Non-equivalence between all and canonical elaborations

In a minimal diversity game, I show that the set of potential maximizers is robust to canonical elaborations, but not to all elaborations. This is the first example to demonstrate that the set-valued notion of robustness to canonical elaborations is strictly weaker than the set-valued notion of robustness to all elaborations.

1 3 canonical elaborations is a strictly weaker notion than robustness to all elaborations (Ui 2001, footnote 6). Morris and Ui (2005) extend the two notions of robustness-robustness to all elaborations and to canonical elaborations-to the respective set-valued notions, and show that the set of (generalized) potential maximizers is robust to canonical elaborations. Nonetheless, Ui's problem remains open (Morris and Ui 2005, footnote 9). Recently, Pram (2019) replaces the solution concept used in elaborations, from Bayesian Nash equilibrium to agent-normal-form correlated equilibrium, and shows that the two notions of robustness are equivalent. 1 In this paper, I show that in a minimal diversity game introduced by Balkenborg and Vermeulen (2016), the set of potential maximizers is robust to canonical elaborations, but not to all elaborations. To the best of my knowledge, this is the first example to demonstrate that the set-valued notion of robustness to canonical elaborations is strictly weaker than the set-valued notion of robustness to all elaborations. This example also implies that Pram's (2019) equivalence result, at least if it is extended to the respective set-valued robustness notions, relies on the use of correlation devices.

Set-valued robustness
A complete-information game consists of a finite set N = {1, 2, … , n} of players, a finite set A = ∏ i∈N A i of action profiles, and a payoff function profile = (g i ) i∈N , g i ∶ A → ℝ for i ∈ N . Suppressing N and A, I denote this game by . A mixed action profile = ( i ) i∈N , i ∈ Δ(A i ) for i ∈ N , is an equilibrium of if for any i ∈ N and a i ∈ A i . 2 Here, the domain of g i is extended to mixed action profiles, and −i = ( j ) j≠i .
An elaboration of is an incomplete-information game consisting of the same sets of players and action profiles as as well as a countable set T = ∏ i∈N T i of type profiles, a bounded type-dependent payoff function profile = (u i ) i∈N , u i ∶ A × T → ℝ for i ∈ N , and a common prior P ∈ Δ(T) . Let T −i = ∏ j≠i T j . Without loss of generality, I assume that P({t i } × T −i ) > 0 for any i ∈ N and t i ∈ T i , and hence the conditional probability P i (t −i |t i ) ∶= P(t i , t −i )∕P({t i } × T −i ) is well defined. I denote this game by ( , P) Given elaboration ( , P) , let for all a ∈ A and t −i ∈ T −i } 1 3 The Japanese Economic Review (2020) 71: [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57] be the set of all types of player i who know that their payoffs are given by g i . Let T = ∏ i∈N T u i i . I say that ( , P) is an -elaboration of if P(T ) = 1 − . I also say that ( , P) is a canonical elaboration of if every type in T i ⧵T u i i has a strictly dominant action.
A behavioral strategy profile = ( i ) i∈N , i ∶ T i → Δ(A i ) , is a Bayesian Nash equilibrium of ( , P) if for any i ∈ N , t i ∈ T i , and a i ∈ A i , where the domain of u i (⋅, t) is extended to mixed action profiles, and −i (t −i ) = ( j (t j )) j≠i . A strategy profile induces an action distribution P ∈ Δ(A) by for each a ∈ A . For action distributions , ∈ Δ(A) and a nonempty and closed set Definition 1 (Morris and Ui 2005) A set of action distributions, E ⊆ Δ(A) , is robust to all (resp., canonical) elaborations in if it is nonempty and closed, and for any > 0 , there exists ̄> 0 such that any (resp., canonical) -elaboration ( , P) of with <̄ admits a Bayesian Nash equilibrium such that d( P , E) ≤ .
With some abuse of terminology, I say that an action distribution ∈ Δ(A) is robust to all (resp., canonical) elaborations when the corresponding singleton set { } is robust to all (resp., canonical) elaborations; a pure action profile a * ∈ A is robust to all (resp., canonical) elaborations when the Dirac measure on a * is robust to all (resp., canonical) elaborations.
Neither a full characterization of robustness to all elaborations nor to canonical elaborations has been known so far. Instead, the literature has identified several sufficient conditions as well as necessary conditions for robustness. (1) Most of the known sufficient conditions, except Ui (2001) and some of the conditions in Morris and Ui (2005), are for robustness to all elaborations, such as a unique correlated equilibrium (Kajii and Morris 1997a, Proposition 3.2), a -dominant equilibrium with ∑ i∈N p i < 1 (Kajii and Morris 1997a, Proposition 5.3), and a monotone potential maximizer with either or the monotone potential being supermodular (Morris and Ui 2005, Proposition 2). Obviously, the two robustness notions, i.e., robustness to all elaborations and to canonical elaborations, are equivalent under any of these conditions. (2) Also, having no other strict -dominant equilibrium with ∑ i∈N p i ≤ 1 is a necessary condition for robustness to canonical elaborations. 3 Thus, the two robustness notions are equivalent under its negation, i.e., having another strict -dominant equilibrium with ∑ i∈N p i ≤ 1 .
(3) For almost every binary-action supermodular game, if an action profile is not a monotone potential maximizer, then it is not robust to canonical elaborations (Oyama and Takahashi 2019). Combined with Morris and Ui (2005, Proposition 2), this result implies that the two robustness notions are generically equivalent in binary-action supermodular games. (4) The two robustness notions are equivalent also for the largest or smallest action profile of a supermodular game. 4 Balkenborg and Vermeulen (2016) introduce the class of minimal diversity games with n players and m actions. 5 Here, I focus on n = 3 and m = 2 . That is, there are three players, N = {1, 2, 3} , and each player i ∈ N has two pure actions,
Denote E ∶= Δ(A⧵{(0, 0, 0), (1, 1, 1)}) . Note that E is the set of all distributions over the set of potential maximizers. Note also that the equilibrium of the first kind, (x 1 , x 2 , x 3 ) = (1∕2, 1∕2, 1∕2) , induces the uniform distribution over A, which does not belong to E . On the other hand, all equilibria of the second kind induce action distributions in E.
Proposition 1 E is robust to canonical elaborations in .
Proof Follows from an extension of Ui (2001) and Morris and Ui (2005, Theorem 4) to non-product-set potential maximizers. ◻ Proposition 2 E is not robust to all elaborations in .
Proof I use the following game = (h , h , * ) among players , , and as a building block of my construction of elaborations: where player chooses a row, player chooses a column, and player chooses a matrix. Player 's payoffs are irrelevant to my construction, and hence are not specified. I denote by x the induced two-player game between players and given player 's mixed action x ∈ [0, 1]: (Recall my convention that "mixed action x" refers to the mixed action that takes pure action 0 with probability 1 − x and 1 with probability x.) I also construct another game ̃ = (h ,h , * ) by relabeling player 's action 0 as action 1, and action 1 as action 0, i.e., I denote by ̃ x the induced two-player game given player 's mixed action x ∈ [0, 1]: Both x and ̃ x are 2 × 2 games with unique mixed action equilibria as follows: Note that in both games x and ̃ x , player 's equilibrium action (1 + x)∕3 changes monotonically with respect to player 's action x, and in particular, player takes action 0 more (resp., less) likely than action 1 if and only if player takes action 0 more (resp., less) likely than action 1. Also note that in game ̃ x , players and take mixed actions with complementary probabilities, i.e., (1 + x)∕3 + (2 − x)∕3 = 1 . These properties will be used later in the proof of Lemma 3.
Inspired by Balkenborg and Vermeulen (2016, Section 7.1), for each , 0 < < 1 , I construct the following elaboration ( , P) of . Each player has five types, where t * i is a "normal" type with the same payoff function as g i , and all other types are divided into the "first generation" (with superscript f) and the "second generation" (with superscript s). 6 The prior P ∈ Δ(T) is given by Since 0 < < 1 , there are ten type profiles that are assigned with strictly positive probabilities. In words, normal type t * i interacts with the other normal types . See Fig. 1.
6 As it will be clear in the example before Lemma 3 and the proofs of Lemmas 3 and 4, I will construct the elaboration ( , P) in such a way that once normal type t * i 's action is given, it uniquely determines the equilibrium actions of first-generation types of superscript i, t

3
The Japanese Economic Review (2020) 71:43-57 Given the prior P, each type has the following posterior: are given as follows: • Each normal type t * i has the complete-information payoff function g i regardless of the other players' types. That is, for any a ∈ A and t −i ∈ T −i . • First-generation types t i,f i+1 and t i,f i+2 play game ̃ in the roles of players and , respectively, when they interact with normal type t * i in the role of player . Also, type t i,f i+1 always receives zero payoff when he interacts with second-generation arbitrary otherwise • Second-generation types t i,s i and t i,s i+2 always receive zero payoff when they interact with normal type t *

i+1
. Also, t i,s i and t i,s i+2 play game in the roles of players and , respectively, when they interact with first-generation type t i,f i+1 in the role of player . That is, for any a ∈ A. Note that ( , P) is an -elaboration of . Thus, to prove Proposition 2, it suffices to show that for any small > 0 , ( , P) has no Bayesian Nash equilibrium that induces an action distribution in a neighborhood of E.

Proof of Lemma 3
Since part (2) follows immediately from part (1), I will show only part (1). By symmetry between actions 0 and 1, I assume without loss of generality where the inequality follows since x < 1∕2 . Lastly, consider the incentive of the next normal type t *
is indifferent between actions 0 and 1 in the interaction with (t 1 3 • By (1) and Lemma 1, type t * i+1 strictly prefers action 1 to action 0 in the interaction with (t i,s i , t i,s i+2 ).
Second, given the equilibrium action of normal types, I can specify, as in the proof of Lemma 3, the equilibrium actions of first-and second-generation types by applying Lemma 2 iteratively. Namely, for each i ∈ N , types t i,f The Japanese Economic Review (2020) 71: [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57] and Note that the state (t * 1 , t * 2 , t * 3 ) occurs with probability 1 − , and at this state, each player i believes with probability (3 − 3 )∕(3 − ) , which converges to 1 as → 0 , that his payoff is the same as g i . Noting that ( � , P � ) is essentially the same as Balkenborg-Vermeulen's game ( ∕(3 − 3 )) , it is easy to show that ( � , P � ) has a unique Bayesian Nash equilibrium i (t Remark 2 In the definition of elaborations, non-normal types may not know their own payoff functions, i.e., their payoffs may depend on other players' types. But this is not essential. I can modify the minimal diversity game and the elaboration in the proof of Proposition 2, and show the non-equivalence between the notions of robustness to known-own-payoff elaborations and to canonical elaborations. A trick is to incorporate "nature" as an explicit player, where nature knows the type profile of the other players and replicates it as her action. 8

Remark 3
If a game admits a potential that is concave in the domain of mixed action profiles, then the set of action distributions over potential maximizers is precisely the set of correlated equilibria (Neyman 1997, Theorem 1), and hence robust to all elaborations. Also, if a game admits a (monotone) potential that is supermodular, then the set of action distributions over potential maximizers is robust to all elaborations (Morris and Ui 2005, Proposition 2). The potential of the minimal diversity game is, however, neither concave nor supermodular (with respect to any total order on each action set).

Remark 4
For any canonical elaboration of a potential game, Ui defines the elaboration potential, and shows that the elaboration potential is indeed a potential of (the normal form of) the elaboration (Ui 2001, Lemma 2). This is a key step toward establishing the robustness of the potential maximizer to canonical elaborations. He otherwise. 8 More precisely, let and ( , P) be the minimal diversity game and the elaboration in the proof of Proposition 2, respectively. Then consider another game ̃ with Ñ = N ∪ {0} , Ã i = A i for i ∈ N , Ã 0 = T , g i (a, t) = u i (a, t) for i ∈ N , and g 0 (a, t) = 1 t=t * . Let Ẽ = Δ((A⧵{(0, 0, 0), (1, 1, 1)}) × {t * }) . Then Ẽ is robust to canonical elaborations in ̃ , but not robust to the elaboration (̃ ,P) with T i = T i for i ∈ N , T 0 = T , P (t, t � ) = P(t)1 t=t � , and ũ i ∶Ã ×T i → ℝ with ũ i ((a, t), t � i ) = u i (a, t) for i ∈ N and ũ 0 ((a, t), t � ) = 1 t=t � . (Note that "t" in ũ i ((a, t), t � i ) is not a type profile in (̃ ,P) , but an action of player 0.) 1 3 also notes that his argument can be used to show the robustness to non-canonical elaborations as long as they admit Bayesian potentials in the sense of van Heumen et al. (1996) (Ui 2001, Remark 2). However, the elaboration ( , P) constructed in the proof of Proposition 2 is based on non-potential games and ̃ , and hence does not admit a Bayesian potential.
Remark 5 Pram (2019) establishes the equivalence between robustness to all elaborations and to canonical elaborations if agent-normal-form correlated equilibrium is used as the solution concept in elaborations. It implies, in particular, that the elaboration ( , P) constructed in the proof of Proposition 2 has an agent-normal-form correlated equilibrium in a neighborhood of E . Indeed, the following is one such example: the three normal types play the uniform distribution over A⧵{(0, 0, 0), (1, 1, 1)} , and independently, all other types randomize between actions 0 and 1 with equal probabilities. More formally, it is ∈ Δ A given by

Conclusion
One of the open questions in the literature is whether robustness to canonical elaborations is a strictly weaker notion than robustness to all elaborations. In this paper, I solved this question for set-valued robustness notions affirmatively by showing that in the three-player two-action minimal diversity game, the set of potential maximizers is robust to canonical elaborations, but not to all elaborations. Of course, many questions remain open. For example, can I show the non-equivalence between the two robustness notions in "generic" games? Can I show the non-equivalence between the singleton-valued robustness notions? Let me elaborate more. Admittedly, the minimal diversity game is "doubly" non-generic; potential games are non-generic in the class of all games, and the minimal diversity game is non-generic even in the class of potential games. In particular, the minimal diversity game has six pure action profiles that tie for potential maximizers. Instead of the minimal diversity game, one can slightly perturb its payoffs and obtain a potential game ′ with a unique potential maximizer a * . By Ui (2001), a * is robust to canonical elaborations in ′ . Then, can I show that a * is not robust to all elaborations in ′ ? Notice that a * is a strict equilibrium of ′ . Thus, along any sequence of k -elaborations ( k , P k ) of ′ with a bounded number of normal types and k → 0 as k → ∞ , for large k, there exists a Bayesian Nash equilibrium where all normal types play a * (Monderer and Samet 1989;Fudenberg and Tirole 1991, Section 14.4). In particular, a * is robust to elaborations with a unique normal type per player, such as the elaboration in the proof of Proposition 2. I have not been able to prove (or disprove) the non-robustness of a * to elaborations with unboundedly many normal types.
Another issue is the use of small incentives in elaborations. At the end of the proof of the first statement of Lemma 3, I concluded that "Overall, t * i+1 strictly prefers action 1 to action 0, and hence i+1 (t * i+1 ) = 1 ." However, note that type t * i+1 believes that he interacts with (t i,s i , t i,s i+2 ) with probability ∕(9 − 7 ) , and hence the incentive margin is of the order of . To make this argument non-sensitive to small incentives, one would like to construct another elaboration that has no approximate Bayesian Nash equilibrium in a neighborhood of E . In fact, Haimanko and Kajii (2016) propose the notion of "approximate robustness" along this line. 10 Once again, I do not know whether E is approximately robust to all elaborations or not in the minimal diversity game. 11 Note that the sensitivity to small incentives is closely related to the non-genericity of the minimal diversity game discussed in the previous paragraph. For example, if one were able to strengthen Ui's (2001) result and show that the unique potential maximizer of a generic potential game is robust (or at least approximately robust) to all elaborations, then by the upper hemicontinuity of approximate robustness (Haimanko and Kajii 2016, Theorem 3), it would imply that E is approximately robust to all elaborations in the minimal diversity game. Conversely, if one 1 3 were able to show that E is not approximately robust to all elaborations in the minimal diversity game, then it would imply that Ui's (2001) result fails without the restriction to canonical elaborations, i.e., even if a game admits a potential with a unique potential maximizer, it may not be robust to all elaborations. t i ∈ T i . 13 By the supermodularity of and the Kakutani-Fan-Glicksberg fixed point theorem, ( , P) has a Bayesian Nash equilibrium in Σ , which plays ā with ex-ante probability at least 1 − . ◻ Proposition 3 easily extends to the respective set-valued robustness notions if the set E ⊆ Δ(A) is an upper (resp. lower) set, i.e., if ∈ E whenever first-order stochastically dominates ′ (resp. ′ first-order stochastically dominates ) and � ∈ E.