Full text of DOI:10.1111/1468-0262.00351

Econometrica, Vol. 70, No. 5 (September, 2002), 1711–1740
COMMUNICATION AND EQUILIBRIUM IN DISCONTINUOUS
GAMES OF INCOMPLETE INFORMATION
By Matthew O. Jackson, Leo K. Simon,
Jeroen M. Swinkels, and William R. Zame1
This paper offers a new approach to the study of economic problems usually modeled as
games of incomplete information with discontinuous payoffs. Typically, the discontinuities
arise from indeterminacies (ties) in the underlying problem. The point of view taken here
is that the tie-breaking rules that resolve these indeterminacies should be viewed as part
of the solution rather than part of the description of the model. A solution is therefore a
tie-breaking rule together with strategies satisfying the usual best-response criterion. When
information is incomplete, solutions need not exist; that is, there may be no tie-breaking
rule that is compatible with the existence of strategy profiles satisfying the usual best-
response criteria. It is shown that the introduction of incentive compatible communication
(cheap talk) restores existence.
Keywords: Auctions, cheap talk, discontinuous games, sharing rules, tie-breaking
rules.
1ꢀ introduction
Economics is replete with situations in which privately informed agents
behave strategically; such situations are usually modeled as games of incom-
plete information. As Harsanyi showed, the equilibrium analysis of such games
is no more complicated than the equilibrium analysis of games of complete
information—provided the set of possible types of agents and the set of actions
available to agents are finite. However, in many familiar situations—including
Bertrand price competition, Cournot quantity competition, Hotelling spatial
competition, games of timing, and auctions—actions are naturally modeled as
continuous variables. Strategic analysis of such situations is difficult because tie-
breaking rules—prescribing behavior of the auctioneer when agents submit the
same bid for instance—lead to payoff functions that are discontinuous in actions.
1 This paper merges “Cheap Talk and Discontinuous Games of Incomplete Information,” by Simon
and Zame, and “Existence of Equilibrium in Auctions and Discontinuous Bayesian Games: Endoge-
nous and Incentive Compatible Sharing Rules,” by Jackson and Swinkels. We are grateful for com-
ments from Kim Border, Martin Cripps, Bryan Ellickson, Preston McAfee, Roger Myerson, John
Nachbar, Phil Reny, John Riley, Larry Samuelson, Mark Satterthwaite, Tianxiang Ye, seminar audi-
ences at Minnesota, Northwestern, Rochester, Stanford, the Stony Brook Game Theory Conference,
Texas, UCLA, and UCSD. We especially thank a co-editor and three referees for helpful comments.
Jackson is grateful for financial support from the National Science Foundation. Zame is grateful for
the hospitality of the UC Berkeley Economics Department in Winter 1996, when much of this work
was done, and for financial support from the Ford Foundation, from the National Science Founda-
tion, and from the UCLA Academic Senate Committee on Research. Swinkels thanks the Boeing
Center for Technology, Information, and Manufacturing for financial support.
1711

1712
m. jackson, l. simon, j. swinkels, and w. zame
Much of the existing analysis of such situations avoids the consequences of dis-
continuity by imposing conditions (such as private values, symmetric information,
nonatomicity of prior distributions, etc.) that guarantee ties do not occur at equi-
librium and hence that discontinuities do not matter.2 As soon as we leave the
simplest environments, however, we find situations in which ties do occur and
discontinuities do matter—indeed, we find situations in which equilibrium does
not exist.
For contexts in which information is complete, Simon and Zame (1990) (hence-
forth SZ) argued that such situations should be modeled, not as games in which
payoffs are discontinuous, but rather as games in which payoffs are only partially
determined, and that the tie-breaking rule that leads to discontinuities in payoffs
should be viewed as part of the solution, rather than as part of the data. SZ show
that (with natural conditions), such a solution (a tie-breaking rule together with
a strategy profile satisfying the usual best-response criteria) always exists. In this
paper, we extend this point of view and result to situations in which information
is incomplete.
It might seem at first glance that this extension would be routine, following
Harsanyi’s method of analyzing a game ꢁ of incomplete information by trans-
forming it into a game ꢁ ∗ of complete information—but it is not. One difficulty
is that it is not clear how indeterminacies in the game ꢁ should be transformed
into indeterminacies in the game ꢁ ∗; another is that it is not clear what assump-
tions on ꢁ will guarantee that ꢁ ∗ has a solution. Most importantly, it is not clear
how a solution for ꢁ ∗ (if it exists) should be interpreted as a solution to ꢁ .
That these difficulties reflect real problems with existence of a solution, and
not merely with a particular approach, can be seen in a simple example. Consider
a sealed-bid auction with two bidders, whose private valuations v1ꢂv2 for a single
indivisible object are drawn from a joint distribution as follows:
with probability 1/2, v1 = 1 and v2 is drawn from the uniform distribution
•
on [1, 2];
with probability 1/2, v2 = 1 and v1 is drawn from the uniform distribution
•
on [1, 2].
It is easy to see that the only tie-breaking rules that admit any equilibrium
at all have the property that when both bidders bid 1, the object is awarded to
the bidder whose valuation is higher. Given any such tie-breaking rule, it is an
equilibrium for both bidders always to bid 1. Of course, such a tie-breaking rule
cannot be implemented by an auctioneer who does not observe the valuations
of the bidders, and allowing such observation would hardly seem consistent with
the presumption that this information is private.
Suppose, however, we allow the bidders to announce their types (their true
valuations) as well as their bids. If the auctioneer is constrained to sell the object
at the highest bid and breaks ties by awarding the object to the bidder who
announces the higher valuation, then it is an equilibrium for both bidders to bid 1
2 Milgrom (1989) provides excellent background reading; for recent work, see LeBrun (1995, 1999),
Maskin and Riley (2000), Bajari (2001), Reny (1999), Athey (2001), and Bresky (2000).

discontinuous games
1713
and to truthfully announce their types.3 The key insight of this paper is that, in
considerable generality, this communication is necessary and sufficient for the
existence of a solution.
It is instructive to think about this auction in an environment in which bids
must be in multiples of a smallest monetary unit ꢃ; for simplicity assume that
1/ꢃ is an integer. Independently of the tie-breaking rule, it is an equilibrium for
both players to follow the bidding strategy
ꢀ
1−ꢃ if v = 1ꢂ
(1)
bꢄv =
1
if v > 1ꢀ
(Because 1/ꢃ is an integer, 1 is a multiple of ꢃ, hence an admissible bid.) For
every ꢃ > 0 the bids convey the information as to which bidder has the higher
valuation, but in the limit when ꢃ = 0 this information is lost; allowing bidders
to announce their valuations restores this lost information. (This echoes a theme
of Christopher Harris.)
As in this simple example, our approach is to extend the model so that indi-
viduals may announce their private information. Such announcements need not
be truthful and do not directly affect payoffs; their only role is to aid in breaking
ties. In at least one interpretation (discussed further below), these announce-
ments can be viewed as “cheap talk.” Our main result (Theorem 1) is that (with
natural conditions) this extension always has at least one solution (a tie-breaking
rule together with a strategy profile satisfying the usual best-response criterion) in
which individuals truthfully announce their private information. Type announce-
ments are thus incentive compatible. We emphasize that the tie-breaking rule
will be determined as part of the solution, and not prescribed exogenously, that
the tie-breaking rule may prescribe different divisions at different ties, and that
the tie-breaking rule may depend on announcements as well as on actions. As the
previous example and others in the text demonstrate, if we are not satisfied with
such a tie-breaking rule then we will be faced with many situations in which no
solution exists.
Although the proof of our main result is parallel to the proof of the main result
of SZ, it is by no means a routine extension. (Our analysis would be much simpler
if we restricted attention to finite type spaces, but it would seem contrived to
insist on continuous action spaces and discrete type spaces.) The text discusses
the differences between the present argument and that in SZ in some detail.
As in SZ, we might interpret an endogenous tie-breaking rule as a proxy for
“actions taken by unseen agents whose behavior is not modeled explicitly.” For
example, although we would commonly model a sealed-bid auction among N
bidders as a simultaneous-move game with N players, it might also (and perhaps
more properly) be modeled as a two-stage game with N +1 players. In the first
3 The bidder whose value is 1 has no incentive to lie since he derives no surplus from obtaining
the object at that price; the bidder whose value is above 1 has no incentive to lie since by telling the
truth she obtains the object for her bid of 1.

1714
m. jackson, l. simon, j. swinkels, and w. zame
stage of this latter game, the N bidders submit simultaneous bids; in the second
stage the auctioneer chooses the winner. If the auctioneer is constrained (by law,
for instance) to choose among the high bidders, then the auctioneer’s strategy
in the two-stage game corresponds precisely to an endogenous tie-breaking rule
in the simultaneous-move game, and the subgame perfect equilibria of the two-
stage game correspond precisely to solutions of the simultaneous-move game.4ꢂ5
In general, the two-stage game will not admit any (subgame perfect) equilibrium
unless we allow the bidders to communicate their private information. These
communications do not affect payoffs—utilities depend only on private infor-
mation, on bids, and on the auctioneer’s actions—but the auctioneer conditions
his actions on these communications. Hence communications in the two-stage
game—which correspond precisely to announcements in the simultaneous-move
game—are “cheap talk” in a familiar sense. Manelli (1996) provides a very similar
use of cheap talk to guarantee the existence of equilibrium in signalling games.
Alternatively, an endogenous tie-breaking rule might be interpreted as a proxy
for the outcome of an unmodeled second stage game. Thus, in their analysis
of first-price sealed-bid auctions for a single indivisible item, Maskin and Riley
(2000) adjoin to the sealed-bid stage a second stage in which the bidders who
submitted the high bids in the first stage participate in a Vickrey auction. In the
private value setting, it is a dominant strategy for bidders in this second stage
Vickrey auction to bid their true values. Thus the second stage auction induces
a tie-breaking rule that awards the item to the bidder who values it the most.
We should emphasize that our results concern only the existence of solutions
in mixed strategies; we have little to say about the existence of solutions in pure
strategies. For recent work on pure strategy equilibrium in auctions and similar
environments, see Maskin and Riley (2000), Reny (1999), and Athey (2001).
Of course, the difficulties that arise because of discontinuities are a conse-
quence of our insistence on a model in which action spaces are continuous.
Restricting attention to discrete action spaces, either as an assumption about the
situations to be modeled or as a modeling strategy for approximating continuous
action spaces by discrete action spaces, will yield a game to which familiar fixed
point theorems may be applied. However, there are a number of reasons why
models with continuous action spaces may be more satisfactory than models with
discrete action spaces.
(i) The equilibria of the game with discrete action spaces may depend very
sensitively on the particular discretization chosen—but it may not be obvious that
4 Of course, the auctioneer chooses among actions that leave him indifferent. One might ask
why the auctioneer should choose in any particular way, but the answer, to quote SZ, is that “ꢅ ꢅ ꢅ
equilibrium theory never explains why any agents would act in any particular way. Equilibrium theory
is intended to explain how agents behave, not why.”
5 In general, equilibrium play for the N + 1st player might depend on the valuation functions of
bidders and on the distribution of types, data that a real auctioneer might not have. It seems an
important challenge to identify circumstances in which uniform auction rules—not depending on such
information—suffice to guarantee the existence of equilibrium. The two-stage auctions of Maskin
and Riley (2000) are encouraging first steps in such a program.

discontinuous games
1715
any particular discretization is “correct.” When van Gogh’s “Irises” was sold at
auction, bids were required to be in multiples of $100,000, but other auctions
frequently allow bids in multiples of $10 or $1. Indeed, when bids are prices per
unit, they may well be in multiples of $.01 or less. When the strategic variable
is time, the issue is more subtle. Discretization amounts to an assumption that
players can move only at some pre-specified speed, but there may be no reason
to suppose that all players can move at the same speed—especially if a great deal
can be gained by moving just a little more quickly.
Of course, continuous action spaces are an idealization, and would not be of
much interest if equilibria in models with continuous action spaces did not cor-
respond to limits of equilibria in models with discrete action spaces. Our conver-
gence result (Theorem 2) shows that this is the case: if we restore information
lost in the limit, then equilibria of the discrete action games converge to equilib-
ria of the continuous action games.
(ii) The decision to model choice variables as continuous can greatly simplify
the analysis of equilibrium. In private value auctions, for example, it is frequently
the case that modeling bids as discrete variables leads to a multiplicity of equi-
libria, while modeling bids as continuous variables allows the conclusion that
equilibrium is (essentially) unique.6 Moreover, as in Maskin and Riley (2000),
modeling bids as continuous variables allows equilibrium to be characterized as
the solutions to differential equations.
(iii) Game theory usually simplifies the study of strategic interactions by
assuming that choice variables are discrete; general equilibrium theory usually
simplifies the study of markets by assuming that commodities are divisible. If we
want to think about strategic interactions in markets, it seems necessary to accom-
modate continuous choice variables in game theory, just as indivisible goods have
been accommodated in general equilibrium theory.
Applications are largely beyond the scope of the present paper, but we do give
one simple application to private value auctions to show how a solution with
communication may sometimes be used as a starting point from which to derive
a solution without communication. Jackson and Swinkels (1999) and Simon and
Zame (1999) provide more extensive elaborations on the same theme, extending
some results of LeBrun (1995, 1999) and Maskin and Riley (2000).
Following this Introduction, Section 2 presents several examples that illustrate
some of the difficulties we face and the way in which communication resolves
them. Section 3 presents the general model. Section 4 discusses the extension to
allow communication, and discusses our general existence result and a conver-
gence theorem that follows as a straightforward consequence. Section 5 presents
the application to private value auctions. Proofs are collected in Section 6.
6 Our convergence result shows that equilibria of the auction games with discrete bids converge
to a communication equilibrium of the auction game with continuous bids. As in Example 3, we can
show that such communication equilibria are behaviorally equivalent to equilibria without communi-
cation. It follows that the equilibria of the auction games with discrete bids converge to the unique
equilibrium of the auction game with continuous bids. Thus, the multiplicity of equilibria disappears
in the limit.

1716
m. jackson, l. simon, j. swinkels, and w. zame
2ꢀ examples
In the Introduction, we have described a first price auction with private val-
ues which has the property that no tie-breaking rule that is independent of
private information is compatible with any equilibrium. The analysis and conclu-
sion depends crucially on the fact that marginal distributions have atoms (see
Section 5). Lest the reader suspect that atoms play a crucial role in general, we
give here a simple example to show that type-dependent tie-breaking rules may
be required as soon as valuations have a common component.7
Example 1: Consider a sealed-bid first price auction for a single indivisible
object. There are two bidders; each bidder i observes a private signal ti (which
we identify as i’s type). Types are independently and uniformly distributed on
ꢆ0ꢂ1ꢇ. Given types t1ꢂt2, valuations are
(2)
v1ꢄt1ꢂt2 = 5+t1 −4t2ꢂ
v2ꢄt1ꢂt2 = 5−4t1 +t2ꢀ
After observing private signals, bidders simultaneously submit bids; the high bid-
der wins and pays his bid. Ties are resolved according to some specified tie-
breaking rule.
We claim that no type-independent tie-breaking rule is compatible with the
existence of equilibrium. We defer the messy analysis to Section 6, but the intu-
ition is not hard to convey.8 Because types are independently distributed and val-
uations are increasing in own type, we can use standard arguments (see Maskin
and Riley (2000) for instance) to show that if there is any equilibrium at all,
then there is an equilibrium in which bid functions are weakly increasing and
continuous at 0. For intuition, suppose there is an equilibrium in which bid func-
tions b1ꢂb2 are strictly increasing. Suppose b1ꢄ0 < b2ꢄ0 . Then the lowest types
of bidder 2 sometimes win the object, which has an expected value less than 5, so
b2ꢄ0 < 5. But then the lowest types of bidder 1 never win the object and would
prefer to bid slightly above b2ꢄ0 and win against the lowest types of bidder 2;
this would be a contradiction. Hence b1ꢄ0 ≥ b2ꢄ0 ; reversing the roles of bid-
ders 1 and 2 we conclude that b1ꢄ0 = b2ꢄ0 . If b1ꢄ0 = b2ꢄ0 < 5, then the lowest
types of bidder 1 would prefer to bid slightly more than b1ꢄ0 and win more often
when bidder 2’s type is low and bidder 1’s valuation is therefore high; this would
be a contradiction. The only remaining possibility is that b1ꢄ0 = b2ꢄ0 ≥ 5, but
in that case the winning bid is always at least 5, and hence the expected payoff
to the winner is negative. That means that the ex ante expectation of one of the
bidders must be negative; that bidder would prefer to bid 0. Again, this would
be a contradiction.
7 Our example is a close relative of Example 3 of Maskin and Riley (2000), but in our example
information has a continuous distribution, rather than an atomic distribution.
8 Our analysis is aided by the fact that each bidder’s valuation is decreasing in the other bidder’s
type, but our experience with discrete examples suggests to us that similar examples could be con-
structed in which valuations are increasing in both types.

discontinuous games
1717
Suppose, however, that we allow the bidders to announce their types as well as
their bids and allow the auctioneer to use these announcements when breaking
ties. Suppose, for instance, that the auctioneer breaks ties so that a bidder who
announces a type above .5 always wins against a bidder who announces a type
below .5, but randomizes with equal probabilities following all other pairs of
announcements. Given this tie-breaking rule, it is an equilibrium for both bidders
to bid 3.5, independent of their type, and to announce their type truthfully.9
(Verifying that this is an equilibrium is straightforward but illuminating. Say a
bidder is of high type if his signal is above .5, and a low type otherwise. If bidder 1
is a high type, bidding above 3.5 wins more often than bidding 3.5 only when
bidder 2 is also a high type—in which case bidder 1 would (on average) prefer to
lose. Thus, this is not an improving deviation. On the other hand, if bidder 1 is
a low type, bidding above 3.5 guarantees than bidder 1 wins all the time—which
yields negative expected utility. Since the putative equilibrium play yields positive
expected utility to all types of bidder 1, this is not an improving deviation either.
Finally, if bidder 1 bids below 3.5 he never wins, and hence obtains 0 expected
utility. Because putative equilibrium play yields positive expected utility to all
types of bidder 1, this is again not an improving deviation.)
The example above illustrates that without type-dependent tie-breaking rules,
equilibria need not exist. Even when equilibria do exist, however, it may happen
that the only equilibria are trivial or degenerate in some sense. The following
example from Jackson (1999) illustrates the point.
Example 2: Consider a sealed-bid second price auction for a single indivisible
object. There are two bidders. Valuations have both a personal component and a
common component; if i’s personal component is xi and the common component
is q, bidder i’s valuation is axi + ꢄ1 − a q. Note that a = 1 is the case of pure
private values, while a = 0 is the case of pure common values; we assume 0 <
a < 1.
Prior to the auction, bidder i observes a private signal ꢄxiꢂti , which we identify
as his type; xi is i’s personal component of the valuation and ti ∈ ꢈLꢂMꢂHꢉ is
correlated with the common component q:
if ti = L, then q = 0;
•
if ti = H, then q = v;
•
if ti = M, then q = 0ꢂv with equal probability (so this signal is uninforma-
•
tive).
Personal components xi are distributed independently and take on values 0ꢂ1
with equal probability. The true common component q is distributed indepen-
dently of x1ꢂx2 and takes on values 0ꢂv with equal probability. If the common
component q = 0, signals are drawn independently from the distribution that puts
probability .5 on each of the alternatives LꢂM; if the common component q = v,
signals are drawn independently from the distribution that puts probability .5 on
each of the alternatives MꢂH.
9 Neither the cut point .5 nor the common bid 3.5 is uniquely determined.

1718
m. jackson, l. simon, j. swinkels, and w. zame
This auction always admits trivial asymmetric equilibria: one bidder always bids
1 + v while the other bidder always bids 0. But if the tie-breaking rule is type-
independent, then for some values of a and v this auction admits no symmetric
equilibrium; indeed, for some values of a and v it admits only equilibria in which
at least one type of one player bids above his maximum possible valuation (con-
ditional on his information). Equilibria in which such strategies are employed
would be ruled out by any sensible notion of perfection, such as that offered by
Simon and Stinchcombe (1995).
3ꢀ games with indeterminate outcomes
A game with indeterminate outcomes consists of:
a finite set of players N = ꢈ1ꢂꢅ ꢅ ꢅ ꢂnꢉ;
•
•
for each player i, a space Ai of actions; write A = ×Ai for the space of
action profiles;
for each player i, a space Ti of types; write T = ×Ti for the space of type
•
profiles;
a probability measure ꢊ on T ;
a space ꢋ of outcomes;
•
•
•
•
an outcome correspondence ꢌ A → ꢋ;
a utility mapping u: graph ꢌ ×T → ꢀN .
Note that outcomes depend on actions but not on types. On the other hand, util-
ities depend on actions, on outcomes, and on types. The interpretation intended
is that for an action profile a ∈ A, the set ꢌꢄa represents the set of outcomes
that might result if the action profile a is taken. In the usual first price auction
for a single indivisible object, the structure is particularly simple: ꢌꢄa is a sin-
gleton unless a involves ties, in which case ꢌꢄa represents the probability with
which each of the high bidders receives the object. (Assuming that ꢌꢄa assigns
positive probability only to the high bidders represents a natural constraint on
the auctioneer. Of course one might certainly imagine situations in which the
auctioneer was subject to fewer constraints, leading to a different specification
of ꢌ.) In an auction for k (not necessarily identical) divisible objects, actions
might be demand schedules, and ꢌꢄa might represent physical or probabilistic
divisions of each of the various objects and charges to the various bidders.
As usual, we write t for a profile of types of all players other than i, and T
for the space of such type profiles. We adopt similar notation for action profiles,
strategy profiles, marginals, etc.
−i
−i
We assume throughout that:
action spaces Ai and type spaces Ti are compact metric;
ꢊ is a Borel measure;
•
•
ꢊ is absolutely continuous with respect to the product ×ꢊi of its
•
marginals;10
10 So information is absolutely continuous in the sense of Milgrom and Weber (1985).

discontinuous games
1719
ꢋ is a compact convex11 metrizable subset of a locally convex topological
vector space ꢁ;
•
the outcome correspondence ꢌ is upper-hemi-continuous, with nonempty
•
compact values;
the utility mapping is continuous.12
•
The game ꢁ is affine if the correspondence ꢌ has convex values and, for each
action profile a ∈ A and type profile t ∈ T the function
(3)
uꢄaꢂ·ꢂt ꢌꢄa −→ ꢀN
is affine. The meaning of affineness can be seen easily in a simple perfect infor-
mation, first-price auction for a cake of size 1. Suppose each bidder i is a von
Neumann–Morgenstern expected utility maximizer, whose utility for cake and
money is Uiꢄcꢂm = uiꢄc +m; without loss normalize so that uiꢄ0 = 0ꢂuiꢄ1 = 1.
If outcomes are physical divisions of cake and payment, then bidder i’s utility
when both bidders bid b and i gets the physical share & is uiꢄ& − &b. Hence
the game is affine exactly when the functions ui are affine; with our normaliza-
tion this means that uiꢄc = c. On the other hand, if outcomes are probabilistic
divisions of cake and payment, then (because i is an expected utility maximizer)
bidder i’s utility when i bids b and gets the probabilistic share & is &ꢆuiꢄ1 −bꢇ.
Hence the game is always affine. In particular, when outcomes are probabilistic
divisions, affineness is compatible with risk aversion or with any other attitude
toward risk.
If type spaces are singletons, an affine game with indeterminate outcomes is
equivalent to what SZ calls a game with an endogenous sharing rule.
Following Milgrom and Weber (1985), a distributional strategy for player i is a
probability measure 'i on Ti ×Ai whose marginal on Ti is ꢊi. If ' = ꢄ'1ꢂꢅ ꢅ ꢅ ꢂ'n
is a profile of distributional strategies, we write '¯ for the joint distribution on
T × A. If f is the Radon-Nikodym derivative of ꢊ with respect to the product
ꢁ
ꢁ
i ꢊi of its marginals, then '¯ = f i 'i.
is universally measurable,13 and ' = ꢄ'1ꢂꢅ ꢅ ꢅ ꢂ'n is a profile of distributional
If & A → ꢋ is a selection from ꢌ (that is, &ꢄa ∈ ꢌꢄa for each a ∈ A) that
strategies, we define expected utilities
ꢂ
(4)
Euiꢄ'ꢀ& = Euiꢄ'iꢀ' ꢂ& =
uiꢄaꢂ&ꢄa ꢂt d'¯ ꢀ
T ×A
−i
11 Convexity of ꢋ itself is merely a convenient technical assumption; all we need is that the range
of ꢌ be contained in some compact convex metrizable set.
12 Assuming that type spaces are compact metric and that utility is continuous in types—rather
than simply measurable—may involve some small loss of generality.
13 Recall that a set G is universally measurable if it is measurable with respect to the completion of
every Borel measure; i.e., for every Borel measure , there are Borel sets Gꢁ, Gꢁꢁ such that ꢄG\Gꢁ
∪
ꢄGꢁ\G ⊂ Gꢁꢁ and ,ꢄGꢁꢁ = 0. Universal measurability of the selection is the weakest measurability
requirement consistent with the desideratum that expected utility be well-defined for all strategy
profiles. As the reader will see, the selections constructed in Theorems 1 and 2 of this paper will in
fact be Borel measurable.

1720
m. jackson, l. simon, j. swinkels, and w. zame
A solution of ꢁ consists of a universally measurable selection & from ꢌ and
distributional strategies '1ꢂꢅ ꢅ ꢅ ꢂ'n that satisfy the usual best response criterion:
for each i and each distributional strategy 'iꢁ on Ti ×Ai we have
(5)
Euiꢄ'iꢀ' ꢂ& ≥ Euiꢄ'ꢁꢀ' ꢂ& ꢀ
−i
−i
i
We emphasize that we allow the tie-breaking rule & to depend on actions but not
on types—which will typically be unobservable.
Alternatively, given a universally measurable selection & from ꢌ, we define a
Bayesian game ꢁ & by specifying players, action spaces, type spaces, and priors
as for ꢁ , and defining utilities by u&ꢄaꢂt = uꢄaꢂ&ꢄa ꢂt . A solution for ꢁ may
therefore be identified as a universally measurable selection & from ꢌ and a
profile of distributional strategies ' = ꢄ'1ꢂꢅ ꢅ ꢅ ꢂ'n which constitute a Bayesian
Nash equilibrium for ꢁ &. If & is a continuous selection, then ꢁ & has continuous
payoffs and the existence of equilibrium follows from familiar results. However,
the correspondence ꢌ may not admit any continuous selections, so that the game
ꢁ & will typically have discontinuous payoffs.
4ꢀ communication
As our examples show, a game with indeterminate outcomes may not admit
any solutions. In this section we show how to expand the game to allow players to
communicate their private information. In the presence of natural assumptions,
this communication guarantees the existence of equilibrium.
Let ꢁ be a game with indeterminate outcomes. For mnemonic purposes, set
Si = Ti for each i; we will view elements of Si as announcements and elements of
Ti as true types. The communication extension ꢁ c is the game with indeterminate
outcomes defined by:
player set N = ꢈ1ꢂꢅ ꢅ ꢅ ꢂnꢉ;
•
•
•
•
•
•
•
action spaces Si ×Ai;
type spaces Ti;
outcomes ꢋ;
outcome correspondence ꢌc S ×A → ꢋc defined by ꢌcꢄsꢂa = ꢌꢄa ;
prior ꢊ;
utility mapping uc: graph ꢌc × T → ꢀN defined by ucꢄsꢂaꢂ/ꢂt =
uꢄaꢂ/ꢂt .
That is, ꢁ c differs from ꢁ only in that we allow players to announce their
types—and hence allow the auctioneer to condition on these announcements—
but the announcements are not payoff relevant.
Theorem 1: If ꢁ is an affine game with indeterminate outcomes, then the exten-
sion ꢁ c admits a solution in which the tie-breaking rule is Borel measurable and
type announcements are truthful.14
14 That is, equilibrium strategies 'i, which are probability distributions on Si ×Ai ×Ti, are supported
on ꢈꢄsiꢂaiꢂti si = tiꢉ.

discontinuous games
1721
If information is complete, the extension ꢁ c coincides with ꢁ . As we have
noted earlier, when information is complete, a game with indeterminate out-
comes is equivalent to a game with an endogenous sharing rule in the sense of
SZ. In this case, therefore, Theorem 1 reduces to the main result of SZ.
The proof of Theorem 1 is in six steps, paralleling the proof of the main result
of SZ, but with substantial differences, indicated below.
Step 1—Finite Approximations: Given a game ꢁ and the communication
extension ꢁ c, we construct families ꢈꢁ r ꢉ, ꢈꢁ cr ꢉ of games with finite action spaces
(but the same type spaces as ꢁ ) which “approximate” ꢁ , ꢁ c. For each of these
games we choose an arbitrary selection qr from the outcome correspondence, and
use these selections to define a Bayesian game. Each of these Bayesian games
admits a Bayesian Nash equilibrium 'r in distributional strategies, having the
property that type announcements are truthful. Each qr , which is a function on
r
r
˜
A, has trivial extensions & S ×A → ꢋꢂ& S ×A×T → ꢋ that are independent
of announcements and types. (The construction here is more elaborate than in
SZ—because we must take account of outcomes and types—but very similar.)
Step 2—Limits: The strategy profiles 'r correspond to joint distributions '¯ ,
r
r
r
˜
and induce outcome-valued vector measures & '¯ . Passing to a subsequence as
necessary, we show that ('r ) converges to a strategy profile ' for the game ꢁ c,
r
r
˜
˜
and ꢄ& '¯ converges to an outcome-valued vector measure , of the form , = &'¯
c
˜
where & is a selection from the outcome correspondence ꢌ and & is the trivial
extension to S ×A×T that does not depend on types.
Step 3—Convergence of Utilities: Convergence of strategy profiles and selec-
tions implies convergence of utilities. (This step, which is required because we
work in outcome space, has no analog in SZ, which works entirely in utility
space.)
Step 4—Identifying Better Responses: The desired solution strategy profile is ';
the tie-breaking rule will be a perturbation of &. Perturbation may be necessary
˜
because & (hence &) is only determined up to sets of '¯ -measure 0, leaving open
the possibility that there are profitable deviations. For each player i we identify a
set Hi ⊂ Ai where perturbations may be necessary to prevent deviations by that
player, and use the absence of profitable deviations in the games ꢁ r to show
that Hi has measure 0. (The argument here is different than in SZ because the
dependence of utilities on types requires that we be substantially more careful in
the construction of the corresponding deviations in the finite games.)
Step 5—Perturbation: We construct the necessary perturbations on the mea-
sure 0 sets Hi. (The argument here is much different than in SZ and more subtle
in several ways. The perturbations are constructed to punish the potential devi-
ator. In SZ all that is necessary is to choose the worst possible outcome for the
deviator. Here, however, utilities depend on types, so there need be no outcome
that is “worst possible” for all types of the potential deviator—indeed, there
need be no outcome that is uniformly bad for all types of the potential devia-
tor. We therefore use punishment outcomes constructed as limits of outcomes
in the finite games. The argument is subtle because these limits must be taken

1722
m. jackson, l. simon, j. swinkels, and w. zame
in the weak sense of convergence of vector valued measures, and because these
various punishments must be assembled in a measurable way. The construction
relies on an infinite dimensional extension of a measurable selection theorem of
Dellacherie and Meyer (1982).)
Step 6—Equilibrium: We verify that the perturbed selection &ꢁ and strategy
profile ' constitute a solution for ꢁ c.
The same argument that proves Theorem 1 establishes a convergence result
for equilibria of sequences of games. In order to give a precise statement, we
first need to describe the relevant notion of convergence of games.
For r = 0ꢂ1ꢂꢅ ꢅ ꢅ ꢂlet ꢁ r = ꢂNr ꢂꢄAir ꢂꢄTir ꢂꢊr ꢂꢋr ꢂꢌr ꢂur ꢃ be a game with inde-
terminate outcomes. We assume that all the games in question have the same set
of players Nr = N = ꢈ1ꢂꢅ ꢅ ꢅ ꢂnꢉ, that the action spaces Ari lie in a fixed compact
metric space Ai, that the type spaces Tir lie in a fixed compact metric space Ti,
and that the outcome spaces ꢋr lie in a fixed compact metric space ꢋ. We say
that the sequence of games ꢈꢁ r ꢉ converges to ꢁ 0 if:
for each i, Ari → Ai0 in the Hausdorff metric;
•
•
•
•
for each i, Tir → Ti0 in the Hausdorff metric;
ꢋr → ꢋ0 in the Hausdorff metric;
the graph of ꢌr Ar → ꢋr converges to the graph of ꢌ0 A0 → ꢋ0 in the
Hausdroff metric;
ꢊr → ꢊ in the total variation norm (as measures on T = T1 ×···×Tn ;
•
for every 1 > 0 there is a ꢃ > 0 and an index r0 such that if:
(i) r ≥ r0;
•
(ii) ꢄar ꢂ/r ꢂtr ∈ graph ꢌr ;
(iii) ꢄa0ꢂ/0ꢂt0 ∈ graph ꢌ0;
(iv) distꢄꢄar ꢂ/r ꢂtr ꢂꢄa0ꢂ/0ꢂt0 < ꢃ;
then ꢀur ꢄar ꢂ/r ꢂtr −u0ꢄa0ꢂ/0ꢂt0 ꢀ < 1.
Note that we require convergence of priors in the variation norm—not in the
weak∗ topology—and uniform convergence of utilities.
Our convergence theorem can be formulated in the following way.
Theorem 2: Let ꢈꢁ r ꢉ be a sequence of affine games with indeterminate out-
comes, converging to the affine game with indeterminate outcomes ꢁ 0. For every
r ≥ 1, let &r ; '1r ꢂꢅ ꢅ ꢅ ꢂ'nr be a solution for the communication extension ꢁ rc in
which type announcements are truthful. Then there is a solution &0; '10ꢂꢅ ꢅ ꢅ ꢂ'n0 for
the communication extension ꢁ 0c in which the type announcements are truthful and
j
a subsequence ꢈꢁ r ꢉ such that:
j
for each i, 'ir → 'i0 weak∗;
for each i, Euri cꢄ'r ꢂ&r → Eu0i cꢄ'0ꢂ&0 .
•
•
•
rj
0
rj
0
j
if '¯ , '¯ are the joint distributions of actions, then &r '¯ → &0'¯ weak∗;
j
j
j
Some consequences of this convergence theorem are worth noting.
(i) Theorem 1 itself is a direct application of Theorem 2 (as sketched above).
(ii) If ꢁ is a symmetric game with indeterminate outcomes, then the com-
munication extension ꢁ c has a symmetric solution. (Write ꢁ c as the limit of

discontinuous games
1723
symmetric finite games ꢁ r . Each ꢁ r has a symmetric solution, which induces a
symmetric solution of the communication extension ꢁ rc. Some subsequence of
these symmetric solutions converges to a solution of the communication exten-
sion ꢁ c, and any limit of such a subsequence is a symmetric solution.) Note
that symmetry entails that the tie-breaking rule depends on actions and type
announcements but not on names of the players.
(iii) The communication extension of a game with indeterminate outcomes
admits a solution that is “perfect” in the sense of Simon and Stinchcombe (1995).
This is useful because, as we have noted earlier, perfection rules out trivial equi-
libria.
We have focused on solutions in which type announcements are truthful, but
this probably involves little loss. Indeed, if ꢁ is a game with indeterminate out-
comes, ꢁ c is the communication extension, and &, ' is a solution of ꢁ c for which
type announcements are not truthful, we can use a familiar revelation argument
to construct a solution &ꢁ, 'ꢁ of ꢁ c that prescribes the same actions, and truthful
type announcements, and that induces the same outcome distribution.15
5ꢀ private value auctions
Our existence result (Theorem 1) guarantees that the communication exten-
sion of a game admits solutions in which players communicate their private infor-
mation. However, in many circumstances it is possible to turn a solution with
communication into a solution without communication, and hence obtain a solu-
tion to the game without communication. The following simple private value
auction will illustrate the point. For more on private value auctions, see LeBrun
(1995, 1999), Maskin and Riley (2000), Athey (2001), Jackson and Swinkels
(1999), Simon and Zame (1999), and Bresky (2000).
Example 3: Consider a sealed bid first price auction for a single indivisible
item. Risk neutral bidders i = 1ꢂꢅ ꢅ ꢅ ꢂn draw a private value ti according to a joint
distribution ꢊ on T = T1 × ··· × Tn = ꢆ0ꢂ1ꢇn; i’s utility if he wins the item and
pays b is ti −b. Write ꢊi for the marginal of ꢊ on Ti. We assume that each of the
marginals ꢊi is nonatomic, and that ꢊ is absolutely continuous with respect to the
product ꢊ1 × ··· × ꢊn of the marginals. Note that, subject to the limitation that
the joint distribution be absolutely continuous with respect to the product of its
marginals, we allow valuations to be correlated or affiliated to an arbitrary extent.
If bids are constrained to be multiples of a smallest monetary unit, the game
corresponding to this auction admits a perfect equilibrium, and such an equi-
librium has the property that bidders never submit bids above their true values.
Theorem 2 therefore guarantees that when arbitrary bids are allowed, the com-
munication extension of the game has a solution &; '1ꢂꢅ ꢅ ꢅ ꢂ'n with the property
that bidders never submits bids above their true values. We assert that in any
such solution, the probability that a tie for the high bid occurs is 0. Because the
15 We thank Phil Reny for pointing this out.

1724
m. jackson, l. simon, j. swinkels, and w. zame
details are a little fussy, we merely indicate the argument here, referring to Jack-
son and Swinkels (1999) or Simon and Zame (1999) for details.
(i) Whenever a tie for the highest bid occurs, there is at most one bidder
whose value exceeds the bid, and this bidder must win the item with probability 1.
(Otherwise, some bidder who does not win such ties with probability 1 would
gain by bidding a little bit more.)
(ii) Ties for the highest bid occur with probability 0. (To see this, fix a bidder
i and a type ti of bidder i, and condition on bidder i being type ti, bidding below
his true value, being the highest bidder, and winning a tie. In view of (i), this
can only happen if the other bidders who are tied with i are bidding their true
value and i is bidding precisely this value. Absolute continuity of information
and nonatomicity of marginal distributions imply that this is a set of probability
0. Integrating and applying Fubini’s theorem guarantees that ties for the highest
bid occur with probability 0.)
Since ties for the high bid occur with probability 0, the tie-breaking rule is
irrelevant; in particular, if 3 is the tie-breaking rule that randomizes equally
among all high bidders, then 3; ' is also a solution for the game with communi-
cation, whence 3; ' is a solution for the game without communication. (To see
this, suppose that 3; ' were not a solution, so that some bidder, say bidder 1,
would prefer to follow a strategy '1ꢁ = '1. If ties occur with positive probability
when bidders follow '1ꢁ ꢂ'2ꢂꢅ ꢅ ꢅ ꢂ'n, then we could construct another strategy '1ꢁꢁ
for bidder 1, in which 1 bids slightly more than in '1ꢁ , which 1 still prefers to
'1, and which has the property that ties occur with 0 probability when bidders
follow '1ꢁꢁꢂ'2ꢂꢅ ꢅ ꢅ ꢂ'n. But if ties occur with 0 probability when bidders follow
'1ꢁꢁꢂ'2ꢂꢅ ꢅ ꢅ ꢂ'n, then payoffs will be the same as when bidders follow '. That is,
'1ꢁꢁ is not preferred by bidder 1, hence '1ꢁ is not preferred by bidder 1.)16
An additional point is worth noting. Assume in addition that valuations are
independently distributed, and write biꢄti for i’s (perhaps mixed) equilibrium
bidding strategy conditional on observing the valuation ti. It is easily seen that bi
is strictly monotone, in the sense that if ti > tiꢁ, then every bid in the support of
biꢄti is at least as large as every bid in the support of biꢄtiꢁ . It follows that biꢄti
is a pure strategy for almost all valuations ti. Since ties occur with probability 0,
we can change the bidding strategies on a set of measure 0 and obtain a pure
strategy equilibrium.
6ꢀ proofs
6ꢀ1ꢀ Details of Example 1
As promised, we provide here the details that no type-independent tie-breaking
rule is compatible with the existence of any equilibrium. To see this, fix a type-
independent tie-breaking rule, and assume that b1ꢂb2 constitute an equilibrium
in mixed behavioral strategies. Because signals are independent and valuations
16 The existence of such a solution could also be obtained from the results of Reny (1999).

discontinuous games
1725
are strictly increasing in own signal, the proof of Proposition 1 in Maskin and
Riley (2000) is easily adapted to show that there is no loss in assuming the
bidding strategies b1ꢂb2 are monotone, in the sense that if tiꢁ > ti, then every bid
in the support of biꢄtiꢁ is at least as large as every bid in the support of biꢄti . It
follows immediately that there is an at most countable set of signals ti for which
the support of biꢄti is not a singleton. For such ti, replace bi by the infimum of
the support of biꢄti . It is easily checked that the modified bid functions b1ꢂb2
again constitute an equilibrium. Thus we have an equilibrium in monotone, pure
behavioral strategies. Altering bids following signals 0, 1 if necessary, there is no
loss in assuming that b1ꢂb2 are continuous at 0, 1.
Let b = maxꢈb1ꢄ0 ꢂb2ꢄ0 ꢉ, and for each i, let ꢊi = supꢈtꢀbiꢄt ≤ bꢉ. Suppose
that ꢊ2 = 0. Then, b1ꢄ0 wins with probability 0, and so earns 0. But, a bid of
b+1 by 1 wins with positive probability, and for 1 small, does so only when t2 ꢅ 0
(using that ꢊ2 = 0) so that v1 ꢅ 5+0−4ꢄ0 = 5. For there not to be a profitable
deviation of this form, it must thus be that b ≥ 5, and hence that the winning
bid is always at least 5. But, the average value of the object, even if allocated
optimally to the player with larger t, is 5+2/3−4ꢄ1/3 < 5, since 2/3 and 1/3 are
the expected higher and lower values of two draws from the uniform distribution.
So, someone is losing money on average, and would be better off to bid 0 always.
This is a contradiction, and so ꢊ2 > 0. Arguing symmetrically, ꢊ1 > 0.
Assume both players use b with positive probability, and assume that ties at b
are broken with probability p ∈ ꢄ0ꢂ1 in favor of player 1. Let tꢁ and tꢁꢁꢂtꢁ < tꢁꢁ, be
two values of t for which b1ꢄt = b. It follows that 5+tꢁ −4Eꢄt2ꢀb2ꢄt2 = b ≥ b,
else 1 would be better to bid b−1 with tꢁ. But then 5+tꢁꢁ −4Eꢄt2ꢀb2ꢄt2 = b > b,
and so 1 should deviate to b +1 with tꢁꢁ. This is a contradiction. There are thus
two remaining possibilities.
(i) One player, w.l.o.g. player 2, does not use b with positive probability (since
ꢊ2 > 0, this implies b2ꢄ0 < b, and also that player 1 bids b with positive probability
since ꢊ1 > 0, and by definition of b). Now, with t = ꢊ2 −1, player 2 never wins, but
by bidding b +1 wins with positive probability for an expected value of 5 +ꢊ2 −
1−4ꢄꢊ1/2 . So, for 2 not to want to deviate, it must be that b ≥ 5+ꢊ2 −4ꢄꢊ1/2 >
5+ꢊ2 −4ꢊ1.
(ii) Both players use b with positive probability. Then, by the above, one
player, again w.l.o.g. player 2, always has ties at b decided against him. Let
ꢄ51ꢂꢊ1 be the (nonempty) interval over which player 1 bids b. Then, with
t = ꢊ2 −1, player 2 wins only when t1 < 51, while by bidding 1 more, he can also
win when t1 ∈ ꢄ51ꢂꢊ1 . For this not to be a profitable deviation, it must be that
b ≥ 5+ꢊ2 −4ꢄ51 +ꢊ1 /2 > 5+ꢊ2 −4ꢊ1.
Assume that ꢊ2 < 1. Pick t = ꢊ2 + 1, and consider replacing b2ꢄt (which is
by definition greater than b) by any bid in ꢄbꢂb2ꢄt . This bid pays less in the
(positive probability, since ꢊ1 > 0) event that it still wins, and when it changes a
win into a loss, t1 ≥ ꢊ1, and hence v2 is at best 5 +ꢊ2 +1−4ꢊ1 < b. So, this is a
profitable deviation, a contradiction. Thus, ꢊ2 = 1.
Since ꢊ2 = 1, it follows that 1 wins with probability 1 (since 2 does not win when
t2 < 1), and hence that he always bids b (he bids at least this by definition, and

1726
m. jackson, l. simon, j. swinkels, and w. zame
need not bid any more since ꢊ2 = 1). Hence, b ≤ 3 = 5+0−4ꢄ1/2 ; otherwise 1
is better to bid 0 with t1 near 0. But then, player 2 can profitably bid b +1 when
he has t above 0, a contradiction. Thus, there is no equilibrium to this game.
6ꢀ2ꢀ Proofs of Theorems 1 and 2
We need some preliminary results. We begin by establishing some facts about
weak∗ convergence and marginals.
Lemma 1: Let XꢂY be compact metric spaces, let 8 be a positive measure on
X, let f be a positive 8-integrable function on X, and let ꢄ5n be a sequence of
positive measures on Y ×X whose marginals on X are 8. If 5n → 5 in the weak∗
topology, then
(i) the marginal of 5 on X is 8;
(ii) f5n → f5 in the weak∗ topology.
Proof: To see (i), fix an open set U ⊂ X and 1 > 0. Because X is compact
and metrizable, U is the increasing union of compact sets. Thus, we can choose
a compact set K ⊂ U such that
(6)
8ꢄU −8ꢄK < 1 and 5ꢄY ×U −5ꢄY ×K < 1ꢀ
Use Urysohn’s Lemma to choose a continuous function g X → ꢆ0ꢂ1ꢇ that is 1
on K and 0 on the complement of U. Then
ꢂ
(7)
5ꢄY ×K ≤ g d5 ≤ 5ꢄY ×U ꢀ
Because the marginal of 5n on X is 8 and g is independent of Y , it follows that
ꢂ
(8)
8ꢄK = 5nꢄY ×K ≤ g d5n ≤ 5nꢄY ×U = 8ꢄU
for each n. Weak∗ convergence of ꢄ5n to 5 guarantees that for n sufficiently
large
ꢃ
ꢃ
ꢃ
ꢃ
ꢃ
ꢂ
ꢂ
(9)
g d5n − g d5 < 1ꢀ
ꢃ
Combining these inequalities, we conclude that ꢀ5ꢄY ×U −8ꢄU ꢀ < 1. Because
1 > 0 is arbitrary, it follows that 5ꢄY × U = 8ꢄU . Because U is an arbitrary
open set, it follows that the marginal of 5 on X is 8, as asserted.
To see (ii), fix a continuous real-valued function h on X ×Y and 1 > 0. Write
ꢄ
M = supX×Y ꢀhꢀ. Because f is integrable, there is a ꢃ > 0 such that E f d8 < 1
whenever 8ꢄE < ꢃ; without loss we may assume ꢃ < 1. Use Lusin’s Theorem to
choose a compact set K ⊂ X such that 8ꢄX \K < ꢃ and the restriction of f to K
is continuous. Choose an open set U ⊃ K such that 8ꢄU \K < 1/maxK f . Use
Urysohn’s Lemma and the Tietze Extension Theorem to choose a continuous

discontinuous games
1727
¯
¯
function f on X that agrees with f on K, vanishes off U, and for which maxX f =
maxK f .
ꢄ
Recalling that the marginal of 5 on X is 8, that E f d8 < 1 whenever 8ꢄE < ꢃ,
¯
that 8ꢄX \K < ꢃ, and that 8ꢄU \K < 1/maxX f , yields
ꢃ
ꢃ
ꢃ
ꢃ
ꢃ
ꢂ
ꢂ
ꢂ
ꢂ
¯
¯
¯
ꢂ
(10)
hf d5 − hf d5 ≤
ꢀhf −hf ꢀd5 +
ꢀhf −hf ꢀd5
ꢃ
Y ×ꢄU\K
Y ×ꢄX\U
ꢅ
ꢆ
ꢂ
ꢂ
¯
≤ M
f d5 +
f d5 +
f d5
Y ×ꢄU\K
Y ×ꢄU\K
Y ×ꢄX\U
ꢅ
ꢆ
ꢂ
ꢂ
ꢂ
¯
= M
f d8+
f d8+
f d8
U\K
U\K
X\U
≤ 31Mꢀ
Similarly
(11)
ꢃ
ꢃ
ꢃ
ꢃ
ꢃ
ꢂ
ꢂ
¯
hf d5n − hf d5n ≤ 31Mꢀ
ꢃ
Weak∗ convergence of ꢄ5n to 5 entails that for n sufficiently large
ꢃ
ꢃ
ꢃ
ꢃ
ꢂ
ꢂ
ꢃ
¯
¯
(12)
hf d5n − hf d5 < 1ꢀ
ꢃ
Combining these inequalities, we conclude that for n sufficiently large
ꢃ
ꢃ
ꢃ
ꢃ
ꢂ
ꢂ
ꢃ
(13)
hf d5n − hf d5 < 1+61Mꢀ
ꢃ
ꢄ
ꢄ
Because 1 > 0 is arbitrary, it follows that hf d5n → hf d5 . Because h is arbi-
trary, we conclude that f5n → f5 weak∗, as asserted.
Q.E.D.
The proof makes use of the theory of vector measures and integration of
vector-valued functions. An excellent reference is Diestel and Uhl (1977); we
collect the basic information here. Let X be a set and ꢂ a sigma-algebra of
subsets of X. (When X is a compact metric space we take ꢂ to be the sigma-
algebra of Borel sets.) Let ꢁ be a Hausdorff, locally convex topological vector
space and let ꢁ∗ be its dual, the space of continuous linear functionals. For
/ ∈ ꢁꢂ3 ∈ ꢁ∗, we write 3 · / for the value of 3 at /. A vector measure on X
with values in ꢁ (an ꢁ-valued measure) is a (weakly) countably-additive function
< ꢂ → ꢁ.17 For < an ꢁ-valued measure and 3 ∈ ꢁ∗, we write 3 ·< for the real-
valued measure defined by 3 ·<ꢄE = 3 ·ꢄ<ꢄE .
The function f X → ꢁ is weakly measurable if the real-valued composition
3 · f is measurable for each 3 ∈ ꢁ∗. (Equivalently, the inverse image of every
17 Weak countable additivity means that if ꢈEnꢉ is a countable disjoint collection of Borel mea-
ꢇ
surable subsets of X, then <ꢄ∪En
=
<ꢄEn , convergence of the summation being in the weak
ꢇ
topology of ꢁ; equivalently, 3 ·<ꢄ∪En
=
3 ·<ꢄEn for each 3 ∈ ꢁ∗.

1728
m. jackson, l. simon, j. swinkels, and w. zame
weakly open set is measurable.) If 8 is a measure on ꢂ , the weakly measurable
function f is Pettis integrable (or weakly integrable) if for each E ∈ ꢂ there is an
ꢄ
element /E ∈ ꢁ such that 3·/E = E 3·f d8 for all 3 ∈ ꢁ∗. If this is the case, we
ꢄ
define
f d8 = /E to be the Pettis integral of f on E. If f is Pettis integrable
then theEfunction < ꢂ → ꢁ defined by <ꢄE = E f d8 is an ꢁ-valued measure.
In this circumstance, we say that f is the Radon-Nikodym derivative of < with
respect to 8 and write < = f8.
ꢄ
Recall that weak∗ convergence of scalar measures on a compact space X is
defined by
ꢂ
ꢂ
(14)
8= −→ 8 ⇐⇒ f d8= −→ f d8 for all continuous f ꢀ
Weak∗ convergence of ꢁ-valued measures on X is defined by
(15)
<= −→ < ⇐⇒ 3 ·<= −→ 3 ·< (weak∗) for all 3 ∈ ꢁ∗ꢀ
If ꢋ ⊂ ꢁ and ꢌ X → ꢋ is a correspondence, we write MꢄXꢂꢋ for the space
of ꢁ-valued measures < for which <ꢄE ∈ ꢋ for each E ∈ ꢂ , and ACꢄXꢂ ꢌ for
the space of ꢁ-valued measures < for which there exists a probability measure
8 and a Pettis integrable selection z from ꢌ such that < = z8. The first part of
the next lemma is a standard result for which there seems to be no convenient
reference; the second part extends Lemma 2 of SZ to the infinite dimensional
context.
Lemma 2: If X is a compact metric space, ꢁ is a Hausdorff locally convex topo-
logical vector space, ꢋ ⊂ ꢁ is a compact convex metrizable subset, and ꢌ X → ꢋ
is an upper-hemi-continuous correspondence with nonempty compact convex values,
then:
(i) MꢄXꢂꢋ is a compact metric space (in the weak∗ topology);
(ii) ACꢄXꢂ ꢌ is a closed subset of MꢄXꢂꢋ .
Proof: To establish (i), we first show that MꢄXꢂꢋ is compact. To this end,
let ꢄ<= ⊂ MꢄXꢂꢋ be a net. For each 3 ∈ ꢁ∗, ꢄ3·<= is a net of scalar measures,
and hence has a convergent subnet. We may therefore extract a single subnet
ꢄ<@ of ꢄ<= with the property that for each 3 ∈ ꢁ∗ there is a scalar measure
83 such that 3 · <@ → 83 weak∗. For each Borel set E ⊂ X, compactness and
convexity of ꢋ guarantees that we may implicitly define a unique element <ꢄE =
ꢋ by requiring that 3 · <ꢄE = 83ꢄE for every 3 ∈ ꢁ∗. The definitions imply
immediately that < ∈ MꢄXꢂꢋ and <@ → < weak∗.
To see that MꢄXꢂꢋ is metrizable, use compactness and metrizability of X
to choose a countable dense subset ꢈfiꢉ of the space CꢄX of real-valued con-
tinuous functions on X and use compactness and metrizability of ꢋ to choose
a countable family ꢄ3j ⊂ ꢁ∗ of linear functionals that distinguishes points of ꢋ.
(That is, / = /ꢁ if and only if 3j ꢄ/ = 3j ꢄ/ꢁ for each j.) Write
(16)
ꢆfiꢆ = supꢀfiꢄx ꢀꢂ ꢆ3j ꢆꢋ = supꢀ3 ·/ꢀꢀ
x∈X
/∈ꢋ

discontinuous games
Define a distance function on MꢄXꢂꢋ by
1729
ꢃ
ꢃ
ꢃ
ꢃ
ꢃ
ꢃ
ꢂ
ꢂ
2−i−j
ꢆfiꢆ+ꢆ3j ꢆꢋ
ꢈ
(17)
dꢄ<ꢂ<ꢁ =
fidꢄ3j ·< − fidꢄ3j ·<ꢁ
ꢀ
iꢂj
This is easily seen to be a metric and it is easily checked that the metric topology
is weaker than the weak∗ topology. Since the weak∗ topology is compact, the
metric topology coincides with the weak∗ topology. This completes the proof
of (i).
To establish (ii), we must show first that ACꢄXꢂ ꢌ is a subset of MꢄXꢂꢋ .
To see this, let < ∈ ACꢄXꢂ ꢌ and write < = z8 for some probability measure
8 and some selection z of ꢌ. By definition, for each Borel set E ⊂ X and each
ꢄ
3 ∈ ꢁ∗ we have 3·<ꢄE = 3·zd8. This is the integral of a scalar function with
respect to a probability measure, so lies in the closed convex hull of 0 and the
range of 3 ·z, which is a subset of 3 ·ꢋ.18 Since 3 ·<ꢄE ∈ 3 ·ꢋ for each 3 ∈ ꢁ∗
and ꢋ is convex, the Separation Theorem guarantees that <ꢄE ∈ ꢋ. Since E is
arbitrary, it follows that < ∈ ACꢄXꢂ ꢌ as desired.
To complete the proof, let ꢄ<n ⊂ ACꢄXꢂ ꢌ be a sequence converging weak∗
to < ∈ MꢄXꢂꢋ ; we must show < ∈ ACꢄXꢂ ꢌ . For each n, choose a probability
measure 8n and a selection zn from ꢌ such that <n = zn8n. Passing to a subse-
quence if necessary, we may assume that there is probability measure 8 such that
8n → 8 weak∗; we construct a selection z from ꢌ such that < = z8.
It is convenient to imbed ꢋ in ꢀꢇ. To accomplish this, let ꢈ3j ꢉ be the countable
family of linear functionals chosen above and define a linear mapping A ꢁ →
ꢀꢇ by Aꢄx = ꢄ31ꢄx ꢂꢅ ꢅ ꢅ ꢂ . If ꢀꢇ is endowed with the product topology, this
mapping is continuous. Because the collection ꢄ3j distinguishes points of ꢋ,
the restriction of A to ꢋ is one-to-one; because ꢋ is compact, the restriction of
A to ꢋ is a homeomorphism. Because we can now replace ꢋꢂꢌꢂꢁ by Aꢄꢋ ,
A ꢈꢌꢂꢀꢇ, we may assume without loss that ꢁ = ꢀꢇ.
For each positive integer k, let Bk ꢀꢇ → ꢀk be the projection on the first k
coordinates and let Ck ꢀk → ꢀk−1 be the projection on the first k−1 coordinates.
Fix an index k. The composition Bk ꢈ <n is a vector measure with values in
ꢀk and the composition Bk ꢈ zn is a selection from the correspondence Bk ꢈ ꢌ.
Continuity of Bk implies that the sequence ꢄBk ꢈ <n of ꢀk-valued measures
converges weak∗ to Bk ꢈ<, so Lemma 2 of SZ guarantees that there is a selection
zk from Bk ꢈꢌ such that Bk ꢈ< = zk8.
Note that Ck ꢈ Bk ꢈ < = Bk−1 ꢈ <. Uniqueness of the Radon-Nikodym deriva-
tive implies that Ck ꢈ zk = zk−1 almost everywhere (with respect to 8). We can
therefore choose a set X0 ⊂ X such that 8ꢄX \X0 = 0 and Ck ꢈzkꢄx = zk−1ꢄx
for every x ∈ X0 and every index k. For each index k and x ∈ X0, write
ꢌkꢄx = Bk−1ꢄzkꢄx ∩ꢌꢄx . Compactness of ꢌꢄx and continuity of Bk guaran-
tee that ꢌkꢄx is compact, and the construction of X0 guarantees that ꢄꢌkꢄx is
a decreasing sequence of compact sets, so the intersection ∩ꢌkꢄx is not empty.
18 Note that 0 = <ꢄꢉ ∈ ꢋ.

1730
m. jackson, l. simon, j. swinkels, and w. zame
Our construction guarantees that this intersection consists of a single point,
which we define to be zꢄx . By construction, z is a selection from ꢌ on X0. The
graph of z is the intersection of the graphs of the measurable correspondences
Bk−1ꢄzkꢄ· ∩ꢌꢄ· , so z is measurable. Extend z arbitrarily to a measurable selec-
tion on all of X.
Our construction guarantees that, for each k, zk = Bk ꢈ z is the Radon-
Nikodym derivative of Bk ꢈ <. Linearity of Bk guarantees that for every Borel
subset G ⊂ X we have
(18)
Bkꢆ<ꢄG ꢇ = ꢄBk ꢈ< ꢄG
= <kꢄG
ꢂ
=
zkd8
G
ꢂ
=
ꢄBk ꢈz d8
G
ꢉ
ꢊ
ꢂ
= Bk
zd8 ꢀ
G
ꢄ
Since this is true for every k, we conclude that <ꢄG
desired.
=
G zd8, as
Q.E.D.
The next lemma is an extension of a result of Dellacherie and Meyer (1982)
to the present context.19
Lemma 3: Let XꢂY be a compact metric spaces, let ꢁ be a locally convex topo-
logical vector space, let ꢋ be a compact convex metrizable subset of ꢁ, and let
ꢌ X → ꢋ be an upper-hemi-continuous correspondence with nonempty, compact,
convex values. Let y → <y be a weak∗ measurable family of ꢁ-valued measures on
X having the property that for every y ∈ Y there is a selection zy from ꢌ such that
<y = zy8. Then there is a Borel measurable function Z X ×Y → ꢁ such that for
each y ∈ Y ꢂ Zꢄ·ꢂy is a selection from ꢌ and <y = Zꢄ·ꢂy 8. (That is, Zꢄ·ꢂy = zy
almost everywhere with respect to 8.)
Proof: As in the proof of Lemma 2, there is no loss in assuming that ꢁ = ꢀꢇ;
we adopt the notation of that proof. For each kꢂꢈBk ꢈ <y y ∈ Y ꢉ is a weak∗
measurable family of measures on X with values in ꢀk, and for every y ∈ Y the
composition Bk ꢈ zy is a selection from Bk ꢈ ꢌ such that Bk ꢈ <y = ꢄBk ꢈ zy 8.
Applying Theorem V.58 of Dellacherie and Meyer (1982) to each coordinate
separately, we may find a Borel function Zk X × Y → ꢀk such that Bk ꢈ <y =
Zkꢄ·ꢂ·ꢂy 8 for each y ∈ Y . As in the proof of Lemma 2, we can construct a Borel
function Z∗ X ×Y → ꢀꢇ such that Bk ꢈZ∗ = Zk for each k.
The construction guarantees that <y = Z∗ꢄ·ꢂy 8 for every y ∈ Y . However, Z∗
need not be quite the function we want because it need not be a selection from ꢌ;
19 We thank a referee for directing us to Dellacherie and Meyer (1982).

discontinuous games
1731
a perturbation will achieve this. To accomplish this perturbation, fix an arbitrary
Borel measurable selection z0 of ꢌ. Define the Borel function Q X × Y →
X ×ꢋ×Y by Qꢄxꢂy = ꢄxꢂZ∗ꢄxꢂy ꢂy . Define a Borel measurable selection Z
from ꢌ by
ꢀ
Z∗ꢄxꢂy if ꢄxꢂy ∈ Q−1ꢆꢄgraphꢌ ×Y ꢇꢂ
(19)
Zꢄxꢂy =
z0ꢄx
otherwiseꢀ
Uniqueness of the Radon-Nikodym derivative implies that for every y ∈ Y ,
Z∗ꢄxꢂy = zyꢄx for 8-almost all x ∈ X, whence Zꢄxꢂy = Z∗ꢄxꢂy = zyꢄx for
8-almost all x ∈ X. Thus Z is the desired mapping.
Q.E.D.
Finally, it is convenient to isolate a lemma that will be used several times.
If ꢁ is a game with action spaces Ai and type spaces Ti and f is any function
˜
or correspondence defined on A, we write f for the trivial extension of f to
A×T f ꢄaꢂt = f ꢄa .
˜
Lemma 4: Let ꢁ be an affine game with indeterminate outcomes. Let ꢄzr be
a sequence of selections from the outcome correspondence ꢌ and let ꢄ5r be a
sequence of positive measures on A×T .20 If there is a selection z from the outcome
r
correspondence ꢌ and a positive measure 5 on A×T such that 5r → 5 and z˜ 5r →
z˜5 (weak∗), then
ꢂ
ꢂ
r
(20)
uiꢄaꢂz˜ ꢄaꢂt ꢂt d5r −→ uiꢄaꢂz˜ꢄaꢂt ꢂt d5
for each i.
Proof: Fix i. We begin by constructing an approximation to ui by a contin-
uously weighted sum of affine functions. Write ꢁ∗ for the dual space of ꢁ (the
space of continuous linear functionals, equipped with the topology of pointwise
convergence).
Fix 1 > 0. For a ∈ A, t ∈ T , the function uiꢄaꢂ·ꢂt is affine on ꢌꢄa , so it can
be approximated to within 1 by an affine function on ꢁ.21 That is, there are a
constant cat and a linear functional 3at ∈ ꢁ∗ such that
(21)
ꢀuiꢄaꢂ/ꢂt −cat −3at ·/ꢀ < 1
for each / ∈ ꢌꢄa . Compactness of AꢂT ꢂꢋ, continuity of ui and 3at, and upper-
hemi-continuity of ꢌ imply that there are neighborhoods Wꢄaꢂt of ꢄaꢂt in
20 We do not assume that 5r is the joint distribution of any strategy profile.
21 See Phelps (1966). In the infinite dimensional context, there may be no affine function on ꢁ that
coincides with uiꢄaꢂ·ꢂt on ꢌꢄa .

1732
m. jackson, l. simon, j. swinkels, and w. zame
A×T and Wꢁꢄaꢂt of 0 in ꢋ such that
ꢄaꢁꢂtꢁ ∈Wꢄaꢂt ꢂ/ꢁ ∈ꢌꢄaꢁ ⇒∃/∈ꢌꢄa such that ꢄ/−/ꢁ ∈Wꢁꢄaꢂt H
(22)
ꢄaꢁꢂtꢁ ∈Wꢄaꢂt ꢂ/ꢁ ∈ꢌꢄaꢁ ꢂꢄ/−/ꢁ ∈Wꢁꢄaꢂt
⇒ꢀuiꢄ/ꢂaꢂt −uiꢄ/ꢁꢂaꢁꢂtꢁ ꢀ<1ꢂꢀ3at ·ꢄ/−/ꢁ ꢀ<1ꢀ
Combining these facts, we conclude that
(23)
ꢄaꢁꢂtꢁ ∈ Wꢄaꢂt ꢂ/ꢁ ∈ ꢌꢄaꢁ ⇒ ꢀuiꢄ/ꢁꢂaꢁꢂtꢁ −cat −3at ·/ꢁꢀ < 31ꢀ
The family ꢈWꢄaꢂt ꢉ is a cover of A×T by open sets. Choose a finite subcover
ꢈWꢄaj ꢂtj ꢉ and a partition of unity ꢈf j ꢉ subordinate to this cover; i.e., a family
of continuous functions f j A×T → ꢆ0ꢂ1ꢇ such that:
f j ꢄaꢁꢂtꢁ = 0 if ꢄaꢁꢂtꢁ " Wꢄaj ꢂtj ;
•
ꢇ
j f j ≡ 1.
•
To simplify notation, write cj = caj tj and 3j = 3aj tj . Define mappings
(24)
c1 A×T −→ ꢀꢂ 31 A×T −→ ꢋ∗
by
ꢈ
ꢈ
(25)
c1ꢄaꢂt = f j ꢄaꢂt cj ꢂ 31ꢄaꢂt = f j ꢄaꢂt 3j ꢀ
j
j
Because ꢈf j ꢉ is a partition of unity subordinate to the cover ꢈWꢄaj ꢂtj ꢉ, it fol-
lows that c1ꢂ31 are continuous functions and that
(26)
ꢄaꢂ/ꢂt ∈ graph ꢌ ×T ⇒ ꢀuiꢄaꢂ/ꢂt −c1ꢄaꢂt −31ꢄaꢂt ·/ꢀ < 31ꢀ
This is the desired approximation to ui.
It follows from (26) that
ꢃ
ꢃ
ꢃ
ꢃ
ꢃ
ꢂ
ꢂ
(27)
uiꢄaꢂz˜ꢄa ꢂt d5 − ꢆc ꢄaꢂt +3 ꢄaꢂt ·z˜ꢄaꢂt ꢇd5 < 31ꢂ
ꢃ
1
1
ꢃ
ꢃ
ꢃ
ꢃ
ꢃ
ꢂ
ꢂ
r
r
uiꢄaꢂz˜ ꢄaꢂt ꢂt d5r − ꢆc1ꢄaꢂt +31ꢄaꢂt ·z˜ ꢄaꢂt ꢄaꢂt ꢇd5r < 31ꢂ
ꢃ
for every r.

discontinuous games
Now apply weak∗ convergence:
1733
ꢂ
r
(28)
ꢆc1ꢄaꢂt +31ꢄaꢂt ·z˜ ꢄaꢂt ꢇd5r
ꢂ
ꢈ
r
=
=
=
f j ꢄaꢂt ꢆcj +3j ·z˜ ꢄaꢂt ꢇd5r
j
ꢂ
ꢂ
ꢈ
ꢈ
cj f j ꢄaꢂt d5r +
f j ꢄaꢂt 3j ·z˜ ꢄaꢂt d5r
r
j
j
ꢂ
ꢂ
ꢈ
ꢈ
cj f j ꢄaꢂt d5r +
f j ꢄaꢂt dꢄ3j ·z˜ 5r
r
j
j
ꢂ
ꢂ
ꢈ
ꢈ
−→ cj f j ꢄaꢂt d5 +
f j ꢄaꢂt dꢄ3j ·z˜5
j
j
ꢂ
ꢂ
ꢈ
ꢈ
=
=
cj f j ꢄaꢂt d5 +
f j ꢄaꢂt 3j ·z˜ꢄaꢂt d5
j
j
ꢂ
ꢈ
f j ꢄaꢂt ꢆcj +3j ·z˜ꢄaꢂt ꢇd5ꢀ
j
Combining this with (26) and keeping in mind that 1 is arbitrary, we obtain
ꢂ
ꢂ
r
(29)
uiꢄaꢂz˜ ꢄaꢂt ꢂt d5r −→ uiꢄaꢂz˜ꢄaꢂt ꢂt d5ꢂ
which is the desired result.
Q.E.D.
With the preliminaries complete, we turn to the proof of Theorem 1.
Proof of Theorem 1: As indicated, the proof is in six steps. The argument
is a bit fussy because we need to keep track of strategies and selections in several
games. Recall that, as a mnemonic device to distinguish between announcements
and true types, we write Si = Ti, for each i. Without loss of generality, we assume
that 0 ≤ ui ≤ 1 for each i.
Step 1—Finite Approximations: For each r = 1ꢂ2ꢂꢅ ꢅ ꢅ and each player i,
choose and fix finite subsets Sir ⊂ Si, Air ⊂ Ai such that every point of Si is within
1/r of some point in Sir and every point of Ai is within 1/r of some point in Ari .
For each r, let qr A → ꢋ be a Borel measurable selection from ꢌ. For each
r, let ꢁ r be the Bayesian game with player set N, action spaces Ari , type spaces
Ti, prior probability distribution ꢊ, and utility functions uri ꢄaꢂt = uiꢄaꢂqr ꢄa ꢂt .
Milgrom and Weber (1985) show that ꢁ r has an equilibrium =r = ꢄ=r1ꢂꢅ ꢅ ꢅ ꢂ=rn
in distributional strategies.
Let ꢁ cr be the Bayesian game with player set N, action spaces Sir × Ari ,
type spaces Ti, prior probability distribution ꢊ, and utility functions uri ꢄsꢂaꢂt =

1734
m. jackson, l. simon, j. swinkels, and w. zame
uiꢄaꢂqr ꢄa ꢂt . Payoffs in ꢁ cr are independent of announcements, so announce-
ments are cheap talk. Define
(30)
di Ai ×Ti −→ Si ×Ai ×Tiꢂ d A×T −→ S ×A×T
by diꢄaiꢂti = ꢄtiꢂaiꢂti and dꢄaꢂt = ꢄtꢂaꢂt . Let 'ir = di=ir ꢂ '¯ = d=¯ be the
r
r
direct image measures. Note that the marginal of 'ir on T is ꢊi, so 'ir is a
distributional strategy for the game ꢁ cr . Moreover, '¯ is the joint distribution on
r
S ×A×T of the tuple 'r = ꢄ'1r ꢂꢅ ꢅ ꢅ ꢂ'nr . Because payoffs in ꢁ cr do not depend
on announcements, 'r is an equilibrium for ꢁ cr . Write
(31)
I = ꢈꢄsꢂaꢂt ∈ S ×A×T s = tꢉ
for the set of announcement/action/type profiles for which announcements are
r
truthful. By construction, '¯ is supported on I, so gives probability one to truth-
ful announcements.
r
r
r
˜
˜
For each r, define & S × A → ꢋ and & S × A × T → ꢋ by & ꢄsꢂaꢂt =
r
r
c
c
c
ꢋ
ꢋ
& ꢄsꢂa = q ꢄa . Define ꢌ S × A × T → ꢋ by ꢌ ꢄsꢂaꢂt = ꢌ ꢄsꢂa = ꢌꢄa .
r
c
r
c
ꢋ
˜
Note that & is a selection from ꢌ and that & is a selection from ꢌ .
Step 2—Limits: Passing to an appropriate subsequence if necessary, we may
assume that, for each i, the sequence ꢄ'ir of scalar measures converges weak∗
r
r
˜
to a scalar measure 'i on Si × Ai × Ti, and the sequence ꢄ& '¯ of ꢁ-valued
measures converges weak∗ to an ꢁ-valued measure , on S × A × T . Note that
convergence of individual strategies implies convergence of joint distributions;
r
that is, '¯ → '¯ , the joint distribution of ' = ꢄ'1ꢂꢅ ꢅ ꢅ ꢂ'n .
By Lemma 2, there is a Borel measurable selection J from ꢌ such that , = J'¯ .
Define &ꢄsꢂa = Jꢄsꢂaꢂs for every ꢄsꢂa ∈ S ×A. Note that '¯ is supported on
˜
I, the set of truthful profiles, so & = J almost everywhere (with respect to '¯ ),
˜
whence , = &'¯ .
Step 3—Convergence of Utilities: Applying Lemma 4 to the game ꢁ c, we con-
clude that utilities converge. That is, for each i
(32)
Euiꢄ'r ꢀ&r −→ Euiꢄ'ꢀ& ꢀ
Step 4—Identifying Better Responses: The selection J is only determined up to
˜
sets of measure 0, so we may have chosen the wrong selections &ꢂ&. This leaves
open the possibility that players may have profitable deviations. We will construct
˜
perturbations of &ꢂ& to eliminate these deviations. In order to do this, we first
identify the places where perturbation is required.
By assumption, ꢊ is absolutely continuous with respect to the product of its
marginals; let F T → ꢀ be the Radon-Nikodym derivative (which we can assume
is a Borel function), so that ꢊ = F ꢄ×ꢊi . Notice that the conditionals are ꢊꢄ·ꢀti =
F ꢄ·ꢀti ꢄ×ꢊi .
Fix a player i. Consider the maps L Ti → ꢀ, M Ti → L1ꢄꢊ
defined by
Lꢄti = Euiꢄ'ꢀ&ꢂti ꢂMꢄti = F ꢄ·ꢀti . The maps L, M are Borel−mi easurable,22
22 We give L1ꢄꢊ
the norm topology.
−i

discontinuous games
1735
so Lusin’s Theorem allows us to find an increasing sequence ꢄTik of compact
subsets of Ti such that the restrictions of L, M to each Tik are continuous
and ꢊiꢄTi\Tik < 2−k. There is no loss of generality in assuming that the sup-
port of 'iꢀTik is Tik; equivalently, every relatively open subset of Tik has posi-
tive ꢊ-measure. Set Ti∗ = ∪Tik, so that ꢊiꢄTi\Ti∗ = 0. Let Hi be the set of pairs
ꢄsiꢂai ∈ Si ×Ai for which there is a type ti ∈ Ti∗ such that
(33)
Euiꢄsiꢂaiꢀ' ꢂtiꢂ& > Euiꢄ'iꢀ' ꢂtiꢂ& ꢀ
−i
−i
That is, player i of type ti prefers to announce si and play ai rather than to follow
'i, given that others are following ' and that the selection is &. The continuity
properties of L and M and the cont−ini uity of utility functions guarantees that Hi
is a Borel set. We assert that 'iꢄHi ×Ti∗ = 0.
To see that this is so, we suppose not, and construct a profitable deviation in
ꢁ cr for r sufficiently large. To this end, let Ni be the marginal of 'i on Ai × Ti,
so NꢄHi = 'iꢄHi ×Ti = 'iꢄHi ×T ∗ > 0. For each j, let Hij be the set of pairs
i
ꢄsiꢂai ∈ Si ×Ai for which there is a type ti ∈ Ti∗ such that
1
j
(34)
Euiꢄsiꢂaiꢀ' ꢂtiꢂ& > Euiꢄ'¯iꢀ' ꢂtiꢂ& +
ꢀ
−i
−i
Since Hi = ∪Hj , we can find some j so that NiꢄHj > 0.
We can find a Borel measurable map h Hij →iTi∗ such that
i
1
j
(35)
Euiꢄsiꢂaiꢀ' ꢂhꢄsiꢂai ꢂ& > Euiꢄ'iꢀ' ꢂhꢄsiꢂai ꢂ& +
−i
−i
for every ꢄsiꢂai ∈ Hij . Applying Lusin’s Theorem to h and recalling that Ti∗ =
∪Tik, we can find a compact subset H ⊂ Hij such that NiꢄH > 0, the restriction
of h to H is continuous, and hꢄH ⊂ Tik. There is no loss in assuming that supp
ꢄNiꢀH = H, so every relatively open subset of H has positive Ni-measure.
Fix an arbitrary ꢄsi∗ꢂti∗ ∈ H. Continuity of ui and of M on Tik guarantees that
the map
(36)
ꢄsiꢂaiꢂti −→ Euiꢄsiꢂaiꢀ' ꢂtiꢂ&
−i
is continuous on Si × Ai × Tik. Hence there are compact neighborhoods K of
ꢄsi∗ꢂti∗ in H and L of hꢄsi∗ꢂti∗ in Tik so that
1
2j
(37)
Euiꢄsiꢂaiꢀ' ꢂtiꢂ& > Euiꢄ'iꢀ' ꢂtiꢂ& +
−i
−i
whenever ꢄsiꢂai ∈ K, ti ∈ L. Our construction guarantees that NiꢄK > 0 and
ꢊiꢄL > 0. Shrinking K, L if necessary, we may find a real number R such that
(38)
ꢄsiꢂai ∈ Kꢂti ∈ L
⇒ Euiꢄsiꢂaiꢀ' ꢂtiꢂ& > R+
1
2j
> R > Euiꢄ'iꢀ' ꢂtiꢂ& ꢀ
−i
−i

1736
m. jackson, l. simon, j. swinkels, and w. zame
Because the restriction of M to Tik is continuous, shrinking L further if necessary
we may guarantee
1
16j
(39)
tiꢂtiꢁ ∈ L ⇒ ꢆF ꢄ·ꢀti −F ꢄ·ꢀtiꢁ ꢆ <
ꢀ
Finally, continuity of ui guarantees that, shrinking L still further if necessary, we
may guarantee
1
16j
(40)
tiꢂtiꢁ ∈ L ⇒ ꢀuiꢄsꢂaꢂ/ꢂt −uiꢄsꢂaꢂ/ꢂtꢁ ꢀ <
ꢀ
Let 1 > 0. Choose an open set Q with K ⊂ Q ⊂ Si ×Ti such that NiꢄQ\K < 1,
and a continuous function 3 Si × Ai → ꢆ0ꢂ1ꢇ that is identically 1 on K, and
identically 0 off U. Let P be the characteristic function of L in Ti. View 3ꢂP
as functions on S × A × T that are independent of other components. Write
ꢄ
M = 3Pd'¯ , and set A = ꢄ1/M 3P. Note that
ꢂ
(41)
1dꢄA'¯ = 1ꢀ
r
r
r
∗
∗
˜
˜
Lemma 1 guarantees that A'¯ → A'¯ and that & A'¯ → &A'¯ (weak ). Weak
convergence guarantees that
ꢂ
ꢂ
r
(42)
1dꢄA'¯ −→ 1dꢄA'¯ = 1ꢀ
Lemma 4 guarantees that
ꢂ
ꢂ
r
r
˜
˜
(43)
uiꢄsꢂaꢂ& ꢄsꢂa ꢂt dꢄA'¯ −→ uiꢄsꢂaꢂ&ꢄsꢂa ꢂt dꢄA'¯ ꢀ
Together, the inequalities (38), (41) guarantee that
ꢂ
˜
(44)
so
uiꢄsꢂaꢂ&ꢄsꢂa ꢂt dꢄA'¯ > R
ꢂ
r
r
˜
uiꢄsꢂaꢂ& ꢄsꢂa ꢂt dꢄA'¯ > R
(45)
for r sufficiently large.
For each r, the marginal of '¯ on Si × Ai × Ti is 'ir ; let ,r ꢄ·ꢀꢄsiꢂaiꢂti be
r
r
the conditionals. Because A depends only on siꢂaiꢂti, the marginal of A'¯ on
Si ×Ai ×Ti is A'ir and the conditionals are ,r ꢄ·ꢀꢄsiꢂaiꢂti . Hence we can write
ꢂ
r
r
˜
(46)
uiꢄsꢂaꢂ& ꢄsꢂaꢂt ꢂt dꢄA'¯
ꢂ ꢂ
ꢌ
ꢍ
r
r
˜
=
uiꢄsꢂaꢂ& ꢄsꢂaꢂt ꢂt d, dꢄA'ir ꢀ

discontinuous games
1737
ꢄ
r
In view of (42), 1dꢄA'¯ is arbitrarily close to 1 for r sufficiently large. Hence,
for each r sufficiently large, there is some ꢄsir ꢂair ꢂti for which
ꢂ
ꢎ
ꢏ
ꢐ
ꢑ
r
r
i
˜
(47)
ui ꢄsr ꢂs ꢂꢄar ꢂa ꢂ& ꢄs ꢂs ꢂꢄar ꢂa ꢂꢄtiꢂt
ꢂꢄtiꢂt
d,r
−i
−i
−i
−i
−i
−i
i
i
i
1
> R+
ꢀ
4j
Note that this integral is the expected payoff to player i in the game ꢁ cr if he is
type ti, announces sir , and plays air . Taken together, (39) and (40) guarantee that
the expected payoff to player i in the game ꢁ cr if he is type tiꢁ ∈ L, announces si,
and plays ai is at least R+ꢄ1/8j . In particular, the expected payoff to player i
in the game ꢁ cr when his type lies in L, he announces si, and plays ai is at least
ꢆR+ꢄ1/8j ꢇꢊiꢄL .
r
On the other hand, applying Lemma 4 to P'¯ → P'¯ guarantees that the
expected payoff to player i in the game ꢁ cr if his type lies in L and he plays
according to 'ir converges to the expected payoff to player i in the game ꢁ c if
his type lies in L and he plays according to 'ir . In view of (38), this is at most
RꢊiꢄL . Taken together, these last three facts constitute a contradiction (for r
large enough). We conclude that NiꢄHi = 0 as asserted.
Step 5—Perturbation: We now correct the selection & on Hi. Intuitively speak-
ing, the correction is to give player i the limit of what he would obtain in the
games ꢁ cr ; the details are complicated because we must put these limits together
in a manner that is consistent across actions of others and measurable in i’s own
actions.
Write
(48)
I
−i = ꢈꢄs ꢂa ꢂt ∈ S−i ×A−i ×T s−i = t ꢉꢀ
−i
−i −i
−i
−i
Fix ꢄsiꢂai ∈ Hi. Define B I → ꢋ by Bꢄs ꢂa ꢂs = ꢌꢄsiꢂaiꢂs ꢂa . For
−i
−i
−i −i
−i
−i
each r, define @r I → ꢋ by @r ꢄs ꢂa ꢂs =−&ir ꢄs−iiꢂaiꢂs ꢂa ; note that @r
−i
−i
−i
−i
is a selection from B.
r
−i
Write '¯ ꢂ'¯−i for the joint distributions on S−i × A−i × T of announce-
−i
ments/actions/types of players other than i. Because announcements (of players
r
−i
other than i) are truthful, '¯ , '¯−i are supported on I ; we abuse notation and
−i
view them as measures on this space.
Let ꢍꢄsiꢂai be the set of ꢁ-valued measures ꢎ on I for which there is a
−i
sequence of integers ꢄrm and points ꢄsirm ꢂairm ∈ Si ×Ai such that:
ꢄsirm ꢂarim → ꢄsiꢂai ;
•
•
r
@rm '¯−i → ꢎ.
Lemma 2 guarantees that ꢍꢄsiꢂai is a nonempty compact set of ꢁ-valued
measures, and that for each ꢎ ∈ ꢍꢄsiꢂai there is a selection @ from B so that
ꢎ = @'¯−i. It is easily checked that the correspondence ꢄsiꢂai → ꢍꢄsiꢂai is
weak∗ upper-hemi-continuous, so it admits a weak∗ Borel measurable selection
ꢄsiꢂai → ꢎꢄs ꢂa
.
i
i

1738
m. jackson, l. simon, j. swinkels, and w. zame
Lemma 3 guarantees that there is a Borel function Ri Hi ×I−i → ꢋ such that
ꢎꢄs ꢂa = Riꢄsiꢂaiꢂ· '¯−i for each ꢄsiꢂai . Define
i
i



Riꢄsiꢂaiꢂs ꢂa ꢂs
if there is a unique i with
ꢄsiꢂai ∈ Hiꢂ
−i
−i −i
(49)
&ꢁꢄsꢂa =


&ꢄsꢂa
otherwiseꢀ
Note that &ꢁ = & except on
ꢒ
(50)
H = ꢆHi ×Ti ×S−i ×A−i ×T ꢇꢀ
−i
i
The construction of Step 4 guarantees that '¯ ꢄH = 0. In particular, Euiꢄ'ꢀ& =
Euiꢄ'ꢀ&ꢁ for all i.
Step 6—Equilibrium: We assert that the selection &ꢁ and strategy profile '
constitute a solution for the game ꢁ c. To see this, fix a player i. We must show
that for almost all ti ∈ Ti, the strategy 'i is a best response to ' , given that agent
−i
i is type ti. We only have to worry about types ti ∈ T ∗, because the complemen-
tary set of types has measure 0. Let ti ∈ T ∗ and supipose there is an announce-
ment si and action ai so that, given he is itype ti, player i would strictly prefer
to play ꢄsiꢂai rather than follow 'i. By construction, ti ∈ T k for some k. Conti-
nuity of payoffs and information on Tik implies there is a rielatively open subset
W ⊂ Tik such that for every tiꢁ ∈ W, player i of type tiꢁ would strictly prefer to
play ꢄsiꢂai rather than follow 'i. Thus, if we write Euiꢄsiꢂaiꢀ' ꢂWꢂ&ꢁ for the
−i
expected utility of player i when his type is in W, he plays siꢂai, others follow
their components of ', and the tie-breaking rule is &ꢁ (and similarly for 'i in
place of siꢂai), we find
(51)
Euiꢄsiꢂaiꢀ' ꢂWꢂ&ꢁ > Euiꢄ'iꢀ' ꢂWꢂ&ꢁ ꢀ
−i
−i
On the other hand, we can estimate Euiꢄsiꢂaiꢀ' ꢂWꢂ&ꢁ directly from payoffs
−i
in the games ꢁ cr . By definition, ꢄsiꢂai ∈ Hi. By construction, for each tiꢁ ∈ W
ꢁ
ꢁ
˜
(52)
& ꢄsiꢂaiꢂs ꢂa ꢂt ꢂt = Riꢄsiꢂaiꢂs ꢂa ꢂs = @ꢄsiꢂai
−i
−i
−i
−i
−i −i
i
except for ꢄs ꢂa ꢂtꢁꢂt
belonging to a set of '¯−i-measure 0. Hence, using
−i
−i
−i
i
Lemma 4 exactly as in Step 4 above and recalling the equilibrium conditions in
the games ꢁ cr , we see that
(53)
Euiꢄsiꢂaiꢀ' ꢂWꢂ&ꢁ = lim Euiꢄsrm ꢂarm ꢀ'r ꢂWꢂ&rm
−i
i
i
−i
rm−→ ꢇ
≤ lim Euiꢄ'r ꢀ'−iꢂWꢂ&
r
rm
m
i
rm−→ ꢇ
= Euiꢄ'iꢀ' ꢂWꢂ&ꢁ ꢀ
−i
This contradicts (51), so we conclude that &ꢁꢂ' is
desired.
a
solution, as
Q.E.D.

discontinuous games
1739
Proof of Theorem 2: The argument follows by substituting the given solu-
tions for the solutions constructed in Step 1 of the proof of Theorem 1, and
continuing as in Steps 2–6.
Q.E.D.
Humanities and Social Sciences, California Institute of Technology, Pasadena, CA
91125, U.S.A.,
Department of Agricultural and Resource Economics, 207 Giannini Hall, Univer-
sity of California, Berkeley, CA 94720, U.S.A.,
Olin School of Business, Washington University in St. Louis, St. Louis, MO 63130,
U.S.A.,
and
Department of Economics, Bunche Hall, University of California, Los Angeles,
CA 90095, U.S.A.
Manuscript received June, 2001; final revision received January, 2002.
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