#  A Belief-Based Characterization of Reduced-Form Auctions††thanks: The
present paper is a substantially revised and extended version of Lang (2022).
I thank Eric van Damme, Debasis Mishra, Rakesh Vohra for their comments on an
earlier draft, and seminar participants for their helpful feedback on several
occasions.

Xu Lang Southwest University of Finance and Economics (langxu@swufe.edu.cn).

(June 27, 2023)

We study games of chance (e.g., pokers, dices, horse races) in the form of
agents’ first-order posterior beliefs about game outcomes. We ask for any
profile of agents’ posterior beliefs, is there a game that can generate these
beliefs? We completely characterize all feasible joint posterior beliefs from
these games. The characterization enables us to find a new variant of Border’s
inequalities (Border, 1991), which we call a belief-based characterization of
Border’s inequalities. It also leads to a generalization of Aumann’s Agreement
Theorem. We show that the characterization results are powerful in bounding
the correlation of agents’ joint posterior beliefs.

JEL Classification number: D7, D8

Keywords: Games of Chance, Posterior Beliefs, Reduced Form Auctions, Aumann’s
Agreement Theorem, Bayesian Persuasion

##  1 Introduction

Consider a card game with three players, where each player draws a card from
the same deck of cards and hand values are compared to determine the winner. A
natural question is how contradictory can two players 1 and 2’ opinions be
about a common event $A$ (e.g., the event that player 1 wins the game). The
celebrated theorem of Aumann (Aumann, 1976) concludes that the players cannot
agree to disagree: If the two players’ posteriors for $A$ are common
knowledge, then these posteriors must be equal. Suppose now instead of forming
opinions about a common event, players 1 and 2 are invited to form opinions
about two disjoint events $A$ and $B$ (e.g., the events that ones own wins the
game). The question then is whether there is a way to combine their
predictions, i.e., to come up with a forecast on $A\cup B$. A common intuition
suggests that the two players cannot attach to $A$ and $B$ very high
probabilities simultaneously. Then how contradictory can the two players’
opinions be? In this paper, we show that a complete solution to this problem
can be derived from the well-known Maskin-Riley-Matthews-Border condition
(Maskin and Riley, 1984; Matthews, 1984; Border, 1991) in auction theory.

Formally we study a class of games of chance (e.g., pokers, dices, horse
races) in a mechanism design framework. There is a set of agents. Each agent
has a set of possible types (e.g., cards). The set of all type profiles of
agents is associated with a probability space. An outcome function (e.g., the
rule of a card game) assigns each true type profiles of agents a probability
of winning for each agent. Each agent privately observes her type and forms
first-order posterior belief about her payoff-relevant event, i.e., the event
that she wins the game. We call a prior-outcome function pair a game. Given
any game, a distribution of joint posterior beliefs of agents would naturally
arise from the game.

We completely characterize the set of all feasible joint posterior beliefs
that arise from all games. To obtain a necessary condition for feasibility, we
apply Border’s theorem to each game. The key to our analysis is a change of
measures, which enables us to find a new variant of the Border inequalities.
We call such inequalities belief-based Border’s inequalities (the Border∗
inequalities thereafter). To show that the Border∗ inequalities is sufficient
for feasibility, we use a revelation principle to construct a required
underlying game. We next discuss how our main characterization result changes
when the prior is fixed. We show that if the prior is independent and
atomless, which hold for standard independent private value models, the
Border∗ inequalities together with posterior independence is characterization
of all feasible joint posterior beliefs.

To gain some intuition about the Border∗ inequalities, we provide three
alternative interpretations. The first interpretation is a Border’s version of
Aumann’s Agreement Theorem where agents form opinions about disjoint events in
the outcome space. We show that when the agents’ joint posteriors are common
knowledge, these posteriors must lie in the probability simplex of the outcome
space. The result reduces to Aumann’s theorem when there are two agents. The
second interpretation is a no-trade condition. We show that the Border∗
inequalities correspond to a no-trade condition for a new class of bets where
agents bet on disjoint events. For a third interpretation, we show that the
Border∗ inequalities define the core of a coalitional game. In this game,
agents with various beliefs can form blocking coalitions and the worth of a
coalition is defined by the probability measure of the agents with such
beliefs.

We generalize our main characterization result from games of chance to a
general model of information structure design. We assume that there is a state
space and different agents can observe signals and form posteriors about
possibly different events, i.e., their payoff-relevant events. We show that if
agents’ payoff-relevant events are pairwise disjoint, every feasible joint
posterior beliefs problem is essentially equivalent to a reduced-form auction
problem and the Border∗ inequalities give the characterization condition. Our
characterization provides one tractable way to generalize a two-agent
characterization of feasible posterior beliefs over a common event in Arieli,
Babichenko, Sandomirskiy, and Tamuz (2021).111It is well known that in general
the feasibility of joint posterior distributions with many agents is a tough
question.  When there are two agents, the agents’ posteriors about
complementary events reduces to the posteriors about a common event.

We provide two extensions for our results. First, we show that our result
implies a belief-based characterization of auctions in the form of bidders’
first-order posterior beliefs about auction outcomes. For some auctions (e.g.,
knockout auction of a cartel, auctions of arts, all-pay auctions), the auction
outcomes may not be publicly observable but important for the after-auction
market interaction. In other markets, insurance companies may sell insurances
to bidders for losing the bids. In such cases, bidders’ posterior beliefs
about auction outcomes will be relevant for the analysis. We show that for
independent priors, our main characterization result extends to both Bayesian
and dominant strategy incentive compatible auctions. In this case, the testing
sets in the Border∗ inequalities reduce to a smaller class of upper contour
sets in the posterior belief space. A similar characterization holds for a
fixed prior satisfying independence and absolute continuity. Our result
extends a reduced-form equivalence in Manelli and Vincent (2010) and Gershkov,
Goeree, Kushnir, Moldovanu, and Shi (2013) from interim allocation outcomes to
bidders’ posterior beliefs about payoff-relevant events.

For the general model of designing information structures, we show that
Border-like characterization is very powerful in bounding the correlation of
feasible joint posterior beliefs. We find that perfectly positively correlated
beliefs are infeasible. For several classes of classic correlated
distributions (i.e., copulas), positive dependence of beliefs are completely
ruled out by the Border∗ inequalities. While some existing characterization
results (e.g., Arieli et al., 2021) show that the posteriors for a common
event cannot be too negatively correlated, our result shows that the
posteriors for disjoint events cannot be too positively correlated. Hence the
two characterizations implied by Aumann’s theorem and Border’s theorem provide
two ends of the spectrum of beliefs correlation.

In Section 2, we introduce a poker game model and main results. Section 3
discusses the general model. Section 4 generalizes the main characterization
to incorporate incentive compatibility. Section 5 studies feasible correlation
of posterior beliefs. Section 6 discusses the relations of the paper to the
literature. The missing proofs are deferred to Appendix.

##  2 A Poker Game Model

Let $N=\\{1,\dots,n\\}$ with $n\geq 2$ be a set of agents (i.e., gamblers).
Each agent $i$ has a prior probability space $(T_{i},\mathcal{F}_{i},\mu_{i})$
where $T_{i}$ is the type space, with a generic element $t_{i}\in T_{i}$,
$\mathcal{F}_{i}$ is the $\sigma$-algebra and $\mu_{i}$ is the probability
measure. The space of type profiles is $(T,\mathcal{F},\mu)$ where
$T=T_{1}\times T_{2}\times\dots\times T_{n}$, $\mathcal{F}$ is the product
$\sigma$-algebra, and $\mu$ is a probability measure on the product space with
marginals $\mu_{i}$. We define $(T_{-i},\mathcal{F}_{-i},\mu_{-i})$ the space
of type profiles for agents other than $i$ analogously.

An outcome function (i.e., the rule of a poker game) is a measurable function
$a:T\to\Delta(N)$ that assigns each type profile of agents a probability of
winning for each agent. Different from a classical mechanism design model, we
assume outcomes are determined by true type profiles. So strategies play no
role in a game (i.e., a game of chance). An interim outcome function is a
measurable function $Q:T\to[0,1]^{n}$ such that $Q_{i}$ only depends on
$t_{i}$, i.e., $Q_{i}(t)=Q_{i}(t_{i})$. That is, an interim outcome function
assigns each type of each agent $i$ an expected probability of winning. We say
$Q$ is reduced-form implementable if there exists an outcome function $a$ that
implements $Q$, i.e., for all $i\in N$ and $t_{i}\in T_{i}$,

$Q_{i}(t_{i})=\int_{T_{-i}}a(i,t_{i},t_{-i})d\mu(t_{-i}|t_{i})$ (1)

We call a prior-outcome function pair $((T,\mathcal{F},\mu),a)$ a game. Given
a game, we define for each agent $i$ and type $t_{i}$ the interim winning
probability $x_{i}=Q_{i}(t_{i})$ the (first-order) posterior belief of agent
$i$ about the event that agent $i$ wins the game. Then $x_{i}\in[0,1]$. For
each type profile $t=(t_{1},\dots,t_{n})$, we say
$(x_{1},\dots,x_{n})=(Q_{1}(t_{1}),\dots,Q_{n}(t_{n}))$ is a vector of joint
posterior beliefs.

We denote by $\nu$ the joint distribution of posterior beliefs induced by
$\mu$ and $a$. That is for every measurable set $C\subset[0,1]^{n}$,

$\nu(C)=\mu(Q\in C)$ (2)

Then $\nu$ is the pushforward measure of $\mu$ by $Q$ and we write
$\nu=Q_{\\#}\mu$. Let $\nu_{i}$ be the marginal probability distribution of
$\nu$.

######  Definition 1.

$\nu\in\Delta([0,1]^{n})$ is a feasible joint distribution of posterior
beliefs for some game, if there exist a prior probability space
$(T,\mathcal{F},\mu)$ and an outcome function $a$ such that $\nu$ and
$((T,\mathcal{F},\mu),a)$ satisfy (1) and (2).

In the following subsections 2.1-2.2 and 2.3, we provide two characterizations
of all feasible joint distributions of posterior beliefs from all games: a
prior-free characterization and a fixed-prior characterization. In both
models, we assume the agents have a common prior probability $\mu$. While the
fixed-prior characterization is relevant for a game designer whose prior is
the common prior, the prior-free characterization is relevant when the agents
have a common prior but the game designer has no such information, in which
case a prior-free characterization is very desirable.

###  2.1 Main Result

The following Theorem 1 provides a characterization of the set of feasible
joint posterior beliefs from games and is the main result of the paper.

######  Theorem 1.

$\nu\in\Delta([0,1]^{n})$ is a feasible joint distribution of posterior
beliefs for some game if and only if

$\sum_{i\in N}\int_{C_{i}}x_{i}d\nu_{i}(x_{i})\leq\nu\left(\underset{i\in
N}{\cup}(C_{i}\times[0,1]^{n-1})\right)$ (3)

for all measurable sets $C_{i}\subseteq[0,1]$, $i\in N$ and

$\sum_{i\in N}\int_{[0,1]}x_{i}d\nu_{i}(x_{i})=1$ (4)

We call (3) and (4) the Border∗ inequalities. (3) says that for any subsets
$C_{1},\dots,C_{n}$ in the agents’ posterior belief spaces, the sum of the
expectations of beliefs of all agents is no greater than the probability that
at least one agent has such a belief. (4) is a new variant of the martingale
condition (Aumann and Maschler, 1995; Kamenica and Gentzkow, 2011). It says
that the sum of the expectations of posterior beliefs of all agents (i.e., the
sum of the prior probabilities for different outcomes) is equal to 1.

Below we first provide a proof for the Theorem and defer its alternative
interpretations to Section 2.2. The proof of necessity is in essence a
reformulation of Border’s theorem with a change of measures. To show
sufficiency, for any posterior beliefs satisfying the characterization
inequalities, we explicitly construct a prior probability space and an interim
outcome function such that Border’s theorem can be applied. The key to our
construction is a revelation argument, i.e., whenever such a feasible solution
exists, we can choose the prior probability space equal to the posterior
probability space and an interim outcome function equal to the identity map.

###### Proof of Theorem 1.

Border’s theorem (Border, 1991, 2007) provides a necessary and sufficient
condition for an interim outcome function to be implementable and will be
essential for our analysis. We will use a generalization of Border’s theorem
in Che, Kim, and Mierendorff (2013) that allows floor constraint.

######  Lemma 1.

Let $Q:T\to[0,1]^{n}$ be measurable and $Q_{i}(t)=Q_{i}(t_{i})$. $Q$ is
reduced-form implementable if and only if

$\sum_{i\in
N}\int_{E_{i}}Q_{i}(t_{i})d\mu_{i}(t_{i})\leq\mu\left(\underset{i\in
N}{\cup}(E_{i}\times T_{-i})\right)$ (5)

for all measurable sets $E_{i}\subseteq T_{i}$ with $i\in N$, and

$\sum_{i\in N}\int_{T_{i}}Q_{i}(t_{i})d\mu_{i}(t_{i})=1$ (6)

Only If. Suppose $\nu$ is feasible for some game, i.e., there exist a prior
probability space $(T,\mathcal{F},\mu)$ and an outcome function $a$ such that
$\nu$ and $((T,\mathcal{F},\mu),a)$ satisfy (1) and (2). Then by definition
the interim outcome function $Q$ generated by $((T,\mathcal{F},\mu),a)$ is
reduced-form implementable. So by the necessity part of Lemma 1, $Q$ satisfies
Border’s condition (5) and (6).

To obtain a reformulation of Border’s theorem, we use a change of measures. By
characterization of pushforward measures, if $Q$ pushes forward $\mu$ to
$\nu$, then for every $\mu$-integrable functions $f:[0,1]^{n}\to\R$,

$\displaystyle\int_{[0,1]^{n}}f(x)d\nu(x)=\int_{T}f(Q(t))d\mu(t)$ (7)

In particular, pick any measurable set $C_{i}\subseteq[0,1]$. Define
$E_{i}=\\{t_{i}\in T_{i}:Q_{i}(t_{i})\in C_{i}\\}$, then $E_{i}$ is a
measurable set in $T_{i}$. Define $f(x)=x_{i}\cdot\mathbf{1}_{C_{i}}(x_{i})$
where $\mathbf{1}_{C_{i}}$ is the indicator function of $C_{i}$. Then $f$
depends only on $x_{i}$. Substitute $f$ into both sides of (7), we get

$\displaystyle\int_{C_{i}}x_{i}d\nu_{i}(t_{i})=\int_{E_{i}}Q_{i}(t_{i})d\mu_{i}(t_{i})$
(8)

Also note $\nu$ is feasible implies $\nu$ and $\mu$ satisfy (2) for all
measurable sets $C\subset[0,1]^{n}$. In particular, pick $C=\underset{i\in
N}{\cup}(C_{i}\times[0,1]^{n-1})$, we have

$\displaystyle\nu(C)=\mu\left(Q\in\underset{i\in
N}{\cup}(C_{i}\times[0,1]^{n-1})\right)=\mu\left(\underset{i\in
N}{\cup}(E_{i}\times T_{-i})\right)$ (9)

Since $Q$ satisfies Border’s condition for testing sets $(E_{i})_{i\in N}$,
substitute (8) and (9) into (5), we obtain the inequalities (3). Finally, for
testing sets $(C_{i})_{i\in N}$ with $C_{i}=[0,1]$, we have $E_{i}=T_{i}$.
From (6), we get the equality (4).

If. Suppose $\nu$ satisfies (3) and (4). Define a product type space
$T=[0,1]^{n}$ with the Borel $\sigma$-algebra and define a prior probability
measure $\mu$ by: for all measurable $C\subseteq[0,1]^{n}$, $\mu(C)=\nu(C)$.
Define $Q:[0,1]^{n}\to[0,1]^{n}$ by: for all $x\in[0,1]^{n}$,

$\displaystyle Q_{i}(x)=x_{i},\,\,i\in N$ (10)

That is, each agent $i$’s $Q_{i}$ is equal to her type $x_{i}$. Since $Q_{i}$
depends only on $x_{i}$, $Q$ is an interim outcome function. Moreover, by
construction $\nu$ is the pushforward measure of $\mu$ by $Q$ and (2) holds.

Now pick any profile of testing sets $(C_{i})_{i\in N}$ with
$C_{i}\subseteq[0,1]$ measurable. $\nu$ satisfies (3) for $(C_{i})_{i\in N}$
implies that $Q$ satisfies (5) for $(E_{i})_{i\in N}$ with $E_{i}=C_{i}$ for
all $i$. From the sufficiency part of Border’s theorem, there exists an
outcome function $a:T\to\Delta(N)$ such that $Q$ is the reduced form. Hence we
have constructed a prior $\mu$ and an outcome function $a$ such that $\nu$,
$\mu$ and $a$ satisfy (1) and (2), i.e., $\nu$ is feasible for game $(\mu,a)$.
∎

###  2.2 Interpretations of Theorem 1

We present three different interpretations of Theorem 1. The first two
interpretations by Aumann’s agreement theorem and no trade theorem are closely
related to the literature, while the third interpretation by blocking is new
for our setting.

1\. Agree to Disagree. Below we show that Border’s inequalities are sufficient
to provide a generalized version of Aumann’s agreement theorem with $n$
agents. The theorem reduces to the classic Aumann’s agreement theorem when
$n=2$. In this case, there are two outcomes and the agents’ posteriors on
complementary events reduces to posteriors on a common event.

######  Theorem 2.

(Aumann’s agreement theorem, Border’s version) For each agent $i\in N$, let
$E_{i}$ be the set of types that $i$ has posterior $r_{i}$ for the event that
$i$ wins the game.

(1) If $Q_{i}=r_{i}$ for all $i\in N$ is common knowledge, then $\sum_{i\in
N}r_{i}=1$.

(2) For $|N|=2$, if $Q_{1}=r_{1}$ and $Q_{2}=r_{2}$ is common knowledge, then
$r_{1}=1-r_{2}$.

Theorem 2 states that if the event $E_{1}\times\dots\times E_{n}$ is common
knowledge, then the agents’ posteriors about their payoff-relevant events must
lie in the probability simplex over the outcomes. For intuition, suppose there
are three agents and the game designer may wish to persuade each agent that
he/she will win for sure. Can the designer persuade all agents simultaneously?
Theorem 2 suggests that the answer is negative if the agents can communicate,
i.e., if their posteriors are common knowledge.

###### Proof of Theorem 2.

Suppose each $Q_{i}$ is a constant $r_{i}$ on $E_{i}$. If
$E=E_{1}\times\dots\times E_{n}$ is common knowledge, then

$\mu_{i}(E_{i})=\mu(E)=\mu\left(\underset{i\in N}{\cup}(E_{i}\times
T_{-i})\right)$ (11)

First apply Border’s condition (5) to $(E_{i})_{i\in N}$,

$\sum_{i\in N}\mu_{i}(E_{i})r_{i}\leq\mu\left(\underset{i\in
N}{\cup}(E_{i}\times T_{-i})\right)$ (12)

Combine (11) and (12), we have

$\sum_{i\in N}r_{i}\leq 1$ (13)

Next let $E_{i}^{\prime}=T_{i}\setminus E_{i}$. Apply Border’s inequalities
(5) to $(E_{i}^{\prime})_{i\in N}$, and subtract (6) from both sides of (5),
we get

$\sum_{i\in N}\mu_{i}(E_{i})r_{i}\geq\mu(E).$ (14)

We get

$\sum_{i\in N}r_{i}\geq 1$ (15)

Hence we conclude $\sum_{i\in N}r_{i}=1$. ∎

An immediate implication of Theorem 2 is that the joint distribution of
posteriors with perfectly positive correlation is infeasible. Consider the
perfectly positively correlated joint posteriors $(x_{1},\dots,x_{n})$
uniformly distributed on the main diagonal of $[0,1]^{n}$, we have the
following corollary.

######  Corollary 1.

Suppose $\nu\in\Delta([0,1]^{n})$ is uniform on the diagonal
$x_{1}=x_{2}=\cdots=x_{n}$, then $\nu$ is not feasible for any game.

###### Proof.

The agents’ posteriors are common knowledge for all possible realizations
$(x_{1},\dots,x_{n})$, and by Theorem 2 it implies $\sum_{i}r_{i}=1$ for all
$r_{i}\in[0,1]$. But this is impossible for $r_{i}$ sufficiently large. ∎

2\. A No trade condition.  Theorem 1 also formalizes a relationship between
Border’s theorem and No Trade theorem (Milgrom and Stokey, 1982, Morris, 2020,
Arieli et al., 2021). The Border∗ inequalities correspond to a no-trade
condition for a new class of bets where agents bet on pairwise disjoint
events. Suppose $N=\\{1,\dots,n\\}$ consists of all possible race outcomes. We
define a market for bets where each agent $i\in N$ decides whether to buy one
unit of bet $\beta_{i}$ and each bet $\beta_{i}:N\to\\{0,1\\}$ is defined by

$\displaystyle\beta_{i}(j)=\begin{cases}1&\textrm{if}~{}j=i\\\
0&\textrm{otherwise}\end{cases}$

In other words, each agent $i$ can bet over whether outcome $i$ occurs and the
mediator must clear the market. If agent $i$’s posterior for outcome $i$ is
$x_{i}$, then agent $i$ is willing to pay a price of $x_{i}$ for the bet
$\beta_{i}$. Suppose further that each agent $i$’s strategy is to bet if and
only if $x_{i}\in C_{i}$. For the mediator, she is expected to pay a price of
$1$ if some agent bets and wins. Condition (3) requires that the mediator’s ex
ante expected revenue is bound above by the maximum expected payment when at
one least one agent bets, for any strategy profile of agents.

3\. A blocking condition. We next provide an alternative view on Theorem 1
from coalitional games where the core of a game is introduced. We first
rewrite all Border∗ inequalities in a form with floor constraints. Subtract
(6) from both sides of (5), we have (5) can be written as

$\sum_{i\in N}\int_{C_{i}}x_{i}d\nu_{i}(x_{i})\geq\nu(C_{1}\times\dots\times
C_{n})$ (16)

for all measurable sets $C_{i}\subseteq[0,1]$, $i\in N$.

We define a coalitional game problem $(N,N^{*},\nu)$ as follows. Let
$N=\\{1,\dots,n\\}$ be a set of agents. For each agent $i\in N$, let
$N_{i}=[0,1]$ be an set of subagents of agent $i$. Let $N^{*}=\cup_{i\in
N}N_{i}$ be the set of all subagents. We call $C=\cup_{i\in N}C_{i}\subseteq
N^{*}$ a coalition. Let $\nu\in\Delta([0,1]^{n})$ be a joint distribution of
all profiles of subagents. The characteristic function
$w_{\nu}:2^{N^{*}}\to\R$ assigns for each coalition $C\subseteq N^{*}$ a worth
of coalition $w_{\nu}(C)=\nu(C_{1}\times\dots\times C_{n})$. That is the worth
of a coalition is defined by the probability measure of the coalition. Note
that agents are complement in this game: $C_{j}=\emptyset$ for some $j$
implies $w_{\nu}(C)=0$.

A solution of the problem $(N,N^{*},\nu)$ is a utility allocation function
$\pi:N^{*}\to[0,1]$ that assigns for each subagent $x_{i}\in[0,1]$ of each
agent $i$ a utility $\pi(x_{i})$. For each coalition $C$, we define the total
payoff of coalition $C$ from solution $\pi$ by

$\displaystyle\pi_{\nu}(C)=\sum_{i\in N}\int_{C_{i}}\pi(x_{i})d\nu_{i}(x_{i})$
(17)

We say $\pi$ is in the core of the game $(N,N^{*},\nu)$ if for every coalition
$C\subseteq N^{*}$, $\pi_{\nu}(C)\geq w_{\nu}(C)$ and
$\pi_{\nu}(N^{*})=w_{\nu}(N^{*})$. Then Border∗ inequalities (4) and (16) can
be interpreted as that the solution $\pi^{*}(x_{i})=x_{i}$ is in the core of
the game. Alternatively, we can define all core-stable distributions of
subagents by all probability distributions $\nu\in\Delta([0,1]^{n})$ such that
$\pi^{*}$ is the core of the game induced by $\nu$.

###  2.3  Characterization with a given prior

Characterization with a given prior is more involved since a feasible joint
distribution of posterior beliefs $\nu$ is further constrained by the given
prior $\mu$ beyond the reduced form feasibility constraint, i.e., there must
exist an interim outcome function $Q$ such that $\nu=Q_{\\#}\mu$. It is
immediate that if $\mu$ has an atom, then $\nu$ has an atom. Furthermore,
$\nu$ is not arbitrary. For example, if $\mu$ is a single point mass and $\nu$
consists of two point masses, each with mass 1/2, then no map can split $\mu$
in half. Theorem 3 below shows that atoms are almost the only obstruction.
Under an atomless assumption about the prior probability spaces we can obtain
a characterization similar as Theorem 1. Let $(T,\mathcal{\mathcal{F}})$ be a
measure space such that $\\{t\\}\in\mathcal{F}$ for every $t\in T$. We say a
measure $\mu$ on $(T,\mathcal{\mathcal{F}})$ is atomless if for any
$A\in\mathcal{F}$ such that $\mu(A)>0$, there exists $B\in\mathcal{F}$
satisfies $0<\mu(B)<\mu(A)$. An example of atomless measures is that $\mu$ is
a continuous probability distribution on $[0,1]^{n}$.

######  Theorem 3.

Let $\mu$ be an independent and atomless prior. $\nu\in\Delta([0,1]^{n})$ is a
feasible joint distribution of posterior beliefs for some game with prior
$\mu$ if and only if $\nu$ is independent and satisfies the Border∗
inequalities (3) and (4).

Theorem 3 shows that a fixed prior imposes a strong restriction on the set of
feasible posterior beliefs: prior independence implies posterior independence.
On the other hand, it shows that a fixed prior imposes only a weak restriction
on the set of feasible posterior beliefs: as long as an atomless prior is
considered, all posterior beliefs satisfying the Border∗ inequalities are
feasible.

##  3 The general model

Let $N=\\{1,\dots,n\\}$, $n\geq 2$ be a finite set of agents and
$\Omega=\\{\omega_{1},\dots,\omega_{m}\\}$, $m\geq 2$ be a finite state space.
The prior probability of state $\omega\in\Omega$ is denoted by
$p_{0}^{\omega}$. The agents have common prior $p_{0}$ regarding the states.
An information structure $I=((S_{i})_{i\in N},P)$ consists of signal spaces
$S_{i}$ (each equipped with a $\sigma$-algebra) and a distribution
$P\in\Delta(\Omega\times S_{1}\times\dots\times S_{n})$, with the marginal of
$P$ on $\Omega$ equal to $p_{0}$. Let $(\omega,s_{1},\dots,s_{n})$ be a
realization. Then, each agent $i$ observes the signal $s_{i}\in S_{i}$. Let
$q_{i}(s_{i})\in\Delta(\Omega)$ denote agent $i$’s posterior belief about
$\Omega$ after receiving the signal $s_{i}$. The posterior belief attributed
to $A\subset\Omega$ is given by

$q_{i}^{A}(s_{i})=P(A|s_{i}).$ (18)

Denoted by $P^{S}$ the marginal of $P$ on the signal space $S$, and by
$p^{S}_{i}$ the marginal of $P$ on the signal space $S_{i}$.

Payoff-relevant events. Suppose for each agent $i$, there is a set
$A_{i}\subseteq\Omega$ of payoff- relevant states for agent $i$. We call
$A_{i}$ the payoff-relevant event for agent $i$. Below we assume each agent
has one payoff-relevant event, i.e., all states in $A_{i}$ and
$\Omega\setminus A_{i}$ are payoff equivalent for agent $i$ and agent $i$ is
interested in the posterior belief for $A_{i}$. Our result generalizes to the
case where each agent has multiple payoff-relevant events.

Our model cover two important classes of problems: (1) Classical binary states
problem with $\Omega=\\{\omega_{1},\omega_{2}\\}$ and $A_{i}=\\{\omega_{1}\\}$
for all $i\in N$. (2) Poker game-like problem where $A_{i}\cap
A_{j}=\emptyset$ for all $i,j$. That is, the agents’ payoff-relevant events
are pairwise disjoint. In particular, we have $\Omega=N$ and $A_{i}=\\{i\\}$
for a poker game model.

We denote $x_{i}=q_{i}^{A_{i}}(s_{i})\in[0,1]$ agent $i$’s posterior
attributed to $A_{i}$ and $x=(x_{1},\dots,x_{n})$. Denote $[0,1]^{n}$ the
product space of posterior beliefs for $A_{1},\dots,A_{n}$, and denote by
$\psi^{I}\in\Delta([0,1]^{n})$ the joint distribution of posterior beliefs
induced by $I$. That is, for each measurable set $B\subset[0,1]^{n}$, define

$\nu^{I}(B)=P(x\in B).$ (19)

Conversely, we can start with an arbitrary $\nu\in\Delta([0,1]^{n})$ and ask
whether it can be generated by some information structure.

######  Definition 2.

Let $(A_{1},\dots,A_{n})$ be a profile of payoff-relevant events for agents.
$\nu\in\Delta([0,1]^{n})$ is $p_{0}$-feasible if there exists some information
structure $I$ with prior $p_{0}$ such that $\nu=\nu^{I}$.

Below for a probability measure $\nu\in\Delta([0,1]^{n})$, and for agent $i$,
we denote by $\nu_{i}$ the marginal distribution of $\nu$ for agent $i$.

Theorem 4 below characterizes all feasible joint posterior beliefs in the
general model of designing information structures. The proof of the Theorem
also establishes an equivalence between feasibility of reduced-form auctions
and feasibility of joint posterior beliefs. In reduced-form auctions (and
poker games), the information structure is defined by the product of outcome
function and prior, and signals are defined by interim outcome functions.
Theorem 4 shows that when the reduced-form auctions with variable priors are
considered, they generate the same set of feasible joint posterior beliefs as
all information structures in the general model.

######  Theorem 4.

Suppose $(A_{1},\dots,A_{n})$ are pairwise disjoint and
$\cup_{i}A_{i}=\Omega$. Let $\nu\in\Delta([0,1]^{n})$ is $p_{0}$-feasible for
some $p_{0}$ if and only if $\nu$ satisfies (3) and (4).

Remark 1. When there are two states and two agents, our Theorem 4 reduces to
Theorem 2 of Arieli et al. (2021). To see this, notice that when there are two
agents, the agents’ posteriors about complementary events reduce to the
posteriors about a common state.

Remark 2. Theorem 4 can be generalized to a problem where each agent has
multiple payoff-relevant events $A_{i}^{1},\dots,A_{i}^{l_{i}}$ and the events
of all agents are pairwise disjoint, by a generalized Border theorem for
multiple heterogeneous objects (Zheng, 2021; Lang and Yang, 2021), where each
agent’s reduced-form outcomes are multi-dimensional.

Remark 3. Theorem 4 restricts attention to a model of joint posterior beliefs
with $A_{i}\cap A_{j}=\emptyset$. The same technique can be used to establish
an equivalence between feasibility of reduced-form social choice problems
(Goeree and Kushnir, 2011; Lang and Mishra, 2023) and feasibility of joint
posterior beliefs over a complete set of states, i.e.,
$A_{i}^{k}=\\{\omega_{k}\\}$ for all $k=1,\dots,m$ and for all $i$.222It is
worth noting that characterization of reduced-form social choice problems in
Goeree and Kushnir (2011) by supporting functions is similar to a
characterization of common prior with many-states in Morris (2020).

Remark 4. The general model in Section 3 is not equivalent to the fixed-prior
poker game model in Section 2.3. Notice that in the fixed-prior poker game
model, the type distributions (i.e., signal distributions) are fixed while
only the outcome function (the conditional probabilities of the states given
signals) is endogenous.

##  4 Bic-dic equivalence

Our model also provides an approach to auctions in the form of bidders’ first-
order posterior beliefs about auction outcomes. In a traditional auction
design problem, a bidder can calculate her interim allocation probabilities
from different bids in any auction. Since the interim allocation probabilities
influence a bidder’s interim utility, it is often understood as a real
variable, i.e., a rule determines real outcomes. The interim allocation
probabilities, however, can be also interpreted as a belief variable. Indeed,
a bidder’s interim probability of winning is just her posterior belief about
the event that she wins the object.

In this section, we study how incentive compatibility further restricts the
set of feasible joint posteriors in auctions. We define an auction model
similar to a poker model in Section 2, except that we let $N=\\{1,\dots,n\\}$
denote a set of bidders, where bidder 1 is the seller and bidders $2,\dots,n$
are the buyers and the types profiles in type space $T$ may influence the
payoffs of the bidders. Different from a poker game where the outcomes depend
on players’ true types, the outcomes in an auction depends on buyers’ reported
types. We say an outcome function $a:T\to\Delta(N)$ (i.e., an auction) is (1)
Bayesian incentive compatible (BIC) if each $Q_{i}$ is nondecreasing in
$t_{i}$ and (2) dominant strategy incentive compatible (DIC) if each $a(i)$ is
nondecreasing in $t_{i}$. Our model covers auctions with many buyers and
bilateral trade with one seller and one buyer as Myerson and Satterthwaite
(1983).

We first present an extension of Theorem 1 for independent priors. We show
that Border∗ inequalities, together with monotonicity constraint, characterize
all feasible posterior beliefs, for both BIC and DIC auctions. The key to the
proof is using revelation principlem which allows us to construct a BIC
interim outcome function. For constructing a DIC auction, we use a reduced
form equivalence between BIC and DIC auctions (Manelli and Vincent, 2010;
Gershkov et al., 2013).

######  Theorem 5.

Consider all independent priors on $[0,1]^{n}$. $\nu\in\Delta([0,1]^{n})$ is
feasible for some BIC (or DIC) auction and some prior if and only if $\nu$ is
independent and

$\sum_{i\in N}\int_{a_{i}}^{1}x_{i}d\nu_{i}(x_{i})\leq 1-\prod_{i\in
N}\nu_{i}(a_{i})$ (20)

for all $a_{i}\in[0,1]$, $i\in N$, and (4) holds.

Theorem 5 states that for independent priors, there is an equivalence between
BIC and DIC in terms of the set of feasible posterior beliefs. However, this
Theorem needs not to imply that the extreme points of the set of feasible
posterior beliefs are generated by the extreme points of reduced-form
implementable outcome functions. Below we provide an example to show that, for
any two interim outcome functions, the convex combinations of their posterior
beliefs need not to be the posterior beliefs of these functions’ convex
combinations.

######  Example 1.

Suppose there are two bidders 1 and 2 with independent and uniform priors on
$[0,1]$. Consider auction $a$ that always allocates the object to bidder 1.
Then $Q_{1}^{a}=1$ and $Q_{2}^{a}=0$. The bidders’ posteriors are the Dirac
measure $\nu^{a}=\delta_{(1,0)}$. Next consider auction $b$ that efficiently
allocates the object to the high bidder. Then $Q_{i}^{b}(t_{i})=t_{i}$. The
bidders’ posteriors are $\nu^{b}(x_{1},x_{2})=x_{1}x_{2}$. Finally, consider
auction $c$ as the equal randomization of two auctions $a$ and $b$. Then
$Q_{1}^{c}(t_{1})=\frac{1}{2}+\frac{t_{1}}{2}$ and
$Q_{2}^{c}(t_{2})=\frac{t_{2}}{2}$, and
$\nu^{c}(x_{1},x_{2})=2(2x_{1}-1)x_{2}$ with $x_{1}\in[\frac{1}{2},1]$ and
$x_{2}\in[0,\frac{1}{2}]$. Hence $\nu^{c}$ is not a convex combination of
$\nu^{a}$ and $\nu^{b}$.

We next present an extension of Theorem 3 with a fixed prior. When both the
prior and posteriors are fixed, the existence of a monotone interim outcome
function is nontrivial since the revelation argument no longer works. Below we
show that if $\mu$ is an independent and atomless measure on $[0,1]^{n}$, we
can construct a monotone interim outcome function that pushes foward $\mu$ to
$\nu$. The proof is based on Brenier’s theorem in optimal transport theory
(see Lemma 4 in Appendix).

######  Theorem 6.

Suppose $\mu\in\Delta([0,1]^{n})$ is independent and absolutely continuous.
$\nu\in\Delta([0,1]^{n})$ is feasible for some BIC (or DIC) auction given
$\mu$ if and only if $\nu$ is independent and (4) and $\eqref{tm6}$ hold.

##  5 Feasible correlation of beliefs

In this section, we further investigate how the Border-like characterization
of feasibility in Theorem 4 for the general model imply particular forms of
correlation between the agents’ posterior beliefs. Arieli et al. (2021) used
their characterization of feasibility to study possible dependence of beliefs.
They provided a binary signal example and showed that perfect negative
correlation is precluded by their characterization condition. Ziegler (2020)
provided a related belief-dependence bound tighter than the usual Fréchet
bounds. See also Burdzy and Pal (2019) and Burdzy and Pitman (2020) for the
references on coherent opinion of experts, which provided alternative
approaches to tight bounds on the probability that the pair of posteriors
differ by more than a constant delta. Below we examine a series of examples
and show that the Border-like characterization impose strong restrictions on
the bounds and possible forms of dependence structures. Roughly, the Border∗
inequalities require that the posterior beliefs are not too positively
dependent.

###  5.1 Independence

For information design problems, there are important concerns that require
agents’ posterior beliefs to be (unconditional) independent: either privacy
concerns, or the avoidance of complicated strategic reasoning on agents’ side,
or the endogenous choices by a designer (see e.g. Bergemann, Brooks and Morris
(2017), He, Sandomirskiy, and Tamuz (2021), Brooks and Du (2021)). Arieli et
al. (2021) Proposition 3 showed that for binary states, an independent and
symmetric distribution is infeasible when the number of agents is sufficiently
large. Below we present a similar result for the Border∗ inequalities.

We first show that with independence, a further reduction of the inequalities
in Theorem 4 is possible. Using a similar technique as Che et al. (2013), we
establish the following characterization.

######  Proposition 1.

(1) If $\nu$ is independent, $\nu$ is $p_{0}$-feasible for some $p_{0}$ if and
only if (4) and (20) hold.

(2) If $\nu$ is further symmetric (i.e., $\nu=F^{n}$), $\nu$ is
$p_{0}$-feasible for some $p_{0}$ if and only if

$\int_{a}^{1}xdF(x)\leq\int_{a}^{1}F^{n-1}(x)dF(x)$ (21)

for all $a\in[0,1]$ and

$n\int_{0}^{1}xdF(x)=1$ (22)

For independent and symmetric posteriors, Border∗ inequalities (3) and (4)
reduce to one-parameter family of inequalities in Proposition 1. Condition
(22) is a symmetric martingale condition. Symmetric posteriors imply that the
prior probabilities are symmetric for all states, i.e.,
$p_{0}^{\omega_{i}}=\frac{1}{n}$ for all $i\in N$.

One might think conditions in Proposition 1 have a structure of majorization
inequalities characterized by Hart and Reny (2015) and Kleiner, Moldovanu, and
Strack (2021), because Border’s theorem falls into this class and (21) follows
from Border’s theorem. We note that (21), however, is not a majorization
inequality in a usual sense. This is because choice variable $F$ appears on
both sides of (21). It is still unclear whether the theory of majorization can
be applied to this problem.

When all agents have symmetric marginals $F$, from (22) $F$ must vary with
$n$. Below we assume agent $1$ is uninformed (e.g., the seller in an auction).
We can keep the marginals of $n-1$ agents fixed and vary the number of agents.

######  Proposition 2.

Suppose agent $1$ is uninformed, $\nu\in[0,1]^{n-1}$ is independent and has
the symmetric marginals $\nu_{i}=F$ for $i=2,\dots,n$. If the number of agents
is sufficiently large ($n\to\infty$), $\nu$ is not feasible.

To see this result, pick $a_{i}=0$ for all $i=2,\dots,n$ and $a_{1}=1$ in
(20), we obtain $(n-1)\mathbb{E}(x)\leq 1$. The inequality must be violated
for $n$ sufficiently large. In particular, if $F$ is uniform, it is easy to
verify that condition (20) holds when $n=3$ and fails for $n\geq 4$, i.e., the
independent uniform posteriors are feasible only for three agents.

###  5.2 Correlation

We now discuss several classes of joint posterior beliefs allowing
correlation. Interestingly, Proposition 2 with fixed marginals and variable
number of agents generalizes to correlated beliefs.

######  Proposition 3.

Suppose agent $1$ is uninformed and $\nu\in[0,1]^{n-1}$ has the symmetric
marginals $\nu_{i}=F$ for $i=2,\dots,n$. Then

(1) If the number of agents is sufficiently large ($n\to\infty$), $\nu$ is not
feasible.

(2) In particular if $F$ is uniform, $\nu$ is not feasible for $n\geq 4$.

We next consider the class of three-agent problems where agents 1 and 2 has
uniform marginals and agent 3 is uninformed, i.e., copulas (see Nelson, 2007).
We show that the agents’ beliefs cannot be too positively dependent. We
introduce the following classic definition of positive quadratic dependence
for bivariate distributions (Lehman, 1966).

######  Definition 3.

Distribution $\nu\in[0,1]^{2}$ is positive quadratic dependent if for all
$a_{1},a_{2}\in[0,1]$,

$\nu(X_{1}\leq a_{1},X_{2}\leq a_{2})\geq\nu_{1}(X_{1}\leq
a_{1})\nu_{2}(X_{2}\leq a_{2})$ (23)

Consider a class of testing sets which require that for each agent $i$, the
posterior belief for $\omega_{i}$ is greater or equal to $a_{i}\in[0,1]$:

$C_{i}=\\{X_{i}\geq a_{i}\\}$ (24)

Then the Border∗ inequalities in Theorem 4 imply that for all $a_{i}\in[0,1]$,
the joint probability for $C_{1}^{c}\times C_{2}^{c}$ is bounded above, which
gives a bound on the joint distribution of posteriors.

######  Proposition 4.

Suppose $\nu\in\Delta([0,1]^{2})$ has uniform marginals $\nu_{i}$ on $[0,1]$.
If $\nu$ is $p_{0}$-feasible for some $p_{0}$, then for all
$a_{1},a_{2}\in[0,1]$,

$\nu(X_{1}\leq a_{1},X_{2}\leq a_{2})\leq\frac{\sum_{i}a_{i}^{2}}{2}$ (25)

We can show that for several classes of copulas, the bound completely rules
out positive quadratic dependent beliefs. For illustration, consider the
following 4 classes of copulas:

(1) $\nu(x_{1},x_{2})=\min\\{x_{1},x_{2}\\}$. This copula corresponds to the
Fréchet-Hoeffding upper bound, and in our case it assumes that agents’ beliefs
are perfectly positively correlated. Then $\nu$ violates condition (25). This
result verifies that perfect positive correlation is precluded by the no-
agreement argument.

(2) $\nu(x_{1},x_{2})=x_{1}x_{2}+\theta x_{1}x_{2}(1-x_{1})(1-x_{2})$, where
$\theta\in[-1,1]$. These correspond to the Farlie-Gumbel-Morgenstern family
and is known to be positive quadratic dependent if $\theta\geq 0$. Then $\nu$
satisfies condition (25) if and only if $\theta\leq 0$, i.e., $\nu$ is
independent or negatively dependent.

(3) $\nu(x_{1},x_{2})=(x_{1}^{-\theta}+x_{2}^{-\theta}-1)^{-\theta}$, where
$\theta\in[-1,+\infty]$. These correspond to the Clayton family and is known
to be positive quadratic dependent if $\theta\geq 0$. Then $\nu$ satisfies
condition (25) if and only if $\theta\leq 0$, i.e., $\nu$ is independent or
negatively dependent.

(4) $\nu(x_{1},x_{2})=\frac{x_{1}x_{2}}{1-\theta(1-x_{1})(1-x_{2})}$, where
$\theta\in[-1,1)$. These correspond to the the Ali–Mikhail–Haq (survival)
family and is known to be positive quadratic dependent if $\theta\geq 0$. Then
$\nu$ satisfies condition (25) if and only if $\theta\leq 0$, i.e., $\nu$ is
independent or negatively dependent.

For the last example, we consider that agents 1 and 2 have a joint
distribution of posteriors uniform on the upper triangle

$\Delta^{*}=\\{(x_{1},x_{2})\in\R^{2}_{+}:x_{1}+x_{2}\geq 1,x_{1}\leq
1,x_{2}\leq 1\\},$ (26)

with a density function

$f(x_{1},x_{2})=\left\\{\begin{array}[]{l
l}2&\text{if}\,\,(x_{1},x_{2})\in\Delta^{*},\\\ 0&\text{otherwise}.\\\
\end{array}\right.$

The joint support restriction to the upper triangle captures a notion of
positive dependence of beliefs. The next result shows that this distribution
is not feasible.333Notice that the symmetric marginal density of $\nu$ is
given by $f(x)=2x$ for all $x\in[0,1]$. Then the condition in Proposition 4 is
given by $2\int_{a}^{1}2x^{2}dx\leq
1-(\frac{1}{2}(2a-1)^{2})\mathbbm{1}_{a\geq 1/2}$, which is violated for
$a\in(0.5,0.66]$.

######  Proposition 5.

Suppose $\nu$ is uniform on $\Delta^{*}$, then $\nu$ is not $p_{0}$-feasible
for any $p_{0}$.

##  6 Relation to Literature

Our paper contributes to the literature on reduced-form mechanisms. Maskin and
Riley (1984) and Matthews (1984) first studied reduced-form implementability
conditions and Border (1991, 2007) proved a conjecture posed by Matthews,
which is known as Border’s theorem. Goeree and Kushnir (2011, 2016, 2022)
developed a geometric approach for general social choice problems by support-
functions. Che, Kim, and Mierendorff (2013) proposed a network flow approach
to reduced-form auction with paramodular constraints. Vohra (2011)
characterized reduced forms by polymatroid. Hart and Reny (2015) and Kleiner,
Moldovanu, and Strack (2021) characterized symmetric reduced-form auctions by
majorization inequalities. Zheng (2021) and Lang and Yang (2021) generalized
reduced-form auctions to multiple goods and constraints. Lang and Mishra
(2023) characterized symmetric reduced form voting rules with two
alternatives. Toikka, Vohra, and Vohra (2023) studied a reduced form approach
to persuasion where the reduced-form variables are the probabilities with
which the receiver takes each of her actions.

Our model provides a new belief-based characterization to reduced-form
auctions. We derive a new version of Border’s inequalities which characterize
all feasible (first-order) posterior beliefs of agents about their payoff-
relevant events. Our result formalizes a relationship between Border’s theorem
and Aumann’s agreement theorem. In some sense, Border’s theorem is equivalent
to Aumann’s theorem by appropriately choosing payoff-relevant events. While
our model considers auction-type reduced-form outcomes, our results provide a
technique to establish a general equivalence between feasible posterior
beliefs problems and reduced-form mechanisms for non-auction problems.

Our model is closely related to the recent literature on feasible joint
posterior beliefs about a common event. When the state space is binary, Arieli
et al. (2021) provided a linear characterization of feasible (first-order)
posterior beliefs and further proved that these linear inequalities lead to a
quantitative Aumann’s agreement theorem. When the state space contains more
than two states, Arieli et al. (2021) and Morris (2020) provided a no-trade
characterization for the problem of feasible joint posterior beliefs. Lang
(2022) shows that different characterization conditions a la Aumann and Border
correspond to different types of market games of bets, which depend on whether
the agents making bets on the same events or distinct events and the numbers
of buyers and sellers of bets. While no-trade characterization requires a
large number of inequalities, our Border-like characterization is similar to
indicator trading schemes in Arieli et al. (2021) and provide a tractable way
to generalize their indicator trading scheme characterization from two states
to many states. Our paper also contributes to the literature on common prior
and no trade (Milgrom and Stokey, 1982; Morris, 1994; Samet, 1998; Feinberg,
2000). Our result complements the existing result by considering a new class
of bets where agents bet on pairwise disjoint events.

The problem of feasible joint posterior beliefs is closely related to the
growing literature on Bayesian persuasion started by Kamenica and Gentzkow
(2011) and Rayo and Segal (2010). A special class of private information is
that the information available to each agent reveals nothing about the
information available to her opponent. These private signals are literally
private and is called private private information structures (He,
Sandomirskiy, and Tamuz, 2021). Private private signals arise as the worst-
case information structure in some problems of robust auction design (see
Bergemann, Brooks and Morris (2017); Brooks and Du, 2021). In auction
problems, private private signals are very desirable when the preservation of
the buyers’ privacy is a major concern. Our belief-based characterization of
Border’s inequalities implies that even if there is no robustness concern,
private private signals can arise endogenously in some auction-type
information design problems due to feasibility reasons.

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##  Appendix A Appendix

###  A.1 Proof of Theorem 3

To prove Theorem 3, we will use the following lemma (Dudley, 2004, Theorem
13.1.1). Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be measurable spaces. If
$T:X\to Y$ is a measurable map and $\mu$ is a measure on $\mathcal{F}$, then
the pushforward measure is defined by $T_{\\#}\mu(A)=\mu(T^{-1}(A))$ for all
$A\in\mathcal{G}$. We have the following lemma.

######  Lemma 2.

Let $(X,\mathcal{F},\mu)$ be an atomless probability space and let
$(Y,\mathcal{G})$ be a Polish space (i.e., separable and completely
metrizable). Then for any probability measure $\nu$ on $(Y,\mathcal{G})$,
there exists a measurable function $T:X\to Y$ such that $\nu=T_{\\#}\mu$.

###### Proof of Theorem 3.

Only if. Suppose $\mu$ is an independent and atomless measure on some type
space $(T,\mathcal{\mathcal{F}})$. The necessity that $\nu$ satisfies the
Border∗ inequalities follows from the proofs of Theorem 1. To show that $\nu$
is independent, notice that from (2) and independence of $\mu$, for every
measurable sets $C_{i}\subseteq[0,1]$, $i\in N$,

$\displaystyle\nu(C_{1}\times\dots\times C_{n})$ $\displaystyle=\mu(Q\in
C_{1}\times\dots\times C_{n})$ $\displaystyle=\mu(Q_{1}\in
C_{1},\dots,Q_{n}\in C_{n})$ $\displaystyle=\prod_{i\in N}\mu_{i}(Q_{i}\in
C_{i})$ $\displaystyle=\prod_{i\in N}\nu_{i}(C_{i})$

where the second equality follows from $Q_{i}(t)=Q_{i}(t_{i})$. Hence $\nu$ is
independent.

If. Suppose $\nu$ is independent and satisfies the Border∗ inequalities. Then
$\nu=\nu_{1}\times\dots\times\nu_{n}$ where each $\nu_{i}$ is a probability
measure on $([0,1],\mathcal{B})$ where $\mathcal{B}$ is the Borel $\sigma$
-algebra, which is Polish. Since $\mu$ is independent and atomless,
$\mu=\mu_{1}\times\dots\times\mu_{n}$ and each $\mu_{i}$ is atomless. By Lemma
2, for each agent $i$, there exists an $\mathcal{F}_{i}$-measurable function
$Q_{i}:T_{i}\to[0,1]$ that pushes forward $\mu_{i}$ to $\nu_{i}$. We define an
interim outcome function to be $Q=(Q_{1},\dots,Q_{n})$. Since $\nu$ satisfies
the Border∗ inequalities, by a change of measures we have $Q$ satisfies
Border’s inequalities. Then Lemma 1 implies that there exists an outcome
function $a$ such that $Q$ is the reduced form. Hence $\nu$ is feasible for
$(\mu,a).$ ∎

###  A.2 Proof of Theorem 4

Before presenting the proof, the following revelation principle is useful for
our analysis. We say an information structure is direct if for each $i$,
$S_{i}=[0,1]$ and $q_{i}(s_{i})=s_{i}$ for all $s_{i}$.

######  Lemma 3.

If $\nu$ is $p_{0}$-feasible for some information structure $I_{0}$, then
there exists a direct information structure $I$ such that $\nu$ is
$p_{0}$-feasible for $I$.

###### Proof.

If $\nu$ is $p_{0}$-feasible for some $p_{0}$, i.e., there exists an
information structure $I_{0}=(S,P_{0})$ such that $\nu=\nu^{I_{0}}$, then
define a direct information structure $I=([0,1]^{n},P)$ by
$P(\omega,C)=P_{0}(\omega,q\in C)$ for every measurable $C\subset[0,1]^{n}$,
we have $\nu=\nu^{I}$. ∎

###### Proof of Theorem 4.

Only if. Suppose $\nu$ is feasible. From Lemma 3, there exists a direct
information structure $I=([0,1]^{n},P)$ such that $\nu=\nu^{I}$. We have

$\displaystyle\sum_{i\in N}\int_{C_{i}}x_{i}d\nu_{i}(x_{i})=$
$\displaystyle\sum_{i\in N}\int_{C_{i}}x_{i}d\nu^{I}_{i}(x_{i})$ (27)
$\displaystyle=$ $\displaystyle\sum_{i\in
N}\int_{C_{i}}q_{i}^{A_{i}}(x_{i})d\nu^{I}_{i}(x_{i})$ (28) $\displaystyle=$
$\displaystyle\sum_{i\in N}\int_{C_{i}}P(A_{i}|x_{i})dP(x_{i})$ (29)
$\displaystyle\leq$ $\displaystyle\sum_{i\in N}\int_{\cup_{i\in
N}(C_{i}\times[0,1]^{n-1})}dP(A_{i},x)$ (30) $\displaystyle=$
$\displaystyle\nu\left(\cup_{i\in N}(C_{i}\times[0,1]^{n-1})\right)$ (31)

where the second equality follows from that $I$ is direct and the last
equality follows from that $A_{1},\dots,A_{n}$ is a partition of $\Omega$, and
hence $\sum_{i\in N}P(A_{i},B)=P(B)=\nu(B)$ for all measurable
$B\subset[0,1]^{n}$.

If. Suppose $\nu$ satisfies (3). We construct a direct information structure
that gives $\nu$ as posteriors. The proof is based on the construction of an
equivalent reduced-form auction problem. Define $D=\\{A_{1},\dots,A_{n}\\}$ a
set of possible allocations of an object for agents. Define for each agent $i$
type space $T_{i}=[0,1]$ with the Borel $\sigma$-algebra. Define $T$ as the
product type space and the prior probability measure $\mu=\nu$ on $T$. Define
$Q_{i}^{A_{i}}(t)=t_{i}$. $\nu$ satisfies (3) implies that $Q$ satisfies
Border’s condition. By Border’s theorem, there exists an outcome function
$a:T\to\Delta(D)$ such that $Q$ is the reduced form.

Now define an information structure $(S,P)$ where $S=T$ and
$dP(\omega,t)=\frac{1}{|A_{i}|}a(A_{i},t)d\mu(t)$ for every $\omega\in A_{i}$
and every $i$. Then $P\in\Delta(\Omega\times S)$. By construction

$\int_{T_{-i}}dP(A_{i},t_{-i}|t_{i})=\int_{T_{-i}}a(A_{i},t_{-i},t_{i})d\mu_{i}(t_{-i}|t_{i})=Q_{i}^{A_{i}}(t_{i})=t_{i}$
(32)

That is, $Q$ is the posteriors induced by $(S,P)$. By construction for each
$B\subset[0,1]^{n}$,

$\nu(B)=\mu(B)=P(Q\in B)$ (33)

Then $\nu$ is the distribution of posteriors induced by $(S,P)$. ∎

###  A.3 Proof of Theorem 5

###### Proof of Theorem 5.

Only If. Suppose $\mu$ is independent, $m$ is a BIC (or DIC) auction and $\nu$
is feasible for the auction problem $(\mu,m)$. A similar analysis as in
Theorem 1 implies that $\nu$ satisfies Border∗ inequalities (3) and (4). Also
$\mu$ is independent implies that $\nu$ is independent. So $\nu$ satisfies
(20) for all $C_{i}\subseteq[0,1]$ and hence for all
$C_{i}=\\{x_{i}\in[0,1]:x_{i}\geq a_{i}\\}$ with $a_{i}\in[0,1]$.

If. We show that when $\nu$ is independent and satisfies (4) and (20), then
$\nu$ is feasible an for some independent prior and some BIC (or DIC) auction.
The construction is similar to Theorem 1. Given the type space $T=[0,1]^{n}$,
define a prior probability measure $\mu$ by $\mu(C)=\nu(C)$ for all measurable
$C\subset[0,1]^{n}$. Define $Q:[0,1]^{n}\to[0,1]^{n}$ by $Q_{i}(x)=x_{i}$ for
all $x\in[0,1]^{n}$ and $i\in N$. Then $Q$ is an interim outcome function. By
construction $Q$ pushes forward $\mu$ to $\nu$. Moreover, each $Q_{i}$ is
nondecreasing.

Pick any profile of testing sets $(C_{i})_{i\in N}$ with
$C_{i}=\\{x_{i}\in[0,1]:x_{i}\geq a_{i}\\}$. Since $Q_{i}(x_{i})=x_{i}$, $\nu$
satisfies condition (20) for $(C_{i})_{i\in N}$ implies that $Q$ satisfies (5)
for $(E_{i})_{i\in N}$ with $E_{i}=\\{x_{i}\in[0,1]:x_{i}\geq a_{i}\\}$. From
Border’s theorem with independent priors and monotonicity constraints (Che et
al., 2013, Corollary 2), there exists a BIC auction $a:T\to\Delta(N)$ that
implements $Q$. From Theorem 2 in Manelli and Vincent (2010), we can make $a$
into a DIC auction. Hence we find an independent prior $\mu$ and a BIC (or
DIC) auction $a$ that generate $\nu$, i.e., $\nu$ is feasible for a BIC (or
DIC) auction problem. ∎

###  A.4 Proof of Theorem 6

The following theorem in optimal transport establishes the existence of a
pushforward (i.e.,a transport plan) with a monotonicity property.

######  Lemma 4.

Brenier’s theorem (Villani, 2003, Theorem 2.12) If $\mu$ and $\nu$ are two
probability measures on $\R^{n}$ with $\mu$ absolutely continuous with respect
to the Lebesgue measure, then there exists a unique map
$T:[0,1]^{n}\to[0,1]^{n}$ that pushes forward $\mu$ to $\nu$ and
$T=\nabla\phi$ with $\phi$ convex.

###### Proof of Theorem 6.

Only if. Suppose $\mu$ is a measure on $[0,1]^{n}$ that is independent and
absolutely continuous. The necessity that $\nu$ satisfies (4) and (20) and is
independent follows from the proof of Theorem 5.

If. Suppose $\nu$ is independent and satisfies (4) and (20). Since $\mu$ is
independent and absolutely continuous, each $\mu_{i}$ is absolutely
continuous. Using Lemma 4 for $n=1$, for each agent $i$, there exists a
measurable function $Q_{i}:[0,1]\to[0,1]$ that pushes forward $\mu_{i}$ to
$\nu_{i}$ and $Q_{i}$ is non-decreasing. We define an interim outcome function
to be $Q=(Q_{1},\dots,Q_{n})$. Since $\nu$ satisfies (4) and (20), by a change
of variables we have $Q$ satisfies Border’s inequalities. Then Lemma 1 implies
that there exists an outcome function $a$ such that $Q$ is the reduced form.
Hence $\nu$ is feasible for $(\mu,a).$ ∎


&-\int_0^t \int_{\NN} \int_{\tilde{\QQ}^{\xi^\XX}} \DG \left (\sum_{i=1}^{\xi^\XX}f^\XX(q_i^\XX) \right) \zeta_{\xi^{\XX}}(\left . q^\XX\right| \xi^\XX)dq^{\XX} dp_{\XX}(\xi^\XX)  ds\,,\\[5pt]
&\Psi^\YY_{t,f^\YY} (\mathrm{u}^\XX,\mathrm{u}^\YY) = \norm{f^\YY,u^\YY(t)} - \norm{f^\YY,u^\YY} - \int_0^t \norm{\LL^\YY_d f^\YY (x),u^\YY(s)}ds +\\[3pt]
&- \int_0^t \int_{\QQ} \AG(q,u) \int_{\QQ} m^{\AG}(\bar{q}|q) f^\YY(\bar{q})d\bar{q} u^\XX(s)(dq)ds+\\[3pt]
&- \int_0^t\int_{\tilde{\QQ}^2} \int_{\QQ} \PG(q_1,q_2) \BG(q_1,q_2,u) f^\YY(\bar{q}) m^{\BG}(\bar{q}|q_1,q_2)d\bar{q}u^\XX(s)(dq_1)u^\XX(s)(dq_2)ds+\\[3pt]
&-\int_0^t \int_{\NN^2} \int_{\tilde{\QQ}^{\xi^\YY}} \DG \left (\sum_{i=1}^{\xi^\YY} f^\YY(q_i^\YY) \right) \zeta_{\xi^{\YY}}(\left .q^\YY \right|\xi^\YY) dq^\YY dp_{\YY}(\xi^\YY) ds\,.
\end{split}
\]

We aim to show that, for any $t \leq T$,

\[
\mathbb{E}\left | \Psi^\XX_{t,f^\XX} (\mathrm{u}^\XX,\mathrm{u}^\YY) \right| = \mathbb{E}\left |  \Psi^\YY_{t,f^\YY} (\mathrm{u}^\XX,\mathrm{u}^\YY) \right| = 0\,.
\]

Using Lemma <ref> we have that

\[
\MM^\XX(t) = \Psi^\XX_{t,f^\XX} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\,,\quad \MM^\YY(t) = \Psi^\YY_{t,f^\YY} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\,.
\]

Using thus Lemma <ref> together with estimate (<ref>) and Hypothesis <ref>-<ref>-<ref>, we have that

\[
\begin{split}
&\mathbb{E} \left | \MM^{\XX;\KK}(t) \right|^2 = \mathbb{E} \left \langle \MM^{\XX;\KK} \right \rangle (t) \leq \frac{C}{\KK}\,,\\
&\mathbb{E} \left | \MM^{\YY;\KK}(t) \right|^2 = \mathbb{E} \left \langle \MM^{\YY;\KK} \right \rangle (t) \leq \frac{C}{\KK}\,,\\
\end{split}
\]

which goes to 0 as $\KK \to \infty$. Therefore, we have

\[
\begin{split}
&\lim_{\KK \to \infty}\mathbb{E} \left | \Psi^\XX_{t,f^\XX} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\right | = 0\,,\\
&\lim_{\KK \to \infty}\mathbb{E} \left | \Psi^\YY_{t,f^\XX} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\right | = 0\,.
\end{split}
\]

Since $\mathrm{u}$ is a.s. strongly continuous and the functions $f^\XX$ and $f^\YY$ are bounded, then the functionals $ \Psi^\XX_{t,f^\XX}$ and $ \Psi^\YY_{t,f^\YY}$ are a.s. continuous at $\mathrm{u}$. Also, for any $\mathcal{D}\left ([0,T],\MMPP\times \MMPP\right)$ we have

\[
\begin{split}
&\left | \Psi^\XX_{t,f^\XX} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK}) \right | \leq C \sup_{s \in [0,T]}\left(1 + \norm{\unit,u^\XX(s)}^2 + \norm{\unit,u^\YY(s)}^2 \right)\,,\\
&\left | \Psi^\YY_{t,f^\XX} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\right | \leq C \sup_{s \in [0,T]}\left(1 + \norm{\unit,u^\XX(s)}^2 + \norm{\unit,u^\YY(s)}^2 \right)\,.
\end{split}
\]

Therefore the sequences $\left( \Psi^\XX_{t,f^\XX} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\right)_{\KK \in \mathbb{N}}$ and $\left( \Psi^\YY_{t,f^\YY} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\right)_{\KK \in \mathbb{N}}$ are uniformly integrable so that

\[
\begin{split}
&\lim_{\KK \to \infty}\mathbb{E} \left | \Psi^\XX_{t,f^\XX} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\right | = \mathbb{E} \left | \Psi^\XX_{t,f^\XX} (\mathrm{u}^{\XX},\mathrm{u}^{\YY})\right |\,,\\
&\lim_{\KK \to \infty}\mathbb{E} \left | \Psi^\YY_{t,f^\YY} (\mathrm{u}^{\XX;\KK},\mathrm{u}^{\YY;\KK})\right | = \mathbb{E} \left | \Psi^\YY_{t,f^\YY} (\mathrm{u}^{\XX},\mathrm{u}^{\YY})\right |\,,\\
\end{split}
\]

which concludes the proof.

Assume Hypothesis <ref>-<ref>-<ref> hold, then the sequence of laws $\left (\Lambda^\KK\right)_{\KK \in \mathbb{N}}$ on $\mathcal{D}\left([0,T];\MMPP\right)$ is tight when endowed with the weak topology.

Using [41], since the limiting process is continuous, repeating the above calculations with $f^\XX = f^\YY = 1$, we obtain that the sequences $\left ( \norm{\unit,u^{\XX;\KK}} \right )_{\KK}$ and $\left (\norm{\unit,u^{\YY;\KK}} \right )_{\KK}$ converge to $\norm{\unit,u^{\XX}}$ and to $\norm{\unit,u^{\YY}}$ in $\mathcal{D}\left ([0,T];\RR\right)$.

§.§ Step 5: proof of the convergence theorem

Putting together Theorems <ref>-<ref>-<ref>, the claim follows.

§ CONCLUSIONS

In the present paper, we introduced a general stochastic model that describes the formation and repair of radiation-induced DNA damage. The derived model generalizes the previously studied model, [15, 17, 16], including a spatial description, allowing for pairwise interaction of cluster damages that might depend on the distances between damages, as well as a general description of the effect of radiation on biological tissue under a broad range of irradiation condition. We studied the mathematical well-posedness of the system as well as we characterize the large system behavior. Further, we believe that the derived model could play a role in describing an effect of recent interest in radiobiology, called the FLASH effect. In particular, at extremely high rates of particle delivery, it has been empirically seen that the unwanted effects of radiation on healthy tissue decrease while the killing effect on the tumor is maintained. Although numerous studies on the topic, the physical and biological mechanism at the very core of this effect is today largely unknown. It is nonetheless believed that spatial interactions of particles can play a major role in the origin of this effect. Therefore, the model introduced in the present research can have the generality to provide a stochastic description of the effect that spatial interdependence between particles can have on biological tissue.

§ NUMERICAL RESULTS

The present Section is devoted to the implementation of the spatial radiation-induced DNA damage repair model defined above. We will first focus on the model without protracted irradiation defined in Section <ref> and then compare it to the case with protracted irradiation and chemical environment as considered in Section <ref>.

For the sake of simplicity, we will implement the model for a circular domain $\QQ \subset \RR^2$ of radius 5 $\mu$m, the case of a more realistic scenario of a domain in $\RR^3$ of arbitrary shape is straightforward. We assume that the radiation field is perpendicular to the cell nucleus, with a total absorbed dose of 5 Gy. We consider the radiation field generated by a monoenergetic beam of carbon ions at $40$ Mev, with a fluence average specific energy $z_F=0.04$ Gy; a microdosimetric description of such radiation field has been considered for instance in <cit.>. We simulate the number of impinging lesions according to a Poisson distribution of average $\frac{D}{z_F}$. The radiation field is assumed isotropic and uniform so that each tract is thus distributed uniformly on the nucleus. Then for each track, specific energy is sampled from the microdosimetric single-event distribution $f_1(z)$ and distributed spatially around the track according to the amorphous track model as described in [34], which prescribes a radial dose distribution as

\[
D(\rho) =
\begin{cases}
C_{\mbox{c}} & \rho \leq R_{\mbox{c}}\,,\\
C_{\mbox{p}}\frac{1}{\rho^2} & R_{\mbox{c}} < \rho \leq R_{\mbox{p}}\,,\\
0 & \mbox{otherwise}\,,
\end{cases}
\]

where $\rho$ is the distance from the center of the track, $R_{\mbox{c}}$ is the core around the track characterized by a higher energy deposition outside which the dose decrease as $\rho^{-2}$ until the radius of the penumbra $R_{\mbox{p}}$ after which no energy deposition is registered. The constants considered are as defined in [34] for the case of low-energy carbon ions considered. Thus, a random number of sub-lethal lesions, resp. lethal lesions are sampled around each track according to a Poisson random variable of average $\kappa z_i$, resp. $\lambda z_i$, with $\kappa = 50$ Gy$^{-1}$, resp. $\lambda = \kappa 10^{-2}$ Gy$^{-1}$.

Regarding the model, the following parameters have been chosen

\begin{equation}\label{EQN:ParamC}
\begin{split}
\RG(q,v) &= \RG \left (1 + \frac{1}{\norm{\Ind{|q-\bar{q}| < r_d} (\bar{q},\bar{s}),\nu} +1}\right)\,,\\
\AG(q,v) &= \AG \left (1 - \frac{1}{\norm{\Ind{|q-\bar{q}| < r_d} (\bar{q},\bar{s}),\nu} +1}\right)\,,\\
\BG(q_1,q_2,v) &=  \BG \Ind{|q_1 - q_2|<r_d}\,,
\end{split}
\end{equation}

with $\RG = 4\, h^{-1}$ , $\AG = 0.1\, h^{-1}$ and $\BG=0.1\, h^{-1}$. Regarding these last constants, they are in agreement with standard values for instance calibrated in <cit.>. Concerning instead equation (<ref>), we assumed that the repair rate decreases, resp. the death rate increases, as the number of damages within a radius $r_d=0.5 \, \mu$m increases. At last, we assume a constant pairwise interaction only for the lesions within $r_d$ distance. It is worth stressing that such aspects are among the major strengths of the proposed model, where rates depend on the local concentration of damages.

Dose deposited (left panel) and damage distribution (right panel).

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