Network Patterns of Favor Exchange http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1657130

Matthew O. Jackson Tomas Rodriguez-Barraquer Xu Tan

Favors •

• •

Cooperation is fundamental to our dayto-day lives and well-being, and to development and growth Many interactions are not contractible, and need to be self-enforcing How does successful favor exchange depend on/influence social structure?

Social Enforcement • Social capital literature’s (e.g., Coleman, Bourdieu, Putnam...) discussion of enforcement has been interpreted as high clustering/transitivity: • If we model social pressure and enforcement what comes out?

j k i

Game Theoretic Modeling: • Social contagion enforces cooperation: – Greif (1989), Kandori (1992), Ellison (1994), OkunoFujiwara & Postlewaite (1995) random matching: Labels/reputations/social norms, social contagion – Raub & Weesie (1990), Ali & Miller (2009), Lippert & Spagnolo (2009), Mihm, Toth, Lang (2009)... prisoners dilemma played on a network: cliques shorten information travel and quicken punishment – Haag & Lagunoff (2004) heterogeneity favors smaller groups

Outline/contribution •

Characterize equilibrium networks of favor exchange

•

Robust equilibria : suggests measure different from clustering

•

Data: Favor Exchange in Rural India

Setting •

{1,..., n} players

•

{1,...,t, ...} time

•

linked players can do favors for each other - needs arise randomly over time

Favors v value of a favor c cost of a favor,

v>c>0

δ discount factor

1>δ>0

p prob. i needs a favor from j in a period

Favor Exchange Favor exchange between two agents iff: c

<

current cost

δp(v-c)/(1-δ) value of future relationship

Network: Power of Ostracism Three agents (a ``triad’’): Ostracize agent who does not perform a favor 1

only need c< 2 δp(v-c)/(1-δ)

2 3

Game: Period t • • • •

At most one agent it is called upon to perform a favor for jt ∈ Ni (gt ) (p small) it keeps or deletes the link Others can respond: announce which (remaining) links they wish to maintain Links are retained if mutually agree resulting network is gt+1

Lots of Equilibria Consider smallest m such that c < m δp(v-c)/(1-δ) current cost

value per future relationship

Lots of Equilibria c
Look for two properties: Renegotiation-proofness: Minimal punishments: agents should not find another equilibrium continuation better for all agents Robustness: minimal spread of contagion/lost links

m=2 1

1 2

2

4

4 3

3 1 2

favors supported by credible expectation of loss of relationships

4 3

m=2

Not RPN 1 2

4 3

Key Idea: Is a SPE but not renegotiationproof

1 2

4 3

1 2

1 4

3

2

4 3

Renegotiation-Proof:

Too Much Criticality?

favor failure leads to autarchy

favor failure leads to loss of a triad

Robustness Against Social Contagion A Renegotiation-Proof network such that –

starting from any network sustained in some continuation, if i deletes a link ij, then any agent who subsequently loses a link is either i or a neighbor of i

Impact of a deletion/perturbation is local

Social Quilts: union of minimal cliques (completely connected subnetworks of m+1 nodes) such that largest simple cycle has no more than m+1 nodes

Social quilt m=2

Theorem: Robustness If m>1 then a network is robust against social contagion if and only if it is a social quilt.

So Far: Lots of SPE Renegotiation implies a form of criticality + Robustness leads to social quilts To be useful with data need asymmetries: Each agent and relationship has its own δi cij vij pij that could depend on network

Supported links: link ij∈g is supported if there exists k such that ik∈g and jk∈g i k j

Theorem: Robustness Heterogeneous Case If no pair of players could sustain favor exchange in isolation and a network is robust against social contagion, then all of its links are supported.

75 Indian Villages – Networks Data from Banerjee, Chandrasekhar, Duflo and Jackson (2010)

Borrow: •

``Favor’’ Networks: – both borrow and lend money – both borrow and lend kero-rice

• ``Social’’ Networks: – both visit come and go – friends (talk together most)

• Others (temple, medical help…)

•

Support

vs

Clustering i

i

k

k j

predicted by this theory

j

a standard in the literature

Support

Clustering

Favor Support `Social’ Support

Unlinked Support Fraction villages below

Conclusions •

Reneg.-proof gives criticality [implies inefficiency...] + Robustness gives social quilts; Identified measure: Support (which differs significantly from clustering) Support is ``high’’ in the Data

• • • –

favor/advice/business networks show significantly more support than purely social