Minority Game

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Introduction to Minority Game

Introduction [1] Most economics theories are deductive in

origin - Each participant knows what is best for him

given that all other participants are equally intelligent in choosing their best actions

In the real world the actual players do not have the perfect foresight, most often their actions are based on trial-and-error inductive thinking

(continued) El Farol problem [2]( Arthur, 1994) - Approach using bounded rationality and inductive reason

ing - Minority game is originally inspired by the problem - About “the equilibrium” Minority Game [3]( Challet, Zhang, 1997) - A simple evolutionary game with a large number of player

s, i.e. a statistical system - About “fluctuations”

Minority Game (MG)original version (Challet, Zhang, 1997) [1], [3]

Definition - A minority game is a repeated game where N

(odd) players have to choose one out of two alternatives (say A and B) at each time step. Those who happen to be in the minority win.

- They base their decision only on the knowledge of the M (M for memory) last winning alternatives, called histories ; there are

histories. => The players are limited in their analyzing

power

M2

(continued)

Strategy - A strategy is defined to be the

next action (to be in A or B) given a specific signal’s M bits.

- Each player has a finite set of strategies.

- Total number of strategy for M bits signal is

M22

(continued) Virtual value (virtual points) - Each strategy has an intrinsic value, called

virtual value, which is the total number of times the strategy has predicted the right alternative

Inductive reasoning - At the beginning of the game, every player gets

a limited set of S strategies ; he uses them inductively, that is he uses the strategy with the highest virtual value (ties are broken by coin tossing).

Emergence of cooperation and organization in an evolutionary game

D. challet, Y. –C. Zhang, Physica A 246 (1997) 407-418

[3]

“Intelligent” players share the limited resources

Actual number of attendance at side A against time, for a population of 1001 players, having memory size of (a) 6, (b) 8 and (c) 10 bits.

=> the more memory, the less fluctuation (i.e. waste)

No “risk free way” Playing randomly - In a perfect timing, the average gain in the population

would be ½ per play. Waste is proportional to fluctuation’s amplitude, hence the average gain is always below ½ in reality

To stay at A or B hoping to get exactly ½ gain - If this strategy indeed rewards ½ gain on average,

many would imitate. This makes the system be no longer symmetry. The apparent advantage disappears

=>In finance, any obvious advantage will be arbitraged away

Advantage of memory size The more intelligent

players gain more and the spread between the rich and the poor is smaller.

Above a certain size (M 6) the average performance of the population appears to saturate due to the simple structure of the game, i.e. there is no more to gain.

The symmetry for A and B

What happens with a bigger “idea bag”

In general, the more strategies, the worse the players perform.

In an evolutionary game players tend to specialize in a single strategy, even though alternatives exist. (Borkar et al 1997 )

(continued)

The general tendency that the oftener one switches, less successful one would end up.

With many strategies, the players tend to switch strategies often and are more likely to get “confused”

The rich get rich, The poor get poor

(continued)

The gap between the rich and the poor appears to increase linearly with time, though reversion is possible but the poor players in general are doomed to stay poor.

=> Are there really good and bad strategies?

Shown Fig.8, all strategies are equivalent in the t limit. Indeed, it can be analytically shown that all the strategies are equivalent to each other, since the game is symmetrical in A and B.

So the bad players are bad because they have used the strategies inopportunely and are unlucky.

MG generalized to include the Darwinist selection

Darwinist selection - The worst player is replaced by new one after a

finite time steps, the new player is a clone of the best player. (i.e. Being all the strategies same, but virtual values are zero)

Mutation possibility in cloning (for diversity) - One of the strategies of the best players can be

replaced by a new one.

(continued)

(continued) “Learning” has emerged - As shown Fig. 9, the “learning” has emerged in time.

Fluctuations are reduced and saturated, this implies the average gain for everybody is improved but never reaches the ideal limit.

What would happens if no mutations is allowed? - Eventually, population is full of the clone copies of the

best player, each may still differ in their decision since the virtual value in their strategies can be different. But as shown Fig. 10, the performance is very bad.

Memory size and evolution Simulation setup - Starting the population very “simple-minded”, say

M=2, and allowing in the cloning process mentioned previous an additional feature that a bit of memory can be added or subtracted for the cloned new player, with a small probability

“Arm race”- As shown Fig. 11, there is an “arm race” among the

players. However, such an evolution appears to saturate and the “arm race” to settle more at a given level.

(continued)

Real Vs Random histories

what difference does it make?

Introduction The MG has a well-defined deterministic

time evolution, which only depends on the initial distribution of strategies and on the random initial string of m bits necessary to start the game. [3,4]

Variance of attendance A in a given room

t

tt

NtAdt

t0

)2

)'((lim2

2 '1

(continued) The properties of (called volatility in a

finance context) - The larger is, the larger the global waste of

resources by the community of agents. From a financial point of view, it is clear that a low volatility is of great importance in order to minimize the risk.

- If all the agents were choosing randomly, the variance would simply be .

- There is a regime where is smaller than the random value . [5]

4/2 Nr

r

(continued) The memory m is a crucial quantity for the two

following reasons. - First, from a geometrical point of view, m defines the

dimension of the space of strategies and thus it is related to the probability that strategies drawn randomly by different agents could give similar predictions.

- Second, m is supposed to be a real memory. Actually, the whole game is constructed around the role of m as a memory. This memory role of m complicates the nature of the problem, since it induces an explicit dynamical feedback in the evolution of system.

The memory of the agents is irrelevant [4]

Simulation setup - Consider the same model used until now (i.e.

original MG), but with the following important difference.

- At each time step, the past history is just invented; that is, a random sequence of m bits is drawn, to play the role of a fake time history.

=> Simulation shows whether there is any need for an explicit time feedback in order to obtain all the distinctive features of the model.

(continued) The variance as a

function of m - For both the case with

and the case without memory, the two models give the same results, not only qualitatively, but also quantitatively

(continued) The dependence of the

whole function on the individual number of strategies s

- As shown Fig. 2, the larger the value of s is, the shallower the minimum of the curve is.

- The same phenomenon occurs for the model with memory [6]

)(m

)(m

(continued) Once s is fixed, let be the

value of m where the minimum of occurs. In [5], it has been pointed out that for the variance grows as N, where N is the number of agents, while for

grows as . In Fig. 3, the same behavior

as in the model with memory is found.

cm

)(m

cmm

cmm 2/1N

summary There is a regime where the whole population of

agents still behaves in a better-than-random way, even if the information they process is completely random.

The behavior of the volatility depend on the size, not on the history, of the memory.

The crucial thing is that everyone must posses the same information.

- Indeed, if inventing a different past history for each different agent, no coordination emerges at all and the results are the same as if the agents were behaving randomly.

Relevance of memory in the minority game

Introduction - For more generalized MG (e.g. heterogeneous age

nts), the irrelevance of memory in the MG is wrong. Quantities of the system are not indep-endent of the “history” [7,8]

Phase transition of the MG with memory

Basic terminology- is the system’s history at time t.- is the strategy of the agent i at time t- where is the decision of the

agents which has +1 or -1.- is the temporal average of the A(t) condition

al to for all .- , ,

N

i tsi tatAi1 )(, )()( )()(, ta tsi i

)(t

)(tsi

A )(t

2 AH 22 A NP /

(continued)

The Irrelevance is right as long as the system is in the symmetric phase.[9]

(continued)

The MG under goes a second order phase transition with symmetry breaking as the control parameter

where is the number of history of m memory

The system is in the symmetric phase( for all

) if and it is in the asymmetric phase for .

NP / MP 2

0 A

c c

Why the volatility is same for both schemes? [10]

The volatility is insensitive to the memory - since the volatility is a statistical quantity that does

not contain an temporal information, it is insensitive. - Therefore, it is important to figure out how the history

or memory affects the outcome of the game. The frequency of selecting entries - The volatility depends on how many times the agents

use each entry of strategies during the game. - Therefore, a relevant quantity is the frequency of

selecting entries in both schemes.

(continued)

Simulation setup - Simulating the game with m=2 and focus on an

entry ((0,0),*), where the history column is (0,0) and * in the prediction column can be either 0 or 1.

- Keeping track of the number of time steps between successive selections of the same entry during the game.

(continued)

Fig.1. Probabilities of the number of time steps for the same entry to be selected again, for s=2, m=2, and N=101. In Fig. (a), the ordinate is logarithmic scales, and in Fig. (b) the “prediction” column (0,0) is chosen for the simulation.

(continued) From previous Fig. 1., the probability distribution of the ti

me steps for selecting the same entries was different from which scheme was used.

Calculating the expectation values of the time step for two schemes, , the expectation values are same for two scheme

The average value for both schemes are the same, even though their probability distributions are different. That is, on the average, the frequency of choosing an entry is the same whether the game utilizes the mem-ory or not. This implies the same volatility for both.

t

ttPT )(

Is memory in the MG irrelevant?

The power spectrum of time series - The power spectrum of time series is composed of the

number of time steps between successive selections of the same entry. This data was also used to generate the probability distribution of Fig. 1(b).

- The result of power spectrum is shown Fig. 2, in which the frequencies f is normalized such that , so the following relation holds:

where is the corresponding periodic time step.

10 f

ptf

t p1

(continued)

(continued) The power spectrum has a peak at ,thus perio-dicit

y of the time series . From Fig. 1.(b), we can interpret the time steps for selecting the same entry “oscillate” between and ( on the average sense ), with period of

for The period of the oscillation is independent of the choice

of an entry and depends only on the size m

2/1f2pt

11 t 72 t

82 121 mtt 2m

(continued)

(continued) In the case of without memory - Power spectrum shows only the background noise

or randomness, which implies that there is no periodicity, thus no temporal correlation.

In the case of with memory - There are three peaks with distinguishable

strength.

Conclusion The difference between with memory and without memory

is whether the periodic structure exists or not. The memory is irrelevant for the emergence of coord-inate

behavior. The role of memory in the game was responsible for gene

rating a periodic response. Therefore, as long as we are interested in the volatility, the memory in the game is irrelevant. Because the volatility is a statistic-al quantity insensitive of temporal information such as a periodicity.

MG and Financial markets

“Stylized facts of financial markets and market crashes in MGs”

[11]

Introduction Stylized facts - The analysis of financial data by the physicists has

led to the characterization of some empirical statistical regularities, known as “Stylized facts”

MG as toy models of financial market - The MG can be considered as a very crude model

of financial markets, because the minority mechanism is found in markets.

Ingredients of the model Interplay between two types of traders - “Producers” are traders who use the market for exch-angin

g goods. Their trading decisions originate from outside opportunities related to the economic activity and not on the market dynamics itself.

=> random walk of market prices - “Speculators” are adaptive agents with bounded rati-onalit

y. Their aim is to gain from market fluctuations. They provide liquidity for producers

=> emergence of stylized facts

(continued)

Possibility of not trading for speculators - Allowing the speculators for the possibility of not

trading if the market does not contain sufficiently profitable arbitrage opportunities for them.

- In the language of Physics, this makes the model grand canonical.

The model

Basic setup - Considering a set of agents who interact

repeatedly in a market. In each period t = 1,2,…, each agent i choose an action ,which is a real number.

- means that agent i wants to buy $ of an asset whereas implies that he wants to sell.

)(tai0)( tai )(tai

0)( tai

(continued) Market mechanism and information - The excess demand : - The payoffs of agents : => minority win

- The price dynamics : where is the return at time t and is related to the m

arket depth. - The market is characterized by a news arrival process whic

h is modeled by an integer which is drawn ra-ndomly and independently in each period from the in-tegers 1,…P.

N

i i tatA1

)()(

)()( tAatg ii

)(

)(log)()(log)1(logtA

tptrtptp

)(tr

)(t

(continued) Producers - The producers behave in a deterministic way with re-spect

to . i.e. - For each agent i and state we draw randomly from a fi

xed distribution. Here we take the bimodal di-tribution with equal probability.

- The number of producers : - The reduced number :

)(t

)()( tii ta

i

1 i

pN

PNn pp /

(continued) Speculators - The speculators are adaptive : - For strategies s>0 (active), is drawn randomly and indepe

ndently for each s, i and from the bimodal distribution. For s=0, for all

- In order to make decision, they use “virtual value me-thod” from original MG. The virtual value is given as

- can be interpreted as modeling either a risk-free asset which ensures a constant gain or more simply a risk-premium for not trading.

)(,)( tisi ta

is,

0, io

0),(,,, )()1( tssisisi itAaUtU

0

Simulation results Market’s ecology - The interplay between the two types of agents is sh-own in

Fig. 1 for . - The market predictability is defined as following

where is the average of A(t) conditional on . - If this quantity is non-zero, then the market is statist-ically

predictable. H measures predictability and relat-ed to negative entropy.

P

AP

H1

2|1

|A

)(t

(continued)

(continued) For fixed number of the speculators and

increasing the number of producers, the market predictability increases (left panel). When speculators added again to the market (right panel) they exploit predictability and hence reduce it.

The same behavior applies for . In the symmetric phase (i.e. H=0), many speculators refrain from playing

0

(continued)

(continued) Market crashes - A snapshot of the dynamics deep in the

symmetric phase is reported in Fig. 4. The market repeatedly undergoes catastrophic events, i.e. crashes

- The market crashes drives speculators away from the market and dynamics anew.

- The frequency of crashes increases with and it decreases with

sn

(continued)

Stylized facts Volatility and volume clustering - Volatility clustering is kwon that the volatility has alg-ebric

ally decaying auto-correlation, and accordingly that the returns activity is clustered in time.

- As shown by Fig. 5, the deeper one goes in the sym-metric phase-i.e., the larger the the more high the volatility regions appear clustered.

- Fig. 6 shows that the long ranged correlation of vola-tility occurs also in the present model

sn

(continued)

(continued)

(continued) Return and volume histograms - In real market, the probability distribution function (pdf) of

returns is known to have fat tails with exponent -4 on average.

- Fig. 7 shows that the MG presented here reproduce fat tail behavior, but the exponent if the tails decre-ases depends on the parameter , and sn pn

(continued)

Summary

MG is a simple toy model but it is belie-ved to capture some essential and gen-eral features of the competition between adaptive agents, which is found, for in-stance, in financial markets

Reference

[1] http://www.unifr.ch/econophysics/[2] W. B. Arthur, Amer. Econ. Assoc. Papers and Pro

c. 84, 406 (1994)[3] D. Challet and Y.-C. Zhang, Physica A 246, 407

(1997)[4] A. Cavagna, Phys. Rev. E 59, R3783 (1999)[5] R. Savit, R. Manuca, and R. Riolo, e-print adap-

org/9712006

Reference

[6] D. Challet and Y.-C. Zhang, Physica A 256, 514 (1998)

[7] D. Challet, M. Marsili, and R. Zecchina, Phys. Rev. Lett. 84, 1824 (2000)

[8] M. Marsili, D. Challet, and R. Zecchina, Physica A 280, 522 (2000)

[9] D. Challet and M. Marsili, Phys. Rev. E 62, 1862 (2000)

Reference

[10] C.-Y. Lee, Phys. Rev. E 64, 015012 (2001)[11] D. Challet et al. Physica A 294 514 (2001)