Cycles, determinism and persistence in agent-based games and financial time-series

J. B. Satinover; D. Sornette

paper

Cycles, determinism and persistence in agent-based games and financial time-series

arXiv:0805.0428

Abstract

The Minority Game (MG), the Majority Game (MAJG) and the Dollar Game ( $G) a r e im p or t an t an d c l ose l y - r e l a t e d v er s i o n so f ma r k e t - e n t r y g am es d es i g n e d t o m o d e l d i f f er e n t f e a t u r eso f r e a l - w or l df inan c ia l ma r k e t s . I na v a r ian t o f t h ese g am es, a g e n t s m e a s u r e t h e p er f or man ceo f t h e i r a v ai l ab l es t r a t e g i eso v er a f i x e d - l e n g t h r o l l in g w in d o w o f p r i or t im e - s t e p s . T h ese a r e t h eso - c a l l e d T im eH or i z o n M G / M A J G /$ G (THMG, THMAJG, TH $G) . T h e i r p r o babi l i s t i c d y nami cs ma y b eco m pl e t e l y c ha r a c t er i z e d in M a r k o v - c hain f or m u l a t i o n . G am eso f b o t h t h es t an d a r d an d T H v a r ian t s g e n er a t e t im e - ser i es t ha t ma y b e u n d er s t oo d a s a r i s in g f r o ma s t oc ha s t i c a l l y p er t u r b e dd e t er mini s mb ec a u se a co in t oss i s u se d t o b r e ak t i es . T h e a v er a g eo v er t h e bin o mia l l y - d i s t r ib u t e d co in - t ossesy i e l d s t h e u n d er l y in g d e t er mini s m . I n or d er t o q u an t i f y t h e d e g r eeo f t hi s d e t er mini s man d o f hi g h er - or d er p er t u r ba t i o n s, w e d eco m p ose t h es i g n o f t h e t im e - ser i es t h ey g e n er a t e (ana l o g o u s t o ama r k e tp r i ce t im eser i es) in t o a s u p er p os i t i o n o f w e i g h t e d H ami l t o nian cy c l eso n g r a p h s (e x a c tl y in t h e T H v a r ian t s an d a pp r o x ima t e l y in t h es t an d a r d v er s i o n s) . T h ecy c l e d eco m p os i t i o na l so p r o v i d es a ‘‘ d i ssec t i o n^{''} o f t h e in t er na l d y nami cso f t h e g am es an d a q u an t i t a t i v e m e a s u r eo f t h e d e g r eeo f d e t er mini s m . W e d i sc u ss h o w t h eo u tp er f or man ceo f s t r a t e g i esr e l a t i v e t o a g e n t s in t h e T H M G (t h e ‘‘ i l l u s i o n o f co n t r o l^{''}) an d t h er e v er se in t h e T H M A J G an d T H$ G (i.e., genuine control) may be understood on a cycle-by-cycle basis. The decomposition offers as well a new metric for comparing different game dynamics to real-world financial time-series and a method for generating predictors. We apply the cycle predictor a real-world market, with significantly positive returns for the latter.

58 page including 16 figures and 9 tables