Perfect Bayesian balance

The perfect Bayesian equilibrium (short: PBG ) is a solution concept in game theory . It is used to solve dynamic games with incomplete information .

Since implausible Nash equilibria can no longer be ruled out by partial game perfection if the information is incomplete , the equilibrium concept is expanded to include the component of sequential rationality and so-called "beliefs" (assessments or assumptions about the probability of occurrence). This approach was first mentioned in 1991 by Drew Fudenberg and Jean Tirole .

Not to be confused with the perfect Bayesian equilibrium, since the latter is intended for static games.

Incomplete and imperfect information

Games with incomplete information cannot be analyzed or only in special cases. Therefore they are modeled as games with complete, but imperfect (imperfect) information ( Harsanyi transformation). Imperfect information means that at least one player does not know the complete history of the game. In a game in extensive form , this shows up when at least one player has an information set with more than one decision node. So there is only perfect information if all the information in the game is a single element.

While chess is an example of a game with complete information ( assuming complete memory ), poker is an example of a game with imperfect information. Here the cards are distributed randomly. A player knows his own cards, but not those of his fellow players and vice versa. This is the only reason why bluffing can make sense.

Games with imperfect information are modeled with a random move at the beginning of the game. This random move decides on the types or characteristics of the players (in poker, on the cards of the players). In the literature one also often finds the term "natural train", since nature is inserted as an additional player. Assuming complete information is available, every player knows the probability distribution of a random move and thus the entire game tree including the payouts ( common knowledge ).

If the information is incomplete, however, not everyone knows the conditions under which the game is played. For example, a player may not know the payouts or preferences of the other players, but his own. Consequently, this player could not make any assumptions about their strategies . A credible balance cannot be established under these circumstances.

Definition of perfect Bayesian equilibrium

With the perfect Bayesian equilibrium , unbelievable equilibria can be excluded, provided that certain criteria are met. It consists of a profile of strategies and a system of assessments that meet requirements one to four: ${\ displaystyle {\ begin {aligned} (\ sigma, \ mu) \ end {aligned}}}$

Requirement 1:

Each player has to have beliefs about each of his information sets about which node he is at.

Requirement 2:

Given these assessments, the players behave sequentially and rationally . This requires optimal reactions of every player to every amount of information, given that the amount of information is reached and given the strategies of the other players from this move on.

Requirement 3:

In all sets of information on the equilibrium path, the estimates are formed according to Bayes' theorem . A set of information is on the equilibrium path if one of its nodes is reached with a positive probability, given the equilibrium strategies of the players.

Requirement 4:

In amounts of information outside the equilibrium path (“off-equilibrium”), the assessments are formed using Bayesian rule whenever possible (see below). If this is not possible, the assessments can be freely chosen.

Some authors content themselves with requirements 1 to 3 in order to define a perfect Bayesian equilibrium. This is often referred to as a weak perfect Bayesian equilibrium. Nevertheless, requirement 4 is necessary to rule out implausible equilibria.

Example I.

Game 1; This game has no real sub-games. It goes back to the German economist and mathematician Reinhard Selten .

Game 1 shows why a refinement of the equilibrium concept is necessary in order to rule out implausible Nash equilibria in dynamic games with imperfect information . The game is shown in extensive form in order to clarify the chronological sequence of the decisions. The possible payouts are given to the end nodes (a, b, c, d, e). Game 1 has no real sub-games . A sub-game begins with a single-element set of information and includes all subsequent decision nodes, provided that their information sets are completely contained. Consequently, every Nash equilibrium in the entire game is trivially subgame perfect.

First player 1 (red) can choose between the pure strategies O, M and U. Player 2 (blue) then decides between the pure strategies O 'and U'. If player 1 plays O, the game ends with the payout (1.3) . This means that player 1 gets payout 1 and player 2 gets payout 3. If player 1 chooses M or U, the amount of information from player 2 is reached. This now learns that either M or U was played.

In the normal form of the game it can be seen that there are exactly two Nash equilibria in pure strategies. These are and , with the payouts (1.3) and (2.1) respectively . ${\ displaystyle {\ begin {aligned} (O, U ') \ end {aligned}}}$ ${\ displaystyle {\ begin {aligned} (U, O ') \ end {aligned}}}$

Player 1 / Player 2	O'	U '
O	(1, 3)	(1, 3)
M.	(0, 2)	(0, 1)
U	(2, 1)	(0, 0)

Since U 'is dominated , player 2 will never play U', given that the amount of information is reached. The balance is therefore implausible. Incredible equilibria can now be ruled out with the concept of the perfect Bayesian equilibrium. ${\ displaystyle {\ begin {aligned} (O, U ') \ end {aligned}}}$

To do this, player 2 must first use his amount of information to make assessments as to which node he is at: The probability of being at node 1 (above) is now . Being the one at node 2 is . ${\ displaystyle p}$ ${\ displaystyle (1-p)}$

If player 2 plays O ', his expected payout is . The expected payout of the strategy U 'is, however . There , O 'is strictly dominant for player 2 . The equilibrium is therefore not compatible with the first two requirements and consequently not a perfect Bayesian equilibrium. ${\ displaystyle {\ begin {aligned} p * 2 + (1-p) * 1 = p + 1 \ end {aligned}}}$ ${\ displaystyle {\ begin {aligned} p * 1 + (1-p) * 0 = p \ end {aligned}}}$ ${\ displaystyle {\ begin {aligned} p + 1> p \ end {aligned}}}$ ${\ displaystyle {\ begin {aligned} (O, U ') \ end {aligned}}}$

In general, the optimal decision a player makes will depend on his judgment. In game 1, however, it is independent of this, since U 'is strictly dominated by O'.

Example II

Game 2; This game has a real subgame, starting at the knot (2: 1).

Game 2 now has a real sub-game with the Nash equilibrium . The whole game has a clear subgame perfect balance . Together with the assessment , this strategy satisfies requirements 1 to 3. Requirement 4 is trivially fulfilled, since there are no quantities of information outside the equilibrium path. So there is a perfect Bayesian balance . ${\ displaystyle {\ begin {aligned} (S, Y) \ end {aligned}}}$ ${\ displaystyle {\ begin {aligned} (B, S, Y) \ end {aligned}}}$ ${\ displaystyle p = 1}$ ${\ displaystyle {\ begin {aligned} ((B, S, Y), p = 1) \ end {aligned}}}$

The strategy is also a Nash equilibrium as no player has any incentive to deviate. With the assessment , she also meets requirements 1 to 3, although the balance is not subgame perfect. ${\ displaystyle {\ begin {aligned} (A, S, X) \ end {aligned}}}$ ${\ displaystyle p = 0}$

Given these assessments, player 3 then behaves sequentially rationally when he chooses strategy X. However, the assessments are not consistent with the strategy of player 2. Requirement 4 now applies, which states that assessments must be made according to Bayesian rule whenever possible, even outside the equilibrium path. Player 3 must therefore have the assessments if player 2 chooses strategy S. This is a contradiction. therefore does not meet requirements 1 to 4 and is therefore not a perfect Bayesian equilibrium. ${\ displaystyle p = 1}$ ${\ displaystyle {\ begin {aligned} ((A, S, X), p = 0) \ end {aligned}}}$

However, it is debatable when Bayesian rule can be applied outside the equilibrium path. Strictly speaking, this would not be possible here, since the amount of information is reached with a probability of 0 and Bayesian rule would require division by 0.

The meaning of "whenever possible"

Game 3; Player 3 has an information set that includes three nodes.

The very vague formulation of requirement 4 states that the assessments outside the equilibrium path are formed with Bayes' theorem whenever possible . This is particularly evident for games in extensive form.

Game 3 shows that it is not entirely clear how exactly it should be interpreted whenever possible . Player 1 (red) could, for example, pursue strategy B, while player 2 (blue) pursues strategy (B ', B'). This means that player 2 chooses action B 'on both of his decision nodes.

Given these strategies, the amount of information from player 3 (green) is never reached. It is therefore out of the equilibrium path. One could argue that the assessments at points X, Y and Z should be freely selectable, since the amount of information given by the equilibrium strategies is reached with probability 0 and Bayesian rule cannot be applied to it.

However, one could also demand that the assessments should be updated during the course of the game if an affected player receives new information. If player 3 unexpectedly finds that his amount of information has been reached, he must also adjust his assessments accordingly.

Whether X or Y is reached earlier depends on how susceptible the balance is to mistakes by the players. To investigate this, Reinhard Selten proposed the concept of trembling-hand-perfect balance in 1975 . The basic idea of this approach is that every player chooses any action with at least a small positive probability, since his hand could start to "tremble" when choosing his strategy. In principle, every node in the game tree can be reached.

If player 1 inadvertently plays strategy A, node Y is reached because player 2 is following strategy (B ', B'). For X to be reached, both players would have to commit a mistake. According to Bayesian rule, it is in principle more likely that node Y will be reached. However, no statement can be made here about the relations to the assessment of being at node Z.

It turns out that within the framework of the perfect Bayesian equilibrium there is no final definition of “whenever possible” that can be carried over to the general case. Consequently, this must be examined separately and possibly also intuitively for each game.

A possible solution to this problem would be the sequential equilibrium formulated by Kreps and Wilson in 1982 . Similar to trembling-hand-perfect balance, this assumes a sequence of completely mixed strategies . The idea behind this approach is to find a fully mixed strategy that converges against the actual equilibrium strategy. Since in this way all amounts of information can be reached with a strictly positive probability, Bayesian rule would always be applicable. However, since this method is very complicated to use, it is sufficient in most cases to restrict oneself to the perfect Bayesian equilibrium.

Refinement of the perfect Bayesian balance

Game 4; Strategy U is strictly dominated by O.

The concept of perfect Bayesian equilibrium can be further refined by introducing an additional requirement.

Requirement 5:

Every player must have judgment at nodes which are out of the equilibrium path and which can only be reached if another player is playing a strictly dominated strategy , if this is possible . ${\ displaystyle p = 0}$

In the context of the perfect Bayesian equilibrium, it is assumed that the players will never play a strictly dominated strategy, starting from an arbitrary amount of information (requirement 2). On the other hand, it doesn't make sense for a player to believe that another player would choose such a strategy. Claim 3 prevents such inconsistent assessments on the equilibrium path. Outside the equilibrium path, the whole thing is more problematic, since requirement 4 does not always apply here.

In the game shown, there are two perfectly Bayesian equilibria that meet requirements 1 to 4, namely and . Since in the second equilibrium the amount of information lies outside the equilibrium path and requirement 4 does not impose any restrictions, the assessments can be freely selected here. ${\ displaystyle {\ begin {aligned} ((M, O '), p = 1) \ end {aligned}}}$ ${\ displaystyle {\ begin {aligned} ((O, U '), p \ leq \ 0.5) \ end {aligned}}}$

Strategy U is, however, strictly dominated by O. So it doesn't make sense of player 2 to believe that player 1 would play U. If U is played, the assessment should therefore also be at the following node . Conversely, this means that it must be. The only equilibrium that meets requirements 1 to 5 is therefore . ${\ displaystyle (1-p) = 0}$ ${\ displaystyle p = 1}$ ${\ displaystyle {\ begin {aligned} ((M, O '), p = 1) \ end {aligned}}}$

An exception is if both U and M are strictly dominated. In this case, the assessments are again freely selectable, since and cannot be 0 at the same time. ${\ displaystyle p}$ ${\ displaystyle (1-p)}$

Here, too, it can be argued how sensible this fifth requirement really is. Should player 2 find, despite the equilibrium strategy O of player 1, that his amount of information has been reached, he must assume that player 1 has made a mistake. In this case, player 1 could just as well have accidentally chosen the strictly dominated strategy U, which would suggest that the assessments here should be freely selectable. However, this approach goes beyond the explanatory power of the perfect Bayesian equilibrium. In order to come to a final solution here, one would have to fall back on the stricter concepts of trembling-hand-perfect balance or sequential balance .

Signal games

Signal games are Bayesian games in which the signals sent by a player influence the decision of an opponent. In a simple signal game there are two players, namely a “transmitter” and a “receiver”. At the beginning, a natural trait decides the type of transmitter, assuming that the probability distribution is common knowledge . The transmitter then learns its type and can now select a specific signal. The receiver observes the signal from the transmitter, but not its type, and can then also choose an action. There is therefore an asymmetrical distribution of information , since the sender has private information about its type.

The possible strategies of the sender can be divided into pooling and separating . A pooling strategy exists when all types choose the same signal. Separating is a strategy when each type sends out a different signal. If only some types send different signals, the strategy is called semi-separating .

The beer quiche game

Game 5; Beer quiche game based on cho and crepes.

A well-known signal game is the beer quiche game by Cho and Kreps (1987). The idea of the game can be described as follows:

In a bar there is a bat (player 2) who wants to fight another man (player 1). Player 1 knows his type and also knows that player 2 wants to fight him. Whenever possible, regardless of his type, player 1 wants to avoid the duel. Player 1 also knows that his order will have an impact on Player 2's decision. Player 1 must therefore try to credibly signal that he is a macho in order to avoid the duel. Player 2, on the other hand, does not know whether player 1 is a "softie" or a "macho", but he does know the probability distribution. Player 2 only benefits from the duel if player 1 is a softie, otherwise he will be beaten up himself. Player 2 therefore wants to observe what player 1 is ordering at the bar in order to be able to draw conclusions about his type. He knows that machos prefer to order beer, while softies prefer quiche.

Game 5 represents the game with the exemplary probability as a game tree. Player 1 (sender) has the pure strategies (beer, beer), (quiche, quiche), (beer, quiche) and (quiche, beer) . The first (second) action is therefore played if nature chooses the “softie” type (“macho” type) for player 1. The first two strategies of the transmitter are pooling strategies, since both types send out the same signal. The last two are separating strategies. ${\ displaystyle p = (1/2)}$

Player 2 (receiver) has the pure strategies (duel, duel), (duel, no duel), (no duel, duel) and (no duel, no duel) . The first (second) action is chosen if player observes 2 beers (quiche).

The analysis of the game shows that there can be no separating equilibrium here, since player 1, if he is a softie, would always have an incentive to deviate. With a separating strategy, player 2 can deduce his type directly based on the order from player 1. As a result, he would always start a duel with the softie and not with the macho. Since player 1 wants to avoid the duel as much as possible, as a softie he would have an incentive to choose the other order.

However, there are two perfect Bayesian equilibria, which are both pooling equilibria, namely:

{\ displaystyle {\ begin {aligned} \ left (\ left ([{\ text {beer}}, {\ text {beer}}], [{\ text {no duel}}, {\ text {duel}} ] \ right), p \ leq \ 0 {,} 5 \, q \ geq \ 0 {,} 5 \ right) \\\ left (\ left ([{\ text {Quiche}}, {\ text {Quiche }}], [{\ text {duel}}, {\ text {no duel}}] \ right), p \ geq \ 0 {,} 5 \, q \ leq \ 0 {,} 5 \ right) \ end {aligned}}}

${\ displaystyle p}$ is the conditional probability and is . ${\ displaystyle \ operatorname {prob} ({\ text {Softie}} \ mid {\ text {Beer}})}$ ${\ displaystyle q}$ ${\ displaystyle \ operatorname {prob} ({\ text {Softie}} \ mid {\ text {Quiche}})}$

Even if the second pooling equilibrium does not seem very intuitive, it still fulfills requirements 1 to 5. However, this equilibrium can also be eliminated through the intuitive criteria of Cho and Kreps.

literature

Robert Gibbons: A Primer in Game Theory . Financial Times, Harlow 1992, ISBN 978-0-7450-1159-2 .
Ken Binmore: Fun and Games: A Text on Game Theory . DC Heath and Company, Lexington, Massachusetts 1992, ISBN 0-669-24603-4 .
Drew Fudenberg, Jean Tirole: Game Theory . The MIT Press, Cambridge, Massachusetts 1991, ISBN 978-0-262-06141-4 .
Martin J. Osborne, Ariel Rubinstein: A Course in Game Theory . The MIT Press, Cambridge, Massachusetts 1994, ISBN 978-0-262-15041-5 .
Manfred J. Holler , Gerhard Illing: Introduction to game theory . Springer, Berlin Heidelberg 2008, ISBN 978-3-540-69372-7 .
Gonzáles-Días, Julio; Meléndez-Jiménez, Miguel A .: On the Notion of Perfect Bayesian Equilibrium . Departamento de Estadística e Investigación Operativa, Universidad de Santiago de Compostela; Department of Teoría e Historia Económica, Universidad de Málaga. Available at: http://eio.usc.es/pub/julio/papers/Perfect_Bayesian.pdf
Battigalli, Pierpaolo: Strategic Independence and Perfect Bayesian Equilibria . Department of Economics, Princeton University, Princeton, New Jersey 1995. Available from: http://upi-yptk.ac.id/Ekonomi/Battigalli_Strategic.pdf

Web links

Encyclopedia for Game Theory - Project of the University of Munich on the subject of game theory
Lecture notes on game theory - archive of the economics faculty of the LMU Munich

Individual evidence

^ ^A ^b Drew Fudenberg, Jean Tirole: Game Theory . The MIT Press, Cambridge, Massachusetts 1991, ISBN 978-0-262-06141-4 . Pp. 321-323

↑ Ken Binmore: Fun and Games: A Text on Game Theory . DC Heath and Company, Lexington, Massachusetts 1992, ISBN 0-669-24603-4 , pp. 501-503 .

↑ ^a ^b ^c ^d ^e Robert Gibbons: A Primer in Game Theory . Financial Times, Harlow 1992, ISBN 978-0-7450-1159-2 , pp. 175-182 .

↑ ^a ^b ^c Manfred J. Holler, Gerhard Illing: Introduction to game theory . Springer, Berlin, Heidelberg 2008, ISBN 978-3-540-69372-7 , pp. 110-124 .

↑ Gonzáles-Días, Julio; Meléndez-Jiménez, Miguel A .: On the Notion of Perfect Bayesian Equilibrium . Departamento de Estadística e Investigación Operativa, Universidad de Santiago de Compostela; Department of Teoría e Historia Económica, Universidad de Málaga. Available on: Archived copy ( Memento of the original dated August 12, 2011 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / eio.usc.es
↑ Dynamic games with incomplete information. (PDF; 735 kB) (No longer available online.) Formerly in the original ; Retrieved December 12, 2011 . ( Page no longer available , search in web archives ) Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.@1@ 2
↑ The Beer Quiche Game (mathematik.uni-muenchen.de). Retrieved December 17, 2011 .

[FudenbergTiroleGameTheory-1] A ^b Drew Fudenberg, Jean Tirole: Game Theory . The MIT Press, Cambridge, Massachusetts 1991, ISBN 978-0-262-06141-4 . Pp. 321-323

[2] Ken Binmore: Fun and Games: A Text on Game Theory . DC Heath and Company, Lexington, Massachusetts 1992, ISBN 0-669-24603-4 , pp. 501-503 .

[GibbonsGameTheory-3] Robert Gibbons: A Primer in Game Theory . Financial Times, Harlow 1992, ISBN 978-0-7450-1159-2 , pp. 175-182 .

[HollerIllingSpieltheorie-4] Manfred J. Holler, Gerhard Illing: Introduction to game theory . Springer, Berlin, Heidelberg 2008, ISBN 978-3-540-69372-7 , pp. 110-124 .