Algorithmic and game theoretic aspects
of resource allocation problems, social
networks and biological networks
Vangelis Markakis
Athens University of Economics and Business
1
My Research Interests
• Theoretical Computer Science
– Analysis of algorithms
– Approximation algorithms
• Game Theory, Microeconomics, Social Choice
Theory (voting)
– Game theoretic analysis (what happens when
everybody cares only for his own interests?)
– Mechanism Design
– Algorithmic aspects
2
Multi-Agent Systems
• Our focus: Interactions of economic agents
• Motivation:
– E-commerce: Auctions (FCC, eBay, Google Adwords)
– Internet: Network design/Cost sharing, selfish routing,
social networks
– …
3
Outline
1. The concept of Nash equilibrium - Algorithms
for finding approximate Nash equilibria
2. Fair division problems (cake-cutting)
3. Sponsored Search Auctions
4. Social (and biological) Networks
5. Voting protocols
4
What is Game Theory?
• Game Theory aims to help us understand
situations in which decision makers interact
• Goals:
– Mathematical models for capturing the properties of
such interactions
– Prediction (given a model how should/would a rational
agent act?)
Rational agent: when given a choice, the agent always
chooses the option that yields the highest utility
5
Models of Games
Games can be:
• Cooperative or noncooperative
• Repeated or 1-shot
• Simultaneous moves or sequential
• Finite or infinite
• Complete information or incomplete information
6
Noncooperative Games in Normal Form
The Hawk-Dove
game
Column Player
2, 2
0, 4
4, 0
-1, -1
7
Definitions
• 2-player game (R, C):
• n available pure strategies for each player
• n x n payoff matrices R, C
• i, j played  payoffs : Rij , Cij
• Mixed strategy: Probability distribution over [n]
• Expected payoffs :
8
Solution Concept
x*, y* is a Nash equilibrium if no player has a unilateral
incentive to deviate:
(x, Ry*)  (x*, Ry*)  x
(x*, Cy)  (x*, Cy*)  y
[Nash, 1951]: Every finite game has a mixed strategy
equilibrium.
(think of it as a steady state)
Proof: Based on Brouwer’s fixed point theorem.
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Solution Concept
x*, y* is a Nash equilibrium if no player has a unilateral
incentive to deviate:
(x, Ry*)  (x*, Ry*)  x
(x*, Cy)  (x*, Cy*)  y
[Nash, 1951]: Every finite game has a mixed strategy
equilibrium.
(think of it as a steady state)
Proof: Based on Brouwer’s fixed point theorem.
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Example: The Hawk-Dove Game
Column Player
2, 2
0, 4
4, 0
-1, -1
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Complexity issues

m = 2 players, known algorithms: worst case exponential time
[Kuhn ’61, Lemke, Howson ’64, Mangasarian ’64, Lemke ’65]


If NP-hard  NP = co-NP [Megiddo, Papadimitriou ’89]
NP-hard if we add more constraints (e.g. maximize sum of payoffs)
[Gilboa, Zemel ’89, Conitzer, Sandholm ’03]

Representation problems
m = 3, there exist games with rational data BUT irrational equilibria
[Nash ’51]

PPAD-complete even for m = 2
[Daskalakis, Goldberg, Papadimitriou ’06, Chen, Deng, Teng ’06]
Poly-time equivalent to:

finding approximate fixed points of continuous maps on convex and
compact domains
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Approximate Nash Equilibria
• Recall definition of Nash eq. :
(x, Ry*)  (x*, Ry*)  x
(x*, Cy)  (x*, Cy*)  y
• -Nash equilibria (incentive to deviate  ) :
(x, Ry*)  (x*, Ry*) +   x
(x*, Cy)  (x*, Cy*) +   y
Normalization: entries of R, C in [0,1]
13
Subexponential Algorithms
Lipton, Markakis, Mehta ’03: For any  in (0,1), We can
compute an -Nash equilibrium in time
• Applicable also to games with more than 2 players
Tsaknakis, Spirakis ’10: Yet another subexponential
algorithm, complexity depends on eigenvalues of matrices
Question: For what values of ε can we have polynomial
time algorithms?
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Polynomial Time Approximation
Algorithms
Daskalakis, Mehta, Papadimitriou ’07:
in P for  = 1-1/φ = (3-5)/2  0.382 (φ = golden ratio)
- Βased on sampling + Linear Programming
- Running time: need to solve polynomial number
of linear programs
Bosse, Byrka, Markakis (’07 and ’10): a different LP-based
method
1. Algorithm 1: 1-1/φ
2. Algorithm 2: 0.364
Running time: need to solve one linear program
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Polynomial Time Approximation
Algorithms
Spirakis, Tsaknakis (’07 and ’10): 0.339
- Yet another LP-based method
- Running time: need to solve polynomial number
of linear programs
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Remarks and Open Problems
• Is there a polynomial time algorithm for any constant  (PTAS)?
Yes if:
– rank(R) = O(1) & rank(C) = O(1) [Lipton, Markakis, Mehta ’03]
– rank(R+C) = O(1) [Kannan, Theobald ’06]
– anonymous games [Daskalakis, Papadimitriou ’07]
• PPAD-complete for  = 1/n [Chen, Deng, Teng ’06]
• Other notions of approximation:
– -well-supported equilibria (every strategy in the support is an
approximate best response) [Kontogiannis, Spirakis ’07]
– Strong approximation (output is geometrically close to an exact Nash
equilibrium) [Etessami, Yannakakis ’07]
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Outline
1.
The concept of Nash equilibrium - Algorithms for
finding approximate Nash equilibria
2.
Fair division problems (cake-cutting)
3.
Sponsored Search Auctions
4.
Social Networks
5.
Voting protocols
18
Cake-cutting problems
Divide the cake among a set of people in a fair manner
Empirically: since Pharaoh times (land division)
Mathematical approaches:
[Steinhaus, Banach, Knaster ’48]
Fairness measure: Envy,
proportionality, and many others
[Foley ’67, Varian ’74]
Infinitely divisible cakes:
Envy-free partitions exist
Cake-cutting procedures: minimize # cuts, achieve
additional fairness criteria [Brams, Taylor ’96,
Robertson, Webb ’98]
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Set of agents
K = {1, 2, …, k}
Discrete version
Set of indivisible goods
M = {1, 2, …, m}
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Indivisible Goods
Envy-free allocations may not exist
Goal: Produce allocations that are approximately fair
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Indivisible Goods
Envy-free allocations may not exist
Goal: Produce allocations that are approximately fair
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Model
For agent p: utility function
:


(monotone)
Special cases:
 Additive utilities
 Same utility for every agent.
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What is fair?
– Proportionality [Steinhaus - Banach - Knaster ’48]
– Envy-freeness [Foley ’67, Varian ‘74]
– Max-min fairness [Dubins - Spanier ’61]
– Equitability
– …..
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Fairness Concepts
Given an allocation A = (A1,…,Ak):
Envy of p for q:
Envy of A:
 Envy-free allocations may not exist
 Finding allocations with minimum possible envy is
an NP-hard problem
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Approximation Algorithms
How close to optimal can we get in polynomial time?
Definition: An algorithm A, for a minimization problem ,
achieves an approximation factor of  (  1), if for every
instance I of , the solution returned by A satisfies:
SOL(I)   OPT(I)
Similarly we can define additive approximation factors
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Some Results
[Lipton, Markakis, Mossel, Saberi ’04]: Approximation
algorithms for the problem of minimizing envy or envy-ratio for
various families of utility functions
[Markakis, Psomas ’11]: Algorithms for producing approximately
proportional allocations for additive utilities
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Outline
1.
The concept of Nash equilibrium - Algorithms for
finding approximate Nash equilibria
2.
Fair division problems (cake-cutting)
3.
Sponsored Search Auctions
4.
Social Networks
5.
Voting protocols
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What is sponsored search?
Advertising slots
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What is sponsored search?
Advertising slots
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The Model of Sponsored Search
• For a fixed search term (e.g. ipod)
– n advertisers
– k slots (typically k << n)
– An auction is run for every single search
– Each advertiser (bidder) has a private valuation/utility for
occupying one of the slots
– Same auction is run for related keywords (e.g. “buy ipod”,
“cheap ipod”, “ipod purchase”, …)
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The Model of Sponsored Search (2)
– Bidders submit an initial budget which they can refresh weekly
or monthly
– Bidders also submit an initial bid which they can adjust as often
as they wish
– Different charging models exist: Pay Per Click, Pay Per View,
Pay Per Transaction
– Currently, most popular is Pay Per Click
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The Actors
• The Search engine:
– Wants to make as much revenue as possible
– At the same time, wants to make sure users receive
meaningful ads and bidders do not feel that they were
overcharged
• The Bidders:
– Want to occupy a high slot and pay as little as possible
• The Searchers:
– Want to find the most relevant ads with respect to what
they are looking for
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The Generalized Second Price
Mechanism (GSP)
• The initial version:
– Each bidder submits a bid
– The search engine ranks them in decreasing order: b1 
b2  …  bn
– The first k bidders are displayed in the k slots
– Every time there is a click on slot i, bidder i pays bi+1
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The Generalized Second Price
Mechanism (GSP)
• The currently used version:
– Each bidder submits a bid
– The search engine keeps a quality score qi for each bidder i
• Yahoo, Bing: qi is the click-through rate of i
• Google: qi depends on click-through rate, relevance of text and other
factors
– The search engine ranking is in decreasing order of qi  bi
q1b1  q2b2  …  qnbn
– The first k bidders of the ranking are displayed in the k slots
– Every time there is a click on slot i, bidder i pays minimum
bid required to keep his position, i.e. (qi+1bi+1)/ qi
35
Some questions we have been
studying…
• [Gomes, Immorlica, Markakis ’09]:
– Current models assume click-through rates do not depend on the
presence of other bidders (competitors)
– We have proposed a model to capture externalities among bidders
– Validation with real data from MSN Live
– How are the Nash equilibria of the GSP mechanism affected by
externality effects?
• 2 M.Sc. Theses supervised jointly with G. D. Stamoulis
– [Dimitris Drosos, July 2009]
• Given the current form of the GSP mechanism, how should advertisers
bid? Some suggestions: [Cary et al ’07]: Balanced Bidding
• We have proposed an alternative bidding strategy that makes more careful
use of the available budget
– [Maria Sotiri, February 2011]
• Analysis of real data (available from Yahoo)
• Exploring further bidding strategies that make local improvements
36
Outline
1.
The concept of Nash equilibrium - Algorithms for
finding approximate Nash equilibria
2.
Fair division problems (cake-cutting)
3.
Sponsored Search Auctions
4.
Social Networks
5.
Voting protocols
37
What is a social network?
Any graph in which each
edge between two nodes has
a social meaning
(friendship, relationship,
coauthorship,…)
Examples:
• facebook
• e-mail graph
• couchsurfing
• myspace
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Marketing over Social Networks
Suppose a company wants to push a new product in the market
(e.g. Apple is releasing a new ipod)
Suppose also that Apple plans to give a few ipods for free as
part of the advertising campaign
What strategy should Apple use to decide who gets a free ipod?
What if there are other companies who want to promote their
competing products to the network users?
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Modeling the Spread of Influence
Main assumption: Clients are influenced only by their friends,
i.e., their neighbors in the social network
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There exist various models in the literature on how nodes
influence each other
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The Threshold Model
Granovetter ’78: Person i buys a product when the number of his
friends who bought it exceeds a certain threshold θ(i)
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Example 1
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =60%
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Example 1
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =60%
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Example 1
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =60%
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Example 1
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =60%
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Example 1
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =60%
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• Final network (w.r.t. the initial one): No further adoption possible
• In this case a unique outcome of the adoption process
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Example 1 - Part 2
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =40%
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Example 1 - Part 2
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =40%
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Example 1 - Part 2
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =40%
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Example 1 - Part 2
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =40%
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Example 1 - Part 2
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =40%
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Example 1 - Part 2
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =40%
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Example 1 - Part 2
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =40%
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Example 1 - Part 2
• all weights = 1/N(i)
If N(i)  :
• p(i) = {R, G}
• θ(i) =40%
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• A reachable outcome though not unavoidable
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Optimization Questions
Problem 1: [finding the optimal budget]
Find a set of nodes S of minimum cardinality such that by giving
the ipod to S, it spreads to the whole network (or to a fixed
fraction of the network)
Problem 2: [finding optimal allocations under fixed budget]
If we are allowed to give only k free ipods, find a set of k nodes
such that maximum spread is achieved by giving the ipod to these k
nodes
55
Game Theoretic Questions
Problem 3: [Analysis of Nash equilibria]
Suppose we view this as a game, where there are many companies
promoting different products. The strategy of each company is to
select a set of nodes to give the product for free. What do the Nash
equilibria look like in this game? [Apt, Markakis ’11]
Problem 4: [Learning to play Nash equilibria]
Are there natural dynamics that lead players to a Nash equilibrium?
How fast do they converge?
56
Biological Networks
• Many types of biological networks
– Metabolic networks: interactions between enzymes and
metabolites
– Protein-protein interactions: Interactions between
proteins are important for the majority of biological
functions
– Gene-regulatory networks: interactions between DNA
segments
• Available in many databases, e.g. KEGG (Kyoto
Encyclopedia of Genes and Genomes)
57
Biological Networks
• Questions of interest:
• Structure of biological networks
– degree distribution, clustering coefficient, …
– Many of them exhibit a similar structure with
social networks
• Control nodes
– Finding sets of nodes that can control the whole
network
58
Outline
1.
The concept of Nash equilibrium - Algorithms for
finding approximate Nash equilibria
2.
Fair division problems (cake-cutting)
3.
Sponsored Search Auctions
4.
Social Networks
5.
Voting protocols
59
Approval voting
• Multi-winner elections
• Every voter selects a subset of candidates that he
approves of
• Used by numerous organizations (IEEE, Game Theory
Society, INFORMS,…) for selecting committees
Notation:
• n voters
• m candidates
• k: size of the committee to be elected
• Votes  elements of {0,1}m
60
Electing a committee from
approval ballots
m = 5 candidates
n = 4 ballots
What is the best
committee of size k = 2?
11001
11110
11000
00111
61
The minisum solution
Pick the committee that minimizes the sum of the Hamming
distances to the voters
 Pick the k candidates with the highest approval rate
11001
11110
1
2
sum = 8
max = 5
11000
0
11000
5
00111
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Maximum Hamming distance
The preferences of some voters may be completely ignored!
11001
11110
1
2
sum = 8
max = 5
11000
0
11000
5
00111
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The minimax solution
[Brams, Kilgour, Sanver ’07]: Pick the k candidates that
minimize the maximum Hamming distance to a voter
11001
11110
3
2
sum = 10
max = 3
10100
2
11000
3
00111
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Some difficulties…
[LeGrand, Markakis, Mehta ’07]:
• Computing a minimax solution is NP-hard
• Any algorithm that computes a minimax solution
is manipulable
 Resort to approximation algorithms
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Mechanism Design
3 notions of immunity to manipulation
• Strategyproof (SP): no voter can benefit by unilaterally
changing his vote, given any preference profile for the
remaining voters
• Group-strategyproof (GSP): no set of voters can all benefit
by changing their votes
• Strongly group-strategyproof (strongly GSP): no set of
voters can change their votes so that at least one of them
benefits and no-one is worse off
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Current Results
Caragiannis, Kalaitzis, Markakis ’10
Lower Bound
Upper Bound
Approx.
ratio
SP
NP-hard
2
2 – 2/(k+1)
3 – 2/(k+1)
GSP
2 – 2/(k+1)
3 – 2/(k+1)
Strongly
GSP
3 – 2/(k+1)

67
Future work
• What is the best approximation ratio achievable in polynomial
time?  PTAS?
–  PTAS for unrestricted version (no constraints on size of
committee)
• Characterization of (group) strategyproof algorithms
• Investigate weighted version of minimax [Brams, Kilgour &
Sanver, ’07]
• Other concepts in Approval Voting
– [Brams, Kilgour ’10]: Satisfaction Approval Voting (SAV):
Pick committee that maximizes the sum of satisfaction
scores of the voters
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Other Topics
• Combinatorial Auctions (auctions where each bidder
specifies bids for various combinations of bids)
• Cooperative Games (games that allow formation of
coalitions)
• Task allocation problems with temperature constraints
(scheduling), with I. Milis
69
Thank You!
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