Iterated elimination procedures

Abstract

We study the existence and uniqueness (i.e., order independence) of any arbitrary form of iterated elimination procedures in an abstract environment. By allowing for a transfinite elimination, we show a general existence of the iterated elimination procedure. Inspired by the seminal work of Gilboa et al. (OR Lett 9:85–89, 1990), we identify a fairly weak sufficient condition of Monotonicity* for the order independence of iterated elimination procedure. Monotonicity* requires a Monotonicity property along any elimination path. Our approach is applicable to different forms of iterated elimination procedures used in (in)finite games, for example iterated elimination of strictly dominated strategies, iterated elimination of weakly dominated strategies, rationalizability, and so on. We introduce a notion of CD* games, which incorporates Jackson’s (Rev Econ Stud 59:757–775, 1992) idea of “boundedness,” and show the iterated elimination procedure is order independent in the class of CD* games. In finite games, we also formulate and show an “outcome” order-independence result suitable for Marx and Swinkles’s (Games Econ Behav 18:219–245, 1997) notion of nice weak dominance.

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Notes

  1. 1.

    Equilibrium solution concepts are based on circular fixed-point reasoning, but as stressed by Selten (1998), humans have a tendency to avoid circular concepts. A natural way of problem solving is to use step-by-step reasoning processes. In contrast to the fixed-point method, the alternative approach develops solution concepts by using iterative procedures, for example Bernheim’s (1984) and Pearce’s (1984) rationalizability, Dekel and Fudenberg’s (1990) iterative procedure, Borgers’s (1993) iterated pure-strategy dominance, Gul’s (1996) \(\tau \)-theories, Herings and Vannetelbosch’s (2000) weakly perfect rationalizability, Ambrus’s (2006) definition of coalitional rationalizability, Cubitt and Sugden’s (2011) reasoning-based iterative procedure, Halpern and Pass’s (2012) iterated regret-minimization procedure, and Hillas and Samet’s (2018) iterative elimination of flaws of weakly dominated strategies. See also Moulin (1979, 1984), Cho (1994), Borgers (1992), Watson (1998), Tyson (2010), and Barthel and Hoffmann (2019) for fruitful applications in economics.

  2. 2.

    Duggan and Le Breton (2014) modeled a player’s decision as a choice set and analyzed set-valued solution concepts in finite games.

  3. 3.

    See also Arieli (2012), Halpern and Pass (2012), Jara-Moroni (2012), Weinstein and Yildiz (2017), and Yu (2014) for related discussions on (infinitely) iterative elimination procedures.

  4. 4.

    For instance, consider a simple one-person game in which the strategy space is \(X=\left( 0,1\right) \) and the payoff function is \(u(x)=x\) for every strategy \(x\in X\). Obviously, every strategy is strictly dominated and the choice set \(c\left( X\right) =\varnothing \). The IESDS procedure is order dependent in this game, e.g., one can elaborately eliminate all strategies except a particular strategy \(x_{0}\in X\). Observe \(\left\{ x_{0}\right\} =c\left( \left\{ x_{0}\right\} \right) \nsubseteq c\left( X\right) \), violating Monotonicity*.

  5. 5.

    This example is taken from Dufwenberg and Stegeman (2002). Lipman (1994) first demonstrated that in infinite games, there is a nonequivalence between countably infinite iterated elimination of never-best replies and the strategic implication of “common knowledge of rationality.”Lipman (1994, Theorem 2) showed the equivalence can be restored by “removing never best replies as often as necessary”—i.e., by allowing for a transfinitely iterated elimination of never-best replies. The requirement of transfinite eliminations is related to the epistemic assumption of “common knowledge of rationality.”In the case of infinite states of nature, transfinite hierarchies of beliefs/knowledge are generally needed to provide a complete description of the uncertainty facing each agent; see, e.g., Lipman (1991), Fagin et al. (1992), and Heifetz and Samet (1998). An “iterative” formalism of “common knowledge” is more restrictive than the alternative “fixed-point” definition of “common knowledge.” The “fixed-point” definition of “common knowledge” can be equivalent to the “iterative” notion of “common knowledge” possibly by using transfinite levels of mutual knowledge; see Heifetz (1996, 1999) and Fagin et al. (1999) for more discussions.

  6. 6.

    Throughout this paper, we assume sets satisfy the ZF axioms (cf., e.g., Jech 2003, p.3).

  7. 7.

    Because the set S may be infinite, it is natural and necessary for us to consider a transfinite sequence of reduction on \(\left( S,\rightarrow \right) \). Lipman (1994) demonstrated that in infinite games, we need the transfinite induction to deal with the strategic implication of “common knowledge of rationality” [see also Chen et al. (2007, Example 1) and Green (2011)].

  8. 8.

    Apt (2011) considered the class of finite sequences of reduction under a variety of dominance relations in finite games. He showed this result by using Newman’s (1942) Lemma.

  9. 9.

    For \(X,Y\subseteq S\), let \(DOM^{Y}\left( X\right) \equiv \left\{ x\in X:\text { }y\succ _{X}x\text { for some }y\in Y\right\} \).

  10. 10.

    This example is taken from Dufwenberg and Stegeman (2002). They showed that this game is a CD game, but the IESDS procedure fails to be order independent in this CD game. On the contrary, this example is not a CD* game under the strict dominance.

  11. 11.

    The TDI condition requires that whenever the decision maker is indifferent between two profiles that differ only in her action, the indifference is transferred to the other players as well.

  12. 12.

    We denote by \(\varDelta \left( X_{i}\right) \) the probability space on \(X_{i}\) and by \(u_{i}\left( \sigma _{i},x_{-i}\right) \) the expected payoff of player i under a mixed strategy \(\sigma _{i}\in \)\(\varDelta \left( X_{-i}\right) \).

  13. 13.

    We denote by \(\varDelta \left( X_{-i}\right) \) the probability space on \(X_{-i}\) and by \(u_{i}\left( x_{i},\mu _{i}\right) \) the expected payoff of player i under a probabilistic belief \(\mu _{i}\in \)\(\varDelta \left( X_{-i}\right) \).

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Correspondence to Xiao Luo.

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We are grateful to the Editor, an Associate Editor, and an anonymous referee for very helpful comments and suggestions. We thank Geir Asheim, Yi-Chun Chen, Amanda Friedenberg, Yossi Greenberg, Chiu Yu Ko, Shravan Luckraz, Andrés Perea, Yongchuan Qiao, Yang Sun, Yeneng Sun, Satoru Takahashi, Chih-Chun Yang, and seminar participants at National University of Singapore and BI Norwegian Business School for helpful comments and discussions. This paper was presented at the 3rd Microeconomics Workshop at Nanjing Audit University, China, 2016, and the 17th SAET Conference, Faro, Portugal, 2017. Financial supports from National University of Singapore and University of Nottingham Ningbo China are gratefully acknowledged. The usual disclaimer applies.

Appendices

Appendix A: Examples

In this appendix, we demonstrate how to apply our analytical framework to a number of iterated elimination procedures discussed in the literature, including iterated strict dominance, iterated weak dominance, rationalizability, and so on. For simplicity, we restrict attention to a finite game: \(G\equiv (N,\{S_{i}\}_{i\in N},\{u_{i}\}_{i\in N})\). For any subset X of strategy profiles, we can define the choice set \(c\left( X\right) \) in the following ways.

  1. 1.

    [strict dominance]\(c\left( X\right) =X\backslash DOM\left( X\right) \), whereFootnote 12

    $$\begin{aligned} DOM\left( X\right) =\left\{ x\in X:\text { }\exists i\in N\text { } \exists \sigma _{i}\in \varDelta \left( X_{i}\right) \text { s.t. }u_{i}\left( \sigma _{i},x_{-i}^{\prime }\right) >u_{i}\left( x_{i},x_{-i}^{\prime }\right) \text { }\forall x_{-i}^{\prime }\in X_{-i}\right\} \text {.} \end{aligned}$$

    That is, \(c\left( X\right) \) consists of all strategy profiles in X where each player i’s strategy is strictly dominated by no mixed strategy in \(\varDelta \left( X_{i}\right) \). Because every strictly dominated strategy \(x_{i}\) in a finite game has an undominated dominator, remaining in a reduced game after eliminating some of the strictly dominated strategies, which strictly dominates \(x_{i}\) in that reduced game, \(\left( Y\cap DOM\left( X\right) \right) \subseteq DOM\left( Y\right) \) for \(c\left( X\right) \subseteq Y\subseteq X\). Thus, Hereditarity holds. By Corollary 1, IESDS is an order-independent procedure. (Under the strict dominance relation, 1-CD* holds true, but Monotonicity fails to be satisfied; for example, \(c\left( x\right) =x\notin c\left( X\right) \) for \(x\in X\backslash c\left( X\right) \).)

  2. 2.

    [weak dominance] \(c\left( X\right) =X\backslash DOM\left( X\right) \), where

    $$\begin{aligned} DOM\left( X\right) =\left\{ x\in X: \begin{array} [c]{c} \exists i\in N\text { }\exists \sigma _{i}\in \varDelta \left( X_{i}\right) \text { s.t. }u_{i}\left( \sigma _{i},x_{-i}^{\prime }\right) \ge u_{i}\left( x_{i},x_{-i}^{\prime }\right) \text { }\forall x_{-i}^{\prime }\in X_{-i}\\ \text {and }u_{i}\left( \sigma _{i},x_{-i}^{\prime }\right) >u_{i}\left( x_{i},x_{-i}^{\prime }\right) \ \text {for some }x_{-i}^{\prime }\in X_{-i} \end{array} \right\} \text {.} \end{aligned}$$

    That is, \(c\left( X\right) \) consists of all strategy profiles in X where each player i’s strategy is weakly dominated by no mixed strategy in \(\varDelta \left( X_{i}\right) \). The IEWDS procedure may not be order independent in general.

  3. 3.

    [strict dominance*]\(c\left( X\right) =X\backslash DOM\left( X\right) \), where

    $$\begin{aligned} DOM\left( X\right) =\left\{ x\in X:\text { }\exists i\in N\text { }\exists s_{i}^{*}\in S_{i}\text { s.t. }u_{i}\left( s_{i}^{*},x_{-i}^{\prime }\right) >u_{i}\left( x_{i},x_{-i}^{\prime }\right) \text { }\forall x_{-i}^{\prime }\in X_{-i}\right\} \text {.} \end{aligned}$$

    That is, \(c\left( X\right) \) consists of all strategy profiles in X where each player i’s strategy is strictly dominated by no strategy in \(S_{i}\) (see, e.g., Milgrom and Roberts 1990; Ritzberger 2002; Chen et al. 2007). Since every strictly dominated strategy in a finite game has an undominated dominator, which strictly dominates that dominated strategy in each of subgames, \(\left( Y\cap DOM\left( X\right) \right) \subseteq DOM\left( Y\right) \) for \(Y\subseteq X\). Thus, \(c\left( Y\right) \subseteq c\left( X\right) \) if \(Y\subseteq X\). That is, Monotonicity holds. By Theorem 1(b), The IESDS* procedure is order independent and preserves Nash equilibria.

  4. 4.

    [pure-strategy dominance]\(c\left( X\right) =X\backslash DOM\left( X\right) \), where

    $$\begin{aligned} DOM\left( X\right) =\left\{ x\in X: \begin{array} [c]{c} \exists i\in N\text { }\forall Z_{-i}\subseteq X_{-i}\text { }\exists s_{i}^{*}\in S_{i}\text { s.t. }u_{i}\left( s_{i}^{*},z_{-i}\right) \ge u_{i}\left( x_{i},z_{-i}\right) \\ \forall z_{-i}\in Z_{-i}\text { and }u_{i}\left( s_{i}^{*},z_{-i}\right) >u_{i}\left( x_{i},z_{-i}\right) \ \text {for some }z_{-i}\in Z_{-i} \end{array} \right\} . \end{aligned}$$

    That is, \(c\left( X\right) \) consists of all strategy profiles in X where each player i’s strategy is undominated in the sense of Borgers (1993). Under the pure-strategy dominance relation, because every dominated strategy in a finite game is clearly dominated in each subgame, \(\left( Y\cap DOM\left( X\right) \right) \subseteq DOM\left( Y\right) \) for \(Y\subseteq X\). Thus, Hereditarity holds. By Corollary 1, the iterated elimination of pure-strategy dominated strategy is an order-independent elimination procedure.

  5. 5.

    [rationalizability]\(c\left( X\right) =X\cap BR\left( X\right) \), whereFootnote 13

    $$\begin{aligned} BR\left( X\right) =\left\{ s\in S:\text { }\forall i\in N\text { }\exists \mu _{i}\in \varDelta \left( X_{-i}\right) \text { s.t. }u_{i}\left( s_{i},\mu _{i}\right) \ge u_{i}\left( s_{i}^{\prime },\mu _{i}\right) \text { }\forall s_{i}^{\prime }\in S_{i}\right\} \text {.} \end{aligned}$$

    That is, \(c\left( X\right) \) consists of all elements in X where each player i’s strategy is a best response to some probabilistic belief in \(\varDelta \left( X_{-i}\right) \). Since \(BR\left( Y\right) \subseteq BR\left( X\right) \) for \(Y\subseteq X\), \(c\left( Y\right) \subseteq c\left( X\right) \) if \(Y\subseteq X\). That is, Monotonicity holds. By Theorem 1(b), rationalizability is an order-independent elimination of never-best-response strategies, which preserves Nash equilibria.

  6. 6.

    [c-rationalizability] Ambrus (2006) proposed a solution concept of “coalitional rationalizability (c-rationalizability)” in finite games by an iterative procedure of restrictions of strategies. The procedure is analogous to iterative elimination of never-best-response strategies, but operates on implicit agreements by coalitions. More specifically, let X and Z be product-form subsets of strategy profiles. Z is a supported restriction by coalition \(J\subseteq N\) given X if (i) \(Z_{j}\subseteq X_{j}\) for \(j\in J\) and \(Z_{i}=X_{i}\) for \(i\notin J\) and (ii) for \(j\in J\), \(x_{j}\in X_{j}\backslash Z_{j}\) implies

    $$\begin{aligned} \max _{f_{-j}\in \varDelta \left( X_{-j}\right) }u_{j}\left( x_{j},f_{-j}\right) <\max _{s_{j}\in S_{j}}u_{j}\left( s_{j},g_{-j}\right) \text { }\forall g_{-j}\in \varDelta \left( Z_{-j}\right) \text { with }g_{-j}^{-J}=f_{-j} ^{-J}\text {,} \end{aligned}$$

    where \(f_{-j}^{-J}\) and \(g_{-j}^{-J}\) are the marginal distributions of \(f_{-j}\) and \(g_{-j}\) over \(S_{-J}\), respectively. Let \({\mathcal {F}}\left( X\right) \) be the set of all the supported restrictions given X. We define the choice rule c for c-rationalizability by

    $$\begin{aligned} c\left( X\right) =\cap _{Z\in {\mathcal {F}}\left( X\right) }Z\text {.} \end{aligned}$$

    Ambrus (2006) defined c-rationalizability by a (fast) iterated elimination procedure associated with this choice rule c; that is, in each elimination round, the intersection of all supported restrictions is retained (see also Ambrus 2009; Luo and Yang 2009 for more discussions). Ambrus (2006, Proposition 5) showed an order-independence result, under the restriction that each elimination round must be an intersection of some supported restrictions. Because the choice rule c satisfies 1-Monotonicity* (see Lemma 4 in “Appendix A”), by Theorem 2(b), Ambrus’s (2006) notion of c-rationalizability is an order-independent procedure, without the aforementioned restriction.

  7. 7.

    [HS-weak dominance] \(c\left( X\right) =X\backslash DOM\left( X\right) \), where

    $$\begin{aligned} DOM\left( X\right) =\left\{ x\in X: \begin{array} [c]{c} \exists i\in N\text { }\exists \sigma _{i}\in \varDelta \left( S_{i}\right) \text { s.t. }u_{i}\left( \sigma _{i},x_{-i}\right) >u_{i}\left( x\right) \\ \text {and }u_{i}\left( \sigma _{i},x_{-i}^{\prime }\right) \ge u_{i}\left( x_{i},x_{-i}^{\prime }\right) \text { }\forall x_{-i}^{\prime }\in X_{-i} \end{array} \right\} \text {.} \end{aligned}$$

    That is, \(c\left( X\right) \) consists of all strategy profiles in X where each player i’s strategy is not weakly undominated in the sense of Hillas and Samet (2018, Definition 3). Under the HS-weak dominance relation, because every dominated strategy in a finite game has an undominated dominator, which dominates that dominated strategy in each of subgames, \(\left( Y\cap DOM\left( X\right) \right) \subseteq DOM\left( Y\right) \) for \(c\left( X\right) \subseteq Y\subseteq X\). That is, Hereditarity holds. By Corollary 1, the HS-weak dominance is an order-independent procedure; see Hillas and Samet’s (2018) Proposition 1.

  8. 8.

    [RBEU] Cubitt and Sugden (2011) offered an iterative procedure of “reasoning-based expected utility procedure (RBEU)” for solving finite games. RBEU uses a sequence of “accumulation” and “elimination” operations to categorize strategies as permissible and impermissible; some strategies remain uncategorized when the procedure halts. Cubitt and Sugden (2011) demonstrated RBEU can delete more strategies than IESDS, while avoiding the order-dependence problem associated with IEWDS. Formally, the choice rule \(c\left( \cdot \right) \equiv \times _{i\in N}c_{i}\left( \cdot \right) \) is defined for each product subset \(X\times Y\subseteq S\times S\) such that

    $$\begin{aligned} c_{i}\left( X\times Y\right) =\left( S_{i}\backslash S_{i}^{+}\left( X\times Y\right) \right) \times \left( S_{i}\backslash S_{i}^{-}\left( X\times Y\right) \right) \text { for all }i\in N\text {,} \end{aligned}$$

    where

    $$\begin{aligned} S_{i}^{+}\left( X\times Y\right)&=\left\{ s_{i}\in S_{i}:\text { }\forall \mu \in \varDelta _{i}^{*}\left( X\times Y\right) \text {, }u_{i}\left( s_{i},\mu \right) \ge u_{i}\left( s_{i}^{\prime },\mu \right) \text { for all }s_{i}^{\prime }\in S_{i}\right\} \text {,}\\ S_{i}^{-}\left( X\times Y\right)&=\left\{ s_{i}\in S_{i}:\text { }\forall \mu \in \varDelta _{i}^{*}\left( X\times Y\right) \text {, }u_{i}\left( s_{i}^{\prime },\mu \right)>u_{i}\left( s_{i},\mu \right) \text { for some }s_{i}^{\prime }\in S_{i}\right\} \text {, and}\\ \varDelta _{i}^{*}\left( X\times Y\right)&=\left\{ \mu \in \varDelta \left( S_{-i}\right) :\mu \left( \times _{j\ne i}\left( S_{j}\backslash Y_{j}\right) \right) =0\text { and }\mu \left( s_{-i}\right) >0\text { }\forall s_{-i}\in \times _{j\ne i}\left( S_{j}\backslash X_{j}\right) \right\} \text {.} \end{aligned}$$

    The choice rule c can be viewed as the aggregate categorization function in Cubitt and Sugden (2011), with the “permissible” set \(S_{i}^{+}\left( X\times Y\right) \) and “impermissible” set \(S_{i}^{-}\left( X\times Y\right) \). Cubitt and Sugden’s (2011) Lemma implies Monotonicity holds for c. By Theorem 1(b), RBEU is an order-independent elimination procedure.

Appendix B1: Proofs

Proof of Theorem 1

(a) By transfinite recursion (see, e.g., Jech 2003, p.22), we define a sequence \(\left\{ X^{\lambda }\right\} _{\lambda \in Ord}\) (where Ord is the class of all ordinals) by

$$\begin{aligned} X^{0}=S\text {, }X^{\lambda +1}=c\left( X^{\lambda }\right) \text {, and }X^{\lambda }=\cap _{\lambda ^{\prime }<\lambda }X^{\lambda ^{\prime }}\text { for a limit ordinal }\lambda \text {.} \end{aligned}$$
(1)

By the Axiom Schema of Separation (see, e.g., Jech 2003, p.7), \(\left\{ X^{\lambda }:\lambda \in Ord\right\} \) is a set because it is a subclass of the power set of S. Suppose, to the contrary, \(X^{\lambda }\ne X^{\lambda ^{\prime }}\) for any \(\lambda \ne \lambda ^{\prime }\); then, there is a bijection from \(\left\{ X^{\lambda }:\lambda \in Ord\right\} \) to Ord. By the Axiom Schema of Replacement (see, e.g., Jech 2003, p.13), Ord is a set, contradicting the fact that Ord is not a set. By \(\left( 1\right) \), it follows that \(X^{\varLambda }=X^{\varLambda +1}=c\left( X^{\varLambda }\right) \) for some \(\varLambda \in Ord\). Let \(\varLambda ^{0}=\inf \left\{ \varLambda \in Ord:X^{\varLambda }=X^{\varLambda +1}=c\left( X^{\varLambda }\right) \right\} \). Then, the sequence \(\left\{ X^{\lambda }\right\} _{\lambda \le \varLambda ^{0}}\) is a fast IEP on \(\left( S,\rightarrow \right) \).

(b) Let \(Z=c\left( Z\right) \). Obviously, \(Z\subseteq X^{0}\). Assume, by induction, \(Z\subseteq X^{\lambda ^{\prime }}\) for all \(\lambda ^{\prime } <\lambda \). By Monotonicity, \(c\left( Z\right) \subseteq c\left( X^{\lambda ^{\prime }}\right) \) for all \(\lambda ^{\prime }<\lambda \). Therefore, \(Z=c\left( Z\right) \subseteq X^{\lambda }\). That is, \(Z\subseteq X^{\lambda }\) for all \(\lambda \le \varLambda \). Therefore, \(X^{\varLambda }\supseteq \cup _{Z=c\left( Z\right) }Z\). Since \(X^{\varLambda }=c\left( X^{\varLambda }\right) \), \(X^{\varLambda }=\cup _{Z=c\left( Z\right) }Z\). \(\square \)

To prove Theorem 2, we need the following three lemmas.

Lemma 1

If \(S\rightarrow ^{*}X\) and \(S\rightarrow ^{*}Y\) imply there exists T such that \(X\rightarrow ^{*}T\) and \(Y\rightarrow ^{*}T\), the iterated elimination procedure is order independent.

Proof

Assume, by absurdity, two IEPs \(S\rightarrow ^{*}X=c\left( X\right) \) and \(S\rightarrow ^{*}Y=c\left( Y\right) \), but \(X\ne Y\). Then, there exists T such that \(X\rightarrow ^{*}T\) and \(Y\rightarrow ^{*}T\). Therefore, \(X=T=Y\), which is a contradiction. \(\square \)

Lemma 2

If c satisfies Monotonicity*, \(S\rightarrow ^{*}X\rightarrow Y\) implies \(Y\rightarrow c\left( X\right) \).

Proof

Let \(S\rightarrow ^{*}X\rightarrow Y\). Since c satisfies Monotonicity*, \(c\left( Y\right) \subseteq c\left( X\right) \). Since \(X\rightarrow Y\), \(c\left( Y\right) \subseteq c\left( X\right) \subseteq Y\subseteq X\). By the definition of \(\rightarrow \), \(Y\rightarrow c\left( X\right) \). \(\square \)

Lemma 3

Suppose \(S\rightarrow ^{*}X\) via an elimination sequence \(\left\{ X^{\lambda }\right\} _{\lambda \le \varLambda } \). Then, \(c\left( X\right) \subseteq \cap _{\lambda <\varLambda }c\left( X^{\lambda }\right) \) if c satisfies Monotonicity*.

Proof

Since c satisfies Monotonicity*, \(c\left( X\right) \subseteq c\left( X^{\lambda }\right) \) for all \(\lambda <\varLambda \). Therefore, \(c\left( X\right) \subseteq \cap _{\lambda <\varLambda }c\left( X^{\lambda }\right) \). \(\square \)

Proof of Theorem 2

(a) Let \(S\rightarrow ^{*}X\) via an elimination sequence \(\left\{ X^{\lambda }\right\} _{\lambda \le \varLambda }\) and \(S\rightarrow ^{*}Y\) via an elimination sequence \(\left\{ Y^{\lambda }\right\} _{\lambda \le \varLambda }\). We say the “diamond property holds (for \(\left\{ X^{\lambda }\right\} _{\lambda \le \varLambda }\) and \(\left\{ Y^{\lambda }\right\} _{\lambda \le \varLambda }\))” if there exists an \(\varLambda \times \varLambda \)-diamond grid \(\left\{ S^{\alpha \beta }\right\} _{\alpha \le \varLambda ;\text { }\beta \le \varLambda }\) such that

  1. 1.

    for all \(\lambda \le \varLambda \), \(S^{\lambda 0}=X^{\lambda }\) and \(S^{0\lambda }=Y^{\lambda }\);

  2. 2.

    for all \(\alpha ,\beta \le \varLambda \), \(\left\{ S^{\alpha \lambda }\right\} _{\lambda \le \varLambda }\) and \(\left\{ S^{\lambda \beta }\right\} _{\lambda \le \varLambda }\) are elimination sequences.

That is, the diamond structure spreads over a grid of \(\varLambda \times \varLambda \) fractals (cf. Fig. 5).

Fig. 5
figure5

A grid of \(\varLambda \times \varLambda \) fractals

Observe that \(S\rightarrow ^{*}X\) and \(S\rightarrow ^{*}Y\) iff there exists an ordinal \(\varLambda \) such that \(S\rightarrow ^{*}X\) via an elimination sequence \(\left\{ X^{\lambda }\right\} _{\lambda \le \varLambda }\) and \(S\rightarrow ^{*}Y\) via an elimination sequence \(\left\{ Y^{\lambda }\right\} _{\lambda \le \varLambda }\). By Lemma 1, it suffices to show the diamond property holds true. We show it by (transfinite) induction on \(\varLambda \). If \(\varLambda =1\), then \(S\rightarrow X\) and \(S\rightarrow Y\). By Lemma 2, \(X\rightarrow c\left( S\right) \) and \(Y\rightarrow c\left( S\right) \). Now assume the diamond property holds for all \(\lambda <\varLambda \). We distinguish two cases.

Case 1: \(\varLambda \) is a limit ordinal. Define \(S^{\varLambda 0}\equiv X^{\varLambda }\) and \(S^{\varLambda \beta }\equiv \cap _{\alpha <\varLambda }S^{\alpha \beta }\) for all \(\beta <\varLambda \) and \(\beta \ne 0\). Since \(X^{\varLambda }=\cap _{\lambda <\varLambda }X^{\lambda }\), \(S^{\varLambda 0}=\cap _{\alpha <\varLambda }S^{\alpha 0}\). By the induction hypothesis, for all \(\beta <\varLambda \), we have

$$\begin{aligned} \left[ S^{\alpha \beta }\rightarrow S^{\alpha \left( \beta +1\right) }\text { }\forall \alpha<\varLambda \right]&\Leftrightarrow \left[ c\left( S^{\alpha \beta }\right) \subseteq S^{\alpha \left( \beta +1\right) }\subseteq S^{\alpha \beta }\text { }\ \forall \alpha<\varLambda \right] \\&\Rightarrow \left[ \cap _{\alpha<\varLambda }c\left( S^{\alpha \beta }\right) \subseteq \cap _{\alpha<\varLambda }S^{\alpha \left( \beta +1\right) }\subseteq \cap _{\alpha<\varLambda }S^{\alpha \beta }\right] \\&\Leftrightarrow \left[ \cap _{\alpha <\varLambda }c\left( S^{\alpha \beta }\right) \subseteq S^{\varLambda \left( \beta +1\right) }\subseteq S^{\varLambda \beta }\right] \text {.} \end{aligned}$$

By Lemma 3,\(\ c\left( S^{\varLambda \beta }\right) \subseteq \cap _{\alpha <\varLambda }c\left( S^{\alpha \beta }\right) \subseteq S^{\varLambda \left( \beta +1\right) }\subseteq S^{\varLambda \beta }\). Therefore, \(S^{\varLambda \beta }\rightarrow S^{\varLambda \left( \beta +1\right) }\) for all \(\beta <\varLambda \). (If \(\beta \) is a limit ordinal, \(S^{\varLambda \beta }=\cap _{\alpha<\varLambda }S^{\alpha \beta } =\cap _{\alpha<\varLambda }\cap _{\beta ^{\prime }<\beta }S^{\alpha \beta ^{\prime }} =\cap _{\beta ^{\prime }<\beta }\cap _{\alpha<\varLambda }S^{\alpha \beta ^{\prime }} =\cap _{\beta ^{\prime }<\beta }S^{\varLambda \beta ^{\prime }}\).) Define \(S^{\varLambda \varLambda }\equiv \cap _{\beta<\varLambda }S^{\varLambda \beta }=\cap _{\beta<\varLambda } \cap _{\alpha <\varLambda }S^{\alpha \beta }\). We find an elimination sequence \(\left\{ S^{\varLambda \beta }\right\} _{\beta \le \varLambda }\). Similarly, we find an elimination sequence \(\left\{ S^{\alpha \varLambda }\right\} _{\alpha \le \varLambda }\) with \(S^{\varLambda \varLambda }=\cap _{\alpha<\varLambda }\cap _{\beta<\varLambda }S^{\alpha \beta }=\cap _{\alpha <\varLambda }S^{\alpha \varLambda }\).

Case 2: \(\varLambda \) is a successor ordinal. By the induction hypothesis, there exists \(\left( \varLambda -1\right) \times \left( \varLambda -1\right) \)-diamond grid \(\left\{ S^{\alpha \beta }\right\} _{\alpha \le \varLambda -1;\ \beta \le \varLambda -1}\) for \(\left\{ X^{\lambda }\right\} _{\lambda \le \varLambda -1}\) and \(\left\{ Y^{\lambda }\right\} _{\lambda \le \varLambda -1}\). Define \(S^{\varLambda 0}\equiv X^{\varLambda }\) and \(S^{\varLambda \left( \beta +1\right) }\equiv c\left( S^{\left( \varLambda -1\right) \beta }\right) \) (and \(S^{\varLambda \beta }\equiv \cap _{\beta ^{\prime }<\beta }S^{\varLambda \beta ^{\prime }}\) if \(\beta \) is a limit ordinal) for all \(\beta \le \varLambda -1\). Since \(X^{\varLambda -1}\rightarrow X^{\varLambda }\), by the induction hypothesis, \(S\rightarrow ^{*}S^{\left( \varLambda -1\right) 0}\rightarrow S^{\varLambda 0}\) and \(S\rightarrow ^{*}S^{\left( \varLambda -1\right) 0}\rightarrow S^{\left( \varLambda -1\right) 1}\). By Lemma 2, \(S^{\varLambda 0}\rightarrow \)\(S^{\varLambda 1}\) and \(S^{\left( \varLambda -1\right) 1}\rightarrow S^{\varLambda 1}\). Again by induction on \(\beta \le \varLambda -1\), we have \(S^{\varLambda \beta }\rightarrow S^{\varLambda \left( \beta +1\right) }\) for all \(\beta \le \varLambda -1\) and \(S^{\left( \varLambda -1\right) \beta }\rightarrow S^{\varLambda \beta }\) for any \(\beta \le \varLambda -1\) (if \(\beta \) is a limit ordinal, the proof is similar to Case 1). Therefore, \(\left\{ S^{\varLambda \beta }\right\} _{\beta \le \varLambda }\) and \(\left\{ S^{\alpha \beta }\right\} _{\alpha \le \varLambda }\) for any \(\beta \le \varLambda -1\) are elimination sequences. Similarly, we can find an elimination sequence \(\left\{ S^{\alpha \varLambda }\right\} _{\alpha \le \varLambda }\) such that \(\left\{ S^{\alpha \beta }\right\} _{\beta \le \varLambda }\) for any \(\alpha \le \varLambda -1\) is an elimination sequence. That is, there exists an \(\varLambda \times \varLambda \)-diamond grid \(\left\{ S^{\alpha \beta }\right\} _{\alpha \le \varLambda ;\ \beta \le \varLambda }\) for \(\left\{ X^{\lambda }\right\} _{\lambda \le \varLambda }\) and \(\left\{ Y^{\lambda }\right\} _{\lambda \le \varLambda }\). Therefore, the diamond property holds.

(b) Let \(S\rightarrow ^{\kappa }X\) via a finite elimination sequence \(\left\{ X^{k}\right\} _{k\le K}\) and \(S\rightarrow ^{\kappa }Y\) via an elimination sequence \(\left\{ Y^{k}\right\} _{k\le K}\). By 1-Montonicity*, we can similarly show the diamond property for \(\left\{ X^{k}\right\} _{k\le K} \) and \(\left\{ X^{k}\right\} _{k\le K}\). \(\square \)

Proof of Theorem 3

Suppose \(S\rightarrow ^{*} X\rightarrow ^{*}Y\). Then, \(Y\subseteq X\). Thus, we have

$$\begin{aligned} \left[ \left( Y\cap DOM\left( X\right) \right) \subseteq DOM\left( Y\right) \right]&\Leftrightarrow \left[ Y\backslash \left( Y\cap DOM\left( X\right) \right) \supseteq Y\backslash DOM\left( Y\right) \right] \\&\Leftrightarrow \left[ Y\backslash DOM\left( X\right) \supseteq Y\backslash DOM\left( Y\right) \right] \\&\Leftrightarrow \left[ X\backslash DOM\left( X\right) \supseteq Y\backslash DOM\left( Y\right) \right] \text {.} \end{aligned}$$

That is, \(\left( Y\cap DOM\left( X\right) \right) \subseteq DOM\left( Y\right) \) iff \(c\left( X\right) \supseteq c\left( Y\right) \). Therefore, Hereditarity* and Monotonicity* are equivalent. If \(S\rightarrow ^{\kappa }X\rightarrow Y\), we similarly obtain that 1-Hereditarity* and 1-Monotonicity* are equivalent.

Proof of Corollary 1

Suppose \(X\rightarrow Y\). That is, \(c(X)\subseteq Y\subseteq X\). By Hereditarity, we have

$$\begin{aligned} \left[ \left( Y\cap DOM\left( X\right) \right) \subseteq DOM\left( Y\right) \right]&\Leftrightarrow \left[ Y\backslash \left( Y\cap DOM\left( X\right) \right) \supseteq Y\backslash DOM\left( Y\right) \right] \\&\Leftrightarrow \left[ Y\backslash DOM\left( X\right) \supseteq Y\backslash DOM\left( Y\right) \right] \\&\Rightarrow \left[ X\backslash DOM\left( X\right) \supseteq Y\backslash DOM\left( Y\right) \right] \text {.} \end{aligned}$$

That is, \(c\left( Y\right) \subseteq c\left( X\right) \) if \(X\rightarrow Y\). By Theorem 2(b), the finitely iterated elimination procedure for G is order independent. \(\square \)

Proof of Theorem 4.

(a) Suppose \(S\rightarrow ^{*}X\rightarrow ^{*}Y\). Since G is a CD* game, \(\left( Y\cap DOM(X)\right) \subseteq DOM^{c\left( Y\right) }(Y)\subseteq \)DOM(Y). That is, Hereditarity* holds. By Theorem 2(a) and Theorem 3(a), the procedure is order independent in \( {\mathcal {L}} ^{*}\left( S,c\right) \).

Now consider a COUSC game G under the strict dominance relation. Suppose \(S\rightarrow ^{*}X\rightarrow ^{*}Y\) via an elimination sequence \(\{X^{\lambda }\}_{\lambda \le \varLambda }\). Let \(y\succ _{X}x\) for some \(y\in X\) and \(x\in Y\). Then, \(\exists i\in N\) such that \(u_{i}\left( y_{i} ,x_{-i}\right) >u_{i}\left( x_{i},x_{-i}\right) \) for all \(x_{-i}\in X_{-i}\). Since G is a COUSC game, by the proof of Dufwenberg and Stegeman’s (2002) Lemma, \(\exists z^{*}\in S\) such that for all \(y^{\prime }\in Y\), (i) \(u_{i}\left( z_{i}^{*},y_{-i}^{\prime }\right) >u_{i}\left( x_{i} ,y_{-i}^{\prime }\right) \) and (ii) \(u_{j}(z_{j}^{*},x_{-j})\ge u_{j}\left( s_{j},x_{-j}\right) \) for all \(j\in N\) and all \(s_{j}\in S_{j}\). Since \(x\in Y\subseteq X^{\lambda }\), \(\ z^{*}\in X^{\lambda }\) for all \(\lambda <\varLambda \). Thus, \(z^{*}\in \cap _{\lambda <\varLambda }X^{\lambda }=Y\). By (i) and (ii), \(z^{*}\succ _{Y}x\) and \(z^{*}\in c\left( Y\right) \). Therefore, \(\left( Y\cap DOM(X)\right) \subseteq DOM^{c\left( Y\right) }(Y)\); that is, G is a CD* game, and hence, the IESDS procedure is order independent in \( {\mathcal {L}} ^{*}\left( S,c\right) \). By Theorem 1(a), the IESDS procedure defined in Definition 1 exists in \( {\mathcal {L}} ^{*}\left( S,c\right) \).

(b) \(S\rightarrow ^{\kappa }X\rightarrow Y\). Since G is a 1-CD* game, \(\left( Y\cap DOM(X)\right) \subseteq DOM^{c\left( Y\right) }(Y)\subseteq \)DOM(Y). That is, 1-Hereditarity* holds. By Theorem 2(b) and Theorem 3(b), the finitely iterated elimination procedure is order independent. \(\square \)

Proof of Corollary 2

(a) Suppose \(S\rightarrow ^{*}X\rightarrow ^{*}Y\). Then, \(c\left( X\right) \subseteq Y\subseteq X\). Since G is CD* and \(S\rightarrow ^{*}X\rightarrow X\), \(DOM\left( X\right) =DOM^{c\left( X\right) }\left( X\right) \). Thus, \(DOM\left( X\right) =DOM^{Y}\left( X\right) =DOM^{c\left( X\right) }\left( X\right) \). Therefore, we obtain

$$\begin{aligned} \left[ X\rightarrow Y\right]&\Leftrightarrow \left[ Y\subseteq X\text { and }X\backslash Y\subseteq DOM\left( X\right) \right] \\&\Leftrightarrow \left[ Y\subseteq X\text { and }X\backslash Y\subseteq DOM^{Y}\left( X\right) \right] \\&\Leftrightarrow \left[ X\rightarrow ^{\text {GKZ}}Y\right] \text {.} \end{aligned}$$

That is, for any CD* game, the GKZ procedure is equivalent to the iterated elimination procedure in Definition 1.

(b) Suppose \(S\rightarrow ^{\kappa }X\rightarrow Y\). We similarly obtain \(\left[ X\rightarrow Y\right] \Leftrightarrow \left[ X\rightarrow ^{\text {GKZ}}Y\right] \). That is, for any 1-CD* game, the finite GKZ procedure is equivalent to the finitely iterated elimination procedure in Definition 1. \(\square \)

Lemma 4

The choice rule c for c-rationalizability satisfies 1-Monotonicity*.

Proof

Let \(X\searrow _{J}Z\) denote “supported restriction Z by coalition J given X.” Consider \(X\rightarrow Y\). Then, \(X\supseteq Y\supseteq c\left( X\right) =\cap _{Z\in {\mathcal {F}}\left( X\right) }Z\ne \varnothing \) by Ambrus’s (2006) Proposition 1. Since \(Y\cap Z\supseteq Y\cap c\left( X\right) \ne \varnothing \) for \(Z\in {\mathcal {F}}\left( X\right) \), by Ambrus’s (2006) Lemmas 1 and 2, \(Y\searrow _{J}\left( Y\cap Z\right) \). Then, \(Y\cap Z\in {\mathcal {F}}\left( Y\right) \) for all Z in \({\mathcal {F}}\left( X\right) \). Thus, \(c\left( Y\right) =\cap _{Z\in {\mathcal {F}}\left( Y\right) }Z\subseteq \cap _{Z\in {\mathcal {F}}\left( X\right) }\left( Y\cap Z\right) \subseteq \cap _{Z\in {\mathcal {F}}\left( X\right) }Z=c\left( X\right) \). \(\square \)

Appendix B2: “Outcome” order independence

Let \(\sqsubseteq ^{*}\) denote the transitive closure of \(\sqsubseteq \); that is, \(Y\sqsubseteq ^{*}X\) iff \(Y=Y^{0}\sqsubseteq Y^{1}\sqsubseteq \cdots \sqsubseteq Y^{K}=X\) for an integer K. Define the “outcome” equivalence relation \(X\circeq Y\) as \(Y\sqsubseteq ^{*}X\) and \(X\sqsubseteq ^{*}Y\). The following observation asserts that \(X\circeq Y\) implies there is an outcome-invariant bijection between the “strategy” equivalence classes of X and of Y. Let \(Z_{i}^{\simeq _{Z}}\) denote the set of the equivalence classes of \(Z_{i}\) under the “strategy” equivalence relation \(\simeq _{Z}\); that is, \(Z_{i}^{\simeq _{Z}}=\left\{ z_{i}^{\simeq _{Z}}:z_{i}\in Z_{i}\right\} \), where \(z_{i}^{\simeq _{Z}}\equiv \left\{ z_{i}^{\prime }\in Z_{i}:z_{i}^{\prime }\simeq _{Z}z_{i}\right\} \).

ObservationSuppose \(X\circeq Y\). For each \(i\in N\), there is a bijection \(\phi _{i}\) from \(X_{i} ^{\simeq _{X}}\) to \(Y_{i}^{\simeq _{Y}}\ \)such that \(u\left( x\right) =u\left( y\right) \) whenever \(y_{i}^{\simeq _{Y}}=\phi _{i}\left( x_{i}^{\simeq _{X}}\right) \)\(\forall i\in N\). Subsequently, the “outcome” relation \(\circeq \) implies the usual outcome equivalence in Chen and Micali (2013, Definition 5).

In order to show Observation, we need Lemma 5.

Lemma 5

(i) Suppose \(x_{i}\simeq _{X}x_{i}^{\prime }\)\(\forall i\in N\). Then, \(u\left( x\right) =u\left( x^{\prime }\right) \). (ii) Suppose \(Y\sqsubseteq X\). For each \(i\in N\), there is an injection \(\phi _{i}\) from \(Y_{i}^{\simeq _{Y}}\ {to}\)\(X_{i}^{\simeq _{X}}\ \)such that \(u\left( x\right) =u\left( y\right) \) whenever \(x_{i}^{\simeq _{X}}=\phi _{i}\left( y_{i}^{\simeq _{Y} }\right) \)\(\forall i\in N\).

Proof

  1. (i)

    Since \(x_{1}\simeq _{X}x_{1}^{\prime }\), \(u\left( x\right) =u\left( x_{1}^{^{\prime }},\left( x_{j}\right) _{j\ne 1}\right) \). Since \(x_{2}\simeq _{X}x_{2}^{\prime }\), \(u\left( x\right) =u\left( x_{1}^{^{\prime }},x_{2}^{^{\prime }},\left( x_{j}\right) _{j\ne 1,2}\right) \). Continue to do this replacement, we obtain \(u\left( x\right) =u\left( x^{\prime }\right) \).

  2. (ii)

    For all \(i\in N\) and \(y_{i}^{\simeq _{Y}}\in Y_{i}^{\simeq _{Y}}\), consider the representative strategy \(y_{i}\in y_{i}^{\simeq _{Y}}\). Since \(Y\sqsubseteq X\), for the representative \(y_{i}\), there is \(x_{i}\in X_{i}\) such that \(y_{i}\simeq _{X\sqcup Y}x_{i}\). Define \(\phi _{i}\left( y_{i}^{\simeq _{Y} }\right) =x_{i}^{\simeq _{X}}\). For any \({\hat{y}}\in Y\) and \({\hat{x}}\in X\). If \({\hat{y}}_{i}\simeq _{Y}y_{i}\) and \({\hat{x}}_{i}\simeq _{X}x_{i}\)\(\forall i\in N\), by (i), \(u\left( {\hat{x}}\right) =u\left( x\right) =u\left( y\right) =u\left( {\hat{y}}\right) \). Suppose \(\phi _{i}\left( y_{i}^{\simeq _{Y} }\right) =\phi _{i}\left( {\hat{y}}_{i}^{\simeq _{Y}}\right) =x_{i}^{\simeq _{X}}\). Then, \(y_{i}\simeq _{X\sqcup Y}x_{i}\) and \({\hat{y}}_{i}\simeq _{X\sqcup Y}x_{i}\). Thus, \(y_{i}\simeq _{Y}y_{i}^{\prime }\); that is, \(y_{i}^{\simeq _{Y} }={\hat{y}}_{i}^{\simeq _{Y}}\). Therefore, \(\phi _{i}\) is an injection from \(Y_{i}^{\simeq _{Y}}\ \)to \(X_{i}^{\simeq _{X}}\). \(\square \)

Proof of Observation

Suppose \(Y\sqsubseteq ^{*}X\). Then, \(Y=Y^{0}\sqsubseteq Y^{1}\sqsubseteq \cdots \sqsubseteq Y^{K}=X\) for an integer K. Let \(i\in N\) and \(k=1,...,K\). By Lemma 5(ii), there is an injection \(\phi _{i}^{k}\) from \(\left( Y_{i}^{k-1}\right) ^{\simeq _{X^{k-1}}}\) to \(\left( Y_{i}^{k}\right) ^{\simeq _{X^{k}}}\) such that \(u\left( y^{k}\right) =u\left( y^{k-1}\right) \) whenever \(\left( y_{i} ^{k}\right) ^{\simeq _{Y^{k}}}=\phi _{i}\left( \left( y_{i}^{k-1}\right) ^{\simeq _{Y^{k-1}}}\right) \)\(\forall i\in N\). Define \(\phi _{i}=\phi _{i} ^{K}\circ \phi _{i}^{K-1}\circ \cdots \circ \phi _{i}^{1}\). Then, \(\phi _{i}\) is an injection from \(Y_{i}^{\simeq _{Y}}\ \)to \(X_{i}^{\simeq _{X}}\) such that \(u\left( x\right) =u\left( y\right) \) whenever \(x_{i}^{\simeq _{X}}=\phi _{i}\left( y_{i}^{\simeq _{Y}}\right) \)\(\forall i\in N\). Similarly, there is an injection from \(X_{i}^{\simeq _{X}}\) to \(Y_{i}^{\simeq _{Y}}\) because \(X\sqsubseteq ^{*}Y\). Since \(Y_{i}^{\simeq _{Y}}\ \)and \(X_{i}^{\simeq _{X}}\) are finite, \(\phi _{i}\) must be a bijection. \(\square \)

Proof of Theorem 5

We show a stronger result: The finitely iterated procedure (by using the relation \(\vdash \)) is “outcome” order independent. Because 1-Monotonicity* (w.r.t. \(\sqsubseteq \)) holds, by the proof of Theorem 2(a), for two finite “outcome” elimination sequences \(\{X^{k}\}_{k\le K}\) and \(\{Y^{k}\}_{k\le K}\) on system \(\left( S,\vdash \right) \), we have \(K\times K\)-grid \(\{S^{kl}\}_{k\le K;l\le K}\) such that \(X^{K}=S^{K0}=c\left( S^{K0}\right) \) and \(Y^{K}=S^{0K}=c\left( S^{0K}\right) \). Now, consider an auxiliary \(K\times K\)-grid \(\{S^{\left( K+k\right) l}\}_{k\le K;l\le K}\) for two sequences (starting from \(S^{K0}\)) \(\left\{ S^{\left( K+k\right) 0}\right\} _{k\le K}\) and \(\left\{ S^{Kl}\right\} _{l\le K}\), where \(S^{\left( K+k\right) 0}\equiv S^{K0}\) for \(k=1,...,K\). Since \(S^{K0}=c\left( S^{K0}\right) \) and \(S^{\left( K+k\right) k}=c\left( S^{\left( K+k-1\right) \left( k-1\right) }\right) \) for \(k=1,...,K\), \(S^{\left( K+K\right) K}=S^{K0}\). Therefore, \(S^{0K}\vdash ^{*}S^{KK}\vdash ^{*}S^{\left( K+K\right) K}=S^{K0}\) (cf. Fig. 6), and hence \(X^{K}\sqsubseteq ^{*}Y^{K}\). Similarly, \(Y^{K} \sqsubseteq ^{*}X^{K}\). Consequently, \(X^{K}\circeq Y^{K}\). \(\square \)

Fig. 6
figure6

An auxiliary \(K\times K\)-grid

Proof of Lemma NWD

Let \(x_{i},z_{i}\in X_{i}\), \(w\in X\sqcup Y\). Since \(Y\sqsubseteq X\), for each \(j\ne i\), there is \(x_{j}^{*}\in X_{j}\) such that \(w_{j}\simeq _{X\sqcup Y}x_{j}^{*}\). By Lemma 5 (i), \(u\left( x_{i},w_{-i}\right) =u\left( x_{i},x_{-i}^{*}\right) \) and \(u\left( z_{i},w_{-i}\right) =u\left( z_{i},x_{-i}^{*}\right) \), where \(x_{-i}^{*}=\left( x_{j}^{*}\right) _{j\ne i}\in X_{-i}\). Thus, for all \(w\in X\sqcup Y\), \(u_{i}\left( x_{i},w_{-i}\right)<u_{i}\left( z_{i},w_{-i}\right) \Leftrightarrow u_{i}\left( x_{i},x_{-i}^{*}\right) <u_{i}\left( z_{i},x_{-i}^{*}\right) \) and \(u\left( x_{i},w_{-i}\right) =u\left( z_{i},w_{-i}\right) \Leftrightarrow u\left( x_{i},x_{-i}^{*}\right) =u\left( z_{i},x_{-i}^{*}\right) \).

[“only if” part] Suppose \(x_{i}\prec _{X}z_{i}\). Then, (i) \(\forall x_{-i}\in X_{-i}\), either \(u_{i}\left( x_{i},x_{-i}\right) <u_{i}\left( z_{i},x_{-i}\right) \) or \(u\left( x_{i},x_{-i}\right) =u\left( z_{i},x_{-i}\right) \), \(\text {and }\)(ii) \(x_{i}\not \simeq _{X}z_{i}\). Therefore, (i) \(\forall w\in X\sqcup Y\), either \(u_{i}\left( x_{i},w_{-i}\right) =u_{i}\left( x_{i},x_{-i}^{*}\right) <u_{i}\left( z_{i},x_{-i}^{*}\right) =u_{i}\left( z_{i},w_{-i}\right) \) or \(u\left( x_{i},w_{-i}\right) =u\left( x_{i},x_{-i}^{*}\right) =u\left( z_{i},x_{-i}^{*}\right) =u\left( z_{i},w_{-i}\right) \), \(\text {and }\)(ii) \(x_{i}\not \simeq _{X\sqcup Y}z_{i}\). That is, \(x_{i} \prec _{X\sqcup Y}z_{i}\).

[“if” part] Suppose \(x_{i}\prec _{X\sqcup Y}z_{i}\). Then, \(\forall x\in X\), either \(u_{i}\left( x_{i} ,x_{-i}\right) <u_{i}\left( z_{i},x_{-i}\right) \) or \(u\left( x_{i} ,x_{-i}\right) =u\left( z_{i},x_{-i}\right) \text {. Since}\)\(x_{i} \not \simeq _{X\sqcup Y}z_{i}\), \(u_{i}\left( x_{i},w_{-i}\right) <u_{i}\left( z_{i},w_{-i}\right) \) for some \(w\in X\sqcup Y\). Therefore, \(u_{i}\left( x_{i},x_{-i}^{*}\right) =u_{i}\left( x_{i},w_{-i}\right) <u_{i}\left( z_{i},w_{-i}\right) =\)\(u_{i}\left( z_{i},x_{-i}^{*}\right) \) where \(x_{-i}^{*}\in X_{-i}\). Thus, \(x_{i}\not \simeq _{X} z_{i}\). That is, \(x_{i}\prec _{X}z_{i}\). \(\square \)

Proof of Theorem 6

Suppose \(X\vdash Y\). Then, \(c\left( X\right) \sqsubseteq Y\) and \(Y\sqsubseteq X\). Let \(i\in N\). By Theorem 5, it suffices to show that for each \(y_{i}\in c_{i}\left( Y\right) \), there is \(z_{i}\in c_{i}\left( X\right) \) such that \(z_{i}\simeq _{c\left( X\right) \sqcup c\left( Y\right) }y_{i}\). Since \(y_{i}\in c_{i}\left( Y\right) \subseteq Y_{i}\) and \(Y\sqsubseteq X\), \(y_{i} \simeq _{X\sqcup Y}x_{i}\) for some \(x_{i}\in X_{i}\). If \(x_{i}\in c_{i}\left( X\right) \), we are done by letting \(z_{i}=x_{i}\). If \(x_{i}\notin c_{i}\left( X\right) \), by finiteness and transitivity of NWD, \(x_{i} \prec _{X}z_{i}\) for some \(z_{i}\in c_{i}\left( X\right) \). By Lemma NWD, \(x_{i}\prec _{X\sqcup Y}z_{i}\). Therefore, \(y_{i}\prec _{X\sqcup Y}z_{i}\). Since \(c\left( X\right) \sqsubseteq Y\), \(z_{i}\simeq _{c\left( X\right) \sqcup Y}{\widetilde{z}}_{i}\) for some \({\widetilde{z}}_{i}\in Y_{i}\). Since \(y_{i}\in c_{i}\left( Y\right) \) and \(y_{i}\not \prec _{Y}{\widetilde{z}}_{i}\), again by Lemma NWD, \(y_{i}\not \prec _{c\left( X\right) \sqcup Y}{\widetilde{z}}_{i}\). Therefore, \(y_{i}\not \prec _{c\left( X\right) \sqcup Y}z_{i}\). Thus, \(z_{i}\simeq _{c\left( X\right) \sqcup Y}y_{i}\), and hence \(z_{i} \simeq _{c\left( X\right) \sqcup c\left( Y\right) }y_{i}\). \(\square \)

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Luo, X., Qian, X. & Qu, C. Iterated elimination procedures. Econ Theory 70, 437–465 (2020). https://doi.org/10.1007/s00199-019-01215-6

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Keywords

  • Iterated elimination procedures
  • Order independence
  • Monotonicity*
  • CD* games
  • “Outcome” order independence

JEL Classification

  • C70
  • D70