DP2015-28

Robust Predictions under
Finite Depth of Reasoning
R IEB Jun ior Re se ar ch F ellow

Kota MURAYAMA
July 3, 2015

* This Discussion Paper won the Kanematsu Fellowship Prize (FY 2013).

Robust predictions under finite depth of reasoning
Kota Murayama∗June, 2015

Abstract
When players have a finite depth of reasoning, it is usually assumed thateach player has a commonly known anchor behavior. This paper provides a
general framework to examine whether predictions are robust to uncertainty
about other players’ anchors. We give two diﬀerent suﬃcient conditions forthe robustness. The first condition shows that any p-dominant equilibrium isrobust if players put suﬃciently small probability (decreasing in p) on highdepth types. This result highlights a distinction between two prominent finite
depth of reasoning models: a risk dominated equilibrium is robust in the
cognitive hierarchy model, but not in the level-k model. We also show thatequilibria of dominance solvable models are robust.
Keywords: Robustness, iterative reasoning, level-k model, cognitive-hierarchymodel, higher-order belief, bounded rationality.

∗

Department of Economics, Northwestern University. E-mail: murayama0209@gmail.com. I

am most grateful to Takashi Ui for his guidance and feedback that have improved this paperin numerous aspects. Special thanks are to Atsushi Kajii and Daisuke Oyama for their time,
advice, and encouragement. For helpful comments, I also thank Vincent Crawford, Hideshi Itoh,
Michihiro Kandori, Willemien Kets, Takashi Kunimoto, Takeshi Nishimura, Ariel Rubinstein, andSatoru Takahashi, and seminar and conference participants at various places. This research ispartially supported by Research Fellowships of Japan Society for the Promotion of Science for
Young Scientists, and is awarded the 2014 Kanematsu Fellowship from Research Institute for
Economics and Business Administration (RIEB) at Kobe University.

11

Introduction

Consider the 11-20 money request game in Arad and Rubinstein (2012). In thisgame, each of two players requests an amount of money between 11 and 20 shekels.
Each player receives the requested amount and if he requests exactly one number lessthan the other player, he will receive an additional 20 shekels. Players typically solve
this game by using inductive reasoning. If a player’s initial guess of the opponent’s
request is 20, then his best response is 19. But if the opponent knows his initial
guess and knows that he plays a best response to the guess, then she will choose 18.
In turn, if he knows this, he will choose 17 instead of 19.
One of the most prevalent models that features inductive reasoning is a finite
depth of reasoning (FDR) model such as the level-k and cognitive-hierarchy models(Stahl and Wilson, 1994; Nagel, 1995; Camerer et al., 2004). This model makes
two distinctive assumptions. First, each player has an anchor behavior and it iscommonly known.1 Second, each player has a reasoning level as his type, and believes
that other players have lower reasoning levels.2 Then we can define an inductive
solution for each game: level-1 (L1) types best respond to the anchor, level-2 (L2)types best respond to a combination of the anchor and the best responses of L1
types, and so forth.
In the FDR model, each player’s anchor behavior plays an important role as astarting point for other players’ iterative reasoning. Unfortunately, it is rare to havea unique candidate for the anchor behavior even in simple games. For instance, whilea uniformly random choice is commonly regarded as an anchor in the 2/3 guessing
game, a recent paper by Burchardi and Penczynski (2014) shows that players’ beliefs
about the anchor are concentrated around salient numbers such as 50, 66, and 100.
Thus, it is essential for expanding the domain of applicability of FDR models to1

Anchor behavior is called level-0 action in the experimental literature. Our use of terminology

intends to emphasize that the anchor behavior is a characteristic of underlying games, and is adiﬀerent object from player’s FDR.
2
A diﬀerence between level-k and cognitive-hierarchy models is their specification of beliefs
about other players’ reasoning levels. Each type in the level-k model believes that other players have
one lower reasoning level, and in the cognitive-hierarchy model, each type’s belief is a conditional
probability distribution induced by a common probability distribution over reasoning levels.

2

know how player’s behavior diﬀers if the anchor behavior is not common knowledge.
This paper provides a general framework to analyze the robustness of predictions to small uncertainty about an anchor behavior. From the perspective of ananalyst, we have the following scenario in mind. An analyst is almost certain thata particular FDR model describes the relevant strategic environment; however, shesuspects that players might consider other possibilities of anchor behaviors. Under
this situation, if the prediction is known to be robust, any model with uncertainty
has the approximately same prediction as long as possible anchor behaviors are close
to the original one with high probability.
Our main contribution is giving a theoretical foundation for using FDR models ingeneral environments.3 We provide an upper-bound on beliefs about other players’
reasoning levels that makes a given prediction robust. For instance, consider a FDR
model whose underlying game is given by a two-player, two-action game with twostrict Nash equilibria. We show that the risk dominated prediction, in which every
type follows the risk dominated Nash equilibrium, is robust in the cognitive-hierarchymodel but not in the level-k model. This result highlights a distinction between thetwo prominent FDR models in the sensitivity of predictions to the violation of thecommon knowledge assumption.
More formally, a (FDR) model is a combination of a game with anchor behavior(a game for short) in which each player is given an anchor behavior, and a set of
reasoning levels (or types). An equilibrium of this model is described above. Our
robustness test proceeds as follows: Fix a model and its equilibrium s∗ . We embed
this model by introducing incomplete information about its anchor behavior. The
embedding model is said to be close to the original model if the anchor behavior
remains the same and each player knows that with high probability.4 We say thats∗ is robust if each close embedding model has an equilibrium in which s∗ is playedwith high probability.
3

There has been a growing number of papers that introduce FDR to various kinds of games

including auctions (Crawford and Iriberri, 2007; Crawford et al., 2009), global games (Kneeland,2014), and mechanism design (Crawford, 2015; de Clippel et al., 2015).
4
In the embedding model, players have a common prior on anchor behaviors. The closeness ismeasured by this prior.

3

The first suﬃcient condition utilizes a relation between the robustness and beliefs
about other players’ reasoning levels. We define a p = (p1 , . . . , pI )-dominant equilibrium analogously to the standard p-dominance.5 We show that any p-dominantequilibrium is robust if there exists a threshold level c̄ such that any higher-level
type of players than c̄ puts suﬃciently small probability (decreasing in p) on his
opponents having levels no lower than c̄. As a corollary, a p-dominant equilibriumin the level-k model is shown to be robust if pi < 1/I for each player i, and any
p-dominant equilibrium with p ∈ [0, 1)I is robust in the cognitive-hierarchy model
for any common distribution.
For the second result, we say a model is dominance solvable if the model yields aunique action for suﬃciently high-level types through the iterated deletion of neverbest responses. We show that a strict equilibrium of the dominance solvable modelis robust. Particularly, the level-k model is dominance solvable if and only if its
underlying game is dominance solvable in the usual sense. Hence, the level-k modelwith a dominance solvable game gives a robust prediction generically in payoﬀs.
Our notion of robustness is based on Kajii and Morris (1997), who study therobustness of equilibria to a violation of common knowledge about payoﬀs (hereafter, KM robustness). The main diﬀerence is that while they consider a complete
information game and introduce payoﬀ uncertainty, we start from the FDR modeland introduce uncertainty in anchor behavior. As a result, since the anchor works
as a coordination device for high-level types under a certain condition on beliefs, a
weaker suﬃcient condition can be obtained than that for KM robustness.
Strzalecki (2014) also derives a similar bound on beliefs in Rubinstein’s (1989)
email game. He shows that coordination is achieved after exchanging a finite numberof messages if the player’s belief on high-level types is bounded. Thus, a particularspecification of beliefs makes the strategic behavior of high-level types change in a
uniformly continuous way in the specific game. Although our question and method
are diﬀerent, our result further develops this insight. Especially, while Strzalecki’s
bound depends on details of the email game, our bound uniformly holds for general
5

A strategy profile is p-dominant if it is optimal for each type of each player i to play the

specified strategy whenever he believes consistently with his belief about others’ types that theother players will follow the specified strategies with probability at least pi .

4

games, and is expressed as a function of the “strength” of equilibrium and thenumber of players.
The rest of the paper is organized as follows: In Section 2, we discuss the relationsof this paper to other studies. Section 3 introduces our framework. Section 4 presents
a motivating example. In Sections 5 and 6, we provide the two suﬃcient conditions.
Section 7 discusses some limitation and future direction of our research. All proofs
are shown in the Appendix.

2

Related LiteratureThis paper is related to two strands of literature. A role of anchor behavior ina strategic environment is pioneered by Schelling (1960). In the experimental literature, a modeler’s choice of an anchor has received increasing attention.6 Both
Burchardi and Penczynski (2014) and Agranov et al. (2015) identify the anchor
behavior of the 2/3 guessing game, but lead to diﬀerent conclusions. Heap et al.
(2014) test a saliency-based anchor behavior by using games in which one action
has a uniquely diﬀerent label from the others. Based on the observation that players do not necessarily best respond to the salient anchor, they raise a doubt on its
universality.7 In a related context, Kneeland (2015) proposes a way of identifying
player’s depth of reasoning independently of the specification of anchor behaviors.
The present paper studies a behavioral implication of not assuming the common
knowledge of anchor behaviors. For this purpose, we use techniques from the robustness literature, especially from Kajii and Morris (1997). There has been many
follow-up papers that propose a variety of suﬃcient conditions for KM robustness.
Ui (2001) shows that a unique maximizer of the potential function is KM robust.
This approach is further elaborated by Morris and Ui (2005), Oyama and Tercieux(2009), and Nora and Uno (2014). Other suﬃcient conditions include a unique correlated equilibrium (Kajii and Morris, 1997) and p-best response sets (Tercieux,
2006). Investigating corresponding notions of these conditions in the FDR model
might be interesting, but it is beyond the scope of this paper.
6
7

Crawford et al. (2013) and Camerer and Ho (2014) provide an excellent survey of FDR models.
See also comments by Crawford (2014).

5

Finally, Heifetz and Kets (2013) also consider the robustness of predictions whenplayers can have either infinite or finite depth of reasoning. They construct a general
universal type space that contains both the standard universal type space (Mertens
and Zamir, 1985; Brandenburger and Dekel, 1993) and a universal cognitive typespace (Strzalecki, 2014). In contrast to Weinstein and Yildiz (2007), they show thatin generic global games (Carlsson and van Damme, 1993), there are types that robustly have multiple rationalizable actions when players’ infinite depth of reasoning
is “almost” common knowledge. The main diﬀerence between their approach andours is that while they use the weakest solution concept and establish its robustness
focusing on global games, we pay more attention to a refined solution concept, andderive its robustness in general games.

3

Framework

3.1

Game with anchor behavior

A game with anchor behavior (called a game hereafter) is a tuple G = (I, {Ai , ui , fi }i∈I ),where I = {1, 2, ..., I} is a set of players, Ai is a finite set of actions for player i,and ui : A → R is player i’s payoﬀ function, where A = A1 × · · · × AI . Unlike thestandard complete information game, fi ∈ Ai specifies an anchor behavior for player
i.8 This anchor can be interpreted as a focal point or a naive play of the game, whichis commonly known among players. It seems diﬃcult in general to find an anchor if
it exists, however, the following two games have a somewhat clear candidate for theanchor.
Example 3.1. (Salient label in a symmetric coordination game)
Each of two players chooses one of the two locations: the Sears Tower (a recognizable
Chicago landmark) or the little-known AT&T Building. Players will receive $100
if both choose the same location, and will receive nothing if they choose diﬀerent
locations. Payoﬀs are shown in Table 1. In this game, the salience of the Sears

8

While it is conceptually easy to incorporate mixed anchor behaviors, we focus on pure anchor

behaviors to keep our notation as simple as possible.

6Sears Tower

AT & T

100, 100

0, 0

0, 0

100, 100Sears Tower
AT & T

Table 1: Payoﬀ matrix of the symmetric coordination game.
Tower may work as an anchor. In fact, Crawford et al. (2008) report that in theirexperiments, about 90% of subjects chose the Sears Tower.
Example 3.2. (Random play in a 2/3-guessing game)
Each of I players calls an integer ai ∈ [0, 100], and his payoﬀ is given by
1 if ai = [2/3 · ā−i ]ui (a1 , . . . , aI ) =
0 if otherwise,
where ā−i = (a1 + · · · + ai−1 + ai+1 + · · · + aI )/(I − 1), and [a] is the closest integer
to a. Thus, each player has a strict incentive to guess the average of other players’
choices ā−i , and call the closest integer to 2/3 · ā−i . Agranov et al. (2015) findthat naive players tend to choose 50 on average, and thus the random play seems acandidate for the anchor in this game.

3.2

Cognitive type space

To model player’s FDR, we follow the formulation of Strzalecki (2014). A (cognitive)
type space of player i is given by Ti = (Ci , µi ), where Ci = {0, 1, 2, . . . } is a set oftypes, and µi : Ci \ {0} → ∆(C−i ) is player i’s belief about other players’ types,
where C−i = C1 × · · · × Ci−1 × Ci+1 × · · · × CI , and ∆(C−i ) denotes a collection ofprobability measures on C−i . We write µi (ci )(c−i ) for the probability of the singleton
event {c−i } given ci , where c−i = (c1 , . . . , ci−1 , ci+1 , . . . , cI ). We assume that eachtype believes that other players have lower types: µi (ci )({c−i ∈ C−i : cj < ci foreach j ̸= i}) = 1 for each ci ∈ Ci \ {0}. Since each type is naturally identified with areasoning level, a player of type k is called Lk type. Note that L0 types are “seeds”
for reasoning, and used to model a starting point for inductive reasoning. Let us
call T = T1 × · · · × TI a (cognitive) type space. The following two are the leading
examples of T .
7

Example 3.3. (Level-k type space)
The level-k type space, denoted by T k , is a type space in which each type believes
that all the other players have exactly one lower level. Formally, for each i ∈ I andci ∈ Ci \ {0},µi (ci )(c−i ) =
1 if cj = ci − 1 for each j ̸= i
0 if otherwise.

Example 3.4. (Cognitive hierarchy type space)
The cognitive hierarchy (CH) type space, denoted by T c , is a type space in which
each type’s belief is a conditional probability induced by a common distribution.
Formally, there exists λ ∈ ∆(Z+ ) with λ(0) > 0 such that for each i ∈ I andci ∈ Ci \ {0},µi (ci )(c−i ) =
j̸=i λ(cj )
 ∑×
ci −1
(




l=0

λ(l))I−10

if cj < ci for each j ̸= i
if otherwise.

For instance, Camerer et al. (2004) use a Poisson distribution for λ.

3.3

Solution concept

Let us call (G, T ) a (FDR) model. Player i’s (pure) strategy in (G, T ) is a mappingsi : Ci → Ai with si (0) = fi . Let Si denote the set of player i’s strategies for eachi ∈ I, and write S = S1 × · · · × SI and S−i = S1 × · · · × Si−1 × Si+1 × · · · × SI .
In equilibrium of (G, T ), L1 types play a best response to the anchor, and L2 types
play a best response to the mix of the anchor and L1 type’s best response according
to his belief. Equilibrium behavior of higher-level types is similarly defined.
Definition 3.1. A strategy profile s∗ = (s∗1 , . . . , s∗I ) is an equilibrium of (G, T ) if,
for each i ∈ I, ci ∈ Ci \ {0}, and ai ∈ Ai \ {s∗i (ci )},∑ ()
ui (s∗i (ci ), s∗−i (c−i )) − ui (ai , s∗−i (c−i )) µi (ci )(c−i ) ≥ 0,c−i ∈C−iwhere s∗−i (c−i ) = (s∗1 (c1 ), . . . , s∗i−1 (ci−1 ), s∗i+1 (ci+1 ), . . . , s∗I (cI )).
In particular, we say that a strategy profile is a strict equilibrium if it satisfies
the above condition with strict inequalities. By definition, a strict equilibrium is8

always unique if it exists. For instance, in the 2/3-guessing game with the level-k
type space, if the anchor behavior is given by 50, there exists a strict equilibrium inwhich L1 types choose 33, L2 types choose 22, and L3 types choose 15, and so on.

3.4

Robustness

Our robustness concept is based on Kajii and Morris (1997). We first extend amodel to accommodate situations where each player knows other player’s anchor
behavior only with high probability. An embedding game of G is a tuple U =(I, Θ, P, {Ai , ui , Πi , gi }i∈I ), where Θ is a nonempty countable set of states, P is acommon prior on Θ, and for each player i ∈ I, Πi is a set of information partitionsof Θ, and gi : Θ → Ai is a state dependent anchor behavior of player i. We assume
that gi is Πi -measurable, that is, gi (θ) = gi (θ′ ) for each θ, θ′ ∈ πi and πi ∈ Πi .
Note that Θ is uncertainty about the anchor behavior, and hence there is no payoﬀ
uncertainty in U. We write P (θ) for the probability of the singleton event {θ} andπi (θ) for the element of Πi containing θ. Suppose that P (πi (θ)) > 0 for each θ ∈ Θ
and i ∈ I. Let E(G) denote a set of all embedding games of G, and let us call (U, T )an embedding model of (G, T ).
An equilibrium of (U, T ) is defined as each non-L0 type of player maximizing
the expected payoﬀ under his beliefs about other players’ types and states. For
each i ∈ I, let us denote player i’s (pure) strategy in (U, T ) by σi : Ci × Θ → Ai
with σi (0, θ) = gi (θ) for each θ ∈ Θ, and assume that σi (ci , ·) is Πi -measurable foreach ci ∈ Ci \ {0}. We write σ = (σ1 , . . . , σI ), and σ−i = (σ1 , . . . , σi−1 , σi+1 , . . . , σI ).
Then, the expected payoﬀ of a strategy profile σ for the Lk type of player i at θ is∑ ∑ ()
Ui (σ, k, θ) ≡ui σi (k, θ), σ−i (c−i , θ′ ) µi (k)(c−i )P (θ′ | πi (θ)),
c−i ∈C−i θ′ ∈Θ

where σ−i (c−i , θ′ ) = (σ1 (c1 , θ′ ), . . . , σi−1 (ci−1 , θ′ ), σi+1 (ci+1 , θ′ ), . . . , σI (cI , θ′ )).
Definition 3.2. A strategy profile σ ∗ is an equilibrium of (U, T ) if, for each i ∈ I,
ci ∈ Ci \ {0}, θ ∈ Θ, and ai ∈ Ai ,∗, ci , θ) ≥ 0.
Ui (σ ∗ , ci , θ) − Ui (ai , σ−iLet us define an event ΩU in U in which each player’s anchor behavior is specified
as in G, and each player knows that. That is, ΩU ≡ {θ ∈ Θ : g−i (θ′ ) = f−i for each9θ′ ∈ πi (θ) and i ∈ I}, where g−i (θ′ ) = (g1 (θ′ ), . . . , gi−1 (θ′ ), gi+1 (θ′ ), . . . , gI (θ′ )). We
are now able to define a close embedding game to G in terms of the anchor behavior.
Definition 3.3. U is an ε-elaboration of G if U ∈ E(G) and P (ΩU ) = 1 − ε. Let
E(G, ε) be the set of all ε-elaborations of G.
In a 0-elaboration, we have in particular that σi (0, θ) = fi for each θ ∈ Θ andi ∈ I. Hence, for each equilibrium s∗ of (G, T ), each (U, T ) with U ∈ E(G, 0) has
an equilibrium σ ∗ with σi∗ (ci , θ) = s∗i (ci ) for each θ ∈ Θ and ci ∈ Ci .
In words, we say that an equilibrium of a model is robust (to incomplete information in anchor behavior) if any suﬃciently “close” embedding model has anequilibrium whose action distribution is probabilistically close to the original equilibrium. For each strategy profile σ of (U, T ), let Pσ (a|c) denote the probability ofan action profile a = (a1 , . . . , aI ) being taken by a type profile c = (c1 , . . . , cI ), i.e.,Pσ (a|c) = P ({θ ∈ Θ : σi (ci , θ) = ai for each i ∈ I}).
Definition 3.4. An equilibrium s∗ of (G, T ) is robust if for each δ > 0, there existsε̄ > 0 such that any (U, T ) with U ∈ E(G, ε) has an equilibrium σ ∗ that satisfies
Pσ∗ (s∗ (c)|c) ≥ 1 − δ for each c and ε ≤ ε̄, where s∗ (c) = (s∗1 (c1 ), . . . , s∗I (cI )).
There are two remarks on the definition of robustness. First, while we do notallow payoﬀ uncertainty for simplicity, the same results follow as long as each U ∈E(G, ε) has the same payoﬀ structure with G and players know that in ΩU . Second,each type’s belief about others’ types is assumed to be independent of his belief aboutstates. Alternatively, we can endow each type with a belief ρi (ci ) ∈ ∆(C−i × Θ)
whose marginal distribution on C−i coincides with µi (ci ). Allowing such a correlation
makes our robustness test more stringent.

4

Motivating example

We illustrate the intuition of our result by using a coordinated attack game.9 There
are two players, and each player chooses either Attack (A) or N ot Attack (N A).
Payoﬀs are shown in Table 2. Suppose that A is an anchor behavior for both players.
In this game G, NA (A) becomes a unique best response for each player whenever
9

This example is taken from Strzalecki (2014).

10

A

NA

A

1, 1

-2, 0

NA

0, -2

0, 0

Table 2: Payoﬀ matrix of the coordinated attack game.
he believes that his opponent will choose NA (A) with probability more than 1/3(2/3). Thus, (NA, NA) is a risk dominant and (A, A) is a risk dominated Nashequilibrium. Since (A, A) is the anchor, (G, T ) has a unique equilibrium in which
every type plays A regardless of the specification of T . Let s∗ denote this risk
dominated equilibrium.
Case 1: Level-k type spaceFirst, consider the level-k type space T k . We show that s∗ is not robust in this model.
Let us introduce an email game elaboration such that Θ = {(0, 0), (1, 0), (1, 1), (2, 1), ...},
Π1 = {{(0, 0)}, {(1, 0), (1, 1)}, ...}, Π2 = {{(0, 0), (1, 0)}, {(1, 1), (2, 1)}, ...}, and P ((a, b)) =ε(1 − ε)a+b for each (a, b) ∈ Θ.10 Assume that each player’s anchor behavior remains
A except for player 1’s anchor behavior at (0, 0) which is now given by NA. Let U edenote this ε(2 − ε)-elaboration of G. Table 3 shows each player’s posterior belief
about states.
Π1

1

1
2−ε

1−ε
2−ε

1
2−ε

1−ε
2−ε

...

Θ

(0, 0)(1, 0)(1, 1)(2, 1)(2, 2)

...

Π2

1
2−ε

1−ε
2−ε

1
2−ε

1−ε
2−ε

...

Table 3: Posterior beliefs over Θ.
In (U e , T k ), NA is a unique best response for the L1 type of player 2 in{(0, 0), (1, 0)} since he believes with certain that player 1 will play NA with probability 1/(2 − ε). In turn, NA is played in {(1, 0), (1, 1)} by the L2 type of player
1 since he believes that player 2 is of L1 type and will play N A with probability
1/(2 − ε) (see Table 4). Repeating similar arguments, we can show that there exists

10This embedding game has an essentially same information structure with that of the email

game.

11

a type profile that plays (NA, NA) with arbitrarily high probability in a uniqueequilibrium of (U e , T k ). Thus, we have shown that s∗ is not robust in (G, T k ).
c1 = 3

A

A

A

A

...

c1 = 2

NA

NA

A

A

...

c1 = 1

A

A

A

A

...

c1 = 0

NA

A

A

A

...

(0, 0)(1, 0)(1, 1)(2, 1)(2, 2)(3, 2)(3, 3)

c2 = 0

A

A

A

...

c2 = 1

NA

A

A

...

c2 = 2

A

A

A

...

c2 = 3

NA

NA

A

...

...

Table 4: Equilibrium actions of L1, L2, and L3 types in (U e , T k ).

Case 2: CH type space
Next, consider the CH type space T c with a uniform distribution. We can show thats∗ is played with high probability in (U e , T c ). For instance, consider the equilibrium
action of player 1’s L2 type in {(1, 0), (1, 1)}. This type believes that player 2 iseither L0 or L1 type equally likely and that player 2 will play A with probability
1 − 1/2(2 − ε). Hence, his unique best response in {(1, 0), (1, 1)} is now given by A
for suﬃciently small ε (see Table 5). To formally show that A is played by any type
c1 = 3

A

A

A

A

...

c1 = 2

NA

A

A

A

...

c1 = 1

A

A

A

A

...

c1 = 0

NA

A

A

A

...

(0, 0)(1, 0)(1, 1)(2, 1)(2, 2)(3, 2)(3, 3)

c2 = 0

A

A

A

...

c2 = 1

NA

A

A

...

c2 = 2

A

A

A

...

c2 = 3

NA

A

A

...

...

Table 5: Equilibrium actions of L1, L2, and L3 types in (U e , T c ).
with high probability, we use Theorem 4 of Strzalecki (2014):
12

Theorem 4.1. (Theorem 4 of Strzalecki 2014)
Suppose T admits beliefs such that for each i = 1, 2, inf m µi (cm )({cj : cm−1 ≤ cj }) <(2 − ε)/3 for any strictly increasing sequence (cm ) ∈ N∞ . Then there exists n ∈ Nsuch that all types choose to Attack at any θ ≥ (n, n).
Since T c satisfies the above condition (for any common distribution), it follows
that A is played by all types with high probability as long as ε is suﬃciently small.
Thus s∗ is played with high probability in (U e , T c ).
It is important to note that this result does not tell us whether s∗ is robust in(G, T c ), since we must check all the other ε-elaborations to establish the robustness.
Nevertheless, our main result states that a diﬀerent specification of type spaces
yields a diﬀerent suﬃcient condition for the robustness, and in particular, a risk
dominated equilibrium is shown to be robust in (G, T c ).

5

p-Dominance and robustness

5.1

p-DominanceLet us denote p = (p1 , p2 , ..., pI ) ∈ [0, 1]I . We first introduce the standard pdominance in G (Morris et al., 1995; Kajii and Morris, 1997). Let ϕi ∈ ∆(A−i )for each i ∈ I. An action profile a∗ ∈ A is p-dominant in G if, for each i ∈(∑I, ai ∈ Ai , and ϕi ∈ ∆(A−i ) with ϕi (a∗−i ) ≥ pi , we have a−i ∈A−i ui (a∗i , a−i ) −)ui (ai , a−i ) ϕi (a−i ) ≥ 0. Thus, a∗i is a best response for player i if he believes thatthe other players will play a∗−i with probability at least pi .
We analogously define p-dominance in (G, T ). Let λi (ci ) ∈ ∆(C−i × A−i ) denotea belief of type ci about other players’ types and actions, and write λi (ci )(c−i , a−i )for the probability assigned to (c−i , a−i ) ∈ C−i × A−i under λi (ci ). We say that∑λi (ci ) is consistent if a−i ∈A−i λi (ci )(·, a−i ) = µi (ci )(·).
Definition 5.1. A strategy profile s∗ is p-dominant in (G, T ) if, for each i ∈ I,∑
ci ∈ Ci \ {0}, ai ∈ Ai , and consistent λi (ci ) with c−i ∈C−i λi (ci )(c−i , s∗−i (c−i )) ≥ pi ,we have∑∑ ()
ui (s∗i (ci ), a−i ) − ui (ai , a−i ) λi (ci )(c−i , a−i ) ≥ 0.

c−i ∈C−i a−i ∈A−i

13This definition says that it is optimal for ci to play s∗i (ci ) if he believes consistently with µi (ci ) that the other players will play s∗−i with probability at least pi .
For instance, recall the coordinated attack model in Section 4. The model has aunique equilibrium in which all types choose A. Since (A, A) is 2/3-dominant, thisequilibrium is 2/3-dominant for any type space.
The following three properties of our p-dominance are all straightforward, andhence are stated without proofs.11
Proposition 5.1. In (G, T ):(1) s∗ is an equilibrium if and only if it is 1-dominant;(2) if s∗ is p-dominant, it is q-dominant for each p ≤ q;12(3) s∗ is p-dominant if all types of players follow a p-dominant action profile a∗in G, that is, s∗i (ci ) = a∗i for each ci ∈ Ci and i ∈ I.

5.2

Main result

Here is our main result.
Theorem 5.1. A p-dominant equilibrium of (G, T ) is robust if there exists a threshold level c̄ ∈ N such that, for each i ∈ I,supci >c̄

∑µi (ci )(c−i ) <{c−i :c̄≤cj ,∃j̸=i}

I(1 − pi ).
I −1

Proof. See the Appendix.
This condition is satisfied if there is a certain threshold level c̄ such that all
higher-level types than c̄ put suﬃciently small probability (decreasing in p) on someother players having a level higher than or equal to c̄. With the level-k type space,this condition reduces to “pi < 1/I for each i ∈ I.” In contrast, if there is a11

Unlike the standard p-dominance, there does not necessarily exist some p ∈ [0, 1)I that makes

a strict equilibrium p-dominant. To avoid this, we can generalize p-dominance by allowing eachprobability pi to depend on each type ci . With some notational complication, this generalization
gives us a slightly stronger result.
12p ≤ q means pi ≤ qi for each i ∈ I.

14

probability distribution over reasoning levels, the sum of conditional probabilities
on slightly lower-level types vanishes as a player’s reasoning level increases. Thus,
the CH type space satisfies the inequality for each pi ∈ [0, 1). Consequently, the two
prominent type spaces exhibit a great diﬀerence in the existence of robust equilibria.
Corollary 5.1. A p-dominant equilibrium of (G, T k ) is robust if pi < 1/I for eachi ∈ I.
Corollary 5.2. A p-dominant equilibrium of (G, T c ) is robust for each p ∈ [0, 1)I .
As demonstrated in the existing literature, non-robustness is often caused by thecontagion argument through higher-order beliefs. Intuitively, player’s FDR restricts
the dependence of his best response on higher-order beliefs. Hence, the behavior of
low-level types continuously changes in any close model. However, the impact ofhigher-order beliefs can be significant if player’s reasoning level is high because thebehavior of Lk type depends on the behavior of Lk-1 type, which in turn depends
on the behavior of Lk-2 type, and so on. Our main result shows that such higherorder dependence can be (uniformly) restricted by an upper-bound on the first-orderbeliefs of high-level types.
Theorem 5.1 is closely related to Theorem 4 of Strzalecki (2014), which gives anupper-bound on beliefs to make the risk dominated equilibrium continuously playedin the email game. Our result further develops his insight that beliefs about the opponents’ reasoning levels have a significant impact on players’ strategic interaction.
A main diﬀerence is that our bound is uniform to all the close models (including theemail game elaboration), and thus it does not depend on the details of information
structure. Instead, our bound is a function of the “strength” of equilibrium (p)and the number of players (I). As a result, the bound also tells us the thresholdvalue of p that makes an equilibrium robust for each type space. In addition, as thenumber of players increases, the bound becomes tighter so that we need a stronger
restriction on beliefs to have a robust equilibrium.
Theorem 5.1 is also related to Proposition 5.3 of Kajii and Morris (1997), which∑shows that a p-dominant action profile is KM robust if i∈I pi < 1. In particular,when p = (p, . . . , p), their condition is stronger than ours. On the other hand,15the relation is reversed when we have the level-k type space. Hence, there is noclear logical relation between two results. However, we can directly show that a
p-dominant equilibrium is robust under the Kajii and Morris’ condition.
Proposition 5.2. A p-dominant equilibrium of (G, T ) is robust if∑i∈I pi < 1.

Proof. See the Appendix.

6

Dominance solvability and robustness

Our second suﬃcient condition relates dominance solvability to robustness. First,
we define the dominance solvability of (G, T ). For L0 types, set Sin (0) = Ai for eachn ≥ 0. For other types, let Si0 (k) = Ai , and define Sin (k) recursively by ai ∈ Sin (k)if and only if
ai ∈ arg max
′
ai

∑

∑

ui (a′i , a−i )λi (c−i , a−i )c−i ∈C−i a−i ∈A−i

for some λi ∈ ∆(C−i ×A−i ) such that∑a−i ∈A−i λi (·, a−i ) = µi (k)(·) and λi (c−i , a−i ) >0 only if aj ∈ Sjn−1 (cj ) for each j ̸= i. We write Si∞ (k) =∩nn≥1 Si (k).

Definition 6.1. (G, T ) is dominance solvable if there exists n ∈ N such that
|Si∞ (k)| = 1 for each k ≥ n and i ∈ I.
The set Si∞ (k) corresponds to a set of (interim correlated) rationalizable actions
for the Lk type of player i (Bernheim 1984; Pearce 1984; Dekel et al. 2007). Hence,a model is dominance solvable if and only if suﬃciently high-level types have aunique rationalizable action. Since the rationalizability is clearly not a valid solution concept for our model, it is important to note that we introduce dominancesolvability to hypothetically consider the implication of having diﬀerently modeled
anchor behaviors.
By definition, if (G, T ) is dominance solvable, G is dominance solvable in theusual sense. While the converse is also true with the level-k type space, this does
not hold in general as the following example shows.
Example 6.1. Consider the following game given by Table 6 with arbitrarily chosenanchor behavior. First, this game is dominance solvable with a unique rationalizable
16

A2

B2

A1

2, 2

1, 1

B1

1, 2

5, 1

Table 6: Payoﬀ matrix.
action pair (A1 , A2 ). Let us introduce the CH type space with a Poisson distributionof parameter 1.5.13 Since each type of player 1 believes that player 2 is of L0 typewith probability at least 1/5, we have S1∞ (k) = {A1 , B1 } and S2∞ (k) = {A2 } foreach k ≥ 1. Thus, this model is not dominance solvable.
In a dominance solvable model, changes in anchor behavior have no eﬀect on
the equilibrium behavior of high-level types. That is, even if we replace the original anchor behavior with any other action, all the modified models yield the sameequilibrium action for suﬃciently high-level types. This straightforward observation
leads to the following proposition.
Proposition 6.1. A strict equilibrium of a dominance solvable model is robust.
Proof. See the Appendix.
Remark 6.1. The strictness of an equilibrium cannot be dropped because low-leveltypes may change their behavior dramatically otherwise. Consider the 3 × 3 game
given by Table 7 with fi = Ci for each i = 1, 2, and consider the level-k type space. It
A2

B2

C2

A1

2, 2

1, 1

1, 0

B1

1, 2

1, 1

1, 0

C1

0, 1

0, 1

1, 0

Table 7: Payoﬀ matrix.
is easy to check that this model is dominance solvable. This model has an equilibriums∗ such that s∗1 = (C1 , B1 , B1 , A1 , A1 , . . .), and s∗2 = (C2 , B2 , A2 , A2 , . . .). Let us
introduce the following ε-elaboration: Θ = {θ1 , θ2 , θ3 , θ4 }, Π1 = {{θ1 }, {θ2 , θ3 , θ4 }},13

Camerer et al. (2004) report that the median estimate of the Poisson parameter is 1.61 in 24

beauty contest games.

17Π2 = {{θ1 , θ2 }, {θ3 , θ4 }}, P (θ1 ) = P (θ2 ) = ε/2, and P (θ3 ) = P (θ4 ) = (1 − ε)/2.
Suppose that player 1’s anchor behavior is A1 at θ1 instead of C1 . Then A2 is aunique best response in {θ1 , θ2 } for the L1 type of player 2. Therefore, A1 (̸= s∗1 (2))
is a unique best response at any state for the L2 type of player 1 (see Table 8).
Hence, s∗ is not a robust equilibrium.
c1 = 2

A1

A1

A1

C1

c1 = 1c1 = 0θ1

θ2c2 = 0C2

c2 = 1

A2

θ3

θ4
C2

Table 8: Equilibrium actions of low-level types.

In the experimental literature, dominance solvable games such as the 2/3-guessing
game are often used. As long as we stick to such a class of games and the level-k
type space, our robustness has little bite as the following corollary shows.
Corollary 6.1. A strict equilibrium of (G, T k ) is robust if G is dominance solvable.

7

Discussion

This paper considers the robustness of equilibria in FDR models. An equilibrium isrobust if it can be played with high probability in any close model where the anchor
behavior remains the same and players know that with high probability. We provide
two diﬀerent suﬃcient conditions for the robustness: any p-dominant equilibrium
satisfying a certain condition on beliefs, and any strict equilibrium of dominancesolvable models.
One may think that our robustness is less appealing because experimental evidence suggests that people have a very low depth of reasoning. However, notice thefact that lab experiments usually exclude experienced players from their subjects.
In addition, learning from past experiences improves reasoning ability, and often

18

leads to an equilibrium play.14 Hence, beyond the ideally controlled environment,
we have no strong evidence against the existence of players who can reason almost
indefinitely. Our result is especially meaningful when analysts cannot exclude thepossibility of players being in such an equilibration process.
Finally, an implicit assumption of this paper is that each player has no diﬃcultyin modeling a complicated anchor behavior.15 Some may argue that players, whoare bounded rational as in the FDR model, may not describe anchor behavior to be
as complex as those in the email game elaboration. However, we think that player’s
FDR and the limited ability of modeling anchor behavior are two diﬀerent concepts.
This paper focuses on the former to examine its implication as clearly as possible.
Future work may investigate a unified framework that accommodates both conceptsof bounded rationality.

Appendix: Omitted proofs
Proof of Theorem 5.1
Proof. Take any p-dominant equilibrium s∗ of (G, T ). If pi = 1 for some i, then
the theorem vacuously holds. Suppose p ∈ [0, 1)I in the sequel. Fix any ε > 0 and
ε-elaboration U ∈ E(G, ε). The proof consists of four steps.
Step 1. Construct events in which s∗ is played in equilibrium.
Let E(i,0) ≡ ΩU for each i ∈ I, and inductively define E(i,k) as follows:




∑
E(i,k) ≡ θ ∈ Θ :µi (k)(c−i )P (Ec−i | πi (θ)) ≥ pi ∩ E0{c−i :cj <k,∀j̸=i}

for each k > 0 and i ∈ I, where Ec−i =∩j̸=i E(j,cj ) and E0 = ΩU .

The following

lemma shows that this is the event we are looking for.
Lemma A.1. (U, T ) has an equilibrium in which the Lk type of player i plays s∗i (k)in E(i,k) for each k ≥ 0 and i ∈ I.
14

In fact, Ho et al. (2013) and Ho and Su (2013) propose models that allow players in FDR

models to update their reasoning levels and beliefs.
15
I thank Michihiro Kandori for pointing out this issue.

19Proof. This proof proceeds similarly to Lemma 5.2 of Kajii and Morris (1997). SinceE(i,0) = ΩU for each i ∈ I, we have σi (0, θ) = s∗i (0) for each θ ∈ E(i,0) and i ∈ I. Let
us construct the modified model of (U, T ) where each player’s strategy is restricted
as follows: σi (k, θ) = s∗i (k) for each θ ∈ E(i,k) , k > 0, and i ∈ I. That is, in themodified model, the Lk type of player i can freely choose his action in Θ \ E(i,k) ;however, he must choose σi (k, θ) = s∗i (k) at any θ ∈ E(i,k) . Clearly, each restricted
strategy of the modified game can be considered as a strategy of (U, T ). We can findan “equilibrium” σ ∗ of this modified model such that σi∗ (k, θ) satisfies the equilibrium
condition in (U, T ) for each θ ∈ Θ \ E(i,k) , k > 0, and i ∈ I. We show that σ ∗ isalso an equilibrium of (U, T ). Let us consider the Lk type of player i. By definition,∗σi∗ (k, θ) is a best response to σ−i
at any θ ∈ Θ \ E(i,k) . It remains to show the caseof θ ∈ E(i,k) . Let us introduce λi (k) ∈ ∆(C−i × A−i ) such that for each c−i ∈ C−i∗
and a−i ∈ A−i , λi (k)(c−i , a−i ) = µi (k)(c−i ) · P ({θ′ ∈ Θ : σ−i(c−i , θ′ ) = a−i } | πi (θ)).

Then by our construction of σ ∗ , we have∑c−i ∈C−i=λi (ci )(c−i , s∗−i (c−i ))∑∗µi (k)(c−i ) · P ({θ′ ∈ Θ : σ−i(c−i , θ′ ) = s∗−i (c−i )} | πi (θ))

c−i ∈C−i≥∑µi (k)(c−i ) · P (Ec−i | πi (θ)) ≥ pi .

c−i ∈C−i

Since s∗ is p-dominant in (G, T ), it follows that for each ai ∈ Ai ,
∑ ∑

(
)
∗
ui s∗i (k), σ−i(c−i , θ′ ) µi (k)(c−i )P (θ′ | πi (θ))

c−i ∈C−i θ ′ ∈Θ

=

∑

∑

ui (s∗i (k), a−i )λi (k)(c−i , a−i )c−i ∈C−i a−i ∈A−i

≥

∑

∑

ui (ai , a−i )λi (k)(c−i , a−i )c−i ∈C−i a−i ∈A−i

=

∑ ∑

)
(
∗
(c−i , θ′ ) µi (k)(c−i )P (θ′ | πi (θ)).
ui ai , σ−ic−i ∈C−i θ ′ ∈Θ∗in E(i,k) . Therefore, σ ∗ is an equilibrium ofThus, s∗i (k, θ) is a best response to σ−i(U, T ) that satisfies the desired property.
Step 2. Find a lower-bound for each P (E(i,k) ).
First, we introduce the notion of common p-belief following Monderer and Samet20

(1989). Let Fi denote a σ-algebra generated by Πi for each i ∈ I. The p-belief
operator is defined by: Bip (E) ≡ {θ ∈ Θ : P (E | πi (θ)) ≥ p} for each (measurable)
event E. Thus, Bip (E) is a set of states in which player i believes with probability
at least p that the event E occurs. For each p ∈ [0, 1]I , let us write B∗p (E) =
∩
pi
p n
p
p n−1
(E)) for each n ∈ N, where [B∗p ]0 (E) = E.
i∈I Bi (E), and [B∗ ] (E) = B∗ ([B∗ ]We say that an event E is common p-belief at θ if, at θ, E is believed by eachplayer with probability at least pi , it is believed by each player with probability atleast pi that E is believed by each player with probability at least pi , and so on, ad
infinitum.
Definition A.1. An event E is common p-belief at θ if θ ∈ C p (E) ≡An event E is said to be simple if E =∩p nn≥1 [B∗ ] (E).

∩i∈I Ei with Ei ∈ Fi for each i ∈ I.

For instance, the event ΩU is simple. Kajii and Morris (1997) uncover a surprising
relationship between the probability of a simple event and the probability of anevent in which that simple event is common p-belief:Lemma A.2. (Proposition 4.2 of Kajii and Morris 1997)∑
If i∈I pi < 1, any simple event E satisfies
(
)
1
−
min
p
∑ i∈I i .
P (C p (E)) ≥ 1 − (1 − P (E))
1 − i∈I pi
The following lemma gives a loose lower-bound for the probability of an event[B∗p ]n (E).16
Lemma A.3. (Lemma B of Kajii and Morris 1997)
For each p ∈ [0, 1)I and event E, we have(
P ([B∗p ]n (E)) ≥ 1 −

∑

pi
1+
1 − pii∈I)n(1 − P (E)).

When we have the level-k type space, E(i,k) can be written as E(i,k) = {θ ∈ Θ :P (Ec−i | πi (θ)) ≥ pi } ∩ E0 = Bipi (Ec−i ) ∩ E0 , where cj = k − 1 for each j ̸= i. Let usb(i,k) . The following lemma provides three properties of Eb(i,k) .
denote this E(i,k) by E
Lemma A.4. The following three properties hold.
16

This bound is loose since it goes −∞ as n → ∞ whenever P (E) < 1.

21

b(i,k) is decreasing in k;(1) Eb(i,k) ⊆ E(i,k) ;(2) E(3) [B∗p ]k (E0 ) ⊆∩bi∈I E(i,k) .

Proof. All of the three properties are shown by mathematical induction.
b(i,1) = B pi (E0 ) ∩ E0 ⊆ E0 = Eb(i,0) . Suppose Eb(i,k+1) ⊆ Eb(i,k) for each(1) If k = 0, Eipi ∩b(i,k+2) = B pi (∩ Ebbbi ∈ I. Then Eij̸=i (j,k+1) ) ∩ E0 ⊆ Bi ( j̸=i E(j,k) ) ∩ E0 = E(i,k+1) . ∥b(i,0) = E0 . Suppose that the claim holds up to k for(2) By definition, E(i,0) = E
b(j,c ) ⊆ E(j,c ) for each 0 ≤ cj ≤ k and j ̸= i, we haveeach i ∈ I. Then since E
j
j
{
}
∑
bc | πi (θ)) ≥ pi ∩ E0 ⊇ {θ ∈ Θ :E(i,k+1) ⊇ θ ∈ Θ : c−i ∈C−i µi (k + 1)(c−i )P (E−i∩ bb
P ( j̸=i E(j,k) | πi (θ)) ≥ pi } ∩ E0 = E(i,k+1) . The second inclusion follows by (1). ∥∩b(i,k) . Then we have Eb(i,k+1) =(3) This holds if k = 0. Suppose [B∗p ]k (E0 ) ⊆ i∈I E∩b(j,k) )∩E0 ⊇ B pi ([B∗p ]k (E0 ))∩E0 . Since E0 is simple, we have ∩ EbBipi ( j̸=i E
ii∈I (i,k+1) ⊇
∩
pi
p k
p k+1
(E0 ). ∥i∈I Bi ([B∗ ] (E0 )) ∩ E0 = [B∗ ]Combining above results gives a lower-bound for P (E(i,k) ).
Step 3. Find a tighter lower-bound for P (E(i,k) ) with k ≥ c̄.
By our assumption, there exists c̄ ∈ N such that supk>c̄∑{c−i :c̄≤cj ,∃j̸=i} µi (k)(c−i ) <I(1 − pi )/(I − 1) for each i ∈ I. By definition of the supremum, we can take ω > 0∑such that {c−i :c̄≤cj ,∃j̸=i} µi (k)(c−i ) < (I − (I + ω)pi )/(I − 1) for each k > c̄ andi ∈ I. Define φ ≡ I/(I + ω), and τ ≡ φ/I = 1/(I + ω). Then we have 0 < φ < 1,
and 0 < τ < 1/I. By using φ and τ , let us inductively construct an event F(i,n) for
each n = 0, 1, . . . as follows:∩∩ bτ
F(i,0) = F0 ≡ B∗φ ( i∈I E(i,c̄) ), and F(i,n+1) ≡ Bi (Fn ) ∩ F0 where Fn ≡j̸=i F(j,n) .
We show that F(i,n) is a subset of E(i,c̄+n) under our hypothesis.
Lemma A.5. F(i,n) ⊆ E(i,c̄+n) for each n ≥ 0 and i ∈ I.
Proof. We show this lemma by induction. By the definition of F(i,0) , we have F(i,0) ⊆E(i,c̄) for each i ∈ I. Suppose that the lemma holds up to n. Take any θ ∈ F(i,n+1) .

22

Let t = c̄ + n + 1. Then we have∑µi (t)(c−i )P (Ec−i | πi (θ))

c−i ∈C−i∑={c−i :0≤cj <c̄,∀j̸=i}

∑

≥

∑

µi (t)(c−i )P (Ec−i | πi (θ)) +{c−i :c̄≤cj ,∃j̸=i}

((

∑

∩ b
µi (t)(c−i )P ( i∈I E(i,c̄) | πi (θ)) +{c−i :0≤cj <c̄,∀j̸=i}>µi (t)(c−i )P (Ec−i | πi (θ))µi (t)(c−i )P (Fn | πi (θ)){c−i :c̄≤cj ,∃j̸=i}

)
)
((
)
)
(I + ω)pi − 1 /(I − 1) · φ + I − (I + ω)pi /(I − 1) · τ = pi .

The weak inequality follows since F(i,n) is decreasing in n, and hence F(i,n) ⊆ E(i,k)
for each 0 ≤ k ≤ c̄ + n by our induction hypothesis. The strict inequality follows∩b(i,c̄) ) and θ ∈ F(i,n+1) ⊆ B τ (Fn ). Thus, we have θ ∈since θ ∈ F0 ⊆ B∗φ ( i∈I E
i
E(i,c̄+n+1) .
P (F(i,n) ) gives a new lower-bound for P (E(i,c̄+n) ).
Step 4. Combine the obtained results.
Take any δ > 0, and letε≤ (Iφ
1 + 1−φ

)(

1+

δ
∑

)c̄ ( 1−τ ) .
pii∈I 1−pi
1−IτObserve that by Lemmas A.3 and A.4, we have
(
(
)
)(
)c̄
∑ pi
)
∩ b
Iφ
Iφ (
P (F0 ) ≥ 1− 1+
1−P ( i∈I E(i,c̄) ) ≥ 1−ε 1+
1+
.
1−φ
1−φ
1 − pii∈I
Then since Fn is decreasing in n, F(i,0) ⊆ E(i,k) if k ≤ c̄, and∑i∈I τ < 1, it follows

from Lemmas A.2 and A.5 that for each type profile c,( 1−τ )∩P ( i∈I E(i,ci ) ) ≥ P (C τ (F0 )) ≥ 1 − (1 − P (F0 ))≥ 1 − δ.
1 − Iτ
By Lemma A.1, there exists an equilibrium σ ∗ of (U, T ) with Pσ∗ (s∗ (c)|c) ≥ 1 − δfor each type profile c. Since U is arbitrary chosen from E(G, ε), we are done.

Proof of Proposition 5.2∑
Proof. Let s∗ denote a p-dominant equilibrium of (G, T ) with i∈I pi < 1. Fix any∑
δ > 0, and let ε ≤ δ(1 − i∈I pi )/(1 − mini∈I pi ). Take any U ∈ E(G, ε). Fix any
23∑type profile c, and let ĉ = maxi∈I ci . Since i∈I pi < 1, it follows from Lemmas∩∩b(i,ĉ) ) ≥ P (C p (E0 )) ≥ 1 − δ. Then by
A.2 and A.4 that P ( i∈I E(i,ci ) ) ≥ P ( i∈I E
Lemma A.1, there exists an equilibrium σ ∗ of U such that Pσ∗ (s∗ (c)|c) ≥ 1 − δ, as
desired.

Proof of Proposition 6.1
Proof. Let s∗ denote a strict equilibrium of (G, T ). By our assumption, there existsn ∈ N such that Si∞ (k) is a singleton for each k ≥ n and i ∈ I. Since s∗ is a strict
equilibrium, for each i ∈ I and 1 ≤ k < n, we can take pki ∈ [0, 1) such that for∑each ai ∈ Ai and λi (k) ∈ ∆(C−i × A−i ) with a−i ∈A−i λi (k)(·, a−i ) = µi (k)(·) and∑k∗c−i ∈C−i λi (k)(c−i , s−i (c−i )) ≥ pi ,
∑

∑ (

)
ui (s∗i (k), a−i ) − ui (ai , a−i ) λi (k)(c−i , a−i ) ≥ 0.

c−i ∈C−i a−i ∈A−i

Let pi = max1≤k<n pki for each i ∈ I. Fix any δ > 0, and let ε ≤ δ/(1 +∑i∈I pi /(1 −pi ))n−1 . Consider any ε-elaboration U ∈ E(G, ε). By using this pi , construct theb(i,k) for each k > 0 as in the proof of Theorem 5.1. Then, by Lemmas A.1 andevent E∩b(i,n−1)
A.4, there exists an equilibrium σ ∗ of (U, T ) in which s∗i (k) is played in i∈I E
by the Lk type of player i for each 1 ≤ k < n and i ∈ I. By Lemmas A.3 and A.4,∩b(i,n−1) ) ≥ P ([B p ]n−1 (E0 )) ≥ 1 −δ. For the behavior of higher-level
we have P ( i∈I E∗
types than n − 1, we use the following claim.
Claim. σi∗ (k, θ) ∈ Si∞ (k) for each θ ∈ Θ, k > 0, and i ∈ I.
Proof. This follows by translating a distribution over states into a distribution overother players’ action profiles. Consider the Lk type of player i and take any θ ∈ Θ.
Since σ ∗ is an equilibrium of (U, T ), for each ai ∈ Ai ,
∑ ∑ (

)
∗
∗
(c−i , θ′ )) µi (k)(c−i )P (θ′ | πi (θ)) ≥ 0.
(c−i , θ′ )) − ui (ai , σ−iui (σi∗ (k, θ), σ−ic−i ∈C−i θ′ ∈ΘLet us define λ′i ∈ ∆(C−i × A−i ) such that for each c−i ∈ C−i and a−i ∈ A−i ,∗(c−i , θ′ ) = a−i } | πi (θ)). Then the above
λ′i (c−i , a−i ) = µi (k)(c−i ) · P ({θ′ ∈ Θ : σ−iinequality can be written as
∑

∑ (

)
ui (σi∗ (k, θ), a−i ) − ui (ai , a−i ) λ′i (c−i , a−i ) ≥ 0 for each ai ∈ Ai .

c−i ∈C−i a−i ∈A−i

24

Hence, the claim is true for L1 types since Sj∞ (0) = Aj for each j ̸= i. Suppose now
that the claim holds up to k − 1 for each i ∈ I. Then we have λ′i (c−i , a−i ) > 0 only if
aj ∈ Sj∞ (cj ) for each j ̸= i, that is, σi∗ (k, θ) ∈ Si∞ (k). By our induction hypothesis,
the claim follows. ∥Since s∗i (k) ∈ Si∞ (k) for each k ≥ 0 and i ∈ I, the above claim and our assumption∩b(i,n−1) . Consequently,
imply that higher-level types than n−1 must follow s∗ in i∈I E
we have Pσ∗ (s∗ (c)|c) ≥ 1 − δ for each type profile c.

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