cooperative games the shapley value and weighted votingarielpro/15896s16/slides/896s16-11.pdf ·...
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Cooperative Games –The Shapley value and Weighted Voting
Yair Zick
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The Shapley Value
Given a player , and a set , the marginal contribution of to is
How much does contribute by joining ? Given a permutation of players, let the predecessors of in be
We write
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The Shapley Value
Suppose that we choose an ordering of the players uniformly at random. The Shapley value of player is
614 1 129 7 4
49 q = 50
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The Shapley Value
Efficient: Symmetric: players who contribute the same are paid the same.Dummy: dummy players aren’t paid.Additive:
The Shapley value is the only payoff division satisfying all of the above!
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The Shapley Value
Proof: let’s prove it on the board.
Theorem: if a value satisfies efficiency, additivity, dummy and symmetry, then it is the Shapley value.
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Computing Power Indices
The Shapley value has a brother – the Banzhaf value
It uniquely satisfies a different set of axiomsDifferent distributional assumption –more biased towards sets of size
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Voting Power in the EU Council of Members
• The EU council of members is one of the governing members of the EU. – Each state has a number of representatives proportional to its population
– Proportionality: “one person – one vote”
• In terms of voting power ‐
Image: Wikipedia Image: Wikipedia
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• Changes to the voting system can achieve better proportional representation.
• Changing the weights – generally unpopular and politically delicate
• Changing the quota – easier to do, an “innocent” change.
Selecting an appropriate quota (EU ‐ about 62%), achieves proportional representation with a very small error!
Voting Power in the EU Council of Members
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• Changes to the quota change players’ power.
• What is the relation between quota selection and voting power?
Changing the Quota
'i 'i(q)
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A “typical” graph of 'i(q)
The graph converges to some value when quota is 50%…
Lower variation towards the 50% quota
Max at
Min at 1
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Weights are a Fibonacci Series
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Maximizing
Theorem: is maximized at
Proof: two cases: if is pivotal for under then
, but . This implies that is pivotal for when the threshold is as well.
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Maximizing Lemma: let Then
for all
Proof: assume that . We write
, so Need to show that
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Maximizing Need to show that Construct an injective mapping
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Maximizing Second case: Let , then
and .By Lemma
which concludes the proof.
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Minimizing Not as easy, two strong candidate minimizers: or
.
Not always them, not clear which one to choose. For below‐median players, setting is worse.
Deciding whether a given quota is maximizing/minimizing is computationally intractable.
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The expected behavior of It seems that analyzing fixed weight vectors is not very effective…
even small changes in quota can cause unpredictable behavior; worst‐case guarantees are not great.
Can we say something about the likely Shapley value when weights are sampled from a distribution?
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Balls and Bins Distributions
• We have balls, bins.• A discrete probability distribution • is the probability that a ball will land in bin
3 2 5 5 5 4 4
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0
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Huge disparity at
some thresholds
Near Equality at others…
Changing the threshold from 500 to 550 results in a huge shift in voting power
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Balls and Bins: Uniform
• Suppose that the weights are generated from a uniform balls and bins process with m balls and n bins.
• Theorem: when the threshold is near integer multiples of , there is a high disparity in voting power (w.h.p.)
• Theorem: when the threshold is well‐away from integer multiples of , all agents have nearly identical voting power (w.h.p.)
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Balls and Bins: Exponential• There are m voters. A voter votes for player i w.p. pi + 1
• The probability of high‐index players getting votes is extremely low. Most votes go to a few candidates.
• Theorem: if weights are drawn from an exponential balls‐and –bins distribution, then with high probability, the resulting weights are super‐increasing
• A vector of weights (w1,…, wn) is called super‐increasing if
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Balls and Bins: Exponential
• In order to study the Shapley value in the Balls and Bins exponential case, it suffices to understand super‐increasing sequences of weights.
• Suppose that weights are ( )
• Let us observe the (beautiful) graph that results.
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Image: Wikipedia
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Super‐Increasing Weights• ∑ 2∈ : the binary representation of • : the minimal set ⊆ such that • Claim: if the weights are super‐increasing, then
• the Shapley value when the threshold is equals the Shapley value when the weights are powers of 2, and the threshold is
• Computing the Shapley value for super‐increasing weights boils down to computing it for powers of 2!
• Using this claim, we obtain a closed‐form formula of the SV when the weights are super‐increasing.
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Conclusion
• Computation: generally, computing the Shapley value (and the Banzhaf value) is #P complete (counting complexity)
• It is easy when we know that the weights are not too large (pseudopolynomial time)
• It is easy to approximate them through random sampling in the case of simple games.
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Further Reading
• Chalkiadakis et al. “Computational Aspects of Cooperative Game Theory”
• Zuckerman et al. “Manipulating the Quota in Weighted Voting Games” (JAIR’12)
• Zick et al. “The Shapley Value as a Function of the Quota in Weighted Voting Games” (IJCAI’11)
• Zick “On Random Quotas and Proportional Representation in Weighted Voting Games” (IJCAI’13)
• Oren et al. “On the Effects of Priors in Weighted Voting Games” (COMSOC’14)
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