How do you use the concept of Bayesian equilibrium to incorporate beliefs and signals into a game?

2 How to find a Bayesian equilibrium?

A Bayesian equilibrium is a solution concept for a Bayesian game that specifies a strategy and a belief for each player. A strategy is a rule that tells each player what action to choose for each possible type they have. A belief is a probability distribution that tells each player how likely they think the other players have each possible type, given the available signals. A Bayesian equilibrium satisfies two conditions: consistency and optimality. Consistency means that the beliefs are derived from the strategies using Bayes' rule, which is a formula that updates the prior probabilities based on new evidence. Optimality means that the strategies are the best responses to the beliefs, which means that no player can increase their expected payoff by deviating from their strategy, given their belief.

3 What are signals and how do they affect beliefs?

Signals are pieces of information that the players observe or send before or during the game, and that influence their beliefs about the types of the other players. Signals can be either hard or soft, depending on how reliable or verifiable they are. Hard signals are signals that are perfectly correlated with the types, such as certificates, contracts, or test scores. Soft signals are signals that are imperfectly correlated with the types, such as opinions, recommendations, or gestures. Signals can also be either cheap or costly, depending on how much effort or resources they require. Cheap signals are signals that are easy or free to send or observe, such as words, messages, or signals. Costly signals are signals that are difficult or expensive to send or observe, such as actions, investments, or sacrifices.

4 How to use signals strategically in a Bayesian game?

Signals can be used strategically in a Bayesian game to convey or conceal information, to influence or manipulate beliefs, and to coordinate or deter actions. For example, a player can use a signal to reveal their type to the other players, if they have an incentive to do so, such as to gain trust, reputation, or cooperation. Alternatively, a player can use a signal to hide their type from the other players, if they have an incentive to do so, such as to avoid discrimination, exploitation, or competition. A player can also use a signal to affect the beliefs of the other players, if they have an influence on their actions or payoffs, such as to induce them to cooperate, compete, or exit. However, not all signals are credible or effective, as some signals can be ignored, misinterpreted, or counteracted by the other players.

5 What are some examples of Bayesian games in real life?

Bayesian games can be used to model and analyze various situations in real life that involve strategic interactions under incomplete information. For instance, auctions are a market mechanism in which buyers bid for an item sold by a seller. The buyers have private information about their valuations of the item, and the seller may have private information about the quality of the item. In signaling games, one player sends a signal to another player before choosing an action. The sender has private information about their type, and the receiver has to infer the type from the signal and choose a response. Screening games involve one player offering a contract or a menu of contracts to another player who has private information about their type. The agent then has to choose a contract based on their beliefs about the terms and payoff of the principal.