Players may hold private information relevant to the game, meaning that the payoffs are not common knowledge.
Hungarian economist John C. Harsanyi introduced the concept of Bayesian games in three papers from 1967 and 1968:[2][3][4] 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.
, where it consists of the following elements: [5] In a strategic game, a pure strategy is a player's choice of action at each point where the player must make a decision.
There are two important and novel aspects to Bayesian games that were themselves specified by Harsanyi.
Infinite sequential calculations would arise where players try to "get into each other's heads."
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.
For finite Bayesian games (where the action and type spaces are finite), the BNE can be represented in two equivalent ways: 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.
[11] Extensive form games with perfect or imperfect information, have the following elements:[12] An unfilled circle usually denotes nature's node.
Although Nature is generally at the tree's root, it can also move to other points.
For two decision nodes to be in the same information set, they must[13] Information sets are denoted by dotted lines, the most common notation today.
If players do not have private information, the probability distribution over types is known as a common prior.
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:[15] 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.
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,[17] cybersecurity of power plants,[18] autonomous driving,[19] mobile edge computing,[20] self-stabilization in dynamic systems,[21] and misbehavior treating in crowdsourcing IoT.
[24] 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.
Thus, the payoff matrix of this Normal-form game for both players depends on the type of the suspect.
ad infinitum – common knowledge), play in the game will be as follows according to perfect Bayesian equilibrium:[25][26] When the type is "criminal", the dominant strategy for the suspect is to shoot, and when the type is "civilian", the dominant strategy for the suspect is not to shoot; alternative strictly dominated strategy can thus be removed.
The Market for Lemons is related to a concept known as adverse selection.
Player 1 is a potential buyer who is interested in the vehicle.
Player1 will never have complete information about player2, but may be able to infer the probability of type1 and type2 appearing from whether the previous firm entering the market was blocked, it is a Bayesian game.
The reason for these judgments is that there are blocking costs for player2, which may need to make significant price cuts to prevent player1 from entering the market, so it will block player1 when the profit it steals from entering the market is greater than the blocking costs.