Conflict and the emergence of hierarchical social structures: a game-theoretic model

and experimental investigation

Charlotte Rutten Wojtek Przepiorka & Vincent Buskens

Social Norms and Institutions Model Building, Applications and Empirical Corroborations

May 10-15, 2015, International Conference at the Congressi Stefano Franscini (CSF) of ETH Zurich, Monte Verità, Ascona

Background

• Gambetta (2009) – “Why prisoners fight, and Signal” (Ch. 4 “Codes of the Underworld”)

• Gould (2003) – “Collision of Wills: How ambiguity about social

rank breeds conflict” • Benard (2013) – “Reputation systems, aggression, and

deterrence in social interaction”

• Ullmann-Margalit (1977) – “Social hierarchy formation and the emergence of partiality norms”

Research aims

• We develop a game theoretic model to study the emergence of hierarchical social structures and conflict more rigorously and we test this model experimentally.

• We study the effect of information context, that is,

different kinds of • uncertainty about own and others’ rank • opportunities for learning one’s own and others’ rank

• Base-line model which allows for decreasing abstraction

Research question How do different information contexts affect the amount of conflict involved with the emergence of informal social hierarchies?

The game theoretic model

• A linear hierarchical social structure, which is implicitly present in a small group

• ‘Strength’ as an a priori unobserved property of actors;

relative strength determines one’s rank • Dyadic interactions between actors of unequal ‘strength’ in a

modified Chicken Game setting

Modified chicken game with complete information

• Specification of the dyadic interaction

c > w/2

• Stronger actor has a dominant strategy; challenging • Best reply strategy of the weaker actor; shirking

Weaker actor Ch Sh

Stronger actor Ch w-c ; -c 0 ; w Sh 0 ; w w/2 ; w/2

Mdified chicken game with incomplete information

Assumption: player i’s strength Si ∈ {1, 2,... , n), where n is the number of actors in the group

Experimental conditions

i knows j’s interaction history

i kno

ws j

’s in

tera

ctio

n h

isto

ry w

ith i

No

i k

now

s j’s

inte

ract

ion

his

tory

with

i Yes

i knows own rank i knows own rank

No Yes No Yes

No

Context 1

Context 2

No

Context 5

Context 6

Yes

Context 3

Context 4

Yes

Context 7

Context 8

Model predictions

Predictions are based on the modified Chicken Game with incomplete information and w = 5, c = 3 and n = 4

Weaker actor Ch Sh

Stronger actor Ch w-c ; -c 0 ; w Sh 0 ; w w/2 ; w/2

Information context 1 2 5 6

i knows own rank No Yes No Yes

i knows j ’s interaction history with i No No No No

i knows j ’s interaction history No No Yes Yes

Outcome variables t = 0 t = T t = 0 t = T t = 0 t = T t = 0 t = T

challenging .83 .50 .5 0 .50 .83 .50 .50 .50

c onflict ( Ch, Ch) .69 .17 .17 .17 .69 .00 .17 .00

s haring (Sh, Sh) .03 .17 .17 .17 .03 .00 .17 .00

The experiment

• A computerized lab experiment with N = 128 student subjects

• Each session consists of 4 parts (4 information contexts), which last for 15 rounds.

• Before each part starts, subjects are randomly matched with 3 other persons in the lab. These groups remain the same throughout the whole part.

• Subjects are also randomly assigned a number between 1 and 4, which he or she keeps for the entire part.

• In each round subjects are randomly matched with another person in their group and each subjects chooses for Action A or Action B.

The experiment

Some very first descriptive results % challenging in the first / last 3 rounds per rank

0.40

0.49

0.74

0.87

0.21

0.32

0.70

0.96

Rank 1 Rank 2 Rank 3 Rank 40.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00C

halle

nge

rate

Period 1 - 3 Period 13 - 15 95%-CI

Some very first descriptive results % challenging in the first / last 3 rounds per information context

0.75

0.55

0.69

0.510.56 0.54 0.57

0.53

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00C

halle

nge

rate

own rank nohistory no

own rank yeshistory no

own rank nohistory yes

own rank yeshistory yes

Period 1 - 3 Period 13 - 15 95%-CI

Some very first descriptive results % conflict in the first / last 3 rounds per information context

0.55

0.25

0.47

0.21

0.290.24

0.17

0.09

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00C

onfli

ct ra

te

own rank nohistory no

own rank yeshistory no

own rank nohistory yes

own rank yeshistory yes

Period 1 - 3 Period 13 - 15 95%-CI

So, some preliminary conclusions…

• We develop a game theoretic model to study the emergence of hierarchical social structures and conflict more rigorously and we test this model experimentally.

• Subjects take into account information about their and others’ rank when deciding whether to challenge others.

• Subjects acquire and learn from the information that is generated through conflict.

• Information contexts which allow for more information to be generated lead to less conflict over time – albeit to the detriment of the underdogs.

Appendix

Chicken Game with incomplete information

i does not know strength of other players Sj and i does not know own strength Si. Ui (Ch) = p[qj(w – c) + (1 – qj)w] + (1 – p)[qj(–c) + (1 – qj)w]

= w – qj[(1 – p)w + c]

Ui (Sh) = p[qj0 + (1 – qj)0.5w] + (1 – p)[qj0 + (1 – qj)0.5w]

= (1 – qj)0.5w

Ui (Ch) > Ui (Sh) if qj < 0.5w/c

Ui (Sh) > Ui (Ch) if qj > 0.5w/c

Mixed strategy equilibrium (MSE) at qi = 0.5w/c ∀ i

Chicken Game with incomplete information

Chicken Game with incomplete information

Chicken Game with incomplete information