# Zero-Sum Games involving Teams against Teams: Existence of Equilibria, and
Comparison and Regularity in Information ††thanks: Supported by the Natural
Sciences and Engineering Research Council of Canada. A preliminary version of
this paper was presented at the 2021 IEEE Conference on Decision and Control.

Ian Hogeboom-Burr  Serdar Yüksel The authors are with the Dept. of Mathematics
and Statistics, Queen’s University, Kingston K7L 3N6, ON, Canada,
<EMAIL_ADDRESS>
###### Abstract

Many emerging problems involve teams of agents taking part in a game. Such
problems require a stochastic analysis with regard to the correlation
structures among the agents belonging to a given team. In the context of
Standard Borel spaces, this paper makes the following contributions for two
teams of finitely many agents taking part in a zero-sum game: (i) An existence
result will be presented for saddle-point equilibria in zero-sum games
involving teams against teams when common randomness is assumed to be
available in each team with an analysis on conditions for compactness of
strategic team measures to be presented. (ii) Blackwell’s ordering of
information structures is generalized to $n$-player teams with standard Borel
spaces, where correlated garbling of information structures is introduced as a
key attribute; (iii) building on this result Blackwell’s ordering of
information structures is established for team-against-team zero-sum game
problems. (iv) Finally, continuity of the equilibrium value of team-against-
team zero-sum game problems in the space of information structures under total
variation is established.

##  1 Introduction

In this paper we study zero-sum games where two sets of agents play against
one another. For such problems, the informational and correlation properties
of agent policies play a crucial role. We study existence of equilibrium
solutions, present a characterization of comparison of information structures,
and finally establish regularity properties of equilibrium values in
information structures.

The notions of statistical models (which we call information structures) and
comparison of statistical experiments were characterized in a seminal paper by
Blackwell [7]. For finite models, Blackwell’s results provide necessary and
sufficient criteria for comparing information structures given his notion of a
“more informative” information structure, where an information structure is
better for a decision-maker than another if it ensures the player will never
experience a worse expected cost under the former than under the latter, where
the cost is a function of a hidden state variable and the player’s action, and
where the action is only a function (possibly noisy with an independent noise
realization) of the player’s measurement. Blackwell’s results induce a partial
order on the set of all information structures and, accordingly, many pairs of
information structures are incomparable under his criteria. Le Cam generalized
Blackwell’s criteria through what is now known as the Le Cam deficiency, which
allows for the proximity of any two information structures to be defined [10].

While Blackwell and Le Cam studied information structures in the context of
single-player decision problems, information structures can also be studied in
the context of multiplayer games. In zero-sum games, Gossner and Mertens
worked on the problem of comparing information structures in [15], and this
comparison was fully characterized in the finite case by Pęski in [22]; these
results were extended to a standard Borel setup in [17]. This directly extends
Blackwell’s comparison theorem to zero-sum games. Information structures in
team problems (or identical interest games) were studied in a finite-space
setting by Lehrer, Rosenberg, and Shmaya [12], where Blackwell’s theorem was
extended to two-player team problems in finite spaces, as well as in [30,
Chapter 4].

###  1.1 Subtleties of Information Structures in Stochastic Games

The study of information structures for multi-player setups inevitably
necessitates a tedious geometric and topological analysis. Indeed, in the
context of team and game problems, non-convexity properties on correlation
structures (or strategic measures) and associated non-existence of equilibrium
results, (see in particular [1] for non-existence of equilibria and also [11,
25] for convexity properties) and compactness/non-compactness properties [25]
(see also [16, 32]), require a careful stochastic analysis with many
counterexamples and sufficient conditions presented.

In addition to such geometric and topological challenges, in the following we
emphasize a number of information structure dependent qualitative attributes
of equilibrium solutions in stochastic games. Unlike teams, the dependence on
information is rather intricate in stochastic game theory:

  1. 1.

More information to a given player does not imply better performance; see e.g.
[5] for a dynamic model counterexample. For completeness, we present an
example with static information, in Example 2.1, where we will see that more
information can hurt players.

  2. 2.

The team theoretic or stochastic control argument of a player choosing to
ignore the additional information as a validation of positive value of
information does not apply in games, even in zero-sum games, due to
equilibrium behaviour.

  3. 3.

As shown in [18], for general non-zero sum games, equilibrium values are not
continuous in information structures even under total variation unlike the
setup in team theory.

On the other hand, team theory and zero-sum games have many crucial properties
in common: They both admit well-defined values (though with possibly non-
unique equilibrium policies) and they both possess strong continuity
properties in information, as studied recently in [18] (see also [31]). Some
general properties of information structures in game theory are studied in
detail in [21, 2, 28, 26]. Accordingly, for zero-sum games, several positive
attributes regarding informational structure-dependent properties of
equilibrium solutions arise:

  1. 1.

Under mild conditions (to be reviewed later, e.g. not requiring absolute
continuity conditions on information structures), every zero-sum game has a
Nash equilibrium solution (also called a saddle-point solution).

  2. 2.

A complete theory of ordering of information structures, generalizing that of
Blackwell’s, is possible; see [22] and [17] for finite and standard Borel
setups, respectively. An implication is that, no additional information can
hurt a player, though this argument is quite subtle in the game case: As
noted, the choose to ignore the additional information and thus no information
cannot hurt argument of stochastic control is not applicable, even though the
implication that more information cannot hurt turns out to be correct via a
more delicate (geometric) argument.

  3. 3.

Similar to team problems, equilibrium values for zero-sum games are continuous
in information structures under total variation and share similar (though
slightly refined) semi-continuity properties under weak/setwise convergences
of information structures.

###  1.2 Contributions

In this paper in the context of zero-sum games where two sets of agents play
against one another, we make the following contributions.

  * •

An existence result will be presented for equilibria in zero-sum games
involving teams against teams when common randomness will be assumed to be
available in Theorem 4. As a further technical contribution, a key compactness
condition is provided in Theorem 2 (which will also be used to establish
compactness of correlated garblings of information structures).

  * •

Blackwell’s ordering of information structures will be extended to $n$-player
teams with standard Borel spaces in Theorem 12, as will Le Cam’s deficiency
results. In this context, correlated garbling of information structures will
be introduced as a defining attribute. Our result generalizes [19], which
considered finite models.

  * •

Blackwell’s ordering of information structures will be generalized to team-
against-team zero-sum game problems in Theorem 14. In particular, we obtain a
partial order on the space of information structures, with an implication
being that more information given to a team of decision makers cannot hurt the
team, generalizing a corresponding zero-sum game result given in [22] and
[17].

  * •

The continuity of value of team-against-team zero-sum game problems in the
space of information structures under total variation will be established in
Theorem 15.

##  2 Model and some Supporting Results

###  2.1 Value of Information can be Negative in Stochastic Games: An Example
with Static Information Structure

Let us give an example on the negative value of information in general
stochastic games. While there exist many examples involving dynamic
information structures (e.g. [5]), the following example shows that even when
the information at an agent only depends on exogenous randomness, having more
information can hurt an agent in a stochastic game.

######  Example 2.1.

Let $\mathbb{X}=[0,1]$, with prior distribution $\zeta$ defined by the
continuous uniform distribution. Let there be two teams of decision makers,
each consisting of $N$ individual decision makers, and each with measurement
spaces $\mathbb{Y}^{i}_{1}=[0,1]$, $1\leq i\leq N$ and
$\mathbb{Y}^{j}_{2}=[0,1]$ for $1\leq j\leq N$. We also define the action
spaces of the respective DMs as $\mathbb{U}^{i}_{1}=\mathbb{U}^{j}_{2}=[0,1]$
again with $1\leq i,j\leq N$, and the teams’ cost functions as, with ${\bf
u}_{i}=\\{u_{i}^{1},\cdots,u_{i}^{N}\\}$, $i=1,2$:

$c_{1}(x,{\bf u}_{1},{\bf
u}_{2})=\begin{cases}(x-\frac{1}{N}\sum_{j=1}^{N}u^{j}_{1})^{2}+2,&\frac{1}{N}\sum_{j=1}^{N}u^{j}_{2}=x\\\
2(x-\frac{1}{N}\sum_{j=1}^{N}u^{j}_{1})^{2}+(\frac{1}{N}\sum_{j=1}^{N}u^{j}_{1}-\frac{1}{N}\sum_{j=1}^{N}u^{j}_{2})^{2},&\frac{1}{N}\sum_{j=1}^{N}u^{j}_{2}\neq
x\end{cases}$

$c_{2}(x,{\bf u}_{1},{\bf
u}_{2})=\begin{cases}(x-\frac{1}{N}\sum_{j=1}^{N}u^{j}_{2})^{2}+2,&\frac{1}{N}\sum_{j=1}^{N}u^{j}_{1}=x\\\
2(x-\frac{1}{N}\sum_{j=1}^{N}u^{j}_{2})^{2}+(\frac{1}{N}\sum_{j=1}^{N}u^{j}_{1}-\frac{1}{N}\sum_{j=1}^{N}u^{j}_{2})^{2},&\frac{1}{N}\sum_{j=1}^{N}u^{j}_{1}\neq
x\end{cases}$

Under measurement channels where neither of the team has any information
regarding $x$, (which could be achieved for instance by measurement channels
returning $y^{i}_{m},m=1,2;1\leq i\leq N$ which are independent of $x$, via a
channel with zero information theoretic capacity), there exists an essentially
unique Nash equilibrium solution given with
$\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}=\frac{1}{2}=\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}$.
This uniqueness can be seen as follows: when the DMs have no information
regarding $x$, the event that $\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}=x$ or
$\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}=x$ has zero measure and therefore the
complementary conditions are active for defining the cost in the above. Note
though that in this case the cost functions essentially turn the problem into
a team problem since minimization over $\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}$ of

$2(x-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1})^{2}+(\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2})^{2}$

will lead to the same solution as the minimization of

$2(x-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1})^{2}+(\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2})^{2}+2(x-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2})^{2}.$

The same applies for $\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}$, and so we can view
the DMs as solving essentially the same problem. Therefore, we have a standard
static quadratic team problem which admits an essentially unique optimal
solution (where the cost is strictly convex in the average team action, though
not on the collection of actions) [30, Theorem 2.6.3] with
$\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}=\frac{1}{2}=\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}$.
This leads to an expected cost of $\frac{1}{6}$ for each team.

However, when the DM measurement channels are such that all DMs have perfect
information regarding $x$ (e.g. through $y^{i}_{1}=x=y^{j}_{2},\qquad 1\leq
i,j,\leq N$), we first see that another equilibrium solution arises: with
$\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}=x=\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}$. One
can observe this is a Nash equilibrium by noting that if DM $2$ holds their
strategy constant as $\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}=x$, then DM 1 will
play the game with cost function
$(x-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1})^{2}+2$, which is minimized by playing
$\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}=x$. The same holds true for DM 2 in the
reverse case where DM 1’s strategy is fixed. We now make the point that this
is the essentially unique deterministic Nash equilibrium:

If the condition
$\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}=x=\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}$ is
not active, we would again reduce to a team problem and in this case, we would
have

$\gamma^{i}_{1}(x)=\frac{2}{3}x+\frac{1}{3}\frac{1}{N}\sum_{i=1}^{N}\gamma^{i}_{2}(x),\qquad\gamma^{i}_{2}(x)=\frac{2}{3}x+\frac{1}{3}\frac{1}{N}\sum_{i=1}^{N}\gamma^{i}_{1}(x)$

leading to, by (essential) uniqueness [30, Theorem 2.6.3],
$\frac{1}{N}\sum_{i=1}^{N}\gamma^{i}_{1}(x)=x=\frac{1}{N}\sum_{i=1}^{N}\gamma^{i}_{2}(x)$,
which however would take us back to the inefficient equilibrium given above.
Thus, the (essentially) unique pure strategy Nash equilibrium for the full-
information problem is at
$\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}=x=\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}$.
This results in an expected cost of $2$ for each of the 2 teams of DMs.

This example demonstrates a static team-against-team game in which more
information hurts all the DMs. We note that by adding individual costs such as
$(u^{i}_{1})^{2}$ so that the costs are strictly convex, the essential
uniqueness can be strengthened to uniqueness (however the goal is just to show
that more information can be detrimental in game problems). For example, if
the costs are modified as follows, then uniqueness under the former
information structure can be attained and a similar analysis is applicable for
the latter if a strict convexity is imposed on the costs in the first cases of
both functions:

$c^{1}(x,{\bf u}^{1},{\bf
u}^{2})=\begin{cases}(x-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1})^{2}+2,&\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}=x\\\
2(\frac{3}{2}x-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1})^{2}+\frac{1}{N}\sum_{i=1}^{N}(u^{i}_{1})^{2}+(\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2})^{2},&\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2}\neq
x\end{cases}$ $c^{2}(x,{\bf u}^{1},{\bf
u}^{2})=\begin{cases}(x-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2})^{2}+2,&\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}=x\\\
2(\frac{3}{2}x-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2})^{2}+\frac{1}{N}\sum_{i=1}^{N}(u^{i}_{2})^{2}+(\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}-\frac{1}{N}\sum_{i=1}^{N}u^{i}_{2})^{2},&\frac{1}{N}\sum_{i=1}^{N}u^{i}_{1}\neq
x\end{cases}$

Note that in this case, $u^{j}_{1}=\frac{1}{2}$ is the unique solution in the
no information case and $u^{i}_{m}=x,m=1,2;1\leq i\leq N$ is the essentially
unique solution in the full information case.

We will see, however, that in the context of zero-sum games involving teams of
DMs, a partial ordering is possible and more information cannot hurt a DM (or
team).

###  2.2 Stochastic Teams and a Supporting Result

Consider $n\in\mathbb{Z}_{\geq 1}$ players on a team in a single-stage game.
Let $\Omega_{0}$ denote a standard Borel state space, with $x\sim\zeta$ an
$\Omega_{0}$-valued random variable known as the state of nature. Recall that
a standard Borel space is a Borel subset of a complete, separable, metric
(Polish) space. Let $\zeta$ be the prior probability distribution on a hidden
state, which is common knowledge to all players. We denote by $\mathbb{Y}^{i}$
Player $i$’s standard Borel measurement space, for $i\in\\{1,\dots,n\\}$. We
use $-i$ to denote all players other than Player $i$. For each player, we
define their measurement $y^{i}$, which is a $\mathbb{Y}^{i}$-valued random
variable, where: $y^{i}=g^{i}(x,v^{i})$ for some noise variable $v^{i}$ (and
which, without any loss, can be taken to be $[0,1]$-valued). By stochastic
realization arguments [9], the above formulation is equivalent to viewing
$y^{i}$ as being generated by a measurement channel $Q^{i}$, which is a Markov
transition kernel from $\Omega_{0}$ to $\mathbb{Y}^{i}$. We denote the joint
probability measure on
$\Omega_{0}\times\mathbb{Y}^{1}\times\dots\times\mathbb{Y}^{n}$ by
$\mu(d\omega_{0},dy^{1},\dots,dy^{n})$, and define this as the information
structure.

For $P\in\mathcal{P}(\Omega_{0})$ and kernel $Q$, we let $PQ$ denote the joint
distribution induced on
$(\Omega_{0}\times\mathbb{Y},\mathcal{B}(\Omega_{0}\times\mathbb{Y}))$ by
channel $Q$ with input distribution $P$:

$PQ(A)=\int_{A}Q(dy|x)P(d\omega_{0}),\quad
A\in\mathcal{B}(\Omega_{0}\times\mathbb{Y}).$

Each player also has a standard Borel action space $\mathbb{U}^{i}$. The team
has a common cost function
$c:\Omega_{0}\times\mathbb{U}^{1}\times\dots\times\mathbb{U}^{n}\rightarrow\mathbb{R}$.

For fixed prior, state space, and measurement spaces $\zeta$, $\Omega_{0}$,
$\mathbb{Y}^{1},\dots,\mathbb{Y}^{n}$, a game for the team is a $n+1$-tuple
consisting of a measurable and bounded cost function and an action space for
each player, $G=(c,\mathbb{U}^{1},\dots,\mathbb{U}^{n})$.

The team’s goal is to minimize the expected cost functional for a given game
$G$:

$\displaystyle
J(G,\mu,\gamma^{1},\dots,\gamma^{n}):=E^{\mu,\bar{\gamma}}[c(x,\gamma^{1}(y^{1}),\dots,\gamma^{n}(y^{n}))]$

where the players select their policies from the set of all admissible
policies $\Gamma^{i}:=\\{\gamma:\mathbb{Y}^{i}\rightarrow\mathbb{U}^{i}\\}$,
which are measurable functions from a player’s measurement space to their
action space. We refer to $u^{i}=\gamma^{i}(y^{i})$ as the action of the
player, and $\gamma^{i}$ as their policy.

In team problems, equilibrium solutions are team policies
$\bar{\gamma}^{*}:=(\gamma^{1,*},\dots,\gamma^{n,*})$ which minimize the value
function. The value function at equilibrium is denoted by $J^{*}(c,\mu)$.
General existence results for team-optimal policies can be found in [29,
Section 5].

To ensure existence of team-optimal policies, throughout this paper we will
assume the player action spaces $\mathbb{U}^{i}$ are compact and the cost
function is continuous and bounded in actions, but not necessarily in
$\omega_{0}$.

We also recall the notion of a strategic measure, which is a probability
measure on the state, measurement, and action spaces induced by a policy.
Given a fixed team policy $\bar{\gamma}=(\gamma^{1},\dots,\gamma^{n})$, a
strategic measure is the joint probability measure induced by $\bar{\gamma}$
on $\Omega_{0}\times\prod^{n}_{k=1}(\mathbb{Y}^{k}\times\mathbb{U}^{k})$. We
now define and revisit various spaces of strategic measures.

We define $L_{C}(\mu)$, as in [32]:

$L_{C}(\mu)=\left\\{P\in\mathcal{P}\left(\Omega_{0}\times\prod_{k=1}^{n}(\mathbb{Y}^{k}\times\mathbb{U}^{k})\right):P(B)=\int\eta(dz)L_{A}(\mu,\tilde{\gamma}(z))(B),\eta\in\mathcal{P}(\mathcal{W})\right\\}.$

Where $\mathcal{W}=[0,1]^{n}$, $\tilde{\gamma}$ represents a collection of
team policies measurably parameterized by $z\in\mathcal{W}$ so that the map
$L_{A}(\mu,\tilde{\gamma}(\cdot)):\mathcal{W}\rightarrow L_{A}(\mu)$ is Borel
measurable, and where $L_{A}(\mu)$ is:

$\displaystyle L_{A}(\mu)=$
$\displaystyle\bigg{\\{}P\in\mathcal{P}\left(\Omega_{0}\times\prod_{k=1}^{n}(\mathbb{Y}^{k}\times\mathbb{U}^{k})\right)$
$\displaystyle:P(B)=\int_{B^{0}\times\prod_{k}A^{k}}\mu(d\omega_{0},d\textbf{y})\prod_{k}1_{u_{k}=\gamma^{k}(y^{k})\in
B^{k}},\gamma^{k}\in\Gamma^{k},B\in\mathcal{B}\left(\Omega_{0}\times\prod_{k=1}^{n}(\mathbb{Y}^{k}\times\mathbb{U}^{k})\right)\bigg{\\}}.$

Note that $L_{A}(\mu)$ is Borel measurable under the weak convergence topology
[25], and we use $L_{A}(\mu,\bar{\gamma})$ to denote the strategic measure
induced by a particular team strategy $\bar{\gamma}$. We define $L_{CCR}(\mu)$
as:

$\displaystyle L_{CCR}(\mu)$ $\displaystyle:=$
$\displaystyle\bigg{\\{}P\in\mathcal{P}\left(\Omega_{0}\times\prod_{k=1}^{n}(\mathbb{Y}^{k}\times\mathbb{U}^{k})\right)$
$\displaystyle\qquad:P(B)=\int_{B\times\mathcal{W}}\eta(dz)\mu(d\omega_{0},d\bar{y})\prod_{k}\Pi^{k}(du^{k}|y^{k},z),\eta\in\mathcal{P}(\mathcal{W})\bigg{\\}},$

where $\Pi^{k}$ is a stochastic kernel from
$(\mathbb{Y}^{k}\times\mathcal{W})$ to $\mathbb{U}^{k}$, for $k=1,\dots,n$. It
was shown in [25, Theorem 3.1] that $L_{C}(\mu)=L_{CCR}(\mu)$.

We will also define $L_{R}(\mu)$ as:

$L_{R}(\mu)=\left\\{P\in\mathcal{P}\left(\Omega_{0}\times\prod_{k=1}^{n}(\mathbb{Y}^{k}\times\mathbb{U}^{k})\right):P(B)=\int_{B}\mu(d\omega_{0},d\bar{y})\prod_{k=1}^{n}\kappa^{k}(du^{k}|y^{k})\right\\},$

where $\kappa^{k}$ is taken from the set of stochastic kernels from
$\mathbb{Y}^{k}$ to $\mathbb{U}^{k}$ for each $k=1,\dots,n$.

Essentially, $L_{A}(\mu)$ is the set of all strategic measures induced by
admissible measurable policies, $L_{R}(\mu)$ is the set induced by allowing
independently randomized admissible policies for each player, and $L_{C}(\mu)$
is the set induced by allowing randomized admissible policies with potentially
common randomness among players.

It was shown in [32] that $L_{C}(\mu)$ is convex, which will be crucial in the
extension of Blackwell’s theorem. Closedness and compactness of $L_{C}(\mu)$
will be required for some of the results presented here. In general,
$L_{C}(\mu)$ is not closed under the weak convergence topology [25, Theorem
4.3]. However, we can relate sufficient conditions for the compactness of the
space $L_{R}(\mu)$ to $L_{C}(\mu)$.

For the following result, we assume that the measurements of the players are
either independent or by a change of measure argument, under an absolute
continuity condition, have been turned into a static information structure
with independent measurements setup, see [29, Section 2.2] for details. In
particular, we assume that

######  Assumption 2.1.

$\displaystyle
P(d\omega_{0},dy^{1},\cdots,dy^{n})\ll\mu_{0}(d\omega_{0})\prod_{i=1}^{n}\bar{Q}^{i}(dy^{i})$
(1)

for some reference measures $\mu_{0}$ and $\bar{Q}^{i}$ so that the Radon-
Nikodym derivative

$\frac{d(P(d\omega_{0},dy^{1},\cdots,dy^{n}))}{d(\mu_{0}(d\omega_{0})\prod_{i=1}^{n}\bar{Q}^{i}(dy^{i})))}=:f(\omega_{0},y^{1},\cdots,y^{n})$

exists.

This condition is important in establishing that $L_{R}(\mu)$ is compact under
the weak convergence topology, as we make formal in the following.

######  Theorem 1.

[29, Theorem 5.2] Consider a static or a dynamic team that satisfies
Assumption 2.1 (and thus, admits an independent static reduction). The set
$L_{R}(\mu)$ is closed under the weak convergence topology. Furthermore, if
the action sets are compact (or under a tightness condition), the set
$L_{R}(\mu)$ is compact under the weak convergence topology.

We will now state the following theorem. This result will be critical both for
existence of equilibria and for comparison of information structures.

######  Theorem 2.

If $L_{R}(\mu)$ is compact, then $L_{C}(\mu)$ is compact.

###### Proof.

Let $\\{\phi_{k}\\}$ be an arbitrary sequence of information structures in
$L_{C}(\mu)$ defined with
$B\in\mathcal{B}\left(\Omega_{0}\times\prod_{k=1}^{n}(\mathbb{Y}^{k}\times\mathbb{U}^{k})\right)$
as

$\phi_{k}(B)=\int_{B}\left(\int\nu_{k}(d\theta)\theta(dz)\right).$

Following the relationship between $L_{CCR}(\mu)$ and $L_{R}(\mu)$, as well as
[25, Theorem 3.1], we can view $L_{C}(\mu)$ as the mixture of elements of
$L_{R}$. The convergence properties of the sequence $\\{\phi_{k}\\}$ can be
studied by the convergence of the family of measures:

$\phi_{k}(B)=\int_{B}f_{m}(z)\left(\int\nu_{k}(d\theta)\theta(dz)\right),$

for a fixed countable collection of weak-convergence determining functions
$f_{m}:\Omega\times(\prod_{i=1}^{n}\mathbb{X}^{i}\times\mathbb{Y}^{i})\rightarrow\mathbb{R}$
[13, Theorem 3.4.5].

Now, using Fubini’s Theorem, we can rewrite the above as:

$\int\nu_{k}(d\theta)\int f_{m}(z)\theta(dz),$

and can note that the inner integral is continuous and bounded in $\theta$.

Thus we can write this as:

$\int\nu_{k}(d\theta)F_{m}(\theta),$

for the continuous and bounded $F_{m}$ on $L_{R}(\mu)$ given with
$F_{m}(\theta):=\int f_{m}(z)\theta(dz)$.

Since $L_{R}(\mu)$ is weakly compact, the set of probability measures on
${\cal B}(L_{R}(\mu))$ is weakly compact [6]; and accordingly there exists a
subsequence of $\\{\nu_{k}\\}$, $\\{\nu_{k_{l}}\\}$, which converges weakly to
some $\nu$ in ${\cal P}(L_{R}(\mu))$. Since $F_{m}$ is continuous and bounded
on $L_{R}(\mu)$, it follows then that $\int\nu_{k_{l}}(d\theta)F_{m}(\theta)$
will converge to $\int\nu(d\theta)F_{m}(\theta)$. This applies for each $m$,
and therefore, we conclude that $\\{\phi_{k}\\}$ has a convergent subsequence
in $L_{C}(\mu)$. Thus, $L_{C}(\mu)$ is compact. $\diamond$

This gives us sufficient conditions to ensure compactness of $L_{C}(\mu)$.
This result will be crucial in our analysis both for the existence of
equilibria as well as on a key closedness property involving a set of
correlated garblings in a team of agents.

###  2.3 Team-against-Team Zero-Sum Games

For teams against teams, we consider two teams. Team 1 (the minimizing team)
consists of $N_{1}$ DMs and Team 2 (the maximizing team) consists of $N_{2}$
DMs. The two teams compete in a single-stage zero-sum game with private
information. Here we use a subscript $i$ to denote the team to which a DM
belongs.

The teams share a common cost function
$c:\Omega_{0}\times\mathbb{U}^{1}_{1}\times\dots\times\mathbb{U}^{N_{1}}_{1}\times\mathbb{U}^{1}_{2}\times\dots\times\mathbb{U}^{N_{2}}_{2}\rightarrow\mathbb{R}$.
Team 1’s goal is to minimize the expected cost and Team 2’s goal is to
maximize it.

The information structure for the game $\mu$ is a joint probability measure on
$\Omega_{0}\times\mathbb{Y}^{1}_{1}\times\dots\times\mathbb{Y}^{N_{1}}_{1}\times\mathbb{Y}^{1}_{2}\times\dots\times\mathbb{Y}^{N_{2}}_{2}$.
We use $\mu_{i}$ to denote the marginal of $\mu$ on
$\Omega_{0}\times\mathbb{Y}^{i}_{i}\times\dots\times\mathbb{Y}^{N_{i}}_{i}$
for $i=1,2$ (i.e. $\mu_{i}$ captures the information available to team $i$).

For teams-against-teams we use the notation $V$ and $V^{*}$, rather than $J$
and $J^{*}$, to denote the value and equilibrium value of a game,
respectively.

######  Definition 3.

Given an information structure $\mu$, let ${\bf\Gamma}_{i}$ denote the set of
team policies of Team $i$, $i=1,2$. We say that
$\bar{\gamma}^{*}_{1},\bar{\gamma}^{*}_{2}$ is an equilibrium for a team-
against-team game if

$\inf_{\bar{\gamma}_{1}\in{\bf\Gamma}_{1}}V(\mu,\bar{\gamma}_{1},\bar{\gamma}^{*}_{2})=V(\mu,\bar{\gamma}^{*}_{1},\gamma^{*}_{2})=\sup_{\bar{\gamma}_{2}\in{\bf\Gamma}_{2}}V(\mu,\gamma^{*}_{1},\bar{\gamma}_{2}).$

##  3 Team-against-Team Zero-Sum Games: Existence of Saddle Point Equilibria

First, we establish an existence result for saddle-point equilibria. As
emphasized explicitly in [1], lack of convexity in the set of team policies
can lead to games without a saddle-point. We will allow each team to select a
team policy in which players may share common randomness. This randomness is
independent between teams. Allowing this, rather than only independently
randomized policies for each team, ensures that the team policy spaces are
convex, which is crucial in demonstrating the existence of an equilibrium.

[4, Theorem 3.1] (and [17, Theorem 3.1]) study existence of saddle-point
equilibria in two-player zero-sum games under an absolute continuity condition
and these have been relaxed in [20] (also [4, Theorem 3.4], [17, Theorem
3.2]). In the following, we present a generalization of the relaxed version
where absolute continuity among the different team-players is not required
(thus Assumption 3.2 presented below does not necessarily apply, but does
apply within the team under Assumption 3.1 below).

For $i=1,2$, for fixed information structure $\mu$, we define $\mu_{i}$ as the
the marginal of $\mu$ on
$\Omega_{0}\times\prod_{j=1}^{N_{i}}\mathbb{Y}^{j}_{i}$. $\mu_{i}$ represents
the portion of the information structure $\mu$ that is relevant for Team $i$’s
action selection (as they can not see the opposing team’s measurements). We
then we define $L_{C}(\mu_{i})$ following our usual definition for a single
team, allowing for actions to be selected using independent (but possibly
shared between players) randomness.

We will assume that $L_{C}(\mu_{i})$ is closed for every fixed policy of the
opponent team, a sufficient condition for this is given below in Assumption
3.1.

######  Assumption 3.1.

Equation (1) holds for both $\mu_{1}$ and $\mu_{2}$.

######  Theorem 4.

For a given team-against-team game, assume the following hold.

  1. (i)

Team policies are selected such that the strategic measures belong to
$L_{C}(\mu_{1}),L_{C}(\mu_{2})$, respectively.

  2. (ii)

Assumption 3.1 applies (the implication, to be shown in the proof, being that
$L_{C}(\mu_{1})$ (when Team 2’s policy is fixed) and $L_{C}(\mu_{2})$ (when
Team 1’s policy is fixed) are compact under weak convergence).

  3. (iii)

The cost function $c$ is bounded and continuous in players’ actions for any
$\omega_{0}\in\Omega_{0}$.

Then an equilibrium exists.

###### Proof.

In what follows, we always assume a team policy
$\bar{\gamma}_{i}=(\gamma^{1}_{i},\dots,\gamma^{N_{i}}_{i})$ is such that
$u^{k}_{i}=\gamma^{k}_{i}(y^{k}_{i},z_{i})$, for some independent randomness
$z_{i}\in[0,1]^{N_{i}}$ for each team, for $k=1,\dots,N_{i}$, $i=1,2$. I.e.,
policies are selected such that the induced strategic measures are in
$L_{C}(\mu_{1})$ and $L_{C}(\mu_{2})$. Note that the measure on
$(\Omega\times\prod_{j=1}^{N_{1}}\mathbb{Y}^{j}_{1}\times\prod_{k=1}^{N_{2}}\mathbb{Y}^{k}_{2})$
is fixed for both teams by the information structure $\mu$, and so
$L_{C}(\mu_{i})$ has a fixed marginal on the state and measurement spaces.

Let $L_{C}(\mu_{i},\bar{\gamma}_{i})$ be team $i$’s strategic measure induced
by team policy $\bar{\gamma}_{i}$, for $i=1,2$. Denote by $S_{i}$ the
projection of the space $L_{C}(\mu_{i})$ onto
$\prod_{k=1}^{N_{i}}(\mathbb{Y}^{k}_{i}\times\mathbb{U}^{k}_{i})$, eliminating
the state space from the measure and simplifying notation for convenience.
Since the measure on the exogenous variables is fixed by the information
structure $\mu$, $S_{i}$ will inherit convexity (as conditional independence
will be preserved under convex combinations given that the opponent team
policy is fixed) and weak compactness from $L_{C}(\mu_{i})$ (following from
the analysis in the proof of [17, Theorem 3.2]); see [25, Theorem 5.2] which
shows that conditional independence is preserved when only one player perturbs
the strategic measure (unlike the case where multiple players make
simultaneous perturbations in [25, Theorem 4.2]). This applies in this context
by viewing each team as a large single-agent.

Now, without loss of generality, we will fix Team 2’s policy as
$\bar{\gamma}_{2}^{*}$ (which is an arbitrary policy that induces a strategic
measure $s_{2}^{*}$ in $L_{C}(\mu_{2})$). We then have:

$\displaystyle V_{G}^{\mu}(s_{1},s_{2}^{*})$ $\displaystyle=\int
s_{1}(dy^{1}_{1},\dots,dy^{N_{1}}_{1},du^{1}_{1},\dots,du^{N_{1}}_{1})\dots$
(2)
$\displaystyle\quad\quad\quad\quad\left(\int\mu^{{\bar{\gamma}_{2}^{*}}}(d\omega_{0},dy^{1}_{2},\dots,dy^{N_{2}}_{2},du^{1}_{2},\dots,du^{N_{2}}_{2}|y^{1}_{1},\dots,y^{N_{1}}_{1})c(\omega_{0},u^{1}_{1},\dots,u_{1}^{N_{1}},u_{2}^{1},\dots,u_{2}^{N_{1}})\right),$
(3)

where $\mu^{{\bar{\gamma}_{2}}^{*}}$ denotes the strategic measure induced by
Team 2’s policy ${\bar{\gamma}_{2}}^{*}$. Define the bracketed term as
$\bar{c}(y^{1}_{1},\dots,y_{1}^{N_{1}},u^{1}_{1},\dots,u_{1}^{N_{1}})$. We can
observe that, by assumption, $\bar{c}$ is continuous in
$u^{1}_{1},\dots,u_{1}^{N_{1}}$, and furthermore it is measurable in
$y^{1}_{1},\dots,y_{1}^{N_{1}}$.

Recall the $w$-$s$ topology [27] on the set of probability measures ${\cal
P}(\Omega_{0}\times\mathbb{U})$; this is the coarsest topology under which
$\int f(x,u)\nu(d\omega_{0},du):{\cal
P}(\Omega_{0}\times\mathbb{U})\to\mathbb{R}$ is continuous for every
measurable and bounded $f$ which is continuous in $u$ for every $x$ (but
unlike weak topology, $f$ does not need to be continuous in $x$). Now, since
the exogenous variables are fixed, weak convergence in this setting is
equivalent to $w$-$s$ convergence, and continuity in the exogenous variable is
not needed here: Consider a sequence of measures $\\{s_{1}\\}_{k}$ which
converges to $s_{1}$ weakly. We have that
$\bar{c}(u^{1}_{1},\dots,u^{1}_{N_{1}},y^{1},\dots,y^{1}_{N_{1}})$ is
continuous in $u^{1}_{1},\dots,u^{1}_{N_{1}}$ by assumption. Since $\mu$ is
fixed, the marginals on $\prod_{k=1}^{N_{2}}\mathbb{Y}^{2}_{k}$ are fixed.
Therefore, by [27, Theorem 3.10] (or [3, Theorem 2.5] and [29]), we can use
the $w$-$s$ topology on the set of probability measures
$\mathcal{P}(\prod_{k=1}^{N_{2}}\mathbb{Y}_{2}^{k}\times\mathbb{U}_{2}^{k})$.
And so we have continuity of $V^{\mu}_{G}$ in $s_{1}$ in the $w$-$s$ topology
and, by the equivalence in this setting, the weak topology.

This also holds for continuity in $s_{2}$ in the reverse case were we fix Team
1’s strategic measure as an arbitrary $s_{1}^{*}$.

Following compactness and convexity of $S_{1}$ and $S_{2}$, by the continuity
(and linearity in each entry) of $V_{G}^{\mu}(s_{1},s_{2})$, by Fan’s minimax
theorem, [14, Theorem 1], we have that an equilibrium exists:

$\min_{S_{1}}\max_{S_{2}}V_{G}^{\mu}(s_{1},s_{2})=\max_{S_{2}}\min_{S_{1}}V_{G}^{\mu}(s_{1},s_{2})$

$\diamond$

The setup where absolute continuity applies also between the teams would be a
special case, which would make the proof presentation slightly more direct, as
we note below.

######  Assumption 3.2.

$\mu(d\omega_{0};dy^{1}_{1},\dots,dy^{N_{1}}_{2};dy^{1}_{2},\dots,dy^{N_{2}}_{2})\ll\zeta(d\omega_{0})\mu_{1}(dy^{1}_{1},\dots,dy^{N_{1}}_{2})\mu_{2}(dy^{1}_{2},\dots,dy^{N_{2}}_{2}).$

In Theorem 4, if we had also imposed Assumption 3.2, the analysis following
(3) would be more direct. However, strictly speaking, this assumption is not
needed as absolute continuity across the teams is not required (only within
members of a team).

##  4 Comparison of Information Structures in Team-against-Team Zero-Sum Games

We now establish a comparison result on information structures for stochastic
teams. Before we do this, we present a brief review.

###  4.1 Brief Review on Comparison of Information Structures

We now recall the fundamental results of Blackwell and Le Cam in the
comparison of information structures for single-player decision problems.
First, we recall the standard notion of garbling.

######  Definition 5.

An information structure induced by some channel $Q_{2}$ is garbled (or
stochastically degraded) with respect to another one, $Q_{1}$, if there exists
a channel $Q^{\prime}$ on $\mathbb{Y}\times\mathbb{Y}$ such that

$Q_{2}(B|x)=\int_{\mathbb{Y}}Q^{\prime}(B|y)Q_{1}(dy|x),\;B\in{\cal
B}(\mathbb{Y}),\;P\;a.s.\;x\in\mathbb{X}.$

We also define the following notion:

######  Definition 6.

An information structure $\mu$ is more informative than another information
structure $\nu$ if

$\inf_{\gamma\in\Gamma}E^{\nu,\gamma}_{P}[c(x,u)]\geq\inf_{\gamma\in\Gamma}E^{\mu,\gamma}_{P}[c(x,u)]$

for all single player decision problems $(c(x,u),\mathbb{U})$.

We now recall Blackwell’s classical result:

######  Theorem 7.

[Blackwell [8]] Let $\mathbb{X},\mathbb{Y}$ be finite spaces. The following
are equivalent:

  * (i)

$Q_{2}$ is weakly stochastically degraded with respect to $Q_{1}$ (that is, a
garbling of $Q_{1}$).

  * (ii)

The information structure induced by channel $Q_{1}$ is more informative than
the one induced by channel $Q_{2}$ for all single player decision problems
with finite $\mathbb{U}$.

The implication (i)$\rightarrow$(ii) is direct and is applicable to general
spaces, as noted in Theorem 4.3.2 of [30]; the converse direction is more
involved: the finite case was studied by Blackwell, the zero-sum games with
finite spaces was studied in [22], and the standard Borel case was studied in
[17].

Le Cam developed a further extension of Blackwell’s theorem, and defined the
notions of Le Cam deficiency and Le Cam distance.

######  Definition 8.

The Le Cam deficiency of an information structure $\mu$ with respect to
another information structure $\nu$ is $\delta(\mu,\nu)=:\inf_{\kappa\in
K}\|\kappa\mu-\nu\|_{TV}$

We now recall the following theorem due to Le Cam, which holds in the standard
Borel setup:

######  Theorem 9 (Le Cam [10]).

Let $\epsilon>0$ be fixed, $\delta(\mu,\nu)<\epsilon$ if and only if for any
policy $\gamma_{\nu}$ under $\nu$ and any game with a bounded cost function
$\|c\|_{\infty}\leq 1$, there exists a policy $\bar{\gamma}_{\mu}$ under $\mu$
such that

$E^{\mu}_{P}[c(x,\bar{\gamma}_{\mu}(y))]<E^{\nu}_{P}[c(x,\gamma_{\nu}(y))]+\epsilon$

We also have the following definition from Le Cam:

######  Definition 10.

The Le Cam distance between two information structures $\mu$ and $\nu$ is

$\Delta(\mu,\nu):=\max\\{\delta(\mu,\nu),\delta(\nu,\mu)\\}.$

Le Cam’s results generalize those of Blackwell, and can be used to define a
topology on information structures based on difference in value over all
games; this was done for zero-sum games in [23].

###  4.2 Comparison of Information Structures in Stochastic Teams

Now, we move to extending Blackwell’s results on comparison of information
structures for single-player decision problems to team problems. This was done
for two-player teams by Lehrer, Rosenberg, and Shmaya in [19] for problems
with finite spaces, see also [30, Chapter 4]. Here, these results are extended
to arbitrary $n$-player teams with standard Borel spaces.

We define the notion of garbling with potentially common randomness for a
team, generalizing the notion from Definition 5:

######  Definition 11.

An information structure $\nu$ is a correlated garbing of an information
structure $\mu$ with potentially common randomness if there exists
$\eta\in\mathcal{P}(\mathcal{W})$, where $\mathcal{W}=[0,1]^{n}$, such that:

$\nu(B)=\int_{B\times\mathcal{W}}\eta(dz)\mu(d\omega_{0},d\bar{y})\prod_{k}\Pi^{k}(d\tilde{y}^{k}|y^{k},z),\quad\forall
B\in\mathcal{B}(\Omega\times\prod_{i=1}^{n}\mathbb{Y}^{i})$

for some stochastic kernels $\Pi^{k}$ from $(\mathbb{Y}^{k}\times\mathcal{W})$
to $\mathbb{Y}^{k}$, for $k=1,\dots,n$.

Note that here $\eta$ is an arbitrary probability measure, as in the
definition of $L_{C}(\mu)$. Assume $L_{C}(\mu)$ is closed (see Theorem 2, a
sufficient condition being the independent static reducibility under
Assumption 2.1.

Let $\mathcal{E}$ denote the set of all team problems with compact action
spaces and cost functions that are continuous in player actions for every
$\omega_{0}$.

######  Theorem 12.

Let Assumption 2.1 hold so that $L_{C}(\mu)$ is closed. For team problems with
static reductions, an information structure $\mu$ is more informative than an
information structure $\nu$ (for all games in $\mathcal{E})$ if only if $\nu$
is a garbling of $\mu$ for each player, with potentially common randomness
among the garblings.

###### Proof.

(Sketch) Part 1: First we study the forward direction. Let $\mu$ be an
information structure, and let $\nu$ be another information structure such
that the measurements under $\nu$ are given by
$y^{i}_{\nu}=h^{i}(y^{i}_{\mu},Z)$ for $i=1,\dots,n$ and some independent
randomness $Z$, that may be common between players.

We can clearly observe that any outcome for a game achievable under $\nu$ is
achievable under $\mu$, since outcomes are determined by policies, which are
measurable functions from measurements to actions, and the measurements under
$\nu$ can be simulated using the measurements from $\mu$ and the independent
randomness $Z$, and thus the resulting expected outcomes can also be achieved
under $\mu$.

Part 2: We now establish the converse direction. Let
$\mu,\nu\in\mathcal{P}(\Omega_{0}\times\mathbb{Y}^{1}\times\dots\times\mathbb{Y}^{n})$,
and assume $\mu$ is more informative than $\nu$. Since this means the expected
cost is lower under $\mu$ than under $\nu$ for all games, in particular it is
true for games where $\mathbb{U}^{i}=\mathbb{Y}^{i}$. We will fix
$\mathbb{U}^{i}=\mathbb{Y}^{i}$ and evaluate this class of games for the
remainder of the proof.

Let $K_{Z}\mu$ denote the space of all garblings of $\mu$ with common
randomness allowed. I.e.

$K_{Z}\mu=\\{\nu\in\mathcal{P}(\Omega_{0}\times\mathbb{Y}^{1}\times\dots\times\mathbb{Y}^{n}):y^{i}_{\nu}=h^{i}(y^{i}_{\mu},Z)\\}$

for some independent randomness $Z$ that may be common between players.

We assume that $\nu\in K_{Z}\mu$ and proceed with a proof by contradiction.

Note that since $L_{C}(\mu)$ is closed under Assumption 2.1, via Theorem 2.
Furthermore, this set is convex, and since $\mathbb{Y}^{i}=\mathbb{U}^{i}$
here, we have that $K_{Z}\mu$ will be closed and convex. This follows from
noticing that $L_{C}(\mu)$ is the set of all measures on
$(\Omega\times\prod_{i=1}^{n}(\mathbb{Y}^{i}\times\mathbb{U}^{i}))$ with fixed
marginal $\mu$ on $(\Omega\times\prod_{i=1}^{n}\mathbb{Y}^{i})$, where the
measure on the action spaces is created through applying all possible policies
(which we can view as garblings) with common randomness; since each player’s
action space and measurement space are assumed to be the same, this means that
$K_{Z}\mu$ is the projection of $L_{C}(\mu)$ onto the state and action spaces,
and so $K_{Z}\mu$ is also closed and convex.

We will now use the Hahn-Banach Separation Theorem for Locally Convex Spaces.
To apply this, we require local convexity of
$\mathcal{P}(\Omega_{0}\times\prod_{k=1}^{n}\mathbb{Y}^{k})$, and so we define
the locally convex space of probability measures with the following notion of
convergence: We say that $P_{n}\rightarrow P$ if

$\displaystyle\int
f(\omega_{0},y^{1},\dots,y^{n})P_{n}(d\omega_{0},dy^{1},\dots,dy^{n})\rightarrow\int
f(\omega_{0},y^{1},\dots,y^{n})P(d\omega_{0},dy^{1},\dots,dy^{n})$

for every measurable and bounded function which is continuous in
$(y^{1},\dots,y^{n})$ for every $\omega_{0}$. We note that our measures must
still have fixed marginal $\zeta$ on $\Omega_{0}$.

Since continuous and bounded functions separate probability measures (in the
sense that, if the integrations of two measures with respect to continuous
functions are equal, the measures must be equal), we can represent every
continuous linear map on our space using the form, following [24, Theorem
3.10]:

$\displaystyle F=\int
f(\omega_{0},y^{1},\dots,y^{n})\nu(d\omega_{0},dy^{1},\dots,dy^{n}),$

for some measurable and bounded function $f(\omega_{0},y^{1},\dots,y^{n})$
continuous for every $\omega_{0}$. It also follows that, given this notion of
convergence, $\mathcal{P}(\Omega_{0}\times\prod_{k=1}^{n}\mathbb{Y}^{k})$ is a
locally convex space.

Thus, since $\nu$ is compact and convex and $K_{Z}\mu$ is closed and convex,
following a combination of [24, Theorem 3.4] and [24, Theorem 3.10], we have
that there exists a continuous and bounded function
$f:\Omega_{0}\times\mathbb{Y}^{1}\times\dots\times\mathbb{Y}^{n}\rightarrow\mathbb{R}$
such that there exist constants $D_{2}>D_{1}$ with

$\langle\nu,f\rangle<D_{1},\langle\kappa_{Z}\mu,f\rangle>D_{2}\quad\quad\forall\kappa_{Z}\mu\in
K_{Z}\mu.$

That is to say,

$\int f(\omega_{0},y^{1},\dots,y^{n})\nu(d\omega_{0},dy^{1},\dots,dy^{n})<\int
f(\omega_{0},y^{1},\dots,y^{n})\kappa_{Z}\mu(d\omega_{0},dy^{1},\dots,dy^{n}).$

Now, since $\mathbb{Y}^{i}=\mathbb{U}^{i}$ we can view $f$ as the cost
function for some game, where $\langle\nu,f\rangle$ gives the expected cost
for the game with information structure $\nu$ when each player plays the
identity policy.

We note that because the inequality holds for all $\kappa_{Z}\mu\in K_{Z}\mu$,
in particular it holds for the minimum over all such measures, so we have the
following:

$\displaystyle\langle\tilde{\gamma}\nu,f\rangle$
$\displaystyle<\inf_{\kappa_{Z}\mu\in K_{Z}\mu}\langle\kappa_{Z}\mu,f\rangle$
$\displaystyle=\inf_{\kappa_{Z}\mu\in K_{Z}\mu}\int
f(\omega_{0},y^{1},\dots,y^{n})\kappa_{Z}\mu(d\omega_{0},dy^{1},\dots,dy^{n})$

But, this is equivalent to finding the optimal strategic measure in
$L_{C}(\mu)$ to play.

Thus, we have that, for the cost function $f$, the performance under the
identity policy under information structure $\nu$ is strictly better than the
performance under the optimal policy for information structure $\mu$. This
contradicts the fact that $\mu$ is more informative than $\nu$. Therefore, we
must have that $\nu\in K_{Z}{\mu}$. $\diamond$

In addition to the extension of Blackwell’s results, we can also extend Le
Cam’s theory to team problems.

######  Theorem 13.

Let $\epsilon>0$ be fixed, $\delta(\mu,\nu)<\epsilon$ if and only if for any
team policy $\bar{\gamma}_{\nu}$ under $\nu$ and any team problem in
$G\in\mathcal{E}$ with a bounded cost function $\|c\|_{\infty}\leq 1$, there
exists a team policy $\bar{\gamma}_{\mu}$ under $\mu$ such that

$J(G,\mu,\bar{\gamma}_{\mu})<J(G,\nu,\bar{\gamma}_{\nu})+\epsilon.$

###### Proof.

We note that, when selecting a team policy $\bar{\gamma}$ under an information
structure, we view the selection of the team policy as the selection of both
the independent randomness $z\in[0,1]^{N}$ and the maps
$\gamma^{i}:\mathbb{Y}^{i}\times[0,1]^{N}\rightarrow\mathbb{U}^{i}$ for each
player. Since $z$ is independent of the other random variables, we will, with
a slight abuse of notation, allow the dependence of $z$ to be absorbed into
the $\gamma^{i}$ terms, and thus denote them as maps from $\mathbb{Y}^{i}$ to
$\mathbb{U}^{i}$.

Step 1. First we show the forward direction. If $\delta(\mu,\nu)<\epsilon$,
then by definition $\inf_{\kappa}\|\kappa\mu-\nu\|_{TV}<\epsilon$. Let
$\gamma_{\nu}$ be an arbitrary team policy under $\nu$. Then we have the
following:

$\displaystyle J(G,\mu,\gamma_{\nu})-J(G,\nu,\gamma_{\nu})$
$\displaystyle=\int
c(\omega_{0},\gamma^{1}_{\nu}(y^{1}),\dots,\gamma^{n}_{\nu}(y^{n}))\nu(d\omega_{0},dy^{1},\dots,dy^{n})$
$\displaystyle\quad\quad\quad\quad-\int
c(\omega_{0},\gamma^{1}_{\nu}(y^{1}),\dots,\gamma^{n}_{\nu}(y^{n}))\mu(d\omega_{0},dy^{1},\dots,dy^{n})$
$\displaystyle\leq\int
c(\omega_{0},\gamma^{1}_{\nu}(y^{1}),\dots,\gamma^{n}_{\nu}(y^{n}))\nu(d\omega_{0},dy^{1},\dots,dy^{n})$
$\displaystyle\quad\quad\quad\quad-\inf_{\kappa}\int
c(\omega_{0},\gamma^{1}_{\nu}(y^{1}),\dots,\gamma^{n}_{\nu}(y^{n}))\kappa\mu(d\omega_{0},dy^{1},\dots,dy^{n})$
$\displaystyle=\inf_{\kappa}(\int
c(\omega_{0},\gamma^{1}_{\nu}(y^{1}),\dots,\gamma^{n}_{\nu}(y^{n}))\nu(d\omega_{0},dy^{1},\dots,dy^{n})$
$\displaystyle\quad\quad\quad\quad-\int
c(\omega_{0},\gamma^{1}_{\nu}(y^{1}),\dots,\gamma^{n}_{\nu}(y^{n}))\kappa\mu(d\omega_{0},dy^{1},\dots,dy^{n}))$
$\displaystyle\leq\inf_{\kappa}\|\nu-\kappa\mu\|_{TV}<\epsilon.$

The first inequality follows because the second term can not be made larger
through an additional filtration of the information through $\kappa$, since
the identity garbling could be selected. The second inequality follows from
the definition of total variation distance.

Step 2. For the converse direction, we assume that for any team policy
$\bar{\gamma}_{\nu}$ under $\nu$ and any game with a bounded cost function
$\|c\|_{\infty}\leq 1$, there exists a team policy $\bar{\gamma}_{\mu}$ under
$\mu$ such that

$J(G,\mu,\bar{\gamma}_{\mu})<J(G,\nu,\bar{\gamma}_{\nu})+\epsilon.$

In particular, this is true for every game in which
$\mathbb{U}^{i}=\mathbb{Y}^{i}$ for all players; we denote this by the
subclass $\mathcal{G}_{\mathbb{Y}=\mathbb{U}}$. Since the inequality is true
for any arbitrary policy $\bar{\gamma}_{\nu}$, in particular it is true for
the team identity policy $\gamma^{id}_{i}(y^{i})=y^{i}$, for $i=1,\dots,n$.
Thus, we have that for any game in this class, there exists a policy
$\bar{\gamma}_{\mu}$ such that:

$\displaystyle J(G,\mu,\bar{\gamma}_{\mu})-J(G,\nu,\gamma^{id})<\epsilon.$

This implies that the following equation holds:

$\sup_{G\in\mathcal{G}_{Y=U}}(J^{*}(G,\mu)-J(G,\nu,\gamma^{id}))<\epsilon.$

We can then complete the proof by noting the following:

$\displaystyle\sup_{G\in\mathcal{G}_{Y=U}}(J^{*}(G,\mu)-J(G,\nu,\gamma^{id}))$
$\displaystyle\leq\sup_{G\in\mathcal{G}_{Y=U}}|J^{*}(G,\mu)-J(G,\nu,\gamma^{id})|$
$\displaystyle=\sup_{\|c\|\leq 1}|\inf_{\bar{\gamma}_{\mu}}\int
c(\omega_{0},\gamma^{1}_{\mu}(y^{1}),\dots,\gamma^{n}_{\mu}(y^{n}))\mu(d\omega_{0},dy^{1},\dots,dy^{n})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad-\int
c(\omega_{0},y^{1},\dots,y^{n})\nu(d\omega_{0},dy^{1},\dots,dy^{n})|$
$\displaystyle=\sup_{\|c\|\leq 1}|\inf_{\kappa}\int
c(\omega_{0},y^{1},\dots,y^{n})\kappa\mu(d\omega_{0},dy^{1},\dots,dy^{n})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad-\int
c(\omega_{0},y^{1},\dots,y^{n})\nu(d\omega_{0},dy^{1},\dots,dy^{n})|$
$\displaystyle\leq\inf_{\kappa}\sup_{\|c\|\leq 1}|\int
c(\omega_{0},y^{1},\dots,y^{n})\kappa\mu(d\omega_{0},dy^{1},\dots,dy^{n})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad-\int
c(\omega_{0},y^{1},\dots,y^{n})\nu(d\omega_{0},dy^{1},\dots,dy^{n})|$
$\displaystyle=\inf_{\kappa}\|\kappa\mu-\nu\|_{TV}$

Since the first term is bounded above by $\epsilon$, we have that
$\inf_{\kappa}\|\kappa\mu-\nu\|_{TV}<\epsilon$, completing the proof.
$\diamond$

###  4.3 Comparison of Information Structures in Zero-Sum Games involving
Teams against Teams

Now, we will prove an extension of Blackwell’s comparison of information
structures similar to that in [22] and [17].

Let $\tilde{\mathcal{G}}$ denote the set of all team-against-team zero-sum
games with for which the cost function is continuous and bounded in players’
actions for every state.

######  Theorem 14.

Take fixed $\Omega_{0},P$, fixed and compact measurement spaces, and
information structures $\nu$ and $\mu$ for which Assumption 3.1 holds. Then
$\mu$ is more informative for the maximizing team than $\nu$ over all games in
$\tilde{\mathcal{G}}$ if and only if the following hold:

  * (i).

The measurement of each player in Team 1 under $\nu$ is given by
$\tilde{y}^{i}_{1}({\nu})=h^{i}(y^{i}_{1}(\mu),Z_{1})$ for $i=1,\dots,n$ and
some independent randomness $Z_{1}$ (with possibly dependent coordinate
components). Denote this information structure by $\kappa_{1}\mu$.

  * (ii).

The measurement of each player in Team 2 under $\mu$ is given by
$\tilde{y}^{i}_{2}({\mu})=h^{i}(y^{i}_{2}(\nu),Z_{2})$ for $i=1,\dots,n$ and
some independent randomness $Z_{2}$ (with possibly dependent coordinate
components). Denote this information structure by $\kappa_{2}\nu$.

That is to say, if and only if $\kappa_{1}\mu=\kappa_{2}\nu$ for some
$\kappa_{1}$ and $\kappa_{2}$.

###### Proof.

(Sketch) The proof follows naturally from the argument in [17], where instead
of showing the results by holding one player constant and using Blackwell’s
comparison theorem for single-player decision problems, we can hold one team
constant and use Theorem 12. For this reason, only a sketch of the proof is
presented here.

Step 1: The forward direction follows from the forward direction to Theorem
12. Under an information structure, if team $i$’s information is held
constant, and team $-i$’s information is garbled with possibly common
randomness, then team $-i$ can not perform better under this garbling than
they did under the original information structure. This follows Theorem 12 by
absorbing team $i$, who have constant information across both information
structures, into the cost function for team $-i$ and treating this as a team
problem. Applying this for $i=1,2$ yields the following relationship for any
game $G$ in $\tilde{G}:$

$V^{*}(G,\mu)\leq V^{*}(G,\kappa_{1}\mu)=V^{*}(G,\kappa_{2}\nu)\leq
V^{*}(G,\nu).$

Thus, the equilibrium value is lower under $\mu$ than under $\nu$ for every
game, and so $\mu$ is more informative than $\nu$.

Step 2: For the converse direction, we apply the converse direction to Theorem
12. We can observe that if $V^{*}(G,\mu)\leq V^{*}(G,\nu)$ for all
$G\in\tilde{\mathcal{G}}$, then it follows that:

$V(G,\mu,\bar{\gamma}^{1}_{\mu},\bar{\gamma}^{2}_{\nu})\leq V^{*}(G,\mu)\leq
V^{*}(G,\nu),$

where $(\bar{\gamma}^{1}_{\mu},\bar{\gamma}^{2}_{\mu})$ and
$(\bar{\gamma}^{1}_{\nu},\bar{\gamma}^{2}_{\nu})$ are the team equilibrium
policies under $\mu$ and $\nu$, respectively. This equation holds by
perturbing the maximizing team’s policy to no longer be its optimal policy
under $\mu$ in the first term, decreasing the value of the game.

Then, by comparing the first and third terms above and fixing Team 2’s policy
as constant for each game $G$, we can absorb Team 2’s effect on Team 1 into
the cost function for any game in $\tilde{\mathcal{G}}$. Projecting onto the
space of team problems for Team 1 this way, we can span all possible team
problems, achieving the inequality for Team 1:

$V^{*}(E,\mu)\leq V^{*}(E,\nu),$

for all team problems $E\in\mathcal{E}$.

Thus, we get that $\mu$ is more informative than $\nu$ for Team 1. Applying
the converse direction to Theorem 12, we get that condition $(i)$ of the
theorem statement holds. Applying a similar analysis by swapping the teams
leads to condition $(ii)$. $\diamond$

This theorem gives a general comparison of information structures result for
both teams and zero-sum games. When only cost functions where one team’s
actions are irrelevant to the cost are considered, the result reduces to that
of team problems, which were discussed in Theorem 12 and in [19]. When each
team is restricted to having only one player, the result reduces to a standard
zero-sum games, a case that was explored in [22] and [17]. When both of these
restrictions are applied, the result reduces to the original single-player
decision problem studied by Blackwell.

##  5 Continuity of Equilibrium Values in Information in Team-against-Team
Zero-Sum Games

Recently in [18], it was shown that for both zero-sum games as well as for
stochastic team problems, equilibrium values are continuous when the
information structures are perturbed under the total variation topology.
Therefore, it is not surprising that we are able to obtain the following
regularity results for zero-sum games involving two teams of competing
players.

As noted in Section 2.3, the teams share a common cost function
$c:\Omega_{0}\times\mathbb{U}^{1}_{1}\times\dots\times\mathbb{U}^{N_{1}}_{1}\times\mathbb{U}^{1}_{2}\times\dots\times\mathbb{U}^{N_{2}}_{2}\rightarrow\mathbb{R}$.
Team 1’s goal is to minimize the expected cost and Team 2’s goal is to
maximize it. The information structure for the game $\mu$ is a joint
probability measure on
$\Omega_{0}\times\mathbb{Y}^{1}_{1}\times\dots\times\mathbb{Y}^{N_{1}}_{1}\times\mathbb{Y}^{1}_{2}\times\dots\times\mathbb{Y}^{N_{2}}_{2}$.
We use $\mu_{i}$ to denote the marginal of $\mu$ on
$\Omega_{0}\times\mathbb{Y}^{i}_{i}\times\dots\times\mathbb{Y}^{N_{i}}_{i}$
for $i=1,2$ (i.e. $\mu_{i}$ captures the information available to team $i$).

######  Theorem 15.

Suppose that the hypotheses of Theorem 4 apply so that equilibria exist. Then
the value function $J^{*}(c,\mu)$ is continuous under total variation
convergence of information structures provided that perturbations in
information satisfy the conditions of existence.

###### Proof.

Following the proof of [18, Theorem 3.1] and viewing the common randomness
variable as an exogenous integration term, the result follows from identical
steps. We note again, as in the proof of Theorem 13, when selecting a team
policy $\bar{\gamma}$ under an information structure, we view the selection of
the policy as the selection of both the independent randomness $z\in[0,1]^{N}$
and the maps
$\gamma^{i}:\mathbb{Y}^{i}\times[0,1]^{N}\rightarrow\mathbb{U}^{i}$ for each
player. Since $z$ is independent of the other random variables, we will, with
a slight abuse of notation, allow the dependence of $z$ to be absorbed into
the $\gamma^{i}$ terms, and thus denote them as maps from $\mathbb{Y}^{i}$ to
$\mathbb{U}^{i}$.

Theorem 4 shows that an equilibrium exists under $\mu$ and $\nu$. Assume that
$\mu$ and $\nu$ are such that $\|\mu-\nu\|_{TV}\leq\epsilon$ for some
$\epsilon>0$. Without loss of generality, assume that
$J^{*}(g,\mu)-J^{*}(g,\nu)\geq 0$ for some game $g$ (a symmetric argument can
be applied in the case where $J^{*}(g,\mu)-J^{*}(g,\nu)\leq 0$). Then we have
the following:

$\displaystyle J^{*}(g,\mu)-J^{*}(g,\nu)$ $\displaystyle=\int
c(x,\bar{\gamma}^{1}_{\mu},\bar{\gamma}^{2}_{\mu})\mu(dx;dy^{1}_{1},\cdots,dy^{N_{1}}_{1};dy^{2}_{1},\cdots,dy^{N_{2}}_{2})$
$\displaystyle\qquad\qquad\qquad-\int
c(x,\bar{\gamma}^{1}_{\nu},\bar{\gamma}^{2}_{\nu})\nu(dx;dy^{1}_{1},\cdots,dy^{N_{1}}_{1};dy^{2}_{1},\cdots,dy^{N_{2}}_{2})$
$\displaystyle\leq\int
c(x,\bar{\gamma}^{1}_{\nu}(y^{1}),\bar{\gamma}^{2}_{\mu}(y^{2}))\mu(dx;dy^{1}_{1},\cdots,dy^{N_{1}}_{1};dy^{2}_{1},\cdots,dy^{N_{2}}_{2})$
$\displaystyle\qquad\qquad\qquad-\int
c(x,\bar{\gamma}^{1}_{\nu}(y^{1}),\bar{\gamma}^{2}_{\mu}(y^{2}))\nu(dx;dy^{1}_{1},\cdots,dy^{N_{1}}_{1};dy^{2}_{1},\cdots,dy^{N_{2}}_{2}).$

The inequality comes from perturbing the minimizer team’s equilibrium strategy
in the first term, making the expected cost larger, and perturbing the
maximizer team’s equilibrium strategy in the second term, making the expected
cost smaller. Since the functions to be integrated are identical and by
considering the symmetric argument for the case with
$J^{*}(g,\mu)-J^{*}(g,\nu)\leq 0$, we have that
$\|\mu_{n}-\mu\|_{TV}\rightarrow 0$, then
$|J^{*}(g,\mu_{n})-J^{*}(g,\mu)|\rightarrow 0$.

$\diamond$

##  6 Conclusion

For games with Standard Borel measurement and action spaces involving two
teams of finitely many agents taking part in a zero-sum game, we presented an
existence result for saddle-point equilibria when common randomness is assumed
to be available in each team with an analysis on conditions for compactness of
strategic team measures provided. Then, Blackwell’s ordering of information
structures was generalized to $n$-player teams with standard Borel spaces,
where correlated garbling of information structures is introduced as a key
attribute, and building on this result Blackwell’s ordering of information
structures was defined for team-against-team zero-sum game problems. Finally,
continuity of equilibrium value of team-against-team zero-sum game problems in
the space of information structures under total variation was established.

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