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2015 | OriginalPaper | Chapter

5. Subgame Perfect Equilibrium

Author : Takako Fujiwara-Greve

Published in: Non-Cooperative Game Theory

Publisher: 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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previous chapter Backward Induction

next chapter Bayesian Nash Equilibrium

This shows that after firm 1 has sunk the cost of the capacity, it does not affect decisions in later subgames.

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.

A characterization of something is its necessary and sufficient condition.

Another way to evaluate infinite sequences of payoffs is overtaking criterion. See Rubinstein [19].

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.

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., \(X = \frac{a}{1-r}\).

The same analysis goes through if players maximize the average payoff.

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.

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.

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.

For a complete characterization of the sustainable set, see Sect. 5.7.

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.

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 .

The definition of the maxmin value over mixed actions is analogous to the one given in Sect. 2.6.

This section follows Smith [22]. Kandori [13] also made a similar analysis, independently from Smith [22].

Because this game has imperfect monitoring, the equilibrium notion is not a subgame perfect equilibrium but a sequential equilibrium (see Chap. 8).

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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Title: Subgame Perfect Equilibrium
Author: Takako Fujiwara-Greve
Publisher: Springer Japan
Book: Non-Cooperative Game Theory
Print ISBN: 978-4-431-55644-2

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

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-4-431-55645-9_5