Sequential balance

The sequential equilibrium (short: SG ) is a game theoretical solution concept for dynamic games with incomplete and / or imperfect information .

The concept of sequential equilibrium introduced by Kreps and Wilson (1982) is a refinement of subgame-perfect equilibrium . This refinement is expressed through the belief system and the requirement for sequential rationality and consistency, especially in dynamic games with incomplete and / or imperfect information .

development

In the partial game perfection concept , the equilibrium strategies must be optimal at each decision node. The prerequisite for this is that the players have complete information , i. H. they must be informed about the course of the game so far and thus be able to know which node they are at.

In games with incomplete and / or imperfect information , however , the concept of sub-game perfection does not exclude all implausible Nash equilibria , since in such situations there is often no real sub-game. In such a game, the subgame perfect equilibrium corresponds to the Nash equilibrium, and then subgame perfection does not help.

In order to avoid these weaknesses in partial game perfection , the concept of 'sequential equilibrium' was developed. It requires that the equilibrium strategies of each set of information must be optimal, provided that the belief system is consistent. (With the belief system, for each amount of information, the beliefs are determined that the players, who get a move on this amount of information, have over the previous game run.)

Representation of the sequential equilibrium

${\displaystyle (s,\mu )}$

With

• ${\displaystyle \textstyle s}$: Strategy combination
• ${\displaystyle \textstyle \mu }$: Probability assessment (belief)

Formal definitions

The assessment that is both consistent and sequentially rational. ${\displaystyle \textstyle (s,\mu )}$

Sequential rationality

An assessment is sequentially rational if the strategies chosen by a player ? are optimal for each amount of information given the assessments and continuation strategies of the other players. ${\displaystyle \textstyle (s,\mu )}$ ${\displaystyle \textstyle I_{i}}$

In other words: In a finite extensive game with perfect recall, an assessment is sequentially rational if the following applies to each player and to each of his sets of information : ${\displaystyle \textstyle (s,\mu )}$ ${\displaystyle i\in N}$ ${\displaystyle I_{i}\in \iota _{i}}$

${\displaystyle u_{i}(s,\mu |I_{i})\gtrsim u_{i}(s\!\,'_{i},s_{-i},\mu |I_{i})}$

consistency

A combination is consistent if there is a sequence which converges against the assessment and has the properties that every strategic profile is completely mixed and every belief system is derived from Bayesian rule, so that: ${\displaystyle \textstyle (s,\mu )}$ ${\displaystyle \textstyle {(s^{\varepsilon },\mu ^{\varepsilon })}}$ ${\displaystyle \textstyle (s,\mu )}$ ${\displaystyle \textstyle {s^{\varepsilon }}}$ ${\displaystyle \textstyle {\mu ^{\varepsilon }}}$ ${\displaystyle \textstyle {s^{\varepsilon }}}$

${\displaystyle \lim _{\varepsilon \to 0}(s^{\varepsilon },\mu ^{\varepsilon })=(s,\mu )}$

comment

The concept of sequential equilibrium limits beliefs about amounts of information that are not achieved in equilibrium by introducing the consistency requirement.

The following intuition stands behind this requirement:

"[...] As soon as there is a deviation from the equilibrium path (an - intentional or unintentional - error), then the further course of the game must again represent a sequential equilibrium from this point on - given any assessments as to why the error happened. That is, based on the probability estimates , the players play optimal strategies again for each event; They revise their probabilities in accordance with Bayesian rule, with the estimates now serving as a basis for calculation - unless another event occurs with zero probability. In the latter case, the acting player must again make consistent assessments. " ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle \mu }$

sentences

1. For every finite extensive game there is at least one sequential equilibrium.
2. If is a sequential equilibrium, then s is a subgame perfect equilibrium . ${\displaystyle \textstyle {(s,\mu )}}$
3. if there is a sequential equilibrium, then expanded is subgame perfect . ${\displaystyle \textstyle {(s,\mu )}}$ ${\displaystyle \textstyle {(s,\mu )}}$

Example and solution

An assessment is presented as follows:

${\displaystyle (s,\mu )=((s_{1},s_{2}),\mu )}$

With

• ${\displaystyle \textstyle {s_{1}}}$: Player 1's strategy corresponds to the probability distribution across his strategies, M, L and R;
• ${\displaystyle \textstyle {s_{2}}}$: Player 2's strategy corresponds to the probability distribution over his strategies l and r;
• ${\displaystyle \textstyle {\mu }}$: Belief of player 2, which is due to the fact that the amount of information from player 2 is reached.

Example of sequential equilibrium

Game in matrix form; Payouts and Nash Equilibria

In this example there are two types of sequential equilibria.

First type : Sequential equilibrium if the amount of information from player 2 is reached, i.e. H.

${\displaystyle s_{1}(L)=0}$

The strategy R is strictly dominated by L and M, so that

${\displaystyle s_{1}(R)=0\qquad {\text{und}}\qquad s_{1}(M)>0}$.

Bayesian rule is applicable because player 2's amount of information is on the equilibrium path:

${\displaystyle \mu (M)={\frac {s_{1}(M)}{s_{1}(M)+s_{1}(R)}}=1}$,

${\displaystyle \mu (R)={\frac {s_{1}(R)}{s_{1}(M)+s_{1}(R)}}=0}$.

With the beliefs, player 2 will rationally choose l because strategy l yields a higher payout:

${\displaystyle u_{2}(M,l)>u_{2}(M,r)}$

It follows:

${\displaystyle s_{2}(l)=1}$,

${\displaystyle \textstyle {s_{2}(r)=0}}$.

The assessment is sequentially rational again if and only if

${\displaystyle s_{1}(L)=0}$,

${\displaystyle s_{1}(M)=1}$,

${\displaystyle s_{1}(R)=0}$.

Conclusion

${\displaystyle \textstyle {(s,\mu )=(((0,1,0),(1,0)),(1,0))}}$  is a sequential equilibrium in a case where the information amount of player 2 is reached.

comment

In this case, the solution procedure is identical to that of the perfect Bayesian equilibrium .

Second type : Sequential equilibrium if the amount of information from player 2 is not reached, i.e. H:

${\displaystyle \textstyle {s_{1}(L)=1}}$,

${\displaystyle \textstyle {s_{1}(M)=0}}$,

${\displaystyle \textstyle {s_{1}(R)=0}}$.

The strategy forms part of the sequential rational assessment if and only if player 2 r plays with a high probability, e.g. B. ${\displaystyle \textstyle {s_{1}(L)=1}}$

${\displaystyle \textstyle {s_{2}(r)=1}}$  .

Otherwise player 1 will deviate from strategy L and the requirement of sequential rationality will not be met.

(Sequential equilibrium assumes that the strategy can occur on the non-equilibrium path in the game. Therefore, player 2 needs the beliefs about his amount of information if he gets his turn.)

To be      sequentially rational, the beliefs must: ${\displaystyle \textstyle {s_{2}(r)=1}}$

Given that q is defined as follows:

${\displaystyle \textstyle {\mu (M)=1-q}}$

${\displaystyle \textstyle {\mu (R)=q}}$,

because of

${\displaystyle \textstyle (1-q)\cdot u_{2}(M,l)+q\cdot u_{2}(R,l)\leq (1-q)\cdot u_{2}(M,r)+q\cdot u_{2}(R,r)}$

${\displaystyle \textstyle \Rightarrow (1-q)\cdot 2+q\cdot 1\leq (1-q)\cdot 1+q\cdot 2}$

follows    . ${\displaystyle \textstyle {q\geq {\frac {1}{2}}}}$

To check whether the assessment is consistent with the strategies and beliefs, the following is considered:

There are strategy combinations , ${\displaystyle s^{\varepsilon }=(s_{1}^{\varepsilon },s_{2}^{\varepsilon })}$

where on the one hand is defined as a small positive number, furthermore as follows: ${\displaystyle \varepsilon }$

${\displaystyle s_{1}^{\varepsilon }(L)=1-\varepsilon }$	${\displaystyle \textstyle {s_{2}^{\varepsilon }(l)=\varepsilon }}$
${\displaystyle s_{1}^{\varepsilon }(M)=(1-q)\varepsilon }$	${\displaystyle \textstyle {s_{2}^{\varepsilon }(r)=1-\varepsilon }}$
${\displaystyle s_{1}^{\varepsilon }(R)=q\varepsilon }$

.

Bayesian rule is now applicable and the beliefs are defined:

${\displaystyle \mu (M)={\frac {s_{1}^{\varepsilon }(M)}{s_{1}^{\varepsilon }(M)+s_{1}^{\varepsilon }(R)}}={\frac {1-q}{1-q+q}}=1-q}$,

${\displaystyle \mu (R)={\frac {s_{1}^{\varepsilon }(R)}{s_{1}^{\varepsilon }(M)+s_{1}^{\varepsilon }(R)}}={\frac {q}{1-q+q}}=q}$.

With     this shows that the assessment is consistent. ${\displaystyle \lim _{{\varepsilon }\to {0}}s^{\varepsilon }=s}$

Conclusion

${\displaystyle \textstyle (s,\mu )=(((1,0,0),(0,1)),(1-q,q))}$is a sequential equilibrium with    if the information amount of player 2 is reached. ${\displaystyle q\geq {\frac {1}{2}}}$

comment

(1) Note for given as a sequential rational strategy by player 2       and    ${\displaystyle \textstyle {s_{2}(r)=1}}$ ${\displaystyle \textstyle {s_{1}(L)=1}}$ ${\displaystyle q\geq {\frac {1}{2}}}$

Until then     , player 2 will always rationally choose r. Because this belief implies that he believes that player 1 chose R on his move, so that it is rational for player 2 to play r. Therefore he plays r with a probability of 1. ${\displaystyle q\geq {\frac {1}{2}}}$

(2) Note for the special case    ${\displaystyle q={\frac {1}{2}}}$

${\displaystyle q={\frac {1}{2}}}$  means that the beliefs about the respective decision nodes in the amount of information from player 2 are the same. Then player 2 is indifferent between r and l. Hence, he will mix his strategies :

In that case      the strategy     forms part of the sequential equilibrium if and only if     : ${\displaystyle q={\frac {1}{2}}}$ ${\displaystyle \textstyle {s_{1}(L)=1}}$ ${\displaystyle \textstyle {p\geq {\frac {2}{3}}}}$

Given that p is defined as follows:

${\displaystyle \textstyle {s_{2}(l)=1-p}}$

${\displaystyle \textstyle {s_{2}(r)=p}}$,

because of

${\displaystyle \textstyle (1-p)\cdot u_{1}(M,l)+p\cdot u_{1}(M,r)\leq u_{1}(L,s_{2})}$

${\displaystyle \textstyle \Rightarrow (1-p)\cdot 4+p\cdot 1\leq 2}$

follows    . ${\displaystyle \textstyle {p\geq {\frac {2}{3}}}}$

(Since R is strictly dominated by M, it is only compared to the case where player 1 chooses M.)

Otherwise    sequential is    not rational. ${\displaystyle \textstyle {s_{1}(L)=1}}$

To check whether the assessment is consistent with the strategies and beliefs, the following is considered:

There are strategy combinations   , ${\displaystyle s^{\varepsilon }=(s_{1}^{\varepsilon },s_{2}^{\varepsilon })}$

where on the one hand as a small positive number and is defined and furthermore as follows: ${\displaystyle \varepsilon }$

${\displaystyle s_{1}^{\varepsilon }(L)=1-2\varepsilon }$	${\displaystyle \textstyle {s_{2}^{\varepsilon }(l)=s_{2}(l)}}$
${\displaystyle s_{1}^{\varepsilon }(M)=\varepsilon }$	${\displaystyle \textstyle {s_{2}^{\varepsilon }(r)=s_{2}(r)}}$
${\displaystyle s_{1}^{\varepsilon }(R)=\varepsilon }$.

Graphical representation of the set of all equilibria

And the beliefs are defined using Bayesian rule: ${\displaystyle \textstyle {\mu }}$

${\displaystyle \mu (M)={\frac {s_{1}^{\varepsilon }(M)}{s_{1}^{\varepsilon }(M)+s_{1}^{\varepsilon }(R)}}={\frac {\varepsilon }{\varepsilon +\varepsilon }}={\frac {1}{2}}}$,

${\displaystyle \mu (R)={\frac {s_{1}^{\varepsilon }(R)}{s_{1}^{\varepsilon }(M)+s_{1}^{\varepsilon }(R)}}={\frac {\varepsilon }{\varepsilon +\varepsilon }}={\frac {1}{2}}}$.

With this shows that the assessment is consistent. ${\displaystyle \lim _{\varepsilon \to {0}}s^{\varepsilon }=s}$

This leads to another sequential equilibrium:

${\displaystyle (s,\mu )=\left(((1,0,0),(1-p,p)),\left({\tfrac {1}{2}},{\tfrac {1}{2}}\right)\right)}$  with     in a case in which the amount of information from player 2 is not reached and the beliefs about the respective decision nodes in the amount of information from player 2 are the same. ${\displaystyle \textstyle {p\geq {\frac {2}{3}}}}$

Delimitation of the sequential equilibrium from the perfect Bayesian equilibrium

The two concepts are a refinement of the subgame-perfect balance . They have in common that the strategies given the belief must be sequentially rational. However, the two concepts differ in the following point:

Requirement 4 of the perfect Bayesian equilibrium is

"In amounts of information outside of the equilibrium path, beliefs are determined by players using Bayesian rule and equilibrium strategies whenever possible."

Following this requirement 4, in the perfect Bayesian equilibrium the beliefs are defined by the Bayesian rule whenever possible, while in the sequential equilibrium the Bayesian rule is applied through the consistency of the belief for all paths in the game (for the strategies both on the equilibrium path and also on the non- equilibrium path). Consistency in the sense of Kreps and Wilson (1982) is that beliefs are the limit of beliefs associated with a sequence of fully mixed strategies that converge to s. ${\displaystyle \textstyle \mu }$ ${\displaystyle \textstyle {\mu ^{\varepsilon }}}$ ${\displaystyle \textstyle {s^{\varepsilon }}}$

Since the application of the concept of sequential equilibrium is very complicated, the perfect Bayesian equilibrium is often used in game theory as the solution concept for the games with incomplete information .

criticism

In the example, however, the second type of sequential equilibrium is not plausible:

The assessment   is a sequential equilibrium if and only if . This implies that: ${\displaystyle \textstyle (s,\mu )=(((1,0,0),(0,1)),(1-q,q)}$ ${\displaystyle q\geq {\tfrac {1}{2}}}$

${\displaystyle \mu (\{M,R\})(M)<\mu (\{M,R\})(R)}$.

But since for player 1 the strategy R is strictly dominated by M and L, if the amount of information is reached by player 2 and thus player 2 would get a move, it is reasonable to estimate for player 2 that player 1 rationally chose M Has. So    in reality is not reasonable; thus the sequential equilibria of the second type are not plausible. ${\displaystyle q\geq {\tfrac {1}{2}}}$

Conclusion : The sequential equilibrium does not exclude all implausible equilibria.

See also

• Nash equilibrium
• Partial perfect balance
• Trembling-hand-perfect balance

literature

• Martin J. Osborne, Ariel Rubinstein: A Course in Game Theory . The MIT Press, Cambridge, Massachusetts 1994, ISBN 0-262-15041-7 .
• Manfred J. Holler, Gerhard Illing: Introduction to game theory . Springer, Berlin Heidelberg 2008, ISBN 978-3-540-69372-7 .
• Jurgen Eichberger: Game theory for economists . Emerald Group Publishing Limited, 1993, ISBN 3-540-69372-6 .
• Robert Gibbons: A Primer in Game Theory . Financial Times, Harlow 1992, ISBN 0-7450-1159-4 .
• Siegfried K Berninghaus, Karl-Martin Ehrhart: Strategic Games: An Introduction to Game Theory . Springer, Berlin Heidelberg 2010, ISBN 978-3-642-11650-6 .
• David M Kreps, Robert Wilson: Sequential Equilibria . In: Econometrica . Econometric Society, Jul. 1982: 50 (4): 863-94. ( Http://www.jstor.org/pss/1912767 ).
• Julio Gonzáles-Díaz, Miguel A. Meléndez-Jiménez: On the Notion of Perfect Bayesian Equilibrium . In: TOP . Springer, Berlin / Heidelberg Nov. 2011: 19. ISSN  1863-8279 (online)
• Drew Fudenberg, Jean Tirole: Perfect Bayesian equilibrium and sequential equilibrium . In: Journal of Economic Theory . Elsevier, Apr. 1991: 53 (2): 236-260.

Individual evidence

1. ^ Martin J. Osborne, Ariel Rubinstein: A Course in Game Theory . The MIT Press, Cambridge, Massachusetts 1994, ISBN 978-0-262-15041-5 , pp. 220 .
2. ↑ Manfred J. Holler, Gerhard Illing: Introduction to game theory . Springer, Berlin Heidelberg 2008, ISBN 978-3-540-69372-7 . P. 116
3. ^ David M Kreps, Robert Wilson: Sequential Equilibria. In: Econometrica. Econometric Society, Jul. 1982: 50 (4): 863-94. ( Http://www.jstor.org/pss/1912767 ). P. 876.
4. ^ David M Kreps, Robert Wilson: Sequential Equilibria. In: Econometrica. Econometric Society, Jul. 1982: 50 (4): 863-94. ( Http://www.jstor.org/pss/1912767 ). P. 876
5. ^ David M Kreps, Robert Wilson: Sequential Equilibria. In: Econometrica. Econometric Society, Jul. 1982: 50 (4): 863-94. ( Http://www.jstor.org/pss/1912767 ). P. 877
6. ^ Robert Gibbons: A Primer in Game Theory . Financial Times, Harlow 1992, ISBN 978-0-7450-1159-2 . P. 180
7. ^ Julio Gonzáles-Díaz, Miguel A. Meléndez-Jiménez: On the Notion of Perfect Bayesian Equilibrium . In: TOP. Springer, Berlin / Heidelberg Nov. 2011: 19. ISSN  1863-8279 (Online) p. 5