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Erschienen in:
Games in Game Theory Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

5. Subgame Perfect Equilibrium

verfasst von : Takako Fujiwara-Greve

Erschienen in: Non-Cooperative Game Theory

Verlag: Springer Japan

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Abstract

For general extensive-form games with or without perfect information, subgame perfect equilibrium is defined. Various repeated games are analyzed, and Perfect Folk Theorem is proved.

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Vorheriges Kapitel Backward Induction
Nächstes Kapitel Bayesian Nash Equilibrium
Fußnoten
1
This shows that after firm 1 has sunk the cost of the capacity, it does not affect decisions in later subgames.
 
2
Whether the players observe the other’s action afterwards or not does not matter, because a player who takes X always chooses C and the player who does not take X always chooses D in the subsequent subgames, regardless of the observation.
 
3
A characterization of something is its necessary and sufficient condition.
 
4
Another way to evaluate infinite sequences of payoffs is overtaking criterion. See Rubinstein [19].
 
5
Some authors think that the exponent \(t-1\) is not “neat” and take the time horizon as \(t=0, 1, 2, \ldots \) so that the discounted sum is formulated as \(\sum _{t=0}^{\infty }\delta ^t u_i(t)\). In either way, the idea is the same in that the first period payoff is undiscounted and future payoffs are discounted exponentially.
 
6
The sum of a geometric sequence starting with a and the common ratio r is derived as follows. Let X be the sum, so that \(X = a + a r + a r^2 +a r^3 + \cdots \). Then by multiplying X with r, we obtain \(r \cdot X = a r + a r^2 +a r^3 + \cdots \). Subtracting this from X, we have \(X -r \cdot X =a\), i.e., \(X = \frac{a}{1-r}\).
 
7
The same analysis goes through if players maximize the average payoff.
 
8
Because Friedman’s work appeared before Selten’s subgame perfect equilibrium, the original theorem uses Nash equilibrium as the equilibrium concept. Nowadays, we see that it is also a subgame perfect equilibrium.
 
9
Because we utilize a correlated action profile, it is not easy to “separately” define an individual player’s strategy. The idea is that, in each period, each player chooses an action prescribed by \(\alpha \) and the realization of the randomization device. If a player does not choose an action consistent with \(\alpha \) and the realization of the randomization device, her/his action is a deviation.
 
10
Of course, the efficient outcome for the firms may not be good for the consumers in the market. The definition depends on the relevant stakeholders.
 
11
For a complete characterization of the sustainable set, see Sect. 5.7.
 
12
Any equilibrium notion based on a Nash equilibrium is defined with respect to unitary deviations. Hence for histories in which more than one player has deviated, no punishment is needed.
 
13
In addition, in the game of Table 5.5, the payoffs of the two players are the same for all action combinations. Such a game is called a common interest game .
 
14
The definition of the maxmin value over mixed actions is analogous to the one given in Sect. 2.​6.
 
15
This section follows Smith [22]. Kandori [13] also made a similar analysis, independently from Smith [22].
 
16
Because this game has imperfect monitoring, the equilibrium notion is not a subgame perfect equilibrium but a sequential equilibrium (see Chap. 8).
 
17
For the definition of perturbation, see Sect. 6.​5. Although the OLG game is not a normal-form game, this result is in contrast to Harsanyi’s Purification Theorem (Theorem 6.2).
 
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Bellman R (2003) Dynamic programming. Reprint. Dover Publications, Mineola, NYMATH
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Friedman J (1971) A non-cooperative equilibrium for supergames. Rev Econ Stud 38(1):1–12MATHCrossRef
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Fudenberg D, Maskin E (1986) The folk Theorem in repeated games with discounting or with Incomplete Information. Econometrica 54(3):533–554MATHMathSciNetCrossRef
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Fujiwara-Greve T (2010) The prisoner’s dilemma of neuroeconomics. Keio University, Mimeograph
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Klemperer P (2004) Auctions: theory and practice. Princeton University Press, Princeton, NJ
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Kosfeld M, Heinrichs M, Zak P, Fischbacher U, Fehr E (2005) Oxytocin increases trust in humans. Nature 435:673–676CrossRef
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Kreps D (1990) A course in microeconomics theory. Prentice Hall, Upper Saddle River, NJ
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Milgrom P (2004) Putting auction theory to work. Cambridge University Press, CambridgeCrossRef
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Rubinstein A (1979) Equilibrium in supergames with the overtaking criterion. J Econ Theory 21(1):1–9MATHCrossRef
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Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4(1):25–55MATHMathSciNetCrossRef
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Smith L (1992) Folk Theorems in overlapping generations games. Games Econ Behav 4(3):426–449MATHCrossRef
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Watson J (2007) Strategy: an introduction to game theory, 2nd edn. Norton, New York
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Yoon K (2001) A folk theorem for asynchronously repeated games. Econometrica 69(1):191–200
Metadaten
Titel
Subgame Perfect Equilibrium
verfasst von
Takako Fujiwara-Greve
Copyright-Jahr
2015
Verlag
Springer Japan
Buch
Non-Cooperative Game Theory

Print ISBN: 978-4-431-55644-2

Electronic ISBN: 978-4-431-55645-9

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-4-431-55645-9_5