donigian/Notes on Game Theory

## Notes on Game Theory
Any strategy which involves change is called a Pure Strategy, otherwise Mixed Strategy.

Game is composed of 3 things:
players
strategies
payoffs

Strategy:
Rational Decision Making
A lot of times, the decision you do make are based on the consequences of the decisions we don't make.

Being rational means that I make decision in a way that will lead to my best expected payoff. Doesn't matter what you think of my rationality.

Any pattern in your decision making can be recognized by your competitor, so Mixed Strategies play an important role.

Random choices are not the same as arbitrary choices.

Payoffs:
1. The payoff to a player reflects what that player cares about, not what another thinks they should care about.
2. Payoffs for different players can't be directly compared
3. For finite games, all that matters about a player's payoff is the way which they rank them. How do players rank their outcomes. Ordinal payoffs.
4. In games that are not finite, you have to be careful in what units you measure payoffs.

Payoffs have to reflect the actual preferences of the player. I could care about time I save while you care about how much money you make.

Cardinal Payoffs: You need to know more than the order of your payoffs. Interval scale, i.e. gap between 20 to 30 is same as 30 to 40.

Being rational means that I make my decisions in a way that will lead to my best expected payoff.
Payoffs of different players can not be compared.

Ultimatum game: You make one offer/division in the game and I say Yes or No.

Finite Game:
Game has to end.
There can never be an infinite number of choices.
Finite players.

In sequential games: order matters. Each player makes decision without knowing other decision the other player makes.
depending on the game, it might be best to go first, middle or last. First Mover advantage is not always good for you.
Flexibility of response vs commitment to action!
Cooperative vs non-cooperative games: players don't have to play together.
Rollback Equilibrium: steady state, only solution that is subgame perfect.

Simultaneous Games (Doesn't require an infinite reqsoning...I think you think...)
Mini-max Theorem John Neumann
Zero Sum or Constant Sum
Mini-max : minimize maximum damage, pessimistic strategy...Murphy's law, whatever can go wrong will.
Maxi-min : maximize minimum damage
When you have a nash equilibrium, doesn't help to change your decision. If you don't play your nash equilibrium you leave chance for someone else to take advantage of your choice.

Iterated elimination of dominated strategies: The iterated elimination (or deletion) of dominated strategies is one common technique for solving games that involves iteratively removing dominated strategies. In the first step, at most one dominated strategy is removed from the strategy space of each of the players since no rational player would ever play these strategies.
	Any strategy which involves change is called a Pure Strategy, otherwise Mixed Strategy.

	Game is composed of 3 things:
	players
	strategies
	payoffs

	Strategy:
	Rational Decision Making
	A lot of times, the decision you do make are based on the consequences of the decisions we don't make.

	Being rational means that I make decision in a way that will lead to my best expected payoff. Doesn't matter what you think of my rationality.

	Any pattern in your decision making can be recognized by your competitor, so Mixed Strategies play an important role.

	Random choices are not the same as arbitrary choices.

	Payoffs:
	1. The payoff to a player reflects what that player cares about, not what another thinks they should care about.
	2. Payoffs for different players can't be directly compared
	3. For finite games, all that matters about a player's payoff is the way which they rank them. How do players rank their outcomes. Ordinal payoffs.
	4. In games that are not finite, you have to be careful in what units you measure payoffs.

	Payoffs have to reflect the actual preferences of the player. I could care about time I save while you care about how much money you make.

	Cardinal Payoffs: You need to know more than the order of your payoffs. Interval scale, i.e. gap between 20 to 30 is same as 30 to 40.

	Being rational means that I make my decisions in a way that will lead to my best expected payoff.
	Payoffs of different players can not be compared.

	Ultimatum game: You make one offer/division in the game and I say Yes or No.

	Finite Game:
	Game has to end.
	There can never be an infinite number of choices.
	Finite players.

	In sequential games: order matters. Each player makes decision without knowing other decision the other player makes.
	depending on the game, it might be best to go first, middle or last. First Mover advantage is not always good for you.
	Flexibility of response vs commitment to action!
	Cooperative vs non-cooperative games: players don't have to play together.
	Rollback Equilibrium: steady state, only solution that is subgame perfect.

	Simultaneous Games (Doesn't require an infinite reqsoning...I think you think...)
	Mini-max Theorem John Neumann
	Zero Sum or Constant Sum
	Mini-max : minimize maximum damage, pessimistic strategy...Murphy's law, whatever can go wrong will.
	Maxi-min : maximize minimum damage
	When you have a nash equilibrium, doesn't help to change your decision. If you don't play your nash equilibrium you leave chance for someone else to take advantage of your choice.

	Iterated elimination of dominated strategies: The iterated elimination (or deletion) of dominated strategies is one common technique for solving games that involves iteratively removing dominated strategies. In the first step, at most one dominated strategy is removed from the strategy space of each of the players since no rational player would ever play these strategies.