Game theory
“There is an art to building gametheoretic models that are simple enough to analyze, yet rich enough to capture intuition and serve as sources of new insights.” Watson, p. 3.
“Formal descriptions of interactions and their results.”
“A strategy is a complete contingent plan for a player in the game.” Watson 23 (my underlining)
Each pure strategy for i designates an action at each point in the game when player i has to make a decision
Properties of strategy set or space, S_{i} for each player: continuous or discrete, bounded or infinite
The game may allow mixed strategies. A mixed strategy is a probability distribution over the domain (support) of the pure strategies available to the player. Watson uses the symbol ΔS_{i} for the set of mixed strategies (infinite dimension, continuous) and σ_{i} for an element of that set.
A strategy profile is (s_{1},s_{2}, … s_{n}) where s_{1} is in S_{1}, etc.or a mixed strategy profile is (σ_{1}, σ_{2}, … , σ_{n}) where σ_{1} is in ΔS_{1} and so on.
The vector s_{i} is a vector of dimension of n1 of the pure strategies of all the players except i
An information set contains nonterminal nodes assigned to a particular player characterized by none of the nodes being either an ancestor (preceding) or a successor node to any other of the nodes and all the edges starting at different nodes in the set appear to be equivalent (The player cannot distinguish at which node he or she is at in the information set). Different conventions are used to identify information sets. They appear as a set of nodes connected by dotted lines in most texts but other conventions are used.
Every information set in the game is a singleton
At least one information set after the start of the game is not a singleton.
When a random event affects the course of a game we say that nature has made a decision. Nature is just a random generator of say warm day or a cold day that may influence subsequent decisions of say a farmer of how hard to work or what to plant. In a game of certainty nature does not move after the game begins (Game of uncertainty: Converse is true)
Nature moves first.
A strategy profile implies an outcome, which may be random.
The outcomes, which may be a probability distribution over specific outcomes, map into a measure of preferences or a utility function. The outcomes may be described as utilities or as payoffs.
A game tree is a type of directed graph in which the elements of a set of nodes (vertices) are joined by directed edges. A game tree has a root (starting point) characterized by having no edges terminating at it. There is a unique path to any other node from the root. Each nonterminal node is assigned to a player and at each terminal node, v, there is a payoff vector with an element for each player measuring their payoffs or utilities at this end point of the game. A player may have to make a decision at an information set that includes more than one node. The responsible player does not know which of the nodes in the information set has resulted from preceding decisions. He or she only has the coarse knowledge that he or she is in the set.
Prisoners' dilemma

Suspect B 

Confess 
Don’t confess 

Suspect A 
Confess 
5 yrs., 5yrs 
go free, 20yrs. 
Don’t confess 
20 yrs., go free 
1 yr., 1yr. 

Source: Originally Tucker 
The faresetting game (asymmetric PD structure)

Air Canada 

Fare $500 
Fare $200 

Canada West 
Fare $500 
50,100 
100,200 
Fare $200 
150,200 
10,10 

Aliprantis and Chakrabarti (p. 41) with airline names changed 
Entry deterrence imperfect information game in normal form

Incumbent 

Fight 
Coop 

Entrant 
IN.FIGHT 
15,0 
5,10 
OUT.FIGHT 
0,50 
0,50 

IN.COOP 
5,5 
20,25 

OUT.COOP 
0,50 
0,50 
A belief is a probability distribution over the strategies of the other players. We will usually restrict our attention to a two player game. The belief structure is
_{}. With two players, for example, the belief of player 1 is that player 2 will play his or her strategy 2 with probability μ_{2}(s_{2}). The probabilities have to be nonnegative and add to 1. Later we will be examining situations in which the choices of other players lead a player to revise (update) his or her beliefs in accordance with Bayes’ rule.