[PPT] - Algorithmic Coalitional Game Theory Lecture 13: Games with PowerPoint Presentation

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Oskar Skibski

University of Warsaw

Algorithmic Coalitional Game Theory

Lecture 13: Games with Externalities

26.05.2020

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Games with externalities

In the standard model, the value of a coalition depends only

n its members. But is it always the case?
When companies decide to cooperate, others are affected.
Political alliances cause other political parties lose their

significance.

In multi-agent systems with limited resources there exist

natural restrictions on the number of coalitions. 2

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Games with externalities

A coalition 𝑇 in partition 𝑄 is called embedded coalition and denoted (𝑇, 𝑄). The set of all embedded coalitions: 𝐹𝐷 𝑂 = 𝑇, 𝑄 ∶ 𝑇 ∈ 𝑄, 𝑄 ∈ 𝒬 𝑂 . In games with externalities, every embedded coalition, i.e., every coalition in every partition can have an arbitrary value. 3

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Games with externalities

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

A coalitional game with externalities is a pair (𝑂, 𝑤), where 𝑂 is the set of 𝑜 players, and 𝑤 ∶ 𝐹𝐷 𝑂 → ℝ is a partition

function. We assume 𝑤 ∅, 𝑄 = 0 for every 𝑄 ∈ 𝒬(𝑂).

Coalitional Game with Externalities [Thrall & Lucas 1963]

𝑂 = {1,2,3} 𝑤 1 , 1}, 2 , {3 = 7, 𝑤 1 , 1 , 2,3 = 5, 𝑤 2 , 1}, 2 , {3 = 10, 𝑤 2 , 2 , 1,3 = 8, 𝑤 3 , 1}, 2 , {3 = 13, 𝑤 3 , 1,2 , 3 = 11 𝑤 1,2 , 1,2 , 3 = 19, 𝑤 1,3 , 2 , 1,3 = 22, 𝑤 2,3 , 1 , 2,3 = 25, 𝑤 1,2,3 , 1,2,3 = 30.

E X A M P L E

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Games with externalities

We say that game has positive externalities if for every partition 𝑄 ∈ 𝒬(𝑂), every 𝑇, 𝑈

!, 𝑈" ∈ 𝑄:

𝑤 𝑇, 𝑄 ≤ 𝑤(𝑇, 𝑄 ∖ 𝑈

!, 𝑈" ∪ {𝑈 ! ∪ 𝑈"})

„When two coalitions merge, others gain.” We say that game has negative externalities if for every partition 𝑄 ∈ 𝒬(𝑂), every 𝑇, 𝑈

!, 𝑈" ∈ 𝑄:

𝑤 𝑇, 𝑄 ≥ 𝑤(𝑇, 𝑄 ∖ 𝑈

!, 𝑈" ∪ {𝑈 ! ∪ 𝑈"})

„When two coalitions merge, others lose.” 5

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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CSG with externalities

Finding the optimal coalition structure requires traversing all partitions. 6

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

𝒬

#(𝑂)

𝒬$(𝑂) 𝒬"(𝑂) 𝒬

!(𝑂)

1|2|3|4 12|3|4 13|2|4 14|2|3 1|23|4 1|24|3 1|2|34 124|3 1234 14|23 13|24 123|4 12|34 1|234 134|2

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Core with externalities

The value of a coalition of deviators depends on the partition

f others.

We assume that others form the worst partition for 𝑇. 7

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

The 𝛽-core of a game (𝑂, 𝑤) is the set of payoff vectors 𝑦 ∈ ℝ% s.t. 𝑦 𝑂 = 𝑤 𝑂 and ∑&∈( 𝑦& ≥ min

)∈𝒬 % :(∈) 𝑤(𝑇, 𝑄) for

every 𝑇 ⊆ 𝑂. 𝛽-Core [Aumann & Peleg 1963]

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Core with externalities

The value of a coalition of deviators depends on the partition

f others.

We assume that others form the best partition for 𝑇. 8

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

The 𝜕-core of a game (𝑂, 𝑤) is the set of payoff vectors 𝑦 ∈ ℝ% s.t. 𝑦 𝑂 = 𝑤 𝑂 and ∑&∈( 𝑦& ≥ 𝑤(𝑇, 𝑄) for every 𝑄 ∈ 𝒬 𝑂 , 𝑇 ∈ 𝑄. 𝜕-Core [Shenoy 1979]

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Core with externalities

For a game (N, 𝑤), let (N ∖ 𝑇, 𝑤() be a residual game defined as follows: 𝑤( 𝑇,, 𝑄, = 𝑤(𝑇,, 𝑄, ∪ 𝑇) for every 𝑇,, 𝑄, ∈ 𝐹𝐷(𝑂 ∖ 𝑇). 9

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

The (simple) recursive core of a game (𝑂, 𝑤) is the set of undominated payoff configurations (𝑦, 𝑄) s.t. 𝑦 𝑂 = ∑(∈) 𝑤(𝑇, 𝑄). Configuration (𝑦, 𝑄) is dominated if there exists a coalition 𝑇 s.t. 𝑧(𝑇) > 𝑦(𝑇) for all payoff configurations (𝑧, 𝑇 ∪ 𝑄′) such that 𝑧%∖(, 𝑄′ is in the simple recursive core of game (𝑂 ∖ 𝑇, 𝑤() (if it is not empty). (Simple) Recursive Core [Koczy 2007]

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Shapley value with externalities

How to extend the Shapley value to games with externalities?

1. First approach – extend axioms, i.e., translate Shapley’s

axioms to obtain uniqueness

2. Second approach – extend the formula, i.e., find a way to

apply standard formula for the Shapley value

3. Third approach – extend the process, i.e., translate

random-order coalition formation process to games with externalities 10

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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SV with externalities – axioms

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Efficiency: ∑&∈% 𝜒& 𝑂, 𝑤 = 𝑤 𝑂, 𝑂

.

Symmetry: 𝜒& 𝑂, 𝑤 = 𝜒. & (𝑂, 𝑔 𝑤 ) for every bijection

𝑔: 𝑂 → 𝑂.

Linearity: 𝜒& 𝑂, 𝑑 ⋅ (𝑤! + 𝑤") = 𝑑 ⋅ 𝜒& 𝑂, 𝑤! + 𝑑 ⋅

𝜒&(𝑂, 𝑤") for every constant 𝑑 ∈ ℝ.

Null-player: ???

Shapley’s Axioms for Games with Externalities Notation: games 𝑔 𝑤 and 𝑑 ⋅ 𝑤! + 𝑤" are defined as follows: 𝑔 𝑤 𝑇, 𝑄 = 𝑤 𝑔 𝑇 , 𝑔 𝑈 ∶ 𝑈 ∈ 𝑄 , where 𝑔 𝑇 = 𝑔 𝑗 ∶ 𝑗 ∈ 𝑇 , and 𝑑 ⋅ 𝑤! + 𝑤" 𝑇, 𝑄 = 𝑑 ⋅ 𝑤! 𝑇, 𝑄 + 𝑑 ⋅ 𝑤" 𝑇, 𝑄 .

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SV with externalities – axioms

Who is a null-player?

player who does not have an impact on the game or
player who does not contribute to the value of any

coalition? Notation:

𝑄 𝑗 is the coalition of player 𝑗 in 𝑄
𝜐&

/ 𝑄 = 𝑄 ∖ 𝑄 𝑗 , 𝑈 ∪ 𝑄 𝑗 ∖ 𝑗 , 𝑈 ∪ 𝑗

is the partition

btained from 𝑄 by moving player 𝑗 to coalition 𝑈
𝑤 𝑇, 𝑄 − 𝑤(𝑇 ∖ 𝑗 , 𝜐&

/ 𝑄 ) is called the elementary

marginal contribution of player 𝑗 to (𝑇, 𝑄) 12

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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SV with externalities – axioms

A player 𝑗 is a null-player in game (𝑂, 𝑤) if for every 𝑇, 𝑄 ∈ 𝐹𝐷(𝑂):

𝑤 𝑇, 𝑄 = 𝑤(𝑇 ∖ {𝑗}, 𝜐&

/ 𝑄 ) for every 𝑈 ∈ 𝑄

i.e., all its elementary marginal contributions are zero. For example: 𝑤 1,2 , 1,2 , 3,4 , 5,6,7 = = 𝑤 2 , 2 , 1,3,4 , 5,6,7 = 𝑤 2 , 2 , 3,4 , {1,5,6,7} = 𝑤 2 , 1 , 2 , 3,4 , 5,6,7 13

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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SV with externalities – axioms

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Efficiency: ∑&∈% 𝜒& 𝑂, 𝑤 = 𝑤 𝑂, 𝑂

.

Symmetry: 𝜒& 𝑂, 𝑤 = 𝜒. & (𝑂, 𝑔 𝑤 ) for every bijection

𝑔: 𝑂 → 𝑂.

Linearity: 𝜒& 𝑂, 𝑑 ⋅ (𝑤! + 𝑤") = 𝑑 ⋅ 𝜒& 𝑂, 𝑤! + 𝑑 ⋅

𝜒&(𝑂, 𝑤") for every constant 𝑑 ∈ ℝ.

Null-player: if 𝑗 is a null-player in 𝑂, 𝑤 , then 𝜒& 𝑂, 𝑤 =

0. Shapley’s Axioms for Games with Externalities These axioms do not imply uniqueness.

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SV with externalities – axioms

There are many more definitions of null-players. A player 𝑗 is a mc-null-player in game (𝑂, 𝑤) if for every 𝑇, 𝑄 ∈ 𝐹𝐷(𝑂):

𝑤 𝑇, 𝑄 = 𝑤(𝑇 ∖ {𝑗}, 𝜐&

∅ 𝑄 ) [Pham Do & Norde 2007]

[De Clippel & Serrano 2008]

𝑤 𝑇, 𝑄 = !

) ⋅ ∑/∈ )∖{(} ∪{∅} 𝑤(𝑇 ∖ 𝑗 , 𝜐& / 𝑄 ) [Bolger

1989] It can be shown that they lead to uniqueness combined with the remaining axioms. 15

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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SV with externalities – axioms

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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SV with externalities – formula

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

The value obtained using average approach satisfies Null- player if for every 𝑇, 𝑄 ∈ 𝐹𝐷 𝑂 , 𝑗 ∈ 𝑇: 𝑏 𝑇, 𝑄 = ∑/∈)∖ ( ∪{∅} 𝑏 𝑇 ∖ {𝑗}, 𝜐&

/(𝑄) .

1. Create a game without externalities X

𝑤 by calculating the value of every coalition as the average of its values in the game with externalities: X 𝑤 𝑇 = Y

)∋(

𝑏 𝑇, 𝑄 ⋅ 𝑤(𝑇, 𝑄) , where ∑)∋( 𝑏 𝑇, 𝑄 = 1.

2. Calculate the Shapley value for the average game X

𝑤. Average Approach [Macho-Stadler et al. 2007]

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SV with externalities – formula

Many values can be obtained using average approach:

𝑏 𝑇, 𝑄 = 𝑄 = 𝑇 ∪

𝑗 ∶ 𝑗 ∈ 𝑂 ∖ 𝑇 [De Clippel & Serrano 2008]

𝑏 𝑇, 𝑄 = 𝑄 = 𝑇, 𝑂 ∖ 𝑇

[McQuillin 2009]

𝑏 𝑇, 𝑄 = 1/|𝒬(𝑂 ∖ 𝑇)| [Albizuri 2010] (violates Null-

player)

𝑏 𝑇, 𝑄 =

𝑆 ∈ 𝒬 𝑂 : 𝑄 ∖ 𝑇 = 𝑈 ∖ 𝑇 ∶ 𝑈 ∈ 𝑆 /|𝒬 𝑂 | [Hu & Yang 2009]

𝑏 𝑇, 𝑄 = ∏/∈)∖ (

𝑈 − 1 ! / 𝑂 − 𝑇 ! [Macho-Stadler et al. 2007] Not all values that satisfy Shapley’s axioms can be obtained. 18

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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SV with externalities – process

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Assume that players leave the grand coalition in a random

rder and divide themselves into groups outside.

In each step, one player departs and joins one of the coalitions (or forms a new one) with some probability. As the result of her leaving, the player is assigned a payoff that equals her elementary contribution. Now, the Process Shapley value is her expected payoff over all 𝑜! orders. Process Approach [Skibski et al. 2018]

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SV with externalities – process

How to formalize this description? Let 𝑞 be a probability distribution on 𝒬 1, … , 𝑜 and define 𝑞 𝑄5 = ∑6∈𝒬 !,…,9 ∶6 ;<=>?@ )! 𝑞(𝑆). Assume 𝜌: 𝑂 → {1, … , 𝑜} is the random order and that after 𝑙 steps, partition 𝑄5 has been formed (outside of the remainder

f the grand coalition).

In 𝑙 + 1 -th step, player joins one of the coalitions with the probability 𝑞(𝜌(𝑄5A!))/𝑞(𝜌 𝑄5 ), where 𝑄5A! is the partition

btained by this transfer.

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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SV with externalities – process

Values according to the process approach:

𝑞 𝑄 = [𝑄 = {{1}, … , {𝑜}}] [De Clippel & Serrano 2008]
𝑞(𝑄) = 𝑄 = 𝑂

[McQuillin 2009]

𝑞(𝑄) = 1/|𝒬 𝑂 | [Hu & Yang 2009]
𝑞(𝑄) = ∏/∈) 𝑈 − 1 ! / 𝑂 ! [Macho-Stadler et al. 2007]

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

A value satisfies Efficiency, Symmetry, Linearity and Null- player if and only if it can be obtained using the process approach. Process Approach [Skibski et al. 2018]

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References

[Albizuri 2010] M.J. Albizuri.

Games with externalities: games in coalition configuration function

form. Mathematical Methods of Operations Research 72, 171-186,

2010.

[Aumann & Peleg 1960] R.J. Aumann, B. Peleg.

Von Neumann-Morgenstern solutions to cooperative games without side

payments. Bulletin of the American Mathematical Society 66, 173–179,

1960.

[Bolger 1989] E.M. Bolger.

A set of axioms for a value for partition function games. International Journal of Game Theory 18, 37-44, 1989.

[De Clippel & Serrano 2008] G. De Clippel, R. Serrano.

Marginal contributions and externalities in the value. Econometrica 76, 1413-1436, 2008.

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References

[Koczy 2007] L.Á. Kóczy.

A recursive core for partition function form games. Theory and Decision 63, 41-51, 2007.

[Macho-Stadler et al. 2007] I. Macho-Stadler, D. Pérez-Castrillo, D.
Wettstein. Sharing the surplus: An extension of the Shapley value for

environments with externalities. Journal of Economic Theory 135, 339- 356, 2007.

[McQuillin 2009] B. McQuillin.

The extended and generalized Shapley value: Simultaneous consideration of coalitional externalities and coalitional structure. Journal of Economic Theory 144, 696-721, 2009.

[Pham Do & Norde 2007] K.H. Pham Do, H. Norde.

The Shapley value for partition function form games International Game Theory Review 9, 353-360, 2007.

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References

[Shenoy 1979] P.P. Shenoy

On coalition formation: A game-theoretical approach. International Journal of Game Theory 8(3):133–164, 1979.

[Skibski et al. 2018] O. Skibski, T. Michalak, M. Wooldridge.

The Stochastic Shapley Value for Coalitional Games with Externalities. Games and Economic Behavior 108, 65-80, 2018.

[Thrall & Lucas 1963] R.M. Thrall, W.F. Lucas.

N-Person games in partition function form. Naval Research Logistics Quarterly 10, 281-298, 1963.

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