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Strategic Network Formation Social and Economic Networks Jafar Habibi MohammadAmin Fazli Social and Economic Networks 1 ToC Strategic Network Formation Pairwise Stability Efficient Networks Some Strategic Network Models

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SLIDE 1

Strategic Network Formation

Social and Economic Networks

Jafar Habibi MohammadAmin Fazli

Social and Economic Networks 1

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SLIDE 2

ToC

  • Strategic Network Formation
  • Pairwise Stability
  • Efficient Networks
  • Some Strategic Network Models
  • Readings:
  • Chapter 6 from the Jackson book

Social and Economic Networks 2

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SLIDE 3

Strategic Network Formation

  • Study the formation of a network by individual selfish agents
  • Agents have a tendency to form relationships that are (mutually)

beneficial and to drop relationships that are not

  • The forces behind such incentives can be quite strong and can
  • perate without individuals realizing that they are being influenced in

this way

  • The term strategic carries with it connotations that are not necessary

to its application

Social and Economic Networks 3

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SLIDE 4

Pairwise Stability

  • A utility of payoff function (𝑣𝑗: 𝐻 𝑂 β†’ 𝑆), 𝑣𝑗(𝑕) represents the net

benefit that i receives if network g in in place

  • A network g is pairwise stable if
  • For all π‘—π‘˜ ∈ 𝑕, 𝑣𝑗 𝑕 β‰₯ 𝑣𝑗(𝑕 βˆ’ π‘—π‘˜) and π‘£π‘˜ 𝑕 β‰₯ π‘£π‘˜(𝑕 βˆ’ π‘—π‘˜)
  • For all π‘—π‘˜ βˆ‰ 𝑕, if 𝑣𝑗 𝑕 + π‘—π‘˜ > 𝑣𝑗 𝑕 then π‘£π‘˜ 𝑕 + π‘—π‘˜ < π‘£π‘˜(𝑕)

Social and Economic Networks 4

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SLIDE 5

Efficient Networks

  • Efficiency: a network g in efficient relative to a profile utility functions

𝑣1, 𝑣2, … , π‘£π‘œ if 𝑗 𝑣𝑗 𝑕 β‰₯ 𝑗 𝑣𝑗(𝑕′) for all 𝑕′ ∈ 𝐻(𝑂)

  • Pareto Efficiency: a network g is Pareto efficient relative to a profile

utility functions (𝑣1, 𝑣2, … , π‘£π‘œ) if there does not exist any 𝑕′ ∈ 𝐻 𝑂 such that 𝑣𝑗 𝑕′ β‰₯ 𝑣𝑗(𝑕) for all i, with strict inequality for some i.

  • We say that one network Pareto dominates another if it leads to a weakly

higher payoff for all individuals and a strictly higher payoff for at least one.

  • What is the relationship between these two kinds of efficiency?

Social and Economic Networks 5

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SLIDE 6

Efficient Networks

  • To better understand the relationship between efficiency and Pareto

efficiency, note that if g is efficient relative to (𝑣1, . . . , π‘£π‘œ) then it must also be Pareto efficient relative to 𝑣1, . . . , π‘£π‘œ .

  • However, the converse is not true. What is true is that g is efficient

relative to (𝑣1, . . . , π‘£π‘œ) if and only if it is Pareto efficient relative to all payoff 𝑣1

β€² , 𝑣2 β€² , … , π‘£π‘œ β€²

such that 𝑗 𝑣𝑗 = 𝑗 𝑣𝑗

β€²

  • Proof: See the blackboard

Social and Economic Networks 6

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SLIDE 7

Efficient Networks

Social and Economic Networks 7

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SLIDE 8

Distance-based Utility

  • Define 𝑐: 1,2, … , π‘œ βˆ’ 1 β†’ 𝑆,
  • 𝑐 𝑙 > 𝑐 𝑙 + 1 > 0 for any k, 𝑑 β‰₯ 0
  • Special case for 𝑐 𝑙 = πœ€π‘™
  • Efficiency Theorem: The unique efficient network structure in the distance-

based utility model is:

  • The complete network if 𝑐 2 < 𝑐 1 βˆ’ 𝑑
  • A star encompassing all nodes if 𝑐 1 βˆ’ 𝑐 2 < 𝑑 < 𝑐 1 + π‘œβˆ’2

2 𝑐 2

  • The empty network if 𝑐 1 + π‘œβˆ’2

2 𝑐 2 < 𝑑

  • Proof: see the black-board

Social and Economic Networks 8

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Distance-based Utility

  • Pairwise Stability theorem: In the distance-based utility model:
  • A pairwise stable network has at most one (nonempty) component.
  • For b(2) < b(1) βˆ’ c, the unique pairwise stable network is the complete

network.

  • For b(1) βˆ’ b(2) < c < b(1), a star encompassing all players is pairwise

stable, but for some n and parameter values in this range it is not the unique pairwise stable network.

  • For b(1) < c, in any pairwise stable network each node has either no links
  • r else at least two links. Thus every pairwise stable network is inefficient

when b(1) < c < b(1) +

π‘œβˆ’ 2 2

b(2).

Social and Economic Networks 9

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SLIDE 10

Externalities

  • Externalities occur when the utility or payoffs to one individual are

affected by the actions of others, although those actions do not directly involve the individual in question

  • We say that there are nonnegative externalities under 𝑣

= (𝑣1, . . . , π‘£π‘œ) if 𝑣𝑗 𝑕 + π‘˜π‘™ β‰₯ 𝑣𝑗(𝑕), for all i, g, π‘˜ β‰  𝑗 β‰  𝑙

  • Positive externalities if the inequality is strict in some instances
  • We say that there are nonpositive externalities under 𝑣

= (𝑣1, . . . , π‘£π‘œ) if 𝑣𝑗 𝑕 + π‘˜π‘™ ≀ 𝑣𝑗(𝑕), for all i, g, π‘˜ β‰  𝑗 β‰  𝑙

  • Negative externalities if the inequality is strict in some instances

Social and Economic Networks 10

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SLIDE 11

Dynamics & Reachability of Stable States

  • How can we predict which networks are likely to emerge from a multitude
  • f pairwise stable networks?
  • There are a variety of approaches focusing on either refining the

equilibrium concept or examining some dynamic process.

  • A natural dynamic process:
  • A random ordering over links
  • If the link has not yet been added to the network, and at least one of the two players

involved would benefit from adding it and the other would be at least as well off given the current network then the link is added.

  • If the identified link has already been added, then it is deleted if either player would

benefit from its deletion.

  • If this process comes to rest on a fixed configuration, then it must be at a pairwise

stable network.

Social and Economic Networks 11

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SLIDE 12

Dynamics & Reachability of Stable States

  • Theorem: Consider the symmetric distance-based utility model in the

case b(1) βˆ’ b(2) < c < b(1). As the number of players grows, the probability that the above described dynamic process leads to an efficient network (star) converges to 0.

  • Proof: See the blackboard

Social and Economic Networks 12

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SLIDE 13

Price of Anarchy & Price of Stability

  • The price of anarchy is the ratio largest total cost (in absolute value)

generated by any pairwise stable network compared to the cost of the efficient network.

  • A ratio of 1 indicates that all pairwise stable networks are efficient
  • The price of stability is the ratio of the lowest total cost (in absolute

value) generated by any pairwise stable network to the cost of the efficient network.

  • A price of stability of 1 indicates that the efficient network is stable
  • These prices can differ substantially, and we can keep track of a

anarchy-stability gap.

Social and Economic Networks 13

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Price of Anarchy & Price of Stability

  • To get an idea of the price of anarchy, consider a special case of the

distance-based utility model in which preferences are directly proportional to distance : 𝑣𝑗 𝑕 =

π‘˜β‰ π‘—

βˆ’π‘šπ‘—π‘˜ 𝑕 βˆ’ 𝑒𝑗 𝑕 𝑑

  • Theorem: The diameter of any pairwise stable network in the model

described above is at most 2βˆšπ‘‘ + 1, and such a network contains at most π‘œ βˆ’ 1 +

2π‘œ2 βˆšπ‘‘ links. Thus, the price of anarchy is no more than 17βˆšπ‘‘.

  • Proof: see the blackboard.

Social and Economic Networks 14

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SLIDE 15

The Coauthor Model

  • Let’s study a model in which individuals would rather fewer number of

connections: negative externality

  • Coauthor Model:
  • Consider collaborations in a joint work
  • Beyond the benefit of having the other player put time into the project, there is also

a form of synergy

  • The synergy is proportional to the product of the amounts of time the two

individuals devote to the project.

  • Negative externality: If an individual’s collaborator increases the time spent on other

projects, then the individual sees less synergy with that collaborator.

  • Effectively, each player has a fixed amount of time to spend on projects, and the time

that researcher i spends on a given project is inversely related to the number of projects

Social and Economic Networks 15

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The Coauthor Model

  • In the coauthor model, if n is even, then the efficient network

structure consists of n/2 separate pairs. If a network is pairwise stable and n β‰₯ 4, then it is inefficient and can be partitioned into fully intraconnected components, each of which has a different number of

  • members. Moreover, if m is the number of members of one

component of a pairwise stable network and 𝑛 is the number of members of a different component that is no larger than the first, then m > 𝑛2

  • Proof: see the blackboard.

Social and Economic Networks 16

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The Islands-Connection Model

  • Nodes that are closer find it cheaper to maintain links to each other,

which generates high clustering

  • Consider some geographic structure for costs in the symmetric

distance-based utitliy model

  • If the distance between two nodes in more than D they don’t receive any

value from each other

  • We have K islands each has J players; Forming a link between players i

and j costs i and j each c if they are on the same island, and C otherwise, where C > c > 0.

  • The utility function:

Social and Economic Networks 17

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SLIDE 18

The Islands-Connection Model

  • Theorem: If 𝑑 < πœ€ βˆ’ πœ€2 and 𝐷 < πœ€ + 𝐾 βˆ’ 1 πœ€2, then any network

that is pairwise stable or efficient is such that

  • The players on any given island are completely connected to one another,
  • The diameter and average path length are no greater than D + 1, and
  • if πœ€ βˆ’ πœ€3 < 𝐷, then a lower bound on individual, average, and overall

clustering is (𝐾 βˆ’ 1)(𝐾 βˆ’ 2)/(𝐾2𝐿2).

  • Proof: See on the blackboard.
  • Captures Small-World properties in a strong sense

Social and Economic Networks 18

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SLIDE 19

The General Tensions Between Stability and Efficiency

  • A transfer rule is a function 𝑒: 𝐻 β†’ π‘†π‘œ such that 𝑗 𝑒𝑗 𝑕 = 0 for all g

(balance condition).

  • A transfer rule can thus capture any reallocation of payoff in a given

network.

  • These payments could subsidize or tax certain links or collections of links

and could be due to intervention by an outside authority or to bargaining by the players.

  • In the presence of transfers, player i’s net payoff becomes 𝑣𝑗 𝑕 + 𝑒𝑗(𝑕)

and this is used by the player in decisions regarding the addition or deletion of links

  • The egalitarian transfer rule (denoted by 𝑒𝑓):

Social and Economic Networks 19

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SLIDE 20

The General Tensions Between Stability and Efficiency

  • A transfer rule t is component balanced if there are no net transfers

across components of the network; that is, π‘—βˆˆπ‘‡ 𝑒𝑗(𝑕) for each network g and component S of g.

  • A profile of utility functions u is component-decomposable if 𝑣𝑗 𝑕

= 𝑣𝑗(𝑕 𝑂𝑗

π‘œ 𝑕 )for all i and g.

  • Two players i and j are complete equals relative to a profile of utility

functions u and a network g if

  • 𝑗𝑙 ∈ 𝑕 iff π‘˜π‘™ ∈ 𝑕
  • 𝑣𝑙 𝑕′ = 𝑣𝑙(π‘•β€²π‘—π‘˜) for all 𝑙 βˆ‰ {𝑗, π‘˜} and for all 𝑕′ ∈ 𝐻 𝑂
  • 𝑣𝑗 𝑕′ = π‘£π‘˜ π‘•β€²π‘—π‘˜ and π‘£π‘˜ 𝑕′ = 𝑣𝑗 π‘•β€²π‘—π‘˜ for all 𝑕′ ∈ 𝐻(𝑂)

Social and Economic Networks 20

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SLIDE 21

The General Tensions Between Stability and Efficiency

  • A transfer rule satisfies equal treatment of equals relative to a profile of

utility functions u if 𝑒𝑗 𝑕 = π‘’π‘˜(𝑕) when i and j are complete equals relative to u and g

  • Theorem: There exist component-decomposable utility functions such

that every pairwise stable network relative to any component balanced transfer rule satisfying equal treatment of equals is inefficient.

  • Proof: See the blackboard

Social and Economic Networks 21

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SLIDE 22

The General Tensions Between Stability and Efficiency

  • If we drop equal treatment of equals, but keep component balance,

then a careful and clever construction of transfers ensures that some efficient network is strongly stable for a class of utility functions

  • Theorem: If the profile of utility functions is component-

decomposable and all nonempty networks generate positive total utility, then there exists a component-balanced transfer rule such that some efficient network is pairwise stable. Moreover, while transfers will sometimes fail to satisfy equal treatment of equals, they can be structured to treat completely equal players equally on at least one network that is both efficient and pairwise stable.

Social and Economic Networks 22