Abstract
Extensive-form games are defined, and an equilibrium concept through backward induction is introduced for games with perfect information.
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Notes
- 1.
The payoffs are usually ordered according to the decision order in the extensive-form game.
- 2.
A partition of a (finite) set Y is a collection of subsets of Y, \(\{Y_1, Y_2, \ldots , Y_K\}\) such that for any \(k, k'\), \(Y_k \cap Y_{k'} = \emptyset \) and \(\cup _{k=1}^K Y_k = Y\).
- 3.
Circles and ovals are not the only way to illustrate information sets. In some literature, decision nodes in the same information sets are connected by lines.
- 4.
Notation can vary in the literature. In this book, the capital H is used to indicate a set of histories.
- 5.
A careful reader may notice that the origin is not a singleton information set, but we can fix this problem by adding an artificial origin and an action, such as the moment he leaves a parking lot. We can also add a second player, his wife, and her actions after he gets home to make a multi-person game.
- 6.
In some literature, e.g. Fudenberg and Tirole [7], it is called a behavior strategy.
- 7.
Fudenberg and Tirole [7] gives an “opposite” example (their Fig. 3.13) such that there is a probability distribution over the terminal nodes which is attainable by a mixed strategy but not by any behavioral strategy, under imperfect recall.
- 8.
In many textbooks, this equilibrium concept is only implicit, and they use the concept of (subgame) perfect equilibrium (see Chap. 5) for all extensive-form games with complete information. However, we (maybe influenced by Gibbons [8]) think that, to understand the subgame perfect equilibrium concept fully, it is important to understand backward induction on its own.
- 9.
In general, we should allow for the possibility that Nature (to be introduced in Sect. 4.8) moves at the origin. In this case, the theorem holds as well, in the sense that no player can improve their payoff by choosing a different action than the one in \(b^a\) in each \(\varGamma ^a\).
- 10.
As we have noted in Sect. 3.2, if \(q_1\) is large, it is possible that \( \frac{1}{2}(A - q_1 - c_2)\) is negative. But such a large \(q_1\) will be shown to be not rational, and thus we ignore those cases.
- 11.
When the set of feasible proposals by player 1 is not a continuum but a finite set, there is an optimal strategy. See Problem 5.1(b) for a related game.
- 12.
- 13.
The following is a simplification of the argument in Brandenburger [4].
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Fujiwara-Greve, T. (2015). Backward Induction. In: Non-Cooperative Game Theory. Monographs in Mathematical Economics, vol 1. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55645-9_4
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