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

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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 = { ! ,

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Oskar Skibski

University of Warsaw

Algorithmic Coalitional Game Theory

Lecture 12: Anytime Coalition Structure Generation

19.05.2020

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Coalition Structure Generation

In other words: which coalition structure will form? 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. 𝑀 𝑇! + β‹― + 𝑀(𝑇"), is maximized.

Coalition Structure Generation

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Coalition Structure Generation

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

The exact algorithms requires a lot of time… Let π‘„βˆ— = arg max

$βˆˆπ’¬ ' 𝑀(𝑄).

Can we find a subset 𝒝 βŠ† 𝒬 𝑂 s.t. 𝛾 β‰₯

( $βˆ— )*+ ( $ ∢$βˆˆπ’ for

every game (𝑂, 𝑀)? We define:

bound 𝒝 = min 𝛾 ∈ ℝ ∢ βˆ€ !,# 𝛾 β‰₯ 𝑀 π‘„βˆ— max 𝑀 𝑄 ∢ 𝑄 ∈ 𝒝

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?

?

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Coalition Structure Generation

4

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

𝒬

.(𝑂)

𝒬/(𝑂) 𝒬0(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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Worst-case guarantee

Proof: On the blackboard. 5

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

For 𝒝 = 𝒬

! 𝑂 βˆͺ 𝒬0 𝑂 we have bound 𝒝 = π‘œ, 𝒝 =

212!, and 𝒝 is the minimal set with bound smaller than ∞. Minimal set with a bound [Sandholm et al. 1999]

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Worst-case guarantee

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

6 Sketch of proof: Fix 𝑄3 = arg max

$βˆˆπ’ 𝑀(𝑄) and π‘‡βˆ— = arg max 4βŠ†' 𝑀(𝑇).

  • bound 𝒝 ≀ π‘œ: We know 𝑀 𝑄3 β‰₯ 𝑀 π‘‡βˆ— . Hence,

𝑀 π‘„βˆ— ≀ |π‘„βˆ—| β‹… 𝑀 π‘‡βˆ— ≀ |π‘„βˆ—| β‹… 𝑀 𝑄3 ≀ π‘œ β‹… 𝑀 𝑄3 .

  • bound 𝒝 β‰₯ π‘œ: Assume 𝑀 𝑇 = 1 if 𝑇 = 1 and 𝑀 𝑇 =

0, oth. Then: 𝑀 𝑄3 = 1 = !

1 𝑀

1 , … , π‘œ = !

1 𝑀(π‘„βˆ—).

  • Clearly, 𝒝 =

𝑇 βŠ† 𝑂 ∢ 1 ∈ 𝑇 = 212!.

  • 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 β‰₯ 212!.

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Worst-case guarantee

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

What subset of coalition structures should we search next?

?

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Worst-case guarantee

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

What subset of coalition structures should we search next?

?

For 𝒝 = 𝒬

! 𝑂 βˆͺ 𝒬0 𝑂 βˆͺ 𝒬 1 𝑂 and π‘œ > 3 we have

bound 𝒝 = βŒˆπ‘œ/2βŒ‰. Improving the bound [Sandholm et al. 1999] Proof: On the blackboard.

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Worst-case guarantee

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

9 Sketch of proof: Assume π‘„βˆ— contains 𝑙 singletons { 𝑗! , … , 𝑗" }. If 𝑙 = 0, then π‘„βˆ— ≀

1 0 and 𝑀 π‘„βˆ— ≀ 1 0 β‹… 𝑀 𝑄3 .

Assume 𝑙 > 0. We know that 𝑀 𝑗! , … , 𝑗" ≀ 𝑀 1 , … , π‘œ ≀ 𝑀 𝑄3 . Also, π‘„βˆ—\{ 𝑗! , … , 𝑗" } ≀

12"

≀

12!

. Hence, 𝑀 π‘„βˆ— ≀

12!

+ 1 𝑀 𝑄3 =

1 0 β‹… 𝑀 𝑄3 .

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Worst-case guarantee

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

10 Sketch of proof (continued): Assume 𝑀 𝑇 = 1 if 𝑇 = 2 and 1 βˆ‰ 𝑇, 𝑀 {1} = 1 and 𝑀 𝑇 = 0, oth. Then:

  • 𝑀 𝑄3 = 1 (since π‘œ > 3)
  • and 𝑀

1 , 2 , {3,4} … , {π‘œ βˆ’ 1, π‘œ} = 1

0 if π‘œ is even,

  • and 𝑀

1 , {2,3} … , {π‘œ βˆ’ 1, π‘œ} = 16!

0 oth.

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

How to search through the coalition structures to improve the guarantee over time?

?

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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

𝒝! 𝒝0 𝒝/ 𝒝. 𝒝7

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Anytime CSG

13

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

How to search through the coalition structures to improve the guarantee over time?

?

Divide the search space into subsets: 𝒬 𝑂 = 𝒝! βˆͺ 𝒝0 βˆͺ β‹― βˆͺ 𝒝", such that: bound 𝒝! β‰₯ bound 𝒝! βˆͺ 𝒝0 β‰₯ β‹― β‹― β‰₯ bound 𝒝! βˆͺ β‹― βˆͺ 𝒝" = 1. Anytime Coalition Structure Generation

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

  • 1. Search 𝒬

! 𝑂 .

  • 2. Search 𝒬0 𝑂
  • 3. Search 𝒬

1 𝑂 .

  • 4. Search 𝒬12! 𝑂 .
  • 5. …
  • 6. Search 𝒬/ 𝑂 .

Anytime CSG-99 [Sandholm et al. 1999] 𝒝! = 𝒬

! 𝑂 , 𝒝0 = 𝒬0 𝑂 , and

𝒝" = 𝒬16/2" 𝑂 for 2 < 𝑙 ≀ π‘œ.

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

𝒬!(𝑂)

[1,1,1,1,1,1,1,1] [2,1,1,1,1,1,1] [3,1,1,1,1,1] [2,2,1,1,1,1] [4,1,1,1,1] [3,2,1,1,1] [2,2,2,1,1] [5,1,1,1] [4,2,1,1] [3,3,1,1] [3,2,2,1] [2,2,2,2] [6,1,1] [5,2,1] [4,3,1] [4,2,2] [3,3,2] [7,1] [6,2] [8]

𝒬"(𝑂) 𝒬#(𝑂) 𝒬$(𝑂) 𝒬

%(𝑂)

𝒬&(𝑂) 𝒬'(𝑂) 𝒬

((𝑂)

[5,3] [4,4]

𝒝! 𝒝0 𝒝/ 𝒝. 𝒝7 𝒝; 𝒝< 𝒝=

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Define β„Ž π‘š = (π‘œ βˆ’ π‘š)/2 + 2. After searching 𝒬> 𝑂 for π‘š > 3, the bound is β‰ˆ π‘œ/β„Ž(π‘š). Specifically, for 𝒝 = 𝒬

! 𝑂 βˆͺ 𝒬0 𝑂 βˆͺ 𝒬 1 𝑂 βˆͺ β‹― βˆͺ 𝒬> 𝑂 :

bound 𝒝 = 4 π‘œ/β„Ž(π‘š) 𝑗𝑔 π‘œ ≑ βˆ’1 𝑛𝑝𝑒 β„Ž π‘š , π‘œ ≑ π‘š 𝑛𝑝𝑒 2 , π‘œ/β„Ž(π‘š) π‘π‘’β„Žπ‘“π‘ π‘₯𝑗𝑑𝑓.

Bounds for Anytime CSG-99 [Sandholm et al. 1999] Proof: On the blackboard.

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Anytime CSG

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

17 Sketch of proof: β„Ž π‘š = (π‘œ βˆ’ π‘š)/2 + 2 = π‘œ βˆ’ π‘š βˆ’ 2 /2 + 1 is a number such that partition of the form:

  • [β„Ž π‘š , β„Ž π‘š βˆ’ 1, 1, … , 1] if 2 ∀ π‘œ βˆ’ π‘š (case A) or
  • β„Ž π‘š , β„Ž π‘š βˆ’ 2, 1, … , 1 if 2|π‘œ βˆ’ π‘š (case B)

appears in level π‘š ([…] contains the list of sizes of coalitions). After searching level π‘š we know that disjoint coalitions of sizes 𝑗 and π‘˜ such that 𝑗 + π‘˜ ≀ π‘œ βˆ’ π‘š βˆ’ 2 appeared in one partition.

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Anytime CSG

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

18 Sketch of proof (continued): Lower bound for the bound: Let 𝑠 = π‘œ mod β„Ž π‘š = π‘œ βˆ’ π‘œ/β„Ž(π‘š) β‹… β„Ž π‘š . Consider partition π‘„βˆ— of the form [β„Ž π‘š , β„Ž(π‘š), … , β„Ž π‘š , 𝑠] and game 𝑀 𝑇 = 1 if 𝑇 = β„Ž(π‘š). We have: 𝑀 π‘„βˆ— = π‘œ/β„Ž(π‘š) and 𝑀 𝑄′ = 1. Thus, bound 𝒝 β‰₯ π‘œ/β„Ž(π‘š) . If we have case B (i.e., 2|π‘œ βˆ’ π‘š) and 𝑠 = β„Ž π‘š βˆ’ 1, then by adding 𝑀 𝑇 = 1 for one specific coalition of size 𝑠 we get: bound 𝒝 β‰₯ βŒŠπ‘œ/β„Ž(π‘š)βŒ‹ + 1 = βŒˆπ‘œ/β„Ž(π‘š)βŒ‰

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Anytime CSG

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

19 Sketch of proof (continued): Upper bound for the bound: To see that bound 𝒝 ≀ π‘œ/β„Ž(π‘š) consider the game (𝑂, 𝑀) such that 𝑀 π‘„βˆ— /𝑀(𝑄3) is the highest. We can assume that 𝑀 𝑇 = 0 for every 𝑇 βˆ‰ π‘„βˆ—. Also, we can assume that 𝑄3 ∩ π‘„βˆ— = 1: if 𝑇, 𝑇3 ∈ 𝑄3 ∩ π‘„βˆ—, then replacing 𝑇, 𝑇3 with 𝑇 βˆͺ 𝑇3 and defining game analogously would result in the same value 𝑀 π‘„βˆ— /𝑀(𝑄3).

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Anytime CSG

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

20 Sketch of proof (continued): Now, since 𝑄3 ∩ π‘„βˆ— = 1 it means that in π‘„βˆ— 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 β‰  β„Ž π‘š βˆ’ 1, we get maximum βŒŠπ‘œ/β„Ž π‘š βŒ‹ coalitions for: β„Ž π‘š , β„Ž π‘š , … , β„Ž π‘š , β„Ž π‘š + 𝑠 . In case B with 𝑠 β‰  β„Ž π‘š βˆ’ 1, we get maximum βŒˆπ‘œ/β„Ž π‘š βŒ‰ coalitions for: β„Ž π‘š , β„Ž π‘š , … , β„Ž π‘š , β„Ž π‘š βˆ’ 1 This implies our thesis.

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

  • 1. Search 𝒬

! 𝑂 .

  • 2. Search 𝒬0 𝑂 .
  • 3. Search 𝒬

1 𝑂 .

  • 4. Search 𝒬?120 𝑂 .
  • 5. …
  • 6. Search 𝒬?0 𝑂 .

Anytime CSG-04 [Dang & Jennings 2004] Define: 𝒬?@ 𝑂 = 𝑄 ∈ 𝒬 𝑂 ∢ 𝑄 > 2, max

4∈$ 𝑇 β‰₯ π‘Ÿ .

Since many of this steps will not improve the bound, the authors considered 𝒬?⌈1(@2!)/@βŒ‰ 𝑂 for π‘Ÿ from ⌊(π‘œ + 1)/4βŒ‹ down to 2 and then search the remaining coalition structures.

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

𝒬!(𝑂)

[1,1,1,1,1,1,1,1] [2,1,1,1,1,1,1] [3,1,1,1,1,1] [2,2,1,1,1,1] [4,1,1,1,1] [3,2,1,1,1] [2,2,2,1,1] [5,1,1,1] [4,2,1,1] [3,3,1,1] [3,2,2,1] [2,2,2,2] [6,1,1] [5,2,1] [4,3,1] [4,2,2] [3,3,2] [8]

𝒬"(𝑂) 𝒬#(𝑂) 𝒬$(𝑂) 𝒬

%(𝑂)

𝒬&(𝑂) 𝒬'(𝑂) 𝒬

((𝑂)

[7,1] [6,2] [5,3] [4,4]

𝒝! 𝒝0 𝒝/ 𝒝= 𝒝< 𝒝; 𝒝7 𝒝.

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

For 𝒝 = 𝒬

! 𝑂 βˆͺ 𝒬0 𝑂 βˆͺ 𝒬 1 𝑂 βˆͺ 𝒬?⌈1(@2!)/@βŒ‰ 𝑂

we have: bound 𝒝 ≀ 2π‘Ÿ βˆ’ 1. Bounds for Anytime CSG-04 [Dang & Jennings 2004] Proof: On the blackboard.

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Anytime CSG

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

24 Sketch of proof: Assume π‘„βˆ— = 𝑇!, 𝑇0, … , 𝑇" s.t. 𝑇! β‰₯ 𝑇0 β‰₯ β‹― β‰₯ 𝑇" . As before, we have 𝑀 𝑇! + β‹― + 𝑀 𝑇@2! ≀ π‘Ÿ βˆ’ 1 β‹… 𝑀(𝑄3). Now, consider 𝑇@, 𝑇0@, … , π‘‡βŒŠ"/@βŒ‹@. These coalitions combined have at most π‘œ/π‘Ÿ players. So, they do not contain at least π‘œ βˆ’ π‘œ/π‘Ÿ = π‘œ(π‘Ÿ βˆ’ 1)/π‘Ÿ. Hence, they appeared already in one partition and we have: 𝑀 𝑇@ + β‹― + 𝑀 π‘‡βŒŠ"/@βŒ‹@ ≀ 𝑀(𝑄3). With the same analysis for groups (𝑇@6!, 𝑇0@6!, … , 𝑇 "/@ @6!), (𝑇@60, 𝑇0@60, … , 𝑇 "/@ @60) and so on, we get that: 𝑀 π‘„βˆ— ≀ 2π‘Ÿ βˆ’ 1 𝑀 𝑄3 .

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Anytime CSG

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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Conclusions

We consider the Anytime Coalition Structure Generation problem.

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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References

  • [Dang & Jennings 2004] V.D. Dang & N.R. Jennings.

Generating coalition structures with finite bound from the optimal

  • guarantees. Proceedings of the 3rd International Conference on

Autonomous Agents and Multiagent Systems (AAMAS), 564-571, 2004.

  • [Rahwan et al. 2011] Rahwan, T.; Michalak, T. P. & Jennings, N. R.

Minimum search to establish worst-case guarantees in coalition structure generation. Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI), 338-343, 2011.

  • [Sandholm et al. 1999] T. Sandholm, K. Larson, M. Andersson,
  • O. Shehory, F. TohmΓ©. Coalition structure generation with worst case
  • guarantees. Artificial Intelligence 111, 209-238, 1999.

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory