Strategic Network Formation Social and Economic Networks MohammadAmin Fazli Social and Economic Networks 1
ToC • Strategic Network Formation • Pairwise Stability • Efficient Networks • Some Strategic Network Models • Readings: • Chapter 6 from the Jackson book Social and Economic Networks 2
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 operate 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
Pairwise Stability • A utility of payoff function ( 𝑣 𝑗 : 𝐻 𝑂 → 𝑆 ), 𝑣 𝑗 () represents the net benefit that i receives if network g in in place • A network g is pairwise stable if • For all 𝑗𝑘 ∈ , 𝑣 𝑗 ≥ 𝑣 𝑗 ( − 𝑗𝑘) and 𝑣 𝑘 ≥ 𝑣 𝑘 ( − 𝑗𝑘) • For all 𝑗𝑘 ∉ , if 𝑣 𝑗 + 𝑗𝑘 > 𝑣 𝑗 then 𝑣 𝑘 + 𝑗𝑘 < 𝑣 𝑘 () Social and Economic Networks 4
Efficient Networks • Efficiency: a network g in efficient relative to a profile utility functions 𝑣 1 , 𝑣 2 , … , 𝑣 𝑜 if 𝑗 𝑣 𝑗 ≥ 𝑗 𝑣 𝑗 ( ′ ) for all ′ ∈ 𝐻(𝑂) • Pareto Efficiency: a network g is Pareto efficient relative to a profile utility functions (𝑣 1 , 𝑣 2 , … , 𝑣 𝑜 ) if there does not exist any ′ ∈ 𝐻 𝑂 such that 𝑣 𝑗 ′ ≥ 𝑣 𝑗 () 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
Efficient Networks • To better understand the relationship between efficiency and Pareto efficiency, note that if g is efficient relative to (𝑣 1 , . . . , 𝑣 𝑜 ) then it must also be Pareto efficient relative to 𝑣 1 , . . . , 𝑣 𝑜 . • However, the converse is not true. What is true is that g is efficient relative to (𝑣 1 , . . . , 𝑣 𝑜 ) if and only if it is Pareto efficient relative to all ′ , 𝑣 2 ′ , … , 𝑣 𝑜 ′ ′ such that 𝑗 𝑣 𝑗 = 𝑗 𝑣 𝑗 payoff 𝑣 1 • Proof: See the blackboard Social and Economic Networks 6
Efficient Networks Social and Economic Networks 7
Distance-based Utility • Define 𝑐: 1,2, … , 𝑜 − 1 → 𝑆 , • 𝑐 𝑙 > 𝑐 𝑙 + 1 > 0 for any k, 𝑑 ≥ 0 • Special case for 𝑐 𝑙 = 𝜀 𝑙 • Efficiency Theorem: The unique efficient network structure in the distance- based utility model is: o The complete network if 𝑐 2 < 𝑐 1 − 𝑑 o A star encompassing all nodes if 𝑐 1 − 𝑐 2 < 𝑑 < 𝑐 1 + 𝑜−2 2 𝑐 2 o The empty network if 𝑐 1 + 𝑜−2 2 𝑐 2 < 𝑑 • Proof: see the black-board Social and Economic Networks 8
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 ) − c, the unique pairwise stable network is the complete network. • For b( 1 ) − 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 or else at least two links. Thus every pairwise stable network is inefficient 𝑜− 2 when b( 1 ) < c < b( 1 ) + b( 2 ). 2 Social and Economic Networks 9
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 = (𝑣 1 , . . . , 𝑣 𝑜 ) if 𝑣 𝑗 + 𝑘𝑙 ≥ 𝑣 𝑗 () , for all i, g, 𝑘 ≠ 𝑗 ≠ 𝑙 • Positive externalities if the inequality is strict in some instances • We say that there are nonpositive externalities under 𝑣 = (𝑣 1 , . . . , 𝑣 𝑜 ) if 𝑣 𝑗 + 𝑘𝑙 ≤ 𝑣 𝑗 () , for all i, g, 𝑘 ≠ 𝑗 ≠ 𝑙 • Negative externalities if the inequality is strict in some instances Social and Economic Networks 10
Dynamics & Reachability of Stable States • How can we predict which networks are likely to emerge from a multitude of 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
Dynamics & Reachability of Stable States • Theorem: Consider the symmetric distance-based utility model in the case b(1) − 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
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
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 : 𝑣 𝑗 = −𝑚 𝑗𝑘 − 𝑒 𝑗 𝑑 𝑘≠𝑗 • Theorem: The diameter of any pairwise stable network in the model described above is at most 2√𝑑 + 1 , and such a network contains at most 2𝑜 2 𝑜 − 1 + √𝑑 links. Thus, the price of anarchy is no more than 17√𝑑 . • Proof: see the blackboard. Social and Economic Networks 14
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
The Coauthor Model • Theorem: 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 ≥ 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 m is the number of members of a different component that is no larger than the first, m 2 then m > • Proof: see the blackboard. Social and Economic Networks 16
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 utility 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
The Islands-Connection Model • Theorem: If 𝑑 < 𝜀 − 𝜀 2 and 𝐷 < 𝜀 + 𝐾 − 1 𝜀 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 𝜀 − 𝜀 3 < 𝐷 , then a lower bound on individual, average, and overall clustering is (𝐾 − 1)(𝐾 − 2)/(𝐾 2 𝐿 2 ) . • Proof: See on the blackboard. • Captures Small-World properties in a strong sense Social and Economic Networks 18
Recommend
More recommend
