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Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA

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Lecture Seventeen: Game theory in networks.

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Game theory in networks Example: Should athletes dope? + improves performance - penalities if caught Beneficial to dope if enough competitors dope. (esp. if competitors dope)

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Game theory in networks Example: Should you install (unsecured) wireless internet access? - costs money + you can check email all night long Beneficial to buy if neighbors don’t.

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Normal form games Model as a game with: a set of players {1, …, n} an action space S i for each player i a payoff u i (s ) to each player i for action profile s in S 1 x … x S n

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Graphical games Defn. A graphical game is a normal form game in which the payoff to i depends only on her neighbors in the graph G.

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Graphical games Doping game: (i,j) are neighbors if they are in the same competition Wireless internet game: (i,j) are neighbors if they can get each others’ wireless signals

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Questions When do equilibria exist and what do they look like? If there are multiple equilibria, which is most likely? How does network structure effect equilibria? How can one design the network to produce “optimal” equilibria?

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Language game Example: Having a similar language as neighbors facilitates interaction (improves communication, understanding, etc.). FrenchEnglish French (v French, v French ) ( 0, 0 )(v English, v English ) ( 0, 0 )

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Two nodes If both speak French, get satisfaction v French. If both speak English, get satisfaction v English. If speak different languages, zero satisfaction.

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Many nodes v v Learn one language: Total satisfaction is sum of edge satisfactions Payoff to v of French: 2 (v French ) Payoff to v of English: 4 (v English )

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Equilibria We all speak French We all speak English ?

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Equilibria v v Learn French if: Total payoff( French ) > Total payoff( English ) Suppose node v has d neighbors, of which fraction p use A. Then v will use A if pdv French > (1-p)dv English or p > v English / (v French + v English ) = q Relative quality of English to French

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Hereafter FrenchEnglish French ((1-q), (1-q)) ( 0, 0 )(q, q) ( 0, 0 )

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Dynamic behavior Start from an initial configuration and let players update strategies over time what equilibrium results? how’s it depend on initial configuration? how’s it depend on network structure?

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Choosing languages If at least a ½ fraction of neighbors are blue, then turn blue, else turn yellow

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Question If everyone initially speaks English, can we force some children to learn French such that eventually everyone speaks French? Can we get a French cascade?

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Cascades Experiment: In your envelope is a card indicating if you are a type-1 player or a type-2 player in the following social network. 1 22 1

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Cascades Experiment: We will play the following bimatrix game on each edge of the social network. Your payoff is the sum of the edge-payoffs. FrenchEnglish French ( 0.6, 0.6 ) ( 0, 0 )( 0.4, 0.4 ) ( 0, 0 )

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FrenchEnglish French ( 0.6, 0.6 ) ( 0, 0 )( 0.4, 0.4 ) ( 0, 0 ) 1 22 1 Experiment: Write on your cards your name and whether you want to speak French or English. You will earn a # of points equal to your expected payoff.

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Basic diffusion example 1 Endow group 0 with blue strategy ``If at least a q fraction of neighbors use blue strategy, then use blue strategy.’’ If q < ½, whole graph will turn blue 012 ABBB

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Basic diffusion example 2 Endow any group with blue ``If at least a q fraction of neighbors use blue, then use blue.’’ Need q < ¼ for behavior to spread

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Cascades Higher q makes cascades harder Given initial adopters, max q for which a cascade happens is the contagion threshold Theorem. [Morris, 2000]: The contagion threshold is always at most ½.

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What hurts cascades? Highly connected clusters, i.e., triadic closure

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What helps cascades? Edges between clusters, i.e., weak ties.

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What hurts cascades? A neighborhood with cohesion p(S) is a set S of nodes such that each node has at least a p fraction of its neighbors in S Neighborhood with cohesion 3/4

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Cascades and clusters Given initial adopters S, if remaining nodes have 1. some cohesive cluster, contagion can’t “break in”. 2. no cohesive cluster, contagion happens. Theorem. [Morris, 2000]: The contagion threshold of a graph G with initial adopters S is the largest q such that q < 1 – p(T) for all clusters T outside S.

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Can compatibility help?

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Diffusion with Compatibility 1.Each person can adopt multiple behaviors at an added cost. 2.Can adapt to peers with different behaviors.

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Benefits of Compatibility Without compatibility, v can get 2q … or 3(1 – q) With compatibility, v can get 2q + 3(1 – q) – c where c is cost of choosing both blue and yellow v v v v

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Compatibility Model Let r be additional cost of adopting both behaviors (costs r per-edge). Payoff matrix is: Player 1/ Player 2 ABAB A(1-q,1-q)(0,0)(1-q, 1-q-r) B(0,0)(q,q)(q, q-r) AB(1-q-r, 1-q)(q-r, q) (max(q,1-q)-r, max(q,1-q)-r)

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For which values of (q,r) will new technology become an epidemic?

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Simple Observations For high r, technologies are incompatible. Each node will chose just one, and results of Morris carry over. For low r, it is almost free to have both technologies. All nodes therefore adopt both and then drop worse one, so contagion happens if q < ½. For intermediate r?

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Example If r is low, groups 1 and -1 switch to AB to be able to communicate with all neighbors … but if r is not low enough, groups 2 and -2 may not find it profitable to adopt A since can already communicate with all 6 neighbors on B! For example, q = 5/12 and r = 2/12 012 ABBB

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Example Technology A can spread if q < ½ and either q+r q. 1/2 1 1 q r A spreads A does not spread!

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Interpretation Strategically, an inferior incumbant can defend against a new superior option by adopting a limited level of compatibility (e.g., operating system emulators). Buffers of bi-lingualism can contain pockets of alternative behaviors, ensuring multiple behaviors will co-exist (e.g., Dutch).

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Next time Network exchange.

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