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

• 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
• 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
• 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
• 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
• 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 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
• 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
• 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 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 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 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
• 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