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1

What are Dynamic Games?

Players choose actions sequentially. The sequence in which players move will be explicitly modeled.

2

What are games with perfect vs. imperfect information (in dynamic games)?

• In a game of perfect information, whenever a player is called upon to move, she has observed all previous moves (i.e., the full history of play so far).
• All other games are games of imperfect information.

3

When is a dynamic game also a static game?

static game (simultaneous moves) = a dynamic game of imperfect information where players move sequentially but no player observes any of the prior moves.

4

What does the extensive-form representation of a game specify?

1) the players,
2) strategy space:
-> which player decides on the action,
-> what each player can do at each of his or her opportunities to move,
-> what each player knows at each of his or her opportunities to move,
3) the payoff received by each player for each combination of moves that could be chosen by the players.

5

How are extensive-form games usually represented?

typically described by means of a game tree.

6

How to describe a game tree?

A game tree is directed graph, i.e., a set of nodes connected by branches:
• decision node: a single player chooses from a set of actions, each action corresponds to a branch that leads to a new decision or terminal node.
• terminal node: the game ends and payoffs are realized.

7

How to represent a player’s ignorance of previous moves?

Information sets. Game tree: all nodes of an information set are connected by dotted lines.

8

What does the assumption "perfect recall" mean?

a player does not forget what she once knew,
including her own actions.

9

How to do the procedure of Backwards Induction in game trees?

• First determine optimal actions for moves at the last decision nodes (followed only by terminal nodes).
• Given these actions will be taken at the last decision nodes, determine optimal actions for the next-to-last decision nodes.
• In this manner, continue backwards through the game tree until the initial decision node is reached.
• The determined path of optimal actions from the initial node to some terminal node is a backwards induction solution of the game.

10

How about dynamic games and Nash equilbria?

Every finite game of perfect information has a pure-strategy Nash equilibrium that can be derived through backwards induction.
Moreover, if no player has the same payoffs at any two terminal nodes, then there is a unique Nash equilibrium that can be derived in this manner.

11

How to write down the strategy spaces in extensive form?

For S₁, in each period that player 1 makes a decision, specify a decision for all potential nodes in that period. Go from top to bottom and from left to right.

12

When a game has n=4 phases, how to solve when n = 3?

look at decision tree from the second node on.