Interacting agents with public and private information(statistical mechanics perspective)

A De Martino (SMC, Roma 1)

mostly joint work with M Marsili (ICTP Trieste)

http://chimera.roma1.infn.it/ANDREA

A De Martino and M Marsili, J. Physics A 39 R465 (2006)

Helbing+2 02Selten+8 04

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ate

user-speccorrected

own payoff both similar

P1(n1) = P0

1 ! P1

1 n1

P2(n2) = P0

2 ! P1

2 n2

! feq

1=

P 12

P 11

+ P 12

+1

N

P 01 " P 0

2

P 11

+ P 12

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3

Treatment 1 Treatment 2 Treatment 3 Treatment 4 Treatment 5

0 500 1000 1500 2000 2500

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actio

n

Iteration t

user-speccorrected

own payoff

both similar

“congestion game”

in financial markets (see T Odean’s group work)

people systematically overweight some typesof information and underweight others ; overweight their successes and underweight their failures ; overestimate the precision of their information ; etc.

}overconfidence(increases trading volumes,decreases exp. utility, causesunderreaction to realinformation, etc.)

“hold losers/sell winners”

a. when is convergence to an optimal state possible?b. when there are many, which one is selected?c. can one supply information so as to stabilize decision dynamics?d. how do information schemes affect the dynamics?e. is there an optimal information scheme?

some questions

1 agent , 1resourceN agents , 1 resourceN agents , O(N) resources

learning? (backward-looking induction vs forward-looking deduction)

one agent , choices : s ! {+,"}

learning :

decision : P (s(t) = s) ! ![ys(t)]

ys(t + 1) ! ys(t) = us(t)

ys(t + 1) ! ys(t) = us(t)!s(t),s

(full)

(partial)

payo!s us(t) i.r.v. with Eus = vs (v+ > v!

)

learn the best choice?

full + !(y) = ey

!

(" > 0) ! yes

partial + !(y) = y! (" = 1) ! yes

Rustichini 99

one agent , state a ! {H, L} (well/badly informed)

forecasts an event ! correctly with prob. a

can he learn a from his past success? s(t) =!

t!<t

!!(t!),f(t!)

learning : P [a = H|s(t) = s] ! !!(t|s)

=!sHs(1 ! H)t!s"0

!sHs(1 ! H)t!s"0 + Ls(1 ! L)t!s(1 ! "0)

! = 1 ! rational ; ! > 1 ! underreact to errors

! if a = L the updated posterior converges a.s. as

!!(t|s) !

!

"

#

"

$

1 for " > "!

!0 for " = "!

0 for " < "!

Gervais-Odean 01

correct priors,biased updates

1 to N agents

choices ai ! {"1, 1} , A =!

j

aj

payo!s ui(ai, a!i) =N ! aiA

2

decision : P (ai(t) = a) ! eyia(t)

(full)

learning : yia(t + 1) ! yia(t) =!

N

N ! aA(t)

2

Ui = yi+ ! yi! " Ui(t + 1) ! Ui(t) = !

!

NA(t)

(optimal learning)

0 2 4 6s0

5

10

15!c(s)

2 4 6 8 10!

0

0.5

1

"2 /#2

s=1/2s=1 ! < !c ! !

2" N

! > !c ! !2" N

2

Fluctuations : !2 =

!

i

(!a2

i " # !ai"2) =

!

i

(1 # !ai"2)

yia(0) randomly sampled from G(0, s2)

so the larger the spread of i.c.’s (heterogeneity), the smaller the fluctuations but the steady state is not optimal

Optimal state :

!

N!1

2do a

N+12

do !a

! !2

= 1 NNE " eN!

Ui(t + 1) ! Ui(t) = !!A(t)/N , A(t) =!

j

aj(t) , ! > 0

i is in hereWhy can’t they get to Nash?

remove self-interaction (learn to respond to others, not to yourself)

! ! {0, 1}Ui(t + 1) ! Ui(t) = !

!

N[A(t) ! !ai(t)]

!Ui(t + 1)" # !Ui(t)" = #!

N[!

j

mj # !mi] mi = !ai"= !

!

N

!H

!mi

H =1

2

!

"

i

mi

#2

!

!

2

"

i

m2

i

! = 0 ! mi = 0

! = 1 : H is harmonic ! mi = ±1

! !2

= N

! !2 = 1 (odd N)

minima :

0

1

2

3

Treatment 1 Treatment 2 Treatment 3 Treatment 4 Treatment 5

0 500 1000 1500 2000 2500

Ove

rre

actio

n

Iteration t

user-speccorrected

own payoff

both similar

partial full NE

externally supplied, unrealistic(it is seen as a 1/N effect, “price-taking”)

buy/sell difference

first gap

AZN, LSE 1999-2002

Farmer+2 04

price

old price price

Large price variation =“closing” of the first gap

Agents areheterogeneousselfishinductive

Resources areequivalentscarce(distributed)

No direct interaction

N agents, M resources and a matrix of dependencies {!µi }

N ! "random {!µ

i }Typical properties :

Resources

Load

{Qµ}

Q

N = cM

N drivers on a grid of M streetsone driver’s

feasible routes

another driver’sfeasible routes

A

BA

B

route for i : !"i = {"µi }

c = N/M = density of vehicles

Aµ(t) = Qµ(t) ! Q

Resources

Load

{Qµ}

Q

how evenly are resources used?

how large are fluctuations?

!i(t) !!

µ

aµi (t)Qµ(t) "

!

i

!i(t) !!

µ

Qµ(t)2

!2 =1

M

!

µ

!(Qµ)2" # !Qµ"2

H =1

M

∑

µ

!Aµ"2

two feasible routes per agent : !"is , s ! {+,"}

decision : P (si(t) = s) ! e!Uis(t)

! ! 0

learning : Uis(t + 1) ! Uis(t) = !

1

M

!

µ

!µisQ

µ(t) +1

2(1 ! "s,si(t))zis(t)

Qµ(t) =!

i

!µis"s,si(t)

“congestion game”

information noise

! = 0

! > 0

! < 0

: unbiased information: overestimates unused routes: underestimates unused routes

!!ig(n)" = "

!!ig(n)!jh(m)" = !"ij"gh"nm

! = 0 : same information (biased or not) for all drivers

! > 0 : user-specific information

! = 0

Not optimal butnot that bad

! = ! = 0

! = !

Disaster for c > cc

0.1

1

10

100

!2

/N

1/"=0

"=0

10!2

10!1

100

101

102

c

0.0

0.1

0.2

0.3

0.4

0.5

H/N

cc ! 3

theory (steady state) : the dynamics minimizes H (approx.)

Uncorrelated result : cc = 2.9638 . . .

noiseless case

(high density)

inductive perform worse than random at high densities!

problem : the quenched disorder is spatially correlated ...

10!2

10!1

100

101

c

0.0

0.2

0.4

0.6

0.8

1.0

!2

/N

random drivers

adaptive drivers, "=0

adaptive drivers, "=#2

! = !2

! = 0 ! = 0

`pessimist’ information bias reduces fluctuations

theory (steady state) : the dynamics minimizes fluctuations

biased case

high car density (hard phase)

calibrated user-specific noise reduces fluctuations

10!1

100

101

102

103

104

!

10!1

100

101

"2/#

100

102

104

! teq

1/4

10!1

101

! = 0

noisy case

0

1

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3

Treatment 1 Treatment 2 Treatment 3 Treatment 4 Treatment 5

0 500 1000 1500 2000 2500

Ove

rre

actio

n

Iteration t

user-speccorrected

own payoff

both similar

10!1

100

101

102

103

104

!

10!1

100

101

"2/#

100

102

104

! teq

1/4

10!1

101

! = 0

10!2

10!1

100

101

c

0.0

0.2

0.4

0.6

0.8

1.0

!2

/N

random drivers

adaptive drivers, "=0

adaptive drivers, "=#2

! = !2

! = 0

! = 0

noiseless case = minH

min!2

minH

min!2

0.01 0.1 1 10 100!

0

0.5

1

"

RS

RSB

ERGODIC

NON ERGODIC

1/c

!=

!"/2

min(1 ! !)H + !"2

no PT(freezing, exp. many SS, ...)

PT(no freezing)