Abstract
This chapter focuses on the representation of the objective description of the game—the game tree. It explores the connections between trees as partially ordered sets, like graphs, and trees as collections of subsets of an underlying set of plays or outcomes. In particular, it identifies a canonical set representation for every tree. This leads to the concept of a game tree: A collection of nonempty subsets of the set of plays that satisfies Trivial Intersection, Boundedness, and Irreducibility. The main theorem of this chapter demonstrates that a game tree preserves the freedom to start from plays or nodes as primitives, hence simultaneously generalizing the approaches of Kuhn and von Neumann and Morgenstern.
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Notes
- 1.
- 2.
- 3.
See the Mathematical Appendix A.1.2 for definitions of properties of binary relations.
- 4.
See the Mathematical Appendix A.1.3 for definitions of properties of functions.
- 5.
Similar results are known, for instance, for finite arbitrary ordered sets (Davey and Priestley 1990, Theorem 8.19).
- 6.
This is also known as Kuratowski’s Lemma; see e.g. Hewitt and Stromberg (1965) and the Mathematical Appendix A.1.2.
- 7.
In this specification players who choose simultaneously are active at the same node; see Chap. 4 for details.
- 8.
Suppose the convention in the specification of nodes would be changed such that for two functions to belong to the same node they would have to agree on the closed interval \(\left [0,t\right ]\). Then there would be no “point in time” when the decision actually “becomes effective.”
- 9.
One may use objects other than plays to obtain a set representation, though, as will be shown in Chap. 3.
- 10.
No confusion should arise between the mapping W(⋅ ) assigning to each node x the set of plays passing through x and the set W of all plays.
- 11.
Observe the formal analogy of Strong Irreducibility with the definition of a Hausdorff space in topology.
- 12.
The set representation by subtrees of a decision tree cannot satisfy Irreducibility. For, if x ≠ y and x ≥ y then \(\downarrow \! y \subseteq \downarrow \! x\), so that \(x \in \downarrow \! z\) and \(y \in \downarrow \! z^{{\prime}}\) for \(z,z^{{\prime}}\in N\) implies \(\downarrow \! x \subseteq \downarrow \! z\) and \(\downarrow \! y \subseteq \downarrow \! z^{{\prime}}\) and, therefore, \(y\notin \downarrow \! z^{{\prime}}\setminus \downarrow \! z\).
- 13.
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Alós-Ferrer, C., Ritzberger, K. (2016). Game Trees. In: The Theory of Extensive Form Games. Springer Series in Game Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49944-3_2
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