Dynamic Games of Complete Information

Dynamic Games of Complete Information

Entry Game

An incumbent monopolist faces the possibility of entry by a challenger.
The challenger may choose to enter or stay out.
If the challenger enters, the incumbent can choose either to accommodate or to fight.
The payoffs are common knowledge.

The first number is the payoff of the challenger. The second number is the payoff of the incumbent.

Sequential-move Matching Pennies

Each of the two players has a penny.
Player 1 first chooses whether to show the Head or the Tail.
After observing player 1’s choice, player 2 chooses to show Head or Tail
Both players know the following rules:
If two pennies match (both heads or both tails) then player 2 wins player 1’s penny.
Otherwise, player 1 wins player 2’s penny.

Dynamic (or sequential-move) Games of Complete Information

A set of players
Who moves when and what action choices are available?
What do players know when they move?
Players’ payoffs are determined by their choices.
All these are common knowledge among the players.

strategy: A complete plan of actions at every round.

Extensive-form Representation

The extensive-form representation of a game specifies:
the players in the game
when each player has the move
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
the payoff received by each player for each combination of moves that could be chosen by the players.

Perfect Information

All previous moves are observed before the next move is chosen.
A player knows Who has moved What before she makes a decision.

Game Tree

A game tree has a set of nodes and a set of edges, each edge connects two nodes (these two nodes are said to be adjacent), and for any pair of nodes, there is a unique path that connects these two nodes.

A path is a sequence of distinct nodes y1, y2, y3, …, yn-1, yn such that yi and yi+1 are adjacent, for i=1, 2, …, n-1. We say that this path is from y1 to yn. We can also use the sequence of edges induced by these nodes to denote the path.

The length of a path is the number of edges contained in the path.
Example 1: x0, x2, x3, x7 is a path of length 3.
Example 2: x4, x1, x0, x2, x6 is a path of length 4.

There is a special node x0 called the root of the tree which is the beginning of the game.The nodes adjacent to x0 are successors of x0.
For any two adjacent nodes, the node that is connected to the root by a longer path is a successor of the other node.
Example 3: x7 is a successor of x3 because they are adjacent and the path from x7 to x0 is longer than the path from x3 to x0.
If a node x is a successor of another node y then y is called a predecessor of x. In a game tree, any node other than the root has a unique predecessor. Any node that has no successor is called a terminal node which is a possible end of the game.
Example 4: x4, x5, x6, x7, x8 are terminal nodes.
Any node other than a terminal node represents some player. For a node other than a terminal node, the edges that connect it with its successors represent the actions av