Abstract
We provide approximation results for Nash equilibria in possibly discontinuous games when payoffs and strategy sets are perturbed. We then prove existence results for a new “finitistic” infinite-game generalization of Selten’s (Int J Game Theory 4: 25–55, 1975) notion of perfection and study some of its properties. The existence results, which rely on the approximation theorems, relate existing notions of perfection to the new specification.
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Notes
Some care is required here. For example, the closedness conclusion of Corollary 1 would not hold if we presented our definitions in terms of sequences rather than nets.
We thank an anonymous referee for suggesting this argument.
This leads Simon and Stinchcombe to strengthen the notion of lof perfection to anchored perfection. However, Simon and Stinchcombe (1995) claim that even anchored perfect profiles may fail limit admissibility in continuous games.
Given \(k\), let \(\{a_{1},\ldots ,a_{\ell }\}\) be the finite support of \(\lambda _{i}^{k}\). Since \(Y_{i}^{m}\rightarrow X_{i}\), for each \(l\in \{1,\ldots ,\ell \} \) one can choose a sequence \((\alpha _{l}^{m})\) such that \(\alpha _{l}^{m}\in Y_{i}^{m}\) for each \(m\) and \(\alpha _{l}^{m}\xrightarrow [m \rightarrow \infty ]{}a_{l}\). For each \(m\), define \(\widehat{\nu }_{i}^{m}\) in \(\Delta (X_{i})\) by
$$\begin{aligned} \widehat{\nu }_{i}^{m}(\{\alpha _{l}^{m}\}):=\lambda _{i}^{k}(\{a_{l}\}),\quad \text{ for} \text{ each} l\in \{1,\ldots ,\ell \} \end{aligned}$$and note that \(\text{ supp}(\widehat{\nu }_{i}^{m})=\{\alpha _{1}^{m},..,\alpha _{\ell }^{m}\}\subseteq Y_{i}^{m}\) for each \(m\). Let \(\widetilde{\nu } _{i}^{m}\in \Delta (X_{i})\) be defined by \(\widetilde{\nu } _{i}^{m}(\{y_{i}^{m}\}):=\frac{1}{\#Y_{i}^{m}}\) for each \(y_{i}^{m}\in Y_{i}^{m}\), where \(\#Y_{i}^{m}\) denotes the cardinality of the finite set \( Y_{i}^{m}\). Define, for each \(m\),
$$\begin{aligned} \nu _{i}^{m}:=\left( 1-\frac{1}{m}\right) \widehat{\nu }_{i}^{m}+ \frac{1}{m}\widetilde{\nu }_{i}^{m} \end{aligned}$$and note that \(\text{ supp}(\nu _{i}^{m})=Y_{i}^{m}\) for each \(m\). To see that \(\nu _{i}^{m}\rightarrow \lambda _{i}^{k}\), let \(O_{i}\) be an open set in \(X_{i}\) and define \(I:=\{l:a_{l}\in O_{i}\}.\) Then there exists an \( \widehat{m}\) such that, \(\alpha _{l}^{m}\in O_{i}\) for each \(m>\widehat{m}\) and each \(l\in I.\) Therefore, \(m>\widehat{m}\) implies that
$$\begin{aligned} \lambda _{i}^{k}(O_{i}\cap \{a_{1},\ldots ,a_{\ell }\})=\sum _{l\in I}\lambda _{i}^{k}(\{a_{l}\})=\sum _{l\in I}\widehat{\nu }_{i}^{m}(\{\alpha _{l}^{m}\})\le \widehat{\nu }_{i}^{m}(O_{i}\cap \{\alpha _{1}^{m},\ldots ,\alpha _{\ell }^{m}\}), \end{aligned}$$so for \(m>\widehat{m},\) we have
$$\begin{aligned} \nu _{i}^{m}(O_{i})&= \left( 1-\frac{1}{m}\right) \widehat{\nu } _{i}^{m}(O_{i})+\frac{1}{m}\widetilde{\nu }_{i}^{m}(O_{i}) \\&= \widehat{\nu }_{i}^{m}(O_{i})+\frac{1}{m}\left( \widetilde{\nu } _{i}^{m}(O_{i})-\widehat{\nu }_{i}^{m}(O_{i})\right) \\&\ge \widehat{\nu }_{i}^{m}(O_{i})-\frac{1}{m} \\&= \widehat{\nu }_{i}^{m}(O_{i}\cap \{\alpha _{1}^{m},\ldots ,\alpha _{\ell }^{m}\})-\frac{1}{m} \\&\ge \lambda _{i}^{k}(O_{i}\cap \{a_{1},\ldots ,a_{\ell }\})-\frac{1 }{m} \\&= \lambda _{i}^{k}(O_{i})-\frac{1}{m}, \end{aligned}$$from which it follows that \(\underline{\lim }\,\nu _{i}^{m}(O_{i})\ge \lambda _{i}^{k}(O_{i})\).
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We thank two anonymous referees for helpful comments.
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Carbonell-Nicolau, O., McLean, R.P. Approximation results for discontinuous games with an application to equilibrium refinement. Econ Theory 54, 1–26 (2013). https://doi.org/10.1007/s00199-012-0727-x
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DOI: https://doi.org/10.1007/s00199-012-0727-x
Keywords
- Discontinuous game
- Nash equilibrium correspondence
- Equilibrium refinement
- Finite approximation
- Trembling-hand perfect equilibrium
- Limit-of-finite perfect equilibrium