Transcript Arbitration, Fairness and Stability Revenue Division in Collaborative Settings Yair Zick Advisor: Prof. Edith Elkind.

Arbitration, Fairness
and Stability
Revenue Division in Collaborative Settings
Yair Zick
Advisor: Prof. Edith Elkind
Acknowledgments
Edith Elkind
Acknowledgments
My research collaborators, past and present!
Yoram Bachrach, Nina Balcan, Alejandro
Carbonara, George Chalkiadakis, Amit Datta,
Anupam Datta, Yuval Filmus, Kobi Gal, Nick
Jennings, Ian Kash, Peter Key, Yoad Lewenberg,
Vangelis Markakis, Moshik Mash, Reshef Meir,
Svetlana Obraztsova, Joel Oren, Dima Pasechnik,
Maria Polukarov, Ariel Procaccia, Jeffery S.
Rosenschein, Shayak Sen, Nisarg Shah, Arunesh
Sinha, Alexander Skopalik and Junxing Wang.
Thesis Outline
Goal: find reasonable revenue divisions among
collaborative agents.
Features: agents may belong to more than one
coalition; causes interdependencies.
Key insights:
- Agents’ reaction to deviation strongly governs
stability of payoff division.
- We develop a framework for handling reaction to
deviation: arbitration functions
- New insights about stability of various MAS, and
their computational complexity.
Cooperative Games
Players divide into
coalitions to perform tasks
Coalition members can
freely divide profits.
How should profits be divided?
Cooperative Games
The coalitions players form are disjoint.
Each player is a member of only one
coalition – devotes all of his resources to a
single task.
A set of players - 𝑁 = {1, … , 𝑛}
Characteristic function - 𝑣: 2𝑁 → ℝ
𝑣(𝑆) – how much money can the agents in
𝑆 make.
Induced Subgraph Games
• We are given a weighted graph
1
2
10
5
3
3
2
4
1
1
5
3
4
7
3
6
22
1
7
9
5
6
4
8
2
• Players are nodes; value of a coalition is the
value of the edges in the induced subgraph.
• Applications: markets, collaboration networks.
Network Flow Games
• We are given a weighted, directed graph
s
3
10
7
5
1
3
6
1
7
3
t
5
1
2
4
• Players are edges; value of a coalition is the value
of the max. flow it can pass from s to t.
• Applications: computer networks, traffic flow,
transport networks.
Cost Sharing
Ride sharing: how to fairly split a taxi fare?
Airport
$50
$70
$60
Cost Sharing
Ride sharing: how to fairly split a taxi fare?
Cost Sharing
Ride sharing: how to fairly split a taxi fare?
For every set of riders 𝑆 ⊆ 𝑁,
𝑣 𝑆 =estimated fare if only members
of 𝑆 share the cab (uses TaxiFareFinder)
Rider 𝑖 must pay the Shapley value of
the resulting game.
Cooperative Games
The Shapley value:
1. Pick a permutation uniformly at random
2. The payoff to 𝑖 equals the expected
marginal contribution to her predecessors.
3
6
4
1
2
5
𝑣 3
𝑣 𝑁 − 𝑣({3,6,4,1,2})
𝑣 {3,6} − 𝑣(3)
𝑣 {3,6,4,1} − 𝑣({3,6,4})
𝑣 {3,6,4} − 𝑣({3,6})
𝑣 {3,6,4,1,2} − 𝑣({3,6,4,1})
Cooperative Games
The core: an outcome (𝐶𝑆, 𝐱) is in the core
if
• No outside payments:
𝑖∈𝑆 𝑥𝑖 = 𝑣(𝑆) for all 𝑆 ∈ 𝐶𝑆
• Stability: 𝑖∈𝑇 𝑥𝑖 ≥ 𝑣(𝑇) for all 𝑇 ⊆ 𝑁
No set of players can get more by deviating
from 𝐶𝑆, 𝐱
OCF Games
(Chalkiadakis et al., JAIR 2010)
Agents have divisible
resources
Overlapping Coalition
Formation (OCF)
How should profits be divided?
OCF Games
• A set of agents 𝑁 = {1, … , 𝑛}; each
agent controls a divisible resource.
• A function 𝑣: 0,1 𝑛 → ℝ
𝑣(0.5,0.3,0) = 7
“If agent 1 contributes 50% of his
resource and agent 2 contributes 30%,
they will generate a profit of 7$”
OCF Games
• Each vector in [0,1]n describes how
much each agent contributes; called a
coalition.
• A coalition structure: a list of coalitions
such that the sum of contributions from
each agent is at most 1 (no agent
contributes more than 100%).
OCF Games
Features
1
5
• Simple model
• Applicable to many settings:
2
6
3
7
4
𝑣 𝐜 = 1 × $10
• Matching markets
• Network/multicommodity flows
s
• Agents completing tasks
(20𝐺𝐵, $5)
10𝐺𝐵
9
7
5
100𝐺𝐵
(10𝐺𝐵, $2)
(90𝐺𝐵, $30)
t
6
8
10
3
4
6
5
1
4
3
s
8
5
4
8
30𝐺𝐵
𝑣4,8 (𝑥, 𝑦)
4
5
7
𝑣 𝐜′ =93 × $7 5
7
Red: 8 tons, worth 10$/kg
Blue: 12 tons, worth 7$/kg
3
2
t
𝐶𝑆, 𝐱
5
8
Outcome
3
4
3
74 0 6
9
14
6
2
2
4
5
5
1
17
6
2
10
3 3
4
4
5
8
4
Making them Offers they
Can’t Refuse:
The Arbitration Function
Deviations and Reactions to
Deviation in OCF Games
Deviation in OCF Games
A set deviates from an outcome (𝐶𝑆, 𝐱) by
reallocating resources.
This is profitable if each member of the
deviating set gets more than what it got
under (𝐶𝑆, 𝐱).
This needs to be carefully defined.
5
Total payoff: 14
11
13
8
3
4
3
74 0
21
8 5
4
1
2
Total payoff: 12
11
9
2
6
2
3 3
4
6
2
4
5
4
The Arbitration Function
Given a deviation of a set 𝑆 ⊆ 𝑁 from an
outcome 𝐶𝑆, 𝐱 , the arbitration function
specifies how much each coalition with
non-𝑆 members will give 𝑆.
Profitable Deviation
Current outcome
Deviate; how much do we
get from arbitration fn.?
$5
$0
Form coalitions, divide
revenue so that all profit!
$5
$0
𝑆 can 𝒜-profitably deviate (i.e. has a profitable
deviation given 𝒜), if each member of 𝑆 is getting
strictly more than what they got under (𝐶𝑆, 𝐱).
Optimistic
≺ ≺ ≺
5
Refined
3
4
3
74 0
Sensitive
6
2
Conservative
21
8 5
4
1
2
3 3
4
6
2
4
5
4
“Agent Rights” vs. Stability
The more generous the arbitration
function, the more egalitarian the payoff
division.
However, freedom to deviate causes social
instability.
Stability in OCF Games
The Arbitrated Core
The Arbitrated Core
Given an arbitration function 𝒜, an
outcome 𝐶𝑆, 𝐱 is said to be 𝒜-stable if
no subset can 𝒜-profitably deviate (i.e. has
a profitable deviation given 𝒜).
The 𝒜-core is the set of all 𝒜-stable
outcomes.
The Arbitrated Core
To find an 𝒜-profitable deviation for 𝑆 ⊆
𝑁, we need to:
1. Find some way of dividing 𝑆’s resources to new coalitions
2. Allocate payoffs from newly formed coalitions
3. Allocate payoffs from arbitration function.
Very complicated!
A simpler condition for 𝓐-profitable
deviation: if 𝒜∗ (𝐶𝑆, 𝐱, 𝑆) > 𝑝𝑆(𝐶𝑆, 𝐱) then
𝑆 contains a subset that can deviate.
Theorem: if 𝒜∗ (𝐶𝑆, 𝐱, 𝑆) > 𝑝𝑆(𝐶𝑆, 𝐱) then
We can deviate
𝑆 contains a subset that can
without those guys!
𝒜-profitably deviate. We can deviate without
1. Let us take some deviation that
ensures a total payoff of 𝒜∗ (𝐶𝑆, 𝐱, 𝑆)
2. Divide payoff such that “total
unhappiness” is minimal.
3. Happy deviators are green
Indifferent deviators are white
Unhappy deviators are red
4. Green deviators are not dependent
on non-green deviators: could have
deviated by themselves and strictly
gained!
those guys!
We can deviate
Iwithout
can transfer
thosemoney
guys! to you
if we share a coalition where
I get paid!
… so, we can’t share
coalitions where I get paid
(could reduce unhappiness)!
Some Arbitrated Cores
Conservative Core: if 𝑆 ⊆ 𝑁 deviates, it gets
nothing from non-deviators. 𝒜∗ 𝐶𝑆, 𝑥, 𝑆 is the
most 𝑆 can make on its own: 𝑣 ∗ 𝑆
Theorem: the conservative core is not empty if
and only if the (classic) core of its discrete
superadditive cover is not empty.
Conservative arbitration makes OCF games
“collapse” to classic cooperative games!
Classic CGT assumes conservative
reaction to deviation!
Is it all for Naught?
Refined core: a coalition that is not
changed by deviation will still pay
deviators; otherwise it will pay nothing.
Things get interesting here…
The Refined Core
Interesting observation: it is possible that
one optimal coalition structure can be
stabilized w.r.t. the refined arbitration
function, but another would not!
The Refined Core
𝑣 1,0,0 = 5:
player 1 can make 5 alone.
𝑣 0.5,0.5,0 = 10, 𝑣 1,1,0 = 20:
players 1 and 2 can make 20 together (by doing one
big task, or two small ones)
𝑣 0,0.5,1 = 9:
Must get at least
players 2 and 3 can make 9 together.
9 from each
coalition!
Leaves me with
at most 2!
10
1
10
2
9
3
The Refined Core
𝑣 1,0,0 = 5:
player 1 can make 5 alone.
𝑣 0.5,0.5,0 = 10, 𝑣 1,1,0 = 20:
players 1 and 2 can make 20 together (by doing one
big task, or two small ones)
𝑣 0,0.5,1 = 9:
players 2 and 3 can make 9 together.
Must get at least 9!
Must get at least 5!
1
10
20
10
2
3
The Refined Core
Theorem: if 𝑣 is homogeneous of degree
𝑘 ≥ 1, then the refined core is not empty.
When 𝑣 guarantees high ROI, highly stable
payoff divisions exist.
Computing Solutions to
OCF Games
Finding optimal coalition structures
and payoffs in polynomial time
Computational Aspects
Computing solutions to OCF games is
computationally hard.
(finding an optimal coalition structure,
finding an 𝒜-core allocation…)
Computational Aspects
These problems become easy when:
•
•
•
•
Agents can only form small coalitions
Agent interactions are simple (interaction graph)
Discrete, poly-bounded, agent resources.
Arbitration function is simple (local).
𝑤7
𝑣1,2 𝑥, 𝑦
𝑤2
𝑣3,4 𝑥, 𝑦
𝑤4
𝑤1
𝑣1,5 𝑥, 𝑦
𝑤5
𝑣1,3 𝑥, 𝑦
𝑤3
𝑣5,7 𝑥, 𝑦
𝑣5,8 𝑥, 𝑦
𝑣5,9 𝑥, 𝑦
𝑣3,6 𝑥, 𝑦
𝑤6
𝑤9
𝑤8
Local: the decision of how
much to give depends only
on the effect on the
coalition.
5
Can be 0, all, some…
but independent of
how deviators behave
outside the coalition!
3
Global: may depend on the
effect deviation had on
other coalitions.
4
3
74 0
21
8 5
4
1
2
6
2
3 3
4
6
2
4
5
4
Computational Aspects
In order to find stable outcomes in
polynomial time, stick to the issues!
Specific Classes of OCF
Games
We identify a class of OCF games for
which the optimistic core is not
empty, and for which stable
outcomes can be efficiently
computed.
5
4
s
5
8
6
6
5
4
s
8
10
3
4
3
1
7
3
Construct
LP
t
7
4
2
5
𝑚
𝑗=1 𝐶𝑗 𝜋𝑗
𝑗:𝑖∈𝐴𝑗 𝐶𝑗
≤ 𝑊𝑖 , ∀𝑖 ∈ N
𝐶𝑗 ≥ 0, ∀𝑗 ∈ [𝑚]
5
9
7
Red: 8 tons, worth 10$/kg
Blue: 12 tons, worth 7$/kg
max:
s.t.:
Generate Dual
t
min:
s.t.:
𝑛
𝑖=1 𝑊𝑖 𝛾𝑖
𝑖∈𝐴𝑗 𝛾𝑖
≥ 𝜋𝑗 , ∀𝑗 ∈ [𝑚]
𝛾𝑖 ≥ 0, ∀𝑖 ∈ 𝑁
Given optimal solutions to primal and dual, 𝐶, 𝛾, set payoff to 𝑖 from
task 𝑗 to be 𝑥𝑗𝑖 = 𝐶𝑗 𝛾𝑖
This is a valid payoff division, and is in the optimistic core!
Summary
• We introduce a new concept to the study of OCF games:
the arbitration function.
• We reexamine solution concepts in light of the arbitration
function.
• We examine computational aspects arising in OCF games.
• We identify sufficient conditions for core non-emptiness in
OCF games.
• Also in thesis:
– Iterated revenue sharing (to appear in IJCAI 2015)
– Nucleolus, bargaining set and Shapley value for OCF games (read my
thesis!)
Future Work
• Conceptual:
– how lenient can the arbitration function be without
destabilizing the game?
– what games can be stabilized by a given arbitration
function? (partial answer in this thesis)
• Computational:
– can we find better algorithms with some assumptions on
the valuation function?
– Can we find approximately optimal/approximately stable
solutions?
– Computing other solution concepts?
Future Work
• New Frameworks:
– In application domains (some work on cellular networks)
– In economic domains (many market domains already form
overlapping coalitions; arbitration functions are natural)
– Arbitrators as a benchmark of the desirability of social
strictness/lenience.
A new research paradigm for the study of strategic
interaction (not just in CGT!)
• New Directions:
– Handling uncertainty
– Applying cooperative game theory (and OCF solution
concepts) in ML.
Publications this thesis is based on:
- Yair Zick and Edith Elkind.
Arbitrators in Overlapping Coalition Formation Games (AAMAS’11)
- Yair Zick, Georgios Chalkiadakis and Edith Elkind.
Overlapping Coalition Formation Games: Charting the Tractability Frontier
(AAMAS’12) (see newer version on ArXiv)
- Yair Zick, Evangelos Markakis and Edith Elkind.
Stability via Convexity and LP Duality in Games with Overlapping Coalitions
(AAAI’12)
- Yair Zick, Evangelos Markakis and Edith Elkind.
Arbitration and Stability in Cooperative Games with Overlapping Coalitions (JAIR’14)
- Yair Zick, Yoram Bachrach, Ian Kash and Peter Key.
Non-Myopic Negotiators See What's Best (IJCAI’15)
Thank you! Questions?
My e-mail: [email protected]
My website: http://www.cs.cmu.edu/~yairzick/