Notes on “refinements and higher order beliefs”

The abstract of our 1997 survey paper on “Refinements and Higher Order Beliefs” reads: This paper presents a simple framework that allows us to survey and relate some different strands of the game theory literature. We describe a “canonical” way of adding incomplete information to a complete information game. This framework allows us to give a simple “complete theory” interpretation (Kreps 1990) of standard normal form refinements such as perfection, and to relate refinements both to the “higher order beliefs literature” (Rubinstein 1989; Monderer and Samet

The abstract of our 1997 survey paper on "Refinements and Higher Order Beliefs" reads: This paper presents a simple framework that allows us to survey and relate some different strands of the game theory literature. We describe a "canonical" way of adding incomplete information to a complete information game. This framework allows us to give a simple "complete theory" interpretation (Kreps 1990) of standard normal form refinements such as perfection, and to relate refinements both to the "higher order beliefs literature" (Rubinstein 1989;Monderer and Samet 1989;Rob et al. 1995;Kajii and Morris 1997) and the "payoff uncertainty approach" (Fudenberg et al. 1988;Dekel and Fudenberg 1990).
In particular, this paper provided a unified framework to relate the notion of equilibria robust to incomplete information introduced in Kajii and Morris (1997) [Hereafter, KM1997] to the classic refinements literature. It followed Fudenberg et al. (1988) and Kreps (1990) in relating refinements of Nash equilibria to a "complete theory" where behavior was rationalized by explicit incomplete information about payoffs, rather than depending on action trembles or other exogenous perturbations. It followed Fudenberg and Tirole (1991), chapter 14, in providing a unified treatment of refinements and a literature on higher-order beliefs rather than proposing a particular solution concept.
The primary purpose of the survey paper was to promote the idea of robust equilibria in KM1997 and we did not try to publish it as an independent paper. Since we wrote this paper, there have been many developments in the literature on robust equilibria, fortunately. But there has been little work emphasizing a unified perspective, 1 3 and consequently this paper seems more relevant than ever. We are therefore very happy to publish it 20 years later. Rather than re-writing the paper, we have decided to publish it as it was originally written, and to provide some notes in the following on relevant developments in the literature and how they relate to the survey. These notes assume familiarity with the basic concepts introduced in the paper.

Key developments in robustness to incomplete information
Here are some key developments in the core idea of robustness to incomplete information as a novel refinement:

Existence
KM1997 provided a sufficient condition, which is an extension of risk dominance, for a Nash equilibrium to be robust. Ui (2001) provided a sufficient condition based on the idea of potential games, and Morris and Ui (2005) provided a number of sufficient conditions based on notions of generalized potential games. Since then, a variety of new sufficient conditions for robustness have been developed: examples include Oyama and Tercieux (2009) and Nora and Uno (2014).
KM1997 showed that risk dominance was both necessary and sufficient for robustness to incomplete information in two-player two-action games. The literature mentioned above has provided stronger sufficient conditions but not necessary conditions. Oyama and Takahashi (2019) provided necessary and sufficient conditions for robustness in many player, two-action, asymmetric payoff, supermodular games.
KM1997 also showed that if there is a unique correlated equilibrium, it must be robust. The logic behind it is completely different from the sufficient conditions mentioned above. Einy et al. (2019) showed that a unique correlated equilibrium is fully characterized by a stronger notion of robustness.

Continuity
The solution concept of KM1997 imposed the requirement that an incomplete information elaboration of a complete information game was close to a complete information game if, with high probability, players know that payoffs are given by the complete information game. If one replaces "know" with "believe with high probability", similar results reported in the paper hold, and there was a somewhat arbitrary modeling choice. In this sense, the robust equilibrium is continuous with respect to limiting beliefs for fixed payoffs. Haimanko and Kajii (2016) showed, however, that it is not continuous in payoffs, i.e., the solution correspondence which maps the complete information game's payoffs to robust equilibria is not upper hemicontinuous at some games. An equilibrium strategy profile in an incomplete information game is an -equilibrium of any nearby game, and so demanding an equilibrium in each elaboration was too stringent. Haimanko and Kajii (2016) then showed that if the notion is weakened 1 3 The Japanese Economic Review (2020) 71:35-41 by considering -equilibria in an incomplete information elaboration for arbitrarily small , then this weaker robust equilibrium is upper hemicontinuous in payoffs. Carmona (2018) examined robust equilibria in an abstract setting from the perspective of generic continuity of the solution correspondence.

Set-based robustness
In KM1997, we examined the robustness of singleton Nash equilibria to incomplete information. It is very natural to look at a set-based notion of robustness: do there exist a set of correlated equilibria of a complete information game with the property that every incomplete information game where payoffs are almost always given by that complete information have an equilibrium where play is in that set of correlated equilibria? The set of all correlated equilibria automatically satisfy this property, but the interesting question is whether there are interesting sufficient conditions for a small set to be robust in this sense. Morris and Ui (2005) formally defined a set-based version of robustness and provide a number of sufficient conditions based on notions of generalized potential games.

Canonical versus full elaborations
The canonical elaborations studied in our unified survey were invented for a clean exposition of the role of the infection argument and the critical path result in KM1997. Because of the additional structure, the set of canonical elaborations is smaller than that of general elaborations considered in KM1997, thus the robustness in the survey is in principle a weaker concept. While this does not matter for the results reported in the survey, the distinction turned out to be important in later work. For instance, Ui (2001) established that potentially maximizing Nash equilibria of potential games are robust to canonical elaborations, but not for general elaborations.
Whether or not the robustness to canonical elaborations is equivalent to the robustness in KM1997 in general then remained an open question for many years. Recently, Pram (2019) gave one answer by showing that for the modified correlated equilibrium version of robustness, there is equivalence. On the other hand, Takahashi (2019) provided an example establishing the failure of equivalence under the original definition, although the counterexample works for the set-valued version of robustness.

Unified approach
As we wrote earlier, the survey paper was valuable because of its unified framework. However, the survey was only about normal form games, whereas the majority of the refinements literature is inspired by extensive form games (and thus non-generic normal form games).
An important recent paper of Takahashi and Tercieux (2018) studied-in the language of our survey-"robustness to all limit common knowledge elaborations" and developed a number of important tight characterizations of the connection between robustness and classical perturbation refinements in the tradition of Kohlberg and Mertens (1986). In particular, they show (roughly, translated into our language) that hyperstability is sufficient for robustness to all limit independent elaborations, that essentiality is sufficient for robustness to all canonical singleton elaborations and that stability is equivalent to robustness to all canonical, independent, singleton, known own payoff elaborations.

Related works
We shall provide notes of somewhat more distantly related works: 1. One can think of a "committed type" in a canonical elaboration as those who are unable to make inference about other players' actions. The type who believes their opponents are committed types can then be interpreted as those who can do one more layer of inference, so they think that they are playing only against "committed types". Murayama (2019) elaborated on this idea to study robustness of predictions when players can only reason to a finite number of levels, relating these questions to the robustness to incomplete information question. 2. Robustness to incomplete information is closely related the global games literature initiated by Carlsson and van Damme (1993). In particular, if an action profile is robust to incomplete information, then it will also be the action profile selected in the global game limit as noise goes to zero. Thus robustness to incomplete information is a sufficient condition for being the "noise-independent selection" of a global game (Frankel et al. 2003). Intuitively, global game perturbation sequences are more restrictive than general elaboration sequences but less restrictive than limit common knowledge sequences. Morris and Shin (2003) described the connection heuristically; some of the same sufficient conditions for noise-independent selection and robustness, respectively, appear in Frankel et al. (2003) and Morris and Ui (2005). Formal proofs of versions of the connection appear in the working paper version of Basteck et al. (2013) and Oury and Olivier (2007). Oyama and Takahashi (2011), Basteck and Daniëls (2011) and Honda (2011) together identified a gap between robustness to all elaborations and robustness to global game elaborations: in two-player, three-action supermodular games with symmetric payoffs, there is generically an equilibrium that is the noise independent selection in global games, but it is not in general robust to incomplete information. 3. The survey focussed on common prior information structures. One could examine non-common prior analogues of all the results. Oyama and Tercieux (2010) have examined a non-common prior version of robustness. In generic games, only unique strategies surviving iterated deletion of strictly dominated strategies are robust to all elaborations. They also show a smallest robust set exists and is equal to the set of a posteriori equilibria (a refinement of subjective correlated equilibria). There is a close connection between this result and Weinstein and Yildiz (2007) who examined an interim notion of robustness in the universal type space. In this interim version, the common prior assumption no longer has the bite that it has in the framework of KM1997 and Oyama and Tercieux (2010). See also Germano et al. (2016) and Penta and Zuazo-Garin (2018). Many other results in the survey will also have non-common prior analogues. For example, Frick and Romm (2015) provided an analysis of non-common prior robustness to some limit common knowledge elaboration, showing that it gives the solution concept of S ∞ W (one round of deletion of weakly dominated strategies followed by iterated deletion of strictly dominated strategies) of Borgers (1994) and Dekel and Fudenberg (1990). 4. Chassang and Takashashi (2011) considered a repeated game version of robustness in the sense of KM1997. 5. The technique in KM1997 is to use "belief operators"-introduced by Monderer and Samet (1989)-to characterize equilibrium behavior in arbitrary common prior type spaces and then state global results that hold across all common prior type spaces exploiting that characterization. Morris and Shin (2007) and Morris and Shin (2016) generalized these belief operators-to be state dependent and record beliefs about multiple events at once-and Oyama and Takahashi (2019) use such generalized belief operators to generalize the scope of the critical path result from KM1997. 6. Morris (1997) described a formal equivalence between the analysis of incomplete information games and games played on networks. Under this equivalence, a symmetric action profile being "robust to incomplete information" in the sense of KM1997 translates to that action being uninvadable under best response dynamics in a network game. This is discussed in Morris (1997), lies behind results (but is not emphasized) in Morris (2000) and is developed explicitly in Oyama and Takahashi (2015). 7. The ideas developed in the robustness literature turn out to be very relevant in characterizing the solution to many player information design problems where the information designer anticipates adversarial equilibrium selection: see Bergemann and Morris (2019), section 7.
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