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8.4 The Minority Game 251

is deﬁned with respect to the behavior of the competitors, a posteriori, and

based on success. As a corollary, the deﬁnition of good behavior may change

when the reference behavior of the other agents changes. The choice to take

in the minority game amounts to predict a future event – which depends only

on the choices of all other players [198]. Of course, one might argue that, in a

fundamental perspective, ﬁnancial markets are also inﬂuenced by the arrival

of external information. However, many readers will know from experience

that on certain days, external information strongly moves markets, while on

other days, it is completely ignored by the operators. Most likely, psychology

is at the origin.

However, there are also important diﬀerences. Firstly, with an average

expectation of a winning trade of ideally 50%, it is not clear why agents

trade at all. One tentative answer which has been given to this question is

the presence of non-speculative trades in a market, originated by “producers”

or investors. In a commodity market, there will be producers who sell their

goods, and buyers who need those goods for their utility, rather than for

proﬁt. An investor might buy shares in a stock market for gaining control

over a company, rather than for speculative proﬁts. One can then set up an

argument that these producers would introduce predictable patterns into the

markets which would be exploited by speculators who can adapt much more

quickly to a market situation than producers [198]. It is not clear, however,

to what extent such an argument could explain trading in FX markets, where

more than 90% of the trading volume is speculative in origin, and which are

extremely liquid.

Secondly, one suspects that the aim of the players taking part in the

minority game corresponds to a contrarian trading strategy. “Be part of the

minority!” implies to buy when everybody is selling and vice versa. However,

in ﬁnancial markets, one often ﬁnds extended trending periods where the most

successful strategy would be to buy when the majority buys and sell when the

majority sells (hopefully early enough, though). Is it more appropriate then

to view ﬁnancial markets as the playground for a majority game rather than

a minority game? Remember that the presence of diﬀerent trader populations

and the switching of their trading philosophy in the Lux–Marchesi model (cf.

the preceding section) was essential to produce realistic time series in that

artiﬁcial market.

Thirdly, in a real-world market, operators may not trade in a given time

interval. This is ignored in the minority game. A straightforward general-

ization to a “grand-canonical minority game” would open such an avenue.

In order to decide whether to trade or not, an agent should compare her

strategies to a benchmark. The basic minority game only compares the rela-

tive merits of all her strategies, and trading is done also when all strategies

lose out. To remedy for that deﬁciency, a rule can be introduced that an agent

only trades when at least one of her strategies has a positive score. Here, one

still faces the problem that the success of the strategy at the origin of the

252 8. Microscopic Market Models

decision to trade, is virtual while any loss incurred while being in the market,

would be real. A more realistic benchmark would be to trade only when at

least one strategy is available whose success rate is superior to a threshold

[198].

The minority (and majority) games can be derived from a market mech-

anism [199], once price formation and market clearing are deﬁned. We pro-

ceed as in Sect. 8.3.2, i.e., determine the price at which the market is cleared

from the aggregated demand D(t), the aggregated supply O(t), and the price

quoted in the last time step, according to

S(t)=S(t − 1)

D(t)

O(t)

, (8.36)

D(t)=



i=1

(t)Θ[a

(t)] =

+ A(t)

, (8.37)

O(t)=−



i=1

(t)Θ[−a

(t)] =

− A(t)

. (8.38)

The return on an investment for one time step from t to t +1 [a

(t)=1,

(t +1)=−1] is

δS

(t +1)=ln



S(t +1)

S(t)



≈

S(t +1)

S(t)

− 1 . (8.39)

Of course, the information on S(t + 1) is not available to the players when

they must place their orders. The best they can do is to base their decision

on their expectation for the return on their investment. Assume that the

expectation of player i at time t for the price of the asset at t +1is

(i)

[S(t +1)]=(1− ψ

)S(t)+ψ

S(t −1) . (8.40)

Let each player place an order at t according to that expectation, calculate

the payoﬀ on the investment over one time step, and compare the payoﬀs of

the majority and the minority sides. It turns out that agents with ψ

> 0

are on the winning side when they are in the minority, i.e., they follow a

contrarian investment strategy. They expect that the future price movement

is negatively correlated with the past move. On the contrary, agents with

< 0 are trend followers and play a majority game. Thus it appears that

real markets may be described best as mixed minority–majority games.

8.4.4 Spin Glasses and an Exact Solution

A slightly modiﬁed, “soft minority game”, can be solved exactly using meth-

ods from spin-glass physics in the limit N

→∞[201]. Agents do not simply

8.4 The Minority Game 253

choose the strategy with the highest virtual score, but proceed in a proba-

bilistic manner: a strategy is chosen with a probability which depends expo-

nentially on its virtual score in the game. Moreover, the binary payoﬀ of one

point when the strategy played was successful, is changed into a gain function

linear in the population diﬀerence between minority and majority sides,

(t)=−a

(t)A(t) , (8.41)

i.e., the minority wins points or money, and the majority loses them. By

deﬁnition, this is a negative sum game. The total average loss in the system

then is

−



g

 = σ

. (8.42)

This equation reemphasizes the interpretation of σ

as a measure of the waste

in the system.

The dynamical equations of the minority game then suggest a description

in terms of a Hamiltonian which is reminiscent of disordered spin systems

[200, 201]. To see the essentials, we limit ourselves to N

= 2 strategies

which would correspond to spin 1/2. To distinguish strategies s

∈{↑, ↓}

from actions a

i,s

(the subscript emphasizes that the action a

depends on a

strategy s

), decompose a

i,s

(t)as

h(t)

(t)=ω

h(t)

+ s

(t)ξ

h(t)

,ω

i,↑

+ a

i,↓

,ξ

i,↑

− a

i,↓

. (8.43)

represents a ﬁxed bias in the strategies of agent i,whereasξ

represents

the ﬂexible part. Of course, they depend on the history h(t) of the game. The

time dependence of a

(t) now is attributed to two time-dependent factors:

one is the particular history h(t) realized in the game during the M rounds

preceding t.Thisiswhyω

and ξ

depend on t only through h(t). The second

factor is the time dependence of s

(t), which reﬂects the choice of strategy

made by agent i at time t based on the available history and his strategy

selection rules (probabilistic or deterministic).

Introducing Ω =

, A(t) can be rewritten as

(t)=Ω



i=1

(t) , (8.44)

and its variance becomes

= Ω





+2Ωξ

s







i=j

s

s

 . (8.45)

Here, x denotes the temporal average of a quantity x while

x is the average

over histories. Unless necessary, the history superscript h is dropped under

the history averages. All 2M histories are explored for long enough times.

254 8. Microscopic Market Models

This allows us to decompose a temporal average into one conditioned on

history x

, followed by one over histories, i.e., x = x

. By symmetry,

A = 0. However, for particular histories, there may be a ﬁnite expectation

value A

 = 0. One may then calculate the average over the histories of the

history-dependent expectation values of A,

A



= Ω



Ωξ

s

 +



i,j

s

s

≡H. (8.46)

When the scores of the strategies are updated using a reliability index

s,i

(t +1)=U

s,i

(t) − 2

−M

s,i

(t)A(t) (8.47)

and a probabilistic strategy selection rule P [s

(t)=s] ∼ exp[ΓU

s,i

(t)] is

adopted, the evolution of s

 with time scale τ =2

−M

δt can be cast in the

form

ds



dτ

= −Γ



1 −s







∂H

∂s





. (8.48)

Formally, these are the equations of motion for magnetic moments m

= s



in local magnetic ﬁelds

Ωξ

interacting with each other through exchange

integrals

. H in (8.46) then is a spin-glass Hamiltonian [200].

Such Hamiltonians can be studied using the replica trick familiar from

the theory of spin glasses [185] and it turns out that, under the standard

assumptions, the ground state of the Hamiltonian which describes the sta-

tionary state always is in the replica-symmetric phase [200, 201]. Within the

replica-symmetric phase, there is a transition, however, as a function of the ra-

tio between the information complexity 2

and the number of players. When

this number is small, the probability distribution of the strategies used in the

game is continuous while, for a large ratio, it contains two delta functions at

the positions of static strategies a

= ±1 in addition to a Gaussian distri-

bution. Agents contributing to the delta functions do not switch strategies

while those under the continuous distributions stochastically change strate-

gies. This is the phase transition seen in the dependence of the volatility on

the memory length/agent number discussed in Sect. 8.4.1.

The small-α phase is called “symmetric” because both A =0and

A

 = 0. In the “asymmetric” large-α phase, we have A = 0 but A

 =0

at least for some histories. A

 therefore is akin to an order parameter in

a symmetry-breaking phase transition. Here, it is the symmetry between the

histories which is lost at the critical α

. In the asymmetric phase, for those

histories with A

 = 0, there is a best strategy

best

(t) ≡ a

best

= −sign A

 (8.49)

which allows for a positive gain

|A

| − 1. In this phase, the market is pre-

dictable. The measure of predictability is H ≡

A



, (8.46). Using (8.36)–

(8.38), we have

8.4 The Minority Game 255

A(t)=D(t) − O(t) , (8.50)

i.e., A(t) also is the excess demand in the market. When A

 =0,there

are persistent periods of excess demand/supply where price will move in

one direction. The volatility is somewhat better than coin tossing but not

dramatically so, because of the crowd–anticrowd repulsion. That information

is used not very eﬃciently is evidenced by σ

, which is signiﬁcantly above its

minimum at α

. The game becomes more information-eﬃcient when players

are added who more evenly cover the strategy space.

In the symmetric small-α phase, H = 0, i.e., the market is unpredictable.

Moreover, A(t) = 0, i.e., there is no excess demand on the average, and

prices are stable. When there are very many players at moderate informa-

tion complexity, herding takes place due to the incomplete crowd–anticrowd

screening, and the volatility increases again. The waste of resources/total loss

of the population is minimal at the transition α = α

When the agents include a term into their strategy selection probability

which rewards the strategy actually used by them in the game with respect

to virtual strategies, a replica-symmetry broken solution can be found. The

interesting point is that the replica-symmetry broken solution describes a

Nash equilibrium. Nash equilibria in the minority game correspond to pure

(static) strategies a

= ±1 independent of t. The replica-symmetric solution,

on the other hand, does not correspond to a Nash equilibrium. However, the

trimodal solution for the strategy probability including the delta-function

peaks at pure strategies contain some of its ingredients.

8.4.5 Extensions of the Minority Game

A variety of extensions can be formulated in order to bridge the gap between

the basic game and a model for ﬁnancial markets. The agent population can

be made heterogeneous in various dimensions such as memory size, strategy

diversiﬁcation, evolutionary strategies, etc., and agents may choose to stay

out of the market. When the game is played with mixed memory sizes, players

with longer memories perform better than those with shorter memories [196].

When the payoﬀ function is changed to lottery-type, i.e., the payoﬀ (both in

real and virtual points) increases with decreasing number of winners, the

probability distribution of A(t) becomes bimodal – it is monomodal in the

standard game. This is a remarkable example of self-organization because

the most likely conﬁgurations are avoided by the players at the expense of

somewhat less likely ones.

One can introduce explicitly hedgers who only possess one strategy. They

do not enter the marketplace for speculation but for “fundamental” (exoge-

neous) reasons, cf. Sect. 2.5. They might as well be producers who use the

market for selling or buying goods. In the game, their role is to introduce

information through their trading activity which is supposed to be due to

drivers external to the game [200]. Also, noise traders, who take random de-

cisions, can be included. Further extensions could include insiders and spies.

256 8. Microscopic Market Models

It is particularly interesting that the minority game can be extended to

allow for predictions of moves in actual markets [202]. It is based on the

“grand canonical” extension of the minority game where agents trade or stay

out of the market depending on the comparison of their scores (virtual or

real) with a threshold value. Thus the number of active traders has become

variable. Also, the threshold can be made a dynamic quantity. One restric-

tion is that the threshold should be positive, i.e., a trader should only use

strategies which have won more often than lost. As a second restriction, the

threshold should increase when the player’s scores decrease, i.e., one should

take less risk after losing for some period of time. These rules generate quite

diverse populations of traders. One may further diversify the trader popula-

tion in terms of wealth (initial capital), investment size (wealthy investors will

place big orders), and investment strategy (trend following versus contrar-

ian, or minority versus majority games). The mechanism of price formation

is assumed to be similar to (8.36).

This extended mixed minority–majority game is trained on a ﬁnancial

time series, converted into a binary sequence, e.g., by just recording the signs

of market moves. In other words, the game is fed with a signal where Zipf

analysis, discussed in Sect. 5.6.3, has demonstrated that non-trivial correla-

tions exist [112, 113]. Such correlations have been uncovered speciﬁcally in

the USD/JPY exchange rates [113] which have been used in this experiment.

Players then take their actions based on that signal history h(t). The sign

history is an external signal whereas A(t) in the minority game was generated

internally to the game. The feedback eﬀect included in A(t) has been removed.

However, the game and the time series of aggregated actions A(t) are used

to carry the game forward into the future. When using hourly quotes of ten

years of USD/JPY exchange rates, the game performs much better than ran-

dom, and the accumulated wealth of the total agent population is increasing

steadily. The actual increase, however, depends on the pooling of the agents’

predictions which is not speciﬁed for the best performances [202]. The trad-

ing strategy certainly is somewhat oversimpliﬁed: depending on the minority

game prediction, put the investment on the USD or JPY side and, after

one hour, withdraw it. Neglect transaction costs, slippage, etc. Despite these

simpliﬁcations, the game apparently produces many of the stylized facts of

ﬁnancial markets: fat-tailed return distributions, price–volume correlations,

volatility clustering, ...[202, 203].

More importantly, when run into the future for several time steps, the

game also generates prediction corridors for future prices of the asset [204].

In many cases, large changes can be predicted accurately in the sense that

the probability density function of the returns possesses a large mean and

a narrow variance. In other cases, the prediction of a sign change comes

out correctly although the prediction corridors are rather wide. Large price

movements such as crashes or booms apparently can be predicted with some

8.4 The Minority Game 257

degree of reliability based on the minority game. Johnson et al. have ﬁled a

patent application on these algorithms [204].

As a ﬁnal remark, it has been shown that a winning strategy can be

set up by playing two diﬀerent losing games one after the other (Parrondo’s

paradox) [205]. It would certainly be interesting to include such eﬀects into

the minority game.

9. Theory of Stock Exchange Crashes

Crashes of stock exchanges, and speculative markets more generally, have

occurred ever since trading securities and commodities has become an im-

portant activity. A historical example is the “tulipmania”, the rise and sub-

sequent crash of prices for tulip bulbs on Dutch commodity markets in 1637

[206, 207], or the South Sea bubble in England, where Newton lost much of

his fortune, cf. Chap. 1. Modern ﬁnancial crashes are discussed below. Since

in such events enormous fortunes are at stake, eﬀorts towards an improved

understanding are mandatory.

9.1 Important Questions

In this chapter, we will attempt answers to the following important questions

concerning ﬁnancial crashes:

• What are the origins of stock exchange crashes?

• Are crashes compatible with rational behavior of investors?

• Are they endpoints of “speculative bubbles” and signal the return of market

prices to their “fundamental values”?

• Do crashes signal phase transitions in markets?

• Are there parallels to earthquakes or avalanches?

• Are earthquakes predictable?

• Are crashes part of the normal statistics of asset price ﬂuctuations, or are

they outliers?

• Can crashes be predicted? Are there crashes which have been predicted

successfully in the past?

• Are there examples of anticrashes, i.e., trend reversals from falling to rising

prices which follow patterns established for crashes?

• Can one measure the strength of crashes in the same way as the Gutenberg–

Richter scale measures the strength of earthquakes?

• Are there signals for the end of a crash?

260 9. Theory of Stock Exchange Crashes

9.2 Examples

Here is a list of the more recent examples of ﬁnancial crahes, some of which

readers may well remember.

1. The “Asian crisis” on October 27, 1997 and the “Russian debt cri-

sis” starting in summer 1998, have been discussed brieﬂy in Chap. 1.

Figure 1.1 shows these two events in the variation of the DAX, the Ger-

man stock index, from October 1996 to October 1998. The Asian crisis is

adrawdownofabout−10% on the German stock market on October 27,

1997, with a very quick recovery. Interestingly, the aggregate drawdown

over scales even as short as a week was rather small. Notice, however,

that the DAX stopped its upward trend in July 1997, and one question

we wish to discuss here is to what extent this can be viewed as a kind of

precursor of the crash. Indeed, there have been predictions of this crash

[208].

On the contrary, the drawdowns of the stock markets in Asia was much

stronger. The Hang Seng index of the Hong Kong stock exchange, e.g.,

lost 24% in a week. The index is shown as the dotted line in Fig. 9.1.

The solid line shows the variation of the US S&P500 index in the four

years prior to the 1997 crash. The long-term upward trend is stopped by

94 95 96 97 98

S&P 500

Hang Seng

?? t

Fig. 9.1. Extrema of variation of the S&P500 and the Hong Kong Hang Seng index,

prior to the 1997 crash. Notice that the Hang Seng index has two pronounced

minima not lying on the log-periodic sequence marked by the vertical lines. By

courtesy of J.-P. Bouchaud. Reprinted from L. Laloux, et al.: Europhys. Lett. 45,

1 (1999),

1999 EDP Sciences

9.2 Examples 261

the drawdown in late October 1997. Similar to the European markets,

its amplitude was much smaller than on the Asian markets. However,

unlike the German market, the index continued to increase throughout

summer 1997 although there have been certain periods of local short-time

decrease, marked by the vertical lines. The labels “t

”and“?”onthese

lines will be explained in Sect. 9.4.

The impact of the Russian debt crisis on the German stock market was

very diﬀerent from the Asian crisis. The decrease was much less abrupt –

though much more persistent and of much larger amplitude, at the end.

Over four months, the DAX lost 39% which corresponds to an average

loss of 2.7% per week. It is obvious from Fig. 1.1 that losses of this order

of magnitude occurred regularly, almost every week, between July and

October 1998.

With reference to the discussion in Sect. 5.3.3, notice that stop-loss orders

would not have protected investors from sizable losses in the Asian crisis

while they would have oﬀered protection throughout most of the Russian

debt crisis. On the other hand, due to the quick recovery of the markets

after the Asian crisis, an investor simply holding his assets for a few more

weeks would have wiped out most, if not all of his losses.

2. Figure 9.2 shows the variation of the Dow Jones Industrial Average in the

Wall Street crash of October 1987 [209]. The index lost about 30% in one

day. To put that into perspective, the loss in a single day is comparable

Fig. 9.2. The Dow Jones Industrial Average during the October crash 1987. By

courtesy of N. Vandewalle. Reprinted with permission from Elsevier Science from

N. Vandewalle, et al.: Physica A 255, 201 (1998).

1998 Elsevier Science