Games with Perfect Information

Yiling Chen

September 10, 2012 Non-Cooperative Game Theory

I What is it? Mathematical study of interactions between rational and self-interested agents.

I Non-Cooperative Focus on agents who make their own individual decisions without coalition Perfect Information

I All players know the game structure.

I Each player, when making any decision, is perfectly informed of all the events that have previously occurred. Deﬁnition of Normal-Form Game

I A ﬁnite n-person game, G =< N, A, u >

I N = {1, 2, ..., n} is the set of players.

I A = {A1, A2, ..., An} is a set of available actions. a = (a1, a2, ..., an) ∈ A is an action proﬁle (or a pure strategy proﬁle).

I u = {u1, u2, ...un} is a set of utility functions for n agents.

I A strategy is a complete contingent plan that deﬁnes the action an agent will take in all states of the world. Pure strategies (as opposed to mixed strategies) are the same as actions of agents.

I Players move simultaneously.

I Matrix representation for 2-person games Example: Battle of the Sexes

I A husband and wife want to go to movies. They can select between “Devils wear Parada” and ”Iron Man”. They prefer to go to the same movie, but while the wife prefers “Devils wear Parada” the husband prefers “Iron Man”. They need to make the decision independently. Husband DWP IM

DWP 10,5 0,0 Wife IM 0,0 5, 10 Example: Battle of the Sexes

DWP 10,5 0,0 Wife IM 0,0 5, 10 Best-Response Correspondences

I The best-response correspondence BRi (a−i ) ∈ Ai are the set of strategies that maximizes agent i’s utility given other agents’ strategy a−i .

I Compute every agent’s best-response correspondences. The ﬁxed points of the best-response correspondences, i.e. a∗ ∈ BR(a∗), are the NEs of the game. Example: Continuous Strategy Space

Cournot Competition

I Two suppliers producing a homogeneous good need to choose their production quantity, qi and q2. The demand that they are facing is p(Q) = 1000 − Q, where Q = q1 + q2. Unit cost c > 0.

I Utility: u1(q1, q2) = q1 ∗ [p(q1 + q2) − c]

I Best-response of supplier 1 is

BR1(q2) = arg maxq1 u1(q1, q2) Example: Matching Pennies

I Each of the two players has a penny. They independently choose to display either heads or tails. If the two pennies are the same, player 1 takes both pennies. If they are diﬀerent, player 2 takes both pennies.

Heads Tails

Heads 1, -1 -1,1

Tails -1,1 1, -1 Mixed Strategies

I A mixed strategy of agent i, σi ∈ ∆(Ai ), deﬁnes a probability, σi (ai ) for each pure strategy ai ∈ Ai .

I Agent i’s expected utility of of mixed strategy proﬁle σ is X X ui (σ) = ui (a)Pσ(a) = ui (a) (Πj σj (aj )) a∈A a∈A

I The support of σi is the set of pure strategies {ai : σ(ai ) > 0}.

I Pure strategies are special cases of mixed strategies. Mixed Strategy Nash Equilibrium

∗ ∗ ∗ ∗ I Mixed-strategy proﬁle σ = {σ1, σ2, ..., σn} is a Nash equilibrium in a game if, for all i,

∗ ∗ ∗ ui (σi , σ−i ) ≥ ui (σi , σ−i ), ∀σi ∈ ∆(Ai ).

I Theorem (Nash 1951): Every game with a ﬁnite number of players and action proﬁles has at least one Nash equilibrium.

I All pure strategies in the support of agent i at a mixed strategy Nash Equilibrium have the same expected utility. Finding Mixed Strategy Nash Equilibrium

I Find the ﬁxed point of the Best-Response Heads Tails correspondences.

Heads 1, -1 -1,1

Tails -1,1 1, -1 Fixed Point

Player 1 r r ∗(p) Heads

p∗(r)

Tails Tails Heads Player 2 p Extensive-Form Game with Perfect Information Example: The Sharing Game

I Alice and Bob try to split two indivisible and identical gifts. First, Alice suggests a split: which can be “Alice keeps both”, “they each keep one”, and “Bob keeps both”. Then, Bob chooses whether to Accept or Reject the split. If Bob accepts the split, they each get what the split speciﬁes. If Bob rejects, they each get nothing.

Alice

2-0 1-1 0-2

Bob Bob Bob

A R A R A R

(2,0) (0,0) (1,1) (0,0) (0,2) (0,0) Loosely Speaking...

I Extensive Form

I A detailed description of the sequential structure of the decision problems encountered by the players in a game.

I Often represented as a game tree

I Perfect Information

I All players know the game structure (including the payoﬀ functions at every outcome).

I Each player, when making any decision, is perfectly informed of all the events that have previously occurred. Def. of Perfect-Information Extensive-Form Games

I A perfect-information extensive-form game, G = (N, H, P, u)

I N = {1, 2, ..., n} is the set of players.

N={Alice, Bob}

Alice

2-0 1-1 0-2 Bob Bob Bob

A R A R A R

(2,0) (0,0) (1,1) (0,0) (0,2) (0,0) Def. of Perfect-Information Extensive-Form Games

I A perfect-information extensive-form game, G = (N, H, P, u) I H is a set of sequences (ﬁnite or inﬁnite)

I Φ ∈ H k I h = (a )k=1,...,K ∈ H is a history k I If (a )k=1,...,K ∈ H and L < K, then k (a )k=1,...,L ∈ H k ∞ k I (a )k=1 ∈ H if (a )k=1,...,L ∈ H for all positive L I Z is the set of terminal histories. H = {Φ, 2 − 0, 1 − 1, 0 − 2, (2 − 0, A), (2 − 0, R), (1 − 1, A), (1 − 1, R), (0 − 2, A), (0 − 2, R)} Alice

2-0 1-1 0-2 Bob Bob Bob A R A R A R

(2,0) (0,0) (1,1) (0,0) (0,2) (0,0) Def. of Perfect-Information Extensive-Form Games

I A perfect-information extensive-form game, G = (N, H, P, u)

I P is the player function, P : H\Z → N. P(Φ)=Alice P(2 − 0)=Bob P(1 − 1) = Bob P(0 − 2) = Bob

Alice

2-0 1-1 0-2 Bob Bob Bob

A R A R A R

(2,0) (0,0) (1,1) (0,0) (0,2) (0,0) Def. of Perfect-Information Extensive-Form Games

I A perfect-information extensive-form game, G = (N, H, P, u)

I u = {u1, u2, ...un} is a set of utility functions, ui : Z → R.

u1((2 − 0, A)) = 2 u2((2 − 0, A)) = 0 ... Alice

2-0 1-1 0-2 Bob Bob Bob

A R A R A R

(2,0) (0,0) (1,1) (0,0) (0,2) (0,0) Pure Strategies in Perfect-Information Extensive-Form Games

I A pure strategy of player i ∈ N in an extensive-form game with perfect information, G = (N, H, P, u), is a function that assigns an action in A(h) to each non-terminal history h ∈ H\Z for which P(h) = i.

I A(h) = {a :(h, a) ∈ H}

I A pure strategy is a contingent plan that speciﬁes the action for player i at every decision node of i. Pure Strategies for the Sharing Game

Alice

2-0 1-1 0-2 Bob Bob Bob

A R A R A R

(2,0) (0,0) (1,1) (0,0) (0,2) (0,0)

S = {S1, S2} E.g. s1 = (2 − 0 if h = Φ), s2 = (A if h = 2 − 0; R if h = 1-1; R if h = 0 − 2). Pure Strategies

Alice

A B Bob Bob

C D E F

Alice (3,8) (8,3) (5,5) J K

(2,10) (1,0)

S = {S1, S2} E.g. s1 = (A if h = Φ; J if h = BF ) s2 = (C if h = A; F if h = B) Normal-Form Representation A perfect-information extensive-form game ⇒ A normal-form game Alice

2-0 1-1 0-2 Bob Bob Bob A R A R A R

(2,0) (0,0) (1,1) (0,0) (0,2) (0,0)

(A,A,A) (A,A,R) (A,R,A) (A,R,R) (R,A,A) (R,A,R) (R,R,A) (R,R,R)

2-0 2,0 2,0 2,0 2,0 0,0 0,0 0,0 0,0

1-1 1,1 1,1 0,0 0,0 1,1 1,1 0,0 0,0

0-2 0,2 0,0 0,2 0,0 0,2 0,0 0,2 0,0

A normal-form game ; A perfect-information extensive-form game Normal-Form Representation: Example 2 A perfect-information extensive-form game ⇒ A normal-form game