Optimal Predition Ma rk ets with Optimal Pla y ers Lea rn - - PowerPoint PPT Presentation

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Optimal Predition Ma rk ets with Optimal Pla y ers Lea rn - - PowerPoint PPT Presentation

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Optimal Predition 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 Predition Ma rk ets Lea rn Optimally John Langfo rd With Alina

slide-1
SLIDE 1 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
slide-2
SLIDE 2 Predi tion Ma rk ets Lea rn Optimally John Langfo rd With Alina Beygelzimer and David P enno k
slide-3
SLIDE 3 Denitions Pla y er: Someone with an initial endo wment
  • f
Optimal Pla y er: A pla y er
  • ptimizing
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
  • urs
and $0
  • therwise.
Optimal Predi tion Ma rk et: A Predi tion ma rk et where the p ri e
  • f
the se urit y is su h that supply = demand. Lea rning Ma rk et: A seque e
  • f
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.
slide-4
SLIDE 4 (W ell kno wn) Optimal Pla y ers use Kelly Betting If w = urrent w ealth, ho w mu h should y
  • u
b et?

f = 1 ⇒

lose everything if y
  • u
a re ever wrong

f = 0 ⇒

never win anything. Kelly b etting sa ys:

f∗ = p−pm

1−pm

Whi h is
  • ptimal
fo r maximizing exp e ted log w ealth.
slide-5
SLIDE 5 (W ell Kno wn) Log loss regret
  • ptimized
b y Ba y es Rule Supp
  • se
y
  • u
have exp erts {i} whi h mak e a p redi tion pit
  • n
round t . Ho w an y
  • u
  • mp
ete with the b est? Let wi = initial p rio r
  • n
exp ert i (

i wi = 1

). Ba y es rule ⇒ w eight
  • n
exp ert i is:

wi

T

  • t=1
  • pit

pmt

yt

1 − pit 1 − pmt

1−yt

where pmt = the w ealth w eighted average. Theo rem: F
  • r
all wi , fo r all sequen es
  • f pit
and y :

L( pm, y) ≤ min

i

L( pi, y) + ln 1 wi

where L = log loss
slide-6
SLIDE 6 (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

1−pi 1−pmwi

. if y = 1 , w ealth afterw a rds is

pi pmwi

. No w,
  • nne t
the dots.
slide-7
SLIDE 7 T
  • think
ab
  • ut
What happ ens when the ma rk et designer a res ab
  • ut
  • ther
losses? What happ ens when the ma rk et pla y er a res ab
  • ut
something
  • ther
losses? Are ma rk et
  • ptions
immo ral?