Coordination, Intermittency, and Trends in Generalized Minority Games: The Influence of Information

1. 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?

2. Introduction • 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”

3. We will study 3 different cases: • Without information • With exogenous information • With endogenous information

4. 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.

5. 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 ).

6. 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.

7. ~ = g arg max p ( t ) g ig The Model with information • Each time t, N agents receive an information • Based on the information, agents formulate a binary bid (buy/sell) • Each agent has S strategies mapping information into actions • Each strategy of every agent has an initial valuation updated according to • The excess demand is where

8. In minority game • In majority game • In our model

9. The Observables • Study of the steady state for of the valuation as a function of α=P/N • The volatility (risk) • The predictability (profit opportunities) • The fraction of frozen agentsϕ • The one-step correlation

10. Case ofrandom 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 .

11. 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.

12. 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

13. 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 .

14. 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

15. 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.

16. 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.

17. 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.

18. Numerical simulations: entropy • 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.

19. 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.

20. 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.