4.3 Log-Linear Learning
4.3.1 Analysis of Convergence and Log-Linear Learning
Definition 4.1 (Learning). An agent i is said to learn the truth, if limt→∞µi,t(θ) = 1, Pθ-almost surely.
We begin our analysis of convergence and learning under the update rule in (4.3.2) by considering the case of a single agenti, who starts from a prior beliefνi(·)and sequentially
updates her beliefs according to Bayes rule:
µi,t(^θ) = Xµi,t−1(^θ)`i(si,t|θ^)
˜ θ∈Θ
µi,t−1(θ˜)`i(si,t|θ˜)
,∀θ^∈Θ. (4.3.3) The Bayesian belief update in (4.3.3) linearizes in terms of the log-ratio of beliefs and signal likelihoods,φi,t(·)andλθˇ(·), leading to
φi,t(θˇ) =log νi( ˇ θ) νi(θ) ! + t X τ=0 λθˇ(si,τ)→log νi(θˇ) νi(θ) ! + (t+1)Eθ{λθˇ(si,0)}, (4.3.4) Pθ-almost surely, as t → ∞; by the strong law of large numbers [177, Theorem 22.1]
applied to the sequence ofEθ-integrable, independent and identically distributed variables:
λθˇ(si,t), t∈N0. In particular, ifDKL `i(·|θ)||`i(·|θˇ)
:= −Eθ{λθˇ(si,t)}> 0, thenφi,t(θˇ)→
−∞almost surely and agenti asymptotically rejects the false state ˇθ in favor of the true stateθ, putting a vanishing belief on the former relative to the latter. Therefore, the single Bayesian agent following (4.3.3) learns the truth if and only if DKL `i(·|θ)||`i(·|θˇ)
> 0
for all ˇθ 6= θ and the learning is asymptotically exponentially fast at the rate minθ∈Θˇ \{θ}
DKL `i(·|θ)||`i(·|θˇ)
as shown in [165].1
The preceding result is also applicable to the case of a Bayesian agents with direct (centralized) access to all observations across the network: consider an outside Bayesian agent o^ who shares the same common knowledge of the prior and signal structures with the networked agents; in particular, o^ knows the signal structures `i(·|θ^), for all θ^ ∈ Θ
andi ∈ [n]; thence, making the same inferences as any other agent when given access to the same observations. Consider next a Gedanken experiment where o^ is granted direct access to all the signals of every agent at all times. The analysis leading to (4.3.4) can be applied to the evolution of log belief ratios foro^, whose observations at every timet ∈N0
1Note from the information inequality for the Kullback-Leibler divergence thatD
KL(·||·) ≥ 0and the
inequality is strict whenever`i(·|θˇ)6≡`i(·|θ), i.e.∃s∈ Sisuch that`i(s|θˇ)6=`i(s|θ)[193, Theorem 2.6.3].
Further note that whenever`i(·|θˇ)≡`i(·|θ)or equivalentlyDKL `i(·|θ)||`i(·|θˇ)
=0, then the two states ˇθ andθare statically indistinguishable to agenti: there is no way for agentito distinguish between ˇθandθ, based only on her received signals. This is because bothθand ˇθinduce the same probability distribution on her sequence of observed i.i.d. signals. Since different statesθ^∈Θare distinguished through their different likelihood functions`i(· | θ^); the more refined such differences are, the better the states are distinguished.
is an element of the product spaceQi∈[n]Si. Subsequently, the centralized Bayesian beliefs
concentrate on the true state at the asymptotically exponentially fast rate of
Rn := min ˇ θ∈Θ\{θ} DKL Y i∈[n] `i(·|θ) Y i∈[n] `i(·|θˇ) = min ˇ θ∈Θ\{θ} X i∈[n] DKL `i(·|θ)||`i(·|θˇ) .(4.3.5) Next to understand the evolution of beliefs under the log-linear updates in (4.3.2), consider the network graph structure as encoded by its adjacency matrix A defined as
[A]ij =1 ⇐⇒ (j, i)∈ E, and[A]ij =0otherwise. For a strongly connectedGthe Perron-
Frobenius theory [115, Theorem 1.5] implies thatAhas a simple positive real eigenvalue, denoted byρ > 0, which is equal to its spectral radius. Moreover, the left eigenspace asso- ciated with ρis one-dimensional with the corresponding eigenvectorα = (α1, . . . , αn)T,
uniquely satisfyingPni=1αi =1, αi > 0, ∀i ∈ [n], andαTA=ραT. The entryαiis also
called the centrality of agenti and as the name suggests, it is a measure of how central is the location of agent in the network. Our main result state that almost sure learning cannot be realized in a strongly connected network unless it has unit spectral radius which is the case only of a directed circle.
Theorem 4.4(No Learning when Spectral Radius ρ > 1). In a strongly connected social network and under the memoryless belief updates in(4.3.2), no agents can learn the truth unless the spectral radiusρ=1.
Proof outline: A complete proof is included in Appendix A.14, but here we provide a
description of the mechanism and the interplay between the belief aggregation and infor- mation propagation. To facilitate the exposition of the underlying logic we introduce some notation. We define a global (network-wide) random variable Φt(θˇ) :=
Pn
i=1αiφi,t(θˇ),
where αi is the centrality of agent i andΦt(θˇ) characterizes how biased (away from the
truth and towards ˇθ) the network beliefs and priors are at each point in time. In partic- ular, if any agent is to learn the truth, then Φt(θˇ) → −∞ as t → ∞ for all the false
states ˇθ ∈Θ\ {θ}. To proceed, we define another network-wide random variable Λt(θˇ)
:= Pni=1αiλθˇ(si,t) which characterizes the information content of the observed signals
(received information) for the entire network, at each time t. Moreover, since the re- ceived signal vectors {(s1,t, . . . ,sn,t), t ∈ N0} are i.i.d. over time, ∀θˇ 6= θ, {Λt(θˇ), t ∈
N0} constitutes a sequence of i.i.d. random variables satisfying E
Λt(θˇ) = − Pn i=1 αiDKL `i(·|θ)||`i(·|θˇ)
6 0. In order for the agents to learn the true state of the world based on their observations, it is necessary that at each false state ˇθ 6= θ some agent be able to distinguish ˇθ from the truthθ, in which caseEΛt(θˇ) < 0, and we can refer to
this criterion as global identifiablityfor the true stateθ. This global identifiability condi- tion can be also viewed in the following sense: consider a gedanken experiment where an external fully rational observero^is granted direct access to all the signals of all agents in
the network and assume further that she shares the same common knowledge of the prior and signal structures with the network agents. Theno^learns the truth if, and only if, it is globally identifiable.
In Appendix A.14 we argue that under the update rules in (4.3.2) the global belief ratio statisticsΦt(θˇ)evolves as a sum of weighted i.i.d. variablesρτΛt−τ(θˇ):
Φt(θˇ) = t X τ=0 ρτ Λt−τ(θˇ) + (1−ρ)β(θˇ) , (4.3.6) whereβ(θˇ) := Pni=1αilog νi(θˇ)/νi(θ)
is a measure of bias in the initial prior beliefs. The weights in (4.3.6) form a geometric progressions in ρ; hence, the variables increase unbounded in their variance and convergence cannot hold true in a strongly connected social network, unlessρ=1. This is due to the fact thatρupper bounds the average degree of the graph [194, Chapter 2], and every node in a strongly connected graph has degree greater than or equal to one, subsequentlyρ≥1for all strongly connected graphs.
Remark4.5 (Polarization, data incest and unlearning). The unlearning in the case ofρ > 1
in Theorem 4.4, which applies to all strongly connected topologies except directed cir- cles (whereρ = 1, see Subsection 4.3.2 below), is related to the inefficiencies associated with social learning and can be attributed to the agents’ naivety in inferring the sources of their information, and their inability to interpret the actions of their neighbors rationally [195]. In particular, when ρ > 1the noise or randomness in the agents’ observations is amplified at every stage of network interactions; since the agents fail to correct for the repetitions in the sources of their observations as in the case of persuasion bias argued by DeMarzo, Vayanos and Zwiebel [47], or data incest argued by Krishnamurthy and Hoiles [27]. When ρ > 1 the effect of the agents’ priors is also amplified through the network interactions and those states θ^ for whichβ(^θ) > 0 in (4.3.6), will be asymptotically re- jected asPtτ=0ρτ(1−ρ)β(θˇ)→ −∞, irrespectively of the observed dataΛτ(θˇ), τ∈N0.
This phenomenon arises as agents engage in excessive anti-imitative behavior, compensat- ing for the neighboring priors at every period [97]. It is justified as a case of choice shift toward more extreme opinions [5, 6] or group polarization [7, 8], when like-minded people after interacting with each other and under the influence of their mutually positive feedback become more extreme in their opinions, and less receptive of opposing beliefs.