Notes on Game Theory

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Self-interested agents

Every agent has their own knowledge about the world and they can only make decisions base on their own knowledge (without knowing others’ choices). Every agent has its utility function, which is a mathematical function for qualifying agent’s preference of choices and the impact of uncertainty
Decision-theoretic rationality
Acts to maximize expected (average) utility

Definitions

Players: People (or organizations, thing) who making decisions, represent as $N$
Actions: What the players can do, represent as $A_i$
Payoffs: What is motivating the players, represented as $u_i$

Two Standard Representations of Games

Normal Form (Matrix Form, Strategic Form)

Showing payoffs as a function of actions
Assume players move simultaneously

Player 1 \ Player 2	C	D
C	Benefit, Benefit	Drawback, Benefit
D	Benefit, Drawback	Drawback, Drawback

A Large Collective Action Game
Too large so cannot write down as a simple matrix

Extensive Form

Prisoner’s Dilemma

These people are captured by police, then they’re not allowed to communicate with each other. Police say to each of them that if he cooperates with police and betrays your friend, he can be released. However, if he refuses to cooperate with the police and his friend does, his friend can be freed and he cannot. If everyone betrays their friend, everyone will not be released. If everyone refuses to betray their friend, everyone will be released.

The most rational choice is assuming your friend would betray you, yet it would lead to the worst outcome.

Games of Pure Competition

$u_1(a)+u_2(b)=C$, where C is some constants. Special case: C = 0, zero sum game.
Two players have exactly opposed interests.