Generalized minority games with adaptive trend-followers and

contrarians A. Tedeschi,

A. De Martino, I. Giardina, M.Marsili

• Interaction of different types of agents in market

• N agents formulate a binary bid: (buy/sell)

• The quantity is the excess demand

• When is large/small the risk perceived by the

agents is high/low and they act as fundamentalists/trend-followers.

• If each agent is rewarded with a good choice is

Some initial considerations

1ia

i

iaN

tA1

)(

)(tA

AFap ii 3AAAF

• Contrarians/trend-followers are described by minority/majority game players (rewarded when acting in the minority/majority group)

• Our model allows to switch from one group to the other

• Trend-following behavior dominates when price movements are small, whereas traders turn to a contrarian conduct when the market is chaotic

Introduction

N

igia

NtA

1

~1

0igp

• Each time t, N agents receive an information Pt ,...,1

• Based on the information, agents formulate a binary bid (buy/sell)

• Each agent has S strategies mapping information into actions 1,1iga

• Each strategy of every agent has an initial valuation updated according to

tAFatptp igigig1

• The excess demand is where )(maxarg~ tpg igg

The Model

Our Model

• In minority game

• In majority game

• In our model 3)( AAAF

AAF

AAF )(

-3

-2

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3 4

The ε parameter

• ε is a tool to interpolate between two market regimes: agents change their conduct at some threshold value A* depending on ε

• This threshold value A* can be verified in real markets from order book data by reconstructing

where O=order and dR= price increment

• We neglect the time dependency of ε (being on much larger time scales than ours)

dROdRPi |)sgn(

The Observables

• Study of the steady state for of the valuation as a function of α=P/N

• The volatility (risk)

N

22 A

• The predictability (profit opportunities)2

| AH

• The fraction of frozen agents ϕ

• The one-step correlation 2

)1()(

tAtA

D

Numerical simulations: volatility

• Small ε: pure majority game behavior

• Increasing ε: smooth change to minority game regime

• ε going to infinity: minimum at phase transition for standard min game

Numerical simulations: predictability

• Increasing ε: H <1 at small α as in min game, H→1 for large α as in maj game

• No unpredictable regime with H=0 is detected at low α, even in the limit ε going to infinity

Numerical simulations: frozen agents

• For large α, one finds a treshold separating maj-like regime with all agents frozen from min-like regime where Φ=0

• For large ε, Φ has a min game charachteristic shape

• In the low α, large ε phase, agents are more likely to be frozen than in a pure min game

Theoretical estimate for the large α regime

• We can give a theoretical estimate (that fits with simulations at large α) of the crossover from min to maj regime.

• The ε crossover value can be computed considering that at large α agents strategies are uncorrelated and A(t) can be approximated with a gaussian variable.

• With these assumptions we analytically estimate the crossover value at ε=1/3 for α>>1 (in a consistent manner from both maj and min sides). Numerically we find ε≈0.37.

Numerical simulations: correlation

• For small ε, D is positive, so the market dynamics is dominated by trend-followers

• The contrarian phase becomes larger and larger as ε grows and, for ε>>1, the market is dominated by contrarians

Numerical simulations: probability distribution

• For α=0.05, the distribution of A(t) shows heavy tails. The distribution peak moves as 1/√ε: the system is self-organized around the value of A such that F(A)=0

• For α=2 and A not too large with a weak dependence on ε42)(log bAAAP

Numerical simulations: Single Realization

• Time series of the excess demand A(t): spikes in A(t) occur in coordination with the transmission of a particular infomation pattern

• Time series of price : we observe formation of sustained trends and bubbles

tl

lAtR )()(

Conclusions

• In our model, market-like phenomenology (heavy tails, trends and bubbles) emerges when the competiton between trend-followers and contrarians is stronger

• Further developments for real market models: grand-canonical extensions, real market history and time-dependent ε coupled to the system performance