Bargaining and
social structure
Edoardo Gallo
University of Oxford (Nuffield College)
New Road, Oxford, OX1 1NF, UK
Email: edoardo.gallo@economics.ox.ac.uk
Webpage: http://users.ox.ac.uk/~scro0919/
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Motivation
• Communities play an important role in perfectly competitive markets, e.g.
Greif (AER, 1993), Rauch and Trindade (REStud, 2002), Kumagai (2007).
• Greif (AER, 1993) argues that communities provide enforcement of sanctions
that deter violation of contracts in an uncertain environment.
• Here I argue that communities exist to give an informational advantage: the
social structure of the community is a conduit of information that members
use to learn about the market.
• The paper investigates the role played by the structure of social networks
for pricing in decentralized, perfectly competitive markets characterized by:
o Incomplete information
o Uncertainty on the price of the good
o Private pairwise bargaining
o Absence of a centralized coordination device
• Relevant markets: developing countries, illegal commodities and wholesale.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Related literature
• Bargaining models
– Classical: Nash (Ecta, 1950); Rubinstein (Ecta, 1982); Rubinstein
and Wolinsky (Ecta, 1985); Rubinstein and Wolinsky (RES, 1990).
– Evolutionary: Young (JET, 1993), Binmore et al. (JET, 1998), Young
(RES, 1998), Sáez-Martí and Weibull (JET, 1999).
– On networks: Calvo-Armengol (2001, 2003); Corominas-Bosch
(JET, 2004); Polanski (JET, 2008); Manea (2008); Abreu and Manea
(2008).
• Empirical evidence
– Wholesale markets: Kirman and Vignes (1991); Hardle and
Kirman (JE, 1995); Kirman et al. (JEBO, 2005); Vignes et al. (2008).
– International trade: Rauch (JEL, 2001); Rauch and Trindade
(REStud, 2002; AER, 2003); Kumagai (2007).
– Illegal markets: Levitt and Venkatesh (QJE, 2000; 2007).
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Nash demand game
x0t
y0t
xt + yt >
≤1
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Adaptive play bargaining process
Buyers and sellers: B={1,…,nB} and S={1,…,nS}
• Set-up is the same for buyers and sellers
• b has concave and strictly increasing vN-M utility
u(x), where x  (0,1), u(0)=0
• b has memory m
• b chooses an optimal reply to the cumulative
probability distribution G(y) of the demands yj
made by sellers in his sample
• Denote the utility of seller s by v(y)
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Communication networks
• Poisson information arrival: the probability that buyer b
receives a sample of past offers from buyer j is
determined by a Poisson process with rate gij
• The rates of these Poisson processes form a weighted,
undirected network g represented by a symmetric matrix
[gij]n×n.
• For expositional purposes assume that gii=0 for all i  B,S
• Let gi≡∑j є Li(g) gij be the weighted degree of i
• A network is connected if there is a path connecting any
pair of agents
• A complete network gC is a network where each agent is
connected to all the other agents
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Markov process
s’
s == {v
{v11,…,v
,…,v’bb,…,v
,…,v’
} єS S
s,…,v
s,…,v
n}nє
vbb =
v’
= {y
{yq-m+1
,…,yqq+1
} }
vss =
v’
= {x
{xq-m+1
,…,xqq+1
} }
q-m+2,…,y
q-m+2,…,x
xq+1
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
yq+1
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Convergence
Definition 1. A state is a convention if any vi  s with i 
B is such that vi = (1-x,...,1-x), and any vj  s with j  S
is such that vj = (x,...,x). Hereafter, denote this
convention by x.
Theorem 1. Assume both gB and gS are connected and
they are not complete networks. The bargaining
process converges almost surely to a convention.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Proof: Intuition
1) b
b’
and
are
are
picked
picked
tototo
play
play
the
the
game
game
b’’and
andss’s’’
are
picked
play
the
game
σ andvσ’
respectively
2) they receive samples from
vs’s respectively
respectively
b and
x and
y respectively
1-x
respectively
3) they demand best replies 1-y
and
y
respectively
4) repeat steps (1)-(3) for m-1 periods to obtain
vb = {y,…,y}
vs = {x,…,x}
vb’ = {1-x,…,1-x}
vs’ = {1-y,…,1-y}
vb’’ = {y,…,y}
vs’’ = {1-y,…,1-y}
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Markov process with mistakes
Definition 2. The demand xb(t) by buyer b at time t is a
mistake if it is not a best response to the sample b has
received before playing. A mistake ys(t) by seller s is
defined analogously.
Definition 3. The stochastically stable states are the states that
are most likely to be observed in the long-run when the
random mistakes are small.
Mathematically, let μє be the stationary distribution of the
Markov process (with mistakes), then a state s is
stochastically stable if limє →0 μє(s)>0
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Further assumptions and notation
Two further assumptions are needed to make the model more tractable.
(i) Mean-field assumption: the size of the information sample of the buyer
b is constant and equal to gb, i.e. the sum of the amount of information b
receives in expectation from each one of his neighbors. The same
assumption holds for the seller s.
(ii) Large memory: assume that the individual memory m ≥ max{gb, gs}
Some additional notation:
Let Bmin = {j B|gj ≤ gb ,  b B} be the subset of buyers with the
least integer weighted degree. Let gbmin ≡ gj for j Bmin . Equivalent
definitions apply to the sellers.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Asymmetric Nash bargaining
solution (ANB)
Theorem 2. There exists a unique stochastically stable
division (x*,1-x*) . The division is the asymmetric Nash
bargaining solution which maximizes
uβ(x) vσ(1-x)
where β ≡ gbmin and σ ≡ gsmin
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
ANB: Interpretation
A weighted network
with n=32 players
and two types of
links: strong links (in
bold) with weight 1
and weak links with
weight 0.5. Colorcoded nodes denote
the players belonging
to the subset of least
connected players.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Quasi-regular networks
Definition 4. Consider the set G of undirected networks with n nodes and
at most L links. Let gd,a be the regular network with degree d=2L/n, i.e.
the largest regular network in G, and link strength a. The network g є G
is a quasi-regular network generated by gd,a if it can be obtained by
randomly adding k links of any strength to gd,a be where k  [0, L-nd/2].
Examples of quasi-regular networks for n=5 and L=7.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Quasi-regular networks (cont’d)
Corollary 1. Fix a communication network gS for the sellers.
Consider the set G of all possible communication structures
gB among the nb buyers such that the total number of links
is L< (nb -1) nb /2 and that the strength of each links is in the
[s, s] range where s, s є R+. The subset of networks GB G
that gives the highest share to buyers are the quasi-regular
networks generated by the regular network gd,a be where
d=2L/ nb . The same statement holds reversing the roles of
buyers and sellers.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Changing the network: Definitions
Let ρ(g) denote the weighted degree distribution of network g.
Definition 5. A distribution ρ’ strictly first order stochastically
dominates (FOSD) another distribution ρ if ρ’(d) < ρ(d) (for all d 
{1,...,D}), where ρ(d)=∑d p(d) is the cumulative distribution of
p(d).
Definition 6. A distribution ρ’’ strictly second order stochastically
dominates (SOSD) another distribution ρ if ∑d ρ’’(d) < ∑d ρ(d) (for
all d  {1,...,D}).
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Changing the network and the ANB
Denser and more homogenous social groups obtain a higher share of
the pie in equilibrium.
Theorem 3. Let (x*,1-x*) be the ANB for sets of agents B and S that
communicate through networks gB and gS. Let ρ(g’B) FOSD ρ(gB)
and ρ(g’’B) SOSD ρ(gB).
(i) Let (x’*B, 1- x’*B) be the ANB for sets of agents B and S with
degree distributions ρ(g’B) and ρ(gS). Then x’*B > x*.
(ii) Let (x’’*B, 1- x’*B) be the ANB for sets of agents B and S with
degree distributions ρ(g’’B) and ρ(gS). Then x’’*B > x*.
The same statement holds reversing the roles of buyers and sellers.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
The Fulton fish market (FFM)
•
•
•
•
Graddy (RAND, 1995) tracked all (n=489) transactions of
whiting by one dealer over 19 days, recording: price,
quantity, exact time, type of buyer and quality of fish.
No posted prices and dealer is free to charge a different
price to each customer.
“Spread of prices throughout the day is very high, and
the interday volatility is large” (Graddy, p. 78).
Types of buyers:
–
–
–
Three ethnic groups: whites, Asians and blacks (small
sample).
Locations: Manhattan, Brooklyn, New Brunswick, Princeton.
Establishments: restaurants, stores, shippers, dealers, fry
shops.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
A puzzling finding
•
•
•
•
Key finding: white sellers charge white buyers
significantly (~7%) more than Asian buyers for the same
homogeneous product.
Graddy (p. 87) concludes that “the reason behind the
price discrimination is less clear.”
Not a typical setting for 3rd degree price discrimination:
competitive industry, no search costs, homogeneous
products, no barriers to entry, no significant difference
in elasticity for Asians vs white buyers.
Graddy (1995) shows that difference is not due to
differences in purchase times, product quality, mode of
payment and volume of transactions.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Applying the model to the FFM
A potential explanation: Asian buyers’ communication
network is denser/more homogeneous than white buyers’.
Therefore, the group of Asian buyers is better at sharing
information on today’s price and this informational
advantage leads to the observed price difference.
–
–
–
–
–
Graddy (p. 84): “very little social contact appears to take place
between groups of Asian buyers and groups of white buyers”
Graddy (p. 87): “Asian buyers appear to be more organized than
white buyers”
Graddy: “Asian buyers certainly spoke to one another and
congregated much more frequently than white buyers”
Homophily is a powerful determinant of social networks, and
racial/ethnic homophily is much stronger than other types (e.g.,
McPherson et al., 2001)
Evidence that Asian immigrant groups form very close-knit
networks (e.g., Sanders et al., 2002; McCabe, 2006)
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
A look at the FFM dataset (1)
Asians obtain a better price
only after the first 1-2 hours
of the market, presumably
due to learning.
Regression analysis shows that the
“Asian” dummy is negatively
correlated (p=0.01) with prices in the
6-7am time period, but it is
statistically insignificant (and
positively correlated) in the 4-5am
time period.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
A look at the FFM dataset (2)
■Asians □Whites
The variability of prices
paid by Asians decreases
faster than the variability
of prices paid by Whites
pointing to faster learning
among Asians of the
current value of fish.
•A two-sample variance comparison test rejects (99% c.f.) the null
hypothesis that VarASIAN(4-5)=VarASIAN(6-7).
•But the same test cannot reject (90% c.f.) the null hypotheses that
VarWHITE(4-5)=VarWHITE(6-7) and VarASIAN(4-5)=VarWHITE(4-5).
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Evidence on Asians’ social networks
Social connections play a key role in business transactions in the overseas Asian
community:
• Redding, Overseas Chinese Networks: Understanding the Enigma, 1995:
–
"[p]ersonalism does in Asia what law does in the West [...] [w]ithout [what is termed
guanxi or connections] nothing can be made to happen [...] the instinct of the Overseas
Chinese to trust friends but no-one else is very deep-rooted.“
– “For the Overseas Chinese the uncertainties of the business environment mean that
playing fields are not level. […] So the Chinese rules are: put your trust primarily in
'your own' people; seek the opportunities by trading rare information; share that
information to build allegiances”
• Xie, Asian Americans: A Demographic Portrait, 2004:.
– “Asian American communities offer many practical resources to immigrants,
including [...] information in native languages, and entrepreneurial opportunities.“
See, e.g., additional references in Rauch and Trindade (REStud, 2002), Rauch and
Casella (EJ,2003), Kumagai (2007).
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Extension
Assume that the two groups share the same
network, i.e. buyers receive information from other
buyers and sellers about past sellers’ demands, then:
– The stochastically stable division is unchanged.
– Core-periphery networks maximize the share for a group.
– A more homogeneous network narrows down the
difference between the two groups.
– In a regular network with homogeneous agents 50-50 is
the stable division.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Further research
• Theoretical
–
–
How sensitive are the results to the
assumptions of a very small ε?
Can we say anything on the speed to
convergence?
• Empirical
–
How do we test the model empirically?
• Field experiment?
• Lab experiment?
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009
Motivation and related literature
Model
Bargaining solution
Comparative statics
Application
Extension and conclusions
Conclusions
Main results:
•The unique stochastically stable division is the ANB with weights
determined by the players with the least weighted degree in each
group.
•Quasi-regular networks maximize the share for a group.
•Denser and more homogeneous networks fare better.
•An empirical analysis of the observed price differential between Asian
and white buyers in the FFM is consistent with these predictions
If the two groups share the same network, then:
•The stochastically stable division is unchanged.
•Core-periphery networks maximize the share for a group.
•A more homogeneous network narrows down the difference between
the two groups.
•In a regular network with homogeneous agents 50-50 is the stable
division.
Evolution and Market Behavior Workshop 2009
Bargaining and social structure
Edoardo Gallo
Date: October 4th, 2009