課程名稱︰賽局理論
課程性質︰選修
課程教師：呂學一
開課學院：電資學院
開課系所︰資工所、網媒所
考試日期（年月日）︰2006.06.27
考試時限（分鐘）：180
試題 :
Game Theory
close-book final exam
June 27, 2006
You may answer the questions in any order. Dishonest behaviors and attempts
will be most seriously punished.
Problem 1 (30 points)
Explain the following terms.
(1) Coalitional game with transferable payoff.
(2) Weak sequential equilibrium.
(3) Subgame perfect equilibrium.
(4) Trembling-hand perfect equilibrium of a strategic game.
(5) Correlated equilibrium.
(6) Behavioral strategy.
Problem 2 (15 points)
Give two equivalent definitions of the core of a coalitional game with
transferable payoff (8 pts). Prove that they are indeed equivalent? (7 pts)
Problem 3 (15 points)
Consider the following game between Alice and Bob. First Alice receives a card
that is either H or L with equal probabilities. Bob does not see the card.
Alice may announce that her card is L, in which case Alice must pay $1 to Bob,
or may claim that the card is H, in which case Bob may choose to concede or to
insist on seeing the card. If Bob concedes, then he must pay $1 to Alice. If
Bob insists on seeing the card, then Alice must pay Bob $4 if the card is L
and Bob must pay $4 to Alice if the card is indeed H.
‧ (5 pts) Formulate the game as an extensive game
(with imperfect information).
‧ (5 pts) Give a mixed strategy Nash equilibrium for the game.
‧ (5 pts) Justify your answer for the previous bullet.
Problem 4 (20 points)
‧ (10 pts) What is the one-deviation property of a strategy profile for an
extensive game with perfect information.
‧ (10 pts) Prove that if a strategy profile satisfies the (above)
one-deviation property, then it has to be a subgame perfect equilibrium
of the finite game.
Problem 5 (20 points)
Consider a 2-player Bayesian game with the following three states of games.
┌─┬──┬──┐ ┌─┬──┬──┐ ┌─┬──┬──┐
│α│ Ｌ │ Ｒ │ │β│ Ｌ │ Ｒ │ │γ│ Ｌ │ Ｒ │
├─┼──┼──┤ ├─┼──┼──┤ ├─┼──┼──┤
│Ｔ│2, 1│0, 0│ │Ｔ│2, 1│0, 0│ │Ｔ│1, 2│3, 0│
├─┼──┼──┤ ├─┼──┼──┤ ├─┼──┼──┤
│Ｂ│3, 0│2, 1│ │Ｂ│0, 0│1, 2│ │Ｂ│3, 0│1, 2│
└─┴──┴──┘ └─┴──┴──┘ └─┴──┴──┘
Each player has types (a) and (b). The information partition of player 1 is
{α}, {β, γ} and the information partition of player 2 is {α, β}, {γ}.
Their (nontrivial) beliefs are as follows.
‧ Player 1(b): (β, γ) = (2/3, 1/3).
‧ Player 2(a): (α, β) = (1/3, 2/3).
What are the (pure strategy) Nash equilibria of this Bayesian game? (Just show
the answer. We won't check the calculation for your answer.)