Multi-asset Minority Games

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1 Multi-asset Minority Games A De Martino (Roma) Joint work with G Bianconi and M Marsili (Trieste) 1. Minority Games with one asset a. Economists : competition under uncertainty, inductive reasoning b. Physicists : origin of anomalous fluctuations in stock markets, transitions between predictable and unpredictable regimes c. Mathematicians : exact solution 2. Minority Games with many assets a. how do traders distribute their investment on different assets when the assets are correlated and the amount of information available varies from one asset to the other? b. can they identify the information-rich assets? c. how does risk affect the composition of their portfolios?

Physicists : origin of anomalous fluctuations in stock markets, transitions between predictable and unpredictable regimes c. Mathematicians : exact solution 2.

2 An elementary model of speculative trading One asset traders (Challet and Marsili, 2002) N s (i = 1,..., N s ) P information patterns (µ = 1,..., P ) Fixed random trading strategies a i = {a µ i }P µ=1 { 1, 1} P At each time step : information µ(t) Dynamical rules for individual choices active/inactive return learning n i (t) = θ[u i (t)] {0, 1} A(t) = N s a µ(t) i n i (t) U i (t + 1) U i (t) = a µ(t) i A(t) {µ(t)} t=1,2,... {{a µ(t) i } N s } t=1,2,...

.., P ) Fixed random trading strategies a i = {a µ i }P µ=1 { 1, 1} P At each time step : information µ(t)

3 An elementary model of speculative trading (Challet and Marsili, 2002) <n act > H = 1 P P A µ µ=1 0.0 H/P s 2 /P H = 0 H > e+01 Nn s s /P H > 0 return statistically predictable from the information H = 0 zero predictability (market informationally efficient)

4 An elementary model of speculative trading Two assets traders P with two assets { 1, 1} N s (i = 1,..., N s ) information patterns (µ = 1,..., P ) Fixed random trading strategies a i = {a At each time step : information µ(t) = (µ + (t), µ (t)) µ i }P µ =1 { 1, 1}P Assume traders may only invest in one asset at each time step which asset? return learning s i (t) = arg max U i (t) A (t) = N s a µ (t) i δ si (t), U i (t + 1) U i (t) = a µ (t) i A (t)

.., P ) Fixed random trading strategies a i = {a At each time step : information µ(t) = (µ + (t), µ (t)) µ i }P

5 y i (t) = { 1,1} Solution I : assumptions U i (t + 1) U i (t) = a µ (t) i A (t) A (t) = U i (t) Assumptions y i (t + 1) y i (t) = s i (t) = arg max U i (t) = sign[y i (t)] { 1,1} 1. µ(t) is a Bernoulli process 2. Batch learning in place of on-line learning y i (t + 1) y i (t) = { 1,1} P N s a µ (t) i A (t) P µ =1 a µ i N s j=1 3. N s, lim P /N s = α finite N s 4. Trading strategies are i.i.d. Bernoulli random vectors a µ (t) i δ si (t), a µ j δ s j (t),

Batch learning in place of on-line learning y i (t + 1) y i (t) = { 1,1} P N s a µ (t) i A (t) P µ =1 a µ i N s

6 Solution II : `polarization y i (t + 1) y i (t) = { 1,1} P P µ =1 a µ i N s j=1 a µ j δ s j (t), What should one calculate? m m(t) = lim N s 1 N s N s 1 s i (t) m = lim t t t <t m(t ) { > 0 if agents trade pref. asset +1 in the steady state < 0 if agents trade pref. asset 1 in the steady state In the end : m m(α + α )

7 Solution III : paths = Prob{path} paths path = {s(t)} t 0 s = {s i } N s { 1, 1}N s µ(t) Bernoulli Prob{path} = P (s(0)) t 0 W (s(t) s(t + 1)) W (s(t) s(t+1)) = = = 5. P (s(0)) = N s N s P (s(0)) t 0 N s δ y i (t + 1) y i (t) + + P (s i (0)) dŷ i (t) 2π N s [ + dy i (t) { 1,1} dŷ i (t) 2π P e iby i(t)[y i (t+1) y i (t)+...] P µ =1 a µ i N s a µ j δ s j (t), j=1 ] e iby i(t)[y i (t+1) y i (t)+...]

P (s(0)) = N s N s P (s(0)) t 0 N s δ y i (t + 1) y i (t) + + P (s i (0)) dŷ i (t) 2π N s [ + dy i

8 Solution IV : generating functionals Dynamical generating functionals (MSR 1973, De Dominicis 1978, Coolen 2001) Z[ψ] = 1 = δ e i P P N s t 0 ψ i(t)s i (t) m(t) = i lim N s ( N s m(t) Z[ψ] = N s s i (t) ) dm(t) = d m(t)dm(t) 2π 1 N s N s [ lim ψ 0 ] Z[ψ] ψ i (t) e i bm(t) (N s m(t) P Ns s i(t) e N s[ψ(m,cm)+f (m,cm)] t 0 dm(t)d m(t) Ψ(m, m) = i t 0 m(t) m(t) F (m, m) = 1 N s N s log exp i m(t)s i (t) + dy i (t) t 0 t 0 = log exp m(t)s(t) + dy(t) t 0 i t 0 Note : from a Markovian multi-agent dynamics to a single (representative) agent (with a more complicated dynamics)

s i(t) e N s[ψ(m,cm)+f (m,cm)] t 0 dm(t)d m(t) Ψ(m, m) = i t 0 m(t) m(t) F (m, m) = 1 N s N s log exp i m(t)s i (t) + dy i (t) t 0 t 0 = log

9 Solution V : infinite system limit Z[ψ] = N s, lim N s P /N s = α finite e N s[ψ(m,cm)+f (m,cm)] t 0 dm(t)d m(t) e N s[ψ(m,cm )+F (m,cm )] m (t) : bm(t) (Ψ + F ) = 0 m (t) = s(t) = exp [ i t 0 m (t)s(t) + ] t 0 dy(t) exp [ i t 0 m (t)s(t) + ] t 0 dy(t) m (t) : m(t) (Ψ + F ) = 0 m (t) = i F m(t) Self-consistent problem

10 Solution VI : representative agent dynamics Λ = y(t + 1) y(t) = [ ( 1 + n ) 1 ( 2 G (n D) 1 + n ) ] 1 2 G t [ 1 + n ] 1 2 G tt n = 1/α z(t)z(t ) = Λ(t, t ) G tt = s(t ) + + z(t) 2 C tt = s(t)s(t ) s(t) z(t ) D tt = 1 4 [1 + m(t) + m(t ) + C tt ] Ergodic steady states : y(t) ỹ = lim t t γ = n χ C tt = C(t t ), G tt = G(t t ) G(t ) = χ < (χ > 0) t lim t ỹ = z 2 = γ s z n (1 + 2m + c) (2 + n χ) 2

z(t ) D tt = 1 4 [1 + m(t) + m(t ) + C tt ] Ergodic steady states : y(t) ỹ = lim t t γ = 2 2 + n χ C

11 Solution VII : finale ỹ = γ s z γ = n χ ỹ = lim t y(t) t 1. ỹ > 0 s = 1 z > γ + 2. ỹ < 0 s = 1 z < γ 3. ỹ = 0 s = s 2z P γ P γ γ < z < γ + m = θ(z γ + ) + s θ(z + γ )θ(γ + z) θ( γ z) c = θ(z γ + ) + (s ) 2 θ(z + γ )θ(γ + z) + θ( γ z) α χ 2α + χ = θ(z + γ )θ(γ + z)

12 Comparison with simulations m H Not very smart α + + α = α + α α H = H=0 Non-ergodic Ergodic H> α + α = P /N H H = 1 P µ A µ 2

5-0.4-0.2 0 0.2 0.4 α + α α H = 0.5 0.4 0.3 0.2 0.1 H=0 Non-ergodic Ergodic H>0 0 0 0.

13 Introducing risk profiles Traders may abstain from investing (risk profiles) U i (t + 1) U i (t) = a µ i A (t) ɛ i A (t) = N s +N p a µ (t) i φ i (t) φ i (t) = δ si (t),θ[u i (t)] Risk profile ɛ i > 0 risk-averse ɛ i < 0 risk-prone N s N p speculators producers always invest

14 Results I : polarization 0,6 m = 1 ( φ i,+ φ i, ) N s spec. 0,4 0,2 + - α s + α s =1 ɛ < 0 (risk prone) : as before ɛ > 0 (risk averse) : better m 0-0,2-0,4 α = P /N s -0,6-1 -0,5 0 0,5 1 + α s - α s-

15 Results II : phase diagram H = H H = 1 P µ A µ a s - H>0 0.2 H + =0, H - >0 Non-Ergodic Ergodic 0.1 Non-Ergodic H=0 H + >0, H - =0 Non-Ergodic a s

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