1. WINE 2011 Manipulating Tournaments WINE 2011 Manipulating Tournaments Manipulating Stochastically Generated Single Elimination Tournaments for Nearly All Players Isabelle Stanton UC Berkeley Virginia Vassilevska Williams UC Berkeley & Stanford University

2. WINE 2011 Manipulating Tournaments Agenda Control In an election protocol, how much power does the election organizer have in affecting the outcome?

3. WINE 2011 Manipulating Tournaments Why does agenda control matter? Good mechanisms are good Our faith in outcomes shouldn’t rely on the morality of the organizer

4. WINE 2011 Manipulating Tournaments Computational Agenda Control Bartholdi, Tovey and Trick added the idea of computational complexity The organizer can always try brute force If it is NP-hard to manipulate, maybe we’re ok? If we can manipulate in polynomial time, the mechanism is definitely broken

5. WINE 2011 Manipulating Tournaments Single Elimination Tournaments (SETs)

6. WINE 2011 Manipulating Tournaments Agenda Control for SETs The organizer chooses the bracket, given the match outcomes Can not be manipulated – the grenade always wins

7. WINE 2011 Manipulating Tournaments Previous Work – Probabilistic Setting 51% 60% 70% 50% 60% Task: find a seeding of the teams maximizing the probability that favorite wins

8. WINE 2011 Manipulating Tournaments Previous Work – Probabilistic Setting 51% 60% 70% 50% 60%50% 40% For this seeding, wins with probability 20%.

9. WINE 2011 Manipulating Tournaments Previous Work – Probabilistic Setting 51% 60% 70% 50% 60% Task: find a seeding of the teams maximizing the probability that favorite wins [Lang. et al’07, Hazon et al.’08]: NP-hard to find a seeding that maximizes the probability that the favorite player will win [Vu et al.’08]: NP-hard even if the probabilities are 0, 100% or 50%

10. WINE 2011 Manipulating Tournaments Previous Work – Deterministic Setting We’ve shown that we can always manipulate in polynomial time for strong enough players [VW‘10], [S,VW‘11] Complexity of manipulation is unknown!

11. WINE 2011 Manipulating Tournaments Our Approach Model the average case Find sufficient combinatorial conditions Show these occur in the average case for many players

12. WINE 2011 Manipulating Tournaments Kings A king is a player who, for every other player, either beats them or beats a player who beats them Kings always exist in tournament graphs

13. WINE 2011 Manipulating Tournaments VW’10 Result - Kings

14. WINE 2011 Manipulating Tournaments Proof Technique Use recursion 1) Find a maximal matching from the Jacks to the Twos 2) Find an arbitrary matching of remaining Jacks 3) Find an arbitrary matching of remaining Twos 4) Make this matching Round 1. The King is still a king who beats half the remaining graph. Repeat until only the King remains

15. WINE 2011 Manipulating Tournaments Our New Result

16. WINE 2011 Manipulating Tournaments Proof Technique 1) Find a maximal matching from the Jacks to the Aces 2) Find a maximal matching of remaining Jacks to the Twos 3) Find an arbitrary matching of remaining Twos+Aces and of the remaining Jacks Aces are the stronger players

17. WINE 2011 Manipulating Tournaments Our New Result

18. WINE 2011 Manipulating Tournaments Condorcet-Random Model A natural model for real noisy tournaments!

19. WINE 2011 Manipulating Tournaments Condorcet-Random Model Question: What can we say about manipulation in the Condorcet-Random Model as a function of p?

20. WINE 2011 Manipulating Tournaments Previous CR Results

21. WINE 2011 Manipulating Tournaments Solution! When N = 512, p is 0.19, when N = 8192, p is 0.02…

22. WINE 2011 Manipulating Tournaments Further Results

23. WINE 2011 Manipulating Tournaments How? [Erdös & Rényi’64] says we have perfect matchings with high probability. Recursively apply for your favorite player 1 2 3 Round 1 Round 2

24. WINE 2011 Manipulating Tournaments Summary We’ve identified some natural instances where we can manipulate easily We’ve shown these instances often appear in natural tournament models These results hint that, if manipulation is NP- hard, the difficult case is the weak players