UNIK4950 - Multiagent systems Lecture 7 Cooperative game ... · Highlights lecture 7 – Cooperative game theory* • Cooperative games – forming coalitions – Characteristic function

UNIK4950 - Multiagent systems Lecture 7

Cooperative game theory Jonas Moen

Highlights lecture 7 – Cooperative game theory*

• Cooperative games – forming coalitions – Characteristic function – The core – The Shapley value

• Simple games – ‘yes/no’ games – Weighted voting games

• Coalition structure formation – ‘central planner’

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*Wooldridge, 2009: chapter 13

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Forming coalitions

It can be claimed that agents in Prisoner’s dilemma are prevented from cooperation due to: • Binding agreements are not possible. (The issue of trust.) • Utility is given directly to individuals as a result of individual

action.

If we drop these assumptions we can open up for modelling coalitions of cooperating agents or cooperative games.

Cooperative games

Cooperative game theory attempts to answer two basic questions: 1. Which coalition will be formed by self-interested rational

agents? 2. How is the utility divided among members in this coalition?

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Cooperative games

Given a set of 𝑁𝑁 agents that can form coalitions 𝐶𝐶,𝐶𝐶𝐶,𝐶𝐶1, … 𝐶𝐶 ∈ 𝐴𝐴𝐴𝐴 is a subset of 𝐴𝐴𝐴𝐴 𝐶𝐶 = 𝐴𝐴𝐴𝐴 is the Grand Coalition 𝐶𝐶 = 𝑖𝑖 is singelton coalition of one singel agent 𝑖𝑖

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𝐴𝐴𝐴𝐴 = 1,2, … ,𝑁𝑁

Cooperative games

A cooperative game (or a coalitional game) is a pair where 𝐴𝐴𝐴𝐴 is the set of agents 𝑣𝑣:𝟐𝟐𝐴𝐴𝐴𝐴 ⟶ ℝ is the characteristic function The characteristic function assigns a numerical value to all possible coalitions. 𝑣𝑣 𝐶𝐶 = 𝑘𝑘 means that the value 𝑘𝑘 is assigned to coalition 𝐶𝐶 and distributed among its members. 12.10.2017 7

𝐺𝐺 = 𝐴𝐴𝐴𝐴, 𝑣𝑣

Cooperative games

Note: • The characteristic function is given • The game does not explicitly say how to distribute utility

among members

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Cooperative games

The cooperation lifecycle – Three stages of cooperative action, [Sandholm et al., 1999]

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Image: Figure 13.1, Wooldridge 2009

The cooperation lifecycle

1. Coalition structure generation Agents maximize utility by looking at the characteristic function of all possible coalitions. Which coalition is stable? This is answered by using the notion of the core.

2. Solving the optimization problem for each coalition This is assumed to be solved given the utility of the characteristic function.

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The cooperation lifecycle

3. Dividing the utility of the solution for each coalition The Shapley value is used for ‘fair’ distribution of the utility among coalition members.

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The core – coalitional stability

We say that the core is the set of outcomes for a coalition, that no other coalition objects to. Formally, a coalition 𝐶𝐶′ ⊆ 𝐴𝐴𝐴𝐴 objects to an outcome 𝐱𝐱 𝑥𝑥1, … , 𝑥𝑥𝑁𝑁 of the coalition 𝐶𝐶 if there exist some outcome 𝐱𝐱′ 𝑥𝑥1′ , … , 𝑥𝑥𝑁𝑁′ in 𝐶𝐶𝐶 such that 𝑥𝑥𝑖𝑖′ > 𝑥𝑥𝑖𝑖 for all 𝑖𝑖 ∈ 𝐶𝐶′. Meaning 𝑥𝑥𝑖𝑖′ is strictly better for all 𝑖𝑖. The outcomes satisfies the characteristic function 𝑣𝑣. 𝑣𝑣 𝐶𝐶 = ∑ 𝑥𝑥𝑖𝑖𝑖𝑖∈𝐶𝐶 12.10.2017 12


The coalition 𝐶𝐶 is stable if the core is non-empty.

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Example of a cooperative game 𝐺𝐺 = 𝐴𝐴𝐴𝐴, 𝑣𝑣 where 𝐴𝐴𝐴𝐴 = 1,2 𝑣𝑣 1 = 5 𝑣𝑣 2 = 5 𝑣𝑣 1,2 = 20

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The grand coalition of game 𝐺𝐺 = 𝐴𝐴𝐴𝐴, 𝑣𝑣 is stable for outcomes 𝑣𝑣 1,2 = 0,20 = 20 → 𝑣𝑣 1 = 5: agent 1 objects 𝑣𝑣 1,2 = 1,19 = 20 → 𝑣𝑣 1 = 5: agent 1 objects 𝑣𝑣 1,2 = 5,15 = 20 ⋮ 𝑣𝑣 1,2 = 15,5 = 20 𝑣𝑣 1,2 = 20,0 = 20 → 𝑣𝑣 2 = 5: agent 2 objects

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The Core


The grand coalition of game 𝐺𝐺 = 𝐴𝐴𝐴𝐴, 𝑣𝑣 is stable for outcomes 𝑣𝑣 1,2 = 5,15 = 20 ⋮ 𝑣𝑣 1,2 = 15,5 = 20 How to divide the utility among coalition members?

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The Core

The Shapley value

The Shapley value is based on the idea that agents should get the average marginal contribution it makes to a coalition, estimated over all possible positions that it would enter the coalition. The Shapley value is the unique value that satisfies the ‘fairness’ axioms.

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The Shapley value

The Shapley value of agent 𝑖𝑖 is given by: Where ∏ 𝐴𝐴𝐴𝐴 is set of all possible ordering of coalition 𝐶𝐶 𝑜𝑜 is an ordering of an coalition 𝜇𝜇𝑖𝑖 𝐶𝐶 = 𝑣𝑣 𝐶𝐶 ∪ 𝑖𝑖 − 𝑣𝑣 𝐶𝐶 , given that 𝐶𝐶 ⊆ 𝐴𝐴𝐴𝐴\ 𝑖𝑖 . is the marginal contribution of agent 𝑖𝑖 to 𝐶𝐶

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𝑆𝑆𝑆𝑖𝑖 =1𝐴𝐴𝐴𝐴 !

� 𝜇𝜇𝑖𝑖 𝐶𝐶𝑖𝑖 𝑜𝑜𝑜𝑜∈∏ 𝐴𝐴𝐴𝐴

The Shapley value

The Shapley value satisfies 3 ‘fairness’ axioms: 1. Symmetry

Agents that make the same contribution to the coalition should get the same utility. Formally, if 𝜇𝜇𝑖𝑖 𝐶𝐶 = 𝜇𝜇𝑗𝑗 𝐶𝐶 for 𝐶𝐶 ⊆ 𝐴𝐴𝐴𝐴\ 𝑖𝑖, 𝑗𝑗 then 𝑖𝑖 and 𝑗𝑗 are interchangeable and 𝑠𝑠𝑆𝑖𝑖 = 𝑠𝑠𝑆𝑗𝑗.

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The Shapley value

The Shapley value satisfies 3 ‘fairness’ axioms: 2. Dummy player

Agents that do not contribute to coalitions should only receive what they can earn on their own. If 𝜇𝜇𝑖𝑖 𝐶𝐶 = 𝑣𝑣 𝑖𝑖 for 𝐶𝐶 ⊆ 𝐴𝐴𝐴𝐴\ 𝑖𝑖 then 𝑠𝑠𝑆𝑖𝑖 = 𝑣𝑣 𝑖𝑖 .

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The Shapley value

The Shapley value satisfies 3 ‘fairness’ axioms: 3. Additivity

Agents that play two games get the sum of the two games. The agent does not benefit from playing the game more than once.

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The Shapley value

The Shapley value satisfies 3 ‘fairness’ axioms: 3. Additivity

If 𝐺𝐺1 = 𝐴𝐴𝐴𝐴, 𝑣𝑣1 and 𝐺𝐺2 = 𝐴𝐴𝐴𝐴, 𝑣𝑣2 with 𝑠𝑠𝑆𝑖𝑖,1 and 𝑠𝑠𝑆𝑖𝑖,2 for player 𝑖𝑖 ∈ 𝐴𝐴𝐴𝐴 then 𝐺𝐺1+2 = 𝐴𝐴𝐴𝐴, 𝑣𝑣1+2 such that 𝑣𝑣1+2 𝐶𝐶 =𝑣𝑣1 𝐶𝐶 + 𝑣𝑣2 𝐶𝐶 then 𝑠𝑠𝑆𝑖𝑖,1+2 = 𝑠𝑠𝑆𝑖𝑖,1+ 𝑠𝑠𝑆𝑖𝑖,2.

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The Shapley value

Example of a cooperative game 𝐺𝐺 = 𝐴𝐴𝐴𝐴, 𝑣𝑣 where 𝐴𝐴𝐴𝐴 = 1,2,3 𝑣𝑣 1 = 𝑣𝑣 2 = 𝑣𝑣 3 = 5 𝑣𝑣 1,2 = 𝑣𝑣 1,3 = 10 𝑣𝑣 2,3 = 20 𝑣𝑣 1,2,3 = 25

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The Shapley value

Permutations of {1,2,3} Marginal contribution of {1,2,3} 1,2,3 → 5,5,15 1,3,2 2,1,3 2,3,1 3,1,2 3,2,1 12.10.2017 24

The Shapley value

Permutations of {1,2,3} Marginal contribution of {1,2,3} 1,2,3 → 5,5,15 1,3,2 → 5,15,5 2,1,3 2,3,1 3,1,2 3,2,1 12.10.2017 25

The Shapley value

Permutations of {1,2,3} Marginal contribution of {1,2,3} 1,2,3 → 5,5,15 1,3,2 → 5,15,5 2,1,3 → 5,5,15 2,3,1 → 5,5,15 3,1,2 → 5,15,5 3,2,1 → 5,15,5 12.10.2017 26

The Shapley value

Permutations of {1,2,3} Marginal contribution of {1,2,3} 1,2,3 → 5,5,15 1,3,2 → 5,15,5 2,1,3 → 5,5,15 2,3,1 → 5,5,15 3,1,2 → 5,15,5 3,2,1 → 5,15,5

Shapley value ⇒ 306

, 606

, 606

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The Shapley value

The Shapley value is hard to compute and represent for large number of players (exponential in 𝑁𝑁). Is it possible to represent the different coalition permutations in a more succinct and tractable way? 1. Induced subgraph representation is succinct but not

complete 2. Marginal contribution nets are complete and succinct

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Induced subgraph

The characteristic function can be defined by an undirected, weighted graph, in which nodes in the graph are members of 𝐴𝐴𝐴𝐴 and the edges are the weight 𝑤𝑤𝑖𝑖,𝑗𝑗 of the edge from node 𝑖𝑖 to node 𝑗𝑗 in the graph [Deng and Papadimitriou, 1994].

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Induced subgraph

To compute the characteristic value of a coalition 𝐶𝐶 ⊆ 𝐴𝐴𝐴𝐴 we simply sum the weights 𝑤𝑤𝑖𝑖,𝑗𝑗 over all the edges in the graph whose components are all contained in 𝐶𝐶: 𝜐𝜐 𝐶𝐶 = ∑ 𝑤𝑤𝑖𝑖,𝑗𝑗𝑖𝑖,𝑗𝑗 ⊆𝐶𝐶

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Induced subgraph

The Shapley value of each player is then given by:

𝑠𝑠𝑆𝑖𝑖 = 12∑ 𝑤𝑤𝑖𝑖,𝑗𝑗𝑗𝑗≠𝑖𝑖

This comes directly from the symmetry axiom, where agents are interchangeable in a two player game.

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Induced subgraph

Example: 𝑣𝑣 1 = 𝑣𝑣 2 = 𝑣𝑣 3 = 5 𝑣𝑣 1,2 = 𝑣𝑣 1,3 = 10, 𝑣𝑣 2,3 = 20 𝑣𝑣 1,2,3 = 25

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0

10

0

5

5

5

1 2

3

Induced subgraph

Example: 𝑠𝑠𝑆1 = 5 + 0 + 0 = 5

𝑠𝑠𝑆2 = 5 + 0 +102 = 10

𝑠𝑠𝑆3 = 5 + 102

+ 0 = 10

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0

10

0

5

5

5

1 2

3

Marginal contribution nets

A marginal contribution net is an extension to the induced subgraph. The characteristic function of a game is represented by a set of rules 𝑟𝑟𝑠𝑠:

𝑟𝑟𝑠𝑠𝐶𝐶 = 𝜙𝜙 → 𝑥𝑥 ∈ 𝑟𝑟𝑠𝑠 𝐶𝐶 = 𝜙𝜙 where 𝜙𝜙 → 𝑥𝑥 is a rule 𝐶𝐶 is a coalition 𝐶𝐶| = means that the rule applies to coalition 𝐶𝐶

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The characteristic function 𝑣𝑣𝑟𝑟𝑟𝑟 associated with the rule set 𝑟𝑟𝑠𝑠 is defined as follows:

𝑣𝑣𝑟𝑟𝑟𝑟 𝐶𝐶 = � 𝑥𝑥𝜙𝜙→𝑥𝑥∈𝑟𝑟𝑟𝑟𝐶𝐶

where 𝜙𝜙 → 𝑥𝑥 is a rule in the rule set 𝑟𝑟𝑠𝑠𝐶𝐶

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Example: 𝐴𝐴𝐴𝐴 = 1,2,3 1 → 5 2 → 5 3 → 5 2 ∧ 3 → 10 12.10.2017 36

0

10

0

5

5

5

1 2

3


Example: 1 → 5 2 → 5 3 → 5 2 ∧ 3 → 10 𝑣𝑣 1 = 5 𝑣𝑣 2 = 5 𝑣𝑣 3 = 5

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0

10

0

5

5

5

1 2

3


Example: 1 → 5 2 → 5 3 → 5 2 ∧ 3 → 10 𝑣𝑣 1,2 = 5 + 5 = 10 𝑣𝑣 1,3 = 5 + 5 = 10 𝑣𝑣 2,3 = 5 + 5 + 10 = 20

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0

10

0

5

5

5

1 2

3


Example: 1 → 5 2 → 5 3 → 5 2 ∧ 3 → 10 𝑣𝑣 1,2,3 = 5 + 5 + 5 + 10 = 25

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0

10

0

5

5

5

1 2

3


Example: 1 → 5 2 → 5 3 → 5 2 ∧ 3 → 10 𝑣𝑣 1 = 𝑣𝑣 2 = 𝑣𝑣 3 = 5 𝑣𝑣 1,2 = 𝑣𝑣 1,3 = 10 𝑣𝑣 2,3 = 20, 𝑣𝑣 1,2,3 = 25 12.10.2017 40

0

10

0

5

5

5

1 2

3

Simple games

Simple games are games with coalition values of either 0 (‘losing’) or 1 (‘winning’). Simple games are representative of ‘yes/no’ voting systems often used in politics. Which coalition is ‘winning’?

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Simple games

Formally, a simple game is a pair of Υ = 𝐴𝐴𝐴𝐴,𝑊𝑊 where 𝐴𝐴𝐴𝐴 = 1,2, … ,𝑁𝑁 is 𝑁𝑁 agents/voters 𝑊𝑊 ⊆ 𝟐𝟐𝐴𝐴𝐴𝐴 is the set of winning coalitions such that 𝑣𝑣 𝑊𝑊 = 1 𝑣𝑣 𝐶𝐶 ∉ 𝑊𝑊 = 0 12.10.2017 42

Simple games

Some properties of simple games:

1. Non-triviality There are some winning coaltions, but not all coalitions are winners, ∅ ⊂ 𝑊𝑊 ⊂ 2𝐴𝐴𝐴𝐴.

2. Monotonicity If 𝐶𝐶1 ⊆ 𝐶𝐶2 and 𝐶𝐶1 ∈ 𝑊𝑊 then 𝐶𝐶2 ∈ 𝑊𝑊. Meaning that if 𝐶𝐶 wins then all supersets of 𝐶𝐶 also wins.

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Simple games

Some properties of simple games:

3. Zero-sum if 𝐶𝐶 ∈ 𝑊𝑊 then 𝐴𝐴𝐴𝐴\𝐶𝐶 ∉ 𝑊𝑊 If coalition 𝐶𝐶 wins then the agents outside 𝐶𝐶 do not win.

4. Empty coalition lose, ∅ ∉ 𝑊𝑊 5. Grand coalition wins, 𝐴𝐴𝐴𝐴 ∈ 𝑊𝑊

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Simple games

Explicitly listing all coalitions will be exponential in number of players. Weighted voting games are natural extensions of simple games possibly reducing number of players in these games.

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Weighted voting games

Weighted voting games are a more concise way of representing many simple games. For instance, instead of representing 100 US senators explicitly we represent the different ‘blocks’ of voters and evaluate those against the criteria of winning, called the quota. This can greatly reduce the number of players/voters

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Given a set of agents 𝐴𝐴𝐴𝐴 = 1,2, … ,𝑁𝑁 . For each agent 𝑖𝑖 we define a weight 𝑤𝑤𝑖𝑖 and a overall quota 𝑞𝑞 such that every coalition 𝐶𝐶 exceeding quota 𝑞𝑞 is a winning coalition 𝑊𝑊.

𝑣𝑣 𝐶𝐶 = �1 if ∑ 𝑤𝑤𝑖𝑖𝑖𝑖∈𝐶𝐶 ≥ 𝑞𝑞 0

where 𝑣𝑣 is the characteristic function of coalition 𝐶𝐶

𝑞𝑞 = 𝐴𝐴𝐴𝐴 +12

in simple majority voting 12.10.2017 47


The weighted voting game can be written on the form 𝑞𝑞;𝑤𝑤1,𝑤𝑤1, … ,𝑤𝑤𝑁𝑁 where 𝑞𝑞 is the quota 𝑤𝑤𝑖𝑖 is the weight of player 𝑖𝑖 ∈ 𝐴𝐴𝐴𝐴 The Shapley value calculates the power of voters in the game.

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Example from the book 100; 99,99,1 Let us calculate the Shapley values for the Grand coalition in order to determine the power of the 3 voter blocks.

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How many permutations of 𝐶𝐶 = 100; 99,99,1 is possible?

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How many permutations of 𝐶𝐶 = 100; 99,99,1 is possible? 991, 992, 1 991, 1, 992 992, 991, 1 992, 1, 991 1,991, 992 1, 992, 992 12.10.2017 51


What is the marginal contribution of voter in each permutation? {1,2,3} 991, 992, 1 → 0,1,0 991, 1, 992 992, 991, 1 992, 1, 991 1,991, 992 1, 992, 992 12.10.2017 52


Permutations of {1,2,3} Marginal contribution of {1,2,3} 991, 992, 1 → 0,1,0 991, 1, 992 → 0,0,1 992, 991, 1 → 1,0,0 992, 1, 991 → 0,0,1 1,991, 992 → 1,0,0 1, 992, 991 → 0,1,0 12.10.2017 53


Permutations of {1,2,3} Marginal contribution of {1,2,3} 991, 992, 1 → 0,1,0 991, 1, 992 → 0,0,1 992, 991, 1 → 1,0,0 992, 1, 991 → 0,0,1 1,991, 992 → 1,0,0 1, 992, 991 → 0,1,0

Shapley value ⇒ 13

, 13

, 13

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The Shapley value is NP-hard to compute, [Deng and Papadimitriou, 1994]

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The core is much easier to compute, i.e. to check if there are non-empty coalitions. This can be done in polynomial time. The core is non-empty, iff there is an agent present in every winning coalition. Formally, for a coalition 𝐶𝐶 ∑ 𝑤𝑤𝑗𝑗 < 𝑞𝑞𝑗𝑗∈𝐶𝐶 and ∑ 𝑤𝑤𝑗𝑗 ≥ 𝑞𝑞𝑗𝑗∈𝐶𝐶∩ 𝑖𝑖 where 𝑞𝑞 is quota, 𝑤𝑤𝑗𝑗 is weight and 𝑖𝑖 is voter 𝑖𝑖. 12.10.2017 56

𝑘𝑘-weighted voting games

Weighted voting games are not complete (representation of simple games). 𝑘𝑘-weighted voting games are complete. We define a number 𝑘𝑘 of different weighted voting games with the same set of players the overall winner is the coalition that wins in all 𝑘𝑘 coalition games.

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European Union three-weight voting game: Germany, the UK, France, Italy, Spain, Poland, Romania, the Netherlands, Greece, Czech Republic, Belgium, Hungary, Portugal, Sweden, Bulgaria, Austria, Slovak Republic, Denmark, Finland, Ireland, Lithuania, Latvia, Slovenia, Estonia, Cyprus, Luxembourg, Malta

𝐴𝐴1 = 255; 29,29,29,29,27,27,14,13,12,12,12,12,12,10,10,10,7,7,7,7,7,4,4,4,4,4,3 𝐴𝐴2 = 14; 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 𝐴𝐴3 = 620; 170,123,122,120,82,80,47,33,22,21, 21,21,21,18,17,17,11,11,11,8,8,5,4,3,2,1,1 (where 𝐴𝐴1 is number of commissioners, 𝐴𝐴2 simple majority, 𝐴𝐴3 is population size)

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How many 𝑘𝑘-dimensions are needed in order to be a complete representation of a simple game? Upper bound is 2𝑁𝑁, but a check if games could be represented by a smaller number of components is NP-complete. A game of non-reducible 𝑘𝑘 is called ‘minimal’.

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Network flow games

Directed graph with edges representing capacities between nodes as routers.

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Network flow games

Now, what coalitions of nodes allow for a capacity 𝑏𝑏 between 𝑙𝑙1 and 𝑙𝑙2 in the graph? We get

𝑣𝑣 𝐶𝐶 = �1 if 𝑁𝑁 ↓ 𝐶𝐶 allows a flow of 𝑏𝑏 from 𝑙𝑙1 to 𝑙𝑙20 otherwise

where 𝑁𝑁 ↓ 𝐶𝐶 is the network flow of coalition 𝐶𝐶.

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Coalitional games with goals

Qualitative Coalitional Games (QCG) In QCG the numerical valued payoffs are replaced with goals. The player has a set of goals it wants to achieve and they do not prioritize among them, they only want one of them to be achieved. The coalition is successful if it can cooperate in such a way that all of its members are satisfied.

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Coalitional games with goals

Coalitional Resouce Games (CRG) To achieve a goal requires some resources that agents are endowed with. The coalition will pool resources in order to achieve some mutual set of goals.

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Coalition structure formation

Central forming of coalition 1. All nodes owned by a single designer 2. Maximizing social welfare The problem is to find the coalition structure that maximizes the aggregated outcome. This is a search through a partition of the overall set of agents 𝐴𝐴𝐴𝐴 into mutually disjoint coalitions.

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Example using 𝐴𝐴𝐴𝐴 = 1,2,3 Coalitions 1 , 2 , 3 , 1,2 , 1, 3 , 2,3 , 1,2,3 Coalition structures 1,2,3 , 1 2,3 , 2 1,3 , 3 1,2 , 1 2 3

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The social choice is:

𝐶𝐶𝑆𝑆∗ = arg max𝐶𝐶𝐶𝐶∈𝑝𝑝𝑝𝑝𝑟𝑟𝑝𝑝𝑖𝑖𝑝𝑝𝑖𝑖𝑜𝑜𝑝𝑝 𝑜𝑜𝑜𝑜 𝐴𝐴𝐴𝐴

𝑉𝑉 𝐶𝐶𝑆𝑆

where 𝑉𝑉 𝐶𝐶𝑆𝑆 = ∑ 𝑣𝑣 𝐶𝐶𝐶𝐶∈𝐶𝐶𝐶𝐶 is the outcome of 𝐶𝐶𝑆𝑆 The problem is that the number of possible coalition structures is exponentially more than coalitions over 𝐴𝐴𝐴𝐴. 12.10.2017 66


In the coalition graph the highest 𝐶𝐶𝑆𝑆1,2∗ of the two lowest levels

are no worse than 1/𝑁𝑁 ⋅ 𝐶𝐶𝑆𝑆∗.

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[Sandholm et al., 1999] propose a search algorithm: 1. Search the 2 lowest levels for 𝐶𝐶𝑆𝑆1,2

∗ . 2. If time, search rest of coalition structure graph, using

breadth-first search from top until exhausted or time is up. 3. Return coalition structure with highest value seen.

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Summary lecture 7 – Cooperative game theory*

• Cooperative games – forming coalitions – Characteristic function (value of coalition) – The Core (stability, non-empty) – The Shapley value (‘fairness’ axioms, induced subgraphs,

marginal contribution nets) • Simple games – ‘yes/no’ games

– Weighted voting games, k-weighted voting games, network flow games

• Coalition structure formation – ‘central planner’

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*Wooldridge, 2009: chapter 13

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UNIK4950 - Multiagent systems Lecture 7 Cooperative game ... · Highlights lecture 7 – Cooperative game theory* • Cooperative games – forming coalitions – Characteristic function