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LECTURE 26:

GAME THEORY 1

INSTRUCTOR:

GIANNI A. DI CARO

15 - 382 COLLECTIVE INTELLIGENCE – S

ICE-CREAM WARS

http://youtu.be/jILgxeNBK_

15781 Fall 2016: Lecture 22

ELEMENTS OF A GAME

 The players: how many players are there? Does nature/chance

play a role? Players are assumed to be rational

 A complete description of what the players can do: the set of all possible actions.

15781 Fall 2016: Lecture 22

ELEMENTS OF A GAME

 A description of the payoff / consequences for each player for every possible combination of actions chosen by all players playing the game.  A description of all players’ preferences over payoffs

Utility function for each player

MAKING DECISIONS: BASIC DEFINITIONS

 Decision-making can involve choosing:  one single action or  a sequence of actions  Action outcomes can be certain or subject to uncertainty  A set 𝐴 of alternative actions to choose from is given, it can be either discrete or continuous  Payoff (for a single agent): function 𝜋: 𝐴 → ℝ that associates a numerical values with every action in 𝐴  Optimal action 𝑎 ∗ (for a single agent scenario): 𝜋(𝑎 ∗ ) ≥ 𝜋 𝑎 ∀𝑎 ∈ 𝐴  Payoff (for a multi-agent scenario): The payoff of the action 𝑎 for agent 𝑖 depends on the actions of the other players! 𝜋: 𝐴 𝑛 → ℝ  Strategy: rule for choosing an action at every point a decision might have to be made (depending or not on the other agents)  The strategy defines the behavior of an agent  The observed behavior of an agent following a given strategy is the outcome of the strategy

PURE VS. RANDOMIZED STRATEGIES

 Pure strategy: a strategy in which there is no randomization , one specific action is selected with certainty at each decision node  All possible pure strategies define the pure strategy set 𝑆  A decision tree can be used to represent a sequence of decisions 1 2 3

 Three action sets (actions may the be same), that result in the pure strategy set: 𝑆 = {𝑎 1 𝑏 1 𝑐 1 , 𝑎 1 𝑏 1 𝑐 2 , 𝑎 1 𝑏 2 𝑐 1 , 𝑎 1 𝑏 2 𝑐 2 , 𝑎 2 𝑏 1 𝑐 1 , 𝑎 2 𝑏 1 𝑐 2 , 𝑎 2 𝑏 2 𝑐 1 , 𝑎 2 𝑏 2 𝑐 2 }

𝐴 1 = 𝑎 1 , 𝑎 2 , 𝐴 2 = 𝑏 1 , 𝑏 2 , 𝐴 3 = 𝑐 1 , 𝑐 2

15781 Fall 2016: Lecture 22

STRATEGIES (POLICIES)

 Strategy: tells a player what to do for every possible situation

throughout the game (complete algorithm for playing the game). It

can be deterministic or stochastic

 Strategy set: what strategies are available for the players to play.

The set can be finite or infinite (e.g., beach war game)

 Strategy profile: a set of strategies for all players which fully

specifies all actions in a game. A strategy profile must include

one and only one strategy for every player

 Pure strategy: one specific element from the strategy set, a single

strategy which is played 100% of the time ( deterministic )

 Mixed strategy: assignment of a probability to each pure strategy.

Pure strategy ≡ degenerate case of a mixed strategy ( stochastic )

15781 Fall 2016: Lecture 22

INFORMATION

 Complete information game: Utility functions, payoffs, strategies and

“types” of players are common knowledge

 Incomplete information game: Players may not possess full

information about their opponents (e.g., in auctions, each player

knows its utility but not that of the other players). “ Parameters ” of the

game are not fully known

 Perfect information game: Each player, when making any decision, is

perfectly informed of all the events that have previously occurred

(e.g., chess) [Full observability]

 Imperfect information game: Not all information is accessible to the

player (e.g., poker, prisoner’s dilemma) [Partial observability]

15781 Fall 2016: Lecture 22 (STRATEGIC-) NORMAL-FORM GAME

 Let’s focus on static games

 There is a strategic interaction among players

 A game in normal form consists of:

o Set of players 𝑁 = { 1 , … , 𝑛}

o Strategy set 𝑆

o For each 𝑖 ∈ 𝑁, a utility function 𝑢𝑖 defined

over the set of all possible strategy profiles ,

𝑛

o If each player 𝑗 ∈ 𝑁 plays the strategy 𝑠𝑗 ∈ 𝑆, the utility

of player 𝑖 is 𝑢𝑖 𝑠 1 , … , 𝑠𝑛 that is the same as player 𝑖 ’ s

payoff when strategy profile (𝑠 1 , … , 𝑠𝑛) is chosen

Payoff matrix

15781 Fall 2016: Lecture 22

• 𝑢 𝑖

𝑖

𝑗

𝑠𝑖+𝑠𝑗 2

𝑖

𝑗

𝑠𝑖+𝑠𝑗 2

𝑖

𝑗 1 2

𝑖

𝑗

THE ICE CREAM WARS

 𝑆 = [ 0 , 1 ]

i

is the fraction of beach

15781 Fall 2016: Lecture 22

THE PRISONER’S DILEMMA (1962)

15781 Fall 2016: Lecture 22 PRISONER’S DILEMMA: PAYOFF MATRIX

• 1,- 1 - 9, 0,- 9 - 6,- 6 Don’t Confess Confess

What would you do?

Don’t confess = Don’t rat out Cooperate with each other Confess = Defect Don’t cooperate to each other, act selfishly! Don’t Confess Confess

B

A

15781 Fall 2016: Lecture 22  Confess (Defection, Acting selfishly) is a dominant strategy for B : no matters what A plays, the best reply strategy is always to confess  (Strictly) dominant strategy : yields a player strictly higher payoff,. no matter which decision(s) the other player(s) choose  Weakly: ties in some cases  Confess is a dominant strategy also for A  A will reason as follows: B ’s dominant strategy is to Confess, therefore, given that we are both rational agents, B will also Confess and we will both get 6 years.

PRISONER’S DILEMMA

15781 Fall 2016: Lecture 22  But, is the dominant strategy (C,C) the best strategy?

PRISONER’S DILEMMA

• 1,- 1 - 9, 0,- 9 - 6,- 6 Don’t Confess Confess Don’t Confess Confess

B

A