How do you use incomplete information in game theory?

1 Types of games

Game theory distinguishes between two types of games based on the level of information available: complete information games and incomplete information games. In complete information games, such as chess or tic-tac-toe, all players know the rules, the possible moves, and the outcomes of each move. In incomplete information games, such as poker or bargaining, some players may have private information that others do not know, such as their cards or their reservation price. Incomplete information games are more realistic and complex, as they involve uncertainty, beliefs, and signals.

2 Bayesian games

A common way to model incomplete information games is to use Bayesian games, which are games where players have beliefs about the unknown aspects of the game, such as the types or actions of other players. These beliefs are represented by probability distributions that reflect the players' subjective expectations. For example, in a poker game, you may have a belief about the probability that your opponent has a certain hand, based on the cards you see and the bets they make. Bayesian games also assume that players update their beliefs according to Bayes' rule, which is a mathematical formula that tells you how to revise your probability estimates based on new evidence.

3 Nash equilibrium

A central concept in game theory is Nash equilibrium, which is a situation where no player can benefit from changing their strategy, given the strategies of the other players. In complete information games, finding a Nash equilibrium is relatively straightforward, as you can compare the payoffs of each possible strategy combination. However, in incomplete information games, finding a Nash equilibrium is more challenging, as you have to take into account the beliefs and signals of the other players. One way to find a Nash equilibrium in Bayesian games is to use the concept of Bayesian Nash equilibrium, which is a strategy profile that maximizes each player's expected payoff, given their beliefs and the strategies of the other players.

4 Signaling and screening

One of the main challenges in incomplete information games is how to communicate or extract information from the other players. Signaling and screening are two methods that can help you achieve this. Signaling is when a player with private information sends a message or performs an action that reveals something about their type or intention. For example, in a job market, a worker may signal their ability by acquiring a degree or a certification. Screening is when a player without private information designs a mechanism or a test that induces the other player to reveal their type or intention. For example, in an insurance market, a company may screen the risk level of a customer by offering different premiums or deductibles.

5 Applications and examples

Incomplete information games are widely used in various fields and domains, such as economics, political science, sociology, or biology. They can help you understand and predict the behavior and outcomes of strategic interactions in different contexts and scenarios. For example, incomplete information games can help you analyze the effects of asymmetric information, which is when one party has more or better information than the other, on market efficiency, contract design, or incentive provision. They can also help you study the role of reputation, credibility, or commitment in bargaining, negotiation, or conflict resolution.

6 Here’s what else to consider

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