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Algoritmi per Sistemi Distribuiti Strategici

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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?

- 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

- 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!

- 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

=

+

- 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

- 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 siSi (nobody can observe others’ move)

- 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

Strategy

Set

Payoffs

Strategy Set

- 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

- 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

- 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 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

- 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

- (Stadium,Stadium) is a NE: Best responses to each other
- (Cinema, Cinema) is a NE: Best responses to each other

- 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)