a general model of boundedly rational observational

A General Model ofBoundedly Rational Observational Learning:

Theory and Evidence

Claudia Neri (University of St.Gallen)and

Manuel Mueller-Frank (IESE)

International Meeting on Experimental and Behavioral Social SciencesIMEBESS

April 14, 2016

Motivation

social learning: we learn from others, in many ways!

observational learning: we make inferences on the information that others hold,based on the observation of their behavior

economists ask:

how do individuals behave after observing the behavior of others?

what are the long-run aggregate outcomes?

policy makers ask:

how effective is social advertising? (Mueller-Frank & Pai 2015)how effective is a microfinance information campaign? (Banerjee et al. 2013)etc

Mueller-Frank and Neri Boundedly Rational Observational Learning 1 / 21

Motivation (cont.)

Modeling observational learning:

repeated decision-making under uncertainty

individuals observe their own private information

individuals observe the choices of others

Two approaches:

Bayesian updating:

Bikhchandani, Hirshleifer & Welch 1992; Banerjee 1992; Smith & Sorensen2000; Gale & Kariv 2003; ....individuals learn rationally: they make inferences on the private informationof all agents based on the interaction structure and the observed actionspros: useful benchmark

cons: unrealistic, requires computational sophistication

Boundedly rational updating:

DeGroot 1973; DeMarzo, Vayanos & Zwiebel 2003; Golub & Jackson2010&2012; Acemoglu, Ozdaglar & ParandehGheibi 2010pros: tractablecons: arbitrary, requires an infinite real-numbered action space


Motivation (cont.)





Two approaches:

Bayesian updating:

Bikhchandani, Hirshleifer & Welch 1992; Banerjee 1992; Smith & Sorensen2000; Gale & Kariv 2003; ....individuals learn rationally: they make inferences on the private informationof all agents based on the interaction structure and the observed actions

pros: useful benchmark





Motivation (cont.)





Two approaches:

Bayesian updating:






Motivation (cont.)





Two approaches:

Bayesian updating:




DeGroot 1973; DeMarzo, Vayanos & Zwiebel 2003; Golub & Jackson2010&2012; Acemoglu, Ozdaglar & ParandehGheibi 2010

pros: tractablecons: arbitrary, requires an infinite real-numbered action space


Motivation (cont.)





Two approaches:

Bayesian updating:






What we do

we propose a general model of boundedly rational observational learning, based onthe concept of Quasi-Bayesian updating

reduced complexity compared to Bayesian updating:not necessary to consider how each observed action might have been affectedby other actionsrationally foundednot arbitraryapplicable to any environment: no restriction on utility function, state space,action space or signal space

we provide experimental evidence on Quasi-Bayesian updating


What we do (cont.)

we apply Quasi-Bayesian updating to a model of repeated interaction in socialnetworks with binary actions

consensus and information aggregation

generally coincidehard to achieve, since they require highly asymmetric environmentsachievable in finite networks, not in infinite networks

we provide experimental evidence on consensus and information aggregation


The ModelObservational learning

Simplified setting (general setting in the paper):

finite set of agents

binary states, binary actions, binary signals Ω = A = S = 0, 1agents share a common prior over the state space Ω

each agent observes an iid private signal from the signal space S

the distribution over the signal space depends on the realized state

utility = 1 if chosen action matches the realized state, = 0 otherwise

each agent chooses an EU-maximizing action from the action space A

each agent observes a subset of other agents and the actions they chose, andupdates his own action


The ModelQuasi-Bayesian updating

Quasi-Bayesian updatingChoosing an action which is Bayes-optimal conditional on the observed actionsassuming that each observed action was selected by the corresponding agent based onlyon his private signal.

in other models of observational learning:... based also on information inferred from the actions chosen by others

Quasi-Bayesian updating abstracts away from:

structure of interactionindirect inferences on the private information of all agents


The ModelQuasi-Bayesian updating

its specific functional form and its general properties vary among environments

we analyze its implications for aggregate behavior in a model of repeatedinteraction in social networks


The ModelQuasi-Bayesian updating and repeated interaction in social networks

finite set of agents organized in a strongly connected network

agents face uncertainty as described in the general model

t = 0: the state is drawn

t = 1: each agent observes a private signal, and then chooses an action

t = 2, 3, ..: each agent observes the action chosen by each of his neighbors int − 1, and then updates his action



Under which conditions do consensus and information aggregation occur?

consensusconvergence to agreement on one action(for any strongly connected network, for any initial action profile)

information aggregationconsensus on an optimal action conditional on the initial action profile(for any strongly connected network, for any initial action profile)



Under Quasi-Bayesian updating, assuming binary actions:

Theorem 1: consensusnecessary and sufficient condition: highly asymmetric environment:one action is optimal conditional on every possible profile of observed actions, except forthe profile where all agents choose the other action

Theorem 2: information aggregationnecessary and sufficient condition: same as in Theorem 1

Theorem 3for every environment there exists a finite network size n∗ such that consensus andinformation aggregation fail for all networks of size larger than n∗



Consensus and information aggregation

with Quasi-Bayesian in the DeGroot modelupdating & its generalizations

achievable only in achievable only infinite networks infinite networks

hard to achieve easy to achieve


The experimentTask, matching and treatments

urn-guessing game (Anderson&Holt 1997;Choi,Gale&Kariv 2005;Grimm&Mengel 2015)

198 participants

fixed-group random matching

subjects are assigned with a label-identity (A, B, C, ...)subjects are matched into groups (each with a different label-identity)

treatments:

network size (within-subject): 5- or 7-agent networksnetwork connections (between-subject): 1-4 neighbors for each subjectchoice set (between-subject): 2 or 4 urns

rounds correspond to between-subject treatments

each round consists of turns, when choices can be updated

network size subjects groups rounds turns obs.per round

5 100 20 18 6 108007 98 14 14 8 10976all 198 34 21776


The experimentUrn-guessing game

Figure : 2-urn game

black urnwhite urn

Figure : 4-urn game

red urn yellow urn

green urn blue urn


The experimentTiming

each subject privately

observes a ball drawn

randomly from the urn

1st decision turn: “what urn is

used?”

each subject observes the

previous choices of his

neighbors

round

the computer randomly

selects one urn

next round

2nd decision turn: “what urn is

used?”

last decision turn: “what urn is

used?”


The experimentNetwork structure

Since the theoretical framework holds “for any network structure”,the experiment implements several network structures

Figure : 5-agent networks

B

A

C D

E

complete

B

A

C D

E

linked circle A-‐C

B

A

C D

E

linked circle A-‐D

B

A

C D

E

linked circle B-‐D

B

A

C D

E

linked circle B-‐E

B

A

C D

E

linked circle C-‐E

B

A

C D

E

star (A center)

B

A

C D

E

star (B center)

B

A

C D

E

star (C center)



Since the theoretical framework holds “for any network structure”,the experiment implements several network structures

Figure : 7-agent networks

B

A

C

D E

F

G

small world 1: A-‐C, D-‐F

B

A

C

D E

F

G

small world 2: B-‐D, E-‐G

B

A

C

D E

F

G

small world 3: C-‐E, F-‐A

B

A

C

D E

F

G

small world 4: A-‐D, B-‐G

B

A

C

D E

F

G

small world 5: A-‐D

B

A

C

D E

F

G

small world 6: A-‐D, A-‐E, B-‐G

B

A

C

D

E

F

G

connected complete components



Since the theoretical framework lifts the assumption of (common) knowledge of thenetwork structure, the experiment implements no knowledge of the network structure

each subject knows how many neighbors he has and what their label-identity is,

but ignores how many neighbors other members of the network have


Results1st-turn choices

In the 1st turn subjects choose an action based on their private signal.

24 THE AMERICAN ECONOMIC REVIEW MONTH YEAR

lected round was correct.41 Otherwise they received nothing. Participants’

earnings ranged between CHF 22 and CHF 42.5, with an average of CHF

34 (including a CHF 10 show-up fee).42

IV. Experimental evidence: Quasi-Bayesian updating

In this section we analyze the experimental data in the light of the the-

oretical framework presented in Section II. We begin by describing agents’

behavior in the first round. As one might expect, in the first round agents

choose an action based on their private signal. Table 2 reports the distri-

bution across individuals of the frequency with which first-round choices

coincide with private signals. The mean is 0.97 in 5-agent networks and

0.95 in 7-agent networks, and the median is 1 for both network sizes.

Table 2—: Frequency of 1st-round choice equal to private signal. Distribu-tion across individuals.

obs mean median std5-agent networks

all games 100 0.97 1 0.102-urn games 100 0.97 1 0.114-urn games 100 0.97 1 0.127-agent networks

all games 98 0.95 1 0.132-urn games 98 0.94 1 0.144-urn games 98 0.96 1 0.13

We then inspect agents’ behavior in rounds following the first, when agents

have the opportunity to revise their choice after observing their neighbors’

41The difference in payment per correct-decision (CHF 2 versus CHF 2.5) was gaugedto attain similar expected earnings for subjects irrespective of the session they partici-pated in, in order to comply with the ETH Decision Science Lab rules.

42In sessions with 5-subject networks, the range was CHF 22-42 (average CHF 35). Insessions with 7-subject networks, the range was CHF 22.5-42.5 (average CHF 33).

Figure : Empirical distribution of the frequency of 1st-turn choice coinciding with private signal.


ResultsQuasi-Bayesian updating

Pool all data

5-agent networks 7-agent networksstar linked complete linked connected

circle circle complete2-urn 0.95 0.92 0.91 0.91 0.90

4-urn 0.94 0.90 0.90 0.89 0.91

Table : Percentage of choices consistent with Quasi-Bayesian updating (t = 2, 3, ..).


ResultsQuasi-Bayesian updating

Define for each subject the fraction of choices consistent with Quasi-Bayesian updating.26 THE AMERICAN ECONOMIC REVIEW MONTH YEAR

0.2

.4.6

Frac

tion

0 .2 .4 .6 .8 1Quasi-Bayesian updating (individual score)

(a) 2-urn games, 5-agent networks

0.2

.4.6

Frac

tion


(b) 2-urn games, 7-agent networks

0.2

.4.6

Frac

tion


(c) 4-urn games, 5-agent networks

0.2

.4.6

Frac

tion


(d) 4-urn games, 7-agent networks

Figure 1. : Empirical distribution of participants’ individual consistencywith Quasi-Bayesian updating. A kernel density estimate is also reported,using an Epanechnikov kernel function with optimal half-width.

V. Long-run properties of Quasi-Bayesian updating in social

networks

The two main questions addressed by the literature on learning in so-

cial networks concern the conditions on the environment and the network

structure under which consensus (i.e. asymptotic agreement in actions)

and information aggregation (i.e. optimality of long-run actions conditional

on the pooled private information of all agents) occur.44 We address both

44For an analysis of consensus see DeMarzo, Vayanos and Zwiebel (2003) and Mueller-Frank (2014) for non-Bayesian models, and Gale and Kariv (2003), Rosenberg, Solanand Vieille (2009), and Mueller-Frank (2013a) for Bayesian models. For an analysis oflearning see Golub and Jackson (2010), Mueller-Frank (2013a,2014), Arieli and Mueller-

Figure : Distribution of participants’ individual consistency with Quasi-Bayesian updating. Median > 92%.


ResultsConsensus and information aggregation

Fraction of groups reaching consensus by the last updating turn.

5-agent networks 7-agent networksstar linked complete linked connected

circle circle complete

consensus 0.48 0.59 0.70 0.29 0.50

of whichwith information 0.97 0.98 1 0.96 1aggregation

Table : Two-urn games. Averages across 20 groups for 5-agent networks and across 14 groups for 7-agent networks.


What we did and what is next

what we did

general model of boundedly rational observational learning based on the concept ofQuasi-Bayesian updating

conditions under which Quasi-Bayesian updating yields consensus and informationaggregation in a model of repeated interaction in social networks

support from experimental evidence:

Quasi-Bayesian updatingconsensus is hard to achieveif consensus occurs, it achieves information aggregation

what is nextIs Quasi-Bayesian updating a good description of behavior in other observationallearning environments and in more complex settings?


a general model of boundedly rational observational

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