Engagement, Disengagement or Exit:
A theory of inter-group communication and association
Maggie Penn
Washington University in St. Louis
1
In many well-functioning & plural societies heterogeneous groups
associate with each other
• Groups communicate and interact with each other, and this
communication may or may not be truthful (informative)
• Groups willingly share power and bear externalities
(stemming from divergent preferences)
Groups benefit from association because everyone makes better
decisions when able to learn from others
2
If the externalities borne by a group become too great
it may choose to unilaterally “exit” society
• Group is an extreme preference outlier and cannot endure sharing
power with others
• Group is particularly well-informed on its own, and doesn’t need to
learn from others
In the latter case, the exit of such a group is an obvious loss to others;
(this paper shows) the exit of a preference outlier can be negative too
3
What kinds of societies support voluntary association and information
sharing across groups?
1. Multiple groups with private information
2. Groups can voluntarily associate with other groups or “go it alone”
3. Within the association:
• Groups communicate with each other and then make decisions
• Each group’s decision jointly affects everyone
4
This paper characterizes the “associations” of groups that can be
sustained in equilibrium
These are sets of groups for which:
1. Some information transmission is possible
2. The benefits of this information outweigh the costs stemming
from preference diversity within the association
3. Groups outside the association prefer to stay outside
5
What I am interested in
• Successful societies as groups that voluntarily interact and
willingly share power
• Larger “associations” may not maximize utilitarian welfare, but do
maximize (something akin to) Rawlsian welfare
6
Specific questions
• How to use power-sharing institutions to grow the size of an
association
• How intra-group communication (norms) might affect a group’s
ability to associate with outgroups
• What equilibrium associations look like
7
The Model
• An unknown state of nature, θ, is drawn from Uniform[0, 1]
• A set of groups g ∈ G that differ in:
– Size ng (a positive integer)
– Bias βg (a real number)
Related Models
Galeotti, Ghiglino, & Squintani (2013), Dewan & Squintani (2012),
Dewan et al. (2012), Patty (2013), Gailmard & Patty (2013), Patty &
Penn (2012)...
8
Association and exit
• Prior to any information being revealed, each group chooses
whether to associate (ag = a) or exit (ag = x)
• The set of groups that choose association is denoted R,
“the association”
• The set of groups that choose exit is denoted X
9
Private information
• (Recall the unknown state of nature θ)
• After the association decision is made, each group g privately
observes ng independent {0, 1} signals with
Pr[signal = 1 | θ]
=
θ
Pr[signal = 0 | θ]
=
1−θ
10
Communication
• Each g in association R sends a public message to the groups in R
• g’s private information is sg , the # of “ones” it observed
11
Preferences
• Every group will undertake an activity, yg ∈ R
• If g has chosen exit its payoff is
−(yg − θ − βg )2
• If g has chosen association its payoff is
X
−
αhR (yh − θ − βg )2
h∈R
• αhR is an exogenous weight attached to group h’s actions by each
member of R (“influence”)
12
(Unknown “state of world”)
θ
y1
y2
y3
(Ideal point = θ + known “bias”)
13
(Unknown “state of world”)
θ
y2
y1
y3
(Ideal point = θ + known “bias”)
14
(Unknown “state of world”)
θ
y2
y1
y3
(Ideal point = θ + known “bias”)
15
Summary of what groups do
1. Choose whether or not to associate
2. If associating, choose a message to send other groups in
association
3. Choose an activity based on information at hand
16
Equilibrium
• Focus on messaging strategies that are either separating or pooling
• Equilibria characterized as divisions of G into sets {E, D, X}
– E: ag = a and message= sg (associate & truthful)
– D: ag = a and message= 0 (associate & babble)
– X: ag = x (exit)
17
Messaging equilibrium
A division of association R into disjoint sets {E, D} is a
messaging equilibrium if the following hold:
1. Equilibrium beliefs: messages sent by g ∈ E are taken as equal to
sg and messages sent by g ∈ D are disregarded as uninformative
2. Actions yg∗ are sequentially rational given groups’ own signals and
the messages they receive
3. Groups g ∈ E have no incentive to lie given the (correct) beliefs,
actions and equilibrium messaging strategies of other h ∈ R
18
Societal equilibrium (“stable societies”)
A society σ = {E, D, X} is stable if the following hold:
1. {E, D} is a messaging equilibrium for association R = E ∪ D
2. Groups don’t want to change their association decisions:
• For g ∈ R, EU (association) ≥ EU (exit)
• For g ∈ X, EU (exit) ≥ EU (association at “best” equilibrium)
19
When can truthful messaging by group g to association R be
sustained?
• When the impact of g’s lie shifts the policy choices of the
“listening groups” too much
20
When can 1 message truthfully to 2?
y2
y1
21
Truthful messaging sustained
(1’s lie moves 2 too much)
y2
y2’
y1
22
Truthful messaging can’t be sustained
(1’s lie is beneficial)
y2
y2’
y1
23
It is harder for centrist groups to lie
y2
y1
y3
24
It is harder for centrist groups to lie
bad
good
y2
y2’ y1
y3
25
y3’
Takeaway points about truthful messaging
• “Congestion effect”: too much information can hinder
communication
– More information (signals) makes it easier to tell a “tiny lie”
– Lying less costly as it shifts others less
• Centrist groups more likely to be truthful
– Lying more costly as it necessarily shifts some groups farther
– Similarly, extreme groups less likely to be truthful
26
When can voluntary association be sustained?
• When the improved precision of the decisions of the association
outweighs the costs of power sharing relative to “going it alone”
27
EUg (association)
↓
−
P
k∈R
2
αR
k (βk − βg ) −
EUg (exit)
P
αR
h V (θ|nE ) −
P
↓
αR
j V (θ|nE + nj ) > −V (θ|ng )
j∈D
h∈E
• Negative externality stemming from divergent preferences in R
28
EUg (association)
↓
−
P
k∈R
2
αR
k (βk − βg ) −
EUg (exit)
P
αR
h V (θ|nE ) −
P
↓
αR
j V (θ|nE + nj ) > −V (θ|ng )
j∈D
h∈E
• Precision of decisions for groups in E (nE is # signals in E)
29
EUg (association)
↓
−
P
k∈R
2
αR
k (βk − βg ) −
EUg (exit)
P
αR
h V (θ|nE ) −
P
j∈D
h∈E
• Precision of decisions for groups in D
30
↓
αR
j V (θ|nE + nj ) > −V (θ|ng )
EUg (association)
↓
−
P
k∈R
2
αR
k (βk − βg ) −
EUg (exit)
P
αR
h V (θ|nE ) −
P
↓
αR
j V (θ|nE + nj ) > −V (θ|ng )
j∈D
h∈E
• Group g’s expected utility from exit (precision of own decision)
31
Important point about associations
For a fixed association R, the set of messaging equilibria µ(R) can be
Pareto ranked
Better messaging equilibria (for everyone in R) minimize this term:
−
X
k∈R
αkR (βk
2
− βg ) −
X
αhR V
(θ|nE ) −
X
j∈D
h∈E
32
αjR V (θ|nE + nj )
What do equilibrium associations look like?
Example 1: A third group induces Groups 1 and 2 to associate
Example 2: A third group prevents Groups 1 and 2 from associating
33
Example: Group 3 induces Groups 1 and 2 to associate
1 & 2 do not associate or communicate
{
β1
β2
• 1 & 2 too distant to communicate
34
Example: Group 3 induces Groups 1 and 2 to associate
1
Talk
2
Babble
3
Talk
β3
β1
β3
β2
• If 3 enters at β1 then 3 & 1 message truthfully; 2 listens
35
Example: Group 3 induces Groups 1 and 2 to associate
1
Talk
2
Babble
3
Talk
β3
β1
β3
β2
• As β3 becomes more moderate 2 can be truthful too
36
Example: Group 3 induces Groups 1 and 2 to associate
1
Talk
2
Babble
3
Talk
β3
β1
β2
β3
• As β3 moves toward and past β2 , 1 can no longer be truthful
37
Example: Group 3 induces Groups 1 and 2 to associate
1
Talk
2
Babble
3
Talk
β3
β1
β2
β3
• As β3 becomes more extreme, 3 can no longer be truthful
38
Example: Group 3 induces Groups 1 and 2 to associate
1
Talk
2
Babble
3
Talk
Leave
β3
β1
β2
β3
• 1 exits; 2 & 3 can now communicate truthfully
39
Example: Group 3 induces Groups 1 and 2 to associate
1
Talk
2
Babble
3
Talk
Leave
β3
β1
β2
β3
• 3 becomes too distant from 2; they all part ways
40
Example: Group 3 prevents Groups 1 and 2 from associating
Information transmitted
between 1 & 2
{
β1
β2
• 1 & 2 associate and communicate
41
Example: Group 3 prevents Groups 1 and 2 from associating
1
Talk
2
Talk
3
Talk
β3
β1
β2
β3
• If 3 enters at β1 all are truthful ... for a while
42
Example: Group 3 prevents Groups 1 and 2 from associating
1
Talk
2
Talk
3
Talk
Babble
β3
β1
β3
β2
• As β3 becomes more extreme 3 can no longer be truthful
43
Example: Group 3 prevents Groups 1 and 2 from associating
1
Talk
2
Talk
3
Talk
Babble
β3
β1
β3
β2
• As β3 moves even farther, 1 can no longer be truthful
44
Example: Group 3 prevents Groups 1 and 2 from associating
1
Talk
Leave
2
Talk
Leave
3
Talk
Babble
Leave
β3
β1
β2
β3
• 1 leaves, and 1 & 2 can’t associate because 3 wants to join!
45
Example: Group 3 prevents Groups 1 and 2 from associating
1
Talk
Leave
2
Talk
Leave
3
Talk
Babble
Leave
β3
β1
β3
β2
• 3 no longer wants to join; 1 & 2 can now communicate truthfully
46
Example: Policy discretion to induce association
β1=0
β2=.07
β3=.17
• No pair of groups would associate / communicate
• 1 & 3 would benefit from associating with 2
– Both would prefer α = (0, 1, 0) to exit
– Is there α > 0 for which association is possible?
47
α=(0,1,0)
α=(0,0,1)
α=(1,0,0)
• Association occurs for shaded region defined as:
{(α1 , α2 , α3 ) : 2.95α3 ≤ α1 ≤ 26.12α3 }
48
α=(0,1,0)
2 can’t communicate
(α3 too low)
α=(0,0,1)
α=(1,0,0)
• In this region (where α1 is much higher than α3 ) Group 2 is too
tempted to lie to Group 1
49
α=(0,1,0)
2 can’t communicate
(α1 too low)
α=(0,0,1)
α=(1,0,0)
• In this region (where α3 is high relative to α1 ) Group 2 is too
tempted to lie to Group 3
50
α=(0,1,0)
2 won’t associate
(α3 too high)
α=(0,0,1)
α=(1,0,0)
• In this region Group 2 can communicate truthfully, but would
prefer to be alone; α3 is too high
51
α=(0,1,0)
2 can’t communicate
(α3 too low)
2 won’t associate
(α3 too high)
2 can’t communicate
(α1 too low)
α=(0,0,1)
α=(1,0,0)
• Interior of shaded region leaves everyone strictly better than exit
(even Group 2, which gets no information from Groups 1 and 3!)
52
α=(0,1,0)
Proportional weight
(.29, .57, .14)
Equal weight
(.33, .33, .33)
α=(0,0,1)
α=(1,0,0)
• Neither equal weights nor proportional weights induce association
• Power sharing with Group 3 (a low-information preference outlier)
is necessary to induce association
53
Bringing the individuals back in: a different game
• The group decides whether to associate or not
(via majority rule, consensus, a leader, etc.)
• Each person privately receives a single signal
• People in association each send a public message about own signal
• Group members privately reveal their messages to each other &
set policy
⇒ Pure-strategy, symmetric equilibria to this game are identical to the
equilibria I have considered so far
54
Banning intra-group communication
• If group prohibits intra-group communication after public
messaging stage, information transmission & association much
easier to achieve
• Banning private communication ⇒ a misstated signal biases both
outgroup (good) and ingroup (bad)
• Good for information transmission & association, possibly bad for
other reasons!
55
“[M]aking good use of dispersed knowledge is the original source of
democracy’s strength.”
–Josiah Ober, Democracy and Knowledge:
Innovation and learning in classical Athens, 2008.
• Thoughtful institutional design in ancient Athens consolidated
information that was widely dispersed throughout population
• Aggregation & distribution of knowledge played a causal role in
Athenian success relative to peer polities
56
The reforms of Cleisthenes (507 BC)
• Athenian society composed of numerous disconnected demes
• To grow capacity of the state, a mechanism was needed to induce
individuals to develop bridging ties of communication across demes
57
Artificial tribes
C
U
I
I
U
I
U
C
I
U
C
C
C
C
C
C
U
U
U
U
I
I
I
I
Tribe 1
Tribe 2
• Demes of same type within tribe foster truthful communication
• Diversity of demes enables emergence of “bridging” groups
58
• Successful societies as “equilibrium associations”
– Some beneficial communication is possible across groups, and
benefits outweigh costs of power sharing
• Association / communication decisions interdependent
– A group’s entry can stimulate association or kill preexisting
communication
• Power-sharing mechanisms may induce an informed group to
“talk”
• Intra-group mechanisms (norms?) may make messages more or
less credible to an association
59