Zoltán Toroczkai

György Korniss (Rensellaer Pol. Inst.)

Kevin Bassler (U. Houston)

Marian Anghel (CNLS-LANL)

Effects of Inter-agent Communications on the Collective

Emergence of robust leadership structure and market efficiency

(Complex Systems Group, LANL)

Resource limitations lead in human, and most biological populations to competitive dynamics.

The more severe the limitations, the more fierce the competition.

Amid competitive conditions certain agents may have better venues or strategies to reach the resources, which puts them into a distinguished class of the “few”, or elites.

Elites form a minority group.

In spite of the minority character, the elites can considerably shape the structure of the whole society:

since they are the most successful (in the given situation), the rest of the agents will tend to follow (imitate, interact with) the elites creating a social structure of leadership in the agent society.

Definition: a leader is an agent that has at least one follower at that moment. The influence of a leader is measured by the number of followers it has. Leaders can be following other leaders or themselves.

The non-leaders are coined “followers”.

-- a set of discrete, autonomous entities (individuals, agents, players) with a certain degree of intelligence, adaptability, and flexibility in the choice of their actions in response to external stimulus, or to follow personal goals (maximize or minimize a set of utility functions).

-- there is no (or very little) centralized control

-- there is a globally available world utility function which rates the past performance of the collective (world history function).

-- the choice of response function of an agent couples to: the world utility function information gathered from neighboring agents on the social network, via Reinforcement Learning.

Agent-system (society):Agent-system (society):

[D.H. Wolpert, K. Tumer (2000): COIN]

The El Farol bar problemThe El Farol bar problem

A B

…

[W. B Arthur(1994)]

A binary (computer friendly) version of the El Farol bar problem:

[Challet and Zhang (1997)]

The Minority Game (MG)The Minority Game (MG)

A = “0” (bar ok, go to the bar)

B = “1” (bar crowded, stay home)

World utility(history): (011..101)

latest bit

m bits

l {0,1,..,2m-1}

(Strategies)(i) =

S(i)1(l)

S(i)2(l)

S(i)

S(l)

(Scores)(i) = C (i)(k), k = 1,2,..,S.

(Prediction) (i) =)}({max )(* kCk i

k }1,0{)()( )(

* lSiP i

k

3-bit history 000 001 010 011 100 101 110 111

associated integ.

0 1 2 3 4 5 6 7

Strategy # 1 0 0 0 1 1 0 0 1

Strategy #2 1 1 0 0 1 0 0 0

Strategy #3 1 1 1 0 0 0 1 0

t

A(t)

Attendance time-series for the MG:

World Utility Function:

2)2/( NA

Agents cooperate if they manage to produce fluctuations below (N1/2)/2 (RCG).

Some macroscopic properties

Predictability (Phase transition)

Persistence – Anti-persistence

Unused strategies - freezing

2

1

2|)(

P

)()(|)()()(

)()(

1

tttt

ttP

)1(1

1

N

i

i

S

n

N

NPNSmN //2),,( m

]|sgn[)( bestS

The El Farol bar game on a social networkThe El Farol bar game on a social network

…

A B

The Minority Game on Networks (MGoN)The Minority Game on Networks (MGoN)

Agents communicate among themselves.

Social network:Social network: 2 components:

1) Aquintance (substrate) network: G (non-directed, less dynamic)

2) Action network: A (directed and dynamic)

G

AA G

Communication types (more bounded rationality):

Majority ruleMajority rule Minority ruleMinority rule

Critic’s ruleCritic’s rule: an agent listens to the OPINION/PREDICTION of all neighboring agents on G, scores them (self included) based on their past predictions, and ACTS on the best score.

(not rational)

(not rational)

(more rational, uses reinforcement learning)

(Links)(i) = (Scores)(i) = F (i)(j), j= 1,2,..,K.

(Prediction) (i) =)}({max )(* jFj i

j }1,0{)()( )(

*

lSiP j

k

)(1iL)(

2iL

)(iKi

L

i

Social NetworksSocial Networks

1. Degree distribution (number of acquitances a person has) :

- it is strongly peaked around a mean degree: there is a recurring cost in terms of time and effort for maintaining a connection. This is a resource as well a cognitive limitation. [MEJ Newman, D. Watts, S. Strogatz, PNAS, 99, 2566, (2002) ].

How do they look like?How do they look like?

Data: EpiSims Census data, from Portland Oregon, 1.6 mill. people[H. Guclu, Z. Toroczkai, … (2002)]

[MEJ Newman, D. Watts, S. Strogatz, PNAS, 99, 2566, (2002) ].

2. “Small world-ness”: it takes only a small number of acquaintances to reach almost anyone in the world: D log (N), where D is the number of steps, N is the number of vertices (people) in the graf.

Milgram’s experiment: [J. Travers, S. Milgram, Sociometry 32, 425 (1969).]

D 6-7.

[D.J. Watts et. al. , Science, 296, 1302 (2002)

- search in social networks is effective because of the high dimensionality of the social space (provides shortcuts).

3. Clustering or transitivity:

Very likely!

A

B C

]2/)1([

ii

ii kk

nC

i

ki=5

ni=3

Ci=0.3

Clustering distribution:

N

ikki i

CkN

kC1

,)(

1)(

Average clustering coefficient:

iCC

People-people network is very strongly clustered:

Location networks:

People move around. Two locations are connected by an edge if a person went from A to B.

A B

CNot very likely

- expect much less clustering

[H. Guclu, Z. T., … (2002)]

Power law tail, exponent: –2.4

[H. Guclu, Z. T., … (2002)]

*1) Regular network with node degree k:

*2) Erdös-Rényi Random networks with link probability p.

3) Small-world networks generated from regular networks (Watts, Strogatz, Newman) .

4) Scale-free networks (Albert-Barabási). (irrelevant here)

Network types:

shows the small-world effect: ND log

Critic’s Rule on a regular network

m

10k

5k

2k

1k

0k

Uniform aggregation does not pay off!

)(min mMGm

RN

Critic’s Rule on Erdos-Renyi network.

2 ,3 ,101 SmN 2 ,6 ,101 SmN

Network Effects: Possibility for Improved Market Efficiency

Network Effects: Possibility for Improved Market Efficiency

A networked, low trait diversity system is more effective as a collective than a sophisticated group!

Can we find/evolve networks/strategies that achieve almost perfect volatility given a group and their strategies (or the social network on the group)?

Improved market efficiency

Macroscopic Properties – Network EffectsMacroscopic Properties – Network Effects

Reduced persistence:

Reduced predictability and phase separation: followers and leaders

Unused links – freezing on action network and persistence

t

N

i

ileadersact S

n

N 11

t

N

i i

outi

link k

k

N

1

)(

)1(1

NT

tkpppH t

outi

iii

N

i

)( ; log

)(

1

N

The network is very efficient at removing any arbitrage opportunities!

Emergence of scale-free leadership structure:

Emergence of scale-free leadership structure:

Robust leadership hierarchy

RCG on the ER network produces the scale-free backbone of the leadership structure

1for ,1);,(

);,()();,(

)();,(

);,();,(

0

1

1

mpmNf

pmNfkpapmNN

papmNN

pmNNkpmNN

kk

k

kk

k

iouti

The influence is evenly distributed among all levels of the leadership hierarchy.

m=6

Structural un-evenness appears in the leadership structure for low trait diversity.

The followers (“sheep”) make up most of the population (over 90%) and their number scales linearly with the total number of agents.

)()(1

)(1

)()(

tktkN

CN

i

outi

outi M=2

N=101, S=2

M=3

M=8

M=6 In low m regime, where trait diversity is low (as in a dictatorship) leaders leave longer!

Leadership position: Symmetric-Asymmetric phase transition

SOME CONCLUSIONS:SOME CONCLUSIONS:

We modeled the inter-agent communications across a social network which forms the skeleton for information passing in a competitive game with bounded rationality.

The game evolves the active network by coupling via reinforcement learning on the agent-level. The game is influenced by the inter-agent communications.

A robust leadership structure emerges naturally. The structure is scale-free and evenly distributed for large trait diversities. The more even is the distri- bution the more de-correlated are the agent’s choices in the strategy space.

The leaders’ position is more persistent/stable the lower the trait diversity.

Networking can lead to a more efficient collective for low-trait diversity agents. It is detrimental for large trait diversities.