multiple treatments with strategic interaction · function can be relaxed either when strategic...
TRANSCRIPT
Multiple Treatments with Strategic Interaction
⇤
Sukjin HanDepartment of Economics
University of Texas at [email protected]
First Draft: February 23, 2016This Draft: April 20, 2017
Abstract
We develop an empirical framework in which we identify and estimate the e↵ects oftreatments on outcomes of interest when the treatments are the results of strategic in-teraction (e.g., bargaining, oligopolistic entry, interaction in the presence of peer e↵ects).We consider a model where agents play a discrete game with complete information whoseequilibrium actions (i.e., binary treatments) determine a post-game outcome in a non-separable model with endogeneity. Due to the simultaneity in the first stage, the modelas a whole is incomplete and the selection process fails to exhibit the conventional mono-tonicity. Without imposing parametric restrictions or large support assumptions, thisposes challenges in recovering treatment parameters. To address these challenges, wefirst analytically characterize regions that predict equilibria in the first-stage game withpossibly more than two players, whereby we find a certain monotonic pattern of theseregions. Based on this finding, we derive bounds on the average treatment e↵ects (ATE’s)under nonparametric shape restrictions and the existence of excluded variables. We alsointroduce and point identify a multi-treatment version of local average treatment e↵ects(LATE’s).
JEL Numbers: C14, C35, C57Keywords: Multiple treatments, strategic interaction, endogeneity, heterogeneous treat-ment e↵ects, average treatment e↵ects, local average treatment e↵ects.
1 Introduction
We develop an empirical framework in which we identify and estimate the heterogeneouse↵ects of treatments on outcomes of interest where the treatments are the results of strategicinteraction (e.g., bargaining, oligopolistic entry, interaction in the presence of peer e↵ects orstrategic e↵ects). Treatments are determined as an equilibrium of a game and these strategic
⇤The author is grateful to Tim Armstrong, Steve Berry, Jorge Balat, Aureo de Paula, Phil Haile, KaramKang, Juhyun Kim, Yuichi Kitamura, Konrad Menzel, Francesca Molinari, Azeem Shaikh, Jesse Shapiro,Dean Spears, Ed Vytlacil, Haiqing Xu, and participants in the 2016 Texas Econometrics Camp, the 2016North American Summer Meeting of the Econometric Society, Interactions Conference 2016 at NorthwesternUniversity, and seminars at Yale, Brown, UBC, and UNC for helpful comments and discussions.
1
decisions of players endogenously a↵ect common or player-specific outcomes. For example,one may be interested in the e↵ects of newspaper entry on local political behaviors, the e↵ectsof entry of carbon-emitting companies on local air pollution and health outcomes, the e↵ectsof the presence of potential entrants in nearby markets on pricing or investment decisions ofincumbents, the e↵ects of large supermarkets’ exit decisions on local health outcomes, andthe e↵ects of provision of limited resources where individuals make participation decisionsunder peer e↵ects as well as based on their own gains from the treatment. In these examples,ignoring strategic interaction in treatment selection processes may lead to biased, or at leastless informative, conclusions about the e↵ects of interest.
We consider a model where agents play a discrete game with complete information, whoseequilibrium actions (i.e., a profile of binary endogenous treatments) determine a post-gameoutcome in a nonseparable model with endogeneity. We are interested in various treatmentparameters in this model. In recovering the parameters, the setting of this paper poses severalchallenges. First, the first-stage game posits a structure in which binary dependent variablesare simultaneously determined, thus making the model as a whole incomplete. Second, dueto this simultaneity, the selection process does not exhibit the conventional monotonic prop-erty a la Imbens and Angrist (1994). Furthermore, we make no assumptions on the jointdistributions of the unobservables nor parametric restrictions on the payo↵ function of eachplayer and on how treatments a↵ect the outcome. In nonparametric models with multiplicityor/and endogeneity, identification may be achieved with excluded instruments of large sup-port. Even though such a requirement is met in practice, estimation and inference can stillbe problematic (Khan and Tamer (2010), Andrews and Schafgans (1998)). We thus allowinstruments and other exogenous variables to be discrete and have small supports.
As a crucial first step to address these challenges, we analytically characterize regionsthat predict equilibria in the first-stage game. Under symmetry and strategic substitutabilityrestrictions on the payo↵ functions satisfy, we fully characterize the geometric properties ofthe regions in the space of unobservables and show that these regions exhibit a monotonicpattern in terms of the number of players who choose to take the action—e.g., the number ofentrants in an entry game. Complete analytical characterization of the equilibrium regionshas not been studied in the literature for the case of more than two players.1
Having characterized the equilibrium regions, we show how the model structure andthe data can be informative about treatment parameters, such as the average treatmente↵ects (ATE’s) and the local ATE (LATE’s). We first establish the bounds on the ATEand other related parameters with possibly discrete instruments of small support. We alsoshow that tighter bounds on the ATE can be obtained by introducing (possibly discrete)exogenous variables excluded from the first-stage game. This is especially motivated in theexamples mentioned above where externalities of strategic decisions typically arise. Wheninstruments have a joint support that is rectangular, we can derive sharp bounds as long asthe outcome variable is binary. Further, with continuous instruments of large supports, weshow that multiplicity and endogeneity become irrelevant and the ATE is point identified.To derive informative bounds, we impose nonparametric shape restrictions on the outcome
1To estimate payo↵ parameters, Berry (1992) partly characterizes equilibrium regions. To calculate thebounds on these parameters, Ciliberto and Tamer (2009) simulate their moment inequalities model that areimplied by the shape of these regions, especially the regions for multiple equilibria. While their approachesare enough for the purpose of their analyses, full analytical results are critical for the identification analysisof the current paper.
2
function, such as monotonicity and symmetry. This symmetry assumption on the outcomefunction can be relaxed either when strategic interaction occurs only within subgroups ofplayers, thus allowing for partial symmetry, or when the first-stage equilibrium selection isstable with the change of instruments. The latter is trivially guaranteed when instrumentsvary enough to o↵set the e↵ect of strategic substitutability. We also introduce and pointidentify a multi-treatment version of the LATE. The simultaneity in the selection processdoes not permit the usual equivalence result by Vytlacil (2002) between the specification of athreshold-crossing selection rule and Imbens and Angrist (1994)’s monotonicity assumption.A monotonic pattern found in the equilibrium regions, however, enables us to recover aversion of the LATE for a treatment of “dichotomous states.”
Partial identification in single-agent nonparametric triangular models with binary endoge-nous variables has been studied in Shaikh and Vytlacil (2011) and Chesher (2005), amongothers. Shaikh and Vytlacil (2011) provide bounds on the ATE in this setting. In a slightlymore general model, Vytlacil and Yildiz (2007) achieve point identification with an exogenousvariable that is excluded from the selection equation and has a large support. Our boundanalysis builds on these papers, but we study a multi-agent model with strategic interactionas a key component of the model. A few existing studies have extended a single-treatmentmodel to a multiple-treatment setting, but their models maintain monotonicity in the selec-tion process and none of them allow simultaneity among the multiple treatments resultingfrom agents’ interaction as we do in this paper.2
In interesting recent work, Pinto (2015), Heckman and Pinto (2015), and Lee and Salanie(2016) relax or generalize the monotonicity of the selection process in multi-valued treat-ments settings, but they generally consider di↵erent types of treatment selection mechanismsthan ours. Pinto (2015) and Heckman and Pinto (2015) introduce unordered monotonicity,while Lee and Salanie (2016) consider more general non-monotonicity. The latter paper doesmention entry games as one example of their treatment selection process, but by assumingknown payo↵s they sidestep the multiplicity of equilibria, which is one of the main focuses ofthis paper. Also, Lee and Salanie (2016)’s main focus is on point identification of marginaltreatment e↵ects, in which case continuous instruments are required and the ATE is recov-ered under a large support assumption. Their LATE under discrete instruments also di↵ersfrom ours; see Section 5 for details.
Without triangular structures, Manski (1997), Manski and Pepper (2000) and Manski(2013) also propose bounds on the ATE with multiple treatments under various monotonic-ity assumptions, including an assumption on the sign of treatment response. We take analternative approach that is more explicit about treatments interaction while remaining ag-nostic about the direction of treatment response. Our results suggest that, provided thatthere exist exogenous variation excluded from the selection process, the bounds calculatedfrom this approach can be more informative than those from their approach. Among thesepapers, Manski (2013) is the closest to ours in that it considers multiple treatments andmultiple agents with simultaneous interaction, but with an important di↵erence from ourapproach. The interaction in his setting is through individuals which are the unit of observa-tion. On the other hand, our setting features the interaction through the treatment/playerunit, and the unit of observation is markets or regions in which the first-stage game is played
2Heckman et al. (2006) consider multiple treatments in a framework of the marginal treatment e↵ects andlocal instrumental variables. See also Jun et al. (2011) for vector endogenous regressors in a triangular model.
3
and from which the outcome variable may emerge.Identification in models for binary games with complete information has been studied
in Tamer (2003) and Ciliberto and Tamer (2009), Bajari et al. (2010), among others. Thepresent paper contributes to this literature by considering post-game outcomes in the model,especially those that are not the game players’ direct concerns such as externalities. Asrelated work that considers post-game outcomes, Ciliberto et al. (2016) introduce a modelwhere firms make simultaneous decisions of entry and pricing upon entry. As a result, theirmodel can be seen as a multi-agent extension of a sample selection model. The model consid-ered in this paper, on the other hand, is a multi-agent extension of a model for endogenoustreatments. Unlike in Ciliberto and Tamer (2009) which assume a parametric distributionfor the unobservables in the game, we impose no restrictions for the joint distribution of allunobservables in the game and the outcome equation. Also a di↵erent approach to partialidentification under multiplicity is employed, as their approach is not applicable to the par-ticular setting of this paper. Lastly, it is typically hard to estimate bounds on underlyingpayo↵ parameters in complete information games even with two players (Ciliberto and Tamer(2009, p. 1813)), i.e., partial identification analysis in this setting may not be constructive.In contrast, our parameters of interest are functionals of the primitives (but excluding thegame parameters), and thus can be easily estimated even in a model with many players.
The paper is organized as follows. Section 2 introduces the model, the parameters ofinterest, and motivating examples. Section 3 delivers the main results of this paper. Westart by conducting the bound analysis on the ATE’s for a two-player case and a binarydependent variable as an illustration. Then we extend the results to a many-player case anda more general dependent variable. The analytical characterization of equilibrium regions formany players is presented in this section. Section 4 relaxes the symmetry assumption anddiscusses an extension of the model, point identification under large support, and relationshipto Manski (2013). The LATE parameter is introduced and identified in Section 5. Section 6presents a numerical illustration.
For a generic S-vector v ⌘ (v1, ...vS
), let v�s
denote an (S�1)-vector where s-th elementis dropped from v, i.e., v�s
⌘ (v1, ..., vs�1, vs+1, ..., vS
). When no confusion arises, we some-times change the order of entry and write v = (v
s
,v�s
) for convenience. For a multivariatefunction f(v), the integral
R
A
f(v)dv is understood as a multi-dimensional integral over a setA contained in the space of v. Vectors in this paper are row vectors.
2 Setup and Motivating Examples
Let D ⌘ (D1, ..., DS
) 2 D ✓ {0, 1}S be a S-vector of binary treatments and d ⌘ (d1, ..., dS)be its realization, where S is fixed. We assume that D is predicted as a pure strategy Nashequilibrium of a complete information game with S players who make entry decisions orindividuals who choose to receive treatments.3 Let Y be a post-game outcome that resultsfrom profile D of endogenous treatments. It can be an outcome common to all players or anoutcome specific to each player. Let (X,Z1, ..., ZS
) be exogenous variables in the model. Lets 2 {1, ..., S} be an index for players or interchangeably for treatments. We assume thereis a variable in Z
s
that is excluded from the equation for Y . For the partial identification
3While mixed strategy equilibria are not considered in this paper, it may be possible to extend the setupto incorporate mixed strategies following the argument in Ciliberto and Tamer (2009).
4
analysis, we may also assume there is a variable in X that is excluded from all the equationsfor D
s
. We consider a model of a semi-triangular system:
Y = ✓(D, X, ✏D), (2.1)
Ds
= 1 [⌫s(D�s
, Zs
) � Us
] , s 2 {1, ..., S}, (2.2)
where ✓ : RS+d
x
+1 ! R and ⌫s : RS�1+d
z
s ! R are functions nonseparable in their argu-ments. Implied from the complete information game, player s’s decision D
s
depends on thedecisions of all others D�s
in D�s
, and thus D is determined by a simultaneous system. Theunobservables (✏D, U1, ..., US
) are arbitrarily dependent to one another. Without loss of gen-erality we normalize the scalar U
s
to be distributed as Unif(0, 1) and ⌫s : RS�1+d
z
s ! (0, 1].The scalar ✏D is dependent on D. This unobservable is allowed to be a vector, which wouldbe important in considering Y itself as an equilibrium outcome of strategic interaction amongplayers. In this case, a vector of player-specific unobservables may enter the equation for Y ina reduced-form fashion.4 The unit of observation, indexed by market or geographical regioni, is suppressed in all the expressions. As mentioned, the model (2.1)–(2.2) is incomplete dueto the possible existence of multiple equilibria in the first-stage game of treatment selection.Moreover, the conventional monotonicity in the sense of Imbens and Angrist (1994) is notexhibited in the selection process due to simultaneity; see Section 5. The potential outcomeof receiving D = d can be written as
Yd = ✓(d, X, ✏d), d 2 D,
and ✏D =P
d2D 1[D = d]✏d.We are interested in the ATE and related parameters. With the average structural func-
tion (ASF)E[Yd|X = x] (2.3)
for vector d 2 D, the ATE can be written as
E[Yd � Yd0 |X = x] = E[✓(d, x, ✏d)� ✓(d0, x, ✏d0)], (2.4)
for d,d0 2 D. Another parameter of interest is the average treatment e↵ect on the treated(ATT): E[Yd � Yd0 |D = d
00, Z = z,X = x] for d,d0,d00 2 D. Unlike the ATT or thetreatment of the untreated in the single-treatment case, d00 does not necessarily equal d ord
0 here. One might also be interested in the sign of the ATE, which in this multi-treatmentcase is essentially establishing an ordering among the ASF’s. Lastly, we are interested in theLATE, which will be considered later after necessary concepts are introduced.
As an example of the ATE, we may choose d = (1, ..., 1) and d
0 = (0, ..., 0) to measuresome cancelling-out e↵ect, or we may be interested in more general nonlinear e↵ects. Anotherexample would be choosing d = (1,d�s
) and d
0 = (0,d�s
) for given d�s
. In the latterexample, we can learn interaction e↵ects of treatments, i.e., how much the average gain(ATE) from treatment s is a↵ected by other treatments: suppressing the conditioning on
4In Section 4, we discuss this in the context of player-specific Y . Having a scalar unobserved type Us
in each player’s decision process Ds
may be relatively innocuous since the interaction is explicitly modeledthrough D�s
which contains other players’ unobserved types.
5
X = x,
E⇥
Y1,d�s
� Y0,d�s
⇤
� Eh
Y1,d0�s
� Y0,d0�s
i
,
where Yd is interchangeably written as Yds
,d�s
here. For example with d�s
= (1, ..., 1) andd
0�s
= (0, ..., 0), complementarity between treatment s and all the other treatments can be
represented as E⇥
Y1,d�s
� Y0,d�s
⇤
� Eh
Y1,d0�s
� Y0,d0�s
i
> 0. Suppose we instead want to
focus on learning about complementarity between two treatments, while averaging over theother S � 2 treatments. This can be dealt with a more general framework of defining theASF and ATE.
Define a partial counterfactual outcome as follows: with a partition D = (D1,D2) 2D1 ⇥D2 = D and its realization d = (d1,d2),
Yd1,D2 ⌘
X
d22D2
1[D2 = d
2]Yd1,d2 . (2.5)
This is a counterfactual outcome that is fully observed once D
1 = d
1 is realized. Then foreach d
1 2 D1, the partial ASF can be defined as
E[Yd1,D2 ] =
X
d22D2
E[Yd1,d2 |D2 = d
2] Pr[D2 = d
2] (2.6)
and the partial ATE between d and d
0 as
E[Yd1,D2 � Yd10
,D2 ]. (2.7)
Using this concept, we can consider complementarity concentrated on, e.g., the first twotreatments: E
⇥
Y11,D2 � Y01,D2
⇤
> E⇥
Y10,D2 � Y00,D2
⇤
.In identifying these treatment parameters, suppose we attempt to recover the e↵ect of a
single treatment withD
1 being a scalar in model (2.1)–(2.2) conditional on D
2 = D�s
= d�s
,and then recover the e↵ects of multiple treatments by transitively using these e↵ects ofsingle treatments. This strategy is not valid since D2 is a function of D1 and also due tomultiplicity. Therefore, the approaches in the literature with single-treatment, single-agenttriangular models are not directly applicable and a new theory is demanded in this moregeneral setting.
We provide examples to which model (2.1)–(2.2) may apply. For concreteness, let Zs
=(Z1s,W ) and X = (X1,W ), where variables commonly present to all the equations arecollected in W .
Example 1 (Externality of airline entry). In this example, we are interested in the e↵ectsof airline competition on local air quality and health. Consider multiple airline companiesmaking entry decisions in local market i defined as a route that connects a pair of cities.Let Y
i
denote the air pollution levels or average health outcomes of this local market. LetD
s,i
denote airline s’s decision to enter market i, which is correlated with some unobservedcharacteristics of the local market that a↵ect Y
i
. The parameter E[Yd,i � Yd0,i
] captures thee↵ects of a market structure on pollution or health. One interesting question would be whetherthe ATE is nonlinear in the number of airlines as companies may operate more e�ciently
6
when facing more competition. As related work, Schlenker and Walker (2015) documenthow sensitively local health outcomes, such as acute respiratory diseases, are a↵ected by thechange in airline schedules. Economic activity variables, such as population and income, canbe included in W
i
, since they not only a↵ect the outcomes but also the entry decisions. Theexcluded variable X1i can be characteristics of the local market that directly a↵ect pollutionor health levels, such as weather shocks or the share of pollution-related industries in the localeconomy. We assume that, conditional on W
i
, these factors a↵ect the outcome but do notenter the payo↵ functions of the airlines. The instruments Z1s,i are cost shifters that a↵ectentry decisions. When Y
i
is a health outcome, pollution levels can be included in X1i.
Example 2 (Incumbents’ response to potential entrants). In this example, we are interestedin how market i’s incumbents respond to the threat of entry of potential competitors. LetYi
be an incumbent firm’s pricing or investment decision and Ds,i
be an entry decision byfirm s in “nearby” markets, which can be formally defined in each context. For example, inairline entry, nearby markets are defined as city pairs that share the endpoints with the citypair of an incumbent (Goolsbee and Syverson (2008)). That is, potential entrants are airlinesthat operate in one (or both) of the endpoints of the incumbent’s market i, but who havenot connected these endpoints. Then the parameter E[Yd,i � Yd0
,i
] captures the incumbent’sresponse to the threat, specifically whether it responds by lowering the price or making aninvestment. As in Example 1, Z1s,i are cost shifters and X1i are other factors a↵ecting priceof the incumbent, excluded from nearby markets, conditional of W
i
. The characteristics ofthe incumbent’s market can be a candidate of X1i, such as the distance between the endpointsof the incumbent’s market in the airline example.
Example 3 (Media and political behavior). In this example, the question is how media a↵ectspolitical participation or electoral competitiveness. In county or market i, either Y
i
2 [0, 1]can denote voter turnout, or Y
i
2 {0, 1} can denote whether an incumbent is re-elected or not.Let D
s,i
denote a market entry decision by local newspaper type s, which is correlated withunobserved characteristics of the county. In this example, Z1s,i is the neighborhood counties’population size and income, which is common to all players (Z11,i = · · · = Z1S,i). In theAppendix, we show that how the analysis in this paper can incorporate instruments that arecommon to all players. Lastly, X1i can include changes in voter ID regulations. Using alinear panel data model, Gentzkow et al. (2011) show that the number of newspapers in themarket significantly a↵ects the voter turnout but find no evidence whether it a↵ects the re-election of incumbents. More explicit modeling of the strategic interaction among newspapercompanies can be important to capture competition e↵ects on political behavior of the readers.
Example 4 (Food desert). Let Yi
denote a health outcome, such as diabetes prevalence, inregion i, and D
s,i
be the exit decision by large supermarket s in the region. Then E[Yd,i�Yd0,i
]measures the e↵ects of absence of supermarkets on health of the residents. Conditional onother factors W
i
, the instrument Z1s,i can include changes in local government’s zoning plansand X1i can include the region’s health-related variables, such as the number of hospitals andthe obesity rate. This problem is related to the literature on “food desert” (e.g., Walker et al.(2010)).
Example 5 (Ground water and agriculture). In this example, we are interested in the im-pact of access to groundwater on economic outcomes in rural areas (Foster and Rosenzweig
7
(2008)). In each Indian village i, symmetric wealthy farmers (of the same caste) make irri-gation decisions D
s,i
, i.e., whether or not to buy motor pumps, in the presence of peer e↵ectsand learning spillovers. Since ground water is a limited resource that is seasonally rechargedand depleted, other farmers’ entry may negatively a↵ects one’s payo↵. The adoption of thetechnology a↵ects Y
i
, which can be the average of local wages of peasants or prices of agri-cultural products, or a village development or poverty level. In this example, continuous orbinary instrument Z1s,i can be the depth to groundwater, which is exogenously given (Sekhri(2014)), or provision of electricity for pumping in a randomized field experiment. X1,i canbe village-level characteristics that villagers do not know ex ante or do not concern about.5
3 Partial Identification of the ATE
To characterize the bounds on the treatment parameters, we make the following assumptions.Let Z ⌘ (Z1, ..., ZS
) and U ⌘ (U1, ..., US
). Let X and Z be the supports of X and Z,respectively.
Assumption IN. (X,Z) ? (✏d,U) 8d 2 D.
Assumption E. (✏d,U) are continuously distributed 8d 2 D.
Assumption R. For any d,d0 2 D, either (a) ✏d = ✏d0 = ✏; or (b) F✏d|U = F
✏d0 |U .
Assumption R(a) and (b) are the rank invariance and rank similarity conditions, respec-tively (Chernozhukov and Hansen (2005)). We proceed with (a); we can easily extend theanalysis to case (b). We now impose shape restrictions on the outcome function ✓(d, x, ✏) andthe first-stage payo↵ ⌫s(d�s
, zs
). For the outcome function, we impose shape restrictions on
#(d, x;u) ⌘ E[✓(d, x, ✏)|U = u]
a.e. u instead of ✓(d, x, ✏) a.e. ✏. These restrictions on the conditional mean are weaker thanthose that are directly imposed on ✓(d, x, ✏). Unless otherwise noted, the assumptions belowhold for each s 2 {1, ..., S}, which statement is omitted for brevity. Let Z
s
be the support ofZs
. The next assumption is a monotonicity assumption.
Assumption M. (i) For every x 2 X , either #(1,d�s
, x;u) � #(0,d�s
, x;u) a.e. u 8d�s
2D�s
, or #(1,d�s
, x;u) #(0,d�s
, x;u) a.e. u 8d�s
2 D�s
;(ii) For all z
s
, z0s
2 Zs
, either ⌫s(d�s
, zs
) � ⌫s(d�s
, z0s
) 8d�s
2 D�s
and 8s 2 {1, ..., S}, or⌫s(d�s
, zs
) ⌫s(d�s
, z0s
) 8d�s
2 D�s
and 8s 2 {1, ..., S}.
Assumption M(i) can be stated in twofold: (a) for every x and d�s
, either #(1,d�s
, x;u) �#(0,d�s
, x;u) a.e. u, or #(1,d�s
, x;u) #(0,d�s
, x;u) a.e. u; (b) for every x, each in-equality in (a) holds for all d�s
. For an outcome function with a scalar index, ✓(d, x, ✏) =✓(µ(d, x), ✏), part (a) is implied by E[✓(t, ✏)|U = u] being strictly increasing (decreasing) int a.e. u.6 Functions that satisfy the latter assumption include strictly monotonic functions
5Especially in this example, the number of players/treatments Si
is allowed to vary across villages. Weassume in this case that players/treatments are symmetric (in a sense that becomes clear later) and ⌫1(·) =· · · = ⌫Si(·) = ⌫(·).
6A single-treatment version of the latter assumption appears in Vytlacil and Yildiz (2007) (AssumptionA-4), which is weaker than assuming ✓(t, ✏) is strictly increasing (decreasing) a.e. ✏; see Vytlacil and Yildiz(2007) for related discussions.
8
such as transformation models ✓(t, ✏) = r(t+✏) where unknown r(·) is a strictly increasing andfunctions that are not strictly monotonic such as limited dependent variables models ✓(t, ✏) =1[t � ✏] or ✓(t, ✏) = 1[t � ✏](t � ✏). There can be, however, functions that violate the latterassumption but satisfy part (a). For example, consider a threshold crossing model with a ran-dom coe�cient: ✓(d, x, ✏) = 1[�(✏)d�> � x�>] where �(✏) is nondegenerate. When �
s
� 0,
then E[✓(1,d�s
, x, ✏) � ✓(0,d�s
, x, ✏)|U = u] = Prh
x�
>
�
s
+d�s
�
>�s
�(✏) x�
>
d�s
�
>�s
|U = u
i
and
thus nonnegative a.e. u, and vice versa. Part (a) also does not impose any monotonicity of✓ in ✏ and thus ✏ is allowed to be a vector.
Part (b) of Assumption M(i) imposes mild uniformity. Uniformity is required across dif-ferent values of d�s
but not across s, which means that di↵erent treatments can have di↵erentdirections of monotonicity. More importantly, knowledge on the direction of the monotonicityis not necessary, unlike Manski (1997) or Manski (2013) where the semi-monotone treatmentresponse is assumed for possible multiple treatments. In Assumption M(ii), even though uni-formity is required not only across d�s
but also across s, it is justifiable especially when zs
ischosen to be of the same kind for all players. For example in an entry game, if z
s
is chosento be each player’s entry cost, then the payo↵s would decrease in their costs for all players.Note that this monotonicity is weaker than a conventional monotonicity that ⌫s(d�s
, ·) iseither non-decreasing or non-increasing in z
s
for all d�s
and s.
Assumption SS. For every zs
2 Zs
, ⌫s(d�s
, zs
) is strictly decreasing in each element ofd�s
.
Assumption SS asserts that the agents’ decisions are produced in a game of strategicsubstitutes in the first stage.
Assumption SY. (i) For every x 2 X , #(d, x;u) = #(d, x;u) a.e. u for any permutationd of d;(ii) For every z
s
2 Zs
, ⌫s(d�s
, zs
) = ⌫s(d�s
, zs
) for any permutation d�s
of d�s
.
Assumption SY imposes symmetry (or exchangeability) in the functions as long as theobserved characteristics remains the same. This assumption is useful to make our incompletemodel tractable. Assumption SY(i) is relaxed in Section 4.1. An assumption related to SY(i)is also found in Manski (2013). SY(ii) trivially holds in the two-player case and it becomescrucial with many players. SY(ii) is related to the assumption in classical entry games (e.g.,Berry (1992), Kline and Tamer (2012)), which imposes that the payo↵ of a player is a functionof the number of other entrants,7 or the “anonymity” assumption in large games (e.g., Kalai(2004), Menzel (2016)), which imposes that the payo↵ depends on the empirical distributionof other players’ decisions. In the language of Ciliberto and Tamer (2009), although SY(ii)restricts heterogeneity in the fixed competitive e↵ects (i.e., how each of other entrants a↵ectsone’s payo↵), the nonseparability between d�s
and zs
in ⌫s(d�s
, zs
) allows heterogeneityhow each player is a↵ected by other entrants; this heterogeneity is related to the variablecompetitive e↵ects.
Assumption NU. For each d�s
2 D�s
, ⌫s(d�s
, Zs
)|X is nondegenerate.
7This assumption is imposed as part of a monotonicity assumption (Assumption 3.2) in Kline and Tamer(2012). The “symmetry of payo↵s” has a di↵erent meaning in their paper.
9
Assumption NU is related to the exclusion restriction and the relevance condition of theinstruments Z
s
.
Theorem 3.1. In model (2.1)–(2.2), suppose Assumptions IN, E, R, M, SS, SY and NUhold. Then the sign of the ATE is identified, and the upper and lower bounds on the ASFand ATE with d, d 2 D are
Ld(x) E[Yd|X = x] Ud(x)
and
Ld(x)� Ud(x) E[Yd � Yd|X = x] Ud(x)� Ld(x)
where, for given d
† 2 D,
Ud†(x) ⌘ infz2Z
(
E[Y |D = d
†,Z = z, X = x] Pr[D = d
†|Z = z]
+X
d0 6=d†
infx
02XU
d†(x;d0)
E[Y |D = d
0,Z = z, X = x0] Pr[D = d
0 |Z = z]
)
,
Ld†(x) ⌘ supz2Z
(
E[Y |D = d
†,Z = z, X = x] Pr[D = d
†|Z = z]
+X
d0 6=d†
supx
02XL
d†(x;d0)
E[Y |D = d
0,Z = z, X = x0] Pr[D = d
0 |Z = z]
)
,
with XU
d†(x;d0) and XL
d†(x;d0) specified in (3.32)–(3.34) below.
Heuristically, the following is the idea of the bound analysis. For given d 2 D, consider
E[Yd|X] = E[Yd|Z, X] = E[Y |D = d,Z, X] Pr[D = d|Z]
+X
d0 6=d
E[Yd|D = d
0,Z, X] Pr[D = d
0|Z], (3.1)
where the first equality is by Assumption IN. In this expression, the counterfactual termE[Yd|D = d
0,Z, X] can be bounded as long as Y is bounded by a known interval (Manski(1990)). Instruments in Z that are excluded from the equation for Y can then be used tonarrow the bound. The goal is to derive tighter bounds on the ATT’s E[Yd|D = d
0,Z, X]in (3.1) by fully exploiting the structure of the model. These bounds then can be used toconstruct bounds on the ATE. Suppose S = 2 and ✓(d, x, ✏) = 1[µ(d, x) � ✏] for illustrativepurpose, and let µd(x) ⌘ µ(d, x) for simplicity. Suppose that we know the direction ofmonotonicity in Assumption M(i) that µ10(x) � µ00(x). Then we can derive the upper
10
bound on, e.g., E[Y00|D = (1, 0), Z,X] as
Pr[Y00 = 1|D = (1, 0), Z = z,X = x] = Pr[✏ µ00(x)|D = (1, 0),Z = z, X = x]
Pr[✏ µ10(x)|D = (1, 0),Z = z, X = x] (3.2)
= Pr[Y = 1|D = (1, 0),Z = z, X = x],
which is smaller than one, the upper bound without the knowledge of the direction. Below,in a general setting, we show that the direction of monotonicity can actually be identifiedfrom the data. Moreover, by additionally using variation of X, we exploit more sophisticatedmonotonicity than those involved in Assumption M(i). The initial motivation of the approachis similar to Shaikh and Vytlacil (2011) and Vytlacil and Yildiz (2007) in that it jointlyexploits the selection model and the existence of X excluded from the selection model todetermine how ✓(d, x, ✏) behaves. The procedure, however, di↵ers from theirs as we need todeal with the nonmononic treatments selection process and the incompleteness of the model.
3.1 Analysis with S = 2 and Binary Y
To illustrate how to determine the direction of monotonicity of the outcome function withrespect to the treatments, this section considers the simple case S = 2 with binary Y andscalar ✏; Section 3.2 considers the general case of many players with possibly non-binary Yand vector ✏. Then the model (2.1)–(2.2) is simplified as
Y = 1 [µ(D1, D2, X) � ✏] , (3.3)
D1 = 1⇥
⌫1(D2, Z1) � U1⇤
, (3.4)
D2 = 1⇥
⌫2(D1, Z2) � U2⇤
. (3.5)
We first define quantities that are identified directly from the data. For x 2 X and z, z0 2 Zwhere Z|
X=x
= Z by Assumption NU, define
h(z, z0, x) ⌘ E[Y |Z = z, X = x]� E[Y |Z = z
0, X = x] (3.6)
= Pr[Y = 1|Z = z, X = x]� Pr[Y = 1|Z = z
0, X = x],
which records the change in the distribution of Y as Z changes. Also, define
hD11(z, z0) ⌘ Pr[D = (1, 1)|Z = z]� Pr[D = (1, 1)|Z = z
0],
hD00(z, z0) ⌘ Pr[D = (0, 0)|Z = z]� Pr[D = (0, 0)|Z = z
0].
Let a function sgn{h} take values �1, 0, 1 when h is negative, zero and positive, respectively.Recall µ
d1d2(x) ⌘ µ(d1, d2, x).
Lemma 3.1. Suppose S = 2 with model (3.3)–(3.5). Under the assumptions of Theorem3.1, for z, z0 2 Z such that hD11(z, z
0) > 0 and �hD00(z, z0) > 0 and for x 2 X ,
sgn�
h(z, z0, x)
= sgn {µ11(x)� µ01(x)} = sgn {µ10(x)� µ00(x)} .
Given the result of this lemma, since h, hD11 and hD00 can be recovered from the data, werecover the signs of µ11(x) � µ01(x) and µ10(x) � µ00(x), i.e., the direction of monotonicity
11
in Assumption M(i). Then we can calculate bounds on the unknown conditional mean terms(the ATT’s) as seen in (3.2). Under Assumption NU, the existence of (z, z0) such thathD11(z, z
0) > 0 and �hD00(z, z0) > 0 is guaranteed by Assumption M(ii), i.e., for (z, z0) such
that hD11(z, z0) > 0, it must be �hD00(z, z
0) > 0 by Assumption M(ii) (and vice versa). M(ii)can be tested from the data.8
Under Assumption SS, (1, 0) and (0, 1) are the values of D that can be realized as possiblemultiple equilibria; see, e.g., Figure 1(a). Given this knowledge, we define
hM
(z, z0, x) ⌘ Pr[Y = 1,D 2 {(1, 0), (0, 1)}|Z = z, X = x]
� Pr[Y = 1,D 2 {(1, 0), (0, 1)}|Z = z
0, X = x],
and
h11(z, z0, x) ⌘ Pr[Y = 1,D = (1, 1)|Z = z, X = x]� Pr[Y = 1,D = (1, 1)|Z = z
0, X = x],
h00(z, z0, x) ⌘ Pr[Y = 1,D = (0, 0)|Z = z, X = x]� Pr[Y = 1,D = (0, 0)|Z = z
0, X = x],
so that h(z, z0, x) = h11(z, z0, x) + h00(z, z0, x) + hM
(z, z0, x). Making use of the symmetryassumption (SY(i)), combining D = (1, 0) and D = (0, 1) will manage the multiple equilibriaproblem.
We define regions in the support U ⌘ (0, 1]2 of U = (U1, U2) that predict pure strategyequilibria (1, 1), (0, 0), (1, 0), and (0, 1) for D: with ⌫s
d�s
(zs
) ⌘ ⌫s(d�s
, zs
) for brevity,
R11(z) ⌘�
U : U1 ⌫11(z1), U2 ⌫21(z2)
,
R00(z) ⌘�
U : U1 > ⌫10(z1), U2 > ⌫20(z2)
,
R10(z) ⌘�
U : U1 ⌫10(z1), U2 > ⌫21(z2)
,
R01(z) ⌘�
U : U1 > ⌫11(z1), U2 ⌫20(z2)
.
By Assumption SS, R11 and R00 are regions of a unique equilibrium and R10 [R01 containsregions of multiple equilibria; formal proofs for this argument and other arguments belowconcerning the equilibrium regions can be found in Proposition 3.1 in a general setup ofS � 2. By Assumption IN, suppressing the arguments (z, z0, x) on the l.h.s.,
h11 + h00 = Pr[✏ µ11(x),U 2 R11(z)]� Pr[✏ µ11(x),U 2 R11(z0)]
+ Pr[✏ µ00(x),U 2 R00(z)]� Pr[✏ µ00(x),U 2 R00(z0)], (3.7)
where the equality uses R11 and R00 being disjoint and regions of unique equilibrium. ByAssumption SY(i) that µ10 = µ01, we have
hM
= Pr[✏ µ10(x),U 2 R10(z) [R01(z)]� Pr[✏ µ10(x),U 2 R10(z0) [R01(z
0)]. (3.8)
The main insight to obtain the results of Lemma 3.1 is as follows. By (3.6), h captures howPr[Y = 1|Z = z, X = x] changes in z. By h = h11+h00+h
M
and (3.7)–(3.8), such a changecan be translated into shifts in the regions of equilibria while the thresholds of ✏ in each of
8Obviously, even though Assumption M(ii) is violated, the results of the lemma, hence that of Theorem3.1, will follow as long as there exists (z, z0) that satisfy hD
11(z, z0) > 0 and �hD
00(z, z0) > 0.
12
0 1
1R10(z)
R01(z)
U2
U1
(a) When Z = z
0 1
1
R10(z0)
R01(z0)
U2
U1
(b) When Z = z
0
0 1
1
�+(z, z0)
��(z0, z)
U2
U1
(c) Di↵erence of (a) and (b)
Figure 1: Illustration of hM
in the proof of Lemma 3.1.
0 1
1R00(z)
R11(z)
U2
U1
(a) When Z = z
0 1
1R00(z0)
R11(z0)
U2
U1
(b) When Z = z
0
0 1
1
�+(z, z0)
��(z0, z)
U2
U1
(c) Di↵erence of (a) and (b)
Figure 2: Illustration of h11 + h00 the proof of Lemma 3.1.
h11, h00 and hM
remaining unchanged by the exclusion restriction. Therefore by inspectinghow Pr[Y = 1|Z = z, X = x] changes in z (i.e., the sign of h) relative to the changes inthe equilibrium regions R11 and R00 (i.e., the signs of hD11 and hD00), we recover the signs ofµ11(x) � µ01(x) and µ10(x) � µ00(x). In doing so, we use a crucial fact that the changes inthe region R10 [R01 are o↵set with the changes in R11 and R00.
To be specific, suppose that (z, z0) are chosen such that hD11(z, z0) > 0 and�hD00(z, z
0) > 0.Then by Assumption M(ii), R11(z) � R11(z0) and R00(z) ⇢ R00(z0). Then
�+(z, z0) ⌘ {R10(z) [R01(z)} \
�
R10(z0) [R01(z
0)
= R00(z0)\R00(z), (3.9)
��(z, z0) ⌘
�
R10(z0) [R01(z
0)
\ {R10(z) [R01(z)} = R11(z)\R11(z0), (3.10)
because, as z changes, an inflow of one region is an outflow of a region next to it. This setalgebra is illustrated in Figures 2–1. Then (3.8) becomes
hM
= Pr[✏ µ10(x),U 2 �+(z, z0)]� Pr[✏ µ10(x),U 2 ��(z, z
0)], (3.11)
by the following general rule: for a uniform random vector U and two sets B and B0 containedin U and for a r.v. ✏ and set A ⇢ E ,
Pr[✏ 2 A, U 2 B]� Pr[✏ 2 A, U 2 B0] = Pr[✏ 2 A, U 2 B\B0]� Pr[✏ 2 A, U 2 B0\B].(3.12)
13
Therefore by combining (3.11) with (3.7) applying (3.12) once more, we have
h(z, z0, x) =Pr[✏ µ11(x),U 2 ��(z, z0)]� Pr[✏ µ00(x),U 2 �+(z, z
0)]
� Pr[✏ µ10(x),U 2 ��(z, z0)] + Pr[✏ µ10(x),U 2 �+(z, z
0)]. (3.13)
Now, given Assumption E, Assumption M(i) holds with µ(1, d�s
, x) > µ(0, d�s
, x) for anyd�s
if and only if
h(z, z0, x) = Pr[µ01(x) ✏ µ11(x),U 2 ��(z, z0)] + Pr[µ00(x) ✏ µ10(x),U 2 �+(z, z
0)],
which is positive as is the sum of two probabilities. One can analogously show this for othersigns and we have the result of Lemma 3.1.9 Lastly, to gain e�ciency in determining the signof h(z, z0, x), define the integrated version of h as
H(x) ⌘ E[h(Z,Z 0, x)|hD11(Z,Z 0) > 0, hD00(Z,Z 0) < 0], (3.14)
then sgn{H(x)} = sgn {µ11(x)� µ01(x)} = sgn {µ10(x)� µ00(x)}.Now, consider calculating the upper bound on Pr[Y00 = 1|X = x]. For the chosen
evaluation point x, suppose H(x) � 0. Then by Lemma 3.1, µ00(x) µ10(x), µ00(x) µ01(x), and µ00(x) µ10(x) µ11(x). Recall µ00 µ10 implies that the upper boundon Pr[Y00 = 1|D = (1, 0),Z, X] is Pr[Y = 1|D = (1, 0),Z, X] by (3.2). Likewise, usingµ00 µ01 and µ00 µ11, we can calculate upper bounds on the other unobserved termsPr[Y00 = 1|D = d,Z, X] for d 6= (0, 0) in (3.1). Consequently we have
Pr[Y00 = 1|X = x] Pr[Y = 1|Z = z, X = x].
Likewise, we can derive the lower bounds on Pr[Y00 = 1|X = x] when H(x) 0.10
So far, we have not exploited the variation of X. Now we derive tighter bounds using thisvariation. For (possibly di↵erent) evaluation points x0, x1, x2 2 X , define
h(z, z0;x0, x1, x2) ⌘ h00(z, z0, x0) + h
M
(z, z0, x1) + h11(z, z0, x2). (3.15)
For x0, x1 and x2 to take di↵erent values, it would be important to have the existence of avariable in X that is excluded from the first-stage game, i.e., the ability of varying X givenZ. Note that for x0 = x1 = x2 = x, h(z, z0;x, x, x) = h(z, z0, x). Since Assumption M(i)only compares µ
d
s
,d�s
(x) and µd
0s
,d�s
(x0) for x = x0, the result of Lemma 3.1 and its proofstrategy cannot be applicable using h(z, z0;x0, x1, x2) when x0, x1 and x2 take di↵erent values.It would not be desirable to modify Assumption M(i) to incorporate monotonicity betweenµd
s
,d�s
(x) and µd
0s
,d�s
(x0) for x 6= x0, as it creates an overly stringent assumption. Instead,we propose a procedure that compares the signs of h(z, z0, x) and h(z, z0;x0, x1, x2) for somex, x0, x1 and x2 to infer the signs of µ
d
(x) � µd
(x0) for some x and x0. This procedure isunique to this multi-treatment setting with strategic interaction and thus distinct from howthe variation of X is exploited in a single-treatment setting as in Shaikh and Vytlacil (2011)
9Note that in deriving the result of the lemma, a player-specific exclusion restriction is not crucial and onemay be able to relax it.
10When H(x) � 0, the lower bounds on Pr[Y00 = 1|X = x] is trivially zero.
14
and Vytlacil and Yildiz (2007).11
Lemma 3.2. Suppose S = 2 with model (3.3)–(3.5). Suppose the assumptions of Theorem3.1 holds. For z, z0 2 Z such that hD11(z, z
0) > 0 and �hD00(z, z0) > 0 and for x0, x1, x2 2 X ,
suppose (3.15) is well-defined. For ◆ 2 {�1, 0, 1},(i) if sgn{h(z, z0;x0, x1, x2)} = sgn{µ10(x1)�µ11(x2)} = ◆, then sgn{µ10(x1)�µ00(x0)} = ◆;(ii) if sgn{h(z, z0;x0, x1, x2)} = sgn{µ00(x0)�µ10(x1)} = ◆, then sgn{µ11(x2)�µ10(x1)} = ◆.
This lemma is proved in a general setup later. The rules established in Lemma 3.2 canbe used to narrow the bounds previously obtained with variation of Z only. Analogous to(3.14), define
H(x0, x1, x2) ⌘ E[h(Z,Z 0;x0, x1, x2)|hD11(Z,Z 0) > 0, hD00(Z,Z 0) < 0]. (3.16)
Fix evaluation points x, x0 2 X and suppose H(x, x0, x0) 0. Also suppose H(x0) � 0, whichimplies µ11(x0) � µ10(x0) by Lemma 3.1 and (3.14). Then sgn{H(x, x0, x0)} = sgn{µ10(x0)�µ11(x0)} 2 {0,�1} therefore µ10(x0) µ00(x) by Lemma 3.2(i). Using this result, we canderive an lower bound on E[Y00|D = (1, 0),Z = z, X = x]:
Pr[Y00 = 1|D = (1, 0),Z = z, X = x] = Pr[✏ µ00(x)|D = (1, 0),Z = z, X = x]
� Pr[✏ µ10(x0)|D = (1, 0),Z = z, X = x0] (3.17)
= Pr[Y = 1|D = (1, 0),Z = z, X = x0].
Note that this bound cannot be achieved when x0 = x1 = x2, as the sign conditions in Lemma3.2(i) and (ii) fail. Therefore, X needs to have excluded variation given Z = z and Z = z
0.12
Without exploiting x0 6= x of X, the conditional probability in (3.17) only has a lower boundof zero (when H(x) � 0). Now, we can collect all x0 2 X that yields µ10(x0) µ00(x) andfurther shrink the bound in (3.17) by taking infimum over all x0 in this set. Other boundson Pr[Y00 = 1|D = d,Z, X] for d 6= (0, 0) can be derived in similar manners and can becombined based on (3.1) to calculate the bounds on Pr[Y00 = 1|X]. This will be explored infull generality in the next section.
3.2 General Analysis
In this section we prove the main theorem (Theorem 3.1) with the full model (2.1)–(2.2),in which Y may no longer be binary and the number of players may exceeds two. We firstintroduce a generalized version of the sign matching results (Lemma 3.1). Recall, for z, z0 2 Zand x 2 X ,
h(z, z0, x) ⌘ E[Y |Z = z, X = x]� E[Y |Z = z
0, X = x].
11In these papers, with a scalar treatment D, the sign of a single object h1(x0) + h0(x) (where h
d
(x) ⌘Pr[Y = 1, D = d|X = x, Z = z]�Pr[Y = 1, D = d|X = x, Z = z0]) would conveniently determine the directionmonotonicity between two outcome index functions evaluated at x and x0.
12The candidates for variables that are excluded from all the equations for Ds
are discussed in the contextof Examples 1–5 in Section 2. One can easily extend the analysis of this paper to a setting where there is novariable in X excluded from the D
s
equations but there are exogenous variables common to the Y equationand each D
s
equation (or common to the Y equation and all the Ds
equations); see Remark 3.4.
15
For k = 1, ..., S, let ek
be an S-vector of all zeros except the k-th element being a unity, andlet e0 ⌘ (0, ..., 0). For j = 0, ..., S, define ej ⌘
P
j
k=0 ek, which is an S-vector where the first jelements are unity and the rest are zero. For some positive integers n
s
, define a permutationfunction � : {n1, ..., nS
} ! {n1, ..., nS
}, which has to be a one-to-one function. For example,
✓
n1 n2 n3 n4 n5
�(n1) �(n2) �(n3) �(n4) �(n5)
◆
=
✓
1 2 3 4 52 1 5 3 4
◆
.
Let ⌃ be a set of all possible permutations. Define a set of all possible permutations ofe
j = (ej1, ..., ej
S
) as
Mj
⌘n
d
j : dj = (�(ej1), ...,�(ej
S
)) for any �(·) 2 ⌃o
(3.18)
for j = 0, ..., S. Note Mj
is a set of all equilibria with j treatments selected or j entrants,
andS
S
j=0Mj
= D. There are S!/j!(S � j)! distinct d
j ’s in Mj
. For example with S = 3,
d
2 2 M2 = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} and d
0 2 M0 = {(0, 0, 0)}. Note d
0 = e
0 = (0, ..., 0)and d
S = e
S = (1, ..., 1). Define
hj
(z, z0, x) ⌘ E[Y |D 2 Mj
,Z = z, X = x] Pr[D 2 Mj
|Z = z]
� E[Y |D 2 Mj
,Z = z
0, X = x] Pr[D 2 Mj
|Z = z
0], (3.19)
hDj
(z, z0) ⌘ Pr[D 2 Mj
|Z = z]� Pr[D 2 Mj
|Z = z
0]. (3.20)
Since Mj
’s are disjoint,P
S
j=0 Pr[D 2 Mj
|Z = ·] = 1 and thus h(z, z0, x) =P
S
j=0 hj(z, z0, x).
Let x = (x0, ..., xS) 2 X S+1 be a collection of (possibly di↵erent) evaluation points, i.e., eachevaluation point X = x
j
is in X for j = 0, ..., S, and define
h(z, z0;x) ⌘S
X
j=0
hj
(z, z0;xj
).
Recall #(d, x;u) ⌘ E[✓(d, x, ✏)|U = u], and for succinctness let #j
(x;u) ⌘ #(ej , x;u) as e
j
is the only relevant set of treatments under Assumption SY(i). We state the main lemma ofthis section.
Lemma 3.3. In model (2.1)–(2.2), suppose Assumptions IN, E, R, M, SS, SY and NU hold.For z, z0 2 Z such that
P
S
k=j
0 hDk
(z, z0) > 0 8j0 = 1, ..., S and for x 2 X and x 2 X S+1,suppose h(z, z0;x) is well-defined. For j = 1, ..., S, it satisfies that(i) sgn{h(z, z0, x)} = sgn {#
j
(x;u)� #j�1(x;u)} a.e. u;
(ii) for ◆ 2 {�1, 0, 1}, if sgn{h(z, z0;x)} = sgn{#k�1(xk�1;u)� #
k
(xk
;u)} = ◆ 8k 6= j, thensgn{#
j
(xj
;u)� #j�1(xj�1;u)} = ◆ a.e. u.
The existence of (z, z0) such thatP
S
k=j
0 hDk
(z, z0) > 0 8j0 is guaranteed by AssumptionsM(ii) and NU; see the discussion after Corollary 3.1 below. For Lemma 3.3(ii), variation inX conditional on Z would be important. To show Lemma 3.3, we formally characterize asan important step the regions that predict equilibria of the first-stage strategic interaction.The analytical characterization of the equilibrium regions when S > 2 can generally becomplicated (Ciliberto and Tamer (2009, p. 1800)) and has not been fully studied in the
16
literature. In the case of more than two players, the symmetry of the payo↵ functions inopponents’ decisions (Assumption SY(ii)) plays an important role in the characterization bysimplifying the regions of multiple equilibria.
Recall, ⌫sd�s
(zs
) ⌘ ⌫s(d�s
, zs
). Let ej be a (S � 1)-vector where the first j elements areunity and the rest are zero for j = 0, ..., S � 1. Note that by Assumption SY(ii), ⌫sej (zs) isthe only relevant payo↵ function to define the regions, therefore for notational simplicity, let⌫sj
(zs
) ⌘ ⌫sej (zs). Now, for each equilibrium profile, we define regions of U ⌘ (U1, ..., US
) in
U ⌘ (0, 1]S . These regions are defined as Cartesian products that are subsets of U :
RdS
(z) ⌘S
Y
s=1
�
0, ⌫sS�1(zs)
⇤
,
Rd0(z) ⌘S
Y
s=1
(⌫s0(zs), 1] ,
and, given d
j = (�(ej1), ...,�(ej
S
)) for some �(·) 2 ⌃13 and j = 1, ..., S � 1,
Rdj
(z) =
8
<
:
U : (U�(1), ..., U�(S)) 2
(
j
Y
s=1
⇣
0, ⌫�(s)j�1 (z�(s))
i
)
⇥
8
<
:
S
Y
s=j+1
⇣
⌫�(s)j
(z�(s)), 1
i
9
=
;
9
=
;
.
(3.21)
For example, for �(·) such that d1 = (�(1),�(0),�(0)) = (0, 1, 0),
R010(z) =�
⌫11(z1), 1⇤
⇥�
0, ⌫20(z2)⇤
⇥�
⌫31(z3), 1⇤
.
Lastly, define the region of all equilibria with j treatments selected or j entrants as
R
j
(z) ⌘[
d2Mj
Rd(z). (3.22)
Now we establish the geometric properties of these regions.
Definition 3.1. Sets A and B are neighboring sets when there exists a point in one set whoseopen "-ball has nonempty intersection with the other set for any " > 0.
Two sets with a nonempty intersection are trivially neighboring sets. Two disjoint setscan possibly be neighboring sets when they share a “border”.
Proposition 3.1. Consider the first-stage game (2.2). Under Assumptions SS and SY(ii),the following holds: For every z 2 Z (which is suppressed),(i) R
j
\R
j
0 = ; for j, j0 = 0, ..., S with j 6= j0;(ii) R
j
and R
j�1 are neighboring sets for j = 1, ..., S;(iii) R
j
and R
j�t
are not neighboring sets for j = t, ..., S and t � 2;
(iv)S
S
j=0Rj
= U .13Sometime we use the notation dj
�
to emphasize the permutation function �(·) from which dj is generated.
17
01
11
U1
U2
U3
(a) R0 ("); R3 (#) (b) R1 (c) R2 (d)S3
j=0 Rj = U
Figure 3: Illustration of Proposition 3.1 for S = 3.
This proposition fully characterizes the equilibrium regions. Figure 3 illustrates the resultsof Proposition 3.1 for S = 3 with R0 = R000, R1 = R100[R010[R001, R2 = R110[R101[R011
and R3 = R111; also see Figures 5 and 6 in the Appendix for relevant figures and for a figurethat depicts regions of multiple equilibria for this case.
For concreteness, we henceforth discuss this proposition in terms of an entry game. By (i)and the fact that M
S
and M0 are singleton, one can conclude that RdS
and Rd0 are regionsof unique equilibrium. For j = 1, ..., S � 1, however, Rdj
\ Rdj
is not necessarily empty for
d
j = (�(ej1), ...,�(ej
S
)) and d
j = (�(ej1), ..., �(ej
S
)) with d
j 6= d
j . In particular, Rdj
\Rdj
areregions of multiple equilibria. By (i), there is no multiple equilibria where one equilibriumhas j entrants and another has j0 entrants for j0 6= j. This is reminiscent of Berry (1992) andBresnahan and Reiss (1990, 1991) in that the equilibrium is unique in terms of the numberof entrants. In other words, even if D = d
j for j = 1, ..., S � 1 is not uniquely predicted byU 2 Rdj
(z) as Rdj
(z) contains a region of multiple equilibria, D 2 Mj
is uniquely predictedby U 2 R
j
(z).14 In the present paper, this result is obtained under substantially weakerconditions on the payo↵ function than those in Berry (1992).
In addition to (i), Proposition 3.1(ii)–(iii) are important for the analyses of this paper.Let A ⇠ B denote that A and B are neighboring sets. Note that A ⇠ B implies B ⇠ Aand vice versa. By (i), neighboring sets in (ii) are disjoint neighboring sets. Then (i)–(iii)immediately imply that R
j
’s are disjoint regions that lie in U in a monotonic fashion, whereall possible neighboring relationships are expressed as
R1 ⇠ R2 ⇠ · · · ⇠ R
S�1 ⇠ R
S
. (3.23)
Given this characterization, we can track the inflow and outflow in each R
j
(z) when thevalue of Z changes, as we do in the S = 2 case. This is formalized in the next corollary.
Proposition 3.1(i), (ii) and (iii) are implied by similar statements that satisfy for allindividual pairs between two regions: (i0) Rdj
\ Rdj
0 = ; 8dj 2 Mj
and 8dj0 2 Mj
0 withj 6= j0; (ii0) Rdj
and Rdj�1 are neighboring sets 8dj 2 Mj
and 8dj�1 2 Mj�1; (iii0) Rdj
andRdj�t
are not neighboring sets 8dj 2 Mj
and 8dj�t 2 Mj�t
with t � 2. These results can beshown using properties of sets defined as Cartesian products: e.g., for two Cartesian productsR =
Q
S
s=1 rs and Q =Q
S
s=1 qs with rs
and qs
being intervals in R, it satisfies that R ⇠ Q if
14Therefore, even if Pr[D = dj |Z = z] 6= Pr[U 2 Rdj (z)], it satisfies that Pr[D 2 Mj
|Z = z] = Pr[U 2R
j
(z)].
18
Player s 1 2 3 4 5
Decision djs
1 1 0 1 0
Decision dj�1s
1 0 0 0 1
Table 1: An example of equilibria that di↵er by one entrant with S = 5 and j = 3.
and only if rs
⇠ qs
8s. This property can be used to prove (ii0) and thus (ii) with R = Rdj
and Q = Rdj�1 . To show rs
⇠ qs
8s, we show that each pair of rs
and qs
is associated with thedecision of player s that falls into one of the categories that are exemplified in Table 1. Eachentry of the table is the s-th element of equilibria d
j and d
j�1 with S = 5 and j = 3. For anyS and j in general, there always exists a player s⇤ such that dj
s
⇤ = 1 and dj�1s
⇤ = 0. In thisexample, s⇤ = 2 (or 4). For all other players, it is easy to see that their equilibrium decisionsmust be one of the four pairs displayed in Table 1, i.e., (dj
s
, dj�1s
) 2 {(1, 1), (0, 0), (1, 0), (0, 1)}8s 6= s⇤. Then it can be shown that for each of these pairs, the corresponding r
s
and qs
satisfy rs
⇠ qs
, whether or not they have a nonempty intersection. This is formally shown inthe Appendix as part of the proof of Proposition 3.1.
Proposition 3.1(i) and (iv) imply that R
j
for j = 1, ..., S partition the entire U . Notethat, reversion (or crossing) of the “border” of the partition does not occur, otherwise itviolates (iii). As becomes clear in the proof of the proposition, this result is essentially dueto Assumption SS that ⌫s
j
< ⌫sj�t
for t � 1.For j = 0, ..., S, define the region of all equilibria with at most j entrants as
R
j(z) ⌘j
[
k=0
R
k
(z).
Although this region is hard to express explicitly in general, it has a simple property:
Corollary 3.1. Under the assumptions of Proposition 3.1, for j = 0, ..., S � 1, Rj(z) isexpressed as a union across �(·) 2 ⌃ of Cartesian products, each of which is a product of
intervals that are either (0, 1] or⇣
⌫�(s)j
(z�(s)), 1
i
for some s = 1, ...S.
This corollary asserts that the region which predicts all equilibria with at most j entrantsis solely determined by the payo↵s of players who stay out facing j entering opponents. Sincederiving the explicit expression of Rj can be cumbersome, we infer its form by focusing onthe “border” of Rj and using the results of Proposition 3.1; see the proof in the Appendix.15
Corollary 3.1 establishes a version of monotonicity in the treatment selection process.This corollary plays a crucial role in calculating the bounds on the treatment parameters, inshowing sharpness, and in introducing the LATE later. Its key implication is the following:by Corollary 3.1 and Assumption M(ii), for any given z, z0 2 Z, either
R
j(z) ✓ R
j(z0) or R
j(z) ◆ R
j(z0). (3.24)
This result can be used, among other things, in proving Lemma 3.3 by equating inflows and
15Berry (1992) derives the probability of an event that the number of entrants is less than a certain value,which can be written as Pr[U 2 Rj(z)] using our notation. This result is not su�cient for the purpose ofour paper.
19
outflows of Rj
’s with those of Rj ’s in calculating (3.19) (and thus h(z, z0;x)), which canbe written as
hj
(z, z0, x) = E[Y |U 2 R
j
(z),Z = z, X = x] Pr[U 2 R
j
(z)]
� E[Y |U 2 R
j
(z0),Z = z
0, X = x] Pr[U 2 R
j
(z0)], (3.25)
by Assumption IN. This approach is analogous to the simpler analysis in Section 3.1. ForLemma 3.3, the result (3.24) also guarantees the existence of z, z0 2 Z such that
S
X
k=j
0
hDk
(z, z0) =n
1� Pr[U 2 R
j
0�1(z)]o
�n
1� Pr[U 2 R
j
0�1(z0)]o
> 0
8j0 = 1, ..., S by Assumptions IN and NU. We introduce a lemma that establishes the con-nection between Corollary 3.1 (or Proposition 3.1) and Lemma 3.3.
Lemma 3.4. Based on the results in Proposition 3.1, h(z, z0;x) ⌘P
S
j=0 hj(z, z0, x
j
) satisfies
h(z, z0;x) =S
X
j=1
Z
�j�1(z0,z)
{#j
(xj
;u)� #j�1(xj�1;u)} du, (3.26)
where �j�1(z0, z) ⌘ R
j�1(z0)\Rj�1(z).
As a special case of this lemma, h(z0, z;x, ..., x) = h(z0, z, x) =P
S
j=0 hj(z0, z, x) can be
expressed as
h(z0, z, x) =S
X
j=1
Z
�j�1(z0,z)
{#j
(x;u)� #j�1(x;u)} du. (3.27)
Now we are ready to prove Lemma 3.3. For part (i), suppose that #j
(x;u)� #j�1(x;u) > 0
a.e. u 8j = 1, ..., S. Then by (3.27), h > 0. Conversely, if h > 0 then it should bethat #
j
(x;u) � #j�1(x;u) > 0 a.e. u 8j = 1, ..., S. Suppose not and suppose #
j
(x;u) �#j�1(x;u) 0 with positive measure for some j. Then by Assumption M(i), this implies
that #j
(x;u)�#j�1(x;u) 0 8j a.e. u, and thus h 0 which is contradiction. By applying
similar arguments for other signs, we have the desired result. The proof for Lemma 3.3(ii) isin the Appendix.
Using Lemma 3.3, we can obtain the results in Theorem 3.1. First the sign of the ATEis identified by Lemma 3.3(i) since E[Yd|X = x] = E[#(d, x;U)]. Next, we calculate thebounds on E[Yd|X = x] with d = d
j for a given d
j 2 Mj
for some j = 0, ..., S. Consider
E[Ydj
|X = x] = E[Y |D = d
j ,Z = z, X = x] Pr[D = d
j |Z = z]
+X
d0 6=dj
E[Ydj
|D = d
0,Z = z, X = x] Pr[D = d
0|Z = z]. (3.28)
Note that for d0 2 Mj
,
E[Ydj
|D = d
0,Z = z, X = x] = E[Y |D = d
0,Z = z, X = x] (3.29)
20
by Assumption SY(i). In order to bound E[Ydj
|D = d
0,Z = z, X = x] for d0 /2 Mj
in (3.28),we systematically use the results of Lemma 3.3. Define the integrated version of h(z, z0, x)and h(z, z0;x) as
H(x) ⌘ E
2
4h(Z,Z 0, x)
�
�
�
�
�
�
S
X
k=j
0
hDk
(Z,Z 0) > 0 for all j0 = 1, ..., S
3
5 ,
H(x) ⌘ E
2
4h(Z,Z 0;x)
�
�
�
�
�
�
S
X
k=j
0
hDk
(Z,Z 0) > 0 for all j0 = 1, ..., S
3
5 ,
and define the following sets of evaluation points of X that satisfy the conditions in Lemma3.3: for j = 1, ..., S,
X 0j,j�1(◆) ⌘ {(x
j
, xj�1) : sgn{H(x)} = ◆, x0 = · · · = x
S
},X 1j,j�1(◆) ⌘ {(x
j
, xj�1) : sgn{H(x)} = ◆, (x
k
, xk�1) 2 X 0
k,k�1(�◆) 8k 6= j} [ X 0j,j�1(◆),
...
X t
j,j�1(◆) ⌘ {(xj
, xj�1) : sgn{H(x)} = ◆, (x
k
, xk�1) 2 X t�1
k,k�1(�◆) 8k 6= j} [ X t�1j,j�1(◆).
Note that X t
j,j�1(◆) ⇢ X t+1j,j�1(◆) for any t. Define X
j,j�1(◆) ⌘ limt!1X t
j,j�1(◆).16 Then by
Lemma 3.3,
if (xj
, xj�1) 2 X
j,j�1(◆), then sgn{#j
(xj
;u)� #j�1(xj�1;u)} = ◆ a.e. u. (3.30)
Consider j0 < j for E[Ydj
|D = d
j
0,Z, X] in (3.28). Then, for example, if (x
k
, xk�1) 2
Xk,k�1(�1) [ X
k,k�1(0) for j0 + 1 k j, then #j
(x;u) #j
0(x0;u) where x = xj
andx0 = x
j
0 by transitively applying (3.30). Therefore
E[Ydj
|D = d
j
0,Z = z, X = x] = E[✓(dj , x, ✏)|U 2 Rdj
0 (z),Z = z, X = x]
=1
Pr[U 2 Rdj
0 (z)]
Z
R
dj0 (z)
#j
(x;u)du
1
Pr[U 2 Rdj
0 (z)]
Z
R
dj0 (z)
#j
0(x0;u)du
= E[✓(dj
0, x0, ✏)|U 2 Rdj
0 (z),Z = z, X = x0]
= E[Y |D = d
j
0,Z = z, X = x0]. (3.31)
Symmetrically, for j0 > j, if (xk
, xk�1) 2 X
k,k�1(1) [ Xk,k�1(0) for j + 1 k j0, then
#j
(x;u) #j
0(x0;u) where x = xj
and x0 = xj
0 . Therefore the same bound as (3.31) isderived. Given these results, to collect all x0 2 X that yield #
j
(x;u) #j
0(x0;u), we can
16In practice, the formula for X t
j,j�1 provides a natural algorithm to construct the set Xj,j�1 for the computa-
tion of the bounds. Practitioners can employ truncation t T for some T and use X T
j,j�1 as an approximationfor X
j,j�1.
21
construct a set
x0 2�
xj
0 : (xk
, xk�1) 2 X
k,k�1(�1) [ Xk,k�1(0) for j
0 + 1 k j, xj
= x
[�
xj
0 : (xk
, xk�1) 2 X
k,k�1(1) [ Xk,k�1(0) for j + 1 k j0, x
j
= x
.
Then we can further shrink the bound in (3.31) by taking infimum over all x0 in this set. Thelower bound on E[Ydj
|D = d
j
0,Z = z, X = x] can be constructed by simply choosing the
opposite signs in the preceding argument.In conclusion, for bounds on the ATE E[Ydj
|X = x], we express the sets XL
dj
(x;d0) andXU
dj
(x;d0) for d0 6= d
j introduced in Theorem 3.1 as follows: for d0 2 Mj
0 with j0 6= j,
XL
dj
(x;d0) ⌘�
xj
0 : (xk
, xk�1) 2 X
k,k�1(�1) [ Xk,k�1(0) for j
0 + 1 k j, xj
= x
[�
xj
0 : (xk
, xk�1) 2 X
k,k�1(1) [ Xk,k�1(0) for j + 1 k j0, x
j
= x
, (3.32)
XU
dj
(x;d0) ⌘�
xj
0 : (xk
, xk�1) 2 X
k,k�1(1) [ Xk,k�1(0) for j
0 + 1 k j, xj
= x
[�
xj
0 : (xk
, xk�1) 2 X
k,k�1(�1) [ Xk,k�1(0) for j + 1 k j0, x
j
= x
, (3.33)
and for d0 2 Mj
,XL
dj
(x;d0) = XU
dj
(x;d0) ⌘ {x}, (3.34)
where the last display is by (3.29).
Remark 3.1. When X does not have enough variation, an assumption that Y 2 [Y , Y ] withknown endpoints can be introduced to calculate the bounds. To see this, suppose we do notuse the variation in X and suppose H(x) � 0. Then #
k
(x;u) � #k�1(x;u) 8k = 1, ..., S by
Lemma 3.3(i) and by transitivity, #j
0 � #j
for any j0 > j. Therefore, we have
E[Ydj
|X = x] X
d2Mj
E[Y |D = d,Z, X = x] Pr[D = d|Z]
+X
d02Mj
0 :j0>j
E[Y |D = d
0,Z, X = x] Pr[D = d
0|Z]
+X
d02Mj
0 :j0<j
E[Ydj
|D = d
0,Z, X = x] Pr[D = d
0|Z]. (3.35)
Without using variation in X, we can bound the last term in (3.35) by Y 2 [Y , Y ]. Thiscase is illustrated in Section 3.1 with ✓(d, x, ✏) = 1[µd(x) � ✏] and #
j
(x;u) = F✏|U (µej (x)|u).
Another example is when Y 2 [0, 1] as in Example 3.
Remark 3.2. By using a similar approach as in Vytlacil and Yildiz (2007), it may be possibleto point identify the ATE by extending the result of Theorem 3.1 using X with larger support.For example, if we can find x0 such that #
j
(x;u) = #j
0(x0;u) (j 6= j0) then we can point
22
identify the ATT:
E[Ydj
|D = d
j
0,Z = z, X = x] =
1
Pr[U 2 Rdj
0 (z)]
Z
R
dj0 (z)
#j
(x;u)du
=1
Pr[U 2 Rdj
0 (z)]
Z
R
dj0 (z)
#j
0(x0;u)du
= E[Y |D = d
j
0,Z = z, X = x0].
The existence of such x0 requires su�cient variation of X conditional on Z. This approachis alternative to the identification at infinity that uses the large variation of Z for pointidentification, which is discussed in Section 4.4 below.
Remark 3.3. An alternative approach to partial identification of the ATE is to embracethe framework established in Manski (1997) and Manski and Pepper (2000) under supportconditions on Yd and stronger monotonicity restrictions. This approach will not require thespecification of the first stage (2.2) and hence no issue of multiplicity arises. It, however,requires a semi-MTR assumption that the treatment response is monotonic in a semi-orderedset, which inevitably restricts the sign of some ATE’s. The current approach, on the otherhand, does not require such an assumption in exchange of other shape restrictions, and candeliver tighter bounds even in the presence of multiplicity. This is especially true when onecan exploit the variability of X given Z. Second, we want to explicitly consider a multi-agentmodel and impose a simultaneity structure in the first stage to better fit the economic applica-tions we consider. This enables us to incorporate further knowledge about payo↵ structures,such as strategic substitutability. Lastly, our framework can relax the restriction that Y isbounded between a known interval.
3.3 Sharp Bounds
We derive sharp bounds on the ASF and ATE under a stronger support condition. Sharpbounds on the mean treatment parameters in this model of a triangular structure can only beobtained for binary Y . We first simplify the bounds obtained in Theorem 3.1 using variationin Z only. We consider the case H(x) > 0; the case H(x) < 0 is symmetric and the caseH(x) = 0 is straightforward. For given d
j 2 Mj
, we first calculate (3.32)–(3.34) for the
bounds on E[Ydj
|X = x] = Pr[Ydj
= 1|X = x]. For ease of notation, let Mj ⌘S
j
k=0Mk
and M>j ⌘S
S
k=j+1Mk
= D\Mj . Then the bounds become
Udj
(x) ⌘ infz2Z
(
Pr[Y = 1,D 2 Mj
|Z = z, X = x] +X
d02M>j
Pr[Y = 1,D = d
0|Z = z, X = x]
+X
d02Mj�1
Pr[D = d
0|Z = z, X = x]
)
, (3.36)
23
Ldj
(x) ⌘ supz2Z
(
Pr[Y = 1,D 2 Mj
|Z = z, X = x] +X
d02Mj�1
Pr[Y = 1,D = d
0|Z = z, X = x]
+X
d02M>j
Pr[D = d
0|Z = z, X = x]
)
, (3.37)
where in Udj
(x), we use
X
d02Mj
infx
02XU
dj(x;d0)
Pr[Ydj
= 1,D = d
0|Z = z, X = x0] =X
d02Mj
Pr[Y = 1,D = d
0|Z = z, X = x]
= Pr[Y = 1,D 2 Mj
|Z = z, X = x]
by (3.34), and similarly in Ldj
(x).
Assumption C. µd(·) and ⌫d�s
(·) are continuous.
Assumption ZZ. Z is compact and satisfies that Z =Q
S
s=1Zs
.
The assumption of rectangular support in Assumption ZZ is the key additional supportcondition. Under Assumptions C, M(ii) and the compactness of Z
s
in Assumption ZZ, forgiven s 2 {1, ..., S}, there exist vectors z
s
and zs
such that ⌫sj
0(zs
) = maxz
s
2Zs
⌫sj
0(zs
) and⌫sj
0(zs
) = minz
s
2Zs
⌫sj
0(zs
) 8j0 = 0, ..., S � 1. Define a joint propensity score as pM
(z) ⌘Pr[D 2 M |Z = z]. Note that
pM
>j
0 (z) = Pr[U 2 U\Rj
0(z)] (3.38)
For a given z�s
2 Z�s
, it satisfies Zs
|z�s
= Zs
by Assumption ZZ and thus the vectors zs
and zs
defined above satisfy
pM
>j
0 (zs
, z�s
) = maxz
s
2Zs
pM
>j
0 (zs
, z�s
), pM
>j
0 (zs
, z�s
) = minz
s
2Zs
pM
>j
0 (zs
, z�s
),
by Corollary 3.1. By iteratively applying the same argument w.r.t. other elements in z�s
under Assumption M(ii), the vectors z ⌘ (z1, ..., zS) and z ⌘ (z1, ..., zS) satisfy
pM
>j
0 (z) = maxz2
Qs
Zs
pM
>j
0 (z), pM
>j
0 (z) = minz2
Qs
Zs
pM
>j
0 (z). (3.39)
Now using z and z, we can simplify the bounds in Theorem 3.1 and show their sharpness inmodel (3.3)–(3.5) where X is assumed to be fixed at x in the data generating process (DGP).The bounds where variation in X is additionally exploited may be shown to be sharp with adi↵erent version of the bounds; see a remark below.
Theorem 3.2. Given model (3.3)–(3.5) conditional on X = x, suppose the assumptionsof Theorem 3.1 and Assumptions ZZ and C hold. Also suppose, for z, z0 2 Z such thatP
S
k=j
0 hDk
(z, z0) > 0 8j0 = 1, ..., S, it satisfies h(z, z0, x) � 0. Then the bounds Udj
(x) andLdj
(x) in (3.36) and (3.37) simplify as
Udj
(x) = Pr[Y = 1,D 2 M>j�1|Z = z, X = x] + Pr[D 2 Mj�1|Z = z],
Ldj
(x) = Pr[Y = 1,D 2 Mj |Z = z, X = x] + Pr[D 2 M>j |Z = z],
24
and these bounds and thus the bounds on the ATE are sharp.
Shaikh and Vytlacil (2011) use the propensity score as a scalar conditioning variable,which summarizes all the exogenous variation in the selection process and is convenient insimplifying the bounds. In the context of the current paper, however, this approach is invalidsince Pr[D
s
= 1|Zs
= zs
,D�s
= d�s
] cannot be written in terms of a propensity scoreof player s as D�s
is endogenous. We instead use vector Z as conditioning variables andestablish partial ordering for the relevant conditional probabilities (that define the lower andupper bounds) w.r.t. the joint propensity score (3.38). In proving the sharpness of thebounds, Corollary 3.1 plays an important role. Even though D is a vector that is determinedby simultaneous decisions, Corollary 3.1 combined with the partial ordering above establishes“monotonicity” of the event U 2 R
j(z) (and U 2 U\Rj(z)) w.r.t. z; see the proof for details.Given the sharp bounds established in Theorem 3.2, we can further narrow the bounds
in model (3.3)–(3.5) where X is no longer fixed, using variation of X excluded from thefirst-stage as explored earlier.
In model (3.3)–(3.5) where X is no longer fixed, the bounds that can be derived usingvariation of X given Z is narrower than the sharp bounds established in Theorem 3.2, as weexplored earlier. The resulting bounds, however, are not automatically implied to be sharpfrom Theorem 3.2, since they are based on a di↵erent DGP and the additional exclusionrestriction.
Remark 3.4. Maintaining that Y is binary, sharp bounds on the ATE with variation in Xcan be derived assuming that the signs of #(d, x;u)� #(d0, x0;u) are identified for d,d0 2 Dand x, x0 2 X via Lemma 3.3. To see this, define
XU
d (x;d0) ⌘�
x0 : #(d, x;u)� #(d0, x0;u) 0 a.e. u
,
XL
d (x;d0) ⌘�
x0 : #(d, x;u)� #(d0, x0;u) � 0 a.e. u
,
which are identified by assumption. Then by replacing X i
d(x;d0) with X i
d(x;d0) (for i 2
{U,L}) in Theorem 3.1, we may be able to show that the resulting bounds are sharp in afully general model where excluded variation in X is also in use. Since Lemma 3.3 impliesthat X i
dj
(x;d0) ⇢ X i
dj
(x;d0) but not necessarily X i
dj
(x;d0) � X i
dj
(x;d0), these modified boundsand the original bounds do not coincide. When variation of X is not part of the DGP,Lemma 3.3(i) establishes equivalence between the two signs, and thus X i
dj
(x;d0) = X i
dj
(x;d0)for i 2 {U,L}, which results in Theorem 3.2. Relatedly, we can also exploit variation fromvariables that are common to both X and Z (with or without exploiting excluded variationof X). This is related to the analysis of Chiburis (2010) and Mourifie (2015) in a single-treatment setting. The results of Lemma 3.3 can be extended to accommodate the discussionof this remark, but we do not pursue it in the current paper for succinctness.
4 Discussions
4.1 Relaxing Symmetry
We propose two di↵erent ways of relaxing the symmetry assumption in the outcome function(Assumption SY(i)) introduced in the preceding section.
25
4.1.1 Partial Symmetry: Interactions Within Groups
In some cases, strategic interactions may occur within groups of players (i.e., treatments).In the airline example, it may be the case that larger airlines interact to one another as agroup, so do smaller airlines as a di↵erent group, but there may be no interaction across thegroups.17 In general for K groups of multiple treatments, we consider, with player indexs = 1, ..., S
g
and group index g = 1, ..., G,
Y = ✓(D1, ...,DG, X, ✏D), (4.1)
Dg
s
= 1⇥
⌫s,g(Dg
�s
, Zg
s
) � Ug
s
⇤
, (4.2)
where eachD
g ⌘ (Dg
1, ..., Dg
S
k
) is the treatment vector of group g andD ⌘ (D1, ...,DG). Thismodel generalizes the model (2.1)–(2.2). It can also be seen as a special case of exogenouslyendowing an incomplete undirected network structure, where players interact to one anotherwithin each of complete sub-networks. In this model each group can di↵er in its number(S
g
) and identity of players (under which the entry decision is denoted as Dg
s
). Also, theunobservables U
g ⌘ (Ug
1 , ..., Ug
S
) can be arbitrarily correlated across groups, in addition tothe fact that Ug
s
’s can be correlated within group g and U ⌘ (U1, ...,UG) can be correlatedwith ✏D. This partly relaxes the independence assumption across markets, which is frequentlyimposed in the entry game literature.
To calculate the bounds on the ATE E[Yd � Yd0 |X = x] we apply the results in Theorem3.1, by adapting those assumptions to the current extension. Assumption SY(i) then can berelaxed by assuming that (the conditional mean of) the outcome function is symmetric withineach group but not across groups. In terms of notation, letD�g ⌘ (D1, ...,Dg�1,Dg+1, ...,DG)and its realization be d
�g. Then such an assumption would be stated as:
Assumption SY2. For each g = 1, ..., G, (i) #(dg,d�g, ·;u) = #(dg,d�g, ·;u) a.e. u forany permutation d
g of dg;(ii) ⌫s,g(dg
�s
, ·) = ⌫s,g(dg
�s
, ·) for any permutation d
g
�s
of dg
�s
.
Under this partial symmetry assumption, the bound on the ASF can be calculated byiteratively applying the previous results to each group. Assumptions Y, M, SS and NU canbe modified so that they satisfy for within-group treatments and interaction. In particular,Assumption NU can be modified as follows: for each d
g
�s
2 Dg
�s
, ⌫s,g(dg
�s
, Zg
s
)|X,Z�g isnondegenerate, where Z ⌘ (Zg,Z�g). That is, there must be group-specific instrumentsthat are excluded from other groups.18
We sketch the idea here with binary Y for simplicity. Analogous to the previous notation,let Mg
j
be the set of equilibria with j entrants in group g and let Mg,j ⌘S
j
k=0Mg
k
. Suppose
G = 2, and d
1 2 {0, 1}S1 and d
2 2 {0, 1}S2 . Consider the ASF E[Yd|X] = E[Yd1,d2 |X] with
d
1 2 M1j�1 and d
2 2 M2k�1 for some j = 1, ..., S1 and k = 1, ..., S2. To calculate its bounds,
we can bound E[Yd|D = d
0,Z, X] in (3.1) for d 6= d by sequentially applying the analysis ofSection 3 in each group. First consider d = (d1,d2) with d
1 2 M1j
. We apply Lemma 3.3 for
17We can also easily extend the model so that smaller airlines take larger airlines’ entry decisions as givenand play their own entry game, which may be more reasonable to assume.
18We maintain Assumption R in the current setting since the assumption is equivalent to assuming a rankinvariance within each group, i.e., ✏dg
,d�g = ✏dg,d�g 8dg, dg 2 {0, 1}Sg and g = 1, ..., G.
26
the D
1 portion after holding D
2 = d
2. Suppose
Pr[Y = 1|D2 = d
2,Z1 = z
1,Z2, X]� Pr[Y = 1|D2 = d
2,Z1 = z
10,Z2, X] � 0,
Pr[D1 2 M1,>j�1|Z1 = z
1]� Pr[D1 2 M1,>j�1|Z1 = z
10] > 0,
then we have µd1,d2(x) � µd1
,d2(x). The proof of Lemma 3.3 can be adapted by holding
D
2 = d
2 in this case, because there is no strategic interaction across groups and thereforethe multiple equilibria problem only occurs within each group. Note that this strategy stillallows that there is dependence between D
1 and D
2 even after conditioning on (Z, X) dueto dependence between U
1 and U
2. Then,
Pr[Yd1,d2 = 1|D = (d1,d2),Z, X = x] = Pr[✏ µd1
,d2(x)|D = (d1,d2),Z, X = x]
Pr[✏ µd1
,d2(x)|D = (d1,d2),Z, X = x] (4.3)
= Pr[Y = 1|D = (d1,d2),Z, X = x].
Next, consider d = (d1,d2) and d = (d1, d2) with d
2 2 M2k
and the other elements aspreviously determined. Then by applying Lemma 3.3 this time for the D
2 portion afterholding D
1 = d
1, we have µd1
,d2(x) � µd1
,d2(x) by supposing
Pr[Y = 1|D1 = d
1,Z1,Z2 = z2, X]� Pr[Y = 1|D1 = d
1,Z1,Z2 = z
20, X] � 0,
Pr[D2 2 M2,>j�1|Z2 = z
2]� Pr[D2 2 M2,>j�1|Z2 = z
20] > 0.
Then
Pr[Yd1,d2 = 1|D = (d1, d2),Z, X = x] Pr[✏ µ
d1,d2(x)|D = (d1, d2),Z, X = x]
Pr[✏ µd1
,d2(x)|D = (d1, d2),Z, X = x] (4.4)
= Pr[Y = 1|D = (d1, d2),Z, X = x],
where the first inequality is by (4.3). Note that in deriving the upper bound in (4.4), it isimportant that at least the two groups share the same signs of within-group h’s and hD’s.This is clearly a weaker requirement than imposing Assumption SY(i).
4.1.2 Stable Equilibrium Selection
Assumption SY(i) can also be relaxed when the equilibrium selection is stable with thechange of instruments. As Z changes, the distribution of D changes. This change occursas players change their decisions by facing di↵erent payo↵s of entry and as the equilibriumselection rule changes. Under the symmetry assumption, SY(i), it is enough to concernthe change in the joint propensity score, Pr[D 2 M
j
|Z = z] � Pr[D 2 Mj
|Z = z
0] =Pr[U 2 R
j
(z)]�Pr[U 2 R
j
(z0)] = Pr[U 2 R
j
(z)\Rj
(z0)]�Pr[U 2 R
j
(z0)\Rj
(z)], which ispurely determined by the change in the players’ payo↵s. When the symmetry assumption isdropped, then we need to learn about the change in the individual propensity score, Pr[D =d
j |Z = z] � Pr[D = d
j |Z = z
0] = Pr[U 2 R⇤dj
(z)] � Pr[U 2 R⇤dj
(z0)], where R⇤d(·) is the
27
0 1
1
R10(z0)
R01(z0)
U2
U1
(a)
0 1
1R10(z)
R01(z)R10(z0)
R01(z0)
U2
U1
(b)
Figure 4: Illustration of Assumptions ES and ES⇤.
region that predicts D = d.19 In general, this change is also determined by the equilibriumselection rule that partitions R
j
(z) =S
d2Mj
R⇤d(z) and R
j
(z0) =S
d2Mj
R⇤d(z
0). Nettingthe change in the payo↵s, one can show that the remaining di↵erence of the propensityscores can be attributed to the change in the selection rule within R
j
(z) \ R
j
(z0). Thisis because R
j
(z)\ {Rj
(z)\Rj
(z0)} = R
j
(z0)\ {Rj
(z0)\Rj
(z)} = R
j
(z) \R
j
(z0), where theterms R
j
(z)\Rj
(z0) and R
j
(z0)\Rj
(z) appear in the joint propensity scores above. Theregion R
j
(z) \R
j
(z0) can be seen as a common support of U with j entrants for Z beingz or z0, and hence a relevant region to consider the equilibrium selection as Z changes. Weassume that the equilibrium selection is stable within this region.
Assumption ES. For j = 1, ..., S � 1, there exist z, z0 2 Z such that the region thatpredicts D = d
j is invariant for Z 2 {z, z0} within R
j
(z)\R
j
(z0) 8dj 2 Mj
, i.e., R⇤dj
(z)\{R
j
(z) \R
j
(z0)} = R⇤dj
(z0) \ {Rj
(z) \R
j
(z0)} 8dj 2 Mj
.
Since the portion of R⇤dj
(z) that predicts a unique equilibrium is never a↵ected by theselection rule, the assumption is only relevant to the regions of multiple equilibria (i.e., theunion of Rdj
(z) \ Rdj
(z) across all pairs d
j , dj 2 Mj
and similarly with z
0) that intersectswith R
j
(z) \ R
j
(z0); this region is hatch-patterned in Figure 4(a) with S = 2. Thereforethis assumption trivially holds when Z varies su�ciently enough that this intersection isempty, or equivalently, enough that R
j
(z) \R
j
(z0) does not contain the regions of multipleequilibria. Specifically, this occurs when the following condition holds:
Assumption ES
⇤. For j = 1, ..., S � 1, there exist z, z0 2 Z such that ⌫s
j�1(z0s
) ⌫sj
(zs
) 8s.
Lemma 4.1. Assumption ES⇤ implies Assumption ES.
The su�ciency of Assumption ES⇤ is illustrated in Figure 4(b) with ⌫s0(z0s
) < ⌫s1(zs) fors = 1, 2. Assumption ES⇤ states that the change in Z is large enough to o↵set the e↵ect ofstrategic substitutability. For example in an entry game with Z
s
being entry cost, AssumptionES⇤ may hold with z0
s
> zs
8s. In this example, all players become less profitable with theincrease in cost, while one player becomes unprofitable to enter whose absence does not helpoverturn the decrease of other firms’ profits. A necessary condition for Assumption ES isthat, for d 2 D, R⇤
d(z) is a function of z only through (⌫1d�1(z1), ..., ⌫Sd�S
(zS
)). That is, with
19Unlike Rd(z) which is purely determined by the payo↵s ⌫s
d�s(z
s
), R⇤d(z) is unknown to the econometrician
even if all the players’ payo↵s had been known, since the equilibrium selection rule is unknown.
28
the same primitives and observables, the same equilibrium is selected; see e.g., Bajari et al.(2010) and de Paula (2013) for related discussions.
Under Assumption ES and without Assumption SY(i), we can apply an analogous proofstrategy as in the symmetry case to determine the direction of monotonicity and ultimatelycalculate the bounds on the ATE. Recall #(d, x;u) ⌘ E[✓(d, x, ✏)|U = u].
Lemma 4.2. In model (2.1)–(2.2), suppose Assumptions IN, E, R, Y, M, SS, SY(ii), NUand ES hold. For any (z, z0, x) such that
P
S
k=j
0 hDk
(z, z0) > 0 8j0 = 1, ..., S, it satisfies that
sgn{h(z, z0, x)} = sgn {#(1,d�s
, x;u)� #(0,d�s
, x;u)}
a.e. u 8d�s
2 D�s
and 8s = 1, ..., S.
Again, when Assumption SY(i) holds, then the result of this lemma is satisfied withoutAssumption ES. Suppose S = 2 and Y is binary for illustration of this lemma. In place ofhM
(z, z0, x) that is used to prove Lemma 3.1, introduce
h10(z, z0, x) ⌘ Pr[Y = 1,D = (1, 0)|Z = z, X = x]� Pr[Y = 1,D = (1, 0)|Z = z
0, X = x],
h01(z, z0, x) ⌘ Pr[Y = 1,D = (0, 1)|Z = z, X = x]� Pr[Y = 1,D = (0, 1)|Z = z
0, X = x].
Then h defined in (3.6) satisfies h = h11+h00+h10+h01; in fact, hM
= h10+h01. Denote asR⇤
10 and R⇤01 the regions that predict D = (1, 0) and D = (0, 1), respectively. For (z, z0) such
that hD11(z, z0) > 0 and �hD00(z, z
0) > 0, we have R11(z) � R11(z0) and R00(z) ⇢ R00(z0),respectively, by Corollary 3.1. Since R⇤
10 [ R⇤01 = R10 [ R01 = R1, (3.9) and (3.10) can
alternatively be expressed as
�+(z, z0) ⌘ {R⇤
10(z) [R⇤01(z)} \R1(z
0), (4.5)
��(z, z0) ⌘
�
R⇤10(z
0) [R⇤01(z
0)
\R1(z). (4.6)
Consider partitions �+(z, z0) = �1+(z, z
0)[�2+(z, z
0) and ��(z, z0) = �1�(z, z
0)[�2�(z, z
0)such that
�1+(z, z
0) ⌘ R⇤10(z)\R1(z
0), �2+(z, z
0) ⌘ R⇤01(z)\R1(z
0),
�1�(z, z
0) ⌘ R⇤10(z
0)\R1(z), �2�(z, z
0) ⌘ R⇤01(z
0)\R1(z).
That is, �1+(z, z
0) and �1�(z, z
0) are regions of R⇤10 exchanged with the regions for D = (0, 0)
and D = (1, 1), respectively, and �2+(z, z
0) and �2�(z, z
0) are for R⇤01.
By Assumption IN and suppressing the argument (z, z0, x) on the l.h.s.,
h10 =Pr[✏ µ10(x),U 2 R⇤10(z)]� Pr[✏ µ10(x),U 2 R⇤
10(z0)]
=Pr[✏ µ10(x),U 2 R⇤10(z)\R⇤
10(z0)]� Pr[✏ µ10(x),U 2 R⇤
10(z0)\R⇤
10(z)]
=Pr[✏ µ10(x),U 2 �1+(z, z
0)]� Pr[✏ µ10(x),U 2 �1�(z, z
0)],
29
where the second equality is by (3.12) and the last equality is by the following derivation:
R⇤10(z)\R⇤
10(z0) =
⇥�
R⇤10(z) \R1(z
0)c
\R⇤10(z
0)c⇤
[⇥�
R⇤10(z) \R1(z
0)
\R⇤10(z
0)c⇤
=⇥�
R⇤10(z) \R1(z
0)c ⇤
[⇥�
R⇤10(z
0) \R1(z)
\R⇤10(z
0)c⇤
= �1+(z, z
0),
where the first equality is by the distributive law and U = R1(z0)c [ R1(z0), the secondequality is by R
c
1 = R⇤c10 \ R⇤c
01 (the first term) and by Assumption ES (the second term),and the last equality is by the definition of �1
+(z, z0) and {R⇤
10(z0) \R1(z)}\R⇤
10(z0)c being
empty. Analogously, one can show that R⇤10(z
0)\R⇤10(z) = �1
�(z, z0) using Assumption ES
and the definition of �1�(z, z
0). Likewise,
h01 =Pr[✏ µ01(x),U 2 R⇤01(z)]� Pr[✏ µ01(x),U 2 R⇤
01(z0)]
=Pr[✏ µ01(x),U 2 R⇤01(z)\R⇤
01(z0)]� Pr[✏ µ01(x),U 2 R⇤
01(z0)\R⇤
01(z)]
=Pr[✏ µ01(x),U 2 �2+(z, z
0)]� Pr[✏ µ01(x),U 2 �2�(z, z
0)].
Also, by the definitions of the partitions,
h11 = Pr[✏ µ11(x),U 2 ��(z, z0)]
= Pr[✏ µ11(x),U 2 �1�(z, z
0)] + Pr[✏ µ11(x),U 2 �2�(z, z
0)]
and
h00 = �Pr[✏ µ00(x),U 2 �+(z, z0)]
= �Pr[✏ µ00(x),U 2 �1+(z, z
0)]� Pr[✏ µ00(x),U 2 �2+(z, z
0)].
Now combining all the terms yields
h(z, z0, x) =Pr[✏ µ11(x),U 2 �1�(z, z
0)]� Pr[✏ µ10(x),U 2 �1�(z, z
0)]
+ Pr[✏ µ11(x),U 2 �2�(z, z
0)]� Pr[✏ µ01(x),U 2 �2�(z, z
0)]
+ Pr[✏ µ10(x),U 2 �1+(z, z
0)]� Pr[✏ µ00(x),U 2 �1+(z, z
0)]
+ Pr[✏ µ01(x),U 2 �2+(z, z
0)]� Pr[✏ µ00(x),U 2 �2+(z, z
0)].
Then by Assumption M(i), µ1,d�s
(x) � µ0,d�s
(x) share the same signs for all s and 8d�s
2{0, 1} and therefore sgn{h(z, z0, x)} = sgn
�
µ1,d�s
(x)� µ0,d�s
(x)
.Lastly, to exploit the variation of X, define20
h(z, z0;x0, x1, x1, x2) ⌘Pr[✏ µ11(x2),U 2 �1�(z, z
0)]� Pr[✏ µ10(x1),U 2 �1�(z, z
0)]
+ Pr[✏ µ11(x2),U 2 �2�(z, z
0)]� Pr[✏ µ01(x1),U 2 �2�(z, z
0)]
+ Pr[✏ µ10(x1),U 2 �1+(z, z
0)]� Pr[✏ µ00(x0),U 2 �1+(z, z
0)]
+ Pr[✏ µ01(x1),U 2 �2+(z, z
0)]� Pr[✏ µ00(x0),U 2 �2+(z, z
0)],
20Note that we cannot assign a di↵erent value of x for µ10(x) and µ01(x), otherwise we cannot apply theassertion of Assumption M(i) in the proof.
30
then the results of Lemma 3.2 will hold with h replaced by h.
Remark 4.1. The bounding strategy of Ciliberto and Tamer (2009) does not work in oursetting. To see this, following Ciliberto and Tamer (2009),
h10 + h01 =Pr[✏ µ10(x),U 2 R⇤10(z)] + Pr[✏ µ01(x),U 2 R⇤
01(z)]
� Pr[✏ µ10(x),U 2 R⇤10(z
0)]� Pr[✏ µ01(x),U 2 R⇤01(z
0)]
has a lower bound
Pr[✏ µ10(x),U 2 Rmin
10 (z)] + Pr[✏ µ01(x),U 2 Rmin
01 (z)]
� Pr[✏ µ10(x),U 2 Rmax
10 (z0)]� Pr[✏ µ01(x),U 2 Rmax
01 (z0)],
where Rmin
d and Rmax
d denote the minimal and maximal possible regions that predict d 2{(1, 0), (0, 1)}. Note that the first two terms and the last two terms cannot be combined. Thefirst and the third terms as well as the second and the last terms can be combined but do notyield the di↵erence of regions that correspond to the regions yielded from h11 + h00.
4.2 Player-Specific Outcomes
Henceforth, we considered a scalar Y that may represent an outcome common to all players ina given market or a geographical region. The outcome, however, can also be an outcome thatis specific to each player. In this regard, consider a vector of outcomes Y = (Y1, ..., YS) whereeach element Y
s
is a player-specific outcome. An interesting example of this setting may bewhere Y is also an equilibrium outcome from strategic interaction not only through D butalso through itself. In this case, it would become important to have a vector of unobservableseven after assuming e.g., rank invariance, since we may want to include ✏D = (✏1,D, ..., ✏
S,D),where ✏
s,D is an unobservable directly a↵ecting Ys
.21 We may also want to include a vectorof observables of all players X = (X1, ..., XS
), where Xs
directly a↵ects Ys
. Then interactionamong Y
s
can be modeled via a reduced-form representation:
Ys
= ✓s
(D,X, ✏D), s 2 {1, ..., S}.
In firms’ entry, the first-stage scalar unobservable Us
may represent each firm’s unobservedfixed cost (while Z
s
captures observed fixed cost). The vector of unobservables in the player-specific outcome equation represents multiple shocks, such as the player’s demand shock andvariable cost shock, and other firms’ variable cost shocks and demand shocks. Unlike ina linear model, it would be hard to argue that these errors are all aggregated in a scalarvariable in this nonlinear outcome model, since it is not known in which fashion they enterthe equation.
4.3 Relation to Manski (2013)
Manski (2013) introduces a framework for social interaction where responses (i.e., outcomes)of agents are dependent on one another through their treatments. The framework relaxes thestable unit treatment value assumption (SUTVA) by allowing interaction across the units.
21In this case, Assumption R should be imposed on ✏s,D for each s.
31
Our framework is similar to Manski (2013) in that we also allow interaction among outcomesof players through their treatments, as we discuss in Section 4.2. The di↵erence is that weconsider interaction across treatment/player unit s, whereas he considers interaction acrossobservational unit i. Furthermore, we explicitly model the selection process of how treatmentsare determined simultaneously through players’ strategic interaction. His model, followinghis earlier work (e.g., Manski and Pepper (2000)), stays silent about the process. Despite thedi↵erence, the two settings share a similar spirit of departing from the SUTVA. The shaperestrictions we impose are related to the assumptions of Manski (2013) for the treatmentresponse, which we compare here. First of all, Assumption SY(i) appears in Manski as ananonymity assumption. Also, we find that Assumptions SY(i) and SY2(i) are related to theconstant TR (CTR) assumption in Manski, although he assumes anonymity separate fromthis assumption. The CTR assumption states that, with d = (d
i
)Ni=1,
c(d) = c(d0) =) Yd = Yd0 .
As noted in Manski, c(d) is an e↵ective treatment in that, as long as c(d) stays constant,the response does not change. SY(i) and SY2(i) can be restated using this concept with aparticular choice of c(d): with d = (d
s
)Ss=1 ,
c(d) = c(d0) =) E[Yd|X = x,U = u] = E[Yd0 |X = x,U = u]
for given x 2 X and a.e. u, where c(d) is chosen such that the game for treatment decisionshas a unique equilibrium in terms of c(d). The symmetry assumption (Assumption SY(i))can be seen as one example where the game has a unique equilibrium in terms of c(d) withc(d) =
P
S
s=1 ds, namely, the numeber of players who choose to take the action. Likewise,
SY2(i) corresponds to c(d) = (c1(d), ..., cG(d)) with cg
(d) =P
S
g
s=1 dg
s
. There can certainlybe other choices of c(d) that delivers a unique equilibrium in each game, although we do notexplore this further.
4.4 Point Identification of the ATE
When there exist player-specific excluded instruments of large support, we point identifythe ATEs. In this case, the shape restrictions (especially on the outcome function) are notneeded. The following assumption holds for each s 2 {1, ..., S}.
Assumption NU2. For each d�s
2 D�s
, ⌫s(d�s
, ·)|(X,Z�s
) has an everywhere positiveLebesgue density.
Assumption NU2 is stronger than Assumption NU. It imposes not only the exclusionrestriction of NU but also a player-specific exclusion restriction and large support.
Theorem 4.1. In model (2.1) and (2.2), suppose Assumptions IN, E and NU2 hold. Thenthe ATE in (2.4) is identified.
The identification strategy is to employ the identification at infinity argument basedon Assumption NU2, which simultaneously solves the multiple equilibria problem and theendogeneity problem. Suppose S = 2 and Z
s
is scalar for illustration; the general case can
32
be proved analogously. For example, to identify E[Y11|X], consider
E[Y |D = (1, 1), X = x,Z = z] = E[Y11|D = (1, 1), X = x,Z = z]
= E[✓(1, 1, x, ✏11)|⌫1(1, z1) � U1, ⌫2(1, z2) � U2]
! E[✓(1, 1, x, ✏11)] = E[Y11|X = x],
where the second equation is by Assumption IN, and the convergence is by Assumption NU2with z1 ! 1 and z2 ! 1. Likewise, E[Y00|X = x] can be identified. The identification ofE[Y10|X = x] and E[Y01|X = x] can be achieved by similar reasoning. Note that D = (1, 0)or D = (0, 1) can be predicted as an outcome of multiple equilibria. When either (z1, z2) !(1,�1) or (z1, z2) ! (�1,1) occurs, however, a unique equilibrium is guaranteed as adominant strategy, i.e., D = (1, 0) or D = (0, 1), respectively. Based on these results, we can(point) identify all the ATE’s.
5 The LATE
The results of Proposition 3.1 can be used to establish a framework that defines the LATEparameter for multiple treatments that are generated by strategic interaction. In this section,given model (2.1)–(2.2), we only maintain the assumptions on the equations for D
s
such asthe symmetry in the payo↵ functions, but not the assumptions on the equation for Y . Inparticular, we no longer require Assumptions R, M(i) and SY(i). In the case of a singlebinary treatment, there is well-known equivalence between the LATE monotonicity assump-tion and the specification of a selection equation (Vytlacil (2002)). This equivalence resultis inapplicable to our setting due to the simultaneity in the first stage.22 But Proposition3.1 implies that, under Assumptions SS and SY(ii), there is in fact a monotonic pattern inthe way the equilibrium regions lie in the space of U as written in (3.23). This monotonicity,formalized in Corollary 3.1, allows us to establish equivalence between a version of the LATEmonotonicity assumption and the simultaneous selection model (2.2). We first introduce arelevant counterfactual outcome that can be used in defining the LATE parameter.
For M ✓ D, introduce a selection variable D
M
2 M that selects an equilibrium D
M
= d
when facing a set of equilibria, M . This variable is useful in decomposing the event D = d
into two sequential events: D = d is equivalent to an event that D 2 M and D
M
= d.Trivially, we have DD = D. When M ( D is not a singleton, D
M
is not observed preciselybecause the equilibrium selection mechanism is not observed in general.23 Using D
M
, we
22For example in a two-player entry game, when entry costs Z1 and Z2 increase, it may be the case thatin one market only the first player enters given this increase as her monopolistic profit o↵sets the increasedcost, while in another market only the second player enters by the same reason applied to this player. Thedirection of monotonicty is reversed in these two markets.
23Alternatively, following the notation of Heckman et al. (2006), we can introduce a equilibrium selectionindicator D
M,d that indicates that an equilibrium d is selected among equilibria in a set M :
DM,d =
(1 if d 2 M is selected,
0 o.w.
Then, DM
= d if and only if DM,d = 1.
33
define a joint counterfactual outcome YM
as an outcome had D been an element in M :
YM
=X
d2M1[D
M
= d]Yd. (5.1)
Conditional onD 2 M , YM
is assigned to be one of the usual counterfactual outcome Yd basedon the equilibrium being selected. When M = D, we can write Y = YD =
P
d2D 1[D = d]Yd,which yields the standard expression that relates the observed outcome with the potentialoutcomes. Moreover, for any partition {M
k
}Kk=1 such that
S
K
k=1 Mk
= D, we can express
X
d2D1[D = d]Yd =
K
X
k=1
X
d2Mk
1[D 2 Mk
]1[DM
k
= d]Yd =K
X
k=1
1[D 2 Mk
]YM
k
,
where the first equality is by the equivalence of the events mentioned above and the secondequality is by (5.1). Therefore, we can establish the following relationship:
Y =K
X
k=1
1[D 2 Mk
]YM
k
, (5.2)
that is, YM
k
is observed when D 2 Mk
.Now, consider a treatment of dichotomous states (e.g., dichotomous market structures):
for j = 0, ..., S � 1,
D 2 M>j vs. D 2 Mj ,
where Mj ⌘S
j
k=0Mk
and M>j ⌘S
S
k=j+1Mk
are previously defined; e.g., for S = 2 and
j = 1, M1 = {(1, 0), (0, 1), (0, 0)} and M>1 = {(1, 1)}. Consider a corresponding treatmente↵ect:
YM
>j
� YM
j
,
where Y = 1[D 2 M>j ]YM
>j
+ 1[D 2 Mj ]YM
j
by (5.2). This quantity is the e↵ect ofbeing treated with an equilibrium of at least j + 1 entrants relative to being treated with anequilibrium of at most j entrants. We now establish that a version of the LATE monotonicityassumption for this treatment 1[D 2 M>j ] of dichotomous states is implied by the modelspecification (2.2), using Corollary 3.1. Let D(z) ⌘ (D1(z1), ..., DS
(zS
)) where Ds
(zs
) is thepotential treatment decision had the player s been assigned Z
s
= zs
.
Lemma 5.1. Under Assumptions M(ii), SS and SY(ii), the first-stage game (2.2) impliesthat, for any z, z0 2 Z and j = 0, ..., S � 1,
D(z0) 2 M>j ) D(z) 2 M>j w.p.1or
D(z) 2 M>j ) D(z0) 2 M>j w.p.1(5.3)
Proof. By Corollary 3.1, (0, ⌫sj
(zs
)] 8s are the only relevant intervals in defining R
>j(z) =
D\Rj . For given z, z0 2 Z, suppose without loss of generality that in Assumption M(ii),
34
⌫sd�s
(zs
) � ⌫sd�s
(z0s
) 8d�s
and 8s. Then it follows that R>j(z) ◆ R
>j(z0), and thus w.p.1,
1[D(z) 2 M>j ] = 1[U 2 R
>j(z)] � 1[U 2 R
>j(z0)] = 1[D(z0) 2 M>j ].
Now, Lemma 5.1 allows us to give the IV estimand an LATE interpretation in our model:
Theorem 5.1. Given model (2.1)–(2.2), suppose Assumptions IN, NU, M(ii), SS and SY(ii)hold. Then it satisfies that, for any j = 0, ..., S � 1,
h(z, z0)P
k>j
hDk
(z, z0)=
E[Y |Z = z]� E[Y |Z = z
0]
Pr[D 2 M>j |Z = z]� Pr[D 2 M>j |Z = z
0]
= E[YM
>j
� YM
j
|D(z) 2 M>j ,D(z0) 2 Mj ].
The LATE parameter E[YM
>j
� YM
j
|D(z) 2 M>j ,D(z0) 2 Mj ] is the average oftreatment e↵ect Y
M
>j
� YM
j
for a subgroup of “markets” that form more competitivemarkets (with at least j + 1 entrants) when players face Z = z, but form less competitivemarkets (with at most j entrants) when players face Z = z
0. For concreteness, supposeS = 2, j = 1, Z
s
is each airline company’s entry cost and Y is the pollution level in a market.The LATE
E[Y{(1,1)} � Y{(1,0),(0,1),(0,0)}|D(z) = (1, 1),D(z0) 2 {(1, 0), (0, 1), (0, 0)}]
is the e↵ect of the existence of competition on pollution levels for markets consist of “compli-ers.”24 It is the average di↵erence of potential pollution levels in a duopolistic market (i.e.,competition) versus a monopolistic or non-operating market (i.e., no competition) for thesubgroups of markets that form a duopoly when companies are facing low cost (Z = z) butform a monopoly or do not operate when facing high cost (Z = z
0). Figure 7 depicts this sub-group of markets. In this example, the LATE monotonicity assumption (implied by the entrygame of strategic substitute with symmetric payo↵s) rules out those markets that respond toentry cost as “defiers.” The LATE becomes the ATE when 1 = Pr[D(z) = (1, 1),D(z0) 2{(1, 0), (0, 1), (0, 0)}] = Pr[D = (1, 1)|Z = z]�Pr[D 2 {(1, 0), (0, 1), (1, 1)}|Z = z
0], which isrelated to the identification at infinity argument in Theorem 4.1.
In general, the LATE can be defined with YM
�YM
0 for any two partitioning setsM andM 0
of D (i.e., D = M[M 0 with M\M 0 = ;) as long as 1[D(z) 2 M ] = 1�1[D(z) 2 M 0] satisfiesthe LATE monotonicity assumption. Lemma 5.1 ensures that our simultaneous selectionmodel imposes this monotonicity for a particular partition, M = M>j and M 0 = Mj .
Remark 5.1. Similarly, it may be possible to recover the marginal treatment e↵ect (MTE) ofHeckman and Vytlacil (1999, 2005, 2007). Given our setting, it should be a transition-specificMTE for Y
M
j
� YM
j�1. The identification of this MTE would require continuous variation ofZ. For discrete Z, the approach by Brinch et al. (2017) can be applied by imposing structures
24In this multi-agent multi-treatment scenario, compliers are defined as those players whose behaviors aresuch that market structures are formed in conformance with the LATE monotonicity assumption (5.3). Unlikein the traditional setting (Imbens and Angrist (1994)) where compliers are defined in terms of the subset ofpopulation, the subpopulation in the present setting is the collection of the markets consist of the complyingplayers.
35
on the MTE function. The MTE approach is also related to Lee and Salanie (2016). Notethat the LATE can be expressed as an integrated MTE. The LATE parameter considered inthis paper is di↵erent from their LATE parameter. Instead of considering the ATE with twodi↵erent treatment values for what they call “super-compliers” group, we consider an averagedichotomous treatment e↵ect for compliers group between the dichotomous choices. Whetherone version is more appropriate than theirs depends on applications.
Remark 5.2. The equilibrium selection mechanism may di↵er across di↵erent counterfactualworlds. In terms of our notation, D
M
(z) may di↵er from D
M
(z0), where D
M
(z) is thecounterfactual variable of D
M
. Note that not only the equilibrium being selected is di↵erentbut also the selection mechanism can be di↵erent. This feature may be emphasized by writingD
M
(z) = �z(z,U) where the functional form of the equilibrium selection function may alsochange in z. By considering Y
M
instead of Yd, however, we can be agnostic about the selectionmechanism, i.e., about the specification of �z(·, ·). The definition (5.1) asserts that Yd can bemeaningfully analyzed within the current framework only when the equilibrium being selectedis known.
6 Numerical Studies
To illustrate the main results of this paper, we calculate the bounds on the ATE using thefollowing data generating process:
Yd = 1{µd + �X � ✏},
D1 = 1{�2D2 + �1Z1 � V1},
D2 = 1{�1D1 + �2Z2 � V2},
where (✏, V1, V2) are drawn from a joint normal distribution with zero means and each cor-relation coe�cient being 0.5, and drawn independent of (X,Z). We draw Z
s
(s = 1, 2) andX from multinomial, allowing Z
s
to take two values, Zs
= {�1, 1} and X to take eitherthree values, X = {�1, 0, 1}, or fifteen values, X = {�1,�6
7 ,�57 , ...,
57 ,
67 , 1}. Being consistent
with Assumptions M and SY, we choose µ11 > µ10 = µ01 > µ00, and with Assumption SS,we choose �1 < 0 and �2 < 0. Without loss of generality, we choose positives values for �1,�2, and �. Specifically, µ11 = 0.25, µ10 = µ01 = 0 and µ00 = �0.25. For default values,�1 = �2 ⌘ � = �0.1, �1 = �2 ⌘ � = 1 and � = 0.5.
In this exercise, we focus on the following ATE:
E[Y11 � Y00|X = 0] = �(µ11)� �(µ00),
where �(·) is the CDF of the standard normal distribution. Given the parameter values,E[Y11 � Y00|X = 0] = 0.2. For h(z, z0, x), we consider z = (1, 1) and z
0 = (�1,�1). Notethat H(x) = h(z, z0, x) and H(x, x0, x00) = h(z, z0;x, x0, x00) since Z
s
is binary. Then we canderive the sets XU
d (0;d0) and XL
d (0;d0) for each d 2 {(1, 1), (0, 0)} and d
0 6= d in Theorem3.1.
Based on our design, H(0) > 0 and thus the bounds when we use Z only are, with x = 0,
maxz2Z
Pr[Y = 1,D = (0, 0)|z, x] Pr[Y00 = 1|x] minz2Z
Pr[Y = 1|z, x],
36
and
maxz2Z
Pr[Y = 1|z, x] Pr[Y11 = 1|x] minz2Z
{Pr[Y = 1,D = (1, 1)|z, x] + 1� Pr[D = (1, 1)|z, x]} .
Using bothZ andX, we have narrower bounds. For example when |X | = 3, withH(0,�1,�1) <0, the lower bound on Pr[Y00 = 1|X = 0] becomes
maxz2Z
{Pr[Y = 1,D = (0, 0)|z, 0] + Pr[Y = 1,D 2 {(1, 0), (0, 1)}|z,�1]} .
With H(1, 1, 0) < 0, the upper bound on Pr[Y11 = 1|X = 0] becomes
minz2Z
{Pr[Y = 1,D = (1, 1)|z, 0] + Pr[Y = 1,D 2 {(1, 0), (0, 1)}|z, 1] + Pr[D = (0, 0)|z, 0]} .
For comparison, we calculate the bounds by Manski (1990) using Z. The Manski’s boundsare
maxz2Z
Pr[Y = 1,D = (0, 0)|z, x] Pr[Y00 = 1|x]
minz2Z
{Pr[Y = 1,D = (0, 0)|z, x] + 1� Pr[D = (0, 0)|z]} ,
and
maxz2Z
Pr[Y = 1,D = (1, 1)|z, x] Pr[Y11 = 1|x]
minz2Z
{Pr[Y = 1,D = (1, 1)|z, x] + 1� Pr[D = (1, 1)|z]} .
We also compare the estimated ATE using the specification of a standard linear IV modelwhere the nonlinearity of the true DGP are ignored:
Y = ⇡0 + ⇡1D1 + ⇡2D2 + �X + ✏,✓
D1
D2
◆
=
✓
�10�20
◆
+
✓
�11 �12�21 �22
◆✓
Z1
Z2
◆
+
✓
V1
V2
◆
.
Here the first stage is the reduced-form representation of the linear simulatneous equationsmodel for strategic interaction. Under this specification, the ATE becomes E[Y11 � Y00|X =0] = ⇡1 + ⇡2, which is estimated via two-stage least squares (TSLS).
The bounds calculated for the ATE are shown in Figures 8–11. Figure 8 shows how thebounds on the ATE change as the value of � changes from 0 to 2.5. The larger � is thestronger the instrument Z is. The first conspicuous result is that the TSLS estimate of theATE is biased due the the problem of misspecification. Next, as expected, the Manski’sbounds and our proposed bounds converge to the true value of the ATE as the instrumentbecomes stronger. Overall, our bounds, with or without exploiting the variation of X, aremuch narrower than the Manski bounds.25 Notice that the sign of the ATE is identified inthe whole range of � as predicted by the first part of Theorem 3.1, in contrast to the Manski’s
25Although we do not make a rigorous comparison of the assumptions here, note that the bounds by Manskiand Pepper (2000) under the semi-MTR is expected to be similar to ours. Their bounds, however, need toassume the direction of the monotonicity.
37
bounds. By using the additional variation from X with |X | = 3, we further narrow downthe bounds, in particular the upper bounds on the ATE in this particular simulation design.Figure 9 depicts the bounds using X with |X | = 15, which yields narrower bounds than usingX with |X | = 3.
Figure 10 shows how the bounds change as the value of � changes from 0 to 1.5, where alarger � corresponds to a stronger exogenous variable X. The jumps in the upper bound areassociated with the sudden changes in the signs of H(�1, 0,�1) and H(0, 1, 1). At least inthis simulation design, the strength of X is not a crucial factor to obtain narrower bounds.In fact, based other simulation results (which are omitted in the paper), we conclude thatthe number of values X can take matters more than the dispersion of X (unless we pursuepoint identification of the ATE).
Figure 11 shows how the width of the bounds is related to the extent to which theopponents’ actions D�s
a↵ect one’s payo↵, captured in �. We vary the value of � from �2 to0, and when � = 0, the players solve a single-agent optimization problem. Thus, heuristically,the bound at this point would be similar to the ones that can be obtained when Shaikh andVytlacil (2011) is extended to a multiple-treatment setting with no simultaneity. In thefigure, as the value of � gets smaller, the bounds get narrower.
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A Appendix
A.1 Proof of Proposition 3.1
The following proposition is useful in proving Proposition 3.1:
Proposition A.1. Let R and Q be sets defined by Cartesian products: R =Q
S
s=1 rs and
Q =Q
S
s=1 qs where rs
and qs
are intervals in R. Then the following holds:(i) If r
s
\ qs
= ; for some s, then R \Q = ;;(ii) If r
s
⇠ qs
8s, then R ⇠ Q;(iii) If r
s
⌧ qs
for some s, then R ⌧ Q;(iv) R \Q =
Q
S
s=1 rs \ qs
;
(v) cl(R) =Q
S
s=1 cl(rs) where cl(·) is the closure of its argument.
The proof of this proposition follows directly from the definition of R and Q.Before proving Proposition 3.1(i), we prove (i0) that appears in the text below Proposition
3.1. We first show a simple case as a reference: Rej \Rej�1 = ; for j = 1, ..., S. Note that
Rej (z) =
(
j
Y
s=1
�
0, ⌫sj�1(zs)
⇤
)
⇥
8
<
:
S
Y
s=j+1
�
⌫sj
(zs
), 1⇤
9
=
;
Rej�1(z) =
(
j�1Y
s=1
�
0, ⌫sj�2(zs)
⇤
)
⇥
8
<
:
S
Y
s=j
�
⌫sj�1(zs), 1
⇤
9
=
;
and the j-th coordinates are⇣
0, ⌫jj�1(zj)
i
in Rej and⇣
⌫jj�1(zj), 1
i
in Rej�1 . Since these two
intervals are disjoint, by Proposition A.1(i), we can conclude that Rej \ Rej�1 = ;. Now toprove (i0), we equivalently prove Rdj
\ Rdj�t
= ; for t � 1 and 0 j � t S � t, and drawinsights from the simple case. Note that d
j�t contains S � j + t zeros. Then there exists
s⇤ such that djs
⇤ = 1 but dj�t
s
⇤ = 0, i.e., Us
⇤ 2⇣
0, ⌫s⇤
j�1(zs⇤)i
in Rdj
but Us
⇤ 2⇣
⌫s⇤
j�t
(zs
⇤), 1i
in Rdj�t
. Suppose not. Then 8s such that djs
= 1, it must hold that dj�t
s
= 1. Thisimplies that d
j�t has at least as many elements of unity as d
j , which is contradiction as
t � 1. Therefore since⇣
0, ⌫s⇤
j�1(zs⇤)i
and⇣
⌫s⇤
j�t
(zs
⇤), 1i
are disjoint, Rdj
and Rdj�t
are
disjoint. When t � 2, by Assumption SS, ⌫s⇤
j�t
(zs
⇤) > ⌫s⇤
j�1(zs⇤) and therefore⇣
⌫s⇤
j�t
(zs
⇤), 1i
and⇣
0, ⌫s⇤
j�1(zs⇤)i
are disjoint and thus the same conclusion follows. Also when t = 1,⇣
⌫s⇤
j�1(zs⇤), 1i
and⇣
0, ⌫s⇤
j�1(zs⇤)i
are obviously disjoint. This proves (i0).
For Proposition 3.1(i), one can conclude from (i0) that Rdj
is disjoint to Rdj
0 for any
d
j
0 2 Mj
0 and hence is disjoint toS
d2Mj
0 Rd. This is true 8dj 2 Mj
, and thereforeS
d2Mj
Rd
is disjoint toS
d2Mj
0 Rd.
To prove (ii0) of the text for two cartesian products, by Proposition A.1(ii), one needs toshow that each pair of intervals of the same coordinate are neighboring intervals. This is im-
mediately true forRej andRej�1 above, since (a) for coordinates 1 s j�1,⇣
0, ⌫sj�1(zs)
i
⇠
41
⇣
0, ⌫sj�2(zs)
i
with a nonempty intersection since⇣
0, ⌫sj�1(zs)
i
⇢⇣
0, ⌫sj�2(zs)
i
; (b) for coor-
dinate s = j,⇣
0, ⌫jj�1(zj)
i
⇠⇣
⌫jj�1(zj), 1
i
and they are disjoint; and (c) for coordinates
j + 1 s S,⇣
⌫sj
(zs
), 1i
⇠⇣
⌫sj�1(zs), 1
i
with a nonempty intersection since⇣
⌫sj
(zs
), 1i
�⇣
⌫sj�1(zs), 1
i
. Now consider Rdj
and Rdj�1 . In d
j and d
j�1, there exists s⇤ such that djs
⇤ = 1
but dj�1s
⇤ = 0 by the same argument as above with t = 1. The rest of the elements in d
j and
d
j�1 fall into one of the four types: for s 6= s⇤, (a0) djs
= dj�1s
= 1; (b0) djs
= 1 but dj�1s
= 0;(c0) dj
s
= dj�1s
= 0; and (d0) djs
= 0 but dj�1s
= 1. See Table 1 in the main text for anexample of this result. We want to express the corresponding intervals of U
s
that generatethese values of dj
s
and dj�1s
. By definition, the number of ones (and zeros) in d
j and d
j�1
di↵ers only by one, which happens in each vector’s s⇤-th element. Knowing this, for thesepairs of dj
s
and dj�1s
in (a0)–(d0), we can determine the decision of the opponents of players (i.e., the value of j in ⌫s
j
) which is useful to construct the payo↵ of s, and thus the corre-sponding interval of U
s
. Specifically, we can determine that the corresponding interval pairs
are: (a00)⇣
0, ⌫sj�1(zs)
i
and⇣
0, ⌫sj�2(zs)
i
; (b00)⇣
0, ⌫sj�1(zs)
i
and⇣
⌫sj�1(zs), 1
i
; (c00)⇣
⌫sj
(zs
), 1i
and⇣
⌫sj�1(zs), 1
i
; (d00)⇣
⌫sj
(zs
), 1i
and⇣
0, ⌫sj�2(zs)
i
. It is straightforward that the pairs in
(a00)–(c00) are neighboring sets by the same arguments as for (a)–(c). The pair in (d00) arealso neighboring sets because ⌫s
j
(zs
) < ⌫sj�2(zs) by Assumption SS. Lastly, for coordinate s⇤,
⇣
0, ⌫s⇤
j�1(zs⇤)i
⇠⇣
⌫s⇤
j�1(zs⇤), 1i
as in (b00). Therefore, Rdj
⇠ Rdj�1 .
For Proposition 3.1(ii), one can conclude from (ii0) that Rdj
neighbors Rdj�1 for anyd
j�1 2 Mj�1 and hence neighbors
S
d2Mj�1
Rd. This is true 8dj 2 Mj
, and thereforeS
d2Mj
Rd ⇠S
d2Mj
0 Rd.
The result in Proposition 3.1(iii) follows from the proof of (i’) above that there exists s⇤
such that djs
⇤ = 1 but dj�t
s
⇤ = 0, i.e., Us
⇤ 2⇣
0, ⌫s⇤
j�1(zs⇤)i
in Rdj
but Us
⇤ 2⇣
⌫s⇤
j�t
(zs
⇤), 1i
in
Rdj�t
. When t � 2, by Assumption SS, ⌫s⇤
j�t
(zs
⇤) > ⌫s⇤
j�1(zs⇤) and therefore⇣
0, ⌫s⇤
j�1(zs⇤)i
⌧⇣
⌫s⇤
j�t
(zs
⇤), 1i
which implies that, by Proposition A.1(iii), their Cartesian products are not
neighboring sets.
Lastly, we prove Proposition 3.1(iv). We consider a S-dimensional hyper-grid for (0, 1]S
that runs through all possible values of ⌫sj
across j = 0, ..., S for each s = 1, ..., S. Specifically,under Assumption SS and by conveniently letting ⌫s
S
= 0 and ⌫s�1 = 1, the hyper-grid is aCartesian product of 1-dimensional grids defined by 0 = ⌫s
S
< ⌫sS�1 < · · · < ⌫s0 < ⌫s�1 = 1 for
each coordinate s. Let each hyper-cube in this hyper-grid be represented as
r1(j1)⇥ r2(j2)⇥ · · ·⇥ rS
(jS
) ⌘�
⌫1j1, ⌫1
j1�1
⇤
⇥�
⌫2j2, ⌫2
j2�1
⇤
⇥ · · ·⇥�
⌫Sj
S
, ⌫Sj
S
�1
⇤
,
where rs
(·) are intervals implicitly defined in the equation and labeled with js
= 0, ..., S.
42
Using these notations, Rej for j = 0, ..., S can be expressed as
Rej =
8
<
:
U : (U1, ..., US
) 2
8
<
:
j
Y
s=1
S
[
k=j
rs
(k)
9
=
;
⇥
8
<
:
S
Y
s=j+1
j
[
k=0
rs
(k)
9
=
;
9
=
;
(A.1)
=
8
<
:
U : (U1, ..., US
) 2S
[
j1=j
· · ·S
[
j
j
=j
·j
[
j
j+1=0
· · ·j
[
j
S
=0
r1(j1)⇥ · · ·⇥ rS
(jS
)
9
=
;
, (A.2)
where the second equality is by iteratively applying the following: for sets A, B and C beingCartesian products (including intervals as a trivial case),
(A [B)⇥ C = (A⇥ C) [ (B ⇥ C).
More generally, Rdj
�
for some �(·) 2 ⌃ can be defined as
Rdj
�
=
8
<
:
U : (U�(1), ..., U�(S)) 2
8
<
:
j
Y
s=1
S
[
k=j
r�(s)(k)
9
=
;
⇥
8
<
:
S
Y
s=j+1
j
[
k=0
r�(s)(k)
9
=
;
9
=
;
(A.3)
=
8
<
:
U : (U�(1), ..., U�(S)) 2
S
[
j1=j
· · ·S
[
j
j
=j
·j
[
j
j+1=0
· · ·j
[
j
S
=0
r�(1)(j1)⇥ · · ·⇥ r
�(S)(jS)
9
=
;
.
(A.4)
Below we show that any hyper-cube r1(j1)⇥ r2(j2)⇥ · · ·⇥ rS
(jS
) is contained in one of Rdj
�
’s
for some j and �(·). We first proceed by showing that there are hyper-cubes that are con-tained in Rej ’s. We then show that any hyper-cube can be transformed using a permutationfunction into a hyper-cube contained in Rej , which means that the original hyper-cube iscontained in some Rdj
which is a “permutated version” of Rej .
Claim 1: For j1 � j2 � · · · � jS
, r1(j1)⇥ r2(j2)⇥ · · ·⇥ rS
(jS
) is contained in Rej for somej j1.Claim 2: For any {j1, ..., jS}, r1(j1) ⇥ r2(j2) ⇥ · · · ⇥ r
S
(jS
) is contained in Rdj
for j max{j1, ..., jS}.
Proof of Claim 1: Start with a hyper-cube at a corner:
r1(0)⇥ r2(0)⇥ · · ·⇥ rS
(0) ⌘�
⌫10 , 1⇤
⇥�
⌫20 , 1⇤
⇥ · · ·⇥�
⌫S0 , 1⇤
.
This hyper-cube is contained in Re0 as the two in fact coincide. Consider the next hyper-cubeon the grid along the 1-st coordinate:
r1(1)⇥ r1(0) · · ·⇥ rS
(0).
43
This hyper-cube is contained in Re1 as
Re1 =S
[
j1=1
·1[
j2=0
· · ·1[
j
S
=0
r1(j1)⇥ · · ·⇥ rS
(jS
).
We move to the 2-nd coordinate holding the 1-st coordinate fixed. Then r1(1)⇥r2(1)⇥r3(0)⇥· · ·⇥ r
S
(0) is still contained in Re1 . Likewise, from r1(1)⇥ r2(1)⇥ r3(1)⇥ r4(0)⇥ · · ·⇥ rS
(0)all the way to r1(1)⇥ · · ·⇥ r
S
(1), these hyper-cubes are contained in Re1 .Now consider the next hyper-cube along the 1-st coordinate, i.e., r1(2)⇥r2(0)⇥· · ·⇥r
S
(0).This is contained in Re1 . We move to the next coordinates holding the 1-st coordinatefixed. Then r1(2)⇥ r2(1)⇥ r3(0)⇥ · · ·⇥ r
S
(0), r1(2)⇥ r2(1)⇥ r3(1)⇥ r4(0)⇥ · · ·⇥ rS
(0) tor1(2)⇥r1(1)⇥· · ·⇥r
S
(1) are still contained inRe
1 . But the next r1(2)⇥r2(2)⇥r3(0)⇥· · ·⇥rS
(0)is no longer contained in Re1 but is contained in
Re2 =S
[
j1=2
·S
[
j2=2
·2[
j3=0
· · ·2[
j
S
=0
r1(j1)⇥ r2(j2)⇥ · · ·⇥ rS
(jS
).
Likewise, following the same sequential rule, r1(2) ⇥ r2(2) ⇥ r3(1) ⇥ r4(0) ⇥ · · · ⇥ rS
(0),r1(2)⇥r2(2)⇥r3(1)⇥r4(1)⇥r5(0)⇥ · · ·⇥r
S
(0) to r1(2)⇥ · · ·⇥rS
(2) are all contained in Re2 .This argument can iteratively be applied to all other hyper-cubes r1(j1)⇥r2(j2)⇥ · · ·⇥r
S
(jS
)generated by the same sequential rule maintaining j1 � j2 � · · · � j
S
. This proves Claim 1.
Proof of Claim 2: In general, consider r1(j1)⇥ · · ·⇥rS
(jS
) for given j1, ..., jS . There existspermutation �(·) and a sequence {k
s
}Ss=1 such that j
s
= k�(s) and k1 � k2 � · · · � k
S
. Thena hyper-cube
r�(1)(j1)⇥ · · ·⇥ r
�(S)(jS) = r�(1)(k�(1))⇥ · · ·⇥ r
�(S)(k�(S))
in the space of (U�(1), ..., U�(S)) or equivalently
r1(k1)⇥ · · ·⇥ rS
(kS
)
in the space of (U1, ..., US
) satisfies the condition in Claim 1 and thus is contained in Rej
for some j kS
by Claim 1. Let ��1(·) be the inverse of �(·). Note that ��1(·) itself is apermutation function. In general, for permutation �(·), if r1(k1)⇥ · · ·⇥r
S
(kS
) is contained inRej for some j, then r
�(1)(k1)⇥ · · ·⇥ r�(S)(kS) is contained in Rdj
�
by definition. Therefore,
since r�
�1(�(s))(js) = rs
(js
) 8s, we can conclude that r1(j1) ⇥ · · · ⇥ rS
(jS
) is contained inRdj
�
�1for j k
S
= j�
�1(S). This proves Claim 2.
A.2 Proof of Corollary 3.1
First, consider a pair of Rdj+1(z) and Rdj
(z) (for dj+1 2 Mj+1 and d
j 2 Mj
) in R
j+1(z) andR
j
(z), respectively. From the proof of Proposition 3.1(ii), we know that the elements in d
j+1
and d
j fall into one of the four types (a0)–(d0) (including s⇤), and thus the corresponding pairs
of intervals fall into one of the four types: (a†)⇣
0, ⌫sj
(zs
)i
and⇣
0, ⌫sj�1(zs)
i
; (b†)⇣
0, ⌫sj
(zs
)i
44
and⇣
⌫sj
(zs
), 1i
; (c†)⇣
⌫sj+1(zs), 1
i
and⇣
⌫sj
(zs
), 1i
; (d†)⇣
⌫sj+1(zs), 1
i
and⇣
0, ⌫sj�1(zs)
i
.
Definition A.1. For two Cartesian products R and Q such that R ⇠ Q and R \ Q = ;,their border R k Q is a set that satisfies R k Q ⌘ cl(R) \ cl(Q). Also, the border R k Q is ahyper-surface that is common to cl(R) and cl(Q).
By Proposition 3.1, Rdj+1(z) ⇠ Rdj
(z) and Rdj+1(z)\Rdj
(z) = ;, and thus their bordercan be properly defined. Given (a†)–(d†), we show that Rdj+1(z) k Rdj
(z) is a Cartesian
product of⇣
0, ⌫sj
(zs
)i
k⇣
⌫sj
(zs
), 1i
= {⌫sj
(zs
)} (for some s) and other intervals. Specifically,
by applying Proposition A.1(iv) and (v) with R = cl(Rdj+1(z)), Q = cl(Rdj
(z)), and rs
andqs
being the closures of the intervals in (a†)–(d†), we have
Rdj+1(z) k Rdj
(z) = {⌫sj
(zs
)}⇥Y
k 6=s
rk
\ qk
, (A.5)
for some s, where each rk
\qk
is one ofh
0, ⌫kj
(zk
)i
, {⌫kj
(zk
)},h
⌫kj
(zk
), 1i
, andh
⌫kj+1(zk), ⌫
k
j�1(zk)i
.
Observe that Rdj+1(z) k Rdj
(z) is therefore a lower-dimensional Cartesian product (with di-mension less than S), which is consistent with the notion of a border or a hyper-surface. Also,observe that this hyperspace is located at ⌫s
j
(zs
) in the s-coordinate. Likewise, (A.5) holds forany Rdj+1(z) and Rdj
(z) pair with a di↵erent value of s and di↵erent choice for each rk
\ qk
.But, since cl(A[B) = cl(A)[ cl(B) for any sets A and B, cl(R
j+1(z)) =S
d2Mj+1
cl(Rd(z))
and cl(Rj
(z)) =S
d2Mj
cl(Rd(z)), and thus
R
j+1(z) k R
j
(z) =[
dj+12Mj+1
[
dj2Mj
(Rdj+1(z) k Rdj
(z)) . (A.6)
Now, let R
>j(z) ⌘S
d2M>j
Rd(z) = U\Rj(z) where M>j ⌘S
S
k=j+1Mk
. Note that
R
j(z) ⇠ R
>j(z) and R
j(z) \R
>j(z) = ; by Proposition 3.1. Then R
j(z) k R
>j(z) =R
j+1(z) k R
j
(z) by the discussions around (3.23). Since Rj(z)[R
>j(z) = U by definition,R
j(z) k R
>j(z) is the only nontrivial hyper-surface of cl(Rj(z)) (and of cl(R>j(z))),i.e., a surface that is not part of the surface of cl(U). Therefore by (A.5) and (A.6), wecan conclude that cl(Rj(z)) and hence R
j(z) is a function of z only through ⌫sj
(zs
) 8s.Moreover, in the expression of Rdk
(z) in (3.21) with k j�1 (and hence in the expression ofR
j�1(z)), there is no interval with ⌫sj
(zs
) in its endpoint by definition.26 Also, the intervalin the expression of Rdj
(z) in (3.21) (and hence in the expression of Rj
(z)) that has ⌫sj
(zs
) in
its endpoints is⇣
⌫sj
(zs
), 1i
8s. Consequently, Rj(z) = R
j�1(z) [R
j
(z) is only expressed
with⇣
⌫sj
(zs
), 1i
8s and (0, 1]. If Rj(z) is expressed using other intervals whose endpoints
are functions of zs
, then it contradicts the fact that Rj(z) is a function of z only through⌫sj
(zs
). This completes the proof.
26That is, the payo↵ ⌫s
j
(zs
) is not relevant in defining markets with fewer than j entrants.
45
A.3 Proof of Lemma 3.4
We prove Lemma 3.4 by drawing on the results of Proposition 3.1. As discussed in the text
in relation to (3.23), for z and z
0 such thatP
S
k=j
0 hDk
(z, z0) =n
1� Pr[U 2 R
j
0�1(z)]o
�n
1� Pr[U 2 R
j
0�1(z0)]o
> 0 for j0 = 1, ..., S, we have
R
j(z) ✓ R
j(z0) (A.7)
for j = 0, ..., S, including R
S(z) = R
S(z0) = U as a trivial case. For those z and z
0, introducenotations27
�j,+(z, z
0) ⌘ R
j
(z)\Rj
(z0), (A.8)
�j,�(z, z
0) ⌘ R
j
(z0)\Rj
(z), (A.9)
and
�j(z0, z) ⌘ R
j(z0)\Rj(z). (A.10)
Note that, for j = 1, ..., S,
R
j
(·) = R
j(·)\Rj�1(·), (A.11)
since R
j(z) ⌘S
j
k=0Rk
(z). Fix j = 1, ..., S. Consider
�j,+(z, z
0) =�
R
j(z) \R
j�1(z)c�
\�
R
j(z0) \R
j�1(z0)c�
c
=�
R
j(z) \R
j�1(z)c�
\�
R
j(z0)c [R
j�1(z0)�
=�
R
j(z) \R
j�1(z)c \R
j(z0)c�
[�
R
j(z) \R
j�1(z)c \R
j�1(z0)�
=��
R
j(z)\Rj(z0)�
\R
j�1(z)c
[��
R
j�1(z0)\Rj�1(z)�
\R
j(z)
= �j�1(z0, z) \R
j(z),
where the first equality is by plugging in (A.11) into (A.8), the third equality is by thedistributive law, and the last equality is by (A.7) and hence
�
R
j(z)\Rj(z0)�
\R
j�1(z)c = ;.But
�j�1(z0, z) \R
j(z) = �j�1(z0, z)\�
�j�1(z0, z)\Rj(z)�
.
27Note that �+(z, z0) and ��(z, z
0) defined in Section 3.1 for the S = 2 are simplified versions of thesenotations: �+(z, z
0) = �1,+(z, z0) and ��(z, z
0) = �1,�(z, z0).
46
Symmetrically, by changing the role of z and z
0, consider
�j,�(z
0, z) =�
R
j(z0) \R
j�1(z0)c�
\�
R
j(z) \R
j�1(z)c�
c
=�
R
j(z0) \R
j�1(z0)c�
\�
R
j(z)c [R
j�1(z)�
=�
R
j(z0) \R
j�1(z0)c \R
j(z)c�
[�
R
j(z0) \R
j�1(z0)c \R
j�1(z)�
=��
R
j(z0)\Rj(z)�
\R
j�1(z0)c
[��
R
j�1(z)\Rj�1(z0)�
\R
j(z0)
= �j(z0, z) \R
j�1(z0)c,
where the last equality is by (A.7) that Rj�1(z) ⇢ R
j�1(z0). But
�j(z0, z) \R
j�1(z0)c = �j(z0, z)\�
�j(z0, z) \R
j�1(z0)�
.
Note that
�j�1(z0, z)\Rj(z) = �j(z0, z) \R
j�1(z0) ⌘ A⇤,
because
�j�1(z0, z)\Rj(z) = R
j�1(z0) \R
j�1(z)c \R
j(z)c
= R
j�1(z0) \R
j(z)c
= R
j(z0) \R
j(z)c \R
j�1(z0)
= �j(z0, z) \R
j�1(z0),
where the second equality is by R
j�1(z) ⇢ R
j(z) and the third equality is by R
j�1(z0) ⇢R
j(z0). In sum,
�j,+(z, z
0) = �j�1(z0, z)\A⇤, (A.12)
�j,�(z, z
0) = �j(z0, z)\A⇤. (A.13)
(A.12) and (A.13) show how the outflow (�j,+(z, z0)) and inflow (�
j,�(z, z0)) of Rj
can bewritten in terms of the inflows of Rj�1 and R
j , respectively. And figuratively, A⇤ adjusts forthe “leakage” when the change from z to z
0 is relatively large. Now, with #j
(u) ⌘ #j
(x;u) ⌘E[✓(ej , x, ✏)|U = u], (3.25) can be expressed as
Z
Rj
(z)#j
(u)du�Z
Rj
(z0)#j
(u)du
=
Z
�j,+(z,z0)
#j
(u)du+
Z
Rj
(z)\Rj
(z0)#j
(u)du�Z
�j,�(z,z0)
#j
(u)du�Z
Rj
(z)\Rj
(z0)#j
(u)du
=
Z
�j,+(z,z0)
#j
(u)du�Z
�j,�(z,z0)
#j
(u)du, (A.14)
47
where the last equality is derived by IN and SY(i). First, for j = 1, ..., S, by (A.12)–(A.13),
Z
�j,+(z,z0)
#j
(u)du�Z
�j,�(z,z0)
#j
(u)du =
Z
�j(z0,z)\A⇤
#j
(u)du�Z
�j(z0,z)\A⇤
#j
(u)du
=
Z
�j�1(z0,z)\A⇤
#j
(u)du+
Z
A
⇤#j
(u)du�Z
A
⇤#j
(u)du
�(
Z
�j(z0,z)\A⇤
#j
(u)du+
Z
A
⇤#j
(u)du�Z
A
⇤#j
(u)du
)
=
Z
�j�1(z0,z)
#j
(u)du�Z
�j(z0,z)
#j
(u)du, (A.15)
where the last equality is because �j�1(z0, z) � A⇤ and �j(z0, z) � A⇤ by the definition ofA⇤.
For j = 0,Z
�0,+(z,z0)#0(u)du�
Z
�0,�(z,z0)#0(u)du = �
Z
�0(z0,z)
#0(u)du, (A.16)
since �0,+(z, z0) = ; by the choice of (z, z0) and �0,�(z, z0) = �0(z0, z). For j = S,
Z
�S,+(z,z0)
#S
(u)du�Z
�S,�(z,z0)
#S
(u)du =
Z
�S�1(z0,z)
#S
(u)du, (A.17)
since �S,�(z, z0) = ; by the choice of (z, z0) and �
S,+(z, z0) = �S�1(z0, z). Then combining(3.25) and (A.14)–(A.17) evaluated at x = x
j
,
h(z, z0;x) ⌘S
X
j=0
hj
(z, z0, xj
) =S
X
j=1
Z
�j�1(z0,z)
{#j
(xj
;u)� #j�1(xj�1;u)} du.
A.4 Proof of Lemma 3.3(ii)
By Lemma 3.4, h(z, z0;x) =P
S
j=1
R
�j�1(z0,z) {#j
(xj
;u)� #j�1(xj�1;u)} du with�j�1(z0, z) ⌘
Rj�1(z0)\Rj�1(z), which can be rewritten as
h(z, z0;x)�X
k 6=j
Z
�k�1(z0,z)
{#k
(xk
;u)� #k�1(xk�1;u)} du
=
Z
�j�1(z0,z)
{#j
(xj
;u)� #j�1(xj�1;u)} du. (A.18)
We prove the case ◆ = 1; the proof for the other cases follows symmetrically. For k 6= j, when#k�1(xk�1;u)�#
k
(xk
;u) > 0 a.e. u, it satisfies�R
�k�1(z0,z) {#k
(xk
;u)� #k�1(xk�1;u)} du >
0. Combining with h(z, z0;x) > 0 implies that the l.h.s. of (A.18) is positive. This impliesthat #
j
(x;u)�#j�1(x;u) > 0 a.e. u. Suppose not and suppose #
j
(xj
;u)�#j�1(xj�1;u) 0
with positive probability. Then by Assumption Y, #j
(x;u)� #j�1(x;u) 0 a.e. u, which is
contradiction.
48
A.5 Proof of Theorem 3.2
Recall Mj ⌘ M j and M>j ⌘S
S
k=j+1Mk
. Then the bounds (3.36) and (3.37) can berewritten as
Udj
(x) = infz2Z
{pM
>j�1(z, x) + pM
j�1(z)} ,
Ldj
(x) = supz2Z
{pM
j
(z, x) + pM
>j
(z)} .
where for a set M ⇢ D, pM
(z, x) ⌘ Pr[Y = 1,D 2 M |Z = z, X = x] and pM
(z) ⌘Pr[D 2 M |Z = z]. Since D = Mj [ M>j for some j, note that p
M
>j
(z, x) = Pr[Y =
1|Z = z, X = x] � pM
j
(z, x). Using this result, for z, z0 such thatP
S
k=j
0+1 hD
k
(z, z0) =pM
>j
0 (z)� pM
>j
0 (z0) > 0 (j0 = 0, ..., S � 1), observe that each term in Udj
(x) satisfies
pM
>j�1(z, x)� pM
>j�1(z0, x) = �pM
j�1(z, x) + pM
j�1(z0, x)
= Pr[✏ µD(x),U 2 �j(z0, z)]
pM
j�1(z)� pM
j�1(z0) = �Pr[U 2 �j(z0, z)]
by (A.7) and (A.10), and thus
pM
>j�1(z, x) + pM
j�1(z)��
pM
>j�1(z0, x) + pM
j�1(z0)
= �Pr[✏ > µD(x),U 2 �j(z0, z)] < 0.
Then this relationship creates a partial ordering of pM
>j�1(z, x) + pM
j�1(z) as a functionof z in terms of p
M
>j
0 (z) (for any j0). According to this ordering, pM
>j�1(z, x) + pM
j�1(z)takes its smallest value as p
M
>j
0 (z) takes its largest value. Therefore, by (3.39),
Udj
(x) = infz2Z
{pM
>j�1(z, x) + pM
j�1(z)} = pM
>j�1(z, x) + pM
j�1(z).
By a symmetric argument, Ldj
(x) = supz2Z {pM
j
(z, x) + pM
>j
(z)} = pM
j
(z, x)+pM
>j
(z).To prove that these bounds on E[Ydj
|X = x] are sharp, it su�ces to show that for anygiven x 2 X and s
j
2 [Ldj
(x), Udj
(x)], there exists a density function f⇤✏,U such that the
following claims hold:(A) f⇤
✏|U is strictly positive on R.(B) The proposed model is consistent with the data: 8j = 0, ..., S
Pr[D 2 Mj |X = x,Z = z] = Pr[U⇤ 2 R
j(z)],
Pr[Y = 1|D 2 Mj , X = x,Z = z] = Pr[✏⇤ µD(x)|U⇤ 2 R
j(z)],
Pr[Y = 1|D 2 M>j , X = x,Z = z] = Pr[✏⇤ µD(x)|U⇤ 2 R
>j(z)].
(C) The proposed model is consistent with the specified values of E[Ydj
|X = x]:
Pr[✏⇤ µdj
(x)] = sj
.
Corollary 3.1 combined with the partial ordering above establishes monotonicity of theevent U 2 R
j(z) (and U 2 R
>j(z)) w.r.t. z. For example, for z, z0 such that pM
>j
(z) >
49
pM
>j
(z0), Corollary 3.1 implies that Rj(z) ⇢ R
j(z0) and hence
1[U 2 R
j(z0)]� 1[U 2 R
j(z)] = 1[U 2 R
j(z0)\Rj(z)]. (A.19)
Given 1[D 2 Mj ] = 1[U 2 R
j(Z)], (A.19) is analogous to a scalar treatment decisionD = 1[D = 1] = 1[U P ] with a scalar instrument P , where 1[U p0] � 1[U p] = 1[p U p0] for p0 > p. Based on this result and the results for the first part of Theorem 3.2, wecan modify the proof of Theorem 2.1(iii) in Shaikh and Vytlacil (2011) to show (A)–(C).
A.6 Proof of Lemma 4.1
For a given j = 1, ..., S�1, suppose there exist z0, z such that ⌫sj�1(zs) ⌫s
j
(z0s
) 8s (Assump-
tion ES⇤). For any d
j and d
j (dj 6= d
j), the expression of the region of multiple equilibriaRdj
(z) \ Rdj
(z) can be inferred as follows. First, there exists s⇤ such that djs
⇤ = 1 and
djs
⇤ = 0, otherwise it contradicts d
j 6= d
j . That is, Us
⇤ 2⇣
0, ⌫s⇤
j�1(zs⇤)i
in Rdj
(z) and
Us
⇤ 2⇣
⌫s⇤
j
(zs
⇤), 1i
in Rdj
(z). For other s 6= s⇤, the pair is realized to be one of the four
types: (i) djs
= 1 and djs
= 0; (ii) djs
= 0 and djs
= 1; (iii) djs
= 1 and djs
= 1; (iv) djs
= 0 anddjs
= 0. Then the corresponding pair of intervals for Rdj
(z) and Rdj
(z), respectively, falls
into one of the four types: (i)⇣
0, ⌫sj�1(zs)
i
and⇣
⌫sj
(zs
), 1i
; (ii)⇣
⌫sj
(zs
), 1i
and⇣
0, ⌫sj�1(zs)
i
;
(iii)⇣
0, ⌫sj�1(zs)
i
and⇣
0, ⌫sj�1(zs)
i
; (iv)⇣
⌫sj
(zs
), 1i
and⇣
⌫sj
(zs
), 1i
. Then by Proposition
A.1(iv), Rdj
(z) \Rdj
(z) is a product of⇣
⌫s⇤
j
(zs
⇤), ⌫s⇤
j�1(zs⇤)i
(by Assumption SS) and some
of⇣
⌫sj
(zs
), ⌫sj�1(zs)
i
,⇣
0, ⌫sj�1(zs)
i
and⇣
⌫sj
(zs
), 1i
.
Now we show that Rdj
(z)\Rdj
(z) and Rdj†(z0) for any d
j† has empty intersection. Note
that for s⇤ such that [djs
⇤ = 1 and djs
⇤ = 0] or [djs
⇤ = 0 and djs
⇤ = 1], it holds that dj†s
⇤ = 0.If dj† = d
j or d
j† = d
j , this is trivially true. If dj† 6= d
j and d
j† 6= d
j then this is true by
contradiction. But dj†s
⇤ = 0 corresponds to Us
⇤ 2⇣
⌫s⇤
j
(z0s
⇤), 1i
. Then by Proposition A.1(i),�
Rdj
(z) \Rdj
(z)�
\ Rdj†(z0) = ; as long as⇣
⌫s⇤
j
(zs
⇤), ⌫s⇤
j�1(zs⇤)i
\⇣
⌫s⇤
j
(z0s
⇤), 1i
= ;, wherethe latter is implied by ⌫s
⇤j�1(zs⇤) ⌫s
⇤j
(z0s
⇤).
A.7 Proof of Theorem 5.1
For given j = 0, ..., S � 1, consider
E[Y |Z = z]� E[Y |Z = z
0]
= E⇥
YM
j
+ 1[D(z) 2 M>j ] {YM
>j
� YM
j
}⇤
� E⇥
YM
j
+ 1[D(z0) 2 M>j ] {YM
>j
� YM
j
}⇤
= E⇥�
1[D(z) 2 M>j ]� 1[D(z0) 2 M>j ]
{YM
>j
� YM
j
}⇤
= E[YM
>j
� YM
j
|D(z) 2 M>j ,D(z0) 2 Mj ] Pr[D(z) 2 M>j ,D(z0) 2 Mj ]
� E[YM
>j
� YM
j
|D(z) 2 Mj ,D(z0) 2 M>j ] Pr[D(z) 2 Mj ,D(z0) 2 M>j ]
= E[YM
>j
� YM
j
|D(z) 2 M>j ,D(z0) 2 Mj ] Pr[D(z) 2 M>j ,D(z0) 2 Mj ], (A.20)
50
where the first equality plugs in Y = 1[D 2 M>j ]YM
>j
+�
1� 1[D 2 M>j ]
YM
j
and appliesAssumption IN, and the last equality is by supposing that the result of Lemma 5.1 is satisfiedwith
Pr[D(z) 2 Mj ,D(z0) 2 M>j ] = 0. (A.21)
But note that
Pr[D(z) 2 M>j ,D(z0) 2 Mj ] = Pr[D(z) 2 M>j ]� Pr[D(z) 2 M>j ,D(z0) 2 M>j ],
where Pr[D(z) 2 M>j ,D(z0) 2 M>j ] = Pr[D(z0) 2 M>j ] by (A.21). Combining this resultwith (A.20) yields the desired result.
51
01
11
U1
U2
U3
(a) R000 (b) R100 (c) R010 (d) R001
(e) R101 (f) R011 (g) R110 (h) R111
0
1
1
1
U1
U2
U3
(⌫100, ⌫200)
(⌫110, ⌫210)
(⌫101, ⌫201)
(⌫111, ⌫211)
⌫310 = ⌫301
⌫311
⌫300
(i)S
0j3
n
S
d2MjRd
o
= U ⌘ (0, 1]3
Figure 5: Illustration of Proposition 3.1 for S = 3.
52
0
1
1
1
U1
U2
U3
(⌫100, ⌫200)
(⌫110, ⌫210)
(⌫101, ⌫201)
(⌫111, ⌫211)
⌫310 = ⌫301
Figure 6: Depicting the Regions of Multiple Equilibria for S = 3.
0 1
1
(⌫11(z1), ⌫20(z2))
(⌫10(z1), ⌫21(z2))
(⌫11(z01), ⌫
20(z
02))
(⌫10(z01), ⌫
21(z
02))
��(z0, z)
U2
U1
Figure 7: LATE Subgroup for S = 2.
53
Figure 8: Bounds under Di↵erent Strength of Z with |X | = 3.
Figure 9: Bounds under Di↵erent Strength of Z with |X | = 15.
54
Figure 10: Bounds under Di↵erent Strength of X with |X | = 15.
Figure 11: Bounds under Di↵erent Strength of Interaction with |X | = 3.
55