa- PA(, θ) R = m + PA(, θ) PA(, ) Namey: [ PA θ s ] ( θ ) =0 ( θ )( ) ( ) PA(, ) (, ) (, ) θ PA θ PA R (, )- A = m A2 MIS Quartery Vo. 4 No. 2/June 207
Chen et a./modeng Fxed Odds Bettng ( θ )( ) ( ) PA(, ) (, ) (, ) θ PA θ PA R m [ PA(, θ )- sa] = m =0 + m + m m PA(, θ) Accordng to Theorem 4, >0, we obtan m [ PA(, θ )- sa] =0. # m + Theorem 5: I c (α (),β ()) s strcty decreasng n, where I c (α (),β ()) s the reguarzed ncomete Beta functon 0 c B eta ( x; α, β ) dx. Ic( α, β ) Ic( α, β ) α Ic( α, β) β Proof: Based on the chan rue of mutvarabe cacuus, = +. α β k I c ( α, β ) Γ( α) Γ ( α + β ) α ( α) k( α) k( β ) kc = [og( c) ϕ ( α) + ϕ ( α + β )] Ic ( α, β ) c α Γ( β ) k! Γ ( k + + α ) Γ ( k + + β ) where ( ) k s the Pochhammer symbo secfed as (x) 0 = ; (x) n = x(x + )(x + 2) (x + n-). Snce ϕ( α) = ( ) r, ϕ ( α + β) ϕ ( α) = ( ) < 0 when α > 0 and β > 0. k = k k + α k = k + α + β k + α Snce c<, og (c) < 0 and [og(c) φ(α) + φ(α + β)] < 0. Snce I c (α,β) > 0 f 0 < c <, we have [og(c) φ(α) + φ(α + β)]i c (α, β) < 0. k = 0 Snce Γ(x)>0 f x>0, we have k Γ( α) Γ ( α + β) α ( α) k( α) k( β) kc c > 0 Γ( β) k! Γ ( k + + α) Γ ( k + + β) k = 0 I Thus, ( c α, β ) < 0 when 0 < c<. α I Smary, we can rove ( c α, β ) > 0 when 0 < c <. β α β It s cear > 0 and < 0 when 0 < <. Ic( α, β ) Thus, < 0,.e., I c (α(),β()) s strcty decreasng n. # Formua for AIC The vaue of AIC crtera s comuted as AICc = 2t + 2 t( t + )/( N t ) 2Lc θ where N denotes the number of data nstances and t denotes the number of arameters, whch s ( + Z) > t = Z = accordng to equatons () and (2). MIS Quartery Vo. 4 No. 2 Aendx/June 207 A3
Chen et a./modeng Fxed Odds Bettng Inut: {o A, o B, R, s A, o B,}, 0 {,, H}, k max, ε. Outut: estmated otma arameter θ*.. Comute baance beef c, c 2,, c H for each bettng game accordng to (9), usng a root fndng agorthm. 2. Intaze a smex SX whch conssts of J + sets of arameters θ n the arameter sace Θ, where J s the dmenson of the arameter sace. 3. Whe: 4. For each θ : 5. Sove A from each bettng accordng to (0) usng a root fndng agorthm. 6. Comute the Lc( ) accordng to (2). ( max( Lc ) mn( ) ) * θ Lc * θ ε 7. If or IteratonCount < k max : 8. Udate vertces n SX usng Refect, Exand, Outsde contracton, Insde contracton, or Shrnk oeratons. 9. If oeraton <> Shrnk : IteratonCount = IteratonCount + 0. Ese: *. Return the arameter set corresondng to max( Lc ( θ )) as θ*. Fgure A. Maxmum Lkehood Estmaton Usng a Neder Mead Method etaed Beef strbuton Estmaton Procedure Sna 2008 Oymc Games ataset For each settng of and Z, varyng from to 3, resectvey, we numercay obtaned the otma arameters θ* 0 Θ. Tabe A reorts the og kehood and AIC vaues for these modes. Generay, the mode s kehood converges when and Z are arger than 2. The mode wth = 2 and Z = 2 s the mode wth the maxmum kehood. The estmated beef dstrbuton functon s gven as: f( x; ) = 0.74* Beta( x;3.76* +0.*,3.76*( )+0.*( ) ) + 0.26* Beta( x;0.* + 66.32*,0.*( ) + 66.32*( ) ) For the AIC crtera, we combned the three comonents wth smaest AICc ( =, Z = ; =, Z = 2; and =, Z = 3). The estmated beef dstrbuton functon s gven as f ( x; ) = 0.67* Beta( x;5.98*,5.98*( )) + + + + 0.09* Beta( x;5.95* + 0.02*,5.95*( ) + 0.02*( ) ) 0.24* Beta( x;5.69* 2 0.6*,5.69* 0.6*( 2 ) ) 3 3 Tabe A. Log Lkehood and AIC Vaues (Sna 2008 Oymc Games) Z = Z = 2 Z = 3 Log kehood AIC Log kehood AIC Log kehood AIC -0.04.26-0. 206.29-0.08.38 2-0.03 20.30-00.79 24. -0.05 29.0 3-0.03 24.59-00.80 220.74-00.827.67 A4 MIS Quartery Vo. 4 No. 2/June 207
Chen et a./modeng Fxed Odds Bettng Sohu Entertanment ataset Tabe A2 shows the resuts on the Sohu entertanment event dataset, varyng and Z from to 3. The mode ( = 3 and Z = 3) s the mode wth the maxmum kehood. The estmated beef dstrbuton functon s as foows: f x Beta x 2 3 2 3 ( ; ) = 0.84* ( ;0.* +0.* + 000*,0.*( )+0.*( ) + 000*( ) ) + + + + + 2 3 2 3 0.6* Beta( x;7.54* 0.* 0.*,7.54* 0.* 0.* ) For the AIC crtera, we combned the three comonents wth smaest AIC ( =, Z = and =, Z = 2 and =, Z = 3). The estmated beef dstrbuton functon s gven as f ( x; ) = 0.62* Beta( x;2.28*, 2.28*( )) + 0.29* Beta( x;.29* + 3.3*,.29*( ) + 3.3*( ) ) 2 3 2 3 + 0.09* Beta( x; + 2.6* +,( ) + 2.6*( ) + ( ) ) Tabe A2. Lkehood and AIC Vaues (Sohu Entertanment) Z = Z = 2 Z = 3 Log kehood AIC Log kehood AIC Log kehood AIC -22.56 47.25-22.20 48.79-22.20 5.20 2-22.26 53.90-2.32 57.75-2.32 64.40 3-22.26 59.63-2.32 68.4-2.20 8.26 Sohu 204 FIFA ataset Tabe A3 shows the resuts on the Sohu 204 FIFA dataset, varyng and Z from to 3. The mode ( = and Z = ) s the mode wth the maxmum kehood. The estmated beef dstrbuton s gven as ( ; ) = ; 428., 428. ( ) f x Beta x For the AIC crtera, we combne the three comonents wth smaest AIC ( =, Z = ; =, Z = 2; and =, Z = 3). The estmated beef dstrbuton functon s gven as f ( x; ) = 0.67* Beta( x;42.8*, 42.8*( )) + 0.24* Beta( x;4.88*, 4.88*( )) + 0.09* Beta( x;4.98*,4.98*( )) Tabe A-3: Lkehood and AIC Vaues (Sohu 204 FIFA) Z = Z = 2 Z = 3 Log kehood AIC Log kehood AIC Log kehood AIC -9.36 240.74-9.36 242.79-9.36 244.86 2-9.36 246.95-9.36 25.2-9.36 255.56 3-9.36 25.2-9.36 257.78-9.36 264.59 MIS Quartery Vo. 4 No. 2 Aendx/June 207 A5
Chen et a./modeng Fxed Odds Bettng Wth the estmated beef dstrbuton functon, we can cacuate the ercentage of eoe who consder the event may haen f there are no odds (or equa odds on both sdes) n the redcton market, as ustrated n Fgure A2. In a three datasets, the bettors beefs are more extreme than the actua event robabty. If the event robabty s ess than 0.5, the fna bet rato w be ower than the event robabty. If the event robabty s hgher than 0.5, the fna bet rato s hgher than event robabty. Peoe wth a Beef >.5 Sna 2008 Oymc Games Sohu Entertanment Sohu 204 FIFA Fgure A2. Bettor Beef Wthout Odds Infuence A6 MIS Quartery Vo. 4 No. 2/June 207