Bayesian game

A Bayesian game called in game theory a game with incomplete information, which is named after the English mathematician Thomas Bayes. The Bayestheorem, by which one can calculate conditional probabilities, forms the basis for solutions to this brand.

Bayesian games can be modeled as games with imperfect information.

• 3.2.1 Perfect Bayesian equilibrium
• 3.2.2 Examples 3.2.2.1 Example 1
• 3.2.2.2 Example 2

Definition

In a game with incomplete information there is at least one player who has not all, of the game crucial information ( iA payoff functions ) about the other players. Bayesian games are thus a priori not be analyzed. It must conjectures about the strategies and decisions of the other players in the form of probability distributions ( beliefs ) are set up.

In a model of Harsanyi Bayesian games are shown with the help of such beliefs as games with imperfect information and analyzed. In such a game, there is at least one player, which not every decision earlier ( both other players as well as random decisions ) is known. For all non- random decisions taken by players is a new player ( nature called ) was introduced, which meets these above all other decisions. Thus, the same solution concepts can be applied as for games with imperfect information.

For example, in the distribution of playing cards and card random order unknown in most cases. Looking at the distribution of the cards as the first train of the player nature, so the players have imperfect information about the decisions taken so far. In contrast, Chess is a classic example of games with perfect information.

Where formal for a game:

An equilibrium consists of a strategy of each player and their beliefs. A strategy is a set of actions for each possible type.

Mathematical Foundations ( Bayestheorem )

With the help of Bayestheorem (also called Bayes' Theorem ) can be calculated conditional probabilities:

Where conditional on the probability of:

Clearly, therefore - as in the picture on the right - is the probability that (red) arrives, arrived at the knowledge that is ( yellow area), just the proportion of the over to leading branch leading to at all branches.

Non - sequential games and sequential games

Sequential games are turn- based games where the payoffs of the players are just the sums of the payoffs of the players in each round. In each round are the strategy sets and payoff functions are identical, one also speaks of repeated games. However, this is not a requirement for sequential games.

Depending on the variety suitable different forms of representation. Usually one uses the normal form only for non-sequential games. The extensive form is, however, used for both non - sequential and sequential games.

Signal Games denote a special type of sequential Bayesian games, which are usually represented in a variant of the extensive form.

Non - sequential games

Bayesian Nash equilibrium

The Bayesian Nash equilibrium refers to a combination of strategies (if any), in which no player can improve its expected payoff by a change in strategy. He must take into account the strategies of the other players his assumptions about the probability distribution ( beliefs ). This concept is analogous to the Nash equilibrium in the case of games with perfect and complete information.

Example

In this game, two work colleagues planning their leisure activity. The ways in which the swimming pool and cinema are available. Player 1 prefers in the cinema, player 2 would prefer the swimming pool. Of course, both spend their free time with a friend rather than an enemy. However, Player 2 knows not how much one player like him.

Player 1 is either of type friend or enemy and he himself knows his type. The payouts are dependent on their relationship to each other and the choice of leisure activity: Are they friends and attend the same institution, they receive a payoff of 3 Are they enemies they would rather avoid and get a payoff of 3 if they are not the same place are. In addition, they receive the payment 2 for the preferred leisure activity.

The probability distribution of the types of player 1 is common knowledge and evenly distributed. Since the players decide ( no consultation ) at the same time, players must establish two assumptions about the choice of player 1.

There are two Bayesian Nash equilibria in which players can not make better by deviating from his strategy.

In the first equilibrium with ( [ cinema, swimming pool ]; cinema) just P ( Cinema | Friend ) = 1 and P ( Pool | enemy ) = 1 This Bayesian P follows ( friend | Cinema ) = P ( enemy | pool ) = 1 A similar procedure in the second equilibrium with ( [ swimming pool, cinema ]; swimming pool) provides a total of:

Sequential Games

Perfect Bayesian equilibrium

The Perfect Bayesian equilibrium is a refinement of subgame Perfect equilibrium for games with incomplete information.

A lot of strategies and assumptions about probability distributions ( beliefs ) is called Perfect Bayesian equilibrium if the following conditions are met:

Note: These requirements define strictly speaking, only a weak perfect Bayesian equilibrium. For a perfect Bayesian equilibrium requires a further condition.

Examples

Example 1

In the so-called Beer - Quiche game (first treated by Cho and Kreps ), there are two players:

Player 1 is either a softie or a macho. From which type of player 1 is dictated by the nature and is only known to himself. Which both players know the probability distribution: and. He has the choice for breakfast beer or quiche to order. As a softie he receives a payoff of 1 by quiche and beer 0, as a macho vice versa.

Player 2 Player 1 hits later and has the ability to duel. However, he just wants to duel, if player 1 is a softie. However, knows player 2 of player 1 is not of type, but only the breakfast order ( signal). Player 1 wants a duel in no case and for no duel always gets payout 2nd player 2 gets payoff 1 if he dueled with a softie or the duel avoids a macho. In all other cases he receives 0 payoff

The diagram on the right illustrates this signal game with accumulated sum payout.

Possible strategies for player 1 are:

Possible strategies for player 2 are:

There are two perfect Bayesian equilibria:

In the first equilibrium with ( [ quiche, quiche ], [ duel duel No ] ), P is true ( Quiche | Softie ) = P ( Quiche | Macho ) = 1 With the Bayestheorem results in accordance with claim 3 of the amounted of Player 2:

.

Analogously in the second equilibrium ( [ Beer, Beer ], [ No duel duel ] ) the beliefs.

Overall, therefore, the following applies:

Although both equilibria satisfy the conditions of a perfect Bayesian equilibrium, Cho and Kreps but expand the concept of equilibrium to another claim. In general, this is called an intuitive criterion and ensures that the first balance is excluded.

Example 2

This example illustrates how beliefs change over a repeated game with the help of Bayestheorems. There is only one player who bets on the outcome of a possibly rigged coin toss. A correct prediction brings a payoff of 1, otherwise 0 The possible types of coins are given by nature and are equally likely: Always head ( IK ), Fair Coin ( FM) and Always number (IZ ).

(1/3 1/3;; 1/3) for the first round of the player places his beliefs according to the probability distribution of types ( IK, FM; IZ ). First strategy head brings the same expected payoff as strategy number. The player is thus freely on heads or tails, and then throws the coin. Before it sets another time, he adjusts his beliefs using the Bayestheorems to:

Head ( K) is thrown:

Number (Z ) is thrown:

Go on the game, to adapt the beliefs above. It is only to note that change the probabilities or after each throw. Assuming that in the first throw head has fallen, the following applies:

Overall, the player does so in the first train random ( uniformly distributed ) on heads or tails. Then it is put on as long as the result of the first round until the Converse occurs. If this case ever eventually occurs, it is from this time choose his strategy again by chance, as it is certainly a fair coin.