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

Oskar Skibski

University of Warsaw

Algorithmic Coalitional Game Theory

Lecture 11: Coalition Structure Generation

12.05.2020

Coalition Structure Generation

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

Find a partition of players 𝑄 = {𝑇", … , 𝑇#} such that the sum

• f values of coalitions, i.e. 𝑤 𝑇" + ⋯ + 𝑤(𝑇#), is maximized.

Coalition Structure Generation

Coalition Structure Generation

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

Find a partition of players 𝑄 = {𝑇", … , 𝑇#} such that the sum

• f values of coalitions, i.e. 𝑤 𝑇" + ⋯ + 𝑤(𝑇#), is maximized.

Coalition Structure Generation

Coalition Structure Generation

4

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

Dynamic Programming (DP)

[Yeh 1986]

5

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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 6: 𝑔 𝑇 ← 𝑔 𝐵 + 𝑔 𝐶 ; 7: 𝑢 𝑇 ← 𝑢 𝐵 ∪ 𝑢 𝐶 ; 8: return 𝑢 𝑂 ;

Dynamic Programming (DP)

[Yeh 1986]

𝑻 𝒈 𝑻 {1} 𝑤 1 = 2 {2} 𝑤 2 = 4 {3} 𝑤 3 = 3 {4} 𝑤 4 = 5 {1,2} 𝑔 1 + 𝑔 2 = 6 {1,3} 𝑤 1,3 = 5 {1,4} 𝑤 1,4 = 7 {2,3} 𝑤 2,3 = 7 {2,4} 𝑔 2 + 𝑔 4 = 9 {3,4} 𝑤 3,4 = 8

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

𝑻 𝒈 𝑻 {1,2,3} 𝑔 1 + 𝑔 2,3 = 9 {1,2,4} 𝑔 1 + 𝑔 2,4 = 11 1,3,4 𝑤 1,3,4 = 10 {2,3,4} 𝑔 2 + 𝑔 3,4 = 12 {1,2,3,4} 𝑔 1 + 𝑔 2,3,4 = 14 For 𝑤 𝑇 = ∑#∈% 𝑗 + ∑#∈% 𝑗 𝑛𝑝𝑒 3

Dynamic Programming (DP)

[Yeh 1986]

Proof: For every coalition 𝑇 ⊆ 𝑂, 𝑇 ≠ ∅, DP checks 2 $ S" − 1

• splits. So:

<

$⊆T,$U∅

2 $ S" − 1 = <

WX" !

𝑜 𝑡 2WS" − 1 = = 1 2 + <

WXY !

𝑜 𝑡 2WS" − 1 = 1 2 3! + 1 − 2!. 7

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

DP runs in time 𝒫 3! . DP Complexity

Dynamic Programming (DP)

[Yeh 1986]

8

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

OPTIMAL

12|3|4 123|4 124|3 12|34 1234

Improved DP (IDP)

[Rahwan & Jennings 2008]

9

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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 .

Improved DP (IDP)

[Rahwan & Jennings 2008]

10

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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 6: 𝑔 𝑇 ← 𝑔 𝐵 + 𝑔 𝐶 ; 7: 𝑢 𝑇 ← 𝑢 𝐵 ∪ 𝑢 𝐶 ; 8: return 𝑢 𝑂 ;

Improved DP (IDP)

[Rahwan & Jennings 2008]

11

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

Improved DP (IDP)

[Rahwan & Jennings 2008]

12

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

We removed 12 arrows

• ut of 31 arrows.

We removed 12 splits

• ut of 25 splits.

Improved DP (IDP)

[Rahwan & Jennings 2008]

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

IDP is correct, i.e., finds the optimal coalition structure. IDP Correctness

Improved DP (IDP)

[Rahwan & Jennings 2008]

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

• nly 38.7% splits of DP.

14

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

IDP runs in time 𝒫 3! . IDP Complexity

Optimal DP (ODP) [Michalak et al. 2016]

15

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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).

Optimal DP (ODP) [Michalak et al. 2016]

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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 6: 𝑔 𝑇 ← 𝑔 𝐵 + 𝑔 𝐶 ; 7: 𝑢 𝑇 ← 𝑢 𝐵 ∪ 𝑢 𝐶 ; 8: return 𝑢 𝑂 ;

Optimal DP (ODP) [Michalak et al. 2016]

17

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

Optimal DP (ODP) [Michalak et al. 2016]

18

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

We removed 17 arrows out

• f 31 arrows.

We removed 12 splits

• ut of 25 splits.

Optimal DP (ODP) [Michalak et al. 2016]

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

There exists a unique path from 𝑂 to every coalition structure which implies ODP is correct. ODP Correctness

Optimal DP (ODP) [Michalak et al. 2016]

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! − 2) and 𝑜

3 =

" ` 3! − 3 ⋅ 2! + 3 = " ) (3!S" − 2! + 1).

20

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

ODP performs 𝑜 2 + 𝑜 3 = "

) 3!S" − 1 splits and less splits

cannot be performed. ODP Minimality

Optimal DP (ODP) [Michalak et al. 2016]

Proof (continue): To see that less splits cannot be performed we show that:

• to reach 𝐵, 𝐶 ∈ 𝒬)(𝑂) from 𝑂 we need an edge from it,

i.e., split of 𝑂 into 𝐵 ∪ 𝐶

• to reach 𝐵, 𝐶, 𝐷 ∈ 𝒬( 𝑂 we need at least one of the

splits 𝐵 ∪ 𝐶, 𝐵 ∪ 𝐷, 𝐶 ∪ 𝐷 and for a different 𝐵′, 𝐶′, 𝐷′ ∈ 𝒬( 𝑂 splits 𝐵′ ∪ 𝐶′, 𝐵′ ∪ 𝐷′, 𝐶′ ∪ 𝐷′ are different. 21

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

ODP performs 𝑜 2 + 𝑜 3 = "

) 3!S" − 1 splits and less splits

cannot be performed. ODP Minimality

Optimal DP (ODP) [Michalak et al. 2016]

22

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

ODP runs in time 𝒫 3! . ODP Complexity Proof: Corollary from the previous statement.

Conclusions

We saw 3 algorithms for Coalition Structure Generation:

• Dynamic Programming (DP)
• Improved DP (IDP)
• Optimal DP (ODP)

There are several heuristic algorithms; they often work faster, but have worse worst-case time complexity. Also, there are several theoretical algorithms that have better worst-case time complexity, but work much slower in practice. 23

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

References

• [Michalak et al. 2016] T. Michalak, T. Rahwan, E. Elkind, M. Wooldridge,

N.R. Jennings. A hybrid exact algorithm for complete set partitioning. Artificial Intelligence 230, 14-50, 2016.

• [Rahwan & Jennings 2008] T. Rahwan, N.R. Jennings.

Coalition structure generation: Dynamic programming meets anytime

• ptimization. Proceedings of the 23rd AAAI Conference on Artificial

Intelligence (AAAI), 156-161, 2008.

• [Yeh 1986] D.Y. Yeh.

A dynamic programming approach to the complete set partitioning

• problem. BIT Numerical Mathematics, Springer 26, 467-474, 1986.

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory