On the Effects of Priors in Weighted Voting GamesJoel Oren, Yuval Filmus, Yair Zick and Yoram Bachrach

1

Motivation• Weighted voting systems are a common decision making

method.• Parliaments• The EU Council of Members• The US Electoral College

• Each member (a state, a political party etc.) has a weight; in order to pass a bill, the total number of votes for the bill must exceed a given threshold.

2

Image: Wikipedia Image: Wikipedia

Motivation• A Central Question: how influential is a member? • Voting power is not equal to voting weight. • Example: there are three parties; need 60 votes to pass a bill.

Party A – 58 votesParty B – 31 votesParty C – 31 votes

Although party A is much bigger than the other two, it has the same decision making ability as the other two (any party needs the help of at least one other party in order to pass a bill).

3

The Shapley Value and WVGs• A Weighted Voting Game (WVG) G on n players is defined as

follows:• For all , an integer weight.• A coalition of players is winning if .

• Given an ordering (permutation) of players, player i is pivotal for an ordering if his predecessors are losing, but if he joins, they win. We write if is pivotal for , and 0 otherwise.

614 1 129 7 4

49 q = 50

4

The Shapley Value and WVGs• The Shapley value of player i is the probability that she is pivotal for

a randomly chosen ordering of the players.

It is denoted .

5

Our Contribution• Previous work shows that when we make no assumptions on

weight distribution, voting power can be highly sensitive to changes in the threshold .

• Our work: study the expected effects on voting power when weights are sampled from various distributions• Binomially distributed weights• Balls and Bins distribution (uniform)• Balls and Bins distribution (exponential)• i.i.d. bounded distributions.

What can we say about voting power (as a function of the threshold) for weights sampled from a given distribution X? 6

Balls and Bins Distributions• We have balls, bins.• A discrete probability distribution • is the probability that a ball will land in bin

7

3 2 5 5 5 4 4

Balls and Bins: Uniform • We have m voters, each votes for one of n candidates with

probability ; number of votes for a candidate = his weight• Intuitively: as m grows, weights are nearly identical. We should

not expect to see much difference in voting power. • Let’s observe the average (over 200 repetitions) voting power

of players when: • n = 10• m = 1000

8

0 45 90 1351802252703153604054504955405856306757207658108559009459900

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Agent 1

Agent 2

Agent 3

Agent 4

Agent 5

Agent 6

Agent 7

Agent 8

Agent 9

Agent 10

threshold

Ave

rage

Sh

aple

y V

alu

eHuge

disparity at some

thresholds

Near Equality at others…

Changing the threshold from 500 to 550 results in a huge shift in voting power

9

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.)

Take home message: changes to the threshold matter, even if weights are very similar to one another.

10

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

11

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.

12

0 45 90 135 180 225 270 315 360 405 450 495 540 585 630 675 720 765 810 855 900 945 9900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Agent 1Agent 2Agent 3Agent 4Agent 5Agent 6Agent 7Agent 8Agent 9Agent 10

13

Image: Wikipedia

Super-Increasing Weights• : 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.

14

i.i.d distributed weights• When the threshold is close to 50%, the Shapley values of the

weakest and strongest players do not fluctuate much. • Proof uses tools from renewal theory: given a weight , and a

list of random variables , set

• What is the expected number of ’s that lie in the interval ?• Theorem: when weights are sampled i.i.d. from the uniform

distribution, the Shapley value of the strongest player is very close to , and the Shapley value of the weakest player is , when the threshold is not too close to or .

15

Conclusions and Future Work• We study how the Shapley value changes as a function of the

threshold, under different weight distributions.• Our results answer several open questions from

Zick et al. (2011), Zuckerman et al. (2012) and Zick (2013). • Future work:• Proportional representation• Probabilistic analysis of other cooperative game models? • Have you seen this function?

0 54 108 162 216 270 324 378 432 486 540 594 648 702 756 810 864 918 972

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Agent 1

Agent 2

Agent 3

Agent 4

Agent 5

Agent 6

Agent 7

Agent 8

Agent 9

Agent 10

16

Thank you! Questions?