# Bayesovske igre

Mihetec, Marija (2014) Bayesovske igre. Diploma thesis, Faculty of Science > Department of Mathematics.

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## Abstract

Bayesian games are games with incomplete information, i.e. such games in which some players have private information about the game that other players do not know and that private information are called the type of the player. Bayesian games consist of a set of players, \$\{1, 2, . . . , N\}\$, the set of types of players, the probability (belief) function which specifies a probability distribution representing what player would believe about the types of the other players, and the payoff function of players. The exact type of a player is not known deterministically to the other players who however have a probabilistic guess of what this type is. The model of game is simplifies by the constistency of beliefs and beliefs are consistent if they are derived from the same probability distribution by conditioning on each players’ type. Speaking differently, if the beliefs are consistent, the only source of differences in beliefs is difference in information. To calculate the equilibrium and payoff in Bayesian games, we use its equivalent Selten game which enables a Bayesian game to be transformed to a strategic form game with complete information. The idea used in formulating a Selten game is to have type agents; in fact, each player in the original Bayesian game is replaced with a number of type agents. Therefore, while the Nash equilibrium is a strategies profile chosen by players as a best response considering strategies of other players, a pure strategy Bayesian Nash equilibrium is a pure strategy Nash equilibrium of the equivalent Selten game. A Stochastic Game, \$\Gamma\$, consists of a finite set of positions or states, \$\{1, 2, . . . , N\}\$, one of which is specified as the starting position. We denote by \$\Gamma^k\$ the game in which \$k\$ is the starting position and the term stochastic game refers to the collection \$\Gamma = \{\Gamma^k|k = 1, 2, . . . , N\}\$. Game \$\Gamma^k\$ associates with a matrix game , \$A^k = (a_{ij}^k)\$. If the stochastic game is in state \$k\$, the players simultaneously choose a row and column of \$A^k\$, say \$i\$ and \$j\$. As a result, two things happen. First, Player 1 wins the amount \$a_{ij}^k\$ from Player 2. Second, with probabilities that depend on \$i,j\$ and \$k\$, the game either stops, or it moves to another state (possibly the same one). The payoffs accumulate throughout the game until it stops. To make sure the game eventually stops, we make the assumption that all the stopping probabilities are positive. Under this assumption, the probability is one that the game ends in a finite number of moves. This assumption also makes the expected accumulated payoff finite no matter how the game is played. Player 1 wishes to maximize the total accumulated payoff and Player 2 to minimize it. A payoff does not end the game but after a payoff is made, it is then decided at random whether the game ends and, if not, which state should be played next. Since no upper bound can be placed on the length of the game, this is an infinite game. In stochastic games, the value and optimal strategies for the players exist for every starting position. Strategies which prescribe for a player the same probabilities for his choice every time the same position is reached, by whatever route, are called stationary strategies. The Shapley’s theorem states that the value and stationary optimal strategies exist there.

Item Type: Thesis (Diploma thesis) Čaklović, Lavoslav 2014 23 NATURAL SCIENCES > Mathematics Faculty of Science > Department of Mathematics Iva Prah 05 Jun 2015 11:26 05 Jun 2015 11:26 http://digre.pmf.unizg.hr/id/eprint/4016

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