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

Monday’s game AB C D x1 x1 Game one: AB C D x1 x1 Game two: 0 26 people played, 7 took A->C->B: 1.7 points 19 took A->D->B: 1.3 points 30 people played, 7 took A->C->B: 1 point 23 took A->C->D->B: 1.2 points Payoff = 3 - delay

Lecture Eighteen: Network exchange.

Networked exchange theory Network represents potential trades what prices result?

Splitting a dollar Experiment: You have one point to share with a random opponent. Write on your card your name and what fraction of the point you wish to keep for yourself. If you ask for x and your opponent y, you get: 1. 2(x) points if x + y ≤ 1, 2. ZERO POINTS otherwise.

What are the equilibria of this game? Raise your hand if you wrote 0.50.

Nash bargaining How to split a dollar? David (0.5) If negotiations fail, you get nothing. Christine (0.5)

Nash bargaining How to split a dollar? If negotiations fail, Mark gets $0.60, Hongyu gets $0.20. Mark (0.65)Hongyu (0.35)

Nash bargaining Equilibria? Any division in which each agent gets at least the outside option is an equilibrium. Yet …. agents usually agree to split the surplus.

Nash bargaining Assumption: If when negotiation fails, - A gets $a - B gets $b Then when succeed, - A gets $(a + s/2) - B gets $(b + s/2) s = (1 – a – b) is the surplus

Nash bargaining Nash: “Agents will agree to split the surplus.” Motivated by axiomatic approach, optimization approach, and outcome of particular game- theoretic formulations.

Bargaining in networks Important special case: graph is bipartite. There are two groups of nodes and all edges are between nodes.

Bargaining in networks Important special case: graph is bipartite, and each agent is in at most one trade (i.e., set of trades form a matching).

Bargaining in networks Value of outside option arises as result of network structure.

Bargaining in networks Tyler (0.50) Matthew (0.50) Eric (0) Elif (0.75) Stephen (0.25) Transactions worth $1. Only one transaction per person!

Other experiments Almost all the money.

Outcomes A solution for a network G is a matching M and a set of values ν u for each node u s.t., - For (u,v) in M, ν u + ν v = 1 - For unmatched nodes u, ν u = 0

Outcomes

Stable outcomes Node u could negotiate with unmatched neighbor v and get (1 - ν v ). Outside option of u is α u = maximum over unmatched neighbors v of (1 - ν v ).

Outcomes ® 1 = 0 ® 3 = 0 ® 4 = 0.5 ® 5 = 0 ® 2 = 1

Stable outcomes Defn. An outcome is stable if for all u, ν u ≥ α u.

A stable outcome ® 1 = 0 ® 3 = 0 ® 4 = 0 ® 5 = 0 ® 2 = 1

Another stable outcome ® 1 = 0 ® 3 = 0 ® 4 = 0 ® 5 = 0 ® 2 = 1

Stable outcomes Notice there are many stable outcomes, so which one should we expect to find?

Balanced outcomes Each individual bargaining outcome should agree with the Nash bargaining solution. s uv = 1 - α u - α v ν u = α u + s/2 And similarly for ν v.

A balanced outcome ® 1 = 0 ® 3 = 0 ® 4 = 0 ® 5 = 0 ® 2 = 1

Current research: understand when balanced outcomes exist and how earnings depend on network structure.

Next time Games of incomplete information.