Optimal Predi tion Ma rk ets with Optimal Pla y ers Lea rn Optimally fo r Log Loss John Langfo rd With Alina Beygelzimer and David P enno k
Predi tion Ma rk ets Lea rn Optimally John Langfo rd With Alina Beygelzimer and David P enno k
De�nitions Pla y er: Someone with an initial endo wment of Optimal Pla y er: A pla y er optimizing exp e ted log w ealth after after T rounds. Predi tion Ma rk et: A ma rk et fo r se urities that pa y $1 if an event o urs and $0 otherwise. Optimal Predi tion Ma rk et: A Predi tion ma rk et where the p ri e of the se urit y is su h that supply = demand. Lea rning Ma rk et: A seque e of ma rk ets where the se urit y p ri e has small regret in log loss with resp e t to all pla y ers.
(W ell kno wn) Optimal Pla y ers use Kelly Betting If w = urrent w ealth, ho w mu h should y ou b et? lose everything if y ou a re ever wrong never win anything. f = 1 ⇒ Kelly b etting sa ys: f = 0 ⇒ Whi h is optimal fo r maximizing exp e ted log w ealth. f ∗ = p − p m 1 − p m
(W ell Kno wn) Log loss regret optimized b y Ba y es Rule Supp ose y ou have exp erts { i } whi h mak e a p redi tion p it on round t . Ho w an y ou omp ete with the b est? Let w i = initial �p rio r� on exp ert i ( � ). Ba y es rule ⇒ w eight on exp ert i is: i w i = 1 where p mt = the w ealth w eighted average. � 1 − y t � y t � T � 1 − p it p it Theo rem: F o r all w i , fo r all sequen es of p it and y : � w i 1 − p mt p mt t =1 where L = log loss y ) + ln 1 L ( � y ) ≤ min L ( � p m , � p i , � w i i
(new) If every agent b ets a o rding to Kelly , w ealth is redistributed a o rding to Ba y es la w. If y = 0 , w ealth afterw a rds is . if y = 1 , w ealth afterw a rds is . 1 − p i 1 − p m w i No w, onne t the dots. p i p m w i
T o think ab out What happ ens when the ma rk et designer a res ab out other losses? What happ ens when the ma rk et pla y er a res ab out something other losses? Are ma rk et options immo ral?
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