Abstract
What is the minimal structure that is needed to perform equilibrium analysis in large extensive form games? To answer this question, this paper provides conditions that are simultaneously necessary and sufficient for the existence of a subgame perfect equilibrium in any well-behaved perfect information game defined on a large game tree. In particular, the set of plays needs to be endowed with a topology satisfying two conditions. (a) Nodes are closed as sets of plays; and (b) the immediate predecessor function is an open map.
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Notes
Definition 1 is equivalent to Definition 5 of AR3 plus the added property that \(\left\{ w\right\} \in N\) for all \(w\in W\), which is called completeness in that work and means that (the singleton sets of) plays/outcomes are included in the formalism as nodes, even if they are “reached” only after infinitely many decisions. AR3 (Proposition 4) shows that assuming completeness is without loss of generality.
A nonempty subset of a partially ordered set is called a chain if it is completely ordered.
Maximum, minimum, and infimum of a chain are with respect to set inclusion. Henceforth, \(\subseteq \) denotes weak set inclusion, and \(\subset \) denotes proper inclusion.
Even though the same symbol serves for the map and its codomain, no confusion can arise, because the argument will always be specified.
A slice may contain terminal nodes. This is the reason to avoid the more popular term “stage.”
A node y is an immediate successor of x if and only if x is the immediate predecessor of y. The set of immediate successors of y is hence given by \(p^{-1}(x)=\left\{ y\in N\;\left| \; \; x=p(y)\right. \right\} \). Note that immediate successors might include terminal nodes.
A preference relation \(\precsim \) on a topological space \(\left( W, \tau \right) \) upper resp. lower semi-continuous if the upper contour set \(\left\{ w^{\prime }\in W\vert w\precsim w^{\prime }\right\} \) resp. the lower contour set \(\left\{ w^{\prime }\in W\vert w^{\prime }\precsim w\right\} \) is closed in \(\tau \) for all \(w\in W\). The relation \(\precsim \) on W is continuous if it is both upper and lower semi-continuous.
Example 3 of Solan and Vieille (2003) shows that a pure subgame perfect \(\varepsilon \)-equilibrium (\(\varepsilon \ge 0\)) may not exist if the players’ payoff functions are upper semi-continuous, but not lower semi-continuous. Purves and Sudderth (2011) analyze \(\varepsilon \)-equilibrium existence (\(\varepsilon >0\)) when only upper semi-continuity of payoff functions is assumed and Flesch et al. (2010) when only lower semi-continuity is assumed.
That is, for a set \(U\subseteq Y_{t}\) of nodes \(W\left( U\right) \in \tau \left| W\left( Y_{t}\right) \right. \) if and only if there is \(u\in \tau \) such that \(W\left( U\right) =u\cap W\left( Y_{t}\right) \).
A topology \(\tau \) is perfectly normal if it is \(\hbox {T}_1\) (i.e., for each pair of distinct points, each has a neighborhood that does not contain the other) and any two disjoint closed sets can be precisely separated by a continuous function; that is, if A, B are disjoint closed sets, there exists a continuous function with range [0, 1] such that \(A=f^{-1}(0)\) and \(B=f^{-1}(1)\). Every metric space is perfectly normal.
A correspondence \(\varPhi :\left( X,\tau _{1}\right) \twoheadrightarrow \left( Y,\tau _{2}\right) \) between topological spaces \(\left( X,\tau _{1}\right) \) and \(\left( Y,\tau _{2}\right) \) is upper resp. lower hemi-continuous if for every open set \(u\in \tau _{2}\) in Y the upper preimage \(\varPhi ^{+}\left( u\right) =\left\{ x\in X\left| \varPhi \left( x\right) \subseteq u\right. \right\} \) resp. the lower preimage \(\varPhi ^{-}\left( u\right) =\left\{ x\in X\left| \varPhi \left( x\right) \cap u\ne \varnothing \right. \right\} \) is open in X (belongs to \(\tau _{1}\)).
A topological space is separated (T\(_2\) or Hausdorff) if any two distinct points can be separated by disjoint neighborhoods.
A topological space is connected if its only subsets which are both open and closed are the empty set and the full space. A subset that is a connected space with its relative topology is called connected.
The predecessor \(p\left( x\right) \) also fails to be a \(\mathcal {G}_{\delta }\) (a countable intersection of open sets), because any open set containing \(\left\{ -1,1\right\} \) is cofinite and, therefore, contains uncountably many nodes of the form \(\left\{ -w,w\right\} \). The terminal nodes \(\left\{ 1\right\} \) and \(\left\{ -1\right\} \) are also not \(\mathcal {G}_{\delta }\)s.
In order not to (further) complicate the discussion, we focus on equilibrium existence for continuous payoff functions, rather than for continuous preference relations. Technically speaking, the implication discussed here would remain open if we defined and focused on “ordinally pseudocompact spaces.”
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Acknowledgments
The authors gratefully acknowledge helpful comments and suggestions by two anonymous referees, Larry Blume, Egbert Dierker, Michael Greinecker, Josef Hofbauer, Johannes Kern, Martin Meier, Karl Schlag, Satoru Takahashi, and Walter Trockel, by seminar participants at Cornell, Princeton, and Yale University, and at the Institute for Advanced Studies in Vienna, by participants in a semi-plenary session at the World Congress of the Game Theory Society 2012, and by participants at the Workshop in honor of Harold Kuhn in Vienna 2012. We also thank the German Research Foundation (DFG) and the Austrian Science Fund (FWF) for financial support under Projects Al1169/1 and I338-G16, respectively.
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Alós-Ferrer, C., Ritzberger, K. Characterizing existence of equilibrium for large extensive form games: a necessity result. Econ Theory 63, 407–430 (2017). https://doi.org/10.1007/s00199-015-0937-0
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DOI: https://doi.org/10.1007/s00199-015-0937-0
Keywords
- Backwards induction
- Subgame perfection
- Equilibrium existence
- Large extensive form games
- Perfect information