R Ravi

ravi+iitd@andrew.cmu.edu

CSV 886 Social Economic and Information Networks

Lecture 7: Bargaining in Networks

Power

• Imbalance in social exchange, taking place in a

graph

– Dependence

– Exclusion/Gatekeeping

– Satiation

– Betweenness

2

Experimental Network Exchange Studies

• Choose small graph with person at each node

• Endow a resource pool in each edge (say 1 $)

• Each node limited to number of exchanges she

can be part of (1-exchange = only one of her

edges)

• Talk to all edge neighbors and pick one to

split the edge money

• Repeat experiment for multiple rounds

3

Class Experiment

Configurations

Stopwatch4

Results?

• 2-node path

• 3-node path

• 4-node path

• 5-node path

5

Stem Graph

6

Unstable networks?

7

Two-person interactions

What model should we use to capture the

interaction between two people who actually

decide to exchange?

• Nash bargaining

• Ultimatum Game

8

Nash Bargaining

• Two nodes A and B with outside options of x

and y

• x + y < 1 (Else no way to agree)

• Surplus s = 1 – (x+y)

• Nash bargaining outcome splits the surplus

evenly:

– A gets x + ½ s

– B gets y + ½ s

9

Examples

10

Status

• Equi-dependent outcomes only when both

players have equal status

• Differential status is known to

– Change negotiating strategy (over/understate

outside offer)

– Change perception (discounting of offers made by

lower status partner)

11

Ultimatum Game

• A proposes division of 1 $ to B

• B can accept or reject

• If B accepts, both keep the splits that A

proposed; If B rejects, neither make any money

If you are A, how much would you propose to give B ?

If you are B, how low will you accept ?

12

Results

Rational Outcome

Reality [Güth, Schmittberger & Schwarze ’82]

13

Outcome

• Matching on the graph (every node exchanges

with at most one other)

• A value for each node, such that the sum of

values for every edge in the matching is 1

14

Instability

• Given an outcome, an instability is an edge

not in the outcome matching for which the

endpoints have value less than 1 (so they can

now get together and do better for

themselves)

• Stable outcome is one with no instability

15

Examples

16

Application

Qualitatively explain network power

17

Application

“Proof” that triangle has no stable outcome

18

Balanced Outcomes

• Take into account “outside options”

• Try to define a outcome that balances the

outside option in the context of the negotiation

19

What is the outside option?

For a node in the negotiation (outcome

matching), outside option is the best value it

can get after stealing away any of its

unmatched edges

Balanced when matched edges split surplus

evenly (by Nash bargaining)

20

Balanced Outcome

• For each edge in the matching, the split of

money represents the Nash Bargaining

outcome for the two nodes involved, given the

best outside option provided by values in the

rest of the network

21

Four node path: which are balanced?

22

Balanced outcome for Stem

23

Example

24

Example

25

Stable versus Balanced

• Balanced outcomes are stable

• In fact, whenever there is a stable outcome for

a network, it also has a balanced outcome

Other notions of balance can be defined to

reconcile observed outcomes

26

SECOND HOMEWORK

1) Problem on Network Effects:

Exercise 2 from Chapter 17

2) Problem on Bargaining:

Exercise 6 from Chapter 12

27