1. Dynamic Weighted Voting Games Edith Elkind Dmitrii Pasechnik Yair Zick AAMAS 2013

2. Cooperative TU Games Agents form coalitions; generate profit. 1 6 4 5 3 2 Coalition members can freely divide profits. How should profits be divided? $5 $3 $2

3. Reminder - TU Cooperative Games A TU game G : agents N = {1,…, n }, and a characteristic function v : 2 N → R Weighted Voting Games (WVGs): Each agent has a weight w i ; coalition weight: w ( S ) A coalition has value 1 (winning), if w ( S ) ≥ q, otherwise has value 0 (losing). q is called the threshold, or quota.

4. Reminder – Payoffs in Cooperative Games We assume that the coalition N is formed. Agents may freely distribute profits. An outcome is a vector x = ( x 1,…, x n ) such that Σ i 2 N x i = v ( N ).

5. Solution Concepts in Cooperative Games Intuition: an outcome is stable if no coalition wants to deviate core ( G ) = { x |  i  C x i ≥ v ( C ) 8 C µ N } each coalition gets at least what it can get alone. Similar notions – the " -core and the least core. Fair payoff distribution – the Shapley value © i ( G ) = ∑ | C |!( n - | C | - 1)!/ n ![ v ( C [ { i }) – v ( C )] C µ N \ { i }

6. Dynamics in Cooperative Games Classic cooperative games are static, one shot games. It is often the case that the value of coalitions changes over time. Changes to agent abilities, available resources, etc.

7. Case Study: EU Voting We study the EU voting game between 2012 and 2061 Each state has a weight corresponding to population size. Generated linear population estimates based on last decade Result - significant changes in voting power: Ireland: +39%; Spain: +31%; Germany: -22%

8. Our Model: Dynamic Cooperative Games Coalition values change as a function of time. Characteristic function: v t : 2 N → R v t ( S ) – the value of S at time t. The game at time t : G t = h N, v t i Dynamic Weighted Voting Games: Agent weights and threshold are a function of time. agent weight at time t : w i ( t ) Quota at time t : q ( t )

9. Three Questions what are the properties of G t for a given t ? (e.g. is the core non-empty at a given time) does G t preserve some property as t changes from t 0 to t 1 ? does a given payoff vector belong to some solution concept for all t between t 0 and t 1 ?

10. Our Contribution We focus on DWVGs : weights are polynomials of degree at most k with integer coefficients between – W and W Algorithms for: computing solution concepts at a given time t deciding whether a player’s Shapley value, the value of the least core, etc. remains in [ l, u ] between t 0 and t 1 running time: polynomial in || t ||, n and W k +1

11. Interpretation (static) WVGs: the same problems can be solved in time poly( n, W ), where W is the maximum weight dynamic WVGs are as easy as static WVGs Why weights = low-degree polynomials? tractability such weights result naturally from interpolation But such predictions may be inaccurate? future work: robustness analysis

12. Techniques The weight w S ( t ) of every coalition S is a polynomial of deg ≤ k with coefficients in [- nW, nW ] Coalition’s signature: its weight polynomial there are poly( n, W k ) signatures value of S changes when w S ( t ) = q ( t ) at most k times

13. Techniques (cont) Key idea: only need to keep track of winning signatures at all time points. Sweeping line algorithm: Main challenge: polynomial root representation sig 1 ( t ) - q ( t ) sig 2 ( t ) - q ( t ) sig 3 ( t ) - q ( t )

14. Conclusions and Future Work We provide the first algorithmic analysis of dynamic cooperative games. Other types of coalitional games where v t can be compactly represented: network flow games induced subgraph games Dealing with uncertainty: instead of assuming that we know v t, we assume that we know u t and l t such that l t ≤ v t ≤ u t vtvt

15. Thank you! Questions? P.S.: I am on the job market!