Cooperation, Power and Conspiracies

Yoram Bachrach

High Level VisionArtificial Intelligence

John McCarthy: “making a machine behave in ways that would be called intelligent if a human we so behaving” (1955)

Coordinating NegotiatingStrategizing

Agenda

Cooperation in Game Theory

Manipulating Power

Collusion in Auctions

UK Elections 2010Conservatives Labour Lib-Dems306 258 57

Seats

ConservativesLabourLib-Dems

Required:326

Alternate Universe ElectionsConservatives Labour Liberals Democrats306 258 28 29

Seats

ConservativesLabourLiberalsDemocrats

Required:326

Treasure Island

$200 $1000

Coalition: C Value: v(C)

Cooperative Games

Cooperation Competition

Cannot achieve goal aloneCoordination

Maximize rewardsIncrease influence

Sharing Rewards

– Stable or Shaky?

– Is it Fair?• requires

• very valuable

$1000

p1 p2 p3

$50 $50 $900

Imputation: A payoff vector such that

Dummy agents Equivalent agentsGame composition

The Shapley Value

• Average contribution across all permutations𝜙𝑖 (𝑣 )= 1

𝑛 ! ∑𝜋∈Π [𝑣 (𝑠𝜋 (𝑖 )∪ {𝑖 } )−𝑣 (𝑠𝜋 (𝑖)¿)]¿

Before Including Contribution

$0 $1000 $1000

$0 $200 $200

𝑺𝝅 (𝒊)𝑺𝝅 (𝒊)

266.66 366.66 366.66

Weighted Voting Games• Agent has weight • Quota • A coalition C wins if • Shorthand: • A simple game

[Power Weight

Power in the UK Elections

• Game 1: [306, 258, 57; 326]

• Game 2: [306, 258, 28, 29; 326]

• Split makes the Labour less powerful– But the power goes to the Conservatives…– … not the Lib-Dems

Conservatives Labour Lib-Dems306 258 5766.66% 16.66% 16.66%

Conservatives Labour Liberals Democrats306 258 28 2975% 8.33% 8.33% 8.33%

Split Merge

False-Name Power ManipulationsA B

2 2

1/2 1/2

A B B’

2 1 1

1/3 1/3 1/3

q = 4

A B

2 2

1/2 1/2

A B B’

2 1 1

4/6 1/6 1/6

q = 3

Power Increase

Power Decrease

Effects of False-Name Manipulation

Manipulator loss bound An agent can decrease her power by a factor of . The bound is tight.

Hardness of manipulability It is a hard computational problem to test if a beneficial manipulation exists.

=?

Manipulation Gain Bound An agent can increase her power by a factor of . The bound is tight.

Quota manipulations: Bounds on quota perturbations influence on power. Hardness of testing which quota is better for a player’s power.

(Bachrach & Elkind, AAMAS 2008; Bachrach et al., AAAI 2008)

Manipulation Heuristics

Heuristic algorithm: try integer splits and approximate power. Tested on random weighted voting games.

95% Manipulabilit

y

(Bachrach et al., JAIR 2011)

Control in Firms19

95m419

95m919

96m219

96m719

96m1219

97m519

97m1019

98m319

98m819

99m119

99m619

99m11

2000m4

2000m9

2001m2

2001m7

2001m12

2002

m520

02m1020

03m320

03m820

04m120

04m620

04m11

2005m4

2005m9

2006m2

2006m7

2006m12

2007m

520

07m1020

08m320

08m820

09m1

2009m6

65

67

69

71

73

75

77

79

81

83

85

perc_controlled_SS1 perc_controlled_B05 perc_controlled_20 perc_controlled_SS_05

The “Rip-off” Game(Bachrach, Kohli, Graepel, AAMAS 2011)

AuctionsValuation / Auction

$900 $500 $400 $300

Sealed bid(1st price)

English (ascending)

Vickrey (2nd Price)

Speculations

$500+𝜖

$500+𝜖Long (increasing) bidding

Truthful bidding

$500

Truthful Efficient allocationVCG

Collusion

Collusion: an agreement between several agents to limit competition by manipulating or defrauding to obtain an unfair advantage

$900 $500 $400 $300

Truthful $900 $500 $400 $300

Collusion $900 $400 $400 $300

Sponsored Search AuctionsSelling advertisements on search engines.Tailored to users and search queries.

Model:

Key part of the online business model. Uses:

Google, Yahoo, Microsoft Key players:

Microsoft – $2 Billion/year (Bing ads)Google - $25 Billion/year (AdWords, AdSense)

Revenue:(Extrapolation, Q1 2010)

What Blocks Agreements?

$50 $50 $900

Value v(C) Payment p(C) Coalition

200<

The Core [Gillies 55’]: Unblocked agreements

p1 p2 p3

$1000

$200 $1000Potential Blockers:

Make sure get at least $200 (1,1,998)

Collusion in Auctions

3 8 105 7 92 4 6

3 8 105 7 92 4 6

3 8 105 7 92 4 6

3 8 105 7 92 4 6

3 8 105 7 92 4 6

3 8 105 7 92 4 6

Definition VCG rule PropertyOptimal according to reports Allocation

Impact on others Payments

(Bachrach, AAMAS, 2010; Bachrach, Key, Zadimoghaddam, WINE 2010)

Multi-Unit Auctions

3 8 105 7 92 4 6

𝑝1=5+4=9

T=5

Multi-Unit Auctions

3 8 105 7 92 4 6 𝑝2=4+3=7

T=5

Collusion in Auctions

1 8 8 108 1 1 98 1 1 90 1 1 1

T=3

Collusion in Auctions

3 1 1 1 93 1 1 1 92 2 3 4 10

T=4

Collusion in Auctions

5 0 0 9 90 0 0 0 00 2 3 4 10

T=4Optimal scheme for diminishing marginals:

Proxy agent bids for all colluders

The Collusion Game

1 8 8 108 1 1 98 1 1 90 1 1 1

T=3

Coalition: C Value:

v(C) = welfare under optimal collusion

Games with Diminishing Marginals

Fairness and Stability with diminishing marginals Always have non-empty cores (stable imputations). The Shapley value is in the core (fair and stable imputation).

Proof sketch:• Marginal contribution vectors

• Centroid is the Shapley value• Convex hull is the Weber set

• Contains the core• Weber set identifies with the core for convex games

• Adding an agent helps more for large coalitions

• The game is convex• Smaller coalitions incur higher payments for the additional player’s items• Denote j’s contribution to is • Show -)• Convexity: • Manipulations of marginal valuation vectors

C’

C

𝑺𝝅 (𝒊)

Non-Diminishing MarginalsCore Payment Marginals Number Type

(H,H,0,…,0) a=b+1 A(0,…,0) b B(1,1,1,L,0,…0), 1 C

Optimal Attack Members

Marginal merging attack (H,H,H,…,H,0), with 2a Hs. All A’s

Same as all A’s. A’s and B’s

False-name marginal splitting: both declare (H,0,0,…,0). (A,B) pair

Type B agents serve as a false-identityHelpful for single A agent, but not for a large set of A’s

Empty core – no stable agreement

2a+2 Items

Non-Diminishing Marginals

Collusion games with arbitrary marginal utility functions – polynomial algorithms: Computing the value (welfare) of a coalition.When all but few agents have identical valuations: compute Shapley value.When there are few valuation functions: test core emptiness.

Proof sketch:

• Coalition value: dynamic programs based on optimal collusion scheme for specific amounts of allocated items

• Core defined by an exponential LP over : • Can maintain a single variable for each agent type (core and equivalent agents)

• Constant number of variables• Coalition profile: number agents of each type

• Less than profiles• Constraint for coalition of profile

Collusion in Sponsored Search Auctions Collusion by advertisers Specific keyword market Top 3 advertiser bids for that keyword Appearances in “mainline” Jointly set bids once for the duration Simulate auction

Feature Change

Appearances (mainline) -3%

Clicks estimate -2%

Revenue -30%

High Level VisionArtificial Intelligence

John McCarthy: “making a machine behave in ways that would be called intelligent if a human we so behaving” (1955)

Coordinating NegotiatingStrategizing

Game Theory Heuristics &Data Analysis

Algorithms

Conclusion

Cooperation Competition

Big Challenges

Incorporating negotiation and agreement modelsUnderstanding human bounded-rational behaviour Designing efficient and attack-resistant mechanisms

Scaling up to real-world systems