Generalized minority games with
adaptive trend-followers and
contrarians
A. Tedeschi,
A. De Martino, I. Giardina, M.Marsili
Some initial considerations
• Interaction of different types of agents in market
• N agents formulate a binary bid: ai  1 (buy/sell)
• The quantity A(t ) 
1
N
a
i
is the excess demand
i
• When A(t ) is large/small the risk perceived by the
agents is high/low and they act as fundamentalists/trendfollowers.
• If each agent is rewarded with pi  ai F  A
choice is F  A  A  A3
a good
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
The Model
• Each time t, N agents receive an information  t  1,..., P
• Based on the information, agents formulate a binary bid (buy/sell)
• Each agent has S strategies mapping information into actions aig   1,1

• Each strategy of every agent has an initial valuation
updated according to
pig 0
pig t 1  pig t   aig F At 
1
• The excess demand is At  
N
N

a
 ig~ where g~  arg max g pig (t )
i 1
Our Model
• In minority game
F  A   A
• In majority game
F ( A)  A
• In our model F ( A)  A  A3
3
2
1
0
-4
-3
-2
-1
0
-1
-2
-3
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 Pi sgn( OdR) | dR
where O=order and dR= price increment

• We neglect the time dependency of ε (being on much
larger time scales than ours)

The Observables
• Study of the steady state for N   of the
valuation as a function of α=P/N
• The volatility (risk)  2  A2
• The predictability (profit opportunities) H  A | 
• The fraction of frozen agents ϕ
• The one-step correlation D 
A(t ) A(t  1)
2
2
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
log P( A)  A2  bA4 with a weak dependence on ε
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 R(t )   A(l ) : we observe formation of sustained
l t
trends and bubbles
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