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

Algorithmic Coalitional Game Theory Lecture 13: Games with Externalities Oskar Skibski University of Warsaw 26.05.2020

Games with externalities In the standard model, the value of a coalition depends only on 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

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

Games with externalities Coalitional Game with Externalities [Thrall & Lucas 1963] 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) . 𝑂 = {1,2,3} 𝑤 1 , 1}, 2 , {3 = 7, 𝑤 1 , 1 , 2,3 = 5, E X 𝑤 2 , 1}, 2 , {3 = 10, 𝑤 2 , 2 , 1,3 = 8, A 𝑤 3 , 1}, 2 , {3 = 13, 𝑤 3 , 1,2 , 3 = 11 M 𝑤 1,2 , 1,2 , 3 = 19, P L 𝑤 1,3 , 2 , 1,3 = 22, E 𝑤 2,3 , 1 , 2,3 = 25, 𝑤 1,2,3 , 1,2,3 = 30. 4 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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

CSG with externalities 𝒬 # (𝑂) 1|2|3|4 𝒬 $ (𝑂) 12|3|4 13|2|4 14|2|3 1|23|4 1|24|3 1|2|34 𝒬 " (𝑂) 123|4 124|3 134|2 1|234 12|34 13|24 14|23 𝒬 ! (𝑂) 1234 Finding the optimal coalition structure requires traversing all partitions. 6 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Core with externalities The value of a coalition of deviators depends on the partition of others. 𝛽 -Core [Aumann & Peleg 1963] The 𝛽 - core of a game (𝑂, 𝑤) is the set of payoff vectors 𝑦 ∈ ℝ % s.t. 𝑦 𝑂 = 𝑤 𝑂 and ∑ &∈( 𝑦 & ≥ )∈𝒬 % :(∈) 𝑤(𝑇, 𝑄) for min every 𝑇 ⊆ 𝑂 . We assume that others form the worst partition for 𝑇 . 7 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Core with externalities The value of a coalition of deviators depends on the partition of others. 𝜕 -Core [Shenoy 1979] 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 . We assume that others form the best partition for 𝑇 . 8 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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) . (Simple) Recursive Core [Koczy 2007] 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). 9 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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

SV with externalities – axioms Shapley’s Axioms for Games with Externalities • 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: ??? 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 . 11 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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 / 𝑄 = 𝑄 ∖ 𝑄 𝑗 , 𝑈 ∪ 𝑄 𝑗 ∖ 𝑗 , 𝑈 ∪ 𝑗 • 𝜐 & is the partition obtained from 𝑄 by moving player 𝑗 to coalition 𝑈 / 𝑄 ) is called the elementary • 𝑤 𝑇, 𝑄 − 𝑤(𝑇 ∖ 𝑗 , 𝜐 & marginal contribution of player 𝑗 to (𝑇, 𝑄) 12 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

SV with externalities – axioms A player 𝑗 is a null-player in game (𝑂, 𝑤) if for every 𝑇, 𝑄 ∈ 𝐹𝐷(𝑂) : / 𝑄 ) for every 𝑈 ∈ 𝑄 • 𝑤 𝑇, 𝑄 = 𝑤(𝑇 ∖ {𝑗}, 𝜐 & 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

SV with externalities – axioms Shapley’s Axioms for Games with Externalities • 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 . These axioms do not imply uniqueness. 14 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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) : ∅ 𝑄 ) [Pham Do & Norde 2007] • 𝑤 𝑇, 𝑄 = 𝑤(𝑇 ∖ {𝑗}, 𝜐 & [De Clippel & Serrano 2008] / 𝑄 ) [Bolger • 𝑤 𝑇, 𝑄 = ! ) ⋅ ∑ /∈ )∖{(} ∪{∅} 𝑤(𝑇 ∖ 𝑗 , 𝜐 & 1989] It can be shown that they lead to uniqueness combined with the remaining axioms. 15 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

SV with externalities – axioms 16 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

SV with externalities – formula Average Approach [Macho-Stadler et al. 2007] 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: 𝑤 𝑇 = Y X 𝑏 𝑇, 𝑄 ⋅ 𝑤(𝑇, 𝑄) , )∋( where ∑ )∋( 𝑏 𝑇, 𝑄 = 1 . 2. Calculate the Shapley value for the average game X 𝑤 . The value obtained using average approach satisfies Null- player if for every 𝑇, 𝑄 ∈ 𝐹𝐷 𝑂 , 𝑗 ∈ 𝑇 : / (𝑄) . 𝑏 𝑇, 𝑄 = ∑ /∈)∖ ( ∪{∅} 𝑏 𝑇 ∖ {𝑗}, 𝜐 & 17 Oskar Skibski (UW) Algorithmic Coalitional Game Theory