Algorithmic Coalitional Game Theory Lecture 12: Anytime Coalition - PowerPoint PPT Presentation

Algorithmic Coalitional Game Theory Lecture 12: Anytime Coalition Structure Generation Oskar Skibski University of Warsaw 19.05.2020

Coalition Structure Generation Coalition Structure Generation Find a partition of players 𝑄 = {𝑇 ! , … , 𝑇 " } such that the sum of values of coalitions, i.e. 𝑤 𝑇 ! + ⋯ + 𝑤(𝑇 " ) , is maximized. In other words: which coalition structure will form? 2 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Coalition Structure Generation The exact algorithms requires a lot of time… Can we search through only a subset of coalition ? structures and be guaranteed to find a solution that is within a certain bound from the optimum? Let 𝑄 ∗ = arg max $∈𝒬 ' 𝑤(𝑄) . ( $ ∗ Can we find a subset 𝒝 ⊆ 𝒬 𝑂 s.t. 𝛾 ≥ )*+ ( $ ∶$∈𝒝 for every game (𝑂, 𝑤) ? We define: 𝑤 𝑄 ∗ bound 𝒝 = min 𝛾 ∈ ℝ ∶ ∀ !,# 𝛾 ≥ max 𝑤 𝑄 ∶ 𝑄 ∈ 𝒝 3 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Coalition Structure Generation 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 0 (𝑂) 123|4 124|3 134|2 1|234 12|34 13|24 14|23 𝒬 ! (𝑂) 1234 4 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Worst-case guarantee Minimal set with a bound [Sandholm et al. 1999] For 𝒝 = 𝒬 ! 𝑂 ∪ 𝒬 0 𝑂 we have bound 𝒝 = 𝑜 , 𝒝 = 2 12! , and 𝒝 is the minimal set with bound smaller than ∞ . Proof: On the blackboard. 5 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Worst-case guarantee Sketch of proof: Fix 𝑄 3 = arg max $∈𝒝 𝑤(𝑄) and 𝑇 ∗ = arg max 4⊆' 𝑤(𝑇) . bound 𝒝 ≤ 𝑜 : We know 𝑤 𝑄 3 ≥ 𝑤 𝑇 ∗ . Hence, • 𝑤 𝑄 ∗ ≤ |𝑄 ∗ | ⋅ 𝑤 𝑇 ∗ ≤ |𝑄 ∗ | ⋅ 𝑤 𝑄 3 ≤ 𝑜 ⋅ 𝑤 𝑄 3 . bound 𝒝 ≥ 𝑜 : Assume 𝑤 𝑇 = 1 if 𝑇 = 1 and 𝑤 𝑇 = • 0 , oth. Then: 𝑤 𝑄 3 = 1 = ! = ! 1 𝑤(𝑄 ∗ ) . 1 𝑤 1 , … , 𝑜 = 2 12! . Clearly, 𝒝 = 𝑇 ⊆ 𝑂 ∶ 1 ∈ 𝑇 • In every ℬ with bound ℬ ≤ ∞ we have at least 1 • partition for every coalition with player 1 (they do overlap, so cannot be in the same partition), so ℬ ≥ 2 12! . 6 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Worst-case guarantee What subset of coalition structures should we search ? next? 7 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Worst-case guarantee What subset of coalition structures should we search ? next? Improving the bound [Sandholm et al. 1999] For 𝒝 = 𝒬 ! 𝑂 ∪ 𝒬 0 𝑂 ∪ 𝒬 1 𝑂 and 𝑜 > 3 we have bound 𝒝 = ⌈𝑜/2⌉ . Proof: On the blackboard. 8 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Worst-case guarantee Sketch of proof: Assume 𝑄 ∗ contains 𝑙 singletons { 𝑗 ! , … , 𝑗 " } . If 𝑙 = 0 , then 𝑄 ∗ ≤ 1 0 and 𝑤 𝑄 ∗ ≤ 1 0 ⋅ 𝑤 𝑄 3 . Assume 𝑙 > 0 . ≤ 𝑤 𝑄 3 . We know that 𝑤 𝑗 ! , … , 𝑗 " ≤ 𝑤 1 , … , 𝑜 12" 12! Also, 𝑄 ∗ \{ 𝑗 ! , … , 𝑗 " } ≤ ≤ . 0 0 Hence, 𝑤 𝑄 ∗ ≤ + 1 𝑤 𝑄 3 = 0 ⋅ 𝑤 𝑄 3 . 12! 1 0 9 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Worst-case guarantee Sketch of proof (continued): Assume 𝑤 𝑇 = 1 if 𝑇 = 2 and 1 ∉ 𝑇, 𝑤 {1} = 1 and 𝑤 𝑇 = 0 , oth. Then: • 𝑤 𝑄 3 = 1 (since 𝑜 > 3 ) = 1 • and 𝑤 1 , 2 , {3,4} … , {𝑜 − 1, 𝑜} 0 if 𝑜 is even, = 16! • and 𝑤 1 , {2,3} … , {𝑜 − 1, 𝑜} 0 oth. 10 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG How to search through the coalition structures ? to improve the guarantee over time? 11 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG 𝒝 / 1|2|3|4 13|2|4 𝒝 ! 1|23|4 14|2|3 12|3|4 𝒝 . 124|3 1|24|3 1|2|34 𝒝 0 14|23 123|4 134|2 𝒝 7 1|234 1234 12|34 13|24 12 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG How to search through the coalition structures ? to improve the guarantee over time? Anytime Coalition Structure Generation Divide the search space into subsets: 𝒬 𝑂 = 𝒝 ! ∪ 𝒝 0 ∪ ⋯ ∪ 𝒝 " , such that: bound 𝒝 ! ≥ bound 𝒝 ! ∪ 𝒝 0 ≥ ⋯ ⋯ ≥ bound 𝒝 ! ∪ ⋯ ∪ 𝒝 " = 1. 13 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG Anytime CSG-99 [Sandholm et al. 1999] 1. Search 𝒬 ! 𝑂 . 2. Search 𝒬 0 𝑂 3. Search 𝒬 1 𝑂 . 4. Search 𝒬 12! 𝑂 . 5. … 6. Search 𝒬 / 𝑂 . 𝒝 ! = 𝒬 ! 𝑂 , 𝒝 0 = 𝒬 0 𝑂 , and 𝒝 " = 𝒬 16/2" 𝑂 for 2 < 𝑙 ≤ 𝑜 . 14 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG 𝒝 / [1,1,1,1,1,1,1,1] 𝒬 ! (𝑂) 𝒝 . [2,1,1,1,1,1,1] 𝒬 " (𝑂) 𝒝 7 [3,1,1,1,1,1] [2,2,1,1,1,1] 𝒬 # (𝑂) 𝒝 ; [2,2,2,1,1] [4,1,1,1,1] [3,2,1,1,1] 𝒬 $ (𝑂) 𝒝 < [5,1,1,1] [4,2,1,1] [3,3,1,1] [3,2,2,1] [2,2,2,2] 𝒬 % (𝑂) 𝒝 = [6,1,1] [5,2,1] [4,3,1] [3,3,2] [4,2,2] 𝒬 & (𝑂) 𝒝 0 𝒬 ' (𝑂) [7,1] [6,2] [5,3] [4,4] 𝒬 ( (𝑂) 𝒝 ! [8] 15 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG Bounds for Anytime CSG-99 [Sandholm et al. 1999] Define ℎ 𝑚 = (𝑜 − 𝑚)/2 + 2 . After searching 𝒬 > 𝑂 for 𝑚 > 3 , the bound is ≈ 𝑜/ℎ(𝑚) . Specifically, for 𝒝 = 𝒬 ! 𝑂 ∪ 𝒬 0 𝑂 ∪ 𝒬 1 𝑂 ∪ ⋯ ∪ 𝒬 > 𝑂 : bound 𝒝 = 4 𝑜/ℎ(𝑚) 𝑗𝑔 𝑜 ≡ −1 𝑛𝑝𝑒 ℎ 𝑚 , 𝑜 ≡ 𝑚 𝑛𝑝𝑒 2 , 𝑜/ℎ(𝑚) 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓. Proof: On the blackboard. 16 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG Sketch of proof: ℎ 𝑚 = (𝑜 − 𝑚)/2 + 2 = 𝑜 − 𝑚 − 2 /2 + 1 is a number such that partition of the form: • [ℎ 𝑚 , ℎ 𝑚 − 1, 1, … , 1] if 2 ∤ 𝑜 − 𝑚 (case A) or ℎ 𝑚 , ℎ 𝑚 − 2, 1, … , 1 if 2|𝑜 − 𝑚 (case B) • appears in level 𝑚 ([…] contains the list of sizes of coalitions). After searching level 𝑚 we know that disjoint coalitions of sizes 𝑗 and 𝑘 such that 𝑗 + 𝑘 ≤ 𝑜 − 𝑚 − 2 appeared in one partition. 17 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG Sketch of proof (continued): Lower bound for the bound: Let 𝑠 = 𝑜 mod ℎ 𝑚 = 𝑜 − 𝑜/ℎ(𝑚) ⋅ ℎ 𝑚 . Consider partition 𝑄 ∗ of the form [ℎ 𝑚 , ℎ(𝑚), … , ℎ 𝑚 , 𝑠] and game 𝑤 𝑇 = 1 if 𝑇 = ℎ(𝑚) . We have: 𝑤 𝑄 ∗ = 𝑜/ℎ(𝑚) and 𝑤 𝑄′ = 1 . Thus, bound 𝒝 ≥ 𝑜/ℎ(𝑚) . If we have case B (i.e., 2|𝑜 − 𝑚 ) and 𝑠 = ℎ 𝑚 − 1 , then by adding 𝑤 𝑇 = 1 for one specific coalition of size 𝑠 we get: bound 𝒝 ≥ ⌊𝑜/ℎ(𝑚)⌋ + 1 = ⌈𝑜/ℎ(𝑚)⌉ 18 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG Sketch of proof (continued): Upper bound for the bound: To see that bound 𝒝 ≤ 𝑜/ℎ(𝑚) consider the game (𝑂, 𝑤) such that 𝑤 𝑄 ∗ /𝑤(𝑄 3 ) is the highest. We can assume that 𝑤 𝑇 = 0 for every 𝑇 ∉ 𝑄 ∗ . Also, we can assume that 𝑄 3 ∩ 𝑄 ∗ = 1 : if 𝑇, 𝑇 3 ∈ 𝑄 3 ∩ 𝑄 ∗ , then replacing 𝑇, 𝑇 3 with 𝑇 ∪ 𝑇 3 and defining game analogously would result in the same value 𝑤 𝑄 ∗ /𝑤(𝑄 3 ) . 19 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG Sketch of proof (continued): Now, since 𝑄 3 ∩ 𝑄 ∗ = 1 it means that in 𝑄 ∗ there are no two coalitions that appeared in the same partition considered so far. Hence, we get the limit on the number of such coalitions. In case A or case B where 𝑠 ≠ ℎ 𝑚 − 1 , we get maximum ⌊𝑜/ℎ 𝑚 ⌋ coalitions for: ℎ 𝑚 , ℎ 𝑚 , … , ℎ 𝑚 , ℎ 𝑚 + 𝑠 . In case B with 𝑠 ≠ ℎ 𝑚 − 1 , we get maximum ⌈𝑜/ℎ 𝑚 ⌉ coalitions for: ℎ 𝑚 , ℎ 𝑚 , … , ℎ 𝑚 , ℎ 𝑚 − 1 This implies our thesis. 20 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Anytime CSG Define: 𝒬 ?@ 𝑂 = 𝑄 ∈ 𝒬 𝑂 ∶ 𝑄 > 2, max 4∈$ 𝑇 ≥ 𝑟 . Anytime CSG-04 [Dang & Jennings 2004] 1. Search 𝒬 ! 𝑂 . 2. Search 𝒬 0 𝑂 . 3. Search 𝒬 1 𝑂 . 4. Search 𝒬 ?120 𝑂 . 5. … 6. Search 𝒬 ?0 𝑂 . Since many of this steps will not improve the bound, the authors considered 𝒬 ?⌈1(@2!)/@⌉ 𝑂 for 𝑟 from ⌊(𝑜 + 1)/4⌋ down to 2 and then search the remaining coalition structures. 21 Oskar Skibski (UW) Algorithmic Coalitional Game Theory