Arbitration, Fairness and Stability

Revenue Division in Collaborative Settings

Yair ZickAdvisor: 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 OutlineGoal: 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 GamesThe 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 - Characteristic function - – how much money can the agents in make.

Induced Subgraph Games

• We are given a weighted graph

• Players are nodes; value of a coalition is the value of the edges in the induced subgraph.

• Applications: markets, collaboration networks.

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1022

Network Flow Games

• We are given a weighted, directed graph

• 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.

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Cost Sharing

Ride sharing: how to fairly split a taxi fare?

Airport

$50

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$70

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 , estimated fare if only members of share the cab (uses TaxiFareFinder)

Rider must pay the Shapley value of the resulting game.

Cooperative GamesThe Shapley value:1. Pick a permutation uniformly at random2. The payoff to equals the expected marginal

contribution to her predecessors.

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𝑣 (3 )

𝑣 ({3,6 })−𝑣 (3)

𝑣 ({3,6,4 })−𝑣 ({3,6 })

𝑣 ({3,6,4,1})−𝑣({3,6,4 })

𝑣 ({3,6,4,1,2})−𝑣({3,6,4,1 })

𝑣 (𝑁 )−𝑣 ({3,6,4,1,2 })

Cooperative GamesThe core: an outcome is in the core if •No outside payments: for all •Stability: for all

No set of players can get more by deviating from

OCF Games(Chalkiadakis et al., JAIR 2010)

Agents have divisible resources

How should profits be divided?

Overlapping Coalition Formation (OCF)

OCF Games

•A set of agents ; each agent controls a divisible resource.•A function

“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 GamesFeatures•Simple model•Applicable to many settings:

• Matching markets• Network/multicommodity flows• Agents completing tasks

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𝑣1,6 (𝑥 , 𝑦 )𝑣 3,5

(𝑥 ,𝑦 )

𝑣2,8 (𝑥 , 𝑦 )𝑣4,8 (𝑥 , 𝑦 )

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(20𝐺𝐵 , $5)

(90𝐺𝐵 , $30)(10𝐺𝐵 , $2)

30𝐺𝐵10𝐺𝐵

100𝐺𝐵

Red: 8 tons, worth 10$/kgBlue: 12 tons, worth 7$/kg

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𝑣 (𝐜 )=1×$10

𝑣 (𝐜 ′ )=3×$7

(𝐶𝑆 ,𝐱 )Outcome

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Making them Offers they Can’t Refuse:

The Arbitration FunctionDeviations and Reactions to

Deviation in OCF Games

Deviation in OCF Games

A set deviates from an outcome by reallocating resources.This is profitable if each member of the deviating set gets more than what it got under .

This needs to be carefully defined.

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Total payoff: 11

Total payoff: 11

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The Arbitration Function

Given a deviation of a set from an outcome , the arbitration function specifies how much each coalition with non- members will give .

Profitable Deviation

can -profitably deviate (i.e. has a profitable deviation given ), if each member of is getting strictly more than what they got under .

$5 $0

Current outcome Deviate; how much do we get from arbitration fn.?

Form coalitions, divide revenue so that all profit!

$5 $0

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Optimistic

Refined

Sensitive

Conservative

≺≺

≺

“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 , an outcome is said to be -stable if no subset can -profitably deviate (i.e. has a profitable deviation given ).The -core is the set of all -stable outcomes.

The Arbitrated Core

To find an -profitable deviation for , we need to:1. Find some way of dividing ’s resources to new coalitions2. Allocate payoffs from newly formed coalitions3. Allocate payoffs from arbitration function.

Very complicated! A simpler condition for -profitable deviation: if then contains a subset that can deviate.

… so, we can’t share coalitions where I get paid

(could reduce unhappiness)!

Theorem: if then contains a subset that can -profitably deviate.

1. Let us take some deviation that ensures a total payoff of

2. Divide payoff such that “total unhappiness” is minimal.

3. Happy deviators are greenIndifferent deviators are whiteUnhappy deviators are red

4. Green deviators are not dependent on non-green deviators: could have deviated by themselves and strictly gained!

I can transfer money to you if we share a coalition where

I get paid!

We can deviate without those guys!

We can deviate without those guys!

We can deviate without those guys!

Some Arbitrated Cores

Conservative Core: if deviates, it gets nothing from non-deviators. is the most can make on its own: 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: player 1 can make 5 alone.: players 1 and 2 can make 20 together (by doing one big task, or two small ones): players 2 and 3 can make 9 together.

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Must get at least 9 from each

coalition!

Leaves me with at most 2!

The Refined Core: player 1 can make 5 alone.: players 1 and 2 can make 20 together (by doing one big task, or two small ones): players 2 and 3 can make 9 together.

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Must get at least 9!

Must get at least 5! 20

The Refined Core

Theorem: if is homogeneous of degree , then the refined core is not empty.

When 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 -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).

𝑤1𝑤5

𝑤6

𝑤8𝑤2

𝑤9𝑤4

𝑤7

𝑤3

𝑣1,5 (𝑥 , 𝑦 )𝑣1,2 (𝑥 , 𝑦 )

𝑣3,4 (𝑥 , 𝑦 )

𝑣1,3 (𝑥 , 𝑦 )𝑣5,9 (𝑥 , 𝑦 )

𝑣3,6 (𝑥 , 𝑦 )

𝑣5,7 (𝑥 , 𝑦 )

𝑣5,8 (𝑥 , 𝑦 )

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Local: the decision of how much to give depends only on the effect on the coalition.

Global: may depend on the effect deviation had on other coalitions.

Can be 0, all, some… but independent of how deviators behave outside the coalition!

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.

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Red: 8 tons, worth 10$/kgBlue: 12 tons, worth 7$/kg

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max:s.t.:

Construct LP

Generate Dual

min:s.t.:

Given optimal solutions to primal and dual, , set payoff to from task to be

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: yairzick@cmu.eduMy website: http://www.cs.cmu.edu/~yairzick/