# Game Theory

In economics, game theory is the study of interaction between different participants in a market. The objective of game theory is to identify the optimal strategy for each participant.

An economic game represents competition between different economic agents. A game typically has three elements: players, strategies and payoffs. A player is a participant in an economic game. Players are typically firms competing with each other, but they could be consumers or a firm and a consumer depending on the context of the game. A strategy is a course of action which either player can adopt. Payoff refers the net benefit or loss that accrue to each player from carrying out his strategy.

A Nash equilibrium is combination of strategies such that no player has any incentive to change its strategy unilaterally.

## Types of Economic Games

Games are either simultaneous-move or sequential, static or dynamic, one-off or repeated, cooperative or non-cooperative, etc.

### Simultaneous-move Game vs Sequential Game

A **simultaneous-move game** is a game in which both players must choose their strategies at the same time. No player knows for sure what the other player’s course of action would be.

Examples of simultaneous-move games include:

- Two firms in an oligopoly deciding about their output level (in Cournot model) or price level (in Bertrand model).
- Two prisoners deciding about whether to confess or not in the prisoners’ dilemma.

A **sequential game** is a game in which one player choses its strategy before the other. The player which decides first typically has a first-mover advantage.

Following are examples of a sequential game:

- Two firms in a Stackelberg model, an oligopoly in which one firm decides about its output level first. The other firm has no option but to cater to the demand which is left-over.
- Two firms are looking to open a retail store in a town which has demand for only one store. The firm which credibly commits to opening the store first wins all the revenue from the new market.

### Static Games vs Dynamic Games

A **static game** is a game which is both simultaneous-move and one-off i.e. there is no repetition and both firms decide at the same time. A dynamic game, on the other hand, is one which is either repeated or sequential.

A **repeated game** is a game which is played over and over again. Repeated games are further classified into infinitely-repeated games, games which continue forever; and finitely-repeated games, which are exact opposite of infinitely-repeated games.

### Cooperative Games vs Non-cooperative Games

A **cooperative-game** is a game in which both players can communicate and cooperate, and a **non-cooperative game** is a game in which no cooperation is possible.

Most of the significant economic games are non-cooperative games. For example:

- Firms in an oligopoly can’t collude because it is illegal to fix price or output or even to talk about price, etc. Hence, they engage in non-cooperative games when deciding about their output simultaneously.
- Prisoners in a prisoner’s dilemma can’t communicate. Hence, a prisoners’ dilemma is a non-cooperative game.

## Presentation of a Game

A game can be visualized using either a payoff matrix or a decision tree (also called extensive form of a game).

### Normal Form

The **normal form** of a game is a presentation in which the game is visualized using a payoff matrix. A payoff matrix is a table which lists the players of a game, their strategies and their associated payoffs such that the payoff that accrue to the row player is listed first.

Normal form is useful in case of a static game i.e. a game in which both players play the game only once at the same time. Such games are also called one-off simultaneous-move games.

Following is a normal-form presentation of prisoner’s dilemma:

Payoffs in Prison Terms | Prisoner Q | ||
---|---|---|---|

Confess | Do not Confess | ||

Prisoner P | Confess | -4,-4 | -1,-8 |

Do not Confess | -8,-1 | -2,-2 |

### Extensive Form

It is better to present a game using a decision tree (also called a game tree) when the game under consideration is a dynamic game i.e. it is repeated or sequential.

**Extensive form** of a game is a presentation in which the game is visualized using a decision tree.

The following diagram shows the extensive form of the prisoner’s dilemma.

## Types of Strategies

Strategies that players in an economic game can employ are classified into either dominant or dominated, pure or mixed, etc.

### Dominant Strategy vs Dominated Strategy

A dominant strategy is a strategy which is optimal regardless of strategy of the other player. A dominant strategy results in the maximum payoff unconditionally i.e. it always results in the best outcome no matter what the opponent does. Not all players/games have dominant strategies.

A dominated strategy is a strategy which generates the worst outcome in all instances. A strategy is dominated if other strategies always generate a better payoff. When one strategy is dominant, other strategies are dominated. A player can have a dominated strategy in a game even if there is no dominant strategy.

### Pure Strategy vs Mixed Strategy

A pure strategy is a strategy which has two discrete values i.e. 0 or 1. For example, a firm either advertises or it doesn’t advertise. Similarly, a prisoner either confesses or it doesn’t.

A mixed strategy is a strategy which is a combination of two or more pure strategies based on their expected probabilities.

Written by Obaidullah Jan, ACA, CFA and last revised on