Algorithmic Coalitional Game Theory Lecture 11: Coalition Structure - PowerPoint PPT Presentation

Algorithmic Coalitional Game Theory Lecture 11: Coalition Structure Generation Oskar Skibski University of Warsaw 12.05.2020

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? • There are 𝐶𝑓𝑚𝑚(𝑜) partitions. • We need to check values of all coalitions, so perform at least 𝑃(2 ! ) steps. 2 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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. Notation: • 𝒬(𝑂) – the set of all partitions of set 𝑂 • 𝒬 # (𝑂) – the set of all partitions of set 𝑂 of size 𝑙 𝑜 𝑙 – size of 𝒬 # 𝑂 , i.e., Stirling number of the second kind • • 𝑤 𝑄 – value of partition 𝑄 , i.e., ∑ $∈& 𝑤(𝑇) 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 ) (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

Dynamic Programming (DP) [Yeh 1986] Input: Game 𝑂, 𝑤 Output: 𝑄 ∗ ∈ 𝒬(𝑂) s.t. 𝑤 𝑄 ∗ ≥ 𝑤 𝑄 for every 𝑄 ∈ 𝒬(𝑂) 1: for 𝑙 from 1 to 𝑜 do 2: for each 𝑇 ⊆ 𝑂, 𝑇 = 𝑙 do 3: 𝑔 𝑇 ← 𝑤 𝑇 ; 𝑢 𝑇 ← {𝑇} ; 4: for each 𝐵, 𝐶 ∈ 𝒬 " (𝑇) do 5: if 𝑔 𝐵 + 𝑔(𝐶) > 𝑔 𝑇 then 𝑔 𝑇 ← 𝑔 𝐵 + 𝑔 𝐶 ; 6: 7: 𝑢 𝑇 ← 𝑢 𝐵 ∪ 𝑢 𝐶 ; 8: return 𝑢 𝑂 ; 5 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Dynamic Programming (DP) [Yeh 1986] 𝑻 𝒈 𝑻 𝑻 𝒈 𝑻 {1} 𝑤 1 = 2 {1,2,3} 𝑔 1 + 𝑔 2,3 = 9 {2} 𝑤 2 = 4 {1,2,4} 𝑔 1 + 𝑔 2,4 = 11 {3} 𝑤 3 = 3 1,3,4 𝑤 1,3,4 = 10 {4} 𝑤 4 = 5 {2,3,4} 𝑔 2 + 𝑔 3,4 = 12 {1,2} 𝑔 1 + 𝑔 2 = 6 {1,2,3,4} 𝑔 1 + 𝑔 2,3,4 = 14 {1,3} 𝑤 1,3 = 5 {1,4} 𝑤 1,4 = 7 For 𝑤 𝑇 = ∑ #∈% 𝑗 + ∑ #∈% 𝑗 𝑛𝑝𝑒 3 {2,3} 𝑤 2,3 = 7 {2,4} 𝑔 2 + 𝑔 4 = 9 {3,4} 𝑤 3,4 = 8 6 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Dynamic Programming (DP) [Yeh 1986] DP Complexity DP runs in time 𝒫 3 ! . Proof: For every coalition 𝑇 ⊆ 𝑂 , 𝑇 ≠ ∅ , DP checks 2 $ S" − 1 splits. So: ! 𝑜 2 $ S" − 1 = < 2 WS" − 1 = < 𝑡 $⊆T,$U∅ WX" ! = 1 𝑜 2 WS" − 1 = 1 2 3 ! + 1 − 2 ! . 2 + < 𝑡 WXY 7 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Dynamic Programming (DP) [Yeh 1986] 𝒬 ' (𝑂) 1|2|3|4 OPTIMAL 𝒬 ( (𝑂) 12|3|4 12|3|4 13|2|4 14|2|3 1|23|4 1|24|3 1|2|34 𝒬 ) (𝑂) 123|4 123|4 124|3 124|3 134|2 1|234 12|34 12|34 13|24 14|23 𝒬 " (𝑂) 1234 8 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Improved DP (IDP) [Rahwan & Jennings 2008] Main idea: „We do not have to consider all splits as long as we can reach every coalition structure.” Which splits should we use? For 𝑇 ≠ 𝑂 , we consider only splits 𝐵, 𝐶 ∈ 𝒬 ! (𝑇) such that: 𝐵 , 𝐶 ≤ 𝑂 − 𝐵 − 𝐶 . 9 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Improved DP (IDP) [Rahwan & Jennings 2008] Input: Game 𝑂, 𝑤 Output: 𝑄 ∗ ∈ 𝒬(𝑂) s.t. 𝑤 𝑄 ∗ ≥ 𝑤 𝑄 for every 𝑄 ∈ 𝒬(𝑂) 1: for 𝑙 from 1 to 𝑜 do 2: for each 𝑇 ⊆ 𝑂, 𝑇 = 𝑙 do 3: 𝑔 𝑇 ← 𝑤 𝑇 ; 𝑢 𝑇 ← {𝑇} ; 4: for each 𝐵, 𝐶 ∈ 𝒬 " 𝑇 s.t. |𝐵|, 𝐶 ≤ 𝑂 − 𝑇 𝑝𝑠 𝑇 = 𝑂 do 5: if 𝑔 𝐵 + 𝑔(𝐶) > 𝑔 𝑇 then 𝑔 𝑇 ← 𝑔 𝐵 + 𝑔 𝐶 ; 6: 7: 𝑢 𝑇 ← 𝑢 𝐵 ∪ 𝑢 𝐶 ; 8: return 𝑢 𝑂 ; 10 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Improved DP (IDP) [Rahwan & Jennings 2008] 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 11 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Improved DP (IDP) [Rahwan & Jennings 2008] 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 We removed 12 arrows We removed 12 splits out of 31 arrows. out of 25 splits. 12 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Improved DP (IDP) [Rahwan & Jennings 2008] IDP Correctness IDP is correct, i.e., finds the optimal coalition structure. Proof: It is enough to show that we can reach every coalition structure. Fix arbitrary 𝑄 = {𝑇 " , … , 𝑇 # } and assume 𝑇 " ≤ ⋯ ≤ |𝑇 # | . If 𝑙 = 2 , then trivial. Assume 𝑙 > 2 . It is enough to show that 𝑄 has an incoming edge from the lower level. This is true, because for 𝑇 " ∪ 𝑇 ) we have: 𝑇 " ≤ 𝑇 ) ≤ 𝑂 − 𝑇 " − 𝑇 ) . 13 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Improved DP (IDP) [Rahwan & Jennings 2008] IDP Complexity IDP runs in time 𝒫 3 ! . Proof: We can calculate the number of splits that we do not consider and show that it is 𝑝 3 ! . Rahwan and Jennings showed that for 𝑜 = 25 , IDP consider only 38.7% splits of DP. 14 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Optimal DP (ODP) [Michalak et al. 2016] Main idea: „Instead of looking at coalition sizes, we can take a lexicographical order” Let us write 𝐵 ≼ 𝐶 if 𝐵 is lexicographically smaller. E.g., 1,2,3 ≼ 1,4 ≼ {2,3,4,5} . For 𝑇 ≠ 𝑂 , we consider only splits 𝐵, 𝐶 ∈ 𝒬 ! (𝑇) such that: 𝐵, 𝐶 ≼ 𝑂 ∖ (𝐵 ∪ 𝐶). 15 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Optimal DP (ODP) [Michalak et al. 2016] Input: Game 𝑂, 𝑤 Output: 𝑄 ∗ ∈ 𝒬(𝑂) s.t. 𝑤 𝑄 ∗ ≥ 𝑤 𝑄 for every 𝑄 ∈ 𝒬(𝑂) 1: for 𝑙 from 1 to 𝑜 do 2: for each 𝑇 ⊆ 𝑂, 𝑇 = 𝑙 do 3: 𝑔 𝑇 ← 𝑤 𝑇 ; 𝑢 𝑇 ← {𝑇} ; 4: for each 𝐵, 𝐶 ∈ 𝒬 " 𝑇 s.t. 𝐵, 𝐶 ≼ 𝑂 ∖ (𝐵 ∪ 𝐶) 𝑝𝑠 𝑇 = 𝑂 do 5: if 𝑔 𝐵 + 𝑔(𝐶) > 𝑔 𝑇 then 𝑔 𝑇 ← 𝑔 𝐵 + 𝑔 𝐶 ; 6: 7: 𝑢 𝑇 ← 𝑢 𝐵 ∪ 𝑢 𝐶 ; 8: return 𝑢 𝑂 ; 16 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Optimal DP (ODP) [Michalak et al. 2016] 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 17 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Optimal DP (ODP) [Michalak et al. 2016] 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 We removed 17 arrows out We removed 12 splits of 31 arrows. out of 25 splits. 18 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Optimal DP (ODP) [Michalak et al. 2016] ODP Correctness There exists a unique path from 𝑂 to every coalition structure which implies ODP is correct. Proof: Fix arbitrary 𝑄 = {𝑇 " , … , 𝑇 # } and assume 𝑇 " ≼ ⋯ ≼ 𝑇 # . If 𝑙 = 2 , then trivial. Assume 𝑙 > 2 . Now, 𝑄 has exactly one incoming edge from the lower level, specifically from {𝑇 " ∪ 𝑇 ) , 𝑇 ( , … , 𝑇 # } , because only for 𝑇 " ∪ 𝑇 ) we have: 𝑇 " ≼ 𝑇 ) ≼ 𝑂 ∖ (𝑇 " ∪ 𝑇 ) ). 19 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Optimal DP (ODP) [Michalak et al. 2016] ODP Minimality 2 + 𝑜 ODP performs 𝑜 3 = " ) 3 !S" − 1 splits and less splits cannot be performed. Proof: Number 𝑜 2 corresponds to the number of splits of coalition 𝑂 . Number 𝑜 3 corresponds to splits of 𝐵 ∪ 𝐶 into {𝐵, 𝐶} such that 𝐵, 𝐶 ≼ 𝑂 ∖ (𝐵 ∪ 𝐶) . It is easy to check that 𝑜 ) (2 ! − 2) and 𝑜 2 = " 3 = ` 3 ! − 3 ⋅ 2 ! + 3 = " ) (3 !S" − 2 ! + 1) . " 20 Oskar Skibski (UW) Algorithmic Coalitional Game Theory