DeGroot learning

Take a society of $n$ agents where everybody has an opinion on a subject, represented by a vector of probabilities $p(0)\; =\; (p\_1(0),\; \backslash dots,\; p\_n(0)\; )$. Agents obtain no new information based on which they can update their opinions but they communicate with other agents. Links between agents (who knows whom) and the weight they put on each other's opinions is represented by a trust matrix $T$ where $T\_\{ij\}$ is the weight that agent $i$ puts on agent $j$'s opinion. The trust matrix is thus in a one-to-one relationship with a weighted, directed graph where there is an edge between $i$ and $j$ if and only if $T\_\{ij\}\; >\; 0$. The trust matrix is stochastic, its rows consists of nonnegative real numbers, with each row summing to 1.

Formally, the beliefs are updated in each period as

:$$

p(t) = T p(t-1)

so the $t$ th period opinions are related to the initial opinions by

:$$

p(t) = T^t p(0)

An important question is whether beliefs converge to a limit and to each other in the long run.

As the trust matrix is stochastic, standard results in Markov chain theory can be used to state conditions under which the limit

:$$

p(\infty) = \lim_{t \to \infty} p(t) = \lim_{t \to \infty} T^t p(0)

exists for any initial beliefs $p(0)\; \backslash in\; [0,\; 1]^n$. The following cases are treated in Golub and Jackson

(2010).

If the social network graph (represented by the trust matrix) is strongly connected, convergence of beliefs is equivalent to each of the following properties:

• the graph represented by $T$ is aperiodic
• there is a unique left eigenvector $s$ of $T$ corresponding to eigenvalue 1 whose entries sum to 1 such that, for every $p\; \backslash in\; [0,\; 1]^n$, $\backslash left(\; \backslash lim\_\{t\; \backslash to\; \backslash infty\}\; T^t\; p\; \backslash right)\_i\; =\; s\; \backslash cdot\; p$ for every $i\; \backslash in\; \backslash \{1,\; \backslash dots,\; n\; \backslash \}$ where $\backslash cdot$ denotes the dot product.

The equivalence between the last two is a direct consequence from Perron–Frobenius theorem.

It is not necessary to have a strongly connected social network to have convergent beliefs, however,

the equality of limiting beliefs does not hold in general.

We say that a group of agents $C\; \backslash subseteq\; \backslash \{1,\; \backslash dots,\; n\; \backslash \}$ is closed if for any $i\; \backslash in\; C$, $T\_\{ij\}\; >\; 0$ only if $j\; \backslash in\; C$ . Beliefs are convergent if and only if every set of nodes (representing individuals) that is strongly connected and closed is also aperiodic.

A group $C$ of individuals is said to reach a consensus if $p\_i(\backslash infty)=\; p\_j(\backslash infty)$ for any $i,\; j\; \backslash in\; C$. This means that, as a result of the learning process, in the limit they have the same belief on the subject.

With a strongly connected and aperiodic network the whole group reaches a consensus.

In general, any strongly connected and closed group $C$ of individuals reaches a consensus for every initial vector of beliefs if and only if it is aperiodic. If, for example, there are two groups satisfying these assumptions, they reach a consensus inside the groups but there is not necessarily a consensus at the society level.

Take a strongly connected and aperiodic social network. In this case, the common limiting belief is determined by the initial beliefs through

:$$

p(\infty) = s \cdot p(0)

where $s$ is the unique unit length left eigenvector of $T$ corresponding to the eigenvalue 1. The vector $s$ shows the weights that agents put on each other's initial beliefs in the consensus limit. Thus, the higher is $s\_i$, the more influence individual $i$ has on the consensus belief.

The eigenvector property $s\; =\; s\; T$ implies that

:$s\_i\; =\; \backslash sum\_\{j=1\}^n\; T\_\{ji\}\; s\_j$

This means that the influence of $i$ is a weighted average of those agents' influence $s\_j$ who pay attention to $i$, with weights of their level of trust. Hence influential agents are characterized by being trusted by other individuals with high influence.

These examples appear in Jackson (2008).

File:DeGroot Learning Convergent.svg

Consider a three-individual society with the following trust matrix:

:$$

T =

\begin{pmatrix}

0 & 1/2 & 1/2 \\

1 & 0 & 0 \\

0 & 1 & 0 \\

\end{pmatrix}

Hence the first person weights the beliefs of the other two with equally, while

the second listens only to the first, the third only to the second individual.

For this social trust structure, the limit exists and equals

:$$

\lim_{t \to \infty} T^t p(0) = \left(\lim_{t \to \infty} T^t\right) p(0) = \begin{pmatrix}

2/5 & 2/5 & 1/5 \\

2/5 & 2/5 & 1/5 \\

2/5 & 2/5 & 1/5 \\

\end{pmatrix} p(0)

so the influence vector is $s\; =\; \backslash left(\; 2/5,\; 2/5,\; 1/5\; \backslash right)$ and the consensus belief is

$2/5\; p\_1(0)\; +\; 2/5\; p\_2(0)\; +\; 1/5\; p\_3(0)$. In words, independently of the initial beliefs,

individuals reach a consensus where the initial belief of the first and the second person has twice as

high influence than the third one's.

File:DeGroot Learning Non-Convergent.svg

If we change the previous example such that the third person also listens exclusively to the first

one, we have the following trust matrix:

:$$

T =

\begin{pmatrix}

0 & 1/2 & 1/2 \\

1 & 0 & 0 \\

1 & 0 & 0 \\

\end{pmatrix}

In this case for any $k\; \backslash geq\; 1$ we have

:$$

T^{2k - 1} =

\begin{pmatrix}

0 & 1/2 & 1/2 \\

1 & 0 & 0 \\

1 & 0 & 0 \\

\end{pmatrix}

and

:$$

T^{2k} =

\begin{pmatrix}

1 & 0 & 0 \\

0 & 1/2 & 1/2 \\

0 & 1/2 & 1/2 \\

\end{pmatrix}

so $\backslash lim\_\{t\; \backslash to\; \backslash infty\}\; T^t$ does not exist and

beliefs do not converge in the limit. Intuitively, 1 is updating based on 2 and 3's beliefs while

2 and 3 update solely based on 1's belief so they interchange their beliefs in each period.

It is possible to examine the outcome of the DeGroot learning process in large societies,

that is, in the $n\; \backslash to\; \backslash infty$ limit.

Let the subject on which people have opinions be a "true state" $\backslash mu\; \backslash in\; [0,\; 1]$. Assume that individuals

have independent noisy signals $p\_i^\{(0)\}(n)$ of $\backslash mu$

(now superscript refers to time, the argument to the size of the society).

Assume that for all $n$ the trust matrix $T(n)$ is such that the

limiting beliefs $p\_i^\{(\backslash infty)\}(n)$ exists independently from the initial beliefs.

Then the sequence of societies $\backslash left(\; T(n)\; \backslash right)\_\{n\; =\; 1\}^\{\backslash infty\}$ is called wise if

:$$

\max_{i \leq n} | p_i^{(\infty)} - \mu | \xrightarrow{\ p\ } 0

where $\backslash xrightarrow\{\backslash \; p\backslash \; \}$ denotes convergence in probability.

This means that if the society grows without bound, over time they will have a common and accurate

belief on the uncertain subject.

A necessary and sufficient condition for wisdom

can be given with the help of influence vectors. A sequence of societies is wise if and only

if

:$$

\lim_{n \to \infty} \max_{i \leq n} s_i(n) = 0

that is, the society is wise precisely when even the most influential individual's influence vanishes in the

large society limit. For further characterization and examples see Golub and Jackson (2010).

• DeGroot, Morris H. 1974. “[https://www.jstor.org/stable/10.2307/2285509 Reaching a Consensus.]” Journal of the American Statistical Association, 69(345): 118–21.
• French, John R. P. 1956. “A Formal Theory of Social Power” Psychological Review, 63: 181–94.
• Golub, Benjamin & Matthew O. Jackson 2010. "[http://www.aeaweb.org/articles.php?doi=10.1257/mic.2.1.112 Naïve Learning in Social Networks and the Wisdom of Crowds]," American Economic Journal: Microeconomics, American Economic Association, vol. 2(1), pages 112-49, February.
• Jackson, Matthew O. 2008. Social and Economic Networks. Princeton University Press.
• Harary, Frank. 1959. “[https://psycnet.apa.org/record/1960-06701-006 A Criterion for Unanimity in French's Theory of Social Power]” in Dorwin Cartwright (ed.), Studies in Social Power, Ann Arbor, MI: Institute for Social Research.

Category:Social learning theory