Algoritmi per Sistemi Distribuiti Strategici

1. Algoritmi per Sistemi Distribuiti Strategici

2. 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?

3. 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

4. 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!

5. 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 = +

6. 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

7. 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)

8. 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

9. Dominant Strategy Equilibrium: Prisoner’s Dilemma Strategy Set Payoffs Strategy Set

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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)