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.  Both strategic and complexity matter!
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 equilibrium s*= (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
Prisoner I
Prisoner II
Don’t Implicate
Implicate
-1, -1
-10, 0
0, -10
-9, -9
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
Prisoner I
Prisoner II
Don’t Implicate
Implicate
-1, -1
-10, 0
0, -10
-9, -9

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
Prisoner I
Prisoner II
Don’t Implicate
Implicate
-1, -1
-10, 0
0, -10
-9, -9

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
Prisoner I
Prisoner II
Don’t Implicate
Implicate
-1, -1
-10, 0
0, -10
-9, -9

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
Man
Woman
Stadium
Cinema
2, 1
0, 0
1, 2

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)