A. Complete type structures
In this section we construct a canonical type structure \({\mathcal {T}}_{\mathfrak {F}}\) for any focus structure \({\mathfrak {F}}\). This type structure will be complete. For notation simplicity, we prove our claim for two players, but our result can be directly generalized to any finite number of players.
We begin by fixing some \({\mathfrak {F}}=((\Theta _i)_{i\in I},(f_i)_{i\in I},(g_i)_{i\in I})\), and for each \(h\in H_i\) we consider the following sequence of spaces:
Obviously, \(\Omega _i^k(h)\) is a compact metrizable space for every \(k\ge 0\). Then, we define the product spaces
and
. For notation simplicity, let us denote the typical element of \({\tilde{T}}_i^0\) by \(t_i\). Hence, each \(t_i\in {\tilde{T}}_i^0\) is essentially an abbreviation for \((\mu _{t_i}^1(h),\mu _{t_i}^2(h),\dots )_{h\in H_i}\), with \(\mu _{t_i}^k(h)\in \Delta (\Omega _i^{k-1}(h))\) standing for the corresponding coordinate of \(t_i\). As usual, we impose the standard coherency condition, thus restricting attention to collections of conditional beliefs in
$$\begin{aligned} {\tilde{T}}_i^1:=\bigm \{t_i\in {\tilde{T}}_i^0: {{\mathrm{marg}}}_{\Omega _i^k(h)} \mu _{t_i}^{k+2}(h)=\mu _{t_i}^{k+1}(h) \text { for all } k\ge 0 \text { and all } h\in H_i\bigm \}. \end{aligned}$$
Then, it follows from earlier works of Brandenburger and Dekel (1993) and Battigalli and Siniscalchi (1999) that there exists a homeomorphism
, with \({\tilde{\pi }}_i^h(t_i):={{\mathrm{Proj}}}_{\Delta (\Theta _i\times S_j(h)\times {\tilde{T}}_j^0)}{\tilde{\pi }}_i(t_i)\). In fact, this is a direct consequence of Kolmogorov extension theorem. Now, for each \(k>1\), we recursively define
$$\begin{aligned} {\tilde{T}}_i^k:=\bigm \{t_i\in {\tilde{T}}_i^{k-1}: {\tilde{\pi }}_i^h(t_i)(\Theta _i\times S_j(h)\times {\tilde{T}}_j^{k-1})=1 \text { for all } h\in H_i\bigm \} \end{aligned}$$
and we let \({\tilde{T}}_i:=\bigcap _{k\ge 0}{\tilde{T}}_i^k\) be the set of conditional belief hierarchies that satisfy coherency and common certainty in coherency. Then, again from Brandenburger and Dekel (1993) and Battigalli and Siniscalchi (1999) it follows that there exists a homeomorphism
, once again with \(\pi _i^h(t_i):={{\mathrm{Proj}}}_{\Delta (\Theta _i\times S_j(h)\times {\tilde{T}}_j)}\pi _i(t_i)\). Note that \({\tilde{T}}_i\) is a compact metrizable space.
Now, for an arbitrary \(\theta _i\in \Theta _i\), define
$$\begin{aligned} {\hat{T}}_{\theta _i}^1:=\bigm \{t_i\in {\tilde{T}}_i: \pi _i^h(t_i)(\{\theta _i\}\times S_j(h)\times {\tilde{T}}_j)=1 \text { for all } h\in H_i\bigm \} \end{aligned}$$
and let \({\hat{T}}_i^1:=\bigcup _{\theta _i\in \Theta _i}{\hat{T}}_{\theta _i}^1\). Observe that \(\{\theta _i\}\) is closed in \(\Theta _i\), and therefore \({\hat{T}}_{\theta _i}^1\) is closed in \({\tilde{T}}_i\) (Aliprantis and Border 1994, Cor. 15.6). Thus, \({\hat{T}}_i^1\) is also closed in \({\tilde{T}}_i\), as it is the finite union of closed subsets. In fact, \({\hat{T}}_i^1\) is also open, as the complement of the closed subset \(\bigcup _{\theta _i'\in \Theta _i{\setminus }\{\theta _i\}}{\hat{T}}_{\theta _i'}^1\). Then, for every \(\theta _i\in \Theta _i\) and every \(k>1\), we recursively define
$$\begin{aligned} {\hat{T}}_{\theta _i}^k:=\bigm \{t_i\in {\hat{T}}_{\theta _i}^{k-1}: \pi _i^h(t_i)(\{\theta _i\}\times S_j(h)\times {\hat{T}}_j^{k-1})=1\bigm \} \end{aligned}$$
and we let \({\hat{T}}_{\theta _i}:=\bigcap _{k\ge 0}{\hat{T}}_{\theta _i}^k\) and moreover \({\hat{T}}_i:=\bigcup _{\theta _i\in \Theta _i}{\hat{T}}_{\theta _i}\). Using a similar argument as above, it follows from Aliprantis and Border (1994, Cor. 15.6) that \({\hat{T}}_{\theta _i}^k\) is closed. Hence, \({\hat{T}}_{\theta _i}\) is also closed as the intersection of closed subsets, and therefore so is \({\hat{T}}_i\) as the finite union of closed sets.
Now, for an arbitrary \(\theta _i\in \Theta _i\), define
$$\begin{aligned} T_{\theta _i}^1:=\bigm \{t_i\in {\hat{T}}_{\theta _i}: \pi _i^h(t_i)(\{\theta _i\}\times S_j(h)\times {\hat{T}}_{\theta _j})=g_i^h(\theta _i)(\theta _j) \text { for all } \theta _j\in \Theta _j \text { and all } h\in H_i\bigm \}. \end{aligned}$$
Observe that \({\hat{T}}_{\theta _j}\) is clopen, and therefore both \(\{t_i\in {\hat{T}}_{\theta _i}: \pi _i^h(t_i)(\{\theta _i\}\times S_j(h)\times {\hat{T}}_{\theta _j})\ge g_i^h(\theta _i)(\theta _j)\}\) and \(\{t_i\in {\hat{T}}_{\theta _i}: \pi _i^h(t_i)(\{\theta _i\}\times S_j(h)\times {\hat{T}}_{\theta _j})\le g_i^h(\theta _i)(\theta _j)\}\) are closed (Aliprantis and Border 1994, Cor. 15.6), thus implying that so is \(T_{\theta _i}^1\). Then, for every \(\theta _i\in \Theta _i\) and every \(k>1\), we recursively define
$$\begin{aligned} T_{\theta _i}^k:=\bigm \{t_i\in T_{\theta _i}^{k-1}: \pi _i^h(t_i)(\{\theta _i\}\times S_j(h)\times T_j^{k-1})=1\bigm \} \end{aligned}$$
and let \(T_{\theta _i}:=\bigcap _{k\ge 0}T_{\theta _i}^k\) and moreover \(T_i:=\bigcup _{\theta _i\in \Theta _i}T_{\theta _i}\). Following the same steps as above, we show that \(T_{\theta _i}\) and \(T_i\) are clopen and therefore compact metrizable subspaces.
Following Brandenburger and Dekel (1993) and Battigalli and Siniscalchi (1999), we show that there exists a continuous function
such that for every
with \(\nu _i^h(\{\theta _i\}\times S_j(h)\times T_{\theta _j}))=g_i^h(\theta _i)(\theta _j)\) for some \(\theta _i\in \Theta _i\) and for all \(h\in H_i\) and all \(\theta _j\in \Theta _j\), there exists some \(t_i\in T_{\theta _i}\) such that \(\pi _i^h(t_i)=\nu _i^h\) for all \(h\in H_i\). Finally, define the type structure \({\mathcal {T}}_{\mathfrak {F}}=((T_i)_{i\in I},(\phi _i)_{i\in I},(\lambda _i)_{i\in I})\), by letting \(\phi _i(t_{\theta _i}):=\theta _i\) and \(\lambda _i^h(t_{\theta _i}):={{\mathrm{marg}}}_{S_j(h)\times T_j}\pi _i^h(t_i)\) for every \(\theta _i\in \Theta _i\) and every \(h\in H_i\). Obviously, \({\mathcal {T}}_{\mathfrak {F}}\) is complete.
B. Proofs
Proofs of Sect. 5
We first introduce some additional notation and prove some intermediate results that we will use throughout the proof of our main theorem. Throughout the entire section, without loss of generality we consider a given structure \({\mathfrak {F}}\) such that for each \(i\in I\), each \(\theta _i\in \Theta _i\) and each \(h\in H_i\) there exists a unique \(\theta _{-i}\in \Theta _{-i}\) with \(g_i^h(\theta _i)(\theta _{-i})=1\).
Lemma B1
(Optimality principle) Fix a structure \({\mathfrak {F}}\), an arbitrary player \(i\in I\), an arbitrary \(\theta _i\in \Theta _i\), an arbitrary history \(h\in H_i\) and some \(k>0\). Then, a strategy \(s_i\in D_{\theta _i}^{k-1}(h)\) is rational in \((B_{\theta _i}^{k}(h), S_i(h))\) if and only if it is rational in \(( B_{\theta _i}^{k}(h), D_{\theta _i}^{k-1}(h))\).
Proof
Necessity is straightforward, i.e., if \(s_i\) is rational in \((B_{\theta _i}^{k}(h),S_i(h))\), then it is obviously the case that \(s_i\in D_{\theta _i}^{k-1}(h)\) and moreover it is rational in the decision problem \((B_{\theta _i}^{k}(h), D_{\theta _i}^{k-1}(h))\). Now, let us now prove sufficiency. Take an arbitrary \(s_i\in D_{\theta _i}^{k-1}(h)\) and assume that it is rational in \((B_{\theta _i}^{k}(h), D_{\theta _i}^{k-1}(h))\). Then, by definition, there exists some \(\beta _i^h\in \Delta (B_{\theta _i}^{k}(h))\) such that
$$\begin{aligned} U_i^h(s_i,\beta _i^h)\ge U_i^h(s_i',\beta _i^h) \end{aligned}$$
(B.1)
for all \(s_i'\in D_{\theta _i}^{k-1}(h)\). Now, assume—contrary to what we want to show—that \(s_i\) is not rational in \((B_{\theta _i}^{k}(h), S_i(h))\), and take an arbitrary rational strategy \(s_i''\) given \(\beta _i^h\). Thus, it is the case that
$$\begin{aligned} U_i^h(s_i'',\beta _i^h)> U_i^h(s_i,\beta _i^h). \end{aligned}$$
(B.2)
Notice that the last inequality is strict, because otherwise \(s_i\) would have been a rational strategy in \((B_{\theta _i}^{k}(h), S_i(h))\). Moreover, from the previous step it follows that \(s_i''\in D_{\theta _i}^{k-1}(h)\). But then, this contradicts the fact that \(s_i\) is rational in \((B_{\theta _i}^{k}(h), D_{\theta _i}^{k-1}(h))\), thus implying that \(s_i\) must necessarily be rational in \((B_{\theta _i}^{k}(h), S_i(h))\). \(\square \)
Now, let \(T_{\theta _i}^{k}:=\bigcap _{h\in H_i}T_{\theta _i}^{k}(h)\). Then, fix an arbitrary \(G\in {\mathcal {H}}:=2^H{\setminus }\{\emptyset \}\), and define
$$\begin{aligned} D_{\theta _i}^{k}(G):= & {} \{s_i\in S_i: s_i\in D_{\theta _i}^{k}(h) \text { for all } h\in H_i(s_i)\cap G\} \end{aligned}$$
(B.3)
$$\begin{aligned} R_{\theta _i}^{k}(G):= & {} \{s_i\in S_i: \text {there is } t_i\in T_{\theta _i}^{k} \text { such that } (s_i,t_i)\in R_i^h \text { for all } h\in H_i(s_i)\cap G\}\nonumber \\= & {} {{\mathrm{Proj}}}_{S_i} (R_i^G\cap (S_i\times T_{\theta _i}^{k})) . \end{aligned}$$
(B.4)
Then, we define the set of \(\theta _i\)’s strategies that survive \({\mathfrak {F}}\)-ICDP at all histories in G by
$$\begin{aligned} D_{\theta _i}(G):=\bigcap _{k=1}^\infty D_{\theta _i}^{k}(G). \end{aligned}$$
Likewise, we define the set of \(\theta _i\)’s strategies that are rational given some type (in \(T_{\theta _i}\)) that satisfies \({\mathfrak {F}}\)-CSBR at all histories in G by
$$\begin{aligned} R_{\theta _i}(G):=\bigcap _{k=1}^\infty R_{\theta _i}^{k}(G). \end{aligned}$$
Construction of conditional beliefs Fix an arbitrary \(G\in {\mathcal {H}}\), an arbitrary \(\theta _i\in \Theta _i\) and an arbitrary \(s_i\in D_{\theta _i}^{1}(G)\). Then, it follows directly from Pearce (1984, Lem. 3) that for every \(h\in H_i(s_i)\cap G\) there exists at least one conditional belief \(\beta _{\theta _i,s_i,G}^h\in \Delta \bigl (S_{-i}(h)\bigr )\) such that
$$\begin{aligned} U_i^h(s_i,\beta _{\theta _i,s_i,G}^h)\ge U_i^h(s_i',\beta _{\theta _i,s_i,G}^h) \end{aligned}$$
(B.5)
for all \(s_i'\in S_i(h)\). Now, consider the following two cases:
-
Suppose there exists some \(k\in {\mathbb {N}}\) such that \(s_i\in D_{\theta _i}^{k}(G){\setminus } D_{\theta _i}^{k+1}(G)\). Then, it follows by definition that \(s_i\) is rational in \((B_{\theta _i}^{k}(h), D_{\theta _i}^{k-1}(h))\). Hence, it follows from the optimality principle (Lemma B1) that we can choose some \(\beta _{\theta _i,s_i,G}^h\in \Delta \bigl (B_{\theta _i}^{k}(h)\bigr )\) satisfying Eq. (B.5).
-
Suppose that \(s_i\in D_{\theta _i}^{k}(G)\) for all \(k\in {\mathbb {N}}\). Then, it follows by definition that \(s_i\) is rational in \((B_{\theta _i}^{k}(h), D_{\theta _i}^{k-1}(h))\) for every \(k\in {\mathbb {N}}\). Thus we can choose some \(\beta _{\theta _i,s_i,G}^h\in \Delta (B_{\theta _i}^{\mathfrak {F}}(h))\) satisfying Eq. (B.5).
In either of the two cases, complete the collection of conditional beliefs \((\beta _{\theta _i,s_i,G}^h)_{h\in H_i}\) by considering arbitrary conditional beliefs \(\beta _{\theta _i,s_i,G}^{h'}\in \Delta (S_{-i}(h'))\) for every \(h'\in H_i{\setminus }(H_i(s_i)\cap G)\).
Construction of types For each player \(i\in I\) and each \(\theta _i\in \Theta _i\) define the finite set
$$\begin{aligned} \Psi _{\theta _i}:=\{\psi _{\theta _i,s_i,G}\ |\ (s_i,G)\in S_i\times {\mathcal {H}}\}, \end{aligned}$$
and let \(\Psi _i:=\bigcup _{\theta _i\in \Theta _i}\Psi _{\theta _i}\) and
. Now, define the function \(\phi _i:\Psi _i\rightarrow \Theta _i\) by \(\phi _i(\psi _i)=\theta _i\) for each \(\psi _i\in \Psi _{\theta _i}\). Moreover, define the mapping \(\gamma _i^h:\Psi _i\rightarrow \Delta (S_{-i}(h) \times \Psi _{-i})\) for each \(h\in H_i\) as follows: For each \(s_i\in D_{\theta _i}^{1}(G)\), let
$$\begin{aligned}&\gamma _i^h(\psi _{\theta _i,s_i,G})(s_{-i},\psi _{-i})\nonumber \\&\quad := {\left\{ \begin{array}{ll} \beta _{\theta _i,s_i,G}^h(s_{-i}) &{} \text {if }\,\, \psi _j=\psi _{\theta _j,s_j, F_{\theta _i}(h)} \text { for all } j\ne i \text { and } g_i^h(\theta _i)\bigl ((\theta _j)_{j\ne i}\bigr )=1\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$
On the other hand, if \(s_i\notin D_{\theta _i}^{1}(G)\), let \(\gamma _i^h(\psi _{\theta _i,s_i,G})\) be an arbitrary probability measure over \(S_{-i}(h)\times \Psi _{-i}\) such that
for all \((\theta _j)_{j\ne i}\in \Theta _{-i}\). Now, observe that \(((\Psi _i)_{i\in I},(\phi _i)_{i\in I},(\gamma _i)_{i\in I})\) is a finite type structure, implying that each \(\psi _i\in \Psi _i\) is associated with a hierarchy of conditional beliefs.
Recall that we have assumed \({\mathcal {T}}_{\mathfrak {F}}=((T_i)_{i\in I},(\phi _i)_{i\in I},(\lambda _i)_{i\in I})\) to be a complete type structure. Then, it follows from Appendix A that there exists a function \(\xi _i:\Psi _i\rightarrow T_i\) mapping each \(\psi _{\theta _i,s_i,G}\) to a (unique) type \(t_{\theta _i,s_i,G}:=\xi _i(\psi _{\theta _i,s_i,G})\in T_{\theta _i}\) such that (i) \(t_{\theta _i,s_i,G}\) and \(\psi _{\theta _i,s_i,G}\) induce the same conditional belief hierarchy, and moreover (ii) it is the case that \(\phi _i(\psi _{\theta _i,s_i,G})=\phi _i(t_{\theta _i,s_i,G})\). Furthermore, notice that by construction it is the case that \(\lambda _i^h(t_{\theta _i,s_i,G})(s_{-i},t_{-i})= \gamma _i^h(\psi _{\theta _i,s_i,G})(s_{-i},\xi _i^{-1}(t_{-i}))\) for all \((s_{-i},t_{-i})\in S_{-i}\times T_{-i}\). Finally, by construction it is the case that \((s_i,t_{\theta _i,s_i,G})\in R_{\theta _i}^G\) whenever \(s_i\in D_{\theta _i}^{1}(G)\).
Before moving on, for notation simplicity, let us adopt the convention that \(T_{\theta _i}^{0}(h):=T_{\theta _i}\).
Lemma B2
For every \(i\in I\), every \(\theta _i\in \Theta _i\), every \(G\in {\mathcal {H}}\) and every \(k>0\), the following hold:
-
(i)
If \(t_i\in T_{\theta _i}^{k-1}(h)\) then \(b_i^h(t_i)\in \Delta (B_{\theta _i}^{k}(h))\).
-
(ii)
If \(s_i\in D_{\theta _i}^{k}(G)\) then \(t_{\theta _i,s_i,G}\in T_{\theta _i}^{k-1}(h)\) for all \(h\in H_i(s_i)\cap G\).
-
(iii)
\(R_{\theta _i}^{k-1}(G)= D_{\theta _i}^{k}(G)\).
Proof
We prove the result by induction on k.
Initial step First, it is rather trivial to prove the result for \(k=1\). Indeed, observe that by construction it is the case that \(B_{\theta _i}^{{\mathfrak {F}},1}(h)= B_{\theta _i}^{{\mathfrak {F}},0}(h)=S_{-i}(h)\), and therefore \(\Delta (B_{\theta _i}^{{\mathfrak {F}},1}(h))= \Delta (S_{-i}(h))\), thus implying that \(b_i^h(t_i)\in \Delta (B_{\theta _i}^{{\mathfrak {F}},1}(h))\) for all \(t_i\in T_{\theta _i}\), which proves (i). Moreover, recall from our convention that \(T_{\theta _i}^{{\mathfrak {F}},0}(h)=T_{\theta _i}\), thus implying that \(t_{\theta _i,s_i,G}\in T_{\theta _i}^{{\mathfrak {F}},0}(h)\) for all \(h\in H_i(s_i)\cap G\), irrespective of whether \(s_i\in D_{\theta _i}^{{\mathfrak {F}},1}\) or not, which proves (ii). Finally, notice that
$$\begin{aligned} R_{\theta _i}^{{\mathfrak {F}},0}(G)= & {} \{s_i\in S_i: \text {there is } t_i\in T_{\theta _i}^{{\mathfrak {F}},0}(h) \text { such that } (s_i,t_i)\in R_i^h \text { for all } h\in H_i(s_i)\cap G\}\\= & {} \{s_i\in S_i: \text {there is } t_i\in T_{\theta _i} \text { such that } (s_i,t_i)\in R_i^G\}\\= & {} D_{\theta _i}^{{\mathfrak {F}},1}(G) \end{aligned}$$
which proves (iii).
Inductive step We assume that the result holds for an arbitrary \(k>0\). We will refer to this as our “induction assumption (IA)”. Then, we are going to prove it for \(k+1\).
Proof of (i) Fix some \(h\in H_i\), and assume that \(t_i\in T_{\theta _i}^{k}(h)\). Then, by definition it is the case that
$$\begin{aligned} t_i\in SB_{\theta _i}^h(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))). \end{aligned}$$
Then, we consider the following two cases:
-
(a)
Let \(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))\ne \emptyset \).
By the definition of strong belief (at h) it is the case that \(\lambda _i^h(t_i)(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h))))=1\). Now, recall by Eq. (B.4) that
$$\begin{aligned} R_{-\theta _i}^{k-1}(F_{\theta _i}(h))={{\mathrm{Proj}}}_{S_{-i}} (R_{-i}^{F_i(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))), \end{aligned}$$
and therefore it follows that \(b_i^h(t_i)(R_{-\theta _i}^{k-1}(F_{\theta _i}(h)))=1\). Now observe that
with \((\theta _j)_{j\ne i}\in \Theta _{-i}\) being such that \(g_i^h(\theta _i)((\theta _j)_{j\ne i})=1\). Thus, it is the case that
Now, there are two possibilities. According to the first possibility we have \(C_{\theta _i}^{k}(h)\ne \emptyset \), in which case we obtain
$$\begin{aligned} B_{\theta _i}^{k+1}(h)= & {} C_{\theta _i}^{k}(h)\\= & {} S_{-i}(h)\cap R_{-\theta _i}^{k-1}(F_{\theta _i}(h)). \end{aligned}$$
Then, by combining \(b_i^h(t_i)(R_{-\theta _i}^{k-1} (F_{\theta _i}(h)))=1\) with \(b_i^h(t_i)(S_{-i}(h))=1\), it is straightforward to obtain \(b_i^h(t_i)(B_{\theta _i}^{k+1}(h) )=1\). According to the second possibility we have \(C_{\theta _i}^{k}(h)=\emptyset \), in which case we obtain \(B_{\theta _i}^{k+1}(h)=B_{\theta _i}^{k}(h)\). But then, since \(t_i\in T_{\theta _i}^{k}(h)\subseteq T_{\theta _i}^{k-1}(h)\), it follows from the IA that \(b_i^h(t_i)(B_{\theta _i}^{k+1}(h))= b_i^h(t_i)(B_{\theta _i}^{k}(h))=1\), which completes this part of the proof for the first case.
-
(b)
Let \(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))=\emptyset \).
Then, it follows by definition that
$$\begin{aligned} R_{-\theta _i}^{k-1}(F_{\theta _i}(h))\cap S_{-i}(h)\subseteq & {} R_{-\theta _i}^{k-1}\bigl (F_{\theta _i}(h)\bigr ) \nonumber \\= & {} {{\mathrm{Proj}}}_{S_{-i}} \Bigl (R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))\Bigr ) \nonumber \\= & {} \emptyset \end{aligned}$$
(B.8)
Now, using the same reasoning as in Eq. (B.6), combined with Eq. (B.8), we obtain
again with \((\theta _j)_{j\ne i}\in \Theta _{-i}\) being such that \(g_i^h(\theta _i)\bigl ((\theta _j)_{j\ne i}\bigr )=1\). Moreover, using the same argument as in Eq. (B.7), we obtain
$$\begin{aligned} C_{\theta _i}^{k}(h)=S_{-i}(h)\cap R_{-\theta _i}^{k-1}(F_{\theta _i}(h)). \end{aligned}$$
Thus, combining the previous two equations, we conclude that \(C_{\theta _i}^{k}(h)=\emptyset \). Hence, \(B_{\theta _i}^{k+1}(h)=B_{\theta _i}^{k}(h)\). Finally, since \(t_i\in T_{\theta _i}^{k}(h)\subseteq T_{\theta _i}^{k-1}(h)\), it follows from the IA that \(b_i^h(t_i)(B_{\theta _i}^{k+1}(h))= b_i^h(t_i)(B_{\theta _i}^{k}(h))=1\), which completes the proof of part (i).
Proof of (ii): Take an \(s_i\in D_{\theta _i}^{k+1}(G)\), and consider some \(h\in H_i(s_i)\cap G\). Since \(D_{\theta _i}^{k+1}(G)\subseteq D_{\theta _i}^{k}(G)\), it follows by the IA that \(t_{\theta _i,s_i,G}\in T_{\theta _i}^{k-1}(h)\). Hence, it suffices to prove that
$$\begin{aligned} t_{\theta _i,s_i,G}\in SB_{\theta _i}^h(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))). \end{aligned}$$
(B.9)
The latter amounts to proving that
$$\begin{aligned}&\Bigm [R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))\ne \emptyset \Bigm ]\nonumber \\&\quad \Rightarrow \ \Bigm [\lambda _i^h(t_{\theta _i,s_i,G})(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h))))=1\Bigm ] \end{aligned}$$
(B.10)
First, notice that \(t_{\theta _i,s_i,G}\in SB_{\theta _i}^h(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h))))\) is trivially satisfied whenever \(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))=\emptyset \). Hence, we will focus on the case where \(R_{-i}^{F_{\theta _i}(h)}\cap (S_{-i}\times T_{-\theta _i}^{k-1}(F_{\theta _i}(h)))\ne \emptyset \). Recall that \((\theta _j)_{j\ne i}\) is the unique element of \(\Theta _{-i}\) receiving positive probability by \(g_i^h(\theta _i)\). Then, for every \(j\ne i\), there exists some \((s_j^*,t_j^*)\in S_j(h)\times T_{\theta _j}\) such that (i) \((s_j^*,t_j^*)\in R_j^{h'}\) for all \(h'\in H_j(s_j^*)\cap F_{\theta _i}(h)\), and (ii) \(t_j^*\in T_{\theta _j}^{k-1}(h')\) for all \(h'\in H_j\cap F_{\theta _i}(h)\).
Now, we are going to prove that \(s_j^*\in D_{\theta _j}^{k}(h')\) for every \(h'\in H_j(s_j^*)\cap F_{\theta _i}(h)\). To do so, take an arbitrary \(t_j^{k-1}\in T_{\theta _j}^{k-1}\), and define the type \(t_j^{**}\) by
$$\begin{aligned} \lambda _j^{h'}(t_j^{**}):= {\left\{ \begin{array}{ll} \lambda _j^{h'}(t_j^*) &{} \text {for each } h'\in H_j(s_j^*)\cap F_{\theta _i}(h),\\ \lambda _j^{h'}(t_j^{k-1}) &{} \text {for each } h'\in H_j{\setminus } (H_j(s_j^*)\cap F_{\theta _i}(h)). \end{array}\right. } \end{aligned}$$
Notice that since \({\mathcal {T}}_{\mathfrak {F}}\) is a complete type structure, such a type exists. Observe that by construction it is the case that \((s_j^*,t_j^{**})\in R_j^{F_{\theta _i}(h)}\), and moreover \(t_j^{**}\in T_{\theta _j}^{k-1}\). Therefore, we obtain
$$\begin{aligned} s_j^*&\in R_{\theta _j}^{k-1}(F_{\theta _i}(h))\cap S_j(h)&\\&= D_{\theta _j}^{k}(F_{\theta _i}(h))\cap S_j(h)&({\text {by the IA}})\\&=\bigm \{s_j\in S_j(h): s_j\in D_{\theta _j}^{k}(h') \text { for all } h'\in H_j(s_j)\cap F_{\theta _i}(h)\bigm \}&\\&\ne \emptyset .&\end{aligned}$$
The latter implies directly by definition that \(C_{\theta _i}^{k}(h) \ne \emptyset \). Hence, it is—also by definition—the case that
$$\begin{aligned} B_{\theta _i}^{k+1}(h)= C_{\theta _i}^{k}(h). \end{aligned}$$
(B.11)
Now, notice that by construction \(\lambda _i^h(t_{\theta _i,s_i,G})\) put positive probability only to strategy-type pairs \((s_j,t_j)\) such that \(t_j=t_{\theta _j,s_j,F_{\theta _i}(h)}\). Moreover, since \(s_i\in D_{\theta _i}^{k+1}(G)\) it follows from the construction of the beliefs that \(b_i^h(t_{\theta _i,s_i,G})\in \Delta \bigl (B_{\theta _i}^{k+1}(h)\bigr )\). Therefore, it follows from Eq. (B.11) that \({{\mathrm{marg}}}_{S_j\times T_j} \lambda _i^h(t_{\theta _i,s_i,G})\) puts positive probability only to strategy-type pairs \((s_j,t_j)\in S_j(h)\times T_j\) such that \(t_j=t_{\theta _j,s_j,F_i(h)}\) and \(s_j\in D_{\theta _j}^{k}(h')\) for all \(h'\in H_j(s_j)\cap F_{\theta _i}(h)\). Hence, from the IA it follows that \({{\mathrm{marg}}}_{S_j\times T_j}\lambda _i^h(t_{\theta _i,s_i,G})\) assigns probability 1 to
$$\begin{aligned} R_j^{F_{\theta _i}(h)}\cap \bigm \{(s_j,t_j)\in S_j\times T_j: t_j\in T_{\theta _j}^{k-1}(h') \text { for all } h'\in H_j\cap F_{\theta _i}(h)\bigm \} \end{aligned}$$
for every \(j\ne i\). Therefore, by definition, \(t_{\theta _i,s_i,G}\in T_{\theta _i}^{k}(h)\), which completes the proof of part (ii).
Proof of (iii): First, we prove that \(R_{\theta _i}^{k-1}(G)\subseteq D_{\theta _i}^{k}(G)\): Take an arbitrary \(s_i\in R_{\theta _i}^{k-1}(G)\). By definition there exists a type in \(t_i\in T_{\theta _i}^{k-1}\) such that \((s_i,t_i)\in R_i^G\). Now, by part (i) of the result—that we have already proven above—it follows that \(b_i^h(t_i)(B_{\theta _i}^{k}(h))=1\) for all \(h\in H_i(s_i)\cap G\), implying that at all histories \(h\in H_i(s_i)\cap G\), the strategy \(s_i\) is rational in the decision problem \((B_{\theta _i}^{k}(h), D_{\theta _i}^{k-1}(h))\). Thus, we conclude that \(s_i\in D_{\theta _i}^{k}(h)\) for all \(h\in H_i(s_i)\cap G\). The latter directly implies that \(s_i\in D_{\theta _i}^{k}(G)\) which completes this part of the proof.
Second, we prove that \(D_{\theta _i}^{k}(G)\subseteq R_{\theta _i}^{k-1}(G)\): Take an arbitrary \(s_i\in D_{\theta _i}^{k}(G)\). Then, by part (ii) that we have already proven above, it follows that \(t_{\theta _i,s_i,G}\in T_{\theta _i}^{k-1}(h)\) for all \(h\in G\cap H_i(s_i)\). Now, fix an arbitrary type \(t_i^{k-1}\in T_{\theta _i}^{k-1}\), and define the type \(t_{\theta _i,s_i,G}^*\in T_{\theta _i}\) by
$$\begin{aligned} \lambda _i^{h}(t_{\theta _i,s_i,G}^*):= {\left\{ \begin{array}{ll} \lambda _i^{h}(t_{\theta _i,s_i,G}) &{} \text {for each } h\in H_i(s_i)\cap G,\\ \lambda _i^{h}(t_i^{k-1}) &{} \text {for each } h\in H_i{\setminus } (H_i(s_i)\cap G). \end{array}\right. } \end{aligned}$$
Notice that since \({\mathcal {T}}_{\mathfrak {F}}\) is a complete type structure, such a type exists. Then, by construction it is the case that \(t_{\theta _i,s_i,G}^*\in T_{\theta _i}^{k-1}\), and we have that \((s_i,t_{\theta _i,s_i,G}^*)\in R_i^h\) for all \(h\in G\cap H_i(s_i)\). Hence, we conclude that \(s_i\in R_{\theta _i}^{k-1}(G)\), which completes the proof of the lemma. \(\square \)
Proof of Theorem 1
Take an arbitrary \(i\in I\), an arbitrary \(\theta _i\in \Theta _i\) and some \(h\in H_i\).
Proof of (i): It follows directly from Lemma B2.i.
Proof of (ii): Fix an arbitrary \(\beta _i^h\in \Delta (B_{\theta _i}^{{\mathfrak {F}},k}(h))\), and let \(s_i^*\in D_{\theta _i}^{{\mathfrak {F}},k}(h)\) be such that
$$\begin{aligned} U_i^h(s_i^*,\beta _i^h)\ge U_i^h(s_i,\beta _i^h) \end{aligned}$$
(B.12)
for all \(s_i\in D_{\theta _i}^{{\mathfrak {F}},k-1}(h)\). In fact, notice that Eq. (B.12) holds, not only for every \(s_i\in D_{\theta _i}^{{\mathfrak {F}},k-1}(h)\), but for every \(s_i\in S_i(h)\) (see Lemma B1). Now, we define \(\beta _{\theta _i,s_i^*,\{h\}}^h:=\beta _i^h\), and construct the type \(t_{\theta _i,s_i^*,\{h\}}\) like we did above. Then, by Lemma B2.ii, it is the case that \(t_{\theta _i,s_i^*,\{h\}}\in T_{\theta _i}^{{\mathfrak {F}},k-1}(h)\), which—together with the fact that \(\beta _{\theta _i,s_i^*,\{h\}}^h:=b_i^h(t_{\theta _i,s_i^*,\{h\}})\)—completes the proof. \(\square \)
Proof of Theorem 2
Observe that by construction
$$\begin{aligned} R_{\theta _i}^{\mathfrak {F}}(H)= & {} {{\mathrm{Proj}}}_{S_i}(R_i\cap (S_i\times T_{\theta _i}^{\mathfrak {F}}))\\ D_{\theta _i}^{\mathfrak {F}}(H)= & {} \{s_i\in S_i: s_i\in D_{\theta _i}^{\mathfrak {F}}(h) \text { for all } h\in H_i(s_i)\}, \end{aligned}$$
and recall by Lemma B2.iii that \(R_{\theta _i}^{\mathfrak {F}}(H)=D_{\theta _i}^{\mathfrak {F}}(H)\), which completes the proof. \(\square \)
Proofs of Sect. 8
In this section, we focus on structures \({\mathfrak {F}}\) with commonly known \(F\in {\mathcal {F}}\), implying that \(\Theta _i\) is a singleton for each \(i\in I\). Thus, recall that we identify the unique \(\theta _i\) with i, e.g., we write \(F_i(h)\) for \(F_{\theta _i}(h)\).
Proof of Proposition 1
We proceed by induction on k. First, note that \(SB_i^1=\bigcap _{h\in H_i} T_i^{1}(h)\). Then, assume that for every \(i\in I\) it is the case that \(SB_i^{k-1}=\bigcap _{h\in H_i} T_i^{k-1}(h)\). Now, observe that for every \(i\in I\) and \(h\in H_i\), it is the case that
Hence, it is the case that
$$\begin{aligned} SB_i^k= & {} SB_i^{k-1}\cap SB_i\bigl (R_{-i}\cap (S_{-i}\times SB_{-i}^{k-1})\bigr )\\= & {} \Bigl (\bigcap _{h\in H_i} T_i^{k-1}(h)\Bigr )\cap \Bigl (\bigcap _{h\in H_i} SB_i^h \Bigl (R_{-i}^{F_i(h)}\cap \bigl (S_{-i}\times T_{-i}^{k-1}\bigl (F_i(h)\bigr )\bigr )\Bigr )\Bigr )\\= & {} \bigcap _{h\in H_i} \Bigl (T_i^{k-1}(h)\cap SB_i^h\Bigl (R_{-i}^{F_i(h)}\cap \bigl (S_{-i}\times T_{-i}^{k-1}\bigl (F_i(h)\bigr )\bigr )\Bigr )\Bigr )\\= & {} \bigcap _{h\in H_i} T_i^{k}(h) \end{aligned}$$
which completes the proof. \(\square \)
In order to prove Proposition 2, we first recall the formal definition of the backward dominance procedure (BDP), originally introduced by Perea (2014).
Backward dominance procedure For an arbitrary \(i\in I\) and an arbitrary \(h\in H\), consider the following sequence of subsets of \(S_i(h)\):
$$\begin{aligned} Q_i^1(h)&:=S_i(h)\\ Q_i^2(h)&:=\{s_i\in Q_i^1(h):s_i \text { is rational in } \bigl (Q_{-i}^1(h'),Q_i^1(h')\bigr ) \text { at all } h'\in H_i(s_i)\cap {{\mathrm{Fut}}}(h)\}\\&~~\vdots \\ Q_i^k(h)&:= \{s_i\in Q_i^{k-1}(h):s_i \text { is rational in } (Q_{-i}^{k-1}(h'),Q_i^{k-1}(h')) \text { at all } h'\in H_i(s_i)\cap {{\mathrm{Fut}}}(h)\}\\&~~\vdots \end{aligned}$$
for each \(k>0\), where
. We say that a strategy \(s_i\) survives k steps of the procedure at \(h\in H_i\) whenever \(s_i\in Q_i^k(h)\). The idea is that a strategy survives k steps of the procedure at some \(h\in H_i\) whenever it is not strictly dominated in the corresponding normal form game—that has survived so far—at every history following h where i is active. Then, we define
$$\begin{aligned} Q_i(h):=\bigcap _{k=1}^\infty Q_i^k(h), \end{aligned}$$
(B.13)
and we say that a strategy survives the procedure whenever it is the case that \(s_i\in Q_i(h)\) for all \(h\in H_i(s_i)\).
Now, let us prove an intermediate lemma that we will use in the proof of Proposition 2.
Lemma B3
Let the structure \({\mathfrak {F}}\) be such that \((F_i)_{i\in I}\) is commonly known with \(F_i(h)={{\mathrm{Fut}}}(h)\) for all \(i\in I\) and all \(h\in H_i\). Then, for every \(i\in I\), every \(h\in H_i\) and every \(k\ge 1\) the following hold:
-
(i)
\(Q_{-i}^k(h)=B_i^{k}(h)\).
-
(ii)
\(Q_i^{k+1}(h)=\{s_i\in S_i(h):s_i\in D_i^{k}(h') \text { for all } h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\}\).
Proof
We proceed to prove the result by induction on k. The result trivially holds for \(k=1\). We assume it holds for \(k-1\) and we will prove it for k. We begin with part (i). Fix an arbitrary \(i\in I\) and an arbitrary \(h\in H_i\), and observe that
which completes the inductive step of the proof for part (i).
Now, we move to the inductive step for part (ii). Again, fix an arbitrary \(i\in I\) and an arbitrary \(h\in H_i\), and take an arbitrary \(s_i\in Q_i^{k+1}(h)\). Then, by definition, \(s_i\) is rational in \((Q_{-i}^k(h'),Q_i^k(h'))\) for every \(h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\), and by part (i) of the present result, \(s_i\) is rational in \((B_i^{k}(h'),Q_i^k(h'))\) for every \(h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\). Now, notice that for every \(s_i'\in S_i(h')\),
$$\begin{aligned} s_i'\text { is rational in } (B_i^{k}(h'),Q_i^k(h'))\Leftrightarrow & {} s_i'\text { is rational in } (B_i^{k}(h'),S_i(h'))\\\Leftrightarrow & {} s_i'\text { is rational in } (B_i^{k}(h'),D_i^{k-1}(h')). \end{aligned}$$
The first equivalence follows from Perea (2012, Lem. 8.14.6), while the second one follows from Lemma B1 above. Hence, \(s_i\) is rational in \((B_i^{k}(h'),D_i^{k-1}(h'))\) for every \(h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\), thus implying that \(s_i\in D_i^{k}(h')\) for every \(h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\). Therefore,
$$\begin{aligned} Q_i^{k+1}(h)\subseteq \{s_i\in S_i(h):s_i\in D_i^{k}(h') \text { for all } h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\}. \end{aligned}$$
(B.14)
Now, in order to prove the inverse weak inequality, take some \(s_i\) with \(s_i\in D_i^{k}(h')\) for every \(h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\). This implies that \(s_i\) is rational in \((Q_{-i}^k(h'),D_i^{k-1}(h'))\) for every \(h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\), and by the previous sequence of equivalences, \(s_i\) is rational in \((Q_{-i}^k(h'),Q_i^k(h'))\) for every \(h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\). Then, by definition, \(s_i\in Q_i^{k+1}(h)\), thus proving that
$$\begin{aligned} Q_i^{k+1}(h)\supseteq \{s_i\in S_i(h):s_i\in D_i^{k}(h')\quad \text { for all } h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\}. \end{aligned}$$
(B.15)
Then, inequalities (B.14) and (B.15) complete this part of the proof. \(\square \)
Proof of Proposition 2
It follows from Perea (2014, Thm. 5.4) that a strategy can be rationally played under CBFR (in a complete type structure) if and only if it survives the BDP, i.e., formally, \(s_i\in Q_i(h)\) for all \(h\in H_i(s_i)\) if and only if \(s_i\in {{\mathrm{Proj}}}_{S_i}(R_i\cap (S_i\times CFB_i))\). Moreover, from our Theorem 2, a strategy \(s_i\) can be rationally played under \({\mathfrak {F}}\)-CSBR (in a complete type structure) if and only if it survives the \({\mathfrak {F}}\)-ICDP, i.e., formally, \(s_i\in D_i^{\mathfrak {F}}(h)\) for all \(h\in H_i(s_i)\) if and only if \(s_i\in {{\mathrm{Proj}}}_{S_i}(R_i\cap (S_i\times T_i^{\mathfrak {F}}))\). Thus, it suffice to prove that a strategy survives BDP if and only if it survives \({\mathfrak {F}}\)-ICDP.
First, consider an arbitrary strategy \(s_i\) surviving the BDP. Then, it must be the case that \(s_i\in Q_i^k(h)\) for every \(k>0\) and every \(h\in H_i(s_i)\). Thus, by Lemma B3, the latter is true if and only if \(s_i\in \{s_i'\in S_i(h):s_i'\in D_i^{k}(h') \text { for all } h'\in {{\mathrm{Fut}}}(h)\cap H_i(s_i)\}\) for all \(k>0\) and for all \(h\in H_i(s_i)\). Obviously, the latter is equivalent to \(s_i\in D_i^{k}(h)\) for every \(k>0\) and every \(h\in H_i(s_i)\), which by definition means that \(s_i\) survives the \({\mathfrak {F}}\)-ICDP, thus completing the proof. \(\square \)