The algebraic geometry of perfect and sequential equilibrium: an extension

Abstract

We extend the generic equivalence result of Blume and Zame (Econometrica 62:783–794, 1994) to a broader context of perfectly and sequentially rational strategic behavior (including equilibrium and nonequilibrium behavior) through a unifying solution concept of “mutually acceptable course of action” (MACA) proposed by Greenberg et al. (Econ Theory 40:91–112, 2009. https://doi.org/10.1007/s00199-008-0349-5). As a by-product, we show, in the affirmative, Dekel et al.’s (J Econ Theory 89:165–185, 1999) conjecture on the generic equivalence between the sequential and perfect versions of rationalizable self-confirming equilibrium.

Appendices

Appendix 1: Definitions

Consider an extensive-form game $\Gamma \left( u\right)$ with perfect recall.

Definition 3

• (i) [Selten (1975); Osborne and Rubinstein (1994, Definition 251.1)] A strategy profile $y\in {\mathbb {Y}}$ is a perfectly equilibrium if there exists a trembling sequence $y^{t}\rightsquigarrow$ y such that, for all $t\ge 0$, $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}}_{h}}v_{i}\left( \left( y_{h}^{\prime }, \quad y_{-h}^{t}\right) ,u_{i}\right)$ for all $i\in N$ and $h\in H_{i}$.

• (ii) [Kreps and Wilson (1982)] A strategy profile $y\in {\mathbb {Y}}$ is a sequential equilibrium if there exists a consistent assessment $\left( y,\mu \right)$ such that $y_{i}\in \arg \max _{y_{i}^{\prime }\in {\mathbb {Y}}_{i}}v_{i}\left( y_{i}^{\prime },\left( y,\mu \right) ,u_{i}|h\right)$ for all $i\in N$ and $h\in H_{i}$.

• (iii) A strategy profile $y\in {\mathbb {Y}}$ is a “weakly” perfect equilibrium if for each player $i\in N$, there exists a trembling sequence $y^{t}\rightsquigarrow$ y such that, for all $t\ge 0$, $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}} _{h}}v_{i}\left( \left( y_{h}^{\prime },\quad y_{-h}^{t}\right) ,u_{i}\right)$ for all $h\in H_{i}$.

• (iv) A strategy profile $y\in {\mathbb {Y}}$ is a “weakly” sequential equilibrium if for each player $i\in N$, there exists a consistent assessment $\left( y,\mu \right)$ such that $y_{i}\in \arg \max _{y_{i}^{\prime }\in {\mathbb {Y}}_{i}}v_{i}\left( y_{i}^{\prime },\left( y,\mu \right) ,u_{i}|h\right)$ for all information set $h\in H_{i}$.

In the notions of perfect equilibrium and sequential equilibrium, all the players are required to have a (common) trembling sequence. In contrast, in the notions of “weakly” perfect equilibrium and “weakly” sequential equilibrium, different players are allowed to adopt distinct trembling sequences.¹⁵

Definition 4

[Greenberg et al. (2009) and Dekel et al. (1999, 2002)] A product set $Y=\Pi _{i\in N}Y_{i}\subseteq {\mathbb {Y}}$ is a perfectly (or sequentially) rationalizable set if for each player $i\in N$, every $y_{i}\in Y_{i}$ is perfect (or sequential) best response with respect to $\left( \Pi _{j\ne i}co^{e}(Y_{j}),u_{i}\right)$. An element in a perfectly (or sequentially) rationalizable set is said to be a perfectly (or sequentially) rationalizable strategy profile.

That is, Y is a perfectly rationalizable set if, for each player i and $y_{i}\in Y_{i}$, we can find a strategy profile $y_{-i}\in \Pi _{j\ne i}co^{e}(Y_{j})$ and a trembling sequence which converges to $\left( y_{i},y_{-i}\right)$ such that $y_{i}$ is a best response along the trembling sequence. In simultaneous-move games, the perfectly rationalizable set is related to Herings and Vannetelbosch’s (1999) definition of “weakly perfect rationalizability.” Similarly, Y is a sequentially rationalizable set if for each player i and $y_{i}\in Y_{i}$, we can find a consistent assessment $\left( \left( y_{i},y_{-i}\right) ,\mu \right)$ where $y_{-i}\in \Pi _{j\ne i}co^{e}(Y_{j})$ such that $y_{i}$ is a sequential best response. The sequentially rationalizable set is associated with Dekel et al.’s (1999, 2002) notion of sequential rationalizability.

Definition 5

(Dekel et al. 1999, 2002). A strategy profile $\widehat{y}\in {\mathbb {Y}}$ is a sequentially rationalizable self-confirming equilibrium (SRSCE) if there exists $Y=\Pi _{i\in N}Y_{i}\subseteq {\mathbb {Y}}$, such that for each $i\in N$ and $y_{i}\in Y_{i}$, we can find a consistent assessment $\left( y,\mu \right)$, with the restrictions that (i) $y_{-i}\in \Pi _{j\ne i}co^{e}(Y_{j})$, (ii) y and $\widehat{y}$ yield a same distribution over terminal nodes, and (iii) $y_{i}$ is a sequential best response under $\left( y,\mu \right)$.

Appendix 2: Proofs

For each player $i\in N$ and $Y_{-i}\subseteq {\mathbb {Y}}_{-i}$, let

$$\begin{aligned} PB_{i}(Y_{-i},u_{i})\equiv & {} \left\{ y_{i}\in {\mathbb {Y}}_{i}:\quad y_{i} \text { is a perfect best response with respect to }\left( Y_{-i},u_{i}\right) \right\} ; \\ SB_{i}(Y_{-i},u_{i})\equiv & {} \left\{ y_{i}\in {\mathbb {Y}}_{i}:\quad y_{i} \text { is a sequential best response with respect to }\left( Y_{-i},u_{i}\right) \right\} . \end{aligned}$$

Let $cl({\mathcal {R}}_{i})$ and $vcl_{U_{i}}({\mathcal {R}}_{i})$ denote the closure of the perfectly rational state set ${\mathcal {R}} _{i}$ and vertical closure of the perfectly rational state set ${\mathcal {R}}_{i}$ (on $U_{i}$), respectively; that is,

$$\begin{aligned} cl({\mathcal {R}}_{i})\equiv & {} \left\{ \left( u_{i},x,y_{i}\right) \in U_{i}\times {\mathbb {Y}}\times {\mathbb {Y}}_{i}:\exists \left\{ \left( u_{i}^{t},x^{t},y_{i}^{t}\right) \right\} _{t=1}^{\infty }\in {\mathcal {R}}_{i} \text { s.t. }\left( u_{i}^{t},x^{t},y_{i}^{t}\right) \rightarrow \left( u_{i},x,y_{i}\right) \right\} ; \\ vcl_{U_{i}}({\mathcal {R}}_{i})\equiv & {} \left\{ \left( u_{i},x,y_{i}\right) \in U_{i}\times {\mathbb {Y}}\times {\mathbb {Y}}_{i}:\exists \left\{ \left( u_{i},x^{t},y_{i}^{t}\right) \right\} _{t=1}^{\infty }\in {\mathcal {R}}_{i} \text { s.t. }\left( u_{i},x^{t},y_{i}^{t}\right) \rightarrow \left( u_{i},x,y_{i}\right) \right\} . \end{aligned}$$

Let

$$\begin{aligned} Q\left( Y_{-i}\right) \equiv \left\{ \left( u_{i},x,y_{i}\right) \in U_{i}\times {\mathbb {Y}}\times {\mathbb {Y}}_{i}:x_{i}=y_{i}\text { and } x_{-i}\in Y_{-i}\right\} \end{aligned}$$

denote the set of “consistent-belief” states at which player i’s belief about the opponent players’ behavior lies in the given set $Y_{-i}$ and the belief about his own behavior is consistent with his strategy.

To prove Theorem 1, we need the following four lemmas:

Lemma 1

In an extensive form $\Gamma$ with perfect recall, for each player $i\in N$ and $Y_{-i}\subseteq {\mathbb {Y}}_{-i}$ (a) $y_{i}\in PB_{i}(Y_{-i},u_{i})\Leftrightarrow \exists \left( u_{i},x,y_{i}\right) \in vcl_{U_{i}}({\mathcal {R}}_{i})\cap Q\left( Y_{-i}\right)$; (b) $y_{i}\in SB_{i}(Y_{-i},u_{i})\Leftrightarrow \exists \left( u_{i},x,y_{i}\right) \in cl({\mathcal {R}}_{i})\cap Q\left( Y_{-i}\right)$.

Proof

(a) Suppose $y_{i}\in PB_{i}(Y_{-i},u_{i})$. Then, there exist $y_{i}^{t}\rightsquigarrow$ $y_{i}$ and $x_{-i}^{t}\rightsquigarrow x_{-i}$ $\in Y_{-i}$ such that $\left( u_{i},\left( y_{i}^{t},x_{-i}^{t}\right) ,y_{i}\right) \in {\mathcal {R}} _{i}$ for all t. Since $\left( u_{i},\left( y_{i}^{t},x_{-i}^{t}\right) ,y_{i}\right) \rightarrow \left( u_{i},\left( y_{i},x_{-i}\right) ,y_{i}\right)$, $\left( u_{i},x,y_{i}\right) \in vcl_{U_{i}}({\mathcal {R}} _{i})\cap Q\left( Y_{-i}\right)$. Conversely, suppose $\left( u_{i},x,y_{i}\right) \in vcl_{U_{i}}\left( {\mathcal {R}}_{i}\right)$, $x_{-i}\in Y_{-i}$ and $x_{i}=y_{i}$. Then, there exists a sequence $\left( u_{i},x^{t},y_{i}^{t}\right) \in {\mathcal {R}}_{i}$ converging to $\left( u_{i},x,y_{i}\right)$. Since $\Gamma$ is finite and $y_{i}^{t}\rightarrow y_{i}$, there is a sufficiently large T such that, for all $t\ge T$ and $h\in H_{i}$, $a_{h}\in$support $\left( y_{h}\right)$ implies $a_{h}\in$ support $\left( y_{h}^{t}\right)$ and $a_{h}\in \arg \max _{a_{h}^{\prime }\in A_{h}}v_{i}\left( \left( a_{h}^{\prime },\quad x_{-h}^{t}\right) ,u_{i}\right)$. Therefore, $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y }}_{h}}v_{i}\left( \left( y_{h}^{\prime },\quad x_{-h}^{t}\right) ,u_{i}\right)$ for all $t\ge T$ and all $h\in H_{i}$. That is, $y_{i}\in PB_{i}(Y_{-i},u_{i})$.

(b) It suffices to show $y_{i}\in SB_{i}(Y_{-i},u_{i})$ iff there exist $u_{i}^{t}\rightarrow u_{i}$and $\left( y_{i}^{t},x_{-i}^{t}\right) \rightsquigarrow$ $\left( y_{i},x_{-i}\right)$ such that $x_{-i}\in Y_{-i}$, $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}}_{h}}v_{i}\left( \left( y_{h}^{\prime },\quad x_{-h}^{t}\right) ,u_{i}^{t}\right)$ for all t and all $h\in H_{i}$.

“$\Leftarrow$”: Let $x^{t}=\left( y_{i}^{t},x_{-i}^{t}\right) \rightsquigarrow$ $\left( y_{i},x_{-i}\right) =x$ such that $x_{-i}\in Y_{-i}$. Without loss of generality, assume $\left( x^{t},\mu _{i}^{t}\right) \rightarrow \left( x,\mu _{i}\right)$, where $\mu _{i}^{t}$ is derived from $x^{t}$ using Bayes’ rule. Suppose that there exist $u_{i}^{t}\rightarrow u_{i}$ such that $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}}_{h}}v_{i}\left( \left( y_{h}^{\prime },\quad x_{-h}^{t}\right) ,u_{i}^{t}\right)$ for all t and all $h\in H_{i}$. Then, for all $h\in H_{i}$ and all t, $v_{i}\left( (y_{h},\quad x_{-h}^{t}),u_{i}^{t}|h\right) \ge v_{i}\left( (y_{h}^{\prime },\quad x_{-h}^{t}),u_{i}^{t}|h\right)$ $\forall y_{h}^{\prime }\in {\mathbb {Y}}_{h}$.¹⁶ Since $v_{i}(\left( y_{h},\cdot \right) ,\cdot |h)$ is continuous, $v_{i}\left( \left( y_{h},\quad x_{-h}\right) ,u_{i}|h\right) \ge v_{i}\left( \left( y_{h}^{\prime },\quad x_{-h}\right) ,u_{i}|h\right)$. Since $\Gamma$ is perfect recall, every sequential optimal strategy satisfies the one deviation property (see, Perea 2002), for all $h\in H_{i}$, $v_{i}\left( \left( y_{i},\quad x_{-i}\right) ,u_{i}|h\right) \ge v_{i}\left( \left( y_{i}^{\prime },\quad x_{-i}\right) ,u_{i}|h\right)$ $\forall y_{i}^{\prime }\in {\mathbb {Y}}_{i}$. That is, $y_{i}\in SB_{i}(Y_{-i},u_{i}).$

“$\Rightarrow$”: Let $y_{i}\in SB_{i}(Y_{-i},u_{i})$. Then, there is $\left( \left( y_{i}^{t},x_{-i}^{t}\right) ,\mu _{i}^{t}\right) \rightarrow \left( \left( y_{i},x_{-i}\right) ,\mu _{i}\right)$ such that $x_{-i}\in Y_{-i}$ and $y_{i}$ is sequentially optimal against the assessment $\left( \left( y_{i},x_{-i}\right) ,\mu _{i}\right)$. Denote $\left( y_{i},x_{-i}\right) =x$ and $\left( y_{i}^{t},x_{-i}^{t}\right) =x^{t}$. Clearly, $x^{t}\rightsquigarrow$ x. We proceed to construct a payoff sequence $u_{i}^{t}\rightarrow u_{i}$ such that $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}}_{h}}v_{i}\left( \left( y_{h}^{\prime }, \quad x_{-h}^{t}\right) ,u_{i}^{t}\right)$ for all t and all $h\in H_{i}$.

Since $\Gamma$ is finite and perfect recall holds, we can define a (finite) partition $\left\{ H_{i}^{l}\right\} _{l=1}^{L}$ of the set $H_{i}$ as follows: $H_{i}^{0}=\varnothing ,$ $H_{i}^{l}\equiv \left\{ h\in H_{i}\backslash \cup _{\ell<l}H_{i}^{\ell }:\text {no }h^{\prime }\in \left[ H_{i}\backslash \cup _{\ell <l}H_{i}^{\ell }\right] \backslash h\text { is reached by }h\right\}$ for all $l\ge 1$. Therefore, for all t and $l=1,\ldots ,L$, we can define $u_{i}^{t,l}$ recursively as follows: Let $u_{i}^{t,0}\equiv u_{i}$,

$$\begin{aligned} u_{i}^{t,l}(z)\equiv \left\{ \begin{array}{ll} u_{i}^{t,l-1}(z)+\delta _{a_{h}^{*}}^{t}, &{} \hbox {if}\; z\; \hbox{is}\;\hbox{not}\; \hbox{precluded}\;\hbox{by}\; a_{h}^{*}\in \hbox { support}\;\left( y_{h}\right)\; \hbox { from }\; h \\ u_{i}^{t,l-1}(z), &{} \hbox { otherwise } \end{array} \right. , \end{aligned}$$

where $h\in H_{i}^{l}$, support$\left( y_{h}\right) =\left\{ a_{h}\in A_{h}:y_{h}\left( a_{h}\right) >0\right\}$ and

$$\begin{aligned} \delta _{a_{h}^{*}}^{t}=\max _{a_{h}\in A_{h}}v_{i}\left( \left( a_{h}, \quad x_{-h}^{t}\right) ,u_{i}^{t,l-1}|h\right) -v_{i}\left( \left( a_{h}^{*},\quad x_{-h}^{t}\right) ,u_{i}^{t,l-1}|h\right) . \end{aligned}$$

Therefore, for $l=1,\ldots ,L$, $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}}_{h}}v_{i}\left( \left( y_{h}^{\prime }, \quad x_{-h}^{t}\right) ,u_{i}^{t,l}|h\right)$ $\forall h\in H_{i}^{l}$. For all $a_{h}$, $a_{h}^{\prime }\in A_{h}$, $v_{i}\left( \left( a_{h},\quad x_{-h}^{t}\right) ,u_{i}^{t,l+1}|h\right) -v_{i}\left( \left( a_{h}^{\prime },\quad x_{-h}^{t}\right) ,u_{i}^{t,l+1}|h\right) =v_{i}\left( \left( a_{h},\quad x_{-h}^{t}\right) ,u_{i}^{t,l}|h\right) -v_{i}\left( \left( a_{h}^{\prime },\quad x_{-h}^{t}\right) ,u_{i}^{t,l}|h\right)$. By induction on l, we have $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}} _{h}}v_{i}\left( \left( y_{h}^{\prime },\quad x_{-h}^{t}\right) ,u_{i}^{t,l}\right)$ $\forall h\in \cup _{\ell =1}^{l}H_{i}^{\ell }$. Hence, $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}}_{h}}v_{i}\left( \left( y_{h}^{\prime },\quad x_{-h}^{t}\right) ,u_{i}^{t,L}\right)$ $\forall h\in H_{i}$, which implies that $\left( u_{i}^{t},\left( y_{i}^{t},x_{-i}^{t}\right) ,y_{i}\right) \in {\mathcal {R}}_{i}$.

It remains to show $u_{i}^{t,L}\rightarrow u_{i}^{L}=u_{i}$. We prove this by induction on l. Clearly, $u_{i}^{t,0}\rightarrow u_{i}$ trivially holds. Suppose $u_{i}^{t,\ell }\rightarrow u_{i}^{\ell }=u_{i}$ for $\ell \le l-1$. By construction of $u_{i}^{t,l}$, it suffices to show $\delta _{a_{h}^{*}}^{t}\rightarrow 0$ $\forall h\in H_{i}^{l}$. Let ${\hat{a}} _{h}\in \arg \max _{a_{h}\in A_{h}}v_{i}\left( \left( a_{h},\quad x_{-h}^{t}\right) ,u_{i}^{t,l-1}|h\right)$. Because of the continuity of $v_{i}$, for any $\varepsilon >0$ there is a sufficiently large T such that, for all $t>T$,

$$\begin{aligned} v_{i}\left( \left( {\hat{a}}_{h},\quad x_{-h}^{t}\right) ,u_{i}^{t,l-1}|h\right) -v_{i}\left( \left( {\hat{a}}_{h},\quad x_{-h}\right) ,u_{i}^{l-1}|h\right)< & {} \varepsilon ; \\ v_{i}\left( \left( a_{h}^{*},\quad x_{-h}\right) ,u_{i}^{l-1}|h\right) -v_{i}\left( \left( a_{h}^{*},\quad x_{-h}^{t}\right) ,u_{i}^{t,l-1}|h\right)< & {} \varepsilon . \end{aligned}$$

Since $y_{h}\in \arg \max _{y_{h}^{\prime }\in {\mathbb {Y}}_{h}}v_{i}\left( \left( y_{h}^{\prime },\quad x_{-h}\right) ,u_{i}^{t}\right)$ and, by the induction hypothesis, $u_{i}=u_{i}^{l-1}$,

$$\begin{aligned} v_{i}\left( \left( {\hat{a}}_{h},\quad x_{-h}\right) ,u_{i}^{l-1}|h\right) -v_{i}\left( \left( a_{h}^{*},\quad x_{-h}\right) ,u_{i}^{l-1}|h\right) \le 0. \end{aligned}$$

Therefore, $v_{i}\left( \left( {\hat{a}}_{h},\quad x_{-h}^{t}\right) ,u_{i}^{t,l-1}|h\right) -v_{i}\left( \left( a_{h}^{*},\quad x_{-h}^{t}\right) ,u_{i}^{t,l-1}|h\right) <2\varepsilon$, i.e., $\delta _{a_{h}^{*}}^{t}\rightarrow 0$.¹⁷ □

Lemma 2

Let $W\subseteq {\mathbb {R}}^{n+m}$ be a semi-algebraic set. (a) $cl\left( W\right)$ and $vcl_{\mathbb { R}^{n}}\left( W\right)$ are semi-algebraic. (b) There exists a closed, lower-dimensional semi-algebraic subset $E\subset {\mathbb {R}}^{n}$ such that for all $x\in {\mathbb {R}}^{n}\backslash E$, $\{y\in {\mathbb {R}}^{m}:(x,y)\in cl\left( W\right) \}=\{y\in {\mathbb {R}}^{m}:(x,y)\in vcl_{{\mathbb {R}}^{n}}\left( W\right) \}$.

Proof

(a) $vcl_{{\mathbb {R}}^{n}}\left( W\right)$ can be rewritten as

$$\begin{aligned} \left\{ \left( x,y\right) \in {\mathbb {R}}^{n}\times {\mathbb {R}}^{m}:\forall \varepsilon >0, \exists \left( x,y^{\prime }\right) \in W\text { s.t. } \left\| y-y^{\prime }\right\| <\varepsilon \right\} , \end{aligned}$$

where $\left\| \cdot \right\|$ is the Euclidean norm. Since W is semi-algebraic, it follows from Tarski–Seidenberg theorem that $vcl_{\mathbb { R}^{n}}\left( W\right)$ is also semi-algebraic. Similarly, $cl\left( W\right)$ is semi-algebraic.

(b) This proof is similar to the proof of Theorem 4 in BZ94. Define f, g : ${\mathbb {R}}^{n}\times {\mathbb {R}}^{m}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} f\left( x,y\right)\equiv & {} \underset{\left( x^{\prime },y^{\prime }\right) \in W}{\inf }\left\| \left( x,y\right) -\left( x^{\prime },y^{\prime }\right) \right\| \\ g\left( x,y\right)\equiv & {} \underset{\left( x,y^{\prime }\right) \in W}{\inf } \left\| \left( x,y\right) -\left( x,y^{\prime }\right) \right\| , \end{aligned}$$

where x and $x^{\prime }$ belong to ${\mathbb {R}}^{n}$, y and $y^{\prime }$ belong to ${\mathbb {R}}^{m}$.

Define

$$\begin{aligned} E=\left\{ x\in {\mathbb {R}}^{n}:\exists \left( x,y\right) \in W, f\left( x,y\right) =0\text { and }g\left( x,y\right) >0\right\} . \end{aligned}$$

Since f and g are semi-algebraic functions, E is a semi-algebraic set. Note that $(x,y)\in cl\left( W\right)$ if and only if $f(x,y)=0$; and $(x,y)\in vcl_{{\mathbb {R}}^{n}}\left( W\right)$ if and only if $g(x,y)=0$. Thus, $(x,y)\in cl\left( W\right) \backslash vcl_{{\mathbb {R}}^{n}}\left( W\right)$ implies $x\in E$. Therefore, for all $x\in {\mathbb {R}} ^{n}\backslash E$, $\{y\in {\mathbb {R}}^{m}:(x,y)\in cl\left( W\right) \}=\{y\in {\mathbb {R}}^{m}:(x,y)\in vcl_{{\mathbb {R}}^{n}}\left( W\right) \}$. Suppose E is not lower-dimensional in ${\mathbb {R}}^{n}$. Then, there is a semi-algebraic open set $\mathcal {O}\subseteq {\mathbb {R}}^{n}$ and an $\varepsilon >0$ with the property that for any $x\in \mathcal {O}$ there exists $\left( x,y\right) \in W$ such that $\ f\left( x,y\right) =0$ and $g\left( x,y\right) \ge \varepsilon$. The set

$$\begin{aligned} G=\left\{ \left( x,y\right) \in \mathcal {O}\times {\mathbb {R}}^{m}:f\left( x,y\right) =0\;{\text { and}}\;g\left( x,y\right) \ge \varepsilon \right\} \end{aligned}$$

is semi-algebraic, and its projection onto ${\mathbb {R}}^{n}$ is all of $\mathcal {O}$. So we can choose a semi-algebraic selection $\beta :\mathcal {O} \rightarrow {\mathbb {R}}^{m}$ with the property that $\left( x,\beta \left( x\right) \right) \in G$. By BZ94’s (p. 786) Lemma, there exists a semi-algebraic open set $\mathcal {O}^{\prime }\subset \mathcal {O}$ on which $\beta$ is continuous. Since $\left( x,\beta \left( x\right) \right) \in cl(W)$ for any $w\in \mathcal {O}^{\prime }$, there is a sequence $\left\{ \left( x^{t},y^{t}\right) \right\} _{t=1}^{\infty }$ in W with limit $\left( x,\beta \left( x\right) \right)$. From the continuity of $\beta$, $\left\| \beta \left( x^{t}\right) -\beta \left( x\right) \right\| \rightarrow 0$. Thus, for t large enough, $g\left( x^{t},\beta \left( x^{t}\right) \right) <\varepsilon$, which contradicts the construction of $\beta$. Therefore, E is a lower-dimensional in ${\mathbb {R}}^{n}$ and the result follows. □

Lemma 3

Consider an extensive form $\Gamma$ with perfect recall. For each player i, there is a closed, lower-dimensional semi-algebraic subset $U_{i}^{0}\subset U_{i}$ such that, for any $Y_{-i}\subseteq {\mathbb {Y}}_{-i}$, $PB_{i}(Y_{-i},u_{i})=SB_{i}(Y_{-i},u_{i})$$\forall u_{i}\in U_{i}\backslash U_{i}^{0}.$

Proof

Let $Y_{-i}\subseteq {\mathbb {Y}}_{-i}$. By Lemma 1, $SB_{i}(Y_{-i},u_{i})=\{y_{i}\in {\mathbb {Y}}_{i}:\left( u_{i},x,y_{i}\right) \in cl({\mathcal {R}}_{i})\cap Q\left( Y_{-i}\right) \}$ and $PB_{i}(Y_{-i},u_{i})=\{y_{i}\in {\mathbb {Y}}_{i}:\left( u_{i},x,y_{i}\right) \in vcl_{U_{i}}({\mathcal {R}}_{i})\cap Q\left( Y_{-i}\right) \}$. By the Tarski–Seidenberg theorem, ${\mathcal {R}}_{i}$ is semi-algebraic. By Lemma 2, $cl({\mathcal {R}}_{i})$ and $vcl_{U_{i}}({\mathcal {R}}_{i})$ are generically equivalent on $U_{i}$ and hence, $cl({\mathcal {R}}_{i})\cap Q\left( Y_{-i}\right)$ and $vcl_{U_{i}}({\mathcal {R}}_{i})\cap Q\left( Y_{-i}\right)$ are generically equivalent on $U_{i}$. Therefore, there is a closed, lower-dimensional semi-algebraic subset $U_{i}^{0}\subset U_{i}$ such that $PB_{i}(Y_{-i},u_{i})=SB_{i}(Y_{-i},u_{i})$$\forall u_{i}\in U_{i}\backslash U_{i}^{0}.$ □

Lemma 4

Let $F:U\rightrightarrows {\mathbb {R}}^{n}$ and $F^{\prime }:U\rightrightarrows {\mathbb {R}}^{n}$ . Suppose $V^{0}\equiv \left\{ u^{0}\in U:F\left( u^{0}\right) \ne F^{\prime }\left( u^{0}\right) \right\}$ is a lower-dimensional subset of U . Then, $F\left( u\right) \subseteq F^{\prime }\left( u\right)$ for all $u\in U$ at which $F\left( \cdot \right)$ is lower hemi-continuous and $F^{\prime }\left( \cdot \right)$ is upper hemi-continuous.

Proof of Lemma 4

Since $V^{0}$ is lower-dimensional, $V^{0}$ contains no open set in U. Let $u\in U$. Therefore, we can find a sequence $\left\{ u^{t}\right\} _{t=1}^{\infty }$ in $U\backslash V^{0}$ such that $u^{t}\rightarrow u$ and $F\left( u^{t}\right) =F^{\prime }\left( u^{t}\right)$ for all t. If $y\in F\left( u\right)$, by lower hemi-continuity of $F\left( \cdot \right)$, there exists a subsequence $u^{t_{k}}\rightarrow u$ such that $y^{k}\rightarrow y$ and $y^{k}\in F\left( u^{t_{k}}\right) =F^{\prime }\left( u^{t_{k}}\right)$. Since the correspondence $F^{\prime }\left( \cdot \right)$ is upper hemi-continuous, $y\in F^{\prime }\left( u\right)$. That is, $F\left( u\right) \subseteq F^{\prime }\left( u\right).$ □

Proof of Theorem 1

By Lemma 3, for each player $i\in N$, there is a closed, lower-dimensional semi-algebraic subset $U_{i}^{0}\subset U_{i}={\mathbb {R}}^{\left| Z\right| }$ such that for all $u_{i}\in U_{i}\backslash U_{i}^{0}$, $PB_{i}(Y_{-i},u_{i})=SB_{i}(Y_{-i},u_{i})$ for all $Y_{-i}\subseteq {\mathbb {Y}}_{-i}$; hence, $PB_{i}(\Pi _{j\ne i}co^{e}(Y_{j}),u_{i})=SB_{i}(\Pi _{j\ne i}co^{e}(Y_{j}),u_{i})$ for all $\Pi _{j\ne i}Y_{j}\subseteq {\mathbb {Y}}_{-i}$. Therefore, for all the payoffs $u\in U\backslash U^{0}$ where $U^{0}=\cup _{i\in N}\left( U_{i}^{0}\times U_{-i}\right)$, a course of action $\sigma$ is perfectly supported by Y in $\Gamma (u)$ if and only if $\sigma$ is sequentially supported by Y in $\Gamma (u)$. Thus, we find a closed, lower-dimensional semi-algebraic subset $U^{0}\subset U$ such that $\Sigma ^{\text {sequential} }(u)=\Sigma ^{\text {perfect}}(u)$ and $\top _{\sigma }^{\text {sequential} }(u)=\top _{\sigma }^{\text {perfect}}(u)$ for all $u\in U\backslash U^{0}$ and for an arbitrary given course of action $\sigma$ in $\Gamma$.

Because $PB_{i}(\Pi _{j\ne i}co^{e}(Y_{j}),u_{i})=SB_{i}(\Pi _{j\ne i}co^{e}(Y_{j}),u_{i})$ for all $i\in N$ and $u\in U\backslash U^{0}$, $\{u^{0}\in U:\Upsilon ^{\text {sequential}}\left( u^{0}\right) \ne \Upsilon ^{\text {perfect}}\left( u^{0}\right) \} \subseteq U^{0}$ is a lower-dimensional subset of U. By Lemma 4, $\Upsilon ^{\text {sequential}}\left( u\right) \subseteq \Upsilon ^{\text {perfect}}\left( u\right)$ for all the payoffs $u\in U$ at which $\Upsilon ^{\text {sequential}}\left( \cdot \right)$ is lower hemi-continuous and $\Upsilon ^{\text {perfect}}\left( \cdot \right)$ is upper hemi-continuous. But, since any perfect-MACA in $\Gamma (u)$ is also a sequential-MACA in $\Gamma (u)$, $\Upsilon ^{\text { perfect}}\left( u\right) \subseteq \Upsilon ^{\text {sequential}}\left( u\right)$ for all $u\in U$. Hence, $\Upsilon ^{\text {sequential}}\left( u\right) =\Upsilon ^{\text {perfect}}\left( u\right)$ for all the payoffs $u\in U$ at which $\Upsilon ^{\text {sequential}}\left( \cdot \right)$ is lower hemi-continuous and $\Upsilon ^{\text {perfect}}\left( \cdot \right)$ is upper hemi-continuous. Similarly, for an arbitrary given course of action $\sigma$ in $\Gamma$, $\top _{\sigma }^{\text {sequential}}(u)=\top _{\sigma }^{\text {perfect}}(u)$ for all the payoffs $u\in U$ at which $\top _{\sigma }^{\text {sequential}}(\cdot )$ is lower hemi-continuous and $\top _{\sigma }^{\text {perfect}}(\cdot )$ is upper hemi-continuous. □

Proof of Corollary 1

Define

$$\begin{aligned} \overset{o}{{\mathcal {R}}}\mathcal {=}\left\{ \left( u,x,y\right) \in U\times {\mathbb {Y}}\times {\mathbb {Y}}:\quad \left( u_{i},x,y_{i}\right) \in \mathcal { R}_{i}\quad \forall i\right\} , \end{aligned}$$

that is, $\overset{o}{{\mathcal {R}}}$ is the set of “joint” perfectly rational states in which the players have consistently aligned beliefs about the opponent players’ behavior. Clearly, $\left( u,x,y\right) \in vcl_{U}\left( \overset{o }{{\mathcal {R}}}\right)$ iff there exists a common sequence $x^{t}\rightsquigarrow x$ such that for each player i, $y_{i}$ is the best response along the trembling sequence $x^{t}$. By Definition 3(i), $\left( u,y,y\right) \in vcl_{U}\left( \overset{o}{\mathcal {R}}\right)$ iff y is a perfect equilibrium; by Kreps and Wilson’s (1982) Proposition 6, $\left( u,y,y\right) \in cl\left( \overset{o}{\mathcal {R}}\right)$ iff y is a sequential equilibrium strategy profile. By the Tarski–Seidenberg theorem, $\overset{o}{\mathcal {R}}$ is semi-algebraic. By Lemma 2, $cl\left( \overset{o }{\mathcal {R}}\right)$ and $vcl_{U}\left( \overset{o}{\mathcal {R}}\right)$ are generically equivalent. Consequently, there is a closed, lower-dimensional semi-algebraic subset $U^{0}\subset U$ such that for all $u\in U\backslash U^{0}$, any sequential equilibrium strategy profile is a perfect equilibrium in game $\Gamma (u)$.

Now, suppose that the sequential equilibrium correspondence $SE\left( \cdot \right)$ is lower hemi-continuous and the perfect equilibrium correspondence $PE\left( \cdot \right)$ is upper hemi-continuous at $u\in U$. Since $\left\{ u^{0}\in U:SE\left( u^{0}\right) \ne PE\left( u^{0}\right) \right\} \subseteq U^{0}$ is a lower-dimensional subset, by Lemma 4, $SE\left( u\right) \subseteq PE\left( u\right)$. Thus, $SE\left( u\right) =PE\left( u\right).$ □

Proof of Corollary 2

(i) By Definition 3(iii)-(iv), a strategy profile $y\in {\mathbb {Y}}$ is a weakly perfect (or weakly sequential) equilibrium in $\Gamma (u)$ iff for each player i, $y_{i}\in PB_{i}\left( y_{-i},u_{i}\right) =PB_{i}\left( \Pi _{j\ne i}co^{e}(y_{j}),u_{i}\right)$ (or $y_{i}\in SB_{i}\left( y_{-i},u_{i}\right) =SB_{i}\left( \Pi _{j\ne i}co^{e}(y_{j}),u_{i}\right)$ ). By Definition 2, the strategy profile y is a weakly perfect (or weakly sequential) equilibrium in $\Gamma (u)$ iff y is a perfect (or sequential) MACA in $\Gamma (u)$. By Theorem 1, for generic $u\in U\backslash U^{0}$, the set of weakly sequential equilibria in $\Gamma (u)$ coincides with the set of weakly perfect equilibria in $\Gamma (u)$.

(ii) By Definition 4, a product set $Y=\Pi _{i\in N}Y_{j}\subseteq {\mathbb {Y }}$ is a perfectly (or sequentially) rationalizable set in $\Gamma (u)$ iff for each player i, $Y_{i}\subseteq PB_{i}\left( \Pi _{j\ne i}co^{e}(Y_{j}),u_{i}\right)$ (or $Y_{i}\subseteq SB_{i}\left( \Pi _{j\ne i}co^{e}(Y_{j}),u_{i}\right)$). By Definition 2, the set Y is a perfectly (or sequentially) rationalizable set in $\Gamma (u)$ iff Y perfectly (or sequentially) supports the null MACA in $\Gamma (u)$. By Theorem 1, for generic payoffs $u\in U\backslash U^{0}$, the union of sequentially rationalizable sets in $\Gamma (u)$ coincides with the union of perfectly rationalizable sets in $\Gamma (u)$; that is, the set of sequentially rationalizable strategy profiles in $\Gamma (u)$ coincides with the set of perfectly rationalizable strategy profiles in $\Gamma (u)$.

(iii) This result follows directly from the following lemma: □

Lemma 5

A path resulting from an SRSCE in Definition A3is a path sequential-MACA in Definition 2and vice versa.

Proof of Lemma 5

“$\Rightarrow$ ” Suppose $\widehat{y}$ is an SRSCE in $\Gamma (u)$. By Definition 5, there exists $Y=\Pi _{i\in N}Y_{i}\subseteq {\mathbb {Y}}$, for each $i\in N$ and each $y_{i}\in Y_{i}$, we can find $y_{-i}\in \Pi _{j\ne i}co^{e}(Y_{j})$ such that $y_{h}=\widehat{y}_{h}$ if $h\in H$ is reachable under $\widehat{y}$ and $y_{i}\in SB_{i}\left( y_{-i},u_{i}\right)$. Construct a path course of action $\sigma$ (associated with $\widehat{y}$) as follows: For any $h\in H$,

$$\begin{aligned}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sigma _{h}=\left\{ \begin{array}{ll} \widehat{y}_{h}, &{} \text {if }h\text { is reachable under }\widehat{y} \\ \emptyset , &{} \text {otherwise} \end{array} \right. . \end{aligned}$$

By the construction of $\sigma$, if $\sigma _{h}\ne \emptyset$, then $y_{h}=\sigma _{h}$ for all $y\in Y$. Moreover, since $y_{i}\in SB_{i}\left( y_{-i},u_{i}\right)$ and $y_{-i}\in \Pi _{j\ne i}co^{e}(Y_{j})$, $y_{i}\in SB_{i}\left( \Pi _{j\ne i}co^{e}(Y_{j}),u_{i}\right)$. By Definition 2, the path course of action $\sigma$ (associated with $\widehat{y}$) is a path sequential-MACA in $\Gamma (u)$ supported by Y.

“$\Leftarrow$” Suppose $\sigma$ is a path sequential-MACA in $\Gamma (u)$ supported by $Y=\Pi _{i\in N}Y_{i}\subseteq {\mathbb {Y}}$. By Definition 2, for each player i and each $\widehat{y}_{i}\in$ $Y_{i}$, there exists $\widehat{y}_{-i}\in \Pi _{j\ne i}co^{e}(Y_{j})$ such that $\widehat{y}_{i}\in SB_{i}\left( \widehat{y} _{-i},u_{i}\right)$; moreover, if $\sigma _{h}\ne \emptyset$, $y_{h}=\sigma _{h}$ for all $y\in Y$ and hence $\widehat{y}_{h}=\sigma _{h}$. That is, $\widehat{y}_{h}=\sigma _{h}$ if h is reachable under $\widehat{y}$. By Definition 5, $\widehat{y}$ is an SRSCE that results in the same path $\sigma$.

To show the last part of Corollary 2, let $\sigma$ be a course of action in $\Gamma$. Notice that the player-by-player union of the sequentially/perfectly $\sigma$-supporting sets in $\Gamma (u)$ is again a sequentially/perfectly $\sigma$-supporting set in $\Gamma (u)$. Thus, $\Upsilon _{\sigma }^{\text {sequential}}(u)$ is the largest sequentially $\sigma$-supporting set, and $\Upsilon _{\sigma }^{\text {perfect}}(u)$ is the largest perfectly $\sigma$-supporting set. By Theorem 1, the largest sequentially $\sigma$-supporting set coincides with the largest perfectly $\sigma$-supporting set for generic payoffs $u\in U\backslash U^{0}.$ □

Proof of Corollary 3

Consider a finite normal form $\Gamma =(N,\left\{ A_{i}\right\} _{i\in N})$. Let $\left( W^{k}(u)\right) _{k=0}^{K}$ be an IEWDS procedure in $\Gamma (u)$, where $W^{k}(u)=\Pi _{i\in N}W_{i}^{k}(u)$ such that $W_{i}^{0}(u)=A_{i}$ and $W_{i}^{k}(u)\subseteq A_{i}$ includes the set of i’s surviving weakly undominated actions in the $\left( k-1\right)$-th round of elimination for all $k\ge 1$. Let $i\in N$ and $k=1,2,\ldots ,K$. Because $\Gamma$ is a normal form, by Pearce’s (1984) Lemma 3, $a_{i}\in W_{i}^{k}(u)$ is not strictly dominated in $W^{k}(u)$ iff $a_{i}\in SB_{i}\left( \Delta \left( W_{-i}^{k}(u)\right) ,u_{i}\right)$; by Pearce’s (1984) Lemma 4, $a_{i}\in W_{i}^{k}(u)$ is not weakly dominated in $W^{k}(u)$ iff $a_{i}\in PB_{i}\left( \Delta \left( W_{-i}^{k}(u)\right) ,u_{i}\right)$. By Lemma 3, there is a closed, lower-dimensional subset $U_{i}^{0}\subset U_{i}$ such that for all $u_{i}\in U_{i}\backslash U_{i}^{0}$, $SB_{i}\left( Y_{-i},u_{i}\right) =PB_{i}\left( Y_{-i},u_{i}\right)$ $\forall Y_{-i}\subseteq \Delta (A_{-i})$. Letting $Y_{-i}=\{y_{-i}\in \Delta (A_{-i}):y_{-i}$ has full support on $W_{-i}^{k}(u)\}$, we have $SB_{i}\left( \Delta \left( W_{-i}^{k}(u)\right) ,u_{i}\right) =PB_{i}\left( \Delta \left( W_{-i}^{k}(u)\right) ,u_{i}\right)$ for all $u_{i}\in U_{i}\backslash U_{i}^{0}$. Define $U^{0}=\cup _{i\in N}\left( U_{i}^{0}\times U_{-i}\right)$. Therefore, $U^{0}\subset U$ is a closed, lower-dimensional semi-algebraic subset such that for all $u\in U\backslash U^{0}$, $a\in W^{k}(u)\backslash W^{k+1}(u)$ iff $a\in W^{k}(u)$ is strictly dominated in $W^{k}(u)$, and hence, $\left( W^{k}(u)\right) _{k=0}^{K}$ is an IESDS procedure in $\Gamma (u)$. Since IESDS is order-independent in finite games, IEWDS is generically an order-independent procedure. □