WINE 2011Manipulating Tournaments WINE 2011Manipulating Tournaments
Manipulating Stochastically Generated Single Elimination
Tournaments for Nearly All Players
Isabelle StantonUC Berkeley
Virginia Vassilevska WilliamsUC Berkeley & Stanford University
WINE 2011Manipulating Tournaments
Agenda Control
• In an election protocol, how much power does the election organizer have in affecting the outcome?
WINE 2011Manipulating Tournaments
Why does agenda control matter?
• Good mechanisms are good• Our faith in outcomes shouldn’t rely on the
morality of the organizer
WINE 2011Manipulating 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
WINE 2011Manipulating Tournaments
Single Elimination Tournaments (SETs)
WINE 2011Manipulating Tournaments
Agenda Control for SETs
• The organizer chooses the bracket, given the match outcomes
Can not be manipulated – the grenade always wins
WINE 2011Manipulating Tournaments
Previous Work – Probabilistic Setting51%
60%
70%
50% 50%
60%
Task: find a seeding of the teams maximizing the probability that favorite wins
WINE 2011Manipulating Tournaments
Previous Work – Probabilistic Setting51%
60%
70%
50% 50%
60% 50%
40%
For this seeding, wins with probability 20%.
WINE 2011Manipulating Tournaments
Previous Work – Probabilistic Setting51%
60%
70%
50% 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%
WINE 2011Manipulating 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!
WINE 2011Manipulating Tournaments
Our Approach
• Model the average case• Find sufficient combinatorial conditions• Show these occur in the average case for
many players
WINE 2011Manipulating 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
WINE 2011Manipulating Tournaments
VW’10 Result - Kings
• If a king beats:– at least players, or– at least as many players as any player that loses
to,
then one can efficiently find a seeding for a SET that wins.
WINE 2011Manipulating Tournaments
Proof Technique
• Use recursion 1) Find a maximal matching from the Jacks to the Twos2) Find an arbitrary matching of remaining Jacks3) Find an arbitrary matching of remaining Twos4) Make this matching Round 1. The King is still a king who beats half the remaining graph. Repeat until only the King remains
≥𝑛2
WINE 2011Manipulating Tournaments
Our New Result
• : the number of players who beat and who beat more players than beats.
• If beats at least players, then we can always find a seeding t wins
• Strictly stronger result than beating • Counterexamples for beating only
WINE 2011Manipulating Tournaments
Proof Technique1) Find a maximal matching from the Jacks to the Aces
2) Find a maximal matching of remaining Jacks to the Twos3) Find an arbitrary matching of remaining Twos+Aces and of the remaining Jacks4) Make this matching Round 1. The King is still a king who beats at least players in the remaining graph. Repeat until only the King remains
Aces are the stronger players
WINE 2011Manipulating Tournaments
Our New Result
• Corollary: If a player is beaten by at most stronger players, then it can win an SET.
• Corollary: If a player ranked in the top third is a king, then it can win an SET.
WINE 2011Manipulating Tournaments
Condorcet-Random Model
• Assume an inherent underlying ranking of the players:
• With no noise, the higher ranked player always beats the lower ranked player
• In reality, we have upsets. Add a noise parameter, , such that the lower ranked player wins with probability p
A natural model for real noisy tournaments!
WINE 2011Manipulating Tournaments
Condorcet-Random Model
Question: What can we say about manipulation in the Condorcet-Random
Model as a function of p?
WINE 2011Manipulating Tournaments
Previous CR Results
• VVW’10: When we can manipulate almost every tournament generated for ALL players
• Problem: n = 512 implies p > .5303 • Not a valid parameter for the model!
WINE 2011Manipulating Tournaments
Solution!
• We can show that, for lower , many players almost always satisfy our king condition.
• We show: If , then
the top players can be made SET winners.
When N = 512, p is 0.19, when N = 8192, p is 0.02…
Why
WINE 2011Manipulating Tournaments
Further Results
• We can trade off the noise to increase the number of players who can win
If , then the top players
can be made SET winners.
WINE 2011Manipulating Tournaments
How?
• [Erdös & Rényi’64] says we have perfect matchings with high probability. Recursively apply for your favorite player
1
23
𝑛2 𝑛
𝑛2+1
𝑛2+2
𝑛2+3
Round 1
𝑛
𝑛2+1
𝑛2+2
𝑛2+3
3𝑛4
3𝑛4
+1
3𝑛4
+2
3𝑛4
+3
Round 2
WINE 2011Manipulating 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