Introduction to Game Theory

#economics #micro

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.
2024 Spring, instructed by prof. Cho, Wonki.

Main Elements in a Game

Definition (Game Theory).

A Game Theory studies the interaction between a group of rational agents who behave strategically.

In this section we separately look into the key elements of this definition.

interaction between a group of individuals: we consider where the players are two or more.
rationality: we assume that every individuals are rational, meaning they seek to maximize their payoff, and have common knowledge. we will later define the rationality formally.
strategic behavior: one player's payoff depends on the other player's move.

We first start by formally defining the Game:

Definition (Game).

The Static form game is specified by where

: set of players, .
: player 's set of pure strategies, .
: player 's payoff function

Strategies and Payoff

Pure Strategy

Definition (pure strategy).

A pure strategy is a complete contingent plan describing which action a player chooses in each possible situation that she faces along the game.

: player 's pure strategy
: set of player 's pure strategies
: strategy space, Cartesian product of all player's strategy space.
: strategy profile, describing the strategies that each player selects

Definition (payoff function for pure strategy).

A payoff function is a function that assigns each strategy profile to the player 's payoff.

: player 's payoff when is played.
: payoff profile when is played.

Mixed Strategy

Definition (mixed strategy).

Given player 's (finite) pure strategy set , a mixed strategy for player is assigning each pure strategy to a probability that such strategy will be played, where .

: probability that plays .
: set of all player's mixed strategy.
: mixed extension of , which is driven by

Definition (payoff function for mixed strategy).

Using the ^5b8ae6 Definition 5 (mixed strategy), we now extend the definition of a payoff function.
where .

Definition (Mixed extension of the game).

The static game allowing for the mixed strategy is

Rationality

Definition (rationality and common knowledge).

In the following game, the two essential assumptions are based on.

rationality: every player 's choose the strategy that can maximize their expected payoff.
common knowledge of rationality: each player knows that the opponents know that the player is rational, and the opponents know that the player knows that they are rational, and so on.

Representation of the Game

Definition (simultaneous and sequential game).

A game is simultaneous game if every player moves simultaneously, only once. And a game if sequential game if the player moves sequentially, i.e. some moves first, and other move later on.

Definition (normal and extensive form representation).

A normal (or strategic) form specifies the each player 's set of strategies and a payoff function in a matrix form.
An extensive form highlights the sequential effect and the information inside of the game. Extensive forms are represented in a game tree form, starting with an unique root, and

Definition (perfect or imperfect information).

A game is one of perfect information if each information set contains a single decision node. Otherwise, it is a game of imperfect information.

To understand these concepts, lets look into the following ^b2b91c Example 12 (battle of sexes - representation).

Example (battle of sexes - representation).

Two players (1, 2) or (Husband, Wife) are deciding where to date, either Football() stadium or Opera House(). Each player's strategies are and the payoff is Represent this game in both normal and extensive form.

Proof. The game in ^b2b91c Example 12 (battle of sexes - representation) can be represented in two ways.

In the case of simultaneous game, the game is usually represented as a normal form, or strategic form. i.e. in matrix form.

	f	o
F	(3, 1)	(0, 0)
O	(0, 0)	(1, 3)

Here, the two players move simultaneously.

In the case of sequential game, we need to take a different approach. When the player 1 moves first, then the strategy for player 2 is now where for example, means that the player 2 choose after observing that the player 1 has chosen , while choosing after observing that the player 1 has chosen . Therefore, in matrix form, we have

	ff	fo	of	oo
F	(3, 1)	(3, 1)	(0, 0)	(0, 0)
O	(0, 0)	(1, 3)	(0, 0)	(1, 3)

In the case of sequential game (when player 1 moves first), representing the game in extensive form is much more convenient. However, note that every sequential game can also be represented in normal form.

The first node(decision) is called an initial node(root), while the bottom four nodes are called the terminal nodes, which includes the payoffs. And each arrows are called branch(action).

Note that for each node, the following principles must hold:

for each node, at least one arrow must start from.
for every node, only one arrow can ends to.
there exists no circular arrow.

Thus, there is a single starting point, where the proceeding only goes to the one direction.

From now on, it is assumed that player 2 moves after observing the player 1's move. i.e. perfect information. If the player 2 does not know whether the player 1 chose F or O, then we can use a information set to emphasize.

Here, the second figure shows the imperfect information. Note that for the nodes in the same information set, each of them must have the identical actions available. Thus, the player 2 in the above figure cannot differentiate between □