Slide 1Evolution and Market Behavior Workshop 2009 Bargaining and
social structure
Edoardo Gallo Date: October 4th, 2009
Bargaining and
social structure
Edoardo Gallo
New Road, Oxford, OX1 1NF, UK
Email:
[email protected]
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation
Extension and conclusions
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:
Incomplete information
Private pairwise bargaining
Relevant markets: developing countries, illegal commodities and
wholesale.
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Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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).
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Nash demand game
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation and related literature
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)
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Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation and related literature
Extension and conclusions
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 Li(g) gij be the weighted degree of i
A network is connected if there is a path connecting any pair of
agents
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Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Markov process
s’ = {v1,…,v’b,…,v’s,…,vn} S
s’ = {v1,…,v’b,…,v’s,…,vn} S
Comparative statics
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Convergence
Extension and conclusions
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.
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Proof: Intuition
they receive samples σ and σ’ respectively
they demand best replies x and y respectively
repeat steps (1)-(3) for m-1 periods to obtain
vb = {y,…,y}
vs = {x,…,x}
they receive samples from vb and vs respectively
they demand best replies 1-y and 1-x respectively
repeat steps (1)-(3) for m-1 periods to obtain
vs’ = {1-y,…,1-y}
vb’ = {1-x,…,1-x}
they receive samples from vb and vs’ respectively
they demand best replies 1-y and y respectively
repeat steps (1)-(3) for m-1 periods to obtain
vb’’ = {y,…,y}
vs’’ = {1-y,…,1-y}
Comparative statics
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation and related literature
Extension and conclusions
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.
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Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Further assumptions and notation
Extension and conclusions
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.
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Asymmetric Nash bargaining solution (ANB)
Comparative statics
Extension and conclusions
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)
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation and related literature
Bargaining solution
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.
Color-coded nodes denote the players belonging to the subset of
least connected players.
Comparative statics
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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 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].
Motivation and related literature
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Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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 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.
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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}).
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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.
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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:
Locations: Manhattan, Brooklyn, New Brunswick, Princeton.
Establishments: restaurants, stores, shippers, dealers, fry
shops.
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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.
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation and related literature
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)
Comparative statics
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation and related literature
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.
Comparative statics
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation and related literature
A look at the FFM dataset (2)
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).
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.
Comparative statics
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
Motivation and related literature
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).
Comparative statics
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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.
Motivation and related literature
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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
Field experiment?
Lab experiment?
Evolution and Market Behavior Workshop 2009 Bargaining and social
structure
Edoardo Gallo Date: October 4th, 2009
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.
Motivation and related literature