cooperation, power and conspiracies yoram bachrach
TRANSCRIPT
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
95m4
1995m
919
96m2
1996m
719
96m12
1997m
519
97m10
1998m
319
98m8
1999m
119
99m6
1999m
1120
00m4
2000m
920
01m2
2001m
720
01m12
2002m
520
02m10
2003m
320
03m8
2004m
120
04m6
2004m
1120
05m4
2005m
920
06m2
2006m
720
06m12
2007m
520
07m10
2008m
320
08m8
2009m
120
09m6
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 10
5 7 9
2 4 6
3 8 10
5 7 9
2 4 6
3 8 10
5 7 9
2 4 6
3 8 10
5 7 9
2 4 6
3 8 10
5 7 9
2 4 6
3 8 10
5 7 9
2 4 6
Definition VCG rule Property
Optimal according to reports Allocation
Impact on others Payments
(Bachrach, AAMAS, 2010; Bachrach, Key, Zadimoghaddam, WINE 2010)
Multi-Unit Auctions
3 8 10
5 7 9
2 4 6
𝑝1=5+4=9
T=5
Multi-Unit Auctions
3 8 10
5 7 9
2 4 6 𝑝2=4+3=7T=5
Collusion in Auctions
1 8 8 10
8 1 1 9
8 1 1 9
0 1 1 1
T=3
Collusion in Auctions
3 1 1 1 9
3 1 1 1 9
2 2 3 4 10
T=4
Collusion in Auctions
5 0 0 9 9
0 0 0 0 0
0 2 3 4 10
T=4Optimal scheme for diminishing marginals:
Proxy agent bids for all colluders
The Collusion Game
1 8 8 10
8 1 1 9
8 1 1 9
0 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