1. Ioannis Caragiannis, Jason A. Covey, Michal Feldman, Christopher M. Homan, Christos Kaklamanis, Nikos Karanikolask, Ariel D. Procaccia, Je ff rey S. Rosenschein. Presented by Noa Avigdor-Elgrabli

2. Agenda Basic definitions Dodgson score upper bounds Randomized approximation algorithm Definition Analysis Deterministic approximation algorithm What’s better? Lower bounds

3. Basic Definitions Input: Example: m alternatives {a, b, c} n agents {1,2,3} Ranking of each agent Condorcet winner: An alternative that beats every other alternative in pairwise election Dodgson score of a*: The least number of exchanges between adjacent alternatives needed to make a* a Condorcet winner Young score of a*: The size of the largest subset of agents for whom a* is a Condorcet winner Agent 1Agent 2Agent 3 aba bac ccb Condorcet winner: a abc 013 Dodgson score: abc 310 Young score:

4. Remarks Algorithm that Usually computes the Dodgson score (under the assumption that the number of agents is significantly large than the number of)

5. Randomized Algorithm for the Dodgson score of a* Construct an integer linear program for the Dodgson score (not solvable in polynomial time  ). Relax it to a linear program (solvable but the solution is fractional  ). Round the fractional solution to an integer solution using some randomness. Prove that the solution is “good” (O (log m) - approximation) with high probability.

6. Notations def(a) = The number of additional agents that must rank a* above a in order for a* to beat a in a pair wise election. e i ja = 1 iff pushing a* by j positions in the ranking of agent i makes a* gain additional vote against a. (e i ja = 0 otherwise)

7. Linear Program [BTT89] iff a* is “pushed” by j positions in the ranking of agent i The cost of the solution We choose for each i exactly one j (from 0 to m-1) we improve the position of a* against each a in at least def(i) agents

8. Randomized Rounding Algorithm 1. Solve the relaxed LP – We get a solution x 2. For each agent i assign X i  j with probability 3. Repeat (2) t times (t would be defined latter) 4. X i max  the maximum of all t received X i ’s 5. Return ∑ i X i max Theorem For t=8 log m the randomized rounding algorithm returns t-approximation of the Dodgson score with probability at least 1/2

9. Analysis Remainder: Lemma (Jogdeo and Samuels) Let Y 1,..., Y n be independent heterogeneous Bernoulli trials. Then

10. Greedy Algorithm Live = {a | def(a) > 0} While Live is not empty: Do the most “cost- effective” push: Theorem: Greedy is O(log m)- approximation

11. Example Live={a1,a2,a3,a4} Greedy(a*)=7 (=Opt) R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 a4 a3a2a1 a3 b4b9b13b16 a2 b5b10b14b17 a1 b6b11b15a*.b1b7b12a*..b2b8a*...b3a*................... 3/1 4/1 7/4

12. Desirable Properties Algorithmic properties: Running Time, randomness, approximation factor. Social choice properties: Truthfulness – the agents cannot benefit by lying. Condorcet Property – a* is a Condorcet winner  a* wins. Monotonicity Property (weak) – pushing an alternative in the preference of an agent cannot worsen its score. Truthfulness  Dictator or Existence of alternative that can’t win [Gibbard-Satterthwaite]

13. Greedy - Non Monotonic ds R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 a4 a3a2a1 a3 b4b9b13b16 a2 b5b10b14b17 a1 b6b11b15a*.b1b7b12a*..b2b8a*...b3a*................... 3 4 7/4 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 a4 a3a2a1 a3 b4b9b13a* a2 b5b10a*b16 a1 b6a*b14b17.b1a*b11b15..b2b7b12...b3b8....a*............... 1 7/4 Cost(a*)=7 2 7/3 3 7/27 Cost(a*)=1+2+3+4=9 4

14. Rounding or Greedy? Approximation factor Running time RandomnessCondorcet Property Monotonicity Roundinglog (m)BadRandomizedYes Greedylog (m)GoodDeterministicYesNo

15. Lower Bounds Dodgson Score: For any ε > 0, there is no polynomial-time Ω ( (1/2 – ε) ln m)-approximation for the Dodgson score of a given alternative*  Greedy is optimal up to factor 2 Young Score: It is NP-hard to approximate the Young score by any factor *under the assumption that an NP problems can’t be solved in quasi-poly time

16. Thank You!