computational aspects of stability in weighted voting games edith elkind (ntu, singapore) based on...
TRANSCRIPT
![Page 1: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/1.jpg)
Computational aspects of stability in
weighted voting games
Edith Elkind (NTU, Singapore)
Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,
Michael Wooldridge, and Dmitrii Pasechnik
![Page 2: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/2.jpg)
Cooperative vs. non-cooperative games
• Non-cooperative games:– each player can choose an action– payoffs are determined by the action profile
• Cooperative (coalitional) games:– players can form groups (coalitions)– payoff to a group determined by its composition– players have to share group payoff
![Page 3: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/3.jpg)
Coalitional games: formal model
• G = (N, v)• N={1, ..., n}: set of players• v: 2N→R: characteristic function
– v(S): payoff available to S, has to be shared among members of S
• p=(p1, ..., pn) is an imputation if– pi ≥ 0 for i =1, ..., n
– p(N) := S i in N pi = v(N)
![Page 4: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/4.jpg)
Coalitional games with compact representations
• Weighted voting games (subject of this talk):– G = (w1, ..., wn; T)
– each player i has a weight wi
– threshold T– v(S)=1 if w(S) ≥ T, v(S)=0 otherwise
• Network flow games:– players are edges of a network– value of a coalition = size of the flow it can carry
• Minimum spanning tree games, matching games, etc.
![Page 5: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/5.jpg)
Coalitional games: stability
• Which imputations are stable?– no subset of players should want to deviate
• Core: set of all stable imputations(p1, ..., pn) is in the core if p(S) ≥ v(S) for all S N
• Problem: core may be emptyweighted voting game G=(1, 1, 1; 2) suppose wlog p1 > 0
then p({2, 3}) < 1 v({2, 3}) = 1
recall: p(S) = S iS pi
![Page 6: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/6.jpg)
When is the core non-empty?
• Def: G=(N, v) is simple if v(S){0, 1} for all S– WVGs are simple games
• Def: in a simple game, i is a veto player if v(S) = 0 for any S N \ { i }
• Claim: a simple game has a non-empty core iffthere is a veto player.
Also, (p1, ..., pn) is in the core iffpi = 0 for all non-veto players
pi > 0
N
S
![Page 7: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/7.jpg)
e-core and least core
Need to relax the notion of the core: core: p(S) ≥ v(S) for all S N e-core: p(S) ≥ v(S) - e for all S Nleast core: smallest non-empty e-core
– minimizing the worst deficit v(S) - p(S)
G=(1, 1, 1; 2):– 1/3-core is non-empty: (1/3, 1/3, 1/3) 1/3-core– e-core is empty for any e < 1/3– least core = 1/3-core
![Page 8: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/8.jpg)
Can we compute the core,
e-core and the least core of weighted voting games?
![Page 9: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/9.jpg)
Computational issuesOur Results (E., Goldberg, Goldberg, Wooldridge, AAAI’07)
• Is the core non-empty? – poly-time: use the lemma
• For a given e, is the e-core non-empty?
• For a given e, is a given imputation p in the e-core?
• Is a given imputation p in the least core?
• Construct an imputation in the least core. – p
Given a WVG G = (w1, ..., wn; T)
reductions from Partition
![Page 10: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/10.jpg)
Computational issues
• Is the core non-empty? – poly-time: use the lemma
• For a given e, is the e-core non-empty? – coNP-hard
• For a given e, is a given imputation p in the e-core? – coNP-hard
• Is a given imputation p in the least core? – NP-hard
• Construct an imputation in the least core. – NP-hard
Given a WVG G = (w1, ..., wn; T)
reductions from Partition
![Page 11: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/11.jpg)
A pseudopolynomial algorithm?
• Hardness reduction from Partition assumes large weights– recall: wi are given in binary, – poly-time algorithm <=>
runs in time poly (n, log wmax)
• What if weights are small?– e.g., at most poly(n)?– we are happy with algorithms
that run in time poly (n, wmax)
![Page 12: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/12.jpg)
min cp1+…+ pn = 1
pi ≥ 0 for all i = 1, …, n
S iJ pi ≥ 1 - c for any J s.t. w(J) ≥ T
linear program exponentially many ineqs
Claim: least core = c-core
LP for the least core
![Page 13: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/13.jpg)
LPs and separation oracles
• Separation oracle: – input: (p, c)– output: “yes” if (p, c) satisfies the LP,
violated constraint otherwise
• Fact: LPs with poly-time separation oraclescan be solved in poly-time.
• Our case: given (p, c), is there a J with w(J) ≥ T, p(J) < 1-c?– reduces to Knapsack => solvable in time poly (n, wmax)
• Works for other problems listed above
![Page 14: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/14.jpg)
An approximation algorithm
• Back to large weights…• Theorem: suppose least core value = e
Then for any d we can compute e’ s.t. e ≤ e’ ≤ (1+ )e d and e’-core is non-empty in time poly (n, log wmax, 1/d)
(FPTAS) • Proof idea: use FPTAS for Knapsack inside the
separation oracle
![Page 15: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/15.jpg)
min cp1+…+ pn = 1
pi ≥ 0 for all i = 1, …, nS iJ pi ≥ 1-c for any J s.t. w(J) ≥ T
p1+…+ pn = 1
pi ≥ 0 for all i = 1, …, nS iJ pi ≥ kd for any J s.t. w(J) ≥ T
Approximating the least core
LPk, k=1, ..., 1/d
![Page 16: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/16.jpg)
LPk and the least core value
p1+…+ pn = 1
pi ≥ 0 for all i = 1, …, nS iJ pi ≥ kd for any J s.t. w(J) ≥ T
Claim: let k* be the largest value of k such that LPk has a feasible solution. Then the value e of the least core satisfies 1 – e - d ≤ k*d ≤ 1- e
Can we find k*?– maybe not, but we can find k’ {k*, k*-1}
LPk
![Page 17: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/17.jpg)
An “almost” separation oracle• Claim: for each k =1, …, 1/d, there is a procedure SOk
that runs in time poly (n, 1/d) and either correctly implements a separation oracle for LPk or stops and produces a feasible solution to LPk-1
• Try to solve LP1, LP2, … , LP1/d using SO1, SO2, … , SO1/d – k’: the largest value of k for which we find a feasible
solution (reported by SOk’ or SOk’+1)• Claim: k’ {k*, k*-1}
e ≤ 1 – k*d ≤ e+ d implies e ≤ 1 – k’d ≤ e+2 d
![Page 18: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/18.jpg)
Implementing SOk
Input to SOk: p1, …, pn
Need to check: S iJ pi ≥ kd for any J s.t. w(J) ≥ T– p’i = max { jd/n : jd/n ≤ pi }
– |p’i - pi| < d/n– check if there is a J with p’(J) < (k-1)d, w(J) ≥ T
(DP for Knapsack)• if not, p1, …, pn is a feasible solution for LPk-1
• if yes, p(J) ≤ p’(J)+d, so p(J) < kd, and hence J is a violated constraint for LPk
![Page 19: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/19.jpg)
Additive => multiplicative
• We have shown: can compute e’ s.t. e ≤ e’ ≤ + e d
• Need: e’ s.t. e ≤ e’ ≤ (1+ ) e d• Claim: if e > 0 then e ≥ 1/n
– proof: • some player i is paid at least 1/n• N \ { i } is a winning coalition
• Given d, run our algorithm with d’ = d/n: e ≤ e’ ≤ + e d/n ≤ + e de= (1+ )e d
![Page 20: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/20.jpg)
Most stable point?
• Least core may contain more than one point, not all of them equally good
• G = (3, 3, 2, 2, 2; 6)p = ( 1/4, 1/4, 1/6, 1/6, 1/6);q = (1/3, 1/6, 1/6, 1/6, 1/6);– p and q are both in the least core = 1/2-core– under p, only 2 coalitions have deficit 1/2– under q, 5 coalitions have deficit 1/2
![Page 21: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/21.jpg)
Nucleolus: definition
• deficit: d(p, S) = v(S) - p(S)• least core: minimizes worst deficit• nucleolus:
– minimize worst deficit;– given this, minimize 2nd worst deficit, etc.
• deficit vector: d(p) = (d(p, S1), ..., d(p, S2n)), ordered from largest to smallest
• Def: nucleolus is an imputation p withlex-minimal deficit vector d(p)
![Page 22: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/22.jpg)
Nucleolus: properties
• Introduced by Schmeidler (1969)• Nucleolus is unique• Always in the least core• “Most stable” imputation
![Page 23: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/23.jpg)
Can we compute the nucleolus of weighted voting games?
![Page 24: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/24.jpg)
Small vs. large weights
• Binary weights: NP-hard to compute– reduction from Partition (E., Goldberg, Goldberg, Wooldridge, AAAI’07)
• Unary weights?– pseudopolynomial algorithm (E., Pasechnik, SODA’09)
![Page 25: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/25.jpg)
Computing nucleolus: general scheme
Sequence of linear programs:• LP1: min (p, e) e
p(S) ≥ v(S) – e for all S 2N
(p1, e1): interior optimizer for LP1
S1: set of tight constraints for (p1, e1)
• LP2: min (p, e) e
p(S) = v(S) – e1 for all S S1
p(S) ≥ v(S) – e for all S 2N \ S1
• LP3, LP4, etc. – till there is a unique solution
![Page 26: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/26.jpg)
Solving LP1: weighted voting games
Solving LP1 = finding the least core LP1: min (p, e) e p1+…+ pn = 1
pi ≥ 0 for all i = 1, …, n
p(S) ≥ v(S) – e for all S 2N
• Can find (p, e) in poly-time. • S1? cannot list explicitly...
![Page 27: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/27.jpg)
How to solve LP2?
LP 2: min (p, e) e p(S) = v(S) – e1 for all S S1
p(S) ≥ v(S) – e for all S 2N \ S1
Is there a poly-time separation oracle for LP2?– input: p1, ..., pn, e; can assume e < e1 – suppose we have found S with w(S) ≥ T, p(S) < 1-e– this is only useful if S is not in S1
– difficulty: S1 can be exponentially large
![Page 28: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/28.jpg)
How to solve LPj?
LP j: min (p, e) e p(S) = v(S) – e1 for all S S1
... p(S) = v(S) – e j-1 for all S S j-1
p(S) ≥ v(S) – e for all S 2N \(S1 U ... U S j-1) S1 ,..., S j-1 can be exponentially large
![Page 29: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/29.jpg)
Idea: listing → counting
• We know e1, ..., e j-1
• Can assume that we know s1=|S1|, ..., s j-1=|S j-1|• Given a candidate solution (p, e),
suppose we can compute in poly-time– top j distinct deficits m1, ..., mj
– nj = number of coalitions with deficit mj
• Check if – mt = et, nt = st for t=1, ..., j-1 – mj ≤ e
• Thm: answer is “yes” iff (p, e) is feasible for LPj
![Page 30: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/30.jpg)
Missing pieces
• How to implement the counting?– WVGs with unary weights:
dynamic programming• If the answer is “no”, need to identify a
violated constraint– WVGs with unary weights:
more dynamic programming
![Page 31: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/31.jpg)
General recipe?
Meta-theorem: given a coalitional game G, suppose that we can, for any p and j=1, ..., n, identify top j deficits under p andcount how many coalitions have those deficitsin poly-time.Then we can compute the nucleolus of G in poly-time.
Question: for which classes of games can we do this?
![Page 32: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/32.jpg)
Conclusions
• Stability in weighted voting games– core: poly-time computable– e-core, least-core
• weakly NP-hard • pseudopolynomial algorithm• FPTAS
– nucleolus• weakly NP-hard• pseudopolynomial algorithm• approximation???
![Page 33: Computational aspects of stability in weighted voting games Edith Elkind (NTU, Singapore) Based on joint work with Leslie Ann Goldberg, Paul W. Goldberg,](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649d985503460f94a82b3b/html5/thumbnails/33.jpg)
An “almost” separation oracle• k’: max {k : SOk has a feasible solution}• k’’: max {k : our procedure finds a feasible solution}• Claim: k’’ {k’, k’-1}
– for k = 1, …, k’:• if SOk works, it produces a feasible solution for k
• if SOk fails, it produces a feasible solution for k-1– for k = k’+1
• if SOk works, it tells us there is no feasible solution for k
• if SOk fails, it produces a feasible solution for k-1– for k > k’+1
• SOk works and tells us there is no feasible solution for k