some thoughts on belief dynamics

7/27/2019 Some Thoughts on Belief Dynamics

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Some Thoughts on Belief Dynamics

Rajesh Kasturirangan

National Institute of Advanced Studies

Bangalore, India

1. Introduction. When I first started watching American football, I was amazed by the

extent to which the game was ordered plays being transmitted from coach to

quarterback to other players like a functioning army, only much better paid. At the same

time, I was ambivalent about a sport that restricted individual creativity to a minimum (as

I perceived it at that time). Then one day I saw a player who was about to be tackled

make an impromptu toss to another player, who ended up scoring a touchdown. That

spark of spontaneity was celebrated by all from the commentators to the fans and, of

course, the players on the scoring side. Sports thrive on these moments, when unscripted

actions lead to unanticipated outcomes; it is these little moments that can turn the tide,

and conversely, a small error can snatch defeat from the arms of victory. A good sport is

one that allows spontaneous moments like the one described above, especially if its to

retain an emotive fan base. Then, there was that fateful day in 1823 when a bored

schoolboy playing soccer on the grounds of Rugby, the well known public school (i.e.,English-speak for private school), took matters in his hands so to speak. That day,

William Webb Ellis, with fine disregard for the rules of football (soccer) as played in his

time, first took the ball in his arms and ran with it, thus originating the distinctive feature

of the Rugby game.

The curious scientist might ask Why did the first action become an acceptable part of

football, while the second is considered such an outrage that it led to a whole another

sport altogether? Life is not that different either. The religious fundamentalist who

insists on a strict interpretation of a sacred text and enforces social relations consistent

with that interpretation, is often more than happy to call his friends on their cell phone to

tell them about his understanding of the same text. So when is a belief change acceptable

and when is it not? Some new beliefs and actions are so egregious that the originator is


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rebuked, ostracized or worse and others are embraced as the next new thing. Indeed,

beliefs wouldnt change if someone didnt come along and say something different, but,

as we all know, some changes are more acceptable than others. So heres my question Is

it possible to study beliefs theoretically and computationally in a manner thats sensitive

to the circumstances in which beliefs (or as is more likely, a cluster of beliefs) are

altered?

2. Decision Theory and Belief Dynamics. The dominant paradigm for studying the

above question is what I call The Decision-Theoretic Strategy. In this strategy, it is

assumed that beliefs are formed and communicated within a rational framework, i.e., the

agents forming, holding and communicating their beliefs do so on the basis of a rational

(could be bounded rationality) system for evaluating and changing their beliefs. Of

course, there can be several models of rationality Logic, Game theory and Probability

(Bayesian Probability) are all competing models of rationality and can all be used

profitably in modeling the rational norms for holding and justifying a belief. For

example, one could assume that people who advocate the use of violence do so because

they have studied the use of violence in the past or see themselves as playing a game with

the authorities and have a rational means to evaluate the extent to which violence will

further their goals. A game theoretic model may study how the two sides are willing to

make concessions and how one of the sides, by using violence as one of its moves can

bargain for better concessions. In this game theoretic framework, violence as a move

would have an optimal state, and presumably agents using violence would try to attain

this optimal state.

3. Stability versus Decisiveness. A lot of the literature on beliefs is tied to decision

making, where utility based models are useful though the Atran sacred values paper

suggests that simple notions of utility will not be able to capture why people switch

strategies under certain circumstances. In other words, beliefs are tied to theories of

choice, especially in economics. What if most beliefs are removed from any

considerations of choice and are only weakly determined by ultimate goals? Consider the

actions of the football player who threw the ball to his colleague to his fellow player.


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That act was only partially determined by the need to score a touchdown, since that goal

is always present in any football drive. In that situation, there are far too many contingent

factors that are equally if not more important.

Similarly, let us take food practices (eating beliefs) as an example. In a restaurant

situation, utility based considerations may apply (say, price versus taste). However most

of the world doesnt eat in restaurants most of the time. We eat at home where you eat

what your mother has made. End of story. There is no choice. I may have preferences, but

I dont have a choice. Even the person who tells you what to do might not have a choice

in becoming a leader - Mom has to be Mom, no one else can be Mom. In other words,

leaders are rarely selected by "free choice" i.e., unbiased voting.

I would say that in most daily life situations, we acquire and exercise our beliefs in

contexts where there is a pretty stable pattern of behaviour, and furthermore, that

behaviour is heavily over constrained rather than under constrained, which means that

there are multiple solutions to decision making problems that are all roughly the same in

their attractiveness. From a satisficing perspective, there is no choice or decision to be

made since most routes to the solution are roughly equivalent.

In other words, while the current study of belief dynamics focuses on forms of utility

maximisation as the fundamental dynamical principle, I believe that beliefs are learnt,

transmitted and altered during the course of routine activities where efficiency

considerations are part of the dynamics but not necessarily the most important one. In

certain economic situations we might all be rational actors (though Kahnemann and

Tversky would say otherwise), but that only goes to show that we know how to act

appropriately in economic situations. Why should we assume that eating food is like

buyingfood? I suggest that the study of the dynamics of beliefs should turn away from

economic decision making contexts that are highly non-generic and start thinking about

daily life phenomena like eating food, driving to work, saying your prayers, making

phone calls, all of which are highly ritualized and strongly constrained, where most of the

time things work as you expect and you never have to make a decision (if at all that


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option is even available). In other words, the world hangs together just fine for most

people most of the time since it is "deeply regular". The science of beliefs should be

about the study of these deep regularities.

4. A Preliminary Definition. I want to define the theoretical notion of beliefs with these

daily life activities in mind. Consider eating practice. In most western cultures, people eat

with a knife and fork, with the fork on the left of the plate and the knife to the right. In

India, in most places, people eat with their right hands. Both of these eating practices are

beliefs. What is common to them (and to all beliefs in my account) are three things they

are acts that are shared across a community, they are remarkably stable over time

(compared to the time scale of the act of eating) and they come in tightly coupled clusters

(for example, in western cultures, the decision to put the fork to left and the knife to the

right is paired with other decisions about where the soup spoon goes etc). In general, I

define a belief as follows:

Definition. A Belief is astable social actdirected towards a community such that:

(a) Generically, the Believer wants the community to share the contents of that

belief

(b) Generically, the community wants to acceptthat belief.

(c) Beliefs come in clusters, a modal combination of acts that are interrelated

within a larger frame (say eating).

(d) The goal of the community is to make sure that each cluster of beliefs is

transmitted successfully across time.

Given these beliefs about beliefs, I would like to point out one principle and three aspects

of belief networks. The principle, coherence, is simple to state but hard to model. To

understand coherence at the simplest level, one notes that beliefs are fundamentally

public and social, not internal states and therefore, beliefs are meant to be shared.

Therefore, the structure of beliefs should obey the constraint that they are meant to be

shared.


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(a) Since beliefs are social acts, a belief by me is always followed by another belief

of someone else. Therefore, belief sequences are free of loops, in a precise graph

theoretic sense.

(b) Belief change takes place in small groups (is 7 the magic number?). Small groups

can accept and hence share beief sequences far more quickly and modify them as

well. However, strictly speaking belief sequences are multiple scale, i.e., a

successful belief narrative shares structural properties across various scales of

network size and there is a continuity of narrative as one goes from individual to

community to society to nation.

(c) Within a given context beliefs are generic acts everything else being equal one

always puts the fork on the left. Therefore beliefs encode general properties of the

world in that context.

Having made all these arguments against utilities and decisions, I believe that the best

way to start exploring alternative frameworks is to devise a model that is the simplest

deviation from the classical game theoretic model of belief evolution. The similarities

and differences between the dynamics of a pure game and an augmented game will be

instructive.

5. A Simple Model. Here, I want to outline the simplest model of network dynamics that

incorporates both individual utility functions as well as the inertial force that votes for

stability. According to the definition of beliefs in section 4, the primary quality of beliefs

is that they are shared. Belief dynamics is fundamentally a study of changes in the way

that beliefs are shared which can happen in two ways:

(a) I stop sharing beliefs with you.

(b) I change my beliefs in order to be able to share something with you.

Utility considerations nudge individual agents towards (a) while inertial considerations

bias the system as a whole towards (b). From a modelling point of view, the utility


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constraint is straightforward: each agent has certain desires and aversions, and if the price

is too high, he would rather quit playing the game. The stability constraint should enforce

the fact the system as a whole would rather thatsome game be played rather than none. It

cares less about the content of the game than the fact of its being played. The utilities are

individual while the stability constraint is a collective constraint. Let us now see how to

model these two conflicting demands.

Suppose G is a graph with n nodes, where each node represents an agent and the links

represent a shared belief between the agents. Let Ai be the agents and B

i be the beliefs of

agent Ai. For each shared belief (leading to an edge connecting the two) between agent Ai

and agent Aj, let Gij be a game being played between Ai and Aj with a payoff matrix Mij.

The game is very simple: the two agents either affirm their jointly held belief, or deny

the jointly held belief. The payoffs tell us the cost of affirming or denying the belief. Let

us also make the following constraints on the dynamics:

(a) The price of playing: There is a number, C such that for any two agents Ai and Aj,

if the price of affirming the joint belief is C, they will always deny their joint

belief.

(b) Severance of relations due to repeated denial: There is a number, N such that for

any two agents Ai and Aj, if the joint belief is denied N times, the link between the

two is broken.

(c) Collective Inertia: Let LG be the number of edges in the graph. The total number

of possible edges in the graph is nC2. Then, the inertial constraint says: there is a

number I, 0 I 1, such that LG /nC2 I.

(d) Changing beliefs rather than changing friends: There is a number such that when

(LG /nC2 - I) , that agents Ai and Aj playing a game with cost C will shift to a

new belief with cost C (if available) rather than sever relations.


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6. Summary. In this short note, I have outlined some theoretical reasons for modifying

the classical decision theoretic framework for modelling belief dynamics. Then, I stated a

simple model that incorporates inertial forces along with decision making strategies into a

graph evolution model. The next step in this modelling project is to test the dynamics of

the above model under various initial conditions, for C, N, n etc. The goal is to play the

stability and utility constraints against each other.

some thoughts on belief dynamics

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