Coordination, Intermittency and Trends in Generalized Minority

Games

A.Tedeschi, A. De Martino, I. Giardina

(to be published in Physica A)

• How the MG would change if agents were allowed to modify their behavior according to the risk they perceive?

• How the macroscopic properties of MG depend on the type of information supplied to the agents?

• 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

• Here we study the effects of this mechanism in different information structures: both the stationary macroscopic properties and the dynamical features strongly depend on wether the information supplied to the system is “random” or “real”

Introduction

We will study 3 different cases:

• Without information

• With exogenous information

• With endogenous information

The simplest model: without information• Let us consider the following setup: each of N agents (i=1, ..., N) must decide

at each time step whether to buy or sell. The success of agent i at time t is measured by the payoff:

Where is the excess demand and f encodes the type of game and the agents’ expectations.

• The simplest function that allows agents to switch from a trend-follower to a contrarian behaviour is :

where if B is true (and 0 otherwise) and L is a threshold, so that for | A(t) | < L agents perceive the game as a Majority Game, | A(t) | > L agents perceive the game as a Minority Game.

• In order to take decisions agents follow the rules

where is the learning rate of agents and the initial conditions are i.i.d. random variables with zero mean and variance 1.

)]([)()()( tAftAtat ii )()( tatA

ii

)()()( LxLxxf 1)( B

)(cosh2))((Pr

)(

tU

eataob

i

taU

i

i

NtAftUtU ii /)]([)()1(

Average and Second Moment of the Excess Demand

• In the pure MG (inset) <A>= 0, and for small L this scenario should dominate, since agents will be extremely risk-sensitive.

• For large L agents become more risk-prone and a Majority scenario (<A> 0) is expected.

• For small trends are formed (as in Majority Game) even for small L , whereas for large the typical excess demand retutns to zero.

• The second moment recalls the Majority Game behaviour for L/N=1 , and smoothly passes to a Minority Game regime for smaller L/N ( of order N for small and of order for larger ).

2A

2A 2N

Correlation

For intermediate values of excess demands are anticorrelated for low L whereas the correlation turns positive when L increseas, that is as agents becomes less and less risk-sensitive. The former regime is characteristic of contrarians and Minority Games, the latter is typical of trend-followers and Majority Games.

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

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

tAFatptp igigig1

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

The Model with information

• In minority game

• In majority game

• In our model

AAF

AAF )(

-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15

AAAAF )(

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

Case of random external information

• The information is a an integer drawn randomly and independently at each time step from with uniform probability.

• In this case the model is Markovian and the information dynamics covers uniformly the state space .

P,...,1

P,...,1

Numerical simulations: volatility and predictability

• Big : pure majority game behavior

• Decreasing : smooth change to minority game regime

• going to zero): minimum at phase transition for standard min game

• For small the system reproduces a min game (with the unpredictable phase), increasing one sees a clear crossover from min game like system to a maj game one, for large • H has the crossover from the fundamentalist to the trend-followers-dominated regime at =1.4.

Numerical simulations: frozen agents and correlation

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

• For small , has a min game charachteristic shape

• Notice that decreases as decreases: as agents become more sensitive to risk it becomes more difficult for them to identify an optimal strategy.

One-step correlation has the same shape:• For big , agents become more risk prone, so D is positive and the market dynamics is dominated by trend-followers• The contrarian phase becomes larger and larger as decreases and, for α <<1, the market is dominated by contrarians

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 information pattern for long time stretches and then quiesce.

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

tl

lAtR )()(

Case of Endogenous Information• The information pattern encodes in its binary

representation the string of the last m losing actions (the “history”) of the market:

(l=1,....,m) so that . The information dynamics in this case is

deterministic and reads if A(t)>0 if A(t)<0

))(sgn( ltA mP 2

)(t

Ptt mod]1)(2[)1( Ptt mod)](2[)1(

• In this case trend-following behavior is expected to strongly influence the macroscopic properties, because of the bias that trends would impose on the resulting history dynamics: we therefore have to calculate the frequency with which each string is generated in the steady state.

• So we have to modify the predictability

• And we have to introduce the entropy

as a measure of the bias (or of the information content).

Obviously, in the random case: and S=1.

)(

P

1)(

2|)(

AH

)(log)( PS

Numerical simulations: volatility and predictability

Already for small values of the behaviour deviates greatly from the case of endogenous inforrmation:

• while there is no evidence of a simmetric regime, fluctuations for small are much smaller than in the MG-regime.

• A detailed analysis of the volatility as a function of for small suggests, that as soon as agents allow for a small amount of risk-proness, fluctuations decrease sharply.

• This effect, together with the small predictability, indicates that efficient states can be reached.

As increases further, the game became a standard Majority Game.

Numerical simulations: frozen agents and correlation

• For small almost all agents are frozen for the interesting values of , so that even individual agents profit for a small greediness.

• In striking contrast to the observations made for the model with exogenous information, when increases tends to a minority-like behaviour.

• Also the results for the autocorrelaction function D confirm that a very small is sufficient to induce strong herding effects for small , at odd with the previous case.

• For small the scenario of a pure MG is reproduced: the entropy S=1 for and S <1 for . In this case one can see the two different regimes plotting

.

• As increases the entropy drops seriously for small as herding trivializes the history dinamics: only a small fraction of histories are generated per sample.

c

c

)]([1)( PfPfQ

Numerical simulations: entropy

Numerical simulations: Single Realization

For small α and small one observes volatility clustering in the time series of A(t):

• Periods of low volatility correspond to a trend, as shown to the distribution Q(f) relatives to those intervals. The game has the character of a majority game and trend-followers dominate .

• Periods of high volatility correspond instead to a chaotic dynamics where the history frequency distribution is uniform. Here fundamentalists dominate.

• Also studying , the time series of prices, one can clearly see the trend-dominate phase and the chaotic period.

tl

lAtR )()(

Conclusions

• The simple microscopic mechanism introduced above, when coupled to real information, determines significant effects in the macroscopic properties and produces realistic features: creation and destruction of trends and volatility clustering.

• WORK IN PROGRESS:

- Analytical solution

- Risk-threshold ( ) fluctuating in time, coupled to the system performance.