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Algoritmi per Sistemi Distribuiti Strategici. Two Research Traditions. Theoretical Computer Science: computational complexity What can be feasibly computed? Centralized or distributed computational models Game Theory: interaction between self-interested individuals

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two research traditions
Two Research Traditions
  • Theoretical Computer Science: computational complexity
    • What can be feasibly computed?
    • Centralized or distributed computational models
  • Game Theory: interaction between self-interested individuals
    • Which social goals are compatible with selfishness?
    • What is the outcome of the interaction?
different assumptions
Different Assumptions
  • Theoretical Computer Science:
    • Processors are obedient, faulty, or adversarial.
    • Large systems, limited comp. resources
  • Game Theory:
    • Players are strategic(selfish).
    • Small systems, unlimited comp. resources
the internet world
The Internet World
  • Agents often autonomous (users)
    • They have their own individual goals
  • Often involve “Internet” scales
    • Massive systems
    • Limited communication/computational resources

 Both strategic and complexity matter!

fundamental questions
Fundamental Questions
  • What are the computational aspects of a game?
  • What does it mean to design an algorithm for a strategic distributed system (SDS)?

Theoretical

Computer

Science

SDS

Design

Game Theory

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+

game theory
Game Theory
  • Given a game, predict the outcome by analyzing the individual behavior of the players (agents)
  • Game:
    • N players
    • Rules of encounter: Who should act when, and what are the possible actions
    • Outcomes of the game
game theory1
Game Theory
  • Normal Form Games
    • N players
    • Si=Strategy set of player i
    • The strategy combination (s1, s2, …, sN) gives payoff (p1, p2, …, pN) to the N players
    • All the above information is known to all the players and it is common knowledge
    • Simultaneous move: each player i chooses a strategy siSi (nobody can observe others’ move)
equilibrium
Equilibrium
  • An equilibriums*= (s1*, s2*, …, sN*) is a strategy combination consisting of a best strategy for each of the N players in the game
  • What is a best strategy? depends on the game…informally, it is a strategy that a players selects in trying to maximize his individual payoff, knowing that other players are also doing the same
dominant strategy equilibrium prisoner s dilemma1
Dominant Strategy Equilibrium: Prisoner’s Dilemma
  • Prisoner I’s Decision:
    • If II chooses Don’t 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
dominant strategy equilibrium prisoner s dilemma2
Dominant Strategy Equilibrium: Prisoner’s Dilemma
  • Prisoner II’s Decision:
    • If I chooses Don’t 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
dominant strategy equilibrium prisoner s dilemma3
Dominant Strategy Equilibrium: Prisoner’s Dilemma
  • It is best for both I and II to implicate regardless of what other one does
  • Implicate is a Dominant Strategy for both
  • (Implicate, Implicate) becomes the Dominant Strategy Equilibrium
  • Note: It’s beneficial for both to Don’t Implicate, but it is not an equilibrium as both have incentive to deviate
dominant strategy equilibrium prisoner s dilemma4
Dominant Strategy Equilibrium: Prisoner’s Dilemma

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
  • Not all games have a dominant strategy equilibrium
a beautiful mind nash equilibrium
A Beautiful Mind: Nash Equilibrium
  • Nash Equilibrium is a strategy combination s*= (s1*, s2*, …, sN*), such that si* is a best response to (s1*, …,si-1*,si+1*,…, sN*), for each i, namely, for each si

pi(s*)≥ pi(s1*, s2*, …, si, …, sN*)

  • Note: It is a simultaneous game, and so nobody knows a priori the choice of other agents
nash equilibrium the battle of the sexes coordination game
Nash Equilibrium: The Battle of the Sexes (coordination game)
  • (Stadium,Stadium) is a NE: Best responses to each other
  • (Cinema, Cinema) is a NE: Best responses to each other
nash equilibrium
Nash Equilibrium
  • In a NE no agent can unilaterally deviate from its strategy given others’ strategies as fixed
  • There may be no, one or many NE, depending on the game
  • 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 always true)
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