1. Generalized minority games with adaptive trend-followers and contrarians A. Tedeschi, A. De Martino, I. Giardina, M.Marsili

2. 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

3. 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

4. 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 )(maxarg ~ tpg igg  The Model

5. Our Model In minority game In majority game In our model 3 )(AAAF 

6. 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)

7. 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 agents ϕ The one-step correlation

8. 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

9. 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

10. 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

11. 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.

12. 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

13. 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 ε

14. 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

15. 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