coordination, intermittency and trends in generalized minority games a.tedeschi, a. de martino, i....
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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.