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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?


Why does agenda control matter?

• Good mechanisms are good• Our faith in outcomes shouldn’t rely on the

morality of the organizer


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


Single Elimination Tournaments (SETs)


Agenda Control for SETs

• The organizer chooses the bracket, given the match outcomes

Can not be manipulated – the grenade always wins


Previous Work – Probabilistic Setting51%

60%

70%

50% 50%

60%

Task: find a seeding of the teams maximizing the probability that favorite wins



60%

70%

50% 50%

60% 50%

40%

For this seeding, wins with probability 20%.



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%


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!


Our Approach

• Model the average case• Find sufficient combinatorial conditions• Show these occur in the average case for

many players


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


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.


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


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


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


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.


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!


Condorcet-Random Model

Question: What can we say about manipulation in the Condorcet-Random

Model as a function of p?


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!


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


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.


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


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

wine 2011 manipulating tournaments wine 2011 manipulating tournaments manipulating stochastically...

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