Game theory

Cooperative game theory

Enforceable bargains, contracts

Characteristic functions, the core

Role of law in framing the interactions of individuals and groups

Property rights

Contract law

Labour law

Course examples

Contractual relations

Investment

 


Non-cooperative game theory

Strategic interdependence

No outside (third party) enforcement of an agreement

Possibility of tacit collusion in a non-cooperative setting

Cartel behaviour without a secret cartel agreement

Possibility of other self-enforcing agreements

Compliance with standards in a competitive industry

Possibility of wealth dissipation

Many oligopoly models

Common property models

Free riding in public good provision

Strategic uncertainty


Example areas of application

Policy

International arrangements

Mechanism design

Cross-disciplinary

Economics, finance, political science, law, biology

Analytical aspirations

Create an analytically useful model

“There is an art to building game-theoretic models that are simple enough to analyze, yet rich enough to capture intuition and serve as sources of new insights.” Watson, p. 3.

Develop a means of “solving” or predicting the outcome


Describing a game

“Formal descriptions of interactions and their results.”

Players

Actions and strategies

“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, Si 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 ΔSi for the set of mixed strategies (infinite dimension, continuous) and σi for an element of that set.

A strategy profile is (s1,s2, … sn) where s1 is in S1, etc.or a mixed strategy profile is (σ1, σ2, … , σn) where σ1 is in ΔS1 and so on.

The vector s-i is a vector of dimension of n-1 of the pure strategies of all the players except i

Timing of decisions, knowledge when choosing an action, knowledge of what other players know

Information sets

An information set contains non-terminal 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.

Common knowledge

Perfect information

Every information set in the game is a singleton

Imperfect information

At least one information set after the start of the game is not a singleton.

Games of certainty

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)

Games of incomplete information

Nature moves first.

Strategy profiles →outcomes

A strategy profile implies an outcome, which may be random.

Outcomes →utilities registering preferences (expected utility if outcomes are 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.

Organizing and presenting information about a game: the extended form and the normal form

The extended form

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 non-terminal 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.

Examples of extended form

Entry in a game of perfect information

 


Entry deterrence in a game of imperfect information.

 

 

 

The normal (matrix) form

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 fare-setting 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

 

 


Beliefs, mixed strategies, and expected utility

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 μ2(s2). The probabilities have to be non-negative 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.