Algoritmi per sistemi distribuiti strategici
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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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Algoritmi per Sistemi Distribuiti Strategici

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Algoritmi per sistemi distribuiti strategici

Algoritmi per Sistemi Distribuiti Strategici


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

=

+


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 dilemma

Dominant Strategy Equilibrium: Prisoner’s Dilemma

Strategy

Set

Payoffs

Strategy Set


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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