Bayesian game

In game theory, a Bayesian game is a strategic decision-making model which assumes players have incomplete information. Players may hold private information relevant to the game, meaning that the payoffs are not common knowledge. Bayesian games model the outcome of player interactions using aspects of Bayesian probability. They are notable because they allowed the specification of the solutions to games with incomplete information for the first time in game theory.
Hungarian economist John C. Harsanyi introduced the concept of Bayesian games in three papers from 1967 and 1968: He was awarded the Nobel Memorial Prize in Economic Sciences for these and other contributions to game theory in 1994. Roughly speaking, Harsanyi defined Bayesian games in the following way: players are assigned a set of characteristics by nature at the start of the game. By mapping probability distributions to these characteristics and by calculating the outcome of the game using Bayesian probability, the result is a game whose solution is, for [|technical reasons], far easier to calculate than a similar game in a non-Bayesian context.

Normal form games with incomplete information

Elements

A Bayesian game is defined by, where it consists of the following elements:
; Set of players, N: The set of players within the game
; Action sets, a_i: The set of actions available to Player i. An action profile a = is a list of actions, one for each player
; Type sets, t_i: The set of types of players i. "Types" capture the private information a player can have. A type profile t = is a list of types, one for each player
; Payoff functions, u: Assign a payoff to a player given their type and the action profile. A payoff function, u = denotes the utilities of player i
; Prior, p: A probability distribution over all possible type profiles, where p = p is the probability that Player 1 has type t₁ and Player N has type t_N.

Pure strategies

In a strategic game, a pure strategy is a player's choice of action at each point where the player must make a decision.

Three stages

There are three stages of Bayesian games, each describing the players' knowledge of types within the game.

Ex-ante stage game. Players do not know their types or those of other players. A player recognizes payoffs as expected values based on a prior distribution of all possible types.
Interim stage game. Players know their type but only a probability distribution of other players. When considering payoffs, a player studies the expected value of the other player's type.
Ex-post stage game. Players know their types and those of other players. The payoffs are known to players.
Improvements over non-Bayesian games

There are two important and novel aspects to Bayesian games that were themselves specified by Harsanyi. The first is that Bayesian games should be considered and structured identically to complete information games. However, by attaching probability to the game, the final game functions as an incomplete information game. Therefore, players can be essentially modeled as having incomplete information, and the probability space of the game still follows the law of total probability. Bayesian games are also useful because they do not require infinite sequential calculations, which is typical of strategic thinking in repeated games. Infinite sequential calculations would arise where players try to "get into each other's heads." For example, one may ask questions and decide, "If I expect some action from player B, then player B will anticipate that I expect that action, so then I should anticipate that anticipation" ad infinitum. Bayesian games allow for the calculation of these outcomes in one move by assigning different probability weights to different outcomes simultaneously. The effect of this is that Bayesian games allow for the modeling of a number of games that in a non-Bayesian setting would be irrational to compute.

Bayesian Nash equilibrium

A Bayesian Nash Equilibrium is a Nash equilibrium for a Bayesian game, which is derived from the ex-ante normal form game associated with the Bayesian framework.
In a traditional game, a strategy profile is a Nash equilibrium if every player's strategy is a best response to the other players' strategies. In this situation, no player can unilaterally change their strategy to achieve a higher payoff, given the strategies chosen by the other players.
For a Bayesian game, the concept of Nash equilibrium extends to include the uncertainty about the state of nature: Each player maximizes their expected payoff based on their beliefs about the state of nature, which are formed using Bayes' rule. A strategy profile is a Bayesian Nash equilibrium if, for every player, the strategy maximizes player 's expected payoff, given:

Their beliefs about the state of nature,
The strategies played by other players.

Mathematically:
For finite Bayesian games, the BNE can be represented in two equivalent ways:

Agent-Form Game: The number of players is expanded from to, where each type of a player is treated as a separate "player." This is detailed in Theorem 9.51 of the book Game Theory.
Induced Normal Form Game: The number of players remains, but the action space for each player is expanded from to. This means the strategy now specifies an action for every type of player. This representation is discussed in Section 6.3.3 of the book Multiagent Systems.

In both cases, the Nash equilibrium for the game can be computed using these representations, and the BNE can be recovered from the results. A linear program can be formulated to compute the BNE efficiently for two-player Bayesian games with a zero-sum objective.

Extensive form games with incomplete information

Elements of extensive form games

with perfect or imperfect information, have the following elements:

Set of players
Set of decision nodes
A player function assigning a player to each decision node
Set of actions for each player at each of her decision nodes
Set of terminal nodes
A payoff function for each player
Nature and information sets

An unfilled circle usually denotes nature's node. Its strategy is always specified and completely mixed. Although Nature is generally at the tree's root, it can also move to other points.
An information set of player i is a subset of player i's decision nodes that she cannot distinguish between. If player i is at one of her decision nodes in an information set, she does not know which node within the information set she is at.
For two decision nodes to be in the same information set, they must

Belong to the same player; and
Have the same set of actions

Information sets are denoted by dotted lines, the most common notation today.

The role of beliefs

In Bayesian games, players' beliefs about the game are denoted by a probability distribution over various types.
If players do not have private information, the probability distribution over types is known as a common prior.

Bayes' rule

An assessment of an extensive form game is a pair

Behavior Strategy profile; and
Belief system

An assessment satisfies Bayes' rule if μ = Pr / Σ Pr whenever h_i is reached with strictly positive probability according to.

Perfect Bayesian equilibrium

A perfect Bayesian equilibrium in an extensive form game is a combination of strategies and a specification of beliefs such that the following two conditions are satisfied:

Bayesian consistency: the beliefs are consistent with the strategies under consideration;
Sequential rationality: the players choose optimally given their beliefs.

Bayesian Nash equilibrium can result in implausible equilibria in dynamic games, where players move sequentially rather than simultaneously. As in games of complete information, these can arise via non-credible strategies off the equilibrium path. In games of incomplete information, non-credible beliefs are also possible.
To address these issues, Perfect Bayesian equilibrium, according to subgame perfect equilibrium, requires that subsequent play be optimal starting from any information set. It also requires that beliefs be updated consistently with Bayes' rule on every path of play that occurs with a positive probability.

Stochastic Bayesian games

Stochastic Bayesian games combine the definitions of Bayesian games and stochastic game to represent environment states with stochastic transitions between states as well as uncertainty about the types of different players in each state. The resulting model is solved via a recursive combination of the Bayesian Nash equilibrium and the Bellman optimality equation. Stochastic Bayesian games have been used to address diverse problems, including defense and security planning, cybersecurity of power plants, autonomous driving, mobile edge computing, self-stabilization in dynamic systems, and misbehavior treating in crowdsourcing IoT.

Incomplete information over collective agency

The definition of Bayesian games and Bayesian equilibrium has been extended to deal with collective agency. One approach is to continue to treat individual players as reasoning in isolation but to allow them, with some probability, to reason from the perspective of a collective. Another approach is to assume that players within any collective agent know that the agent exists but that other players do not know this, although they suspect it with some probability. For example, Alice and Bob may sometimes optimize as individuals and sometimes collude as a team, depending on the state of nature, but other players may not know which of these is the case.