Institute of Physics Wroclaw University of Technology

28/09/2005

How can statistical mechanics contribute to social sciences?

Piotr Magnuszewski, Andrzej Radosz, Piotr Magnuszewski, Andrzej Radosz, Katarzyna Ostasiewicz, Michał Tyc, Katarzyna Ostasiewicz, Michał Tyc,

Piotr GoliczewskiPiotr Goliczewski

Volatility

• One of the frequently observed features of social systems is a phenomenon called volatility

• Volatility is a notion taken from economical sciences (financial markets) and denote “unpredictable” and extreme events. Such phenomena resemble catastrophic events in SOC systems (e.g., large avalanches in the sandpile model)

Social Interpretation of SOC Conditions

• Individuals are metastable in the sense that they do not change their behaviour until some level of stimulus has been reached.

• Interaction among agents is a dominant feature of the model dynamics.

• Agents influence but do not slavishly imitate each other.

• The system is slowly driven so that most agents are below their threshold (or critical) states a lot of the time.

based on Moss and Edmonds

Threshold Models (Granovetter,...)• Granovetter’s model describe the situation, where

each member of some group has to decide whether to join a certain collective action. Each individual has a certain threshold defined as the proportion of a group he would have to see join before he would do so (conditional decision making). The threshold is the point where the perceived benefits of joining the action exceed the perceived costs. Of course, to initialize the action there must be at least one person with zero threshold.

• The final result (the proportion of a group participating in the action) depends strongly on the exact distribution of threshold in the population. That is, two groups which do not differ in average value may exhibit completely different kinds of behaviour.

Reflexivity about self + empathy + social change plans

Deliberation about “others” cooperation and social actions

Norms (belief, desire, intention)

Cognition about facts (world models) and forecasts

Imitation (of all/of peers)

Externalities (environment)

Individual pure economic rationality

Modeling Decision-Making

Emotions

Habits+

From Nils Ferrand

Binary Choice Model

Iii ,...2,11

Each individual is viewed as making a binary choice,

The choice, assumed to represent the solution of a maximization problem:

ih Measures the deterministic difference in the payoff of the two choices(external influence, economic benefits, habits, externalities, norms)

ii

ijJ

Denotes an individual random component to the payoffs associated with the two choices (uncertainty, emotions)

Interactive aspect of decision-making: each agent has beliefs about the behaviour of others which influence the payoff associated with the agent’s decision (imitation)

iiiii Jhi

max1

1,1i

depends on the parameters of that procedure

Brock-Durlauf Model

ii

jijiijiii JEh

i

2

1 2

1max

 Maximization of the problem

Decision of the agent is expressed as a maximization problem, including three components: deterministic, random and subjective

Subjective expected value E(.) embodies the interactive aspect of decision-making

A common assumption is the logistic distribution of the difference of the

random payoff terms (independent across individuals)

0exp1

111Pr

i

iii z

z

Network with N-1 neighbours B=2.0 J=0.0067

Average choice vs. field strength h

Multiple equilibria (globally undesirable outcomes) can emerge due to social interactions.

Global utility vs. Field strength h

Average choice vs. field strength h

Network with 4 nearest neighbours B=2.0 J=0.25

Global utility vs. Field strength h

Statistical mechanics

i

 Analogies

Probability measures (equilibrium) have the same form as those found in many

statistical mechanics models.In a special case

each person cares only about the average choice of others

distributions of random terms are identical accross individuals

deterministic private incentives are identical accross individuals

hhi

ijI

JJ ij

1expected average choice in a

population is

This is mean field (Currie-Weiss) approximation for ferromagnetism.

Jmhm tanh

Statistical mechanics

0Hh 1J

 Solutions

In a low temperature regime there exists a threshold for a multiple solution

there will exists a single solution.

whereas if

3,2,1mm

0Hh

-2 -1 1 2

-2

-1

1

2

-2 -1 1 2

-2

-1

1

2

= 3, h = 0.1, J = 1 = 3, h = 0.5, J = 1

Statistical mechanics

 Conclusions

2. Multiple equilibria, yet undesirable outcomes, can emerge due to social interactions.

3. The use of logistic assumption for random utility differences means that theoretical model under analysis corresponds to one of the standard econometric models of binary choice.

4. These types of models, by emphasizing the possibility of multiple equilibria, suggest new types of statistics that should be computed in order to understand cross-group behaviour.