- Email: [email protected]

March 19, 2016

1. Introduction

• Individuals are located on nodes of a network. They choose

actions and their rewards depend on these actions along with the actions of others on the network. The key point: • The effect of player 1’s action on player 2’s payoff depends on

where the two players are located in a network.

Introduction: Examples

1. value of learning a language depends on how many friends and colleagues learn the same language. 2. value of acquiring information on market prices depend on how much information friends acquire. 3. firms collaborate but also compete in markets. 4. efforts feed off each other: it pays to work hard on a project if colleagues also work hard.

• There are two basic building blocks: one, the formal

description of the pattern of relationships among individual entities and two, a description of the externalities that an individual’s actions create for other individuals and how these are mediated by the pattern of ties between them. We ask: 1. What are the effects of network location on individual behavior and well being? 2. How does individual behavior and well being respond to changes – the adding of links or the redistribution of links – in a network? 3. Are some networks better for the attainment of socially desirable outcomes?

• Two conceptual issues: 1. A single action or link specific actions? Single action is reasonable is some contexts: such as consumer search about product prices. While in other contexts link specific action is more natural – e.g., effort in research projects, individuals or firms have the choice of putting different amounts of resources in different projects. The single action formulation is simpler and most applications to date assume this. 2. Direct or indirect effects of actions? Both will be studied.

2. Games on fixed networks, Goyal (2007)

• Suppose each player i takes an action si in S, where X is a

compact subset of [0, 1]. The payoff (utility or reward) to player i under the profile of actions s = (s1 , ..., sn ) is given by Πi : S n × G → R. • In the games where the action set S is continuous, it will be

assumed that S is also convex. Define s−i as the profile of all strategies other than player i.

• Networks in the payoff function: A network has a number of

different attributes – such as neighborhood size, average path length, degree distribution, centrality – and it is clear that these attributes will play more or less important roles depending on the particular context under study. • Neighbors and non-neighbors. The simplest way to model this

is to classify other players into two categories, neighbors and non-neighbors, and to treat all members in each group alike. The effects of actions of neighbors are then termed local effects, while the actions of non-neighbors are termed global effects.

• Pure local effects: Define the function φk : S k → R. In this

case: Πi (s |g ) = φηi (g ) (si , {sj }j ∈Ni (g ) ) • Remarks: same payoffs of players with same degree; Two, the

payoff function is anonymous. If {sj0 }j ∈Ni (g ) is a permutation of actions in {sj }j ∈Ni (g ) then φηi (g ) (si , {sj0 }j ∈Ni (g ) ) = φηi (g ) (si , {sj }j ∈Ni (g ) ). • Pure global effects: the actions of all players have the same

effects on payoffs. Πi (s |g ) = φn−1 (si , s−i ).

• A combined model of local and global effects: Define a

function fk : S k → R, where k is the degree of a player i, and define a function hk : S n−k −1 → R. Assume that functions respect anonymity of actions and assume that they are same across players. Πi (s |g ) = Φ(si , fki ({sj }j ∈Ni (g ) ), hk (s )).

(1)

• An important special case of this framework arises when the

payoff depends simply on the sum of neighbors actions and the sum of non-neighbors actions. Π i ( s | g ) = Φ ( si ,

∑

j ∈Ni (g )

sj ,

∑

k∈ /Ni (g )∪{i }

sk ) .

(2)

• Nature of externality: Effects of others actions on payoffs as

well as marginal payoffs. In the polar cases, pure local effects or pure global effects, it is easy to define them.

Definition A game with pure local effects satisfies positive externality if for each φk , and for s, s 0 ∈ S k , if s ≥ s 0 then φk (si , s ) ≥ φk (si , s 0 ). Similarly, the game exhibits negative externality if for each φk , and for s, s 0 ∈ S k , if s ≥ s 0 then φk (si , s ) ≤ φk (si , s 0 ).

Definition A game with pure local effects exhibits strategic complements (substitutes), if for all φk , si > si0 , s, s 0 ∈ S k , if si ≥ si0 then φk (si , s ) − φk (si0 , s ) ≥ (≤)φk (si , s 0 ) − φk (si0 , s 0 ).

Example 1: Local Public Goods (Bramoulle and Kranton 2007) • There are n players. Suppose each player chooses a search

intensity si ∈ S, where S is a compact and convex interval in R+ . The payoffs to a player i, in a network g , faced with a profile of efforts s = {s1 , s2 , ..., sn }, are given by: Π i ( s | g ) = f ( si +

∑

sj ) − csi

(3)

j ∈ Ni ( g )

where c > 0 is the marginal cost of effort. It is assumed that f (0) = 0, f 0 > 0 and f 00 < 0. A game of pure local effects. It is also a game of positive externality and strategic substitutes.

Example 2: Crime (Ballester, Calvo-Armengol, and Zenou 2006) • The role of interaction effects in shaping the level of criminal

activity has been a recurring theme is different literatures such as social psychology and economics. Consider a n player game with linear quadratic payoffs. The payoffs to player i faced under strategy profile s, are given by: 1 Πi (s ) = α.si + ρsi2 + ∑ γgij si .sj 2 j 6 =i

(4)

• Assume that α > 0, ρ < 0 and γ > 0. So this is a game of

pure local effects, positive externalities and complements.

Example 3: R&D Networks (Goyal and Moraga 2001) • Firms increasingly choose to collaborate in research with other

firms. This research collaboration takes a variety of forms and is aimed both at lowering costs of production as well as improving product quality and introducing entirely new products. • Suppose demand is linear and given by Q = 1 − p and that the initial marginal cost of production in a firm is c¯ and assume that nc¯ < 1. Each firm i chooses a level of research effort given by si ∈ S = [0, c¯ ]. Collaboration between firms is at a bilateral level and it allows for firms to share research efforts which lower costs of production. The marginal costs of production of a firm i, in a network g , facing a profile of efforts s, are given by: ci (s |g ) = c¯ − (si +

∑

j ∈Ni (g )

sj ) .

(5)

• The cost of efforts is given by Z (si ) = αsi2 /2, where α > 0.

Given costs c = {c1 , c2 , ...cn }, firms choose quantities ({qi }i ∈N ), with Q = ∑i ∈N qi . Solve for market quantity equilibrium given any cost profile. • Then payoffs of firm i, located in network g , faced with a research profile s is: "

1 − c¯ + si [n − ηi ] + ∑j ∈Ni (g ) sj [n − ηj (g )] − ∑l ∈N \{i }∪Ni sl [1 + ηl (g )] n+1

• This game exhibits local & global effects. Positive externality

across neighbors and negative externality across non-neighbors actions. Actions of neighbors are complements, while the actions of non-neighbors are substitutes.

#

Example 4: Communities and Competitive Exchange (Ghiglino and Goyal (2010) • Consider a pure exchange competitive economy with

individuals located on nodes of an (undirected) network. There are two goods, x and y . Individuals have Cobb-Douglas preferences; the novel feature is that the good y is a relative consumption good. In particular, assume that utility of individual i, facing a consumption profile (xi , yi )i ∈N is: ui (xi , yi , y−i ) = xiσ [yi − αηi (yi −

1 ηi

∑

yj )]1−σ

(6)

j ∈ Ni ( g )

where σ ∈ (0, 1) and α measures the strength of social comparisons, Ni (g ) refers to the set of neighbors, and ηi to the number of neighbors of i.

• Let good x be the numeraire and sets its price equal to be 1. • A competitive equilibrium is defined as a price py (for good y )

that clears all markets given that individuals optimally allocate their budget across x and y . Our interest is in understanding how the structure of the network affects individual consumption and competitive prices.

3 Local public goods, Bramoulle and Kranton, 2007 • Existence of Nash equilibrium: The action set is compact,

the payoffs are continuous in actions of all players are concave in own action, it follows from standard theorem that there is Nash equilibrium in pure strategies. • Networks and equilibrium: A useful first step is a general

property concerning aggregate level of effort – own plus the neighborhood – enjoyed by any individual. Let sˆ be such that f 0 (sˆ ) = c and define sNi (g ) = ∑j ∈Ni sj . From the concavity of f (.), it then follows that if sNi (g ) ≥ sˆ then marginal returns to effort are lower than the marginal cost and so optimal effort is 0, while if sNi (g ) < sˆ , then marginal returns from effort to player are strictly larger than marginal costs c and so optimal effort is positive and in fact given by sˆ − sNi (g ) .

Proposition A profile of actions s ∗ = {s1 , s2 , ..., sn } is a Nash equilibrium if and only if for every player i either (1). sN∗ i (g ) ≥ sˆ or (2). sN∗ i (g ) ≤ sˆ and si∗ = sˆ − sN∗ i (g ) . • There are two types of players: those who receive aggregate

effort in excess of sˆ from their neighbors and exert no effort on their own, and two, players who receive less than sˆ aggregate effort from their neighbors and contribute exactly the difference between what they receive and sˆ .

• Specialized equilibria: profile where some players choose

positive action while others choose 0 action. such profiles illustrate free riding in a specially acute form. • Point 1: In the empty network – there is a unique equilibrium

in which every player chooses sˆ : no free riding. It turns out that this is the only network in which no free riding is possible. How do we prove this?

• An independent set of a network g is a set of players I ⊆ N

such that for any i, j ∈ I , gij 6= 1. A maximal independent set is an independent set that is not contained in any other independent set. Does there exist a maximal independent set in every network? The answer to this is yes. Proof by construction: • First number the players 1, 2, ....n. start by placing player 1 in

I . If player 2 ∈ / Ni (g ), then include her in the independent set,I , if not then include her in the complement set I c . Next consider player 3: if player 3 ∈ / N1 (g ) ∪ N2 (g ), then include her in I , while if she is not then include her in I c . Move next to player 4 and proceed likewise until you reach player n.

• Examples: In the empty network there exists a unique

maximal independent set and this is the set of all players N. In the complete network on the other hand, there are n distinct maximal sets, each of which contain a single player. In the star network, there are two maximal independent sets, one, which contains the central player, and two, which contains all the peripheral players. • Now assign the action sˆ to every member of a maximal

independent set and assign action 0 to every player who is not a member of the maximal independent set. This configuration constitutes an equilibrium in view of the characterization provided in Proposition 1. Such an equilibrium is by construction a non-trivial specialized equilibrium.

• In any non-empty network, a maximal independent set must

be a strict subset of the set of players N.

Proposition There exists a specialized equilibrium in every network. In the empty network the unique equilibrium is specialized and every player chooses sˆ , so there is no free riding. In any non-empty network there exists a specialized equilibrium with free riding.

• Network advantages: Specialized equilibria point to unequal

effort. Does this translate into unequal payoffs? Is there any systematic relation between network position and payoffs? Network location is a rather general idea and there are different aspects of networks that may play a role. In the present context with positive payoff externality, the intuition is that higher degree players should earn more. Is this true? • Example: Star network. The two equilibria are both

specialized and have clearly unequal payoffs. • This shows that it is difficult to say anything definite on

relation between degree and payoff. Why is this? How can we work around this? Come back to this in games with incomplete network knowledge.

• Network structure and social welfare: Aggregate welfare

from a strategy profile s in network g is defined as: W (s |g ) =

∑ [f (si + sN (g ) − csi ]. i

i ∈N

Given a network g , a strategy profile profile s is efficient if there is no other action profile s 0 such that W (s 0 |g ) > W (s |g ).

Proposition Every equilibrium in a non-empty network is inefficient. • This result is a direct consequence of individual actions having

positive externality on others.

• Proof: Fix some non-empty network g , and let s ∗ be an

equilibrium with si > 0, for some i ∈ N, and suppose that for some i and j, gij = 1. We know that if si > 0, then sNi (g ) + si = sˆ and this implies that f 0 (si + sNi (g ) ) − c = 0. Now consider the partial derivative of social welfare with respect to si evaluated at si∗ : ∂W (s ∗ |g ) = ∂si

∑

j ∈{i }∪Ni (g )

f 0 (sj∗ +

∑

k ∈ Nj ( g )

sk∗ ) − c =

∑

j ∈ Ni ( g )

f 0 (sj∗ +

∑

sk∗ )

(7)

k ∈ Nj ( g )

• which is strictly positive since f 0 (·) > 0. So welfare can be

strictly increased by increasing si . Thus the equilibrium is inefficient.

• Effects of adding links: example Start with two stars each

with 3 peripheral players. Fix an equilibrium in which the two centers exert action sˆ while the peripheral players all choose 0. • First add a link between a center and a spoke of the other

star. The old action profile still constitutes an equilibrium. It then follows that aggregate welfare increases on the adding of a link. • Second, add a link between the centers. The two centers do

not constitute a maximal independent set any more. Equilibrium must change. The best equilibrium is one in which the center of one star and spokes of the other star choose sˆ , and all other players choose 0. In this new equilibrium, welfare strictly decreases if 2c sˆ > f (4ˆs ) − f (sˆ ). • Thus effects of adding links depend on subtle details of the

network.

General observations:

• Important role of networks in sustaining specialized equilibria:

creating significant free riding and payoff inequality. • Connections and network advantages: Multiple equilibria exist

with contrasting relationships. • Adding links: increases payoffs in some standard networks but

not for other simple networks! • A key limitation is the multiplicity of equilibria, with very

different properties. • Recent follow up work, see Amours, Bramoulle and Kranton

(2011), Strategic Interaction and Networks.

4 Criminal networks, Ballester et al (2006).

• We start for simplicity with the following payoffs: n 1 Πi (s |g ) = αsi − βsi2 + λ ∑ gij si .sj 2 j =1

(8)

where α, β, λ are all positive. • Differentiating payoffs (8) with respect to si we get: n ∂Πi (s ) = α − βsi + λ ∑ gij sj = 0. ∂si j =1

(9)

• The main result is the relationship between network centrality

and individual behavior. • An individual’s incentives depend on others’ actions via local

pairwise terms. • So any differences in player choices must be due to differences

in pairwise local effects. The effects of pairwise effects are local, but the choices of neighbors reflect the choices of her neighbors, and so on. Thus a player’s incentives will be based on the direct and the indirect connections across the network.

Bonacich Centrality

• Consider the adjacency matrix G of network g ; in this matrix

an entry in a square corresponding to a pair {i, j } signifies the presence or absence of a link. • Let Gk be the k th power of the matrix. The 0 power matrix

G0 = I , the n × n identity matrix. In Gk , an entry gikk measures the ‘number’ of walks of length k that exist between players i and j in network g . The following example illustrates this idea.

• Example 1: Consider a network with 3 players, 1, 2 and 3.

Suppose links take on values of 0 and 1, and let the network consist of two links, g12 = g23 = 1. This network can be represented in an adjacency matrix G as follows. 1 2 3 1 0 1 0 Table 2.1 2 1 0 1 3 0 1 0 • Simple computations now reveal that G2 is given by:

1 2 3

1 1 0 1

2 0 2 0

3 1 Table 2.2 0 1

• Thus there is 1 walk of length 2 between 1 and 1 and between

3 and 3, but 2 walks of length 2 between 2 and 2. There are no other walks of length 2 in this network. • Let a ≥ 0 be a scalar and let I be the identity matrix. Define

the matrix M (g , a) as follows: M (g , a) = [I − aG]−1 =

∞

∑ a k Gk .

(10)

k =0

• This expression is well defined so long as a is sufficiently

k counts the total small. The entry mi,j (g , a) = ∑k∞=0 ak gi,j number of walks in g from i to j, where walks of length k are weighted by factor ak .

• Given parameter a, the Bonacich centrality vector is defined as

Cb (g , a) = [I − aG]−1 · 1

(11)

• where 1 is the (column) vector of 1’s. In particular, the

Bonacich centrality of player i is given by: n

Cb (i; g , a) =

∑ mi,j (g , a).

(12)

j =1

• This measure of centrality counts the total number of

(suitably weighted) walks of different lengths starting from i in network g .

• To see this note that (12) can be rewritten as follows: n

Cb (i; g , a) = mi,i (g , a) + ∑ mi,j (g , a).

(13)

j 6 =i

• Since G0 = I , it follows that mi,i (g , a) ≥ 1 and so for every

player i in any network, Cb (i; g , a) ≥ 1. It is exactly equal to 1 in case a = 0. • The Bonacich centrality of a node can also be expressed as a

function of the centrality of its neighbors. Let λ(a) be the (largest) eigen value of the adjacency matric G. The Bonacich centrality of a node can then be defined as: Cb (i; g , a) =

1 gij Cb (j; g , a). λ j∑ ∈N

(14)

• It is easy to compute Bonacich centrality measures for

different networks. For instance, in the star network with 3 players as represented in the adjacency matrix in example 1, the Bonacich centrality of nodes 1 and 3 is 0.500 while that of the central player is 0.707. • More generally, the ratio of Bonacich centralities between the √ central player and the peripheral player in a star is n − 1, which is an increasing and a concave function of the number of nodes n. • Let G be the adjacency matrix of network ties then the vector

of Bonacich centralities is given by Cb (g , a) = [I − aG]−1 .1, where a is a scalar (small enough to ensure that the inverse is well defined). The Bonacich centrality of a player i is given by: n

Cb (i; g , a) =

∑ mij (g , a).

(15)

j =1

• where mij (g , a) is the total number of weighted walks of all

lengths between players i and j, in network g . Define

• The game Γ has a unique interior Nash equilibrium

s ∗ (Γ) = {s1∗ , s2∗ , ...sn∗ }, which is given by s ∗ (Γ) =

α Cb (g , λ∗ ) β

(16)

• Simple algebra now yields the following expression on

equilibrium efforts: si∗ (Γ) =

Cb (i; g , λ∗ ) ∗ s˜ (ρ) C˜ b (g , λ∗ )

where s˜ ∗ = ∑ si∗ and C˜ b (g , λ) = ∑ C (i; g , λ).

(17)

• Proof: The necessary and sufficient condition for the matrix

[ βI − λG]−1 to be well defined and non-negative follow from Debreu and Herstein (1953). This condition suffices for the existence of a unique interior equilibrium. • Next consider the characterization of equilibrium. From the first order conditions, it follows that an interior si∗ satisfies: n

βsi − λ ∑ gij sj = α.

(18)

j =1

• Using matrix Γ we can rewrite this as follows:

[ βI − λG]s ∗ = α.1.

(19)

• The matrix [ βI − λG] is generically non-singular and so there

is a unique generic solution in Rn .

• Now exploit U.s = s˜ .1 to rewrite the above as:

β[I − λ∗ G]s ∗ = [α].1.

(20)

• Inverting the matrix and using the definition of Cb (.) we can

write this as: βs ∗ = [α]Cb (g , λ∗ )

(21)

• Simple algebra then yields:

s∗ = • This completes the proof.

α Cb ( g , λ ∗ ) . β

(22)

• This expression captures the key insight: equilibrium efforts

are proportional to Bonacich centrality. • Recall, that in a star network the central node has a higher

centrality than the peripheral nodes, and this result implies therefore that the central player will exert higher criminal efforts as well. • By contrast, in a cycle or a complete network all players have

the same centrality and so their efforts will be equal as well.

• The interest now turns to effects of adding links to a network.

Let g 0 be denser than g :

Proposition Suppose β is large and g 0 be denser than g . Then the equilibrium in the denser network exhibits higher aggregate effort. • Intuition: the network changes via the addition of a link.

Adding a link between i and j, adds a new complementarity effect for players i and j. This in turn raises their incentives for higher efforts and, via local complementarities, the incentives of their neighbors, and so on. This leads to an increase in efforts for everyone.

Summary of criminal networks

• Any game with linear quadratic payoffs can be expressed in

terms of three components: an own concavity component, a uniform global effects component and a local pairwise effects components. • Equilibrium efforts of a player are proportional to her

Bonacich centrality in the network. • These results require that complementarity effects be modest:

strong effects lead to multiple equilibria.

Revisiting social comparisons

• Goyal and Ghiglino (2010) show that general equilibrium

prices and consumption are a function of a single network statistic: (Bonacich) centrality. • An individual’s “centrality”’ is given by the weighted sum of

paths of different lengths to all others in a social network. Individual consumption is proportional to its node centrality and the relative price of good y is proportional to the average network centrality of all agents in the network. • Adding links to a network pushes up centralities and in turn

pushes up the price of good y .

5. Pricing in networks: Introduction

• Supply and trading chains are prominent in agriculture, in

transport and communication networks, in international trade, in finance. • Goods and services pass through individuals or firms located

on these chains. • The routing of economic activity, the allocation of surplus and

the efficiency of the system depend on the prices set by these different intermediaries.

Introduction: Applications

1. Transportation and communication Networks: air lines, shipping, Internet, cable TV, trains, tolls. 2. Supply chains: Vertically specialized supply chains traverse the world. 3. Corruption: Single chain of officials/multiple competing chains of official. 4. Intermediation in agriculture: Long chains of intermediation are common in agriculture. 5. Financial Intermediation: Decentralized (OTC) broker-dealer network. profitability.

Pricing in a routing problem • A source node, S , and a destination node, D . • A path between the two is a sequence of interconnected

nodes. The source node and the destination node and all the paths between them together define a network. • Passage from source to destination generates a value. • Intermediaries simultaneously post a price to get a share of

this value; prices determine a cost for every path between S and D . • Good moves along a least cost path and an intermediary earns

payoffs only if she is located on it. • We study Nash equilibrium.

Literature • Empirical work: theoretical foundation for empirical relation

between ‘central’ traders and market power. D’Ignazio and Giovannetti (2006, 2009) Li and Sch¨ urhoff (2012). • Industrial organization: model of pricing in general networks

with asymmetric demand. Kotowski and Leister (2012), Condorelli and Galeotti (2011), Manea (2013), Acemoglu and Ozdagler (2007a, 2007b), Blume et al. (2007) and Gale and Kariv (2009). • Network theory: role of criticality in shaping behavior.

Ballester et al (2006), De Marzo et al (2003), Golub and Jackon (2010). • Experiments: trading games, dictator games and Nash

bargaining.

Benchmark Model: Choi, Galeotti and Goyal (2014)

• There is a source S and a destination D . • There are n traders located in a network g that connects S

and D . • Traders simultaneously post prices: cost of a ‘path’ between s

and b is sum of prices of traders on the path. • The surplus between b and s is 1. • Pick cheapest path if it is less than 1 (randomize). • Seller and buyer split residual surplus equally.

Theory: Benchmark Model

• Given g and p, let Q∗ = {q ∈ Q : c (q, p ) = c (p ), c (p ) ≤ 1}

be the set of feasible least cost paths. • The expected payoff to intermediary is

( Πi (p) =

0 ηi p |Q∗ | i

if i 6∈ q for all q ∈ Q∗ or c ∗ (p) > 1 otherwise,

(23) where ηi is the number of paths in Q∗ that contain trader i. • We study pure strategy Nash equilibrium in prices. • c ∗ (p ) denotes the lowest cost under p.

Essential and critical traders

• We say that a trader is critical if he lies on all paths between

source and destination. • We say that trader i is essential under p if he belongs to every

feasible least-cost path. • Critical nodes vs essential nodes: Criticality is a property of

the network per se, while essentiality is defined by the network and the price profile together. So, a node may be essential even if there are no critical nodes in the network.

Theory: Equilibrium Theorem A. Existence: In every network, there exists an efficient equilibrium. B. Characterization: An equilibrium p ∗ is inefficient (c (p ∗ ) > 1); or intermediaries extract all the surplus (c (p ∗ ) = 1); or they earn nothing (c (p ∗ ) = 0). Moreover, 1. p ∗ is an equilibrium in which intermediaries earn nothing if, and only if, no trader is essential. 2. p ∗ is an equilibrium in which intermediaries earn all the surplus if, and only if, (i) if trader i belongs to the least-cost path, and he sets a positive price then trader i is an essential trader; and (ii) if trader i belongs only to non-least-cost paths, and he belongs to path q then c−i (q, p ∗ ) ≥ 1. 3. p ∗ is an inefficient equilibrium if, and only if, if trader i belongs to path q then c−i (q, p ∗ ) ≥ 1.

Theory: Equilibrium

• The argument for the existence of an efficient equilibrium is

constructive. • consider a network with no critical traders. The 0 price profile

is a Nash equilibrium, as no intermediary can earn positive profits by deviating and setting a positive price. If an intermediary sets a positive price, S and D will circumvent him, as there exists a zero cost path without him. • Next, consider a network with critical traders. It may be checked that a price profile in which critical traders set positive prices that add up to 1 and all non-critical traders set 0 price is an equilibrium.

Theory: Observations • In every efficient equilibrium, intermediation costs take on

extreme values. Unique least cost path: if intermediation costs are below the value of exchange, an intermediary in that path could slightly increase his intermediation price and still guarantee trade. The converse is true with multiple least-cost paths: prices must equal zero. • Critical traders and market power. Observe that a critical

trader is essential. Hence, the presence of critical traders is sufficient to ensure that intermediaries extract all surplus in every efficient equilibrium. • Criticality is sufficient but not necessary. • Multiplicity of equilibria: Consider the ring network with six

traders.

Theory: open issues • Multiplicity of equilibria. Consider the ring network with 6

traders. All three equilibria described by above Theorem are possible. • Motivates equilibrium refinement: trembling hand perfection,

strictness, strong Nash equilibrium, elimination of weakly dominated strategies, and perturbed Nash demand games. • Some refinements are too strong while others are not

effective, e.g., a wide range of equilibrium may be sustained under trembling hand perfection, elimination of weakly dominated strategies, and perturbed bargaining. • To gain a deeper understanding of the relation between

networks and market power, we conduct an experiment.

Experiment: Design

• We consider four different networks. 1. Ring networks (size = 4, 6, 10): competition stable, coordination grows. 2. Ring 6 with hubs and spokes: critical traders plus competition. • We ran two sessions per network treatment; in total, there are

8 sessions. • 60 rounds in each session; • Random matching and random assignment of network

positions.

Experiment: Rings and Ring with Hubs

A B

D

C

F

C

E

RING 4

A

B

C

E

G

RING 10

F

a2

b1

A

B

E

D

e1

d2

b2

a1

D

J

H

D

RING 6

f2

I

B

A

C

F f1 e2

c1 c2 d1

RING with HUBS & SPOKES

Experiment: Treatments and sessions

Session Treatment Ring n = 4 Ring n = 6 Ring n = 10 Ring with hubs

16 18 20 18

1 / 240 / 180 / 120 / 180

2 16 / 240 24 / 240 20 / 120 24 / 240

Total 32 / 480 42 / 420 40 / 240 42 / 420

Finding 1: Efficiency is remarkably high in all networks

Network Ring 4 Ring 6 Ring 10 Ring with Hubs and Spokes

All ( ≥ 2) 1.00 (480) 1.00 (420) 1.00 (240) 0.95 (420)

minimum distance of buyer-sell pair 2 3 4 1.00 --(480) 1.00 1.00 -(289) (131) 1.00 1.00 1.00 (49) (87) (69) 1.00 0.94 0.90 (126) (155) (109)

Note. The number of group observations is reported in parentheses.

5 --1.00 (35) 0.90 (30)

Finding 2: Distribution of surplus is extremal

100 90 80

Intermediation cost

70 60 50 40 30 20 10 No. Cr = 1

No. Cr = 2

No. Cr = 0

0 (2, 2)

(2, 4)

(3, 3)

(2, 8)

(3, 7)

(4, 6)

(5, 5)

(2, --)

(3, --)

No. Paths = 1

Ring 4

Ring 6

Ring 10

(3, 5)

(4, 4)

(4, 6)

(5, 5)

No. Paths = 2

Ring with Hubs and Spokes

(2, 4) or (3, 3)

Finding 3: Criticality yields large payoffs

Network

(#Cr,#Paths, d(q),d(q')) (1, 2, 3, 5)

Ring with Hubs and Spokes

(1, 2, 4, 4) (2, 2, 4, 6) (2, 2, 5, 5)

1 ~ 20 0.56 (20) 0.48 (16) 0.73 (16) 0.65 (8)

Rounds 21 ~ 41 0.68 (26) 0.56 (13) 0.77 (19) 0.67 (8)

41 ~ 60 0.72 (25) 0.67 (10) 0.80 (24) 0.74 (11)

Notes. The number in a cell is the average fraction of costs charged by critical traders. The number of observations is reported in parentheses. #Cr denotes the number of critical intermediaries, #Paths denotes the number of paths connecting buyer and seller, d(q) denotes the length of path q beween buyer and seller.

Pricing 60 Price on a shorter path: Ring or non-critical traders in Ring with Hubs and Spokes Price on a longer path: Ring or non-critical traders in Ring with Hubs and Spokes Price of critical traders in Ring with Hubs and Spokes

50

Price on an equal-distance path: non-critical traders in Ring with Hubs and Spokes

Price

40

30

20

10

0 2

4

2

8

3

7

4

6

Cr

3

5

Cr

4

Cr

4

No. Cr = 1

Ring 6

Ring 10

Ring with Hubs and Spokes

6

Cr

No. Cr = 2

5

Competition on paths

Network

(d(q), d(q'))

|cost1 - cost2|

Ring 4

(2, 2)

3.99

--

(2, 4)

4.45

0.65

Ring 6

Ring 10

Freq. on a shorter path

(3, 3)

4.01

--

(2, 8)

15.20

0.64

(3, 7)

5.30

0.68

(4, 6)

6.82

0.68

(5, 5)

5.01

--

Notes. We report the sample median of absolute differences of two competing paths, using the sample of last 20 rounds. The number in the last column is the frequency of trading on a shorter path.

Summary of benchmark model

1. Efficiency: theory says that efficient equilibrium always exists, but inefficient equilibria are common. Experiments show that subjects select efficient outcomes. 2. Division of surplus: theory says division is extremal. Experiment outcomes consistent with theory. 3. Network structure: Theory is permissive. In experiments intermediaries earn money if and only if they are critical.

Extension 1: unknown surplus

• Source/destination know surplus, intermediaries have

incomplete information. • Surplus v ∈ [0, 1] drawn from a distribution F (·) and density

function f (·). • Intermediary posts a ‘price’ pi ≥ 0. • Define h (x ) = f (x )/[1 − F (x )] to be the hazard rate. • Given g and p, define Q v to be the set of feasible least cost

paths, for a realized value v .

Impact of uncertain demand • In a ring with two paths nothing important changes: prices

are still set at 0. • In a line network with two nodes, there is an outcome where

both intermediaries set a price equal to 1/3, so there is no trade with probability 2/3. • With 3 intermediaries the price will be 1/4 and so the

probability of no trade is 3/4. • Thus individual prices are falling, aggregate price is rising and

the probability of trade is falling too in the number of critical traders. • Choi, Galeotti and Goyal (2014) generalize this insight to all

networks.

Observations • The result on critical nodes is sharp but criticality may be

seen as too demanding: a node that lies on most (but not all) paths has the same status as compared to a node that lies on only one path. Moreover, all critical paths have equal status in the model. • It may be argued that location in the path – upstream or

downstream – should matter. • Related work with alternative pricing protocols develops these

points. • For auctions, see Kotowski and Leister (2014) and Gottardi

and Goyal (2012); for bargaining, see Condorelli, Galeotti and Renault (2015), Gofman (2011), and Manea (2013); for bid-and-ask prices, see Acemoglu and Ozdaglar (2007), Blume et al. (2007) and Gale and Kariv (2009).

Pricing in networks: Concluding observations • This discussion illustrates the ways in which posted prices may

be used to study intermediation in networks. • Similarly, we can use auctions and bargaining to study pricing

in networks. • In all cases, critical traders are key to shaping market power. • The models also suggest that the pricing protocol and the

timing (sequential versus simultaneous) of decision making interact with networks and with private information. • In the posted price model, all prices are set at the same time.

If prices were set in sequence then upstream traders will extract more of the surplus, just as in the bargaining and in the auction models. • The extent of this extraction will be mitigated by private

information downstream.

6. Games on random networks, Galeotti et al (2010).

• Introspection and casual observation both suggest that

individuals have very incomplete information about the network: know their own connections and have information about some statistics of the network. • Aim: present a framework to study behavior in networks when

players have incomplete information about the network.

Main findings: 1. Location in network: behavior and payoffs are monotonic in degree: increasing in games with complements, decreasing in games with substitutes. 2. Payoffs increasing (decreasing) in degree if game has positive (negative) externality. 3. Adding links: model in terms of dominance relations of degree distributions. Show how effects depend on whether game exhibits complements or substitutes.

Basic notation: • Players: N = {1, 2, ..., n } Each player i is a node i of an

undirected network g ; gi,j = 1, 0 indicates presence/absence of link between i and j,, respectively. • Neighbors: Ni (g ) = {j ∈ N : gi,j = 1} , is set of neighbors

and ηi (g ) = |Ni (g )| is her degree in g . • Actions: Player i chooses xi ∈ X , where X is a compact

subset of [0, 1]. We allow for both continuous and discrete actions sets.

• Payoffs: The payoff of player i under x = (x1 , ..., xn ) is:

ui (x, g ) = vki (g ) (xi , xNi (g ) ) where xNi (g ) is the vector of actions taken by the neighbors of i. – note anonymity of neighbors and homogeneity across players with same degree. • Assume that for any xi and k-dimensional vector x:

vk +1 (xi , (x, 0)) = vk (xi , x ). • Thus adding a link to a neighbor who chooses action 0 is

similar to not having an additional neighbor. Rules out payoffs, e.g., product or average of neighbors actions.

(A)

• A game exhibits strict strategic complements if it satisfies

increasing differences. That is, for all k, xi > xi0 , and x > x 0 : vk (xi , x ) − vk (xi0 , x ) > vk (xi , x 0 ) − vk (xi0 , x 0 ). Analogously for substitutes. • A game exhibits positive externalities if for each vk , and for all

x ≥ x 0 , vk (xi , x ) ≥ vk (xi , x 0 ). Negative externalities analogously defined.

• An example where payoffs depend on sum of actions. k

vk

xi , ∑ xj j =1

!

k

= f (xi + λ ∑ xj ) − c (xi )

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j =1

where c (·) is the cost of action and f (·) is the gross return. Clearly satisfies Assumption A. • Bramoulle and Kranton (2007): public goods model if λ = 1,

f (·) is concave, c (·) is linear and increasing. [(strict) strategic substitutes and positive externalities.] • Goyal and Moraga (2001): collaboration among monopolies if

λ = 1, f 00 (·) > 0 and c 00 (·) > f 00 (·) > 0. [(strict) strategic complements and positive externalities].

• Local information is reflected in the knowledge of own degree,

while information concerning the network at large is reflected in knowledge of the aggregate distribution of degrees across the society. • Let P (·) be the unconditional probability that any given node

has a given degree P (ki ). • Let the degrees of the neighbors of a player i of degree ki be

denoted by kN (i ) , which is a vector of dimension ki . • Assume that degrees of neighbors are independent, as in

the classical Erdos-Renyi model of random networks (in the limit, when n is infinite).

• The strategy for player i is a mapping

σi : {0, 1, ..., n − 1} → ∆(X ), where ∆(X ) is the set of distribution functions on X . We say σ is nondecreasing if σ(k 0 ) first order stochastic dominates σ(k ) for each k 0 > k. Similarly, for nonincreasing. Given a player i of degree ki let dψ−i (σ, ki ) denote the probability measure over xNi ∈ X ki induced by the beliefs P (· | ki ) composed with the strategies played via σ. The expected payoff to a player is given by: U (xi , σ, ki ) =

Z xN (i ) ∈X ki

vki (xi , xN (i ) )dψ−i (σ, ki )

• An equilibrium is a (Bayesian) Nash equilibrium of this game,

in the standard fashion.

Proposition There exists a symmetric Bayes-Nash equilibrium in the game. This follows from standard results so long as types of players are finite and strategy sets are either finite or compact convex sets and payoffs are continuous in all actions and concave in own action (if action sets are compact convex sets).

Proposition Suppose assumption A is satisfied and degrees of neighbors are independent. Every symmetric equilibrium is monotone increasing (decreasing) if payoffs satisfy the strict strategic complements (substitutes) property. • Intuition: Consider strategic complements. Given assumption

A, the best response of a k+1 degree player would be the same as a degree k player if the k + 1’th player chooses 0, for sure. However, in a non-trivial equilibrium, the (k + 1)’th neighbor would be choosing, on average, a positive action. Strict complementarities imply that the k + 1 degree player best responds with strictly higher actions than her k degree peers.

• Proof: We present the proof for the case of strategic

complements. The proof for the case of strategic substitutes is analogous and omitted. • Let {σk∗ } be the strategy played in a symmetric equilibrium of

the network game. If {σk∗ } is a trivial strategy with all degrees choosing action 0 with probability 1, the claim follows directly.

• Therefore, from now on, we assume that the equilibrium

strategy is non-trivial and that there is some k 0 and some x 0 > 0 such that x 0 ∈ supp(σk∗0 ).

• Consider any k ∈ {0, 1, ..., n } and let xk = sup[supp(σk∗ )]. If

xk = 0, it trivially follows that xk 0 ≥ xk for all xk 0 ∈ supp(σk∗0 ) with k 0 > k. So let us assume that xk > 0. Then, for any x < xk , Property A and the assumption of (strict) strategic complements imply that vk +1 (xk , xl1 , ., xlk , xs ) − vk +1 (x, xl1 , ., xlk , xs )

≥ vk (xk , xl1 , ., xlk ) − vk (x, xl1 , ., xlk ) • for any xs , with the inequality being strict if xs > 0.

• Then, taking expectations across types, noting that degrees of

any two neighboring nodes are independent, and that there are some players with degree k who choose xk > 0 implies that U (xk , σ∗ , k + 1) − U (x, σ∗ , k + 1) > U (xk , σ∗ , k ) − U (x, σ∗ , k ). • On the other hand, note that from the choice of xk ,

U (xk , σ∗ , k ) − U (x, σ∗ , k ) ≥ 0 for all x. Combining the aforementioned considerations we conclude: U (xk , σ∗ , k + 1) − U (x, σ∗ , k + 1) > 0, for all x < xk . This means that if x ∈ supp(σk∗+1 ) then x ≥ xk , which of course implies that σk∗+1 FOSD σk∗ . Iterating the argument as needed, the desired conclusion follows, i.e., σk∗0 FOSD σk∗ whenever k 0 > k.

Proposition Suppose that payoffs satisfy Assumption A and they are either strict strategic substitutes or complements. Then under positive externalities the expected payoffs are non-decreasing in degree, while under negative externalities the expected payoffs are non-increasing in degree. • Intuition: Positive externality: suppose that k neighbors of

k + 1 degree player follow the equilibrium strategy, but her (k + 1)0th neighbor chooses 0. Assumption A implies that she can earn payoffs as high as a k degree player by imitating this player. • The payoff degree relation depends only on externality and is

independent of complements or substitutes.

• Proof: We present the proof for positive externalities. The

proof for negative externalities is analogous and omitted. The claim is obviously true for a trivial equilibrium in which all players choose the action 0 with probability 1. So, let σ∗ be a (non-trivial) equilibrium strategy. Suppose xk ∈ supp(σk∗ ) and ∗ xk +1 ∈ supp σk +1 . Property A implies that vk +1 (xk , xl1 , ...., xlk , 0) = vk (xk , xl1 , ...., xlk ), for all xl1 , ...., xlk . Since the payoff structure satisfies positive externalities, it follows that for any x > 0, vk +1 (xk , xl1 , ...., xlk , x ) ≥ vk (xk , xl1 , ...., xlk ).

• Looking at expected utilities, we obtain that:

U (xk , σ∗ , k + 1) ≥ U (xk , σ∗ , k ). • Since σk∗+1 is a best response in the network game being ∗

played and xk +1 ∈ supp σk +1 ,

U (xk +1 , σ∗ , k + 1) ≥ U (xk , σ∗ , k + 1) and the result follows.

Revisiting Local public goods 1. In the Bramoulle and Kranton (2007) game of local public goods, payoffs exhibit substitutes and positive externalities. So we apply propositions 5 and 6 to obtain: 2. More connected players choose lower effort and hence provide less public goods. 3. More connected players earn higher payoffs, thus network connections render clear payoff benefits. 4. This is in marked contrast to the original complete information (regarding network) setting, where both actions and payoffs bear no systematic relation to networks. Thus incomplete network knowledge is more plausible and yields sharper equilibrium predictions!

8. Social networks and markets

• The relationship between community and markets remains a

central theme in the social sciences. • The empirical evidence on this subject is wide ranging and

very mixed. • In some instances, markets are associated with the erosion of

social relations, while in other contexts, markets appear to be crucial for their preservation. • The impact of markets on welfare and inequality varies

enormously.

Caste Networks and Globalization

• Munshi and Rosenzweig (2006) explore the impact of

economic liberalization of the Indian economy in the 1990s. • A shift toward the corporate and finance sectors, which

increased the returns to white-collar jobs, for which knowledge of English was necessary. • The authors estimate that in the city of Mumbai, the

premium to English education (compared to education in Marathi) went up by roughly 25% over the 1990s.

Case study 1: Traditional Networks and Globalization Munshi and Rosenzweig 2006, AER.

• Liberalization of the Indian economy in the 1990s. • Shift in Bombay’s economy toward the corporate and finance

sectors −→ Raised returns from English language education (approx. 25%) in 2000. • Working class boys adopted English much less than girls. • Gap in English education between girls of high and low castes

disappeared; the gap for boys remained intact. • Inequality between men and women fell.

Case Study 2: Kerala Fisherman and Mobile Telephony Jensen 2007, QJE • Cellphone adoption by Kerala fishermen in 1990’s. • Fishermen unable to observe prices in different markets. High

transportation costs, limited duration of market, perishability. • By 2001, 65% of fishing boats in Kerala owned a cellphone. • Adopters are well connected large boats. • Cellphone allows call to buyers before deciding on market. • Use of phones −→ increased profits for all fishermen. • Price dispersion and waste fell. • Inequality among fisherman rose.

Introduction • We develop a model where individuals located within a social structure choose a network exchange action (x) and a market exchange action (y ). • Payoffs to action x are increasing in the number of neighbours in the social structure who adopt the same action: this captures the personalized and possibly reciprocal nature of network exchange. • In contrast, market exchange is anonymous and short-term, and agents are price-takers: payoffs to action y are independent of the decisions of others. • Relation between the returns to the network and market actions: allow for both a complements and a substitutes relation. • We study who adopts the network and market actions, respectively, and how this choice affects aggregate welfare and inequality.

Networks, markets and inequality (Gagnon and Goyal, 2016) • Players: N = {1, 2, ...n }, with n ≥ 3. • Networks: g is a graph, where gij ∈ {0, 1} for all j, i ∈ N. • Actions: Player i chooses two actions, “network action” xi

and “market action” yi , where xi ∈ {0, 1} and yi ∈ {0, 1}. • Define

χi (a) =

∑

xj

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j ∈ Ni ( g )

• Given a, i’s payoff is:

Πi (a|g) = Φ (ai , χi (a))

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Model Payoffs: Properties

Property 1: Φ ((0, 0) , χi (a)) = 0. Property 2: Φ ((0, yi ) , m ) = πy ∀m ∈ N+ . The effect of m on marginal returns from x: ∆ (yi , m ) = Φ ((1, yi ) , m ) − Φ ((0, yi ) , m ) Property 3: ∆ (yi , m ) is weakly increasing in m.

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Model Payoffs Properties

Denote effect of action y on her marginal payoffs from x by: ξ (m ) = Φ ((1, 1) , m ) − Φ ((0, 1) , m ) − [Φ ((1, 0) , m ) − Φ ((0, 0) , m )] Property 4 If x and y are complements then ξ (m ) ≥ 0 for m ∈ N+ , and ξ (m ) is increasing in m ∈ N+ . If x and y are substitutes ξ (m ) < 0 for m ∈ N+ , and ξ (m ) is decreasing in m ∈ N+ .

Examples Example Player i’s payoffs are given by: Πi (a|g) = (1 + θyi ) xi χi (a) + yi − px xi − py yi

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θ ∈ [0, 1]: Substitutes for θ < 0 and complements for θ > 0.

Example Player i’s payoffs are given by: Πi (a|g) = (xi χiα (a) + yi )θ − px xi − py yi

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θ ∈ [0, ∞): Substitutes for θ ∈ (0, 1) and complements for θ > 1.

Equilibrium and welfare

Definition An equilibrium (x∗ , y∗ ) is maximal if there does not exist another equilibrium (x0 , y0 ) ∈ An that Pareto-dominates it.

Definition Given a network g and a price vector p, aggregate welfare from a strategy profile (x, y) is given by: W (x, y|p, g) =

∑ Πi (x, y|p, g).

i ∈N

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Equilibrium: Existence and Uniqueness Theorem

Theorem Suppose Properties 1-4 hold. Then there exists a unique maximal equilibrium generically.

Equilibrium: Existence and Uniqueness Intuition

• Steps in proof: 1. Complements: start from a profile with everyone choosing (0, 0). Iterate through best responses: noting that actions are complements, any increase in action x by one individual provokes a further increase (weakly) in others’ actions. As the action set is binary, the process must converge and the limit is an equilibrium. 2. Substitutes: exploit the payoffs structure more directly to construct different types of equilibrium in the cases where the market action alone is attractive and where it isn’t. 3. Existence of maximal equilibrium: the set of strategies, and hence the set of equilibria, is finite.

Equilibrium: Uniqueness Intuition

• The essential idea here is to start with two maximal equilibria

and construct a larger maximal equilibrium in terms of number of individuals choosing action x. • Exploiting the complementarity in payoffs for the network

action, we then show that this new equilibrium Pareto-dominates the two initial maximal equilibria. This contradiction completes the proof of Theorem ??.

Equilibrium characterization Preliminaries: q-core

Individual chooses between xi = 1 and xi = 0: will choose xi = 1 if χi is high enough. Similarly, her neighbors will choose x = 1 if a sufficient number of their own neighbors choose x = 1.

Definition Bollobas, 1984 The q-core of g, denoted by gq , is the largest collection of players that have strictly more than q links to other players in gq . This set is unique. Note that gq +k ⊆ gq , for any q, k ≥ 0.

Example The q-core in an arbitrary network

• Suppose q = 4. • Step 1: eliminate all nodes with k ≤ 4. • Step 2: iterate.

Example 1 The q-core in an arbitrary network

• Suppose q = 4. • Step 1: eliminate all nodes with k ≤ 4. • Step 2: iterate.

Example 1 The q-core in an arbitrary network

• Suppose q = 4. • Step 1: eliminate all nodes with k ≤ 4. • Step 2: iterate.

Example 1 The q-core in an arbitrary network

• Suppose q = 4. • Step 1: eliminate all nodes with k ≤ 4. • Step 2: iterate.

Example 1 The q-core in an arbitrary network

• Suppose q = 4. • Step 1: eliminate all nodes with k ≤ 4. • Step 2: iterate.

Example 1 The q-core in an arbitrary network

• Suppose q = 4. • Step 1: eliminate all nodes with k ≤ 4. • Step 2: iterate.

Example 1 The q-core in an arbitrary network

• Suppose q = 4. • Step 1: eliminate all nodes with k ≤ 4. • Step 2: iterate.

Thresholds for q-core • For simplicity, focus on ‘strong’ substitutes and complements.

This rules out choice of both x and y in substitutes case and solely action x in the complements case. • Recall from Property 3 that Φ ((1, yi ) , m ) is increasing in

m ∈ N+ , unlike Φ ((0, 0) , m ) and Φ ((0, 1) , m ) which both give fixed payoffs (from Properties 1 and 2).

• There thus exists q1 ≥ 0 such that for any m > q1 :

Φ ((1, 0) , m ) > max {Φ (0, yi , m )} yi ∈{0,1}

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• Similarly, there exists q2 such that for m > q2 :

Φ ((1, 1) , m ) > max {Φ (0, yi , m )} yi ∈{0,1}

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Payoffs properties For expositional simplicity, we assume Φ(., .) satisfies a slightly stronger condition. Property 5: Actions x and y are strong substitutes if for all m ∈ N+ : Φ ((1, 1) , m ) 6=

max

xi ,yi ∈{0,1}

{Φ ((xi , yi ) , m)}

while they are strong complements if for all m ∈ N+ : Φ ((1, 0) , m ) 6=

max

xi ,yi ∈{0,1}

{Φ ((xi , yi ) , m)}

Strong substitutes rules out (1, 1) is optimal, while strong complements rules out (1, 0), is optimal.

Equilibrium Characterization

Theorem Assume Properties 1-5 hold. Let a∗ be the maximal equilibrium. 1. Strong Substitutes. ai∗ = (1, 0) if and only if i ∈ g q1 ; for i∈ / g q1 , ai∗ = (0, 0) if πy ≤ 0, and ai∗ = (0, 1) otherwise. 2. Strong Complements. ai∗ = (1, 1) if and only if i ∈ g q2 ; if i∈ / g q2 , ai∗ = (0, 0) if πy ≤ 0, and ai∗ = (0, 1) otherwise.

Connections and markets

• q-core: Intuition that highly connected nodes adopt the

network action, less connected nodes adopt the market action. Our analysis goes beyond this intuition. The connections of neighbors and their neighbors matter... • Strategic structure: In the substitutes case, nodes lying

outside the relevant q-core choose market action, in the complements case it is the nodes within the relevant q-core that choose this action!

Substitutes and complements

Left: (θ = −0.9), py = 0.5 and px = 4.1. Right: (θ = 1), py = 1.1 and px = 5.1.

Core-periphery vs regular networks

Top: Substitutes (with θ = −0.9), py = 0.5 and px = 1.5. Bottom: complements (with θ = 1), py = 1.5 and px = 6.5.

Market Participation

• Receptive to markets: sparse or dense networks? • Individuals and markets: “well” connected or marginalized? • Theorem 1: in any network, there is a unique maximal

equilibrium (x∗ , y∗ ). • We define market penetration

M(g) ≡

∑i ∈N yi∗ (g) N

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Market Participation

Proposition Suppose the payoffs function Φ satisfies Properties 1-4. 1. Market participation is (weakly) lower in denser networks with substitutes, and (weakly) larger in case of complements. 2. Markets adopted by ‘less’ connected in case of substitutes, by ‘well’ connected in case of complements; 3. Market participation (weakly) increases with πy .

Welfare Do markets raise welfare?

Proposition In the case of complements, the introduction of the market always (weakly) increases aggregate welfare. In the case of substitutes, the introduction of the market may lower aggregate welfare. • Intuition: In the case of complements, markets reinforce social

ties and this raises payoffs. In the case of substitutes, marginal poorly connected individuals may move out of social exchange to markets. This weakens social ties, and could lower aggregate welfare.

Markets and Inequality Do markets increase inequality?

Given g, equilibrium inequality is denoted by R(p):

R (g ) ≡

1 + max {Πi (a∗ )}i ∈N 1 + min {Πi (a∗ )}i ∈N

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R (g) are payoffs of the “wealthiest” players compared to those of the “poorest”. It is close to other traditional metrics of inequality, including the range, the 20:20 ratio or the Palma ratio.

Markets and Inequality

Proposition In case of substitutes, the introduction of the market (weakly) decreases inequality. In case of complements if M (g) ∈ (0, 1) then markets strictly increase inequality, while if M (g) = 1, its effect on inequality is ambiguous. Similar findings also obtain for Gini-coefficient.

Markets and Inequality Intuition

• When x and y are substitutes, markets offer an outside option

to those players who benefit the least from x before its introduction. • When x and y are complements, the opposite logic obtains.

Indeed, in many cases, only the best-off players can afford y or both y and x, therefore benefiting from the complementarity between x and y . In such cases, y clearly increases inequality.

The dynamics of markets and networks

• More generally, these discussions suggest that the dynamics

between markets and social networks exhibit interesting non-linearities. • One technology can lead to the relative decline of social

networks, while a subsequent technology can lead to a revival and expansion of social networks. • This suggests that social networks are very malleable.

The Writing on the Wall

• Through much of human history, news was passed on through

private communication. Indeed, The Royal Society was set up in London in 1660, in an attempt to formalize such private communication (of the invisible college) through weekly meetings. • The growth of newspapers, television and radio magazines

through the 19th and 20th century gradually led to a decline of importance of social interaction in the process of communication. • We may well be witnessing a reversal of this movement.

The Writing on the Wall • The explosive growth of online social networks is one of the

defining features of the last decade. The Reuters Institute for the Study of Journalism (RISJ) reports that more than half the population of many countries (e.g. Brazil, Spain, Italy and Finland) use Facebook for news purposes (RISJ, 2014). • This rise of online news has proceeded in tandem with a sharp

decline in traditional newspaper markets (Newman, 2009; Currah, 2009). For an entertaining account of the fall and rise of social networks as vehicles for communication of news, see Standage (2013). • See Katz and Lazerfeld (1955) remains the standard reference

on social interaction in the age of mass media.

9. Conflict in networks • Individuals seek to accumulate more resources. One possible

avenue to obtain additional resources is to appropriate them through conflict. • There is wide ranging evidence that conflict usually takes

place among physically contiguous entities. • When Mr A wins in a fight with Mr B this brings him in

contact – and potentially in conflict – with those who are close to Mr B. When Mr A fights with Mr B, the technology of conflict will map their relative resources onto the respective probabilities of winning. • In this paper, our aim is develop a framework that will help in

understanding the dynamics of conflict and resource accumulation, in structured populations.

9. Conflict in networks: Dziubinski, Goyal and Minarsch, 2016 • Rulers are endowed with resources. A ruler can engage in • • •

• •

conflict with others to enlarge his resources. The set of potential conflicts is defined by a contiguity network. Rulers are farsighted and aim to maximize their resources. Rulers decide on whether to wage war or remain peaceful. The winner of a war takes control of the loser’s node and resources; he then decides on whether to wage war against other neighbors, or to stay peaceful. The game ends when either all players choose to be peaceful or when only one player is left. We study the influence of the technology of conflict and the structure of the network on war and peace, and the fate of kingdoms.

9. Conflict in networks

• Our first result identifies a threshold property in the

technology of conflict: above this threshold, every ruler wishes to wage war and, eventually, there is a hegemon. Below the threshold, resources and networks determine the prospects for peace and the number of kingdoms. We develop sufficient conditions for war and a lasting peace, respectively.

9. Conflict in networks: Model • We consider a set of players who each control sets of nodes of •

•

•

•

a network whose links represent possible conflict pairs. Each node is allocated (non-transferable) resources, which are controlled by the player controlling that node. A player can appropriate resources from other players by engaging in a conflict with them. The outcome of a bilateral conflict depends on the resources controlled by the conflicting players and is given by the Tullock contest function. If Mr A devotes x and Mr B devotes y then the probability of Mr A winning is x γ /(x γ + y γ ), where γ > 0, is reflects the importance of resources and is referred to as the technology of conflict. A player chooses between two courses of action: remaining peaceful (not entering into a conflict with any neighbour) and engaging in a sequence of conflicts with specified neighbours (and potentially the neighbors of neighbors, and so forth).

9. Conflict in networks

• The game proceeds in rounds. • At the beginning of each round, a player is picked at random

and chooses her strategy. • After a conflict between two players, the winner takes over the

nodes and the resources controlled by the loser. • The game continues until either no player wishes to fight

(peace) or only one player is left (hegemony).

9. Conflict in networks

• We identify a threshold property of conflict: independent of

the network and resource configuration, a peace equilibrium cannot be sustained with two or more players if the technology of conflict favours the larger resource contestant – when the Tullock parameter γ > 1. • This result is driven by two features of the contest success

function: when γ > 1 a conflict yields higher expected resources to the stronger player and it is always better to attack than to wait for other players to have engaged in conflict with each other.

9. Conflict in networks

• On the other hand, if technology of conflict favours the

weaker player – i.e., if the Tullock parameter γ < 1, then it is always better to wait and let other players fight. • We develop sufficient conditions on resources and network

configurations for perpetual strife or lasting peace. A lasting peace needs resources to be sufficiently evenly distributed. • If resources are unequal, then perpetual strife and hegemony

is the only possible outcome.

10. Readings Main Readings 1. S. Goyal (2007), Chapter 3. Connections 2. S. Goyal (2016), Networks and Markets, mimeo, Cambridge. Related Articles 1. Ballester, C., A. Calvo-Armengol, and Y. Zenou (2006), Who’s who in networks. Wanted: The Key Player, Econometrica. 2. Bramoull´ e, Y. and R. Kranton (2007), Local public goods in Networks, Journal of Economic Theory. 3. Galeotti, Goyal, Jackson, Vega-Redondo and Yariv (2010), Network Games, Review of Economic Studies. 4. Goyal, S. and J. L. Moraga-Gonzalez (2001), R&D Networks, Rand Journal of Economics. 5. Amours, Bramoulle and Kranton (2014), Strategic interaction in networks. American Economic Review 6. Galeotti, A. and S. Goyal 2010, Law of the few. American Economic Review. 7. Gagnon, J., and S. Goyal (2016), Networks, markets and inequality, mimeo, Cambridge. 8. Ghiglino, C. and S. Goyal (2010), Keeping up with the neighbors, Journal of the European Economic Association. 9. Jackson and Zenou (2014), Games on networks, in Handbook of Game Theory 10. Bramoulle and Kranton (2015), Games played on networks, in Oxford Hanbook of the Economics of Networks.

References Continued

1. Condorelli, D., A. Galeotti and L. Renou (2016), Bilteral trading in networks, Review of Economic Studies. 2. Goyal, S. and F. Vega-Redondo (2007), Structural Holes in Social Networks, Journal of Economic Theory. 3. Manea, M (2013), Intermediation in Networks, mimeo, MIT. 4. Manea, M (2015), Models of Bilateral Trade in Networks, to appear in The Oxford Handbook of the Economics of Networks, edited by Y. Bramoulle, A. Galeotti and B. Rogers. 5. Kotowoski, M. and M. Leister (2014), Trading networks and equilibrium intermediation, mimeo, Berkeley. 6. Choi, S., A. Galeotti and S. Goyal (2014), Trading in Networks: theory and experiments, Cambridge-INET Institute Working Paper 2014-08. 7. Dziubinski, M., S. Goyal and D. Minarsch (2016), War or peace? A dynamic model of conflict. mimeo Cambridge University.

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