Algorithmic Mechanism Design

1
Algorithmic Issues in Strategic Distributed
Systems
2
Suggested readings

• Algorithmic Game Theory, Edited by Noam Nisan,
  Tim Roughgarden, Eva Tardos, and Vijay V.
  Vazirani, Cambridge University Press.
• Algorithmic Mechanism Design for Network
  Optimization Problems, Luciano Gualà, PhD Thesis,
  Università degli Studi dellAquila, 2007.
• Web pages by Éva Tardos, Christos Papadimitriou,
  Tim Roughgarden, and then follow the links
  therein

3
Two Research Traditions

• Theory of Algorithms computational issues
• What can be feasibly computed?
• How long does it take to compute a solution?
• Which is the quality of a computed solution?
• Centralized or distributed computational models
• Game Theory interaction between self-interested
  individuals
• What is the outcome of the interaction?
• Which social goals are compatible with
  selfishness?

4
Different Assumptions

• Theory of Algorithms (in distributed systems)
• Processors are obedient, faulty, or adversarial
• Large systems, limited computational resources
• Game Theory
• Players are strategic (selfish)
• Small systems, unlimited computational resources

5
The Internet World

• Agents often autonomous (users)
• Users have their own individual goals
• Network components owned by providers
• Internet scale
• Massive systems
• Limited communication/computational resources
• ? Both strategic and computational issues!

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Fundamental question

• How the computational aspects of a strategic
  (i.e., non-cooperative) distributed system can be
  addressed?

Theory of Algorithms
Game Theory
Algorithmic Game Theory


7
Basics of Game Theory

• A game consists of
• A set of players
• A set of rules of encounter Who should act when,
  and what are the possible actions (strategies)
• A specification of payoffs for each combination
  of strategies
• A set of outcomes
• Game Theory attempts to predict the outcome of
  the game (solution) by taking into account the
  individual behavior of the players (agents)

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Equilibrium

• Among the possible outcomes of a game, equilibria
  play a fundamental role.
• Informally, an equilibrium is a strategy
  combination in which individuals are not willing
  to change their state.
• When a player does not want to change his state?
  In the Homo Economicus model, when he has
  selected a strategy that maximizes his individual
  payoff, knowing that other players are also doing
  the same.

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Roadmap

• Equilibria Dominant Strategy Equilibrium (DSE)
  versus Nash Equilibrium (NE)
• Computational Aspects of Nash Equilibria
• Does a NE always exist?
• Can a NE be feasibly computed, once it exists?
• What about the quality of a NE?
• Case study Selfish Routing in Internet, Network
  Connection Games
• (Algorithmic) Mechanism Design
• Which social goals can be (efficiently)
  implemented in a non-cooperative distributed
  system?
• Strategy-proof mechanisms VCG-mechanisms
• Case study Mechanism design for some basic
  network design problems (shortest paths and
  minimum spanning tree)

10

• FIRST PART
• (Nash)
• Equilibria

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(Some) Types of games

• Collusive versus non-collusive
• Symmetric versus asymmetric
• Zero sum versus non-zero sum
• Simultaneous versus sequential
• Perfect information versus imperfect information
• One-shot versus repeated

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Normal-Form games
We are interested in simultaneous,
perfect-information interactions between a set of
agents. These are the so-called normalform (or
strategic) games

• N rational players
• For each player i, a strategy set Si
• Payoff the strategy combination (s1, s2, , sN),
  where si?Si, gives payoff (p1, p2, , pN) to the
  N players
• ? S1?S2 ? ? SN payoff matrix

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A famous game the Prisoners Dilemma
Non-collusive, symmetric, non-zero sum,
simultaneous, perfect information, one-shot,
2-player game
Strategy Set
Prisoner I Prisoner II Prisoner II Prisoner II
Prisoner I Dont Implicate Implicate
Prisoner I Dont Implicate 1, 1 6, 0
Prisoner I Implicate 0, 6 5, 5
Payoffs
Strategy Set
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Prisoner Is decision

• Prisoner Is decision
• If II chooses Dont Implicate then it is best to
  Implicate
• If II chooses Implicate then it is best to
  Implicate
• It is best to Implicate for I, regardless of what
  II does Dominant Strategy

Prisoner I Prisoner II Prisoner II Prisoner II
Prisoner I Dont Implicate Implicate
Prisoner I Dont Implicate 1, 1 6, 0
Prisoner I Implicate 0, 6 5, 5
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Prisoner IIs decision

• Prisoner IIs decision
• If I chooses Dont Implicate then it is best to
  Implicate
• If I chooses Implicate then it is best to
  Implicate
• It is best to Implicate for II, regardless of
  what I does Dominant Strategy

Prisoner I Prisoner II Prisoner II Prisoner II
Prisoner I Dont Implicate Implicate
Prisoner I Dont Implicate 1, 1 6, 0
Prisoner I Implicate 0, 6 5, 5
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Hence
Prisoner I Prisoner II Prisoner II Prisoner II
Prisoner I Dont Implicate Implicate
Prisoner I Dont Implicate 1, 1 6, 0
Prisoner I Implicate 0, 6 5, 5

• It is best for both to implicate regardless of
  what the other one does
• Implicate is a Dominant Strategy for both
• (Implicate, Implicate) becomes the Dominant
  Strategy Equilibrium
• Note If they might collude, then its beneficial
  for both to Not Implicate, but its not an
  equilibrium as both have incentive to deviate

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Dominant Strategy Equilibrium

• Dominant Strategy Equilibrium is a strategy
  combination s (s1, s2, , sN), such that
  si is a dominant strategy for each i, namely,
  for each s (s1, s2, , si , , sN)
• pi (s1, s2, , si, , sN) pi (s1, s2, , si,
  , sN)
• Dominant Strategy is the best response to any
  strategy of other players
• It is good for agent as it needs not to
  deliberate about other agents strategies
• Of course, not all games (only very few in the
  practice!) have a dominant strategy equilibrium

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A more relaxed solution concept Nash
Equilibrium 1951

• Nash Equilibrium is a strategy combination
• s (s1, s2, , sN) such that for each i, si
  is a best response to (s1, ,si-1,si1,,
  sN), namely, for any possible alternative
  strategy si
• pi (s) pi (s1, s2, , si, , sN)

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Nash Equilibrium

• In a NE no agent can unilaterally deviate from
  his strategy given others strategies as fixed
• Agent has to deliberate about the strategies of
  the other agents
• If the game is played repeatedly and players
  converge to a solution, then it has to be a NE
• Dominant Strategy Equilibrium ? Nash Equilibrium
  (but the converse is not true)

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Nash Equilibrium The Battle of the Sexes
(coordination game)
Man Woman Woman Woman
Man Stadium Cinema
Man Stadium 2, 1 0, 0
Man Cinema 0, 0 1, 2

• (Stadium, Stadium) is a NE Best responses to
  each other
• (Cinema, Cinema) is a NE Best responses to each
  other
• ? but they are not Dominant Strategy Equilibria
  are we really sure they will eventually go out
  together????

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A crucial issue in game theory the existence of
a NE

• Unfortunately, for pure strategies games (as
  those seen so far, in which each player, for each
  possible situation of the game, selects her/his
  action deterministically), it is easy to see that
  we cannot have a general result of existence
• In other words, there may be no, one, or many NE,
  depending on the game

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A conflictual game Head or Tail
Player I Player II Player II Player II
Player I Head Tail
Player I Head 1,-1 -1,1
Player I Tail -1,1 1,-1

• Player I (row) prefers to do what Player II does,
  while Player II prefer to do the opposite of what
  Player I does!
• ? In any configuration, one of the players
  prefers to change his strategy, and so on and so
  forththus, there are no NE!

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On the existence of a NE

• However, when a player can select his strategy
  randomly by using a probability distribution
  over his set of possible pure strategies (mixed
  strategy), then the following general result
  holds
• Theorem (Nash, 1951) Any game with a finite set
  of players and a finite set of strategies has a
  NE of mixed strategies (i.e., the expected payoff
  cannot be improved by changing unilaterally the
  selected probability distribution).
• Head or Tail game if each player sets
  p(Head)p(Tail)1/2, then the expected payoff of
  each player is 0, and this is a NE, since no
  player can improve on this by choosing a
  different randomization!

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Three big computational issues

1. Finding a NE, once it does exist
2. Establishing the quality of a NE, as compared to
   a cooperative system, namely a system in which
   agents can collude (recall the Prisoners
   Dilemma)
3. In a repeated game, establishing whether and in
   how many steps the system will eventually
   converge to a NE (recall the Battle of the Sexes)

25
On the computability of a NEfor pure strategies

• By definition, it is easy to see that an entry
  (p1,,pN) of the payoff matrix is a NE if and
  only if pi is the maximum ith element of the row
  (p1,,pi-1, p(s)s?Si ,pi1,,pN), for each
  i1,,N.
• Notice that, with N players, an explicit (i.e.,
  in normal-form) representation of the payoff
  functions is exponential in N ? brute-force
  (i.e., enumerative) search for pure NE is then
  exponential in the number of players (even if it
  is still polynomial in the input size, but the
  normal-form representation needs not be a
  minimal-space representation of the input!)
• ? Alternative cheaper methods are sought (we will
  see that for some categories of games of our
  interest, a NE can be found in poly-time w.r.t.
  to the number of players)

26
On the quality of a NE

• How inefficient is a NE in comparison to an
  idealized situation in which the players would
  strive to collaborate selflessly with the common
  goal of maximizing the social welfare?
• Recall in the Prisoners Dilemma game, the DSE ?
  NE means a total of 10 years in jail for the
  players. However, if they would not implicate
  reciprocally, then they would stay a total of
  only 2 years in jail!

27
The price of anarchy

• Definition (Koutsopias Papadimitriou, 1999)
  Given a game G and a social-choice minimization
  (resp., maximization) function f (i.e., the sum
  of all players payoffs), let S be the set of NE,
  and let OPT be the outcome of G optimizing f.
  Then, the Price of Anarchy (PoA) of G w.r.t. f
  is
• Example in the PD game, ?G(f)10/25

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A case study for the existence and quality of a
NE selfish routing on Internet

• Internet components are made up of heterogeneous
  nodes and links, and the network architecture is
  open-based and dynamic
• Internet users behave selfishly they generate
  traffic, and their only goal is to
  download/upload data as fast as possible!
• But the more a link is used, the more is slower,
  and there is no central authority optimizing
  the data flow
• So, why does Internet eventually work is such a
  jungle???

29
Modelling the flow problem

• Internet can be modelled by using game theory it
  is a congestion game in which
• players users
• strategies paths
  over which users can route their traffic
• Non-atomic Selfish Routing
• There is a large number of (selfish) users
• All the traffic of a user is routed over a single
  path simultaneously
• Every user controls a tiny fraction of the
  traffic.

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Mathematical model

• A directed graph G (V,E)
• A set of sourcesink pairs si,ti for i1,..,k
• A rate ri ? 0 of traffic between si and ti for
  each i1,..,k
• A set Pi of paths between si and ti for each
  i1,..,k
• The set of all paths ?Ui1,,k Pi
• A flow vector f specifying a traffic routing
• fP amount of traffic routed on si-ti path P
• A flow is feasible if for every i1,..,k
• ?P?Pi fP ri

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Mathematical model (2)

• For each e?E, the amount of flow absorbed by e
  is
• fe?Pe?P fP
• For each edge e, a latency function le(fe)
• Cost or total latency of a path P
• C(P)?e?Pfe le(fe)
• Cost or total latency of a flow f (social
  welfare)
• C(f)?P?? C(P)

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Flows and NE

• Definition A flow f is a Nash flow if no agent
  can improve its latency by changing unilaterally
  its path.
• QUESTION Given an instance (G,r,l) of the
  non-atomic selfish routing game, does it admit a
  Nash flow? And in the positive case, what is the
  PoA of such Nash flow?

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Example Pigous game 1920

• Latency depends on the congestion (x is the
  fraction of flow using the edge)

Rate 1
s
t
Latency is fixed

• What is the NE of this game? Trivial all the
  fraction of flow tends to travel on the upper
  edge ? the cost of the flow is 11 01 1
• What is the PoA of this NE? The optimal solution
  is the minimum of C(x)xx (1-x)1 ? C (x)2x-1
  ? OPT1/2 ? C(OPT)1/21/2(1-1/2)10.75 ? ?G(C)
  1/0.75 4/3

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The Braesss paradox

• Does it help adding edges to improve the PoA?
• NO! Lets have a look at the Braess Paradox (1968)

Latency of each path 0.50.50.51 0.75
v
1
x
1/2
s
t
1/2
x
1
Latency of the flow 20.751.5 (notice this is
optimal)
w
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The Braesss paradox (2)
To reduce the latency of the flow, we try to add
a no-latency road between v and w. Intuitively,
this should not worse things!
v
1
x
0
s
t
x
1
w
36
The Braesss paradox (3)
However, each user is incentived to change its
route now, since the route s?v?w?t has less
latency (indeed, x1)
If only a single user changes its route, then its
latency decreases approximately to 0.5.
v
x
1
0
s
t
x
1
w
But the problem is that all the users will decide
to change!
37
The Braesss paradox (4)

• So, the new latency of the flow is now
• 1110112gt1.5
• Even worse, this is a NE!
• The optimal min-latency flow is equal to that we
  had before adding the new road! So, the PoA is

Notice 4/3, as in the Pigous example
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Existence of a Nash flow

• Theorem (Beckmann et al., 1956) If xl(x) is
  convex and continuously differentiable, then the
  Nash flow of (G,r,l(x)) exists and is unique, and
  is equal to the optimal min-latency flow of the
  following instance
• (G,r, ?(x)?0 l(t)dt/x).
• Remark The optimal min-latency flow can be
  computed in polynomial time through convex
  programming methods.

x
39
Flows and Price of Anarchy

• Theorem 1 In a network with linear latency
  functions, the cost of a Nash flow is at most 4/3
  times that of the minimum-latency flow.
• Theorem 2 In a network with general latency
  functions, the cost of a Nash flow is at most n/2
  times that of the minimum-latency flow.
• (Roughgarden Tardos, JACM02)

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A bad example for non-linear latencies

• Assume igtgt1

xi
1-?
1
s
t
1
?
0
A Nash flow (of cost 1) is arbitrarily more
expensive than the optimal flow (of cost close to
0)
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Convergence towards a NE(in pure strategies
games)

• Ok, we know that selfish routing is not so bad at
  its NE, but are we really sure this point of
  equilibrium will be eventually reached?
• Convergence Time number of moves made by the
  players to reach a NE from an initial arbitrary
  state
• Question Is the convergence time (polynomially)
  bounded in the number of players?

42
The potential function method

• (Rough) Definition A potential function for a
  game (if any) is a real-valued function, defined
  on the set of possible outcomes of the game, such
  that the equilibria of the game are precisely the
  local optima of the potential function.
• Theorem In any finite game admitting a potential
  function, best response dynamics always converge
  to a NE of pure strategies.
• But how many steps are needed to reach a NE? It
  depends on the combinatorial structure of the
  players' strategy space

43
Convergence towards the Nash flow

• ? Positive result The non-atomic selfish routing
  game is a potential game (and moreover, for many
  instances, the convergence time is polynomial).
• ? Negative result However, there exist instances
  of the non-atomic selfish routing game for which
  the convergence time is exponential (under some
  mild assumptions).