14.126 Game Theory
Haluk Ergin & Muhamet Yildiz
Lecture 4: Normal Form Games (continued) Bayesian Games Rationalizability ai is rationalizable if ∀j ∈ N , ∃ a nonempty Zj ⊂ Aj s.t.
(i) ai ∈ Zi, and
(ii) ∀j ∈ N : Zj ⊂ B(Δ(Z−j )).
Iterated Elimination of Strictly Dominated Strategies:
0 t+1 � � t �� Let A = A. Inductively, let Ai = Bi Δ A−i . ∞ AIESDS = � At. t=0
Proposition: If the game is finite, then AIESDS consists of the set of rationalizable action profiles. Order of Elimination
The outcome of IESDS is independent of the order of elimination of the strictly dominated strategies. (See O&R section 4.2.2).
Is the same true for iterated elimination of weakly domi nated strategies?
a b a b a 0,0 0,0 a 0,0 0,0 b 0,0 1,1 b 0,0 1,1 MK of Rationality & IESDS Example: Simplified Price Competition
High Medium Low High 6,6 0,10 0,8 Medium 10, 0 5,5 0,8 Low 8,0 8,0 4,4
Rationality ⇒
Rationality + Mutual Knowledge of Rationality ⇒ CK of Rationality & IESDS Example: Cournot Duopoly
Two firms produce the same good at 0 marginal cost.
They choose quantities qi, qj ∈ [0, 1].
The inverse demand in the market is given by
p(q1, q2) = max{0, 1 − (q1 + q2)}.
Payoffs: ui(qi, qj ) = p(q1, q2) × qi, for i �= j.
∗ Best reply of: qi (qj ) = (1 − qj )/2.
Rationality + Common Knowledge Rationality ⇒
What about the Cournot Oligopoly with 3 or more firms? Nash Equilibrium
∗ ∗ ∗ A strategy profile σ = (σ1, . . . , σn) ∈ Δ(A1) × . . . × Δ(An) is a Nash Equilibrium (NE) if for any i ∈ N : ∗ ∗ ∗ ∀σi ∈ Δ(Ai) : ui(σi , σ−i) ≥ ui(σi, σ−i).
Note:
∗ ∗ ∗ 1. σ is a NE if σi ∈ Bi(σ−i) for any i ∈ N . 2. A finite normal form game may not have a pure strat egy NE.
3. If σ∗ is a NE then supp(σ∗) ⊂ AIESDS .
Theorem (Nash, 1950) Every finite normal form game has a Nash equilibrium in mixed strategies. Kakutani’s Fixed Point Theorem: Let C be a convex, m compact subset of of R and let f: CC→ be a correspon dence such that:
• For any x ∈ C, f (x) is nonempty and convex
• f has a closed graph (is usc): for all sequences (xk) and (yk) in C such that yk ∈ f (xk) for all k, xk → x, and yk → y, we have y ∈ f (x).
Then there exists x∗ ∈ C such that x∗ ∈ f (x∗).
Proof Let C = Δ(A1) × . . . × Δ(An) & define B: CC→ by
B(σ) = B1(σ−1) × . . . × Bi(σ−i) × . . . × Bn(σ−n) σ ∈ C. C is convex and compact. B is nonempty, convex valued, and has a closed graph. Hence there is σ∗ ∈ B(σ∗). By the ∗ definition of B, σ is a NE. � Correlated Equilibrium
Given a finite normal form game, a correlated equilibrium � � (CE) is a tuple Ω, π, P = (Pi)i∈N , a = (ai)i∈N s.t.: • (Ω, π) is a finite probability space,
• for all i ∈ N , Pi is a partition of Ω,
• for all i ∈ N , ai : Ω → Ai is a Pimeasurable function � � (i.e. if ω, ω ∈ Pi for some Pi ∈ Pi, then ai(ω) = ai(ω )),
• and for all i ∈ N and any Pimeasurable bi: Ω → Ai: � � π(ω)ui(ai(ω), a−i(ω)) ≥ π(ω)ui(bi(ω), a−i(ω)). ω∈Ω ω∈Ω Given a finite normal form game:
Proposition For any CE (Ω, π, P , a), we can construct an � � � � “equivalent” CE (A, π , P , a ) with Pi = {{ai} × A−i | ai ∈ � −1 Ai}, ai(a) = ai, and π = π ◦ a .
Proposition For any mixed strategy NE σ, there is a cor related equilibrium (Ω, π, P , a) s.t. σ = π ◦ a−1.
Proposition The set of correlated equilibrium payoffs: ⎧ ⎫ ⎨ ⎬ u(a) = � π(ω)u(a(ω)) | (Ω, π, P , a) is a CE E π ⎩ ω∈Ω ⎭ n is a closed and convex subset of R .
−1 Proposition For any CE (Ω, π, P, a) and ai s.t. π(ai (ai)) > 0, ai is rationalizable.. Example: Battle of the Sexes
OB O 2,1 0,0 B 0,0 1,2
NE?
CE Payoffs?
OB O p1 p2 B p3 p4 Bayesian Games
(Simultaneous Move Games with Payoff uncertainty) A Bayesian Game is a tuple (N, T , p, A, u) where: • N = {1, . . . , n} is a finite set of players,
• A = A1 × . . . An, where Ai is the set i’s actions, • T = T1 × . . . Tn, where Ti is the set of i’s types,
• p = (pi)i∈N , where pi(·|ti) is a probability distribution over T−i (i’s prior over others’ types conditional on his), • u = (ui)i∈N where ui: A×T → R is player i’s vNM payoff function.
Given a Bayesian Game, a pure strategy Bayesian Nash ∗ ∗ ∗ Equilibrium (BNE) is a profile a = (ai )i∈N where ai : Ti → Ai, s.t. for all i ∈ N , ti ∈ Ti, and ai ∈ Ai: � � ∗ ∗ ui(a (t); t)pi(dt−i|ti) ≥ ui(ai, a−i(t−i); t)pi(dt−i|ti). T−i T−i (BNE is strict if for almost all ti, there is strict inequality above when ai �= ai(ti) .) Harsanyi’s Purification Theorem
Take any: • normal form game G = (N, A, u) and a “regular” mixed strategy NE σ of G, and
• a collection of random variables � = (�i(a))i∈N,a∈A with values in [−1, 1] s.t.: – each �i = (�i(a))a∈A has an absolutely continuous distribution µ, and – (�i)i∈N are independent. Let the perturbed Bayesian game G(γ�) be s.t., the types of player i are the possible realizations of �i, the priors p are γ� derived from the actual distribution µ of �, and ui (a, �) = ui(a) + γ�i(a).
Given any a sequence γk → 0, for every k there is a strict pure strategy BNE ak of G(γk�), s.t. µ ◦ (ak)−1 → σ.