Bargaining and social structure

Evolution and Market Behavior Workshop 2009Bargaining and social structure

Edoardo GalloDate: October 4th, 2009

Bargaining and social structure

Edoardo Gallo

University of Oxford (Nuffield College)New Road, Oxford, OX1 1NF, UK

Email: [email protected]

Webpage: http://users.ox.ac.uk/~scro0919/



Motivation

Motivation and related literatureModelBargaining solution

Comparative staticsApplication

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:

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.



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 ModelBargaining solution





Nash demand game


xt yt

xt + yt ≤ 1

xt ytxt yt

xt + yt > 1

0 0






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)




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



Markov process


xq+1 yq+

1

s = {v1,…,vb,…,vs,…,vn} є S

s = {v1,…,vb,…,vs,…,vn} є Svb = {yq-m+1,

…,yq}vs = {xq-m+1,…,xq}

v’b = {yq-m+2,…,yq+1}

v’s = {xq-m+2,…,xq+1}

s’ = {v1,…,v’b,…,v’s,…,vn} є S

s’ = {v1,…,v’b,…,v’s,…,vn} є S





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.




Proof: Intuition


1) b and s are picked to play the game

2) they receive samples σ and σ’ respectively

3) they demand best replies x and y respectively

4) repeat steps (1)-(3) for m-1 periods to obtain

vb = {y,…,y}vs = {x,…,x}

1) b’ and s’ are picked to play the game

2) they receive samples from vb and vs respectively

3) they demand best replies 1-y and 1-x respectively


vs’ = {1-y,…,1-y}

vb’ = {1-x,…,1-x}

1) b’’ and s’’ are picked to play the game

2) they receive samples from vb and vs’ respectively

3) they demand best replies 1-y and y respectively


vb’’ = {y,…,y}vs’’ = {1-y,…,1-y}






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



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 gb

min ≡ gj for j Bmin . Equivalent definitions apply to the sellers.




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 ≡ gs

min





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.





Quasi-regular networksDefinition 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.



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.






Changing the network: DefinitionsLet 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}).






Changing the network and the ANBDenser 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.






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.






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.







Applying the model to the FFMA 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)






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.






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.






Evidence on Asians’ social networksSocial 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).





ExtensionAssume 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.






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?






ConclusionsMain 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.




Documents

Bargaining and social structure