05 - Dynamic Games of Complete Information

Question 1

What is a subgame of an extensive-form game?

Answer 1

a) begins at a decision node D that is a singleton information set,
b) includes all the decision and terminal nodes following D in the game tree (but no other nodes), and
c) does not cut any information sets (i.e., if a decision node ˜D follows D, then all other nodes in the information set containing ˜D must also follow D).

Question 2

What is a subgame perfect Nash equilibrium (SPNE)?

Answer 2

A strategy profile σ in an extensive-form game is a subgame perfect Nash equilibrium (SPNE) if it induces a Nash equilibrium in every subgame

Question 3

How do subgame perfect Nash equilibrium (SPNE) and NE relate?

Answer 3

Every SPNE is a Nash equilibrium, but the reverse need not hold

Question 4

In a finite game of perfect information, when is a strategy profile subgame perfect?

Answer 4

In a finite game of perfect information, a strategy profile is a SPNE if and only if it is a Nash equilibrium that can be derived through backwards induction.

Question 5

What does Zermelo’s Theorem imply about subgame perfect Nash equilibria?

Answer 5

Every finite game of perfect information has a pure-strategy SPNE. Moreover, if no player has the same payoffs at any two terminal nodes, then there is a unique SPNE.

Question 6

How does a subgame perfect Nash equilibrium look like?

Answer 6

A subgame perfect Nash equilibrium is a strategy profile: it specifies actions for every information set in the game tree.

Question 7

How does a subgame perfect outcome look like?

Answer 7

A subgame perfect outcome describes the path from the initial node to the terminal node that is reached when SPNE strategies are played.

Question 8

How do Backwards Induction (BI) and SPNE relate?

Answer 8

SPNE generalizes backwards induction (BI): whenever BI applicable, SPNE and BI equivalent, but SPNE also applies to

(i) imperfect information and
(ii) infinite-horizon games.

Question 9

How does the one-shot deviation principle help to identify SPNE?

Answer 9

In the above class of extensive-form games for T < 1, a strategy profile s* = (s1, …, sn) is an SPNE if and only if for every player i, there is no profitable one-shot deviation from s*i .
Idea of proof:
-> If there is a profitable one-shot deviation, there is a profitable deviation in some subgame, contradicting SPNE.
-> Suppose there is a profitable deviation; however complicated it is, there is a last subgame where itdeviates (BI intuition).

Question 10

How can you check whether a strategy x fulfills the one-shot-deviation principle?

Answer 10

1) identify all one-shot-deviations (=at one node, choose different strategy)
2) compare that u(x) >= u(each deviation)
Note: the evaluation of the utility takes place at the node of deviation!