11institutetext: Weizmann Institute 11email:
<EMAIL_ADDRESS>Ariel University 22email:
<EMAIL_ADDRESS>
33institutetext: Technion - Israel Institute of Technology 33email:
<EMAIL_ADDRESS>
# Prediction-sharing During Training and Inference

Yotam Gafni 11 Ronen Gradwohl 22 Moshe Tennenholtz 33

###### Abstract

Two firms are engaged in a competitive prediction task. Each firm has two
sources of data—labeled historical data and unlabeled inference-time data—and
uses the former to derive a prediction model, and the latter to make
predictions on new instances. We study data-sharing contracts between the
firms. The novelty of our study is to introduce and highlight the differences
between contracts that share prediction models only, contracts to share
inference-time predictions only, and contracts to share both.

Our analysis proceeds on three levels. First, we develop a general Bayesian
framework that facilitates our study. Second, we narrow our focus to two
natural settings within this framework: (i) a setting in which the accuracy of
each firm’s prediction model is common knowledge, but the correlation between
the respective models is unknown; and (ii) a setting in which two hypotheses
exist regarding the optimal predictor, and one of the firms has a structural
advantage in deducing it.

Within these two settings we study optimal contract choice. More specifically,
we find the individually rational and Pareto-optimal contracts for some
notable cases, and describe specific settings where each of the different
sharing contracts emerge as optimal. Finally, in the third level of our
analysis we demonstrate the applicability of our concepts in a synthetic
simulation using real loan data.

###### Keywords:

Data Sharing Strategic Machine Learning Strategic Classification Information
Sharing.

##  1 Introduction

Machine learning (ML) is becoming a highly distributed endeavor. Data is
spread among different firms, each of whom may have their own ML capabilities
and economic utilities. In many cases, one firm’s data and prediction
capabilities are complemented by those available to a competing firm, and each
firm would benefit from access to the other’s predictions. For example, two
investment banks that attempt to predict loan defaults could each improve
their respective predictions by accessing the other’s predictions. Indeed,
this is in the spirit of one of the most fundamental ideas in ML—aggregating
weak learners into strong ones [11]. However, the distributed nature of firms’
capabilities introduces a major obstacle: Why, and under what conditions,
would firms willingly share their predictions with competitors? And what would
equilibrium behavior look like, given such sharing?

Our main innovation in this paper is the observation that this obstacle
actually consists of two separate questions, corresponding respectively to the
training and inference phases in ML. First, why would firms share the labels
they have (about individuals in their data logs) in the training phase? And
second, why would firms share their predictions (about new, incoming
instances) in the inference phase? As we show in this paper, this distinction
has real bite.

In order to tackle the question of training/inference-stage prediction-
sharing, we proceed on three levels. First, we develop a general Bayesian
model that captures the two kinds of sharing. The Bayesian model specifies the
informational environment, while a utility model specifies the economic
implications. In the Bayesian model, each firm obtains a training signal that
represents the prediction model (a.k.a. classifier) learned by that firm via
its labeled historical data. The firm also obtains an inference-time signal
that represents the classifier’s prediction on unlabeled inference-time data.
In the utility model we associate a real number with each outcome quadrant:
True-positive, true-negative, false-positive, and false-negative predictions.
We moreover assume that if both firms arrive at the same outcome, then the
associated utility is split between them.

In the second level of our analysis, we apply our model to a game-theoretic
study of two natural settings. In the first setting, the accuracy of each
firm’s prediction model is common knowledge, but the correlation between the
respective models is unknown. As for utilities, each firm has a safe
prediction that yields utility zero (whether right or wrong), and a risky
prediction. For example, a firm predicting a customer’s trustworthiness in
order to decide whether or not to issue a loan. If a loan is provided, the
firm’s utility depends on the accuracy of the trustworthiness prediction, and
whether or not the customer has other offers. If no loan is provided, the
firm’s utility is fixed at $0$. In the second setting we study, there are two
hypotheses regarding the optimal predictor, and one of the firms has a
structural advantage in deriving it. Furthermore, firms’ utilities are
symmetric across prediction types (unlike the first setting), and depend only
on the predictions’ correctness—e.g., a firm recommending a movie to a viewer,
where the utility depends on whether or not it accurately predicts the
viewer’s tastes.

Finally, in the third level of our analysis, we demonstrate the applicability
of our ideas in a synthetic simulation using real loan data. This is intended
to provide an accessible, practical recasting of our abstract model’s results.
In broad terms, if we take a single firm’s perspective, the no-sharing
contract allows it to build a classifier based on its own historical data.
Then, based on its assessment (prior) of the competitor, it decides whether or
not to act in accordance with the classifier’s prediction (signal). An example
of choosing to ignore the classifier’s signal would be if the firm knows that
its competitor can perfectly predict whether a loan would be repaid. Then, all
the benefit of issuing a good loan is split (e.g., by the random decision of
the consumer as to which of the offered loans to accept). However, since the
firm knows that its own classifier is imperfect, it knows it will also end up
issuing some bad loans. If the cost of bad loans outweighs the benefit of
splitting the profit from good loans, the firm would decide to ignore its
classifier and not issue any loans. Expanding on this example, the train-
sharing contract can allow the firm to make a more refined decision: Based on
seeing how the other firm predicts on the historical data, it can assess
whether or not to follow its own classifier. The full-sharing contract allows
even more intricate decision rules: They can depend both on what the firm
learns about the competitor’s predictions on historical data, and also on the
competitor’s prediction on each specific consumer. Lastly, the infer-sharing
contract does not see the competitor’s predictions on historical data, so it
must maintain its assessment/prior over the other firm’s classifier, but it
can use the competitor’s prediction on the real-time consumer to decide
whether to follow its own classifier’s prediction. In our practical
implementation of Section 5 we examine the performance of the optimal decision
rules under different contracts, and show that each of no-sharing, full-
sharing, and train-sharing is uniquely optimal for some set of parameters.

The emphasis of our work in game theoretic terms is to require that a contract
is both individually rational and Pareto-optimal (IRPO). This follows the
assumption that the natural state of affairs is that no contract is signed
(no-sharing). Thus, for the firms to agree for any kind of prediction-sharing,
it must be that for each of them, the expected utility under the prediction-
sharing contract is at least as good as under no-sharing. We refer to this
property as the contract being individually rational. Moreover, the contract
must be Pareto-optimal w.r.t. the four possible contracts. E.g., if the
utilities under full-sharing dominate these under train-sharing, even if
train-sharing is by itself individually rational, it would make sense that the
firms choose to sign the Pareto-optimal contract rather than a Pareto
dominated one. As we see, there are different settings so that each of the
contract types may become uniquely IRPO.

Lastly, we note that in order for the firms to share their predictions, they
need a way to match records. Facing this issue is common in the industry and
there are companies that specialize in this task.111E.g., in advertising,
identifying the same user on different devices is called cross-device
targeting, and “attribution providers” companies such as AppsFlyer and
Singular enable this. This type of prediction-sharing is valuable, even if
done for identifiers both firms hold, as different firms may be exposed to
different properties of the same identifier. As an example, think of firms
that know different social and financial features associated with the same
social security number. In this case, there is a difference between sharing
each firm’s binary prediction regarding the user, or the entire data it holds
for that identifier. Importantly, our model assumes that firms share their
training and inference-time signals, and not their entire data. In practice,
in the training stage the signals come in the form of true labels in the
historical data, and in the inference stage in the form of the classifier’s
predictions. The fact that this still proves to be useful is by itself
interesting, as it suggests a path to data sharing that protects both the
firm’s intellectual property (in terms of both data and models used in
training), and possibly the users’ privacy.

###  1.1 Our Contribution

In Section 2, we provide the first model to reason about contracts that may
involve sharing prediction both in the training and inference stage. In
Sections 3,4 we focus on two natural sub-models of the general model we
present:

  1. 1.

A Correlation Model: Both firms know their own and their competitor’s
prediction accuracy, but not the correlation between the two prediction
models. We characterize the uniquely individually rational and Pareto-optimal
contracts for some notable cases. We also show that all contracts except
inference-sharing can be optimal in this setting.

  2. 2.

A Two Hypotheses Model: One firm is able to determine the correct hypothesis
during training, while the other has information about customers that is
valuable during inference. Here, we show that inference-sharing can be the
unique individually rational and Pareto-optimal contract.

Overall, we conclude that each of the four train/inference combination
contracts can be optimal:

  * –

No-sharing is the optimal individually-rational contract when the cost of
making a wrong prediction is equal to the reward of making a correct
prediction. We show this first in Lemma 1, for the case where the prediction
model of each firm (based on its own data) is common knowledge, and then
generalize it in Theorem 3.1 for the general training-phase prior. This
characterization follows from two main insights: (1) Under full-sharing, when
the two firms share their inference-time signals, the firms will simply follow
the primary firm signal. This is because a negative primary firm signal
overshadows a positive secondary firm signal. (2) Given the first insight, the
primary firm is only set to lose by sharing its signal, since the aggregate
utility of the two firms is constant (and equals the accuracy of the primary
firm’s prediction). The secondary firm becomes more informed under full-
sharing, and can extract the same utility as the primary firm.

  * –

Full-sharing is the optimal individually-rational contract when the two firms
have the same prediction accuracy. It is at least as good as no-sharing
because the firms can use both signals to “amplify” or mitigate their
individual signal (Theorem 3.2), in a way that is mutually beneficial w.r.t.
their equilibrium behavior under no-sharing. Full-sharing is also at least as
good as train-sharing, because in both cases the equilibrium behavior is
symmetric, but the full-sharing equilibrium is more informed (Lemma 4). This
is also true w.r.t. the infer-sharing equilibrium, even more generally (Lemma
2), as the symmetry in the infer-sharing case stems not from having the same
prediction accuracy, but from the fact that the infer-phase signals are
shared, and the train-phase signals can not individually teach more about the
correlation than the common prior.

  * –

Train-sharing is the optimal individually-rational contract when the two firms
benefit from reaching different equilibria given a different correlation
between their signals. In particular, the firms may prefer to each follow its
signal when the correlation is low, but have the secondary firm ’yield’ to the
primary firm when the correlation is high and exit the market. Learning ‘when
to quit’ benefits the secondary firm as well, and so can emerge as the optimal
individually-rational contract when full-sharing is ‘too permissive’ for the
primary firm to follow, due to loss in competitive advantage.

  * –

Infer-sharing is harder to come up with a situation where it is the optimal
individually-rational contract. In fact, we show that in our correlation model
it can not be the uniquely optimal individually-rational contract (Lemma 2).
In Section 4 we explore a model we call “the two hypotheses model”, which has
a natural interpretation in health and scientific contexts, and show how
infer-sharing may arise as the uniquely optimal individually-rational contract
there (Theorem 4.1).

Beyond the existence results detailed above, which help provide intuition into
the different types of prediction sharing contracts, the theorems of Section 3
also provide a partial characterization of our correlation model in several
important cases such as symmetric utilities, or symmetric prediction accuracy.
In Section 5, and further in Appendix 0.D, we demonstrate how our abstract
Bayesian model may be put into practice and implemented, using a real loan
dataset.

###  1.2 Related Work

Previous work in ML considered different aspects of strategic prediction. For
example, [1] and [9] study competition in prediction, based on shared and
independent data, respectively. Literature on federated learning [e.g. 5, 10]
considers free-riding by data providers to save costs while still benefiting
from better predictions. [12] and [14] study data aggregation between
competitors: In the former, a firm aims to exploit another firm’s contributed
data but also to mislead it, putting the integrity of the data-sharing
protocol at risk. In the latter, segmentation information about consumers is
split between firms, and firms decide whether or not to share their part of
the data with others during the inference phase. The former work focuses
solely on training models, while the second only on the inference stage of a
known segmentation. Finally, some papers deal with the imbalance between firms
with stronger and weaker models through the lens of fairness, leveraging tools
from cooperative game theory [6, 4]. There is also a growing economics
literature on data markets [see, e.g., the survey of 2]. However, neither this
economics literature, nor work on strategic ML consider strategic sharing of
prediction models between competitors. They also do not contrast sharing
during training and during inference, a distinction we see as crucial for ML
in the distributed economy.

There is some analogy between our work and the fundamental ML idea of
aggregating weak learners into strong ones, and specifically to bagging and
stacking. In bagging [3], the ML algorithm deliberately creates subsets of the
data and learns models for them in parallel; this is somewhat analogous to
how, in our setting, different firms develop their own models. In stacking
[19], there are two stages: first, models (derived, e.g., from bagging)
produce predictions over a data-set, and second, a meta-learning algorithm
learns how to generate a final authoritative prediction from the models’
predictions. In a sense, our work can be viewed as strategic bagging and
stacking.

##  2 Model

#### Informational environment

There are two firms engaged in a competitive prediction task. Each firm
obtains data in two phases: training and inference. In the training phase,
examples with binary labels are drawn at random, and each firm learns a
respective prediction model (i.e., classifier). The training phase may consist
of one example, multiple examples, or “infinitely many” examples. In the
inference phase firms use their learned model in order to predict the label of
a new example. Firm 1’s prediction is either $A$ or $B$ and firm 2’s
prediction is either $a$ or $b$, where the former indicates that the firm’s
prediction model believes the label is 1 and the latter indicates the label is
0. We model this interaction in an abstract Bayesian framework using the rich
signal spaces of [16] and [13]. We next describe the formal model, and then
highlight the main elements and their interpretations.

A world model $w$ consists of a prior distribution $\pi_{w}$ over $\\{0,1\\}$,
as well as two signal spaces, one for each firm. For every true label
$t\in\\{0,1\\}$, each signal space partitions $[0,1]$ into two sets,
representing the probabilities associated with firms’ prediction models, given
true label $t$.222Formally, each signal space is a Lebesgue measurable bi-
partition of $[0,1]\times\\{0,1\\}$. For the first firm, the first set is
denoted $A_{w}^{t}\subset[0,1]$, and the second is denoted
$B_{w}^{t}=[0,1]\setminus A_{w}^{t}$. For the second firm, the two sets are
denoted $a_{w}^{t}$ and $b_{w}^{t}$. Given $w$, a random example is modeled as
a label $t$ drawn from $\\{0,1\\}$ according to $\pi_{w}$, as well as $\zeta$
drawn from $[0,1]$ uniformly at random.333The uniformity assumption here is
without loss of generality. Firm 1’s signal (i.e., its model’s suggested
prediction under $w$) on this example is then $1$ if $\zeta\in A_{w}^{t}$
under $t$, and $0$ otherwise; firm 2’s signal is $1$ if $\zeta\in a_{w}^{t}$
under $t$, and $0$ otherwise. In words, $\zeta$ chooses a “location” on the
interval [0,1]. This location decides some signal for Firm 1 (according to the
way it partitions the interval $[0,1]$), and similarly for Firm 2 (possibly
with a different partition). Sampling $\zeta$ uniformly at random from $[0,1]$
is in a sense similar to sampling a random feature vector that is used to
train the firms’ prediction models / requires a prediction at inference time.

In general, firms may not know the true $w$. Instead, let $W$ be a possibly
infinite set of possible world models, and suppose there is a commonly known
prior $\pi$ over them. An example of this framework is illustrated in Figure
1.

Given this informational environment, the interaction proceeds as follows. In
stage 0, Nature chooses an element $w$ of $W$ according to $\pi$. Then:

  1. 1.

In the training stage, each firm $i$ obtains a training signal $w_{i}$ about
the realized world model $w$. Each $w_{i}$ is a function of firm $i$’s
respective signal space under $w$. Given signal $w_{1}$ (respectively,
$w_{2}$) and the prior over $W$, each firm $i$ uses Bayesian updating to
derive posterior beliefs $\pi_{i}$ over world models $W$.

  2. 2.

In the inference stage, $\zeta$ is drawn from $[0,1]$ uniformly at random, and
a label $t$ is drawn from $\\{0,1\\}$ according to $\pi_{w}$. Firm 1 obtains
the inference-time signal $X\in\\{A,B\\}$ that satisfies $\zeta\in
X_{w^{\prime}}^{t}$, where $w^{\prime}\sim\pi_{1}$; firm 2 obtains the
inference-time signal $x\in\\{a,b\\}$ that satisfies $\zeta\in
x_{w^{\prime}}^{t}$, where $w^{\prime}\sim\pi_{2}$.444Notice that the
inference-time signal is drawn according to the firm’s posterior, rather than
according to some specific true possible world. This is since we are
interested in calculating the firms’ equilibrium behaviors, which follow their
Bayesian perspective.

  3. 3.

In the action stage, each firm $i$ takes an action $a_{i}\in\\{0,1\\}$.
Utilities depend on both firms’ actions, and true label $t$.

Next, we consider different contracts for prediction sharing. Under no-
sharing, the interaction proceeds as above. Under train-sharing, there is an
additional stage between 1 and 2:

  * 1b.

Firms share their respective training signals $w_{1}$ and $w_{2}$.

Under infer-sharing, an additional stage between 2 and 3:

  * 2b.

Firms share respective inference-time signals $X$ and $x$.

Finally, under full-sharing both 1b and 2b take place.

#### Summary and interpretation

We now summarize the model elements and their interpretations:

  * –

The world model $w$ is an information-theoretically optimal pair of
classifiers for the firms.

  * –

The training signal $w_{i}$ implies a posterior $\pi_{i}$ over world models,
which we interpret as the actual classifier firm $i$ is able to train. We
interpret the signal $w_{i}$ as firm $i$’s predictions on its labeled
historical data. In practical terms, the training signal can be interpreted as
the model that best fits the training data, out of all possible models. The
firms can then share these signals (i.e., the functions or code representing
their best models given their data), without sharing the data itself.

  * –

The inference-time signal is the prediction ($X\in\\{A,B\\}$ for firm 1,
$x\in\\{a,b\\}$ for firm 2) made by the trained classifier on an unlabeled
inference-time example.

  * –

Under train-sharing, firms share $w_{1}$ and $w_{2}$, their predictions on
labeled historical data.

  * –

Under infer-sharing, firms share $X$ and $x$, their respective predictions on
the unlabeled inference-time example.

This formulation can capture a wide range of scenarios. The prior over $W$
implies a prior over the relative share $\pi_{w}$ of each label, a prior over
the accuracy of each firm’s model, and a prior over the correlation between
the predictions of firms’ models. The framework is illustrated in Figure 1.
See also Figure 1 and Figure 2 in [13].

Figure 1: There are two world models, represented by the top two and bottom
two pairs of intervals, respectively. For both world models,
$\pi_{w}=Pr[t=1]=\kappa$. In the first world, $A_{w}^{1}=[0,1]$ and
$A_{w}^{0}=[0,\lambda]$. Thus, if $t=1$ firm 1 always obtains signal $A$, and
if $t=0$ firm 1 obtains signal $A$ with probability $\lambda$—i.e., whenever
$\zeta\in[0,\lambda]$—and signal $B$ with probability $1-\lambda$.
Furthermore, $a_{w}^{1}=[0,1]$ and $a_{w}^{0}=[\lambda,\lambda+\mu]$. Thus, if
$t=1$ firm 2 always obtains signal $a$, and if $t=0$ obtains signal $a$ with
probability $\mu$—i.e., whenever $\zeta\in[\lambda,\lambda+\mu]$—and signal
$b$ with probability $1-\mu$. Finally, the bottom two pairs of line segments
represent the firms’ signal spaces in the second world model, which differs
from the first only in firm 1’s signal under $t=1$, namely,
$A_{w}^{1}=\emptyset$ and $B_{w}^{1}=[0,1]$. The interval structure of each of
the firms results in a joint interval structure (and an induced joint
probability over firm 1 signal $A/B$, firm 2 signal $a/b$, and the true
realization $0/1$), shown on the rhs of the figure. In the infinite data
model, where each of the firms learns its own interval structure with
certainty, firm 1 is able to deduce the correct world model just by knowing
its own interval structure. On the other hand, firm 2 does not learn (in a
Bayesian sense) anything from its own interval structure. This example
captures our “Two Hypotheses” model of Section 4.

#### Strategies

A strategy $s_{i}$ of firm $i$ in the action stage is a mapping from the
firm’s signals to a distribution over actions $a_{i}\in\\{0,1\\}$. The firm’s
signals depend on the contract: under no-sharing, the respective signals are
$\sigma_{1}^{ns}=(w_{1},X)$ for firm 1 and $\sigma_{2}^{ns}=(w_{2},x)$ for
firm 2. Under train-sharing, they are $\sigma_{1}^{ts}=(w_{1},w_{2},X)$ and
$\sigma_{2}^{ts}=(w_{1},w_{2},x)$. Under infer-sharing, they are
$\sigma_{1}^{is}=(w_{1},X,x)$ and $\sigma_{2}^{is}=(w_{2},X,x)$. And under
full sharing, both firms obtain signals $\sigma_{i}^{fs}=(w_{1},w_{2},X,x)$.

#### Utility Model

As noted above, utility $u_{i}(p,t,p^{\prime})$ of firm $i$ depends on 3
variables: The firm’s action $p$, the true label $t$, and the other firm’s
action $p^{\prime}$. For a given example, action $p$ is correct if it matches
the example’s label $t$. Given a training signal $w_{i}$, a contract
$ct\in\\{ns,ts,is,fs\\}$, and a pair of strategies $(s_{1},s_{2})$, the
expected utility of firm $i$ is

$u_{i}^{ct}(w_{i},s_{1},s_{2})=E\Big{[}u_{i}\Big{(}s_{1}\left(\sigma_{1}^{ct}\right),t,s_{2}\left(\sigma_{2}^{ct}\right)\Big{)}\Big{]},$
(1)

where the expectation is over the draw of $w$ from $W$ according to
$\pi|w_{i}$, the draw of $t$ according to $\pi_{w}$, the draw of $w_{j}$ under
$w$, the draws of inference-time signals $X$ and $x$ under $w$, and the
distributions of firms’ randomization over actions.

We make some simplifying assumptions about utilities. First, we assume that

$u_{1}=u_{2}$ (2)

. Second, we assume $u_{i}(p,t,p)=\frac{1}{2}u_{i}(p,t,\neg p)$, i.e., that if
the two firms take the same action, the utility (whether positive or negative)
is divided between them, in the sense that:

$\sum_{i=1}^{2}u_{i}(p,t,p)=\sum_{i=1}^{2}\frac{1}{2}u_{i}(p,t,\neg
p)\stackrel{{\scriptstyle\text{Eq.~{}\ref{eq:symmetric_utils}}}}{{=}}u_{1}(p,t,\neg
p).$

To emphasize the notation, $u_{i}(p,t,p)$ is the utility when the other firm’s
prediction $p^{\prime}$ is equal to $p$, and $u_{i}(p,t,\neg p)$ is the
utility when the other firm’s prediction $p^{\prime}$ is different than $p$.

Thus, the ex-post utility is determined by four numbers:
$R_{0}=u_{1}(0,0,1),R_{1}=u_{1}(1,1,0),C_{0}=u_{1}(0,1,1),C_{1}=u_{1}(1,0,0)$,
where for example $R_{0}$ is the reward from correctly taking action $0$ while
the other firm takes action $1$. We assume that $R_{0},R_{1}\geq 0$ and
$C_{0},C_{1}\leq 0$.

In the paper, we largely focus on two specific utility models that capture
important settings. In Section 3, we focus on a utility model we call
significant-action utilities. In this model, there is a significant
action—w.l.o.g., the action $1$. For example, this action may be choosing to
issue a loan. When taking the other, safe action, both reward and cost satisfy
$R_{0}=C_{0}=0.$ If a firm takes a correct significant action exclusively,
meaning that the other firm takes the safe action, it gets the full reward
$R_{1}$. On the other hand, if a firm takes an incorrect significant action
exclusively, it pays a cost $C_{1}$. When $C_{1}=1$, we call this the
symmetric significant-action utility model.

In Section 4, we focus on a utility model we call matching recommendations [as
in, e.g., 15]. In this model, there are no costs to a mistake—formally,
$C_{0}=C_{1}=0$—and there is a symmetric reward for any correct
action—formally, $R_{0}=R_{1}=1$. E.g., consider a firm that chooses between
two possible recommendations to a user, and, if it correctly recommends what
the user is looking for, the user will make a purchase.

#### Equilibrium, individual rationality, and Pareto optimality

Given training signals $w_{1}$ and $w_{2}$ and a contract
$ct\in\\{ns,ts,is,fs\\}$, a pair of strategies $s=(s_{i},s_{\neg i})$ form a
Nash equilibrium at $(w_{1},w_{2})$ if for each $i$ and each strategy
$s_{i}^{\prime}$,

$u_{i}^{ct}(w_{i},s_{i},s_{\neg i})\geq
u_{i}^{ct}(w_{i},s_{i}^{\prime},s_{\neg i}).$ (3)

An equivalent and perhaps more useful formulation of the equilibrium condition
takes the perspective of the agent together with her beliefs [see the
discussion in Chapter 9 of 17]. We define the utility from taking action
$p_{i}\in\\{0,1\\}$ given the collection of signals $\sigma_{i}^{ct}$ and the
other firm’s strategy $s_{\neg i}$ as

$\tilde{u}_{i}^{ct}(\sigma_{i}^{ct},p_{i},s_{\neg
i})=E\Big{[}u_{i}\Big{(}p_{i},t,s_{\neg i}\left(\sigma_{\neg
i}^{ct}\right)\Big{)}\mid\sigma_{i}^{ct}\Big{]},$ (4)

where the expectation is over the conditional draw of $\sigma_{\neg i}^{ct}$
and $t$ given signals $\sigma_{i}^{ct}$. We then say that $s$ is an
equilibrium for firm $i$ if for every belief $\sigma_{i}^{ct}$ and every
possible action $p_{i}^{\prime}\in\\{0,1\\}$,

$\tilde{u}_{i}^{ct}(\sigma_{i}^{ct},s_{i}(\sigma_{i}^{ct}),s_{\neg
i})\geq\tilde{u}_{i}^{ct}(\sigma_{i}^{ct},p_{i}^{\prime},s_{\neg i}).$ (5)

Next, a contract $ct$ Pareto dominates contract $ct^{\prime}$ at
$(w_{1},w_{2})$ if there exists an equilibrium $s$ under $ct$ such that, for
every equilibrium $s^{\prime}$ under $ct^{\prime}$,

$u_{1}^{ct}(w_{1},s)\geq u_{1}^{ct^{\prime}}(w_{1},s^{\prime})~{}~{}~{}\mbox{
and }~{}~{}~{}u_{2}^{ct}(w_{2},s)\geq u_{2}^{ct^{\prime}}(w_{2},s^{\prime}).$
(6)

If at least one of the inequalities is strict then the Pareto dominance is
strict. Contract $ct$ Pareto dominates contract $ct^{\prime}$ if it Pareto
dominates $ct^{\prime}$ at every $(w_{1},w_{2})$, and in this case we write
$ct\succeq ct^{\prime}$. If $ct\succeq ct^{\prime}$ and $ct^{\prime}\succeq
ct$, we write $ct=ct^{\prime}$, and say that the two contracts are equivalent.
Contract $ct$ strictly Pareto dominates $ct^{\prime}$ if $ct\succeq
ct^{\prime}$ but $ct^{\prime}\not\succeq ct$, and in this case we write
$ct\succ ct^{\prime}$.

Contract $ct$ is individually rational (IR) at $(w_{1},w_{2})$ either if it is
the no-sharing contract (which we consider the default contract), or if $ct$
Pareto dominates the no-sharing contract at $(w_{1},w_{2})$. Contract $ct$ is
always IR if it is IR at every $(w_{1},w_{2})$, namely, $ct\succeq ns$.

Contract $ct$ is Pareto optimal if it is not Pareto dominated by any other
contract, Pareto-optimal IR (IRPO) if it is both Pareto optimal and always IR,
and uniquely IRPO if it is the only contract that is both Pareto optimal and
always IR.

We note that although our model is general, and can handle both mixed and pure
Bayesian equilibria, our results in Section 3 onwards are for pure Bayesian
equilibria.

##  3 Contracts for Prediction-Sharing with Unknown Correlation

In this section we focus on the first of two specific settings within our
framework. We assume firms have significant-action utilities and “infinite
data”. The latter assumption means that each firm’s prediction model is in
some sense an optimal classifier given the data features it is able to see. We
believe that this is the most natural assumption to closely approximate
massive data sets.555See also our analysis of a finite data case in Section
4.1. The main caveat is that neither firm knows the correlation between the
firms’ classifiers, even after learning its own classifier. We further assume
that the prediction accuracy of each firm’s classifier—formally,
$Pr_{\pi_{w}}[1|X=A]$ and $Pr_{\pi_{w}}[1|x=a]$—are common knowledge.666We use
$Pr[1]$ as shorthand for $Pr[t=1]$, and may omit the identifiers $\pi_{w},X,x$
when clear from context. Finally, for simplicity we assume that
$Pr[0]=Pr[1]=\frac{1}{2}$, and that the false-positive and false-negative
rates are the same for each of the firms:

$\alpha\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Pr[A|1]=Pr[B|0]~{}~{}~{}\mbox{
and }~{}~{}~{}\beta\stackrel{{\scriptstyle\mathrm{def}}}{{=}}Pr[a|1]=Pr[b|0].$
(7)

Still, the full joint distribution of the firms’ pair of signals together with
the true realizations under $w$ is unknown. As we see later in Section 3.2,
this is equivalent to both firms not knowing how the signals of the two firms
are correlated under the no-sharing contract, regardless of $(w_{1},w_{2})$.
Firms also do not know the true label of the outcome they are trying to
predict in the inference phase. We assume w.l.o.g. that
$\alpha\geq\beta\geq\frac{1}{2}$.

At one extreme, it is possible that the firms’ signals are independent. At the
other extreme, it is possible that they are fully correlated. In Appendix B we
show how this model can be formulated using our general model from Section 2.

###  3.1 Warm-Up: Known Correlation

We start our investigation with a simple model in which the correlation
between the firms is known. For an example in which firms’ signals are known
to be conditionally independent, see Figure 2.

Figure 2: Known correlation: An example of conditionally independent signals
with $\alpha=0.7,\beta=0.6$. With one possible world, both firms know the
joint distribution over true realizations and inference-time signals with
certainty.

When the precision accuracy of both firms is common knowledge, as we assume
throughout this section, then the correlation between the firms’ predictions
fully determines the joint distribution of the pair of signals under label
$t$. We show that formally in Claim 3.1. By correlation we mean the Pearson
correlation of the signals, namely

$\theta_{t}=\frac{Pr[X=A\land
x=a|t]-\alpha\beta}{\sqrt{\alpha(1-\alpha)\beta(1-\beta)}},$

where $t$ is the true label realization. Notice that the two Bernoulli
variables are the two firms’ signals given the true realization. For
simplicity, we assume that $\theta_{1}=\theta_{0}$, and denote the correlation
simply by $\theta$.

###### Claim

In the correlation model, knowing the correlation $\theta$ determines the
joint distribution of Firm1’s signal $A/B$, Firm2’s signal $a/b$, and the true
realization $0/1$.

###### Proof

To see that the correlation determines the joint distribution in our setting,
recall that for the Bernoulli variables in our settings, the Pearson
correlation under label $1$ satisfies $\theta=\frac{Pr[X=A\land
x=a|1]-\alpha\beta}{\sqrt{\alpha(1-\alpha)\beta(1-\beta)}}$. Thus, given
$\theta$, $\alpha$, and $\beta$ we have

$\begin{split}&Pr[X=A\land x=a|1]\\\
&=\sqrt{\alpha\beta}\left(\sqrt{\alpha\beta}+\theta\cdot\sqrt{(1-\alpha)(1-\beta)}\right).\end{split}$
(8)

This then determines $Pr[X=A\land x=b|1]=\alpha-Pr[X=A\land x=a|1],Pr[X=B\land
x=a|1]=\beta-Pr[X=A\land x=a|1]$, and $Pr[X=B\land x=b|1]=1-Pr[X=A\land
x=a|1]-Pr[X=A\land x=b|1]-Pr[X=B\land x=a|1]$. That is, it fully determines
the joint distribution. For example, when $\alpha=\beta$ and $\theta=0$ (i.e.,
the signals are conditionally independent), we have $Pr[X=A\land
x=a|1]=\alpha^{2}$, and when $\alpha=\beta$ and $\theta=1$, we have
$Pr[X=A\land x=a|1]=\alpha$. Finally, a symmetric argument holds under label
0.

When the correlation is known, there is no added value in sharing $w_{i}$,
since the world model $w$ is already known to both firms. Therefore, no-
sharing is equivalent to train-sharing, and infer-sharing is equivalent to
full-sharing. The only question is, which of these contracts, if any, is IRPO?

######  Lemma 1 ()

With known correlation and symmetric significant-action utilities, only no-
sharing and the equivalent train-sharing are IRPO. The unique equilibrium
under these contracts has two regimes: A high $\beta$ regime where both firms
play by their inference-time signals, and a low $\beta$ regime where Firm 2
“gives in” and always takes action $0$, while Firm 1 matches its action to its
inference-time signal.

This matches what we learned to expect in practice: Firms develop their own
classification models, and, assuming they are accurate enough, predict
according to them. In Section 3.2 we show, however, that once the correlation
is not known with certainty, this conclusion may change, and full-sharing or
train-sharing contracts may be uniquely IRPO.

We also note that the threshold that separates the high and low $\beta$ regime
is itself dependent on $\alpha$. The higher $\alpha$ is, the higher the
threshold for the high beta regime, where Firm 2 follows its prediction
signal. I.e., fixing Firm 2’s prediction accuracy $\beta$, the firm is more
likely to give in the higher Firm 1’s prediction accuracy $\alpha$ is.

Lemma 1 deals with symmetric significant-utilities. In the asymmetric case,
with a higher cost for a mistake in the significant action $C_{1}$, but not so
high as to prohibit ever taking a significant action altogether, the firms
would prefer full-sharing, which enables them to take the significant action
only when both receive positive signals. We show this in Appendix 0.A.

###  3.2 Unknown Correlation

So far we have considered known correlations. However, a more natural model is
that the correlation is unknown, and only some distribution over it is known.
As we will see, this model can give rise to train-sharing as uniquely IRPO.

We begin with some preliminary lemmas. First, we show that within the
specification of this subsection, full-sharing always Pareto dominates infer-
sharing.

######  Lemma 2 ()

For any distribution $\pi_{\theta}$ over correlations and any $R_{1}$ and
$C_{1}$, $fs\succeq is$.

###### Proof

In the correlation model, the private signal $w_{i}$ a firm gets during the
training phase does not impact its posterior regarding the correlation
$\theta$, which follows the distribution $\Theta$. Thus under infer-sharing,
where each firm $i$ only sees $w_{i}$, we can ignore it, and we have
$\sigma_{1}^{is}=\sigma_{2}^{is}=Xx$ for some pair of inference-time signal
$X,x$.

Thus, we can conclude that the infer-sharing equilibrium is symmetric between
the firms. That is since as we argue above, the posterior for both firms after
the training phase stays the same as the common prior. In the inference phase,
both firms share their signals, and so both firms end up with the exact same
information. Both firms’ equilibrium strategy is to predict $1$ if and only if

$\begin{split}&E_{\theta\sim\Theta}[Pr[1|X=x_{1},x=x_{2},\theta]\\\
&-C_{1}\cdot Pr[0|X=x_{1},x=x_{2},\theta]]\geq 0.\end{split}$

Under full-sharing, a similar argument shows that for every pair of signals
$Xx$ and correlation $\theta$ (which both firms learn during the training
phase), the symmetric equilibrium strategy is to predict $1$ if and only if
$Pr[1|X=x_{1},x=x_{2},\theta]-C_{1}\cdot Pr[0|X=x_{1},x=x_{2},\theta]$.

We can thus write, for the symmetric equilibrium strategies
$s\stackrel{{\scriptstyle def}}{{=}}s_{1}=s_{2}$ of the infer-sharing
contract,

$\begin{split}&u_{i}^{is}=E[u_{i}(s(X,x),t,s(X,x)]\\\
&=E_{\theta\sim\Theta}[E[u_{i}(s(X,x),t,s(X,x)|\theta]]\\\ &\leq
E_{\theta\sim\Theta}[E[\max_{s}u_{i}(s(X,x),t,s(X,x)|\theta]]=u_{i}^{fs}.\end{split}$

Next, we see that train-sharing and no-sharing contracts are equivalent under
sufficient symmetry.

######  Lemma 3 ()

If $R_{1}=C_{1}$ then $ts=ns$.

###### Proof

Suppose first that, under train-sharing, the firms follow the same equilibrium
strategies $s_{1},s_{2}$ for any realization $\theta\sim\Theta$. Then, it must
be that, under no-sharing, $s_{1},s_{2}$ is also an equilibrium: This is
immediate since the IC conditions of Equation 5 under no-sharing follow
immediately if the more granular IC conditions of the same equation under
train-sharing are satisfied.

Now, we know by Lemma 1 that for any fixed $\theta$ the equilibrium strategies
under train-sharing (which are the same as the equilibrium strategies for no-
sharing given we know that the correlation is $\theta$) depend only on the
values of $\alpha,\beta$, and so are independent of $\theta$. Thus, the same
equilibrium strategies are played for any $\theta$.

######  Lemma 4 ()

If $\alpha=\beta$, then for any distribution $\pi_{\theta}$ over correlations
and any $R_{1}$ and $C_{1}$, $fs\succeq ts$.

Finally, we can use the lemmas above to identify IRPO contracts. The two
theorems below show that, under sufficient symmetry, only full-sharing or no-
sharing are such contracts.

######  Theorem 3.1 ()

If $R_{1}=C_{1}$ then no-sharing is uniquely IRPO.

######  Theorem 3.2 ()

If $\alpha=\beta$ then full-sharing is either uniquely IRPO, or $fs=ns$ are
the only IRPO.

However, outside the symmetries in Theorems 3.1 and 3.2, train-sharing can
emerge as uniquely optimal.

######  Theorem 3.3 ()

Train-sharing is uniquely IRPO for an open subset of parameters
$\pi_{\theta}$, $\alpha$, $\beta$, $R_{1}$, $C_{1}$.

The intuition underlying the construction in the proof of Theorem 3.3 is the
following. Under no-sharing, the firms play the same equilibrium regardless of
their train-phase signals $w_{1}$ and $w_{2}$. Under train-sharing, the
equilibrium may depend on $w_{1}$ and $w_{2}$, and so in some cases may
improve both firms’ utilities relative to the no-sharing equilibrium. This
happens particularly when the firms learn that their signals are highly
correlated, which results in Firm 2 not taking a significant action (e.g., not
issue a loan). This saves Firm 2 from attaining negative utility, and allows
Firm 1 to fully exploit the utility of its predictions.

##  4 A “Two Hypotheses” Model

In Section 3 we showed that all contracts except for infer-sharing can be
uniquely optimal. In this section we complete the picture by describing a
setting where infer-sharing is uniquely optimal. We focus on the second
setting described in the introduction, which is summarized in Figure 1 of
Section 2. We assume that firms have “infinite data” and matching-
recommendations utilities: $C_{0}=C_{1}=0$ and $R_{0}=R_{1}=1$. This setting
captures a variety of natural circumstances, such as multi-factorial genetic
disease and chemical testing. Consider a genetic disease that only manifests
itself as a result of some environmental cause. There are two firms: Firm 1
performs genetic testing and knows (i) what genes cause the disease (ii) for a
specific person, whether these genes are present. Firm 2, on the other hand,
has users’ behavioral data (e.g., credit card histories) and can identify the
environmental cause. However, it does not understand the underlying genes that
enable the disease.

Formally, let $t=1$ denote the presence of the disease, inference-time signals
$A$ and $B$ denote the presence of two different gene mutations in the
population, and inference-time signals $a$ and $b$ denote the presence and
absence of the environment cause, respectively. There are two hypotheses: (I)
the disease is caused by mutation $A$ and the environmental cause, and (II)
the disease is caused by mutation $B$ and the environmental cause. Thus,
Hypothesis I (resp., Hypothesis II) is that firms see inference-time signals
$Aa$ (resp., $Ba$) if and only if $t=1$. The following are common knowledge:

  * –

Hypothesis I is correct with probability $\pi_{I}$, and Hypothesis II with
probability $1-\pi_{I}$;

  * –

without the environmental cause, the disease remains dormant ($Pr[0|b]=1$);

  * –

the incidence rate of the disease in the general population is $Pr[1]=\kappa$;
and

  * –

the incidence rates of the two different gene mutations in the general
population are $Pr[A]=\kappa+(1-\kappa)\cdot\lambda$ and $Pr[B]=1-Pr[A]$.

Our main result is that, within this setting, there are instances where infer-
sharing is uniquely optimal.

######  Theorem 4.1 ()

Infer-sharing is uniquely IRPO for an open subset of parameters $\pi_{I}$,
$\kappa$, $\lambda$, and $\mu$.

The intuition underlying the construction in the proof of Theorem 4.1 is the
following. Generally, in the two hypotheses model, Firm 1 has the ability to
deduce the correct world model during training, even with only its own signal.
In the cases we identify, train/full-sharing makes it lose this advantage, and
thus can not be beneficial for it. We are left with no/infer-sharing as
possible individually rational contracts. Since generally in the two
hypotheses model, the signal of Firm 1 by itself is not enough to decide the
user classification with certainty, infer-sharing helps in that Firm 1 can
both determine the correct hypothesis and has the pair of signals that
determines the true realization, and thus it always predicts correctly. In the
cases we identify, the behavior of Firm 2 remains the same under both
contracts, because of the fact that it can not deduce the correct world model
during training. Hence, infer-sharing allows Firm 1 a “free information meal”,
similar to the example, given for a model that only captures inference stage
sharing, without consideration of the training stage, in [15].

###  4.1 Beyond the Infinite-data Model

So far, we focused on the infinite-data model, where the training signal
allows the firm to deduce the marginal distribution over its signal and the
true realization. We conjecture that with enough data, the results are similar
to the idealized infinite case that we analyze. However, with few samples, the
results may change significantly. To demonstrate how the analysis may lead to
different results when there is only little historical data, we consider the
setting of Section 4, but when only one labeled example of past data is
available to the firms. Thus, after the hypothesis (world) is drawn
(Hypothesis $I$ w.p. $\pi_{I}$, and otherwise Hypothesis $II$), a sample is
drawn from the joint distribution over the pair of signals and true
realizations, and each firm sees its own signal and the true realization.
I.e., if the true hypothesis is Hypothesis $I$, then Firm 1 sees $(A,0)$ w.p.
$\alpha$, $(A,1)$ w.p. $\beta$, and $(B,1)$ w.p. $1-\alpha-\beta$. The firms
then update a Bayesian posterior over the world models. Under train-sharing
and full-sharing, when historical predictions are shared, both firms see the
entire sample, i.e., the pair of signals and the true realization.

In the appendix, we prove that, in the two hypotheses model with the
parameters used for Theorem 4.1 but with a single labeled example, the
statement of Theorem 4.1 breaks down, as do some of the properties of
equilibria derived in the theorem’s proof. In particular:

######  Theorem 4.2 ()

Under the parameters of Theorem 4.1 but with one sample, no-sharing and train-
sharing are not necessarily equivalent, no-sharing is IRPO (rather than infer-
sharing), and Firm 1 has lower equilibrium expected utility than Firm 2.

##  5 Implementation for a Real Data-set

To see how our ideas may be put to practice, we use the peer-to-peer loan data
of LendingClub, popularized by recent research such as [7], and publicly
available at Kaggle [18], to conduct a synthetic simulation. We take a random
subset of 25% of the features and assign it to Firm 1777In the Appendix, we
include robustness tests where we vary the choice of features.. We take
another subset of 10% of the features (possibly overlapping) and assign it to
Firm 2. Vertically, we split the data into train, test and validation sets. We
let each of the firms train a neural net over the training data (that includes
only its features). Each neural net was trained for 20 epochs on a $8$-GB RAM
M1 MacBook Pro, which takes about half an hour. The training signal consists
of the neural net’s predictions on whether loans are good or bad. The firms
use the test data to learn the signal performance, which we assume then
becomes common knowledge. Depending on the contract, the firms choose their
equilibrium strategies based on the performance in the test data: under train-
sharing and full-sharing they also see the other firm’s predictions on the
test data (rather than only knowing the aggregate performance measurements).
The firms then use their models to get a signal for every example in the
validation data. Under infer-sharing and full-sharing they see the other
firm’s signals on the validation set, and may use it to alter their final
actions. They are evaluated using their actions on the validation data, under
significant action utilities with $R_{1}=1$ and cost $C_{1}$.

Recall that our model assumes that firms share their training and inference-
time signals, and not their entire data. We thus compare the performance of
full-sharing—sharing of the firms’ signals both on the historical data (Here:
the test data) and the inference stage data (Here: the validation data)—with
total-sharing—sharing of the firms’ entire data, training a joint neural net
model over the shared data, and dividing the utility that this model achieves
on the validation data.

There are several important aspects in which the practical implementation
deviates from our formal model:

  * –

We do not naturally have a common Bayesian prior over the joint distribution
of signals and true realizations. We make the simplifying assumption that the
signals are independent when relevant, i.e., under no-sharing and infer-
sharing. Under train-sharing and full-sharing, we use the test data to learn
the signals’ correlations.

  * –

Simplifying symmetry assumptions—e.g., that the prediction accuracy is
symmetric across labels—do not naturally appear in the real data-set, and so
our decision rules need to adapt and do not exactly follow the ones given
symmetry.

  * –

The calculation of equilibrium strategies over the test data leads to an
empirical equilibrium that has some error when compared to a theoretical
equilibrium taken in expectation. Moreover, the average utility of different
contracts as calculated on the validation set may also have some error.

We find that the results generally follow the lines of our discussion in
Section 3: Varying by cost (going from $C_{1}=0$ to $C_{1}=2.5$ in $0.05$
steps), as summarized in Figure 4, we find regimes where either full-sharing,
no-sharing, or train-sharing are uniquely IRPO. While full-sharing is almost
always a Pareto optimal contract, there are significant regimes where it is
not IR for firm 1, which results in the no-sharing and train-sharing regimes.
In almost all instances and cost values of the simulation, infer-sharing is
Pareto dominated by full-sharing, as predicted by Lemma 2.

The behavior of no-sharing and train-sharing is of particular interest. With
low values of $C_{1}$, both contracts have the two firms issue a loan
regardless of the signal. Then, with higher values of $C_{1}$, the firms move
to an equilibrium where each acts according to its signal, and later to an
equilibrium where firm 1 predicts its signal while firm 2 does not issue any
loans. At each such equilibrium shift, there is a discontinuity for firm 1’s
utility. For example, moving from each firm predicting its own signal to Firm
2 not issuing loans, allows it to get the full utility of its action instead
of half.

Figure 3: No Sharing, Train Sharing and Full Sharing contracts performance for
both firms and different costs. We do not include the infer sharing contract
utility as they are very similar to (and dominated by) full sharing. We mark
regimes where each contract is the optimal-welfare IR contract.

Figure 4: Full Sharing vs. Total Sharing for different costs

In Figure 4, we compare the performance of full-sharing with total-sharing for
different costs. Importantly, both the full-sharing and total-sharing models
are trained once for the symmetric cost $C_{1}=1$, and are then adapted to
different costs by decision rules decided based on the test data. Somewhat
surprisingly, they achieve very comparable performance, with even a slight
advantage to full-sharing. We believe that this is due to the richer dual
signal in the full-sharing case, which allows for more granular decision
rules, mitigating the generally better prediction accuracy of total-sharing.

##  6 Discussion

The analysis of incentives is a crucial aspect of the general effort to
encourage data sharing, as recognized by the European Commission: “In spite of
the economic potential, data sharing between companies has not taken off at
sufficient scale. This is due to a lack of economic incentives (including the
fear of losing a competitive edge)” [8]. This paper introduces a novel element
of data sharing—the distinction between sharing during training and
inference—and demonstrates its importance to understanding firms’ data-sharing
incentives.

Some natural questions arise as a result of our work:

  * –

We have assumed a common prior over priors for the firms. What if the firms
have different beliefs? How robust is the emergence of uniquely optimal
contracts to small differences in the epistemic models of the firms?

  * –

Our work is set within the framework of mechanism design without money, i.e.,
we suppose that firms share data based on mutual gain, rather than based on
monetary compensation. In some cases it is natural to consider that one of the
firms may compensate the other as part of the data sharing process. This could
be interesting as future work and may build on the framework and insights we
develop.

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##  Appendix 0.A Known Correlation Results and Proofs

See 1

###### Proof

We first emphasize again that when the prior over priors consists of a single
possible prior, the first stage of learning is redundant, and so there is no
difference between “no-sharing” and “train-sharing”, and similarly no
difference between “infer-sharing” and “full-sharing”. Under no-sharing, a
strategy $s^{no}_{i}$ of firm $i\in\\{1,2\\}$ is, given signal $x$, whether to
predict $0$ or $1$. Under full-sharing, a strategy $s^{full}_{i}$ of firm $i$
is, given both firms’ signals $x_{1},x_{2}$, whether to predict $0$ or $1$. To
specify a strategy (or part of a strategy), we sometimes use the notation
$signal\rightarrow prediction$, e.g., under no sharing $A\rightarrow 0$ means
that the primary firm predicts $0$ when it gets the signal $A$.

Under full-sharing, there is a unique symmetric equilibrium where
$Aa\rightarrow 1,Ab\rightarrow 1,Ba\rightarrow 0,Bb\rightarrow 0$. I.e., both
firms follow the primary firm’s signal. This holds by the following argument.
Under full-sharing, we have $\sigma_{1}^{fs}=\sigma_{2}^{fs}$. Moreover, in
the known correlation case, $w_{i}$ can be ignored (as there is only one
possible world model) and $\sigma_{1}^{fs}$ is of the form $Xx$ for some pair
of inference-time signals $X\in\\{A,B\\}$ of Firm1 and $x\in\\{a,b\\}$ of
Firm2. Let $p_{i}=s_{i}(Xx)$ be the prediction of firm $i$ if the pair of
signals is $Xx$, and let $Q^{Xx}_{p_{i},s_{\neg i}}=\begin{cases}1&p_{i}\neq
s_{\neg i}(Xx)\\\ \frac{1}{2}&p_{i}=s_{\neg i}(Xx)\end{cases}$.

When $p_{i}=1$, Equation 4 takes the form:

$\begin{split}\tilde{u}_{i}^{fs}(\sigma_{i}^{fs},1,s_{\neg
i})&=E[u_{i}(1,t,s_{\neg i}(Xx))]\\\ &=Q^{Xx}_{1,s_{\neg i}}\cdot
E[u_{i}(1,t,0)]\\\ &=R_{1}\cdot Pr[t=1|Xx]+C_{1}\cdot Pr[t=0|Xx]\\\
&=Pr[t=1|Xx]-Pr[t=0|Xx],\end{split}$ (9)

where the first transition is since the signal that firm $\neg i$ sees is
fixed to be the same one that firm $i$ sees, and so the expectation is only
over the true realization $t$. The second transition is by the structure of
our utility model. The third transition is by conditional expectation. The
fourth transition is since in the symmetric significant action utility model
$R_{1}=1,C_{1}=-1$.

When $p_{i}=0$, Equation 4 takes the form:

$\begin{split}\tilde{u}_{i}^{fs}(\sigma_{i}^{fs},0,s_{\neg
i})&=E[u_{i}(0,t,s_{\neg i}(Xx))]\\\ &=Q^{Xx}_{0,s_{\neg i}}\cdot
E[u_{i}(0,t,0)]\\\ &=R_{0}\cdot Pr[t=1|Xx]+C_{0}\cdot Pr[t=0|Xx]\\\
&=0,\end{split}$ (10)

where the last transition is since in the significant action utility model
$R_{0}=C_{0}=0$. We conclude that to satisfy the equilibrium condition of
Equation 5, it suffices to show that for every pair of signals $Xx$, both
agents’ strategies choose $s_{i}(Xx)=1$ if and only if
$Pr[t=1|Xx]-Pr[t=0|Xx]\geq 0$.

Moreover, we know by Bayes’ formula and our symmetry assumption
($Pr[0]=Pr[1]$) that:

$Pr[1|Xx]-Pr[0|Xx]=\frac{\frac{1}{2}}{Pr[Xx]}\left(Pr[Xx|1]-Pr[Xx|0]\right),$

and so the left-hand-side expression is non-negative iff the right-hand-side
expression is non-negative.

Let $\rho\stackrel{{\scriptstyle def}}{{=}}Pr[X=A\land x=a|1]$. Then we have

$\begin{split}Pr[Aa|1]-Pr[Aa|0]=\rho-(1-\alpha-\beta+\rho)\stackrel{{\scriptstyle\alpha\geq\beta\geq\frac{1}{2}}}{{\geq}}0,\\\
\qquad
Pr[Ab|1]-Pr[Ab|0]=\left(\alpha-\rho\right)-\left(\beta-\rho\right)\stackrel{{\scriptstyle\alpha\geq\beta}}{{\geq}}0,\end{split}$

and since
$Pr[1|Bb]-Pr[0|Bb]=-\left(Pr[1|Aa]-Pr[0|Aa]\right),Pr[1|Ba]-Pr[0|Ba]=-\left(Pr[1|Ab]-Pr[0|Ab]\right),$
the inequalities are reversed for these signals. The utility for the primary
firm is thus, following Eq. 1,

$\begin{split}&u_{1}^{fs}(s_{1},s_{2})=E_{X,x,t}[u_{1}(s_{1}(Xx),t,s_{2}(Xx))]\\\
&=\sum_{x_{1}\in\\{A,B\\}}\sum_{x_{2}\in\\{a,b\\}}\big{(}Pr[X=x_{1}\land
x=x_{2}]\cdot Pr[1|X=x_{1}\land x=x_{2}]\cdot
u_{1}(s_{1}(x_{1},x_{2}),1,s_{2}(x_{1},x_{2}))\big{)}\\\ &+Pr[0|X=x_{1}\land
x=x_{2}]u_{1}(s_{1}(x_{1},x_{2}),0,s_{2}(x_{1},x_{2}))\big{)}\\\
&\stackrel{{\scriptstyle\text{Bayes
formula}}}{{=}}\sum_{x_{1}\in\\{A,B\\}}\sum_{x_{2}\in\\{a,b\\}}\big{(}Pr[1]\cdot
Pr[X=x_{1}\land x=x_{2}|1]\cdot
u_{1}(s_{1}(x_{1},x_{2}),1,s_{2}(x_{1},x_{2}))\\\ &+Pr[0]\cdot Pr[X=x_{1}\land
x=x_{2}|0]\cdot u_{1}(s_{1}(x_{1},x_{2}),1,s_{2}(x_{1},x_{2}))\big{)}]\\\
&=\frac{1}{2}\sum_{x_{2}\in\\{a,b\\}}\big{(}Pr[1]\cdot Pr[X=A\land
x=x_{2}|1]-Pr[0]\cdot Pr[X=A\land x=x_{2}|0]\big{)}\\\
&=\frac{1}{2}\left(Pr[1]\cdot Pr[X=A|1]-Pr[0]\cdot Pr[X=A|0]\right)\\\
&=\frac{1}{4}\left(\alpha-(1-\alpha)\right)=\frac{2\alpha-1}{4}.\end{split}$
(11)

Under no-sharing, there are two regimes. Consider if the primary firm plays
$A\rightarrow 1,B\rightarrow 0$. We argue that then, the secondary firm always
plays $b\rightarrow 0$, since its utility from $b\rightarrow 1$ is (following
Equation 4):

$\begin{split}\tilde{u}_{2}^{ns}(b,1,s_{1})&=E[u_{2}(1,t,s_{1}(X))|x=b]\\\
&=\frac{1}{2}\cdot Pr[t=1\land X=A|x=b]+Pr[t=1\land X=B|x=b]\\\
&-\frac{1}{2}\cdot Pr[t=0\land X=A|x=b]-Pr[t=0\land X=B|x=b]\\\
&\stackrel{{\scriptstyle\text{Bayes
Formula}}}{{=}}\frac{1}{Pr[x=b]}\bigg{(}\frac{1}{2}Pr[x=b\land t=1\land
X=A]+Pr[x=b\land t=1\land X=B]\\\ &-\frac{1}{2}\cdot Pr[x=b\land t=0\land
X=A]-Pr[x=b\land t=0\land X=B]\bigg{)}\\\
&=2\bigg{(}\frac{\alpha-\rho}{4}+\frac{1-\alpha-\beta+\rho}{2}-\frac{\rho}{2}-\frac{\beta-\rho}{4}\bigg{)}=\frac{1}{2}\left(2-2\rho-\alpha-3\beta\right)\\\
&\stackrel{{\scriptstyle\alpha\geq\beta\geq\frac{1}{2}}}{{\leq}}0=\tilde{u}_{2}^{ns}(b,0,s_{1})\end{split}$

As for the prediction given the signal $a$, the utility of Firm2 from
$a\rightarrow 1$ is

$\begin{split}\tilde{u}_{2}^{ns}(a,1,s_{1})&=E[u_{2}(1,t,s_{1}(X))|x=a]\\\
&=\frac{1}{Pr[x=a]}\big{(}\frac{1}{2}Pr[x=a\land t=1\land X=A]+Pr[x=a\land
t=1\land X=B]\\\ &-\frac{1}{2}\cdot Pr[x=a\land t=0\land X=A]-Pr[x=a\land
t=0\land X=B]\big{)}\\\
&=2\left(\frac{\rho}{4}+\frac{\beta-\rho}{2}-\frac{1+\rho-\alpha-\beta}{4}-\frac{\alpha-\rho}{2}\right)\\\
&=\frac{1}{2}\left(3\beta-1-\alpha\right)\end{split}$

Thus, the secondary firm best response is $a\rightarrow 1$ if and only if
$3\beta-\alpha-1\geq 1$. It is straightforward to verify that in both cases
the strategies then form a unique equilibrium.

In the case that the secondary firm plays $a\rightarrow 0$, the equilibrium
utility for the primary firm is (following Equation 1):

$\begin{split}u_{1}^{ns}&=E[u_{1}(s_{1}(X),t,0)]\\\ &=Pr[X=A\land
t=1]-Pr[X=A\land t=0]\\\
&=\frac{1}{2}\left(\alpha-(1-\alpha)\right)=\frac{2\alpha-1}{2}\geq\frac{2\alpha-1}{4}\stackrel{{\scriptstyle\text{Eq.~{}\ref{eq:u1_fs}}}}{{=}}u_{1}^{fs}.\end{split}$

In the case that the secondary firm plays $a\rightarrow 1$, the equilibrium
utility for the primary firm is

$\begin{split}u_{1}^{ns}&=E[u_{1}(s_{1}(X),t,s_{2}(x))]\\\
&=\frac{1}{2}Pr[X=A\land t=1\land x=a]+Pr[X=A\land t=1\land x=b]\\\
&-\frac{1}{2}Pr[X=A\land t=0\land x=a]-Pr[X=A\land t=0\land x=b]\\\
&=\frac{1}{2}\left(\frac{\rho}{2}+(\alpha-\rho)-\frac{1-\alpha-\beta+\rho}{2}-(\beta-\rho)\right)\\\
&=\frac{1}{4}\left(3\alpha-\beta-1\right)=\frac{2\alpha-1}{4}+\frac{1}{4}(\alpha-\beta)\geq\frac{2\alpha-1}{4}\stackrel{{\scriptstyle\text{Eq.~{}\ref{eq:u1_fs}}}}{{=}}u_{1}^{fs}.\end{split}$

###### Lemma 5

For any $\beta$ there is such $\alpha$ and a utility model (as defined by
$C_{0},C_{1},R_{0},R_{1}$) so that full-sharing is the unique feasible
contract with known independent correlation.

###### Proof

We give the idea of the construction. Consider asymmetric significant-action
utilities, and fix (normalize) $R_{1}=1$.

Under full-sharing, as we know by the proof of Lemma 1, the equilibrium
strategies $s_{1},s_{2}$ both have $s_{i}(Xx)$ for a pair of signals $Xx$ if
and only if $Pr[Xx|1]-C_{1}\cdot Pr[Xx|0]\geq 0$. The expression is monotone
in the pair of signals $Xx$: It is highest for $Aa$, lower for $Ab$, even
lower for $Ba$, and lowest for $Bb$. Thus, the equilibrium strategies are
$Aa\rightarrow 1$ (i.e., the firms predict $1$ upon seeing $Aa$, and $0$
otherwise), if and only if:

$\begin{split}&Pr[Aa|1]-C_{1}Pr[Aa|0]=\alpha\beta-C_{1}(1-\alpha)(1-\beta)\geq
0\\\
&Pr[Ab|1]-C_{1}Pr[Ab|0]=\alpha(1-\beta)-C_{1}\beta(1-\alpha)<0,\end{split}$

where we use the expressions for independent correlation probabilities.

Under no-sharing, we can follow the argument of Lemma 1 (but with a parametric
cost $C_{1}$) to derive inequalities that guarantee an equilibrium where the
firms predict according to their signals ($A\rightarrow 1,a\rightarrow 1$).
Lastly, given that these are the no-sharing equilibrium strategies, assume
that $u_{2}^{ns}\leq
u_{1}^{ns}=\frac{\alpha\beta}{2}+\alpha(1-\beta)-C_{1}\left(\frac{(1-\alpha)(1-\beta)}{2}+\beta(1-\alpha)\right)\leq\frac{1}{2}\left(\alpha\beta-
C_{1}(1-\alpha)(1-\beta)\right)=u_{1}^{fs}=u_{2}^{fs}$. If all these
conditions hold, then full-sharing is the unique IRPO contract. This holds,
for example, when $\alpha=0.9,\beta=0.85,C_{1}=2.5$. More generally, the
conditions hold for any $\beta$ when $\alpha=\beta$ and
$C_{1}=\frac{2\beta^{2}-2\beta-1}{2(\beta-1)(\beta+1)}$.

##  Appendix 0.B Proofs for Unknown Correlation Case

See 4

###### Proof

Under train-sharing, the agents are symmetric, as they share their signals at
the training phase, and have the same prediction accuracy at the inference
phase. Since the agents are symmetric, we can assume a symmetric equilibrium
is being played. Then, as we saw in the proof of Lemma 2, this means the
equilibrium strategy is to take a significant action (predict $1$) if and only
if the expected utility is positive given the correlation learnt in the
training phase (and without knowledge of the other firm’s signal, as we are
under train-sharing). Under full-sharing, the added information of the other
firm’s signal allows for more granular decisions when to take a significant
action, and so $fs\succeq ts$.

See 3.1

###### Proof

We know by Lemma 2 that $full-sharing\geq infer-sharing$ and by Lemma 3 that
$train-sharing=no-sharing$. Thus, the only possibly IRPO contracts are $no-
sharing$ or $full-sharing$. However, in the symmetric case, we know from the
proof of Lemma 1 that regardless of the correlation $\theta$, full-sharing
always results in a symmetric equilibrium $s,s$ where $Aa\rightarrow
1,Ab\rightarrow 1$ (and $0$ otherwise). In essence, this is since these are
the pairs of signals with probability $Pr[1|Xx]>\frac{1}{2}$. Under no-
sharing, also by the proof of Lemma 1, the equilibrium strategies
$s_{1},s_{2}$ do not depend on $\theta$, and $u_{1}^{ns}\geq u_{1}^{fs}$ for
any fixed $\theta$. Thus, the expected utility for the primary firm under
full-sharing and any fixed correlation is $\frac{2\alpha-1}{4}$, and since the
expression does not depend on $\theta$, we have
$u_{1}^{ns}=E_{\theta\sim\Theta}[E[u_{1}(s_{1}(X),t,s_{2}(x)|\theta]]\geq
E_{\theta\sim\Theta}[E[u_{1}(s(X,x),t,s(X,x)|\theta]]=u_{1}^{fs}$.

See 3.2

###### Proof

We consider both symmetric and asymmetric no-sharing equilibria. In a
symmetric no-sharing equilibrium $s_{1},s_{2}$, the symmetry means that
$(s_{1}(A)=1)\leftrightarrow(s_{2}(a)=1),(s_{1}(B)=1)\leftrightarrow(s_{2}(b)=1)$.
Under such a symmetric equilibrium, since $\alpha=\beta$, the utilities of the
firms are the same. The utilities of the firms are the same for full-sharing
as well, under the symmetric equilibrium $s,s$. The sum of utilities of the
firms satisfies

$\displaystyle u_{1}^{ns}+u_{2}^{ns}$
$\displaystyle=E[u_{1}(s_{1}(X),t,s_{2}(x))+u_{2}(s_{1}(X),t,s_{2}(x))]$
$\displaystyle=E_{\theta\sim\Theta}[E[u_{1}(s_{1}(X),t,s_{2}(x))+u_{2}(s_{1}(X),t,s_{2}(x))|\theta]]$
$\displaystyle=E_{\theta\sim\Theta}[E[1[s_{1}(X)=1\lor
s_{2}(x)=1]u_{1}(1,t,0)|\theta]]$
$\displaystyle=E_{\theta\sim\Theta}[E[1[s_{1}(X)=1\lor s_{2}(x)=1](R_{1}\cdot
Pr[t=1|Xx]$ $\displaystyle-C_{1}\cdot Pr[t=0|Xx])|\theta]]$ $\displaystyle\leq
E_{\theta\sim\Theta}[E[(R_{1}\cdot Pr[t=1|Xx]-C_{1}\cdot
Pr[t=0|Xx])^{+}|\theta]]$
$\displaystyle=E_{\theta\sim\Theta}[E[1[s(X,x,\theta)=1](R_{1}\cdot
Pr[t=1|Xx]$ $\displaystyle-C_{1}\cdot Pr[t=0|Xx])|\theta]]$
$\displaystyle=u_{1}^{fs}+u_{2}^{fs},$

where we use the notation $(x)^{+}=\max\\{x,0\\}$. We also use the fact that
full-sharing takes a significant action if and only if $R_{1}\cdot
Pr[t=1|X,x,\theta]-C_{1}\cdot Pr[t=0|Xx,\theta]$ holds, as we have seen in
previous proofs. Therefore, and since $u_{1}^{fs}=u_{2}^{fs}$, each of the
firms has at least as much expected utility under full-sharing than under no-
sharing, which yields the theorem statement.

As for asymmetric no-sharing equilibria, it can be directly calculated that
the only possible such equilibrium when $\alpha=\beta$ is where $s_{1}$ is
$A\rightarrow 1,B\rightarrow 0$, and $s_{2}$ always predicts $0$. Given this
equilibrium, we know that

$\begin{split}&E_{\theta\sim\Theta}[R_{1}\left(\frac{Pr[1|Aa,\theta]}{2}+Pr[1|Ba,\theta]\right)-C_{1}\cdot\left(\frac{[Pr[0|Aa,\theta]}{2}+Pr[0|Ba,\theta]\right)]<0.\end{split}$
(12)

by the equilibrium condition that determines $a\rightarrow 1$.

Thus,

$\begin{split}u_{1}^{ns}&=E_{\theta\sim\Theta}[R_{1}\left(Pr[1|Aa,\theta]+Pr[1|Ab,\theta]\right)-C_{1}\cdot\left([Pr[0|Aa,\theta]+Pr[0|Ab,\theta]\right)]\\\
&=E_{\theta\sim\Theta}[R_{1}\cdot\frac{Pr[1|Aa,\theta]}{2}-C_{1}\cdot\frac{Pr[0|Aa,\theta]}{2}]\\\
&+E_{\theta\sim\Theta}[R_{1}\cdot\frac{Pr[1|Aa,\theta]}{2}+Pr[1|Ab,\theta]-C_{1}\cdot\left(\frac{[Pr[0|Aa,\theta]}{2}+Pr[0|Ab,\theta]\right)]\\\
&\stackrel{{\scriptstyle\text{Eq.~{}\ref{eq:asymmetric_equilibrium_equal_precision_a}}}}{{<}}E_{\theta}[R_{1}\cdot\frac{Pr[1|Aa,\theta]}{2}-C_{1}\cdot\frac{Pr[0|Aa,\theta]}{2}]\\\
&\leq
E_{\theta}[\left(R_{1}\cdot\frac{Pr[1|Aa,\theta]}{2}-C_{1}\cdot\frac{Pr[0|Aa,\theta]}{2}\right)^{+}]\\\
&\leq u_{1}^{fs}.\end{split}$

See 3.3

###### Proof

We describe the general settings that yield our example, and then state such
$\pi_{\theta},\alpha,\beta,R_{1},C_{1}$ values that implement it. Consider two
possible worlds, with probability $w$ and $1-w$, respectively: In the first,
signals are independent, and in the second, signals are totally correlated,
i.e., $\theta$ is the maximal correlation possible between two firms with
prediction accuracies $\alpha,\beta$. Notice that since

$\begin{split}&\beta=Pr[x=a|1]\geq Pr[X=A\land x=a|1]\stackrel{{\scriptstyle
Eq.~{}\ref{eq:Aa1_formula}}}{{=}}\sqrt{\alpha\beta}\left(\sqrt{\alpha\beta}+\theta\cdot\sqrt{(1-\alpha)(1-\beta)}\right),\end{split}$

we have $\theta\leq\sqrt{\frac{\beta(1-\alpha)}{\alpha(1-\beta)}}$, and
$\theta$ is maximized when this inequality holds as an equality. When this
happens, we have $Pr[X=A\land x=a|1]=\beta,Pr[X=A\land
x=b|1]=\alpha-\beta,Pr[X=B\land x=a|1]=0,Pr[X=B\land x=b|1]=1-\alpha$, and the
mirror image of it conditional on $0$: $Pr[X=B\land x=B|0]=\beta,Pr[X=B\land
x=a|0]=\alpha-\beta,Pr[X=A\land x=b|0]=0,Pr[X=A\land x=A|0]=1-\alpha$. See
Figure 6 for an illustration of the case when $\alpha=0.7,\beta=0.6$.

For conciseness, in this proof we specify the strategies by what signals lead
to predicting $1$, and all other signals lead to predicting $0$. Now, consider
a case where the train-sharing equilibrium in the independent correlation case
is $A\rightarrow 1,a\rightarrow 1$, and in the total correlation case is
$A\rightarrow 1$, while under no-sharing the equilibrium is $A\rightarrow
1,a\rightarrow 1$. With probability $w$, $\theta=0$ and the equilibrium the
firms play under both regimes is the same and so are the utilities. But w.p.
$1-w$, the equilibrium played under the two regimes is not the same. This
benefits the secondary firm by the fact that this is the train-sharing
equilibrium: It means that given that $\theta=\theta_{max}$ and the primary
firm plays $A\rightarrow 1$, the secondary firm’s best response is to not take
a significant action given any signal. But this also benefits the primary
firm, since now it does not need to share its reward of $R_{1}\cdot
Pr[1|Aa,\theta_{max}]-C_{1}\cdot Pr[0|Aa,\theta_{max}]$ with the secondary
firm, and this expression is strictly positive as part of the equilibrium
condition for the primary firm. Overall, this establishes ($train-sharing>no-
sharing$).

It is then enough, in order for $train-sharing$ to be the unique feasible
contract, to require that it does not hold that $full-sharing>no-sharing$
(recall that by Lemma 2, $full-sharing\geq infer-sharing$). This holds
whenever the primary firm’s utility is larger under no-sharing than under
full-sharing.

We can require that the full-sharing equilibrium both in the independent case
and the totally correlated case is $Aa\rightarrow 1,Ab\rightarrow 1$, and this
implies

$u_{1}^{fs}=\frac{1}{4}\left(w\left(R_{1}\alpha-
C_{1}(1-\alpha)\right)+(1-w)\left(R_{1}\alpha-C_{1}(1-\beta)\right)\right).$

We also have:

$\begin{split}u_{1}^{ns}&=\frac{1}{2}\bigg{(}w\bigg{(}R_{1}\left(\frac{\alpha\beta}{2}+\alpha(1-\beta)\right)-C_{1}\left(\frac{(1-\alpha)(1-\beta)}{2}+(1-\alpha)\beta\right)\bigg{)}\\\
&+(1-w)\left(R_{1}\left(\frac{\beta}{2}+\alpha-\beta\right)-C_{1}\cdot\frac{1-\beta}{2}\right)\bigg{)},\end{split}$

and we impose the condition $u_{1}^{ns}>u_{1}^{fs}$.

Finally, we note that the open set
$0.72<\alpha<0.721,0.513<\beta<0.514,0.755<C_{1}<0.756,0.999<R_{1}<1.001,0.5<w<0.50001$
satisfies all the above conditions.

###  0.B.1 Representing the Correlation Model of Section 3 in our General
Framework of Section 2

First, we show how to describe our model of unknown correlations within the
framework of our general model for firms’ prediction-sharing. Consider the
following construction. Fix some $\alpha,\beta$ and correlation distribution
$\Theta$. We draw $\eta\sim\mathrm{Uniform}([0,1])$ and $\theta\sim\Theta$,
and describe how the two parameters determine a world $w_{\eta,\theta}$. In a
world $w_{\eta,\theta}$, the interval structure of firm 1 satisfies

$\begin{split}&A_{w}=\\\
&\begin{cases}\\{[\eta,\eta+\alpha]\times\\{1\\}\cup([0,1]\setminus[\eta,\eta+\alpha])\times\\{0\\}\\}\\\
\qquad\qquad\qquad\qquad\eta+\alpha\leq 1\\\
\\{[\eta,1]\cup[0,\eta+\alpha-1]\times\\{1\\}\\\
\cup([0,1]\setminus([\eta,\eta+\alpha]\cup[0,\eta+\alpha-1)))\times\\{0\\}\\}\\\
\qquad\qquad\qquad\qquad\eta+\alpha>1\\\ \end{cases}.\end{split}$

We let
$\rho=\sqrt{\alpha\beta}(\sqrt{\alpha\beta}+\theta\cdot\sqrt{(1-\alpha)(1-\beta)}$
as in Equation 8. Let

$\begin{split}&e_{w}=\\\
&\begin{cases}[\eta+\alpha-\rho,\eta+\alpha-\rho+\beta]\\\
\qquad\qquad\qquad\qquad\eta+\alpha-\rho+\beta\leq 1\\\
[\eta+\alpha-\rho,1]\cup[0,\beta-(1-(\eta+\alpha-\rho))]\\\
\qquad\qquad\qquad\qquad\eta+\alpha-\rho+\beta>1\land\eta+\alpha-\rho\leq 1\\\
[\eta+\alpha-\rho-1,\beta+\eta+\alpha-\rho-1]\\\
\qquad\qquad\qquad\qquad\eta+\alpha-\rho>1\land\eta+\alpha-\rho+\beta\leq 2\\\
[\eta+\alpha-\rho-1,1]\cup[0,\beta-(1-(\eta+\alpha-\rho-1))]\\\
\qquad\qquad\qquad\qquad\eta+\alpha-\rho>1\land\eta+\alpha-\rho+\beta>2\\\
\end{cases}.\end{split}$

The interval structure of firm 2 then satisfies
$a_{w}=\\{e_{w}\times\\{1\\}\cup([0,1]\setminus e_{w})\times\\{0\\}\\}$.

The main technical claim for the construction is that not only firm 1, but
also firm 2, gets a uniform draw over its intervals $a_{w}$. That is, if
$\mathrm{frac}(x)=x-\lfloor x\rfloor$ is the fractional part of a number $x$,
then $\mathrm{frac}(\eta+\alpha-\rho)\sim\mathrm{Uniform}([0,1])$ (up to a
measure zero adjustment at $0$ and $1$). To see this, observe that, for every
$\rho$, $\mathrm{frac}(\eta+\alpha-\rho)|\rho\sim\mathrm{Uniform}([0,1])$, and
so the probability density function
$\tilde{p}(\mathrm{frac}(\eta+\alpha+\rho))=E_{\rho\sim\rho(\Theta)}[\mathrm{frac}(\eta+\alpha+\rho)|\rho]=1$.

We can then verify that the firms’ posterior over the correlation between the
firms’ signals is $\Theta$ regardless of the training signal $w_{i}$—their own
interval structure—that they observe. As we show in Section 3.1, knowing
$Pr[X=A\land x=a|1]$ uniquely determines $\theta$, and vice-versa. In our
construction, any world has $Pr[X=A\land x=a|1]=\rho$, which is a 1-to-1
function of $\theta$. So, firm 1, upon learning $A_{w}$, knows that the
distribution over $\rho$ is determined by $\Theta$, and so concludes that the
distribution over correlations is $\Theta$. Similarly, firm 2, upon learning
$a_{w}$ (which is determined by $\eta+\alpha-\rho$), knows that $\eta$ is
drawn uniformly over $[0,1]$, and so by Bayes’ rule
$\tilde{p}(\rho|\eta+\alpha-\rho)=\tilde{p}(\eta+\alpha-\rho|\rho)\frac{\tilde{p}(\rho)}{\tilde{p}(\eta+\alpha-\rho)}=\tilde{p}(\rho)$,
where the last transition follows from the firms’ uniform prior over world
models.

To further illustrate this construction, let us use the example we repeatedly
use in our proofs, where the correlation (conditioned on the true realization)
between the firms’ signal is either $0$ (conditional independence) or the
maximal possible (total correlation). Figure 5 shows an intuitive but
incorrect way to model this case within our general framework. Figure 6 shows
a simple correct modelling using a finite number of worlds (which is not our
general construction). Figure 7 shows how we model this case using our general
construction.

Figure 5: Incorrect Modelling of the unknown correlation within the general
prediction-sharing framework. In this modelling, we have
$\alpha=0.7,\beta=0.6$, and there are two possible worlds, one appears w.p.
$z$, and results in independent signals $A,a$ and $B,b$ given the true
realization (whether $0$ or $1$). The other appears w.p. $1-z$ and results in
totally correlated signals. However, this modelling does not capture our model
of unknown correlation, since firm 2 can deduce the correlation based only on
its own information.  Figure 6: Correct Modelling of the unknown correlation
within the general prediction-sharing framework. In this modelling, we have
$\alpha=0.7,\beta=0.6$, and there are four possible worlds, appearing
respectively w.p. $z^{2},(1-z)z,(1-z)z,$ and $(1-z)^{2}$. The first and third
possible worlds result in independent signals $A,a$ and $B,b$ given the true
realization (whether $0$ or $1$). The second and fourth possible worlds result
in totally correlated signals. In this modelling, whatever interval structure
firm 1 sees, a Bayesian updating of the posterior would lead it to believe
that the correlation between the firms’ signal is independent w.p. $z$ and
totally correlated w.p. $1-z$. The same holds for firm 2.  Figure 7: General &
Correct Modelling of the unknown correlation within the general prediction-
sharing framework, using our general construction. In this modelling, we have
$\alpha=0.7,\beta=0.6$, and there are infinite possible worlds, drawn either
from the upper type (representing the independent signals case) w.p. $z$ or
the lower type (representing the totally correlated signals case) w.p. $1-z$,
and then the intervals as described in the figure are shifted cyclically with
an offset $\eta\sim\mathrm{Uniform}([0,1])$. In this modelling, whatever
interval structure firm 1 sees, a Bayesian updating of the posterior would
lead it to believe that the correlation between the firms’ signal is
independent w.p. $z$ and totally correlated w.p. $1-z$, and the same holds for
firm 2.

##  Appendix 0.C Two Hypotheses Model Proofs

Figure 8: For the reader’s convenience we include Figure 1 again here.

See 4.1

###### Proof

We describe conditions that yield the stated result.

(Conditions that yield $no-sharing=train-sharing$)

Consider

$(1-\kappa)\mu>2\kappa,\frac{1}{2}(1-\kappa)(1-\lambda-\mu)>\kappa>\frac{(1-\kappa)\lambda}{2}.$
(13)

Then,

Since $w_{1}$ (the signal that Firm1 receives regarding the true model) is
enough to determine the correct hypothesis with certainty, under train-sharing
both firms know the correct hypothesis given $\sigma_{i}^{ts}$. Let
$\sigma_{2}^{ts}=\\{w_{1},w_{2},a\\}$, i.e., Firm2 sees $w_{1},w_{2}$ in the
training phase and the signal $a$ in the inference phase. For any strategy
$s_{1}$ of Firm1, and train-phase signals $w_{1},w_{2}$, we have

$\begin{split}&\tilde{u}_{2}^{ts}(\\{w_{1},w_{2},a\\},0,s_{1})=E\Big{[}u_{2}\Big{(}0,t,s_{1}(\\{w_{1},w_{2},X\\})\Big{)}\mid\\{w_{1},w_{2},a\\}\Big{]}\\\
&\geq
E\Big{[}u_{2}\Big{(}0,t,0\Big{)}\mid\\{w_{1},w_{2},a\\}\Big{]}=\frac{1}{2}Pr[t=0|\\{w_{1},w_{2},a\\}]\\\
&=\frac{1}{2}Pr[t=0|a]=\frac{1}{2}\frac{(1-\kappa)\mu}{\kappa+(1-\kappa)\mu}\\\
&\stackrel{{\scriptstyle\text{Eq.~{}\ref{eq:thm4_ns=ts}}}}{{>}}\frac{\kappa}{\kappa+(1-\kappa)\mu}=Pr[t=1|\\{w_{1},w_{2},a\\}]=E\Big{[}u_{2}\Big{(}1,t,0\Big{)}\mid\\{w_{1},w_{2},a\\}\Big{]}\\\
&\geq
E\Big{[}u_{2}\Big{(}0,t,s_{1}(\\{w_{1},w_{2},X\\})\Big{)}\mid\\{w_{1},w_{2},a\\}\Big{]}\\\
&=\tilde{u}_{2}^{ts}(\\{w_{1},w_{2},a\\},1,s_{1}),\end{split}$

i.e., it is dominant for secondary firm to play $a\rightarrow 0$ regardless of
the hypothesis or the primary’s firm action. Since the signal $b$ always
coincides with true realization $0$, the secondary firm always plays
$b\rightarrow 0$ as well. But since predicting $0$ is the dominant strategy
for any training-phase signal under train-sharing, as we saw in previous
proofs, this is also the dominant strategy under $no-sharing$ for the
secondary firm.

As for the primary firm, because it has a different signal for the true
realization $1$ under each hypothesis, it can determine which hypothesis is
true both in the $no-sharing$ and $train-sharing$ contracts. Overall, this
means that the equilibrium for both contracts is that the secondary firm
always predicts $0$, and the primary firm, given that it finds that hypothesis
I is correct, predicts $A\rightarrow 1,B\rightarrow 0$, and given that it
finds that hypothesis II is correct, always predicts $0$ (by a direct
calculation of the conditional utilities for Firm1 and using Equation 13).

(Condition that prevents $full-sharing$ being always individually rational)

Consider

$\kappa>(1-\kappa)\lambda.$ (14)

Under full-sharing, both firms can determine the true hypothesis and given the
pair of signals can also follow up with predicting the correct true
realization with certainty. Thus, the expected utility under any
$w_{1},w_{2}$, for each firm, is $\frac{1}{2}$. To show that it does not hold
that full-sharing is always individually rational, it thus suffices to show
that $u_{1}^{ns}>\frac{1}{2}$ with some $(w_{1},w_{2})$. There are two
possible $w_{1}$ for the primary firm: Either that the true realization $1$
always coincides with $A$ (which happens if and only if Hypothesis I is true),
or that it always coincides with $B$. If the first holds, and by our
assumption in Equation 14, for the equilibrium strategies we saw in our
analysis of no-sharing,
$u_{1}^{ns}=\kappa+\frac{(1-\kappa)(1-\lambda)}{2}=\frac{2\kappa+1-\kappa-\lambda+\kappa\lambda}{2}=\frac{1+\kappa-(1-\kappa)\lambda}{2}>\frac{1}{2}=u_{1}^{fs}$.

(Conditions for $infer-sharing>no-sharing$)

We note that under infer-sharing, the primary firm can both determine the
correct hypothesis and has the pair of signals that determines the true
realization, and thus always predicts correctly. We wish to find a condition
so that the secondary firm has the same strategy as under $no-sharing$, to
always predict $0$. Under both hypotheses, the secondary firm should predict
$Ab,Bb\rightarrow 0$. As for signal $Aa$, the secondary firm predicts $0$, as
long as $(1-\pi_{I})(1-\kappa)\mu>\pi_{I}\kappa$. Similarly for signal $Ba$,
the secondary firm predicts $0$ as long as
$\pi_{I}(1-\kappa)\mu>(1-\pi_{I})\kappa$. When this holds, the primary firm
gains utility through its ability to differentiate between the $Ba$ and $Bb$
signals under hypothesis II. This “free meal” phenomenon is similar to the
example in the introduction of [15].

We wrap up by noting that all the above conditions are satisfied when
$0.2<\mu<0.38,0.05<\kappa<0.06,0.011<\lambda<0.02$, and $0.65<\pi_{I}<0.75$.

See 4.2

###### Proof

We consider additional conditions, on top of these of Theorem 4, that would
yield the stated result: An example where firm 2 has higher expected utility
in the IRPO contract than firm 1, and that all contracts besides no-sharing
are not individually rational.

(Under train-sharing) In the two hypotheses model with one sample, the pair of
signals $Aa$ and $Ba$, together with their true realization, determines with
certainty the true hypothesis. On the other hand, the signals $Ab$ or $Bb$
together with their true realization (that can only be $0$) maintains the same
posterior as the prior. Thus, we conclude that under the train-sharing and
full-sharing contracts, there is a probability of $\kappa+(1-\kappa)\mu$ that
both firms learn the true hypothesis, and otherwise the firms maintain their
prior.

The secondary firm always predicts $0$ (the dominance argument of Theorem 4.1
generalizes regardless of the training-phase signal), and the primary firm
predicts $A\rightarrow 1,B\rightarrow 0$ if it knows Hypothesis I is correct,
and always predicts $0$ if it knows Hypothesis II is correct. Otherwise, its
posterior is the same as the prior, and so its prediction for $A$ is $0$ if
and only if $\frac{(1-\kappa)(\lambda+(1-\pi_{I})\mu)}{2}\geq\pi_{I}\kappa$,
and its prediction for $B$ is $0$ if and only if
$\frac{(1-\kappa)(1-\lambda-(1-\pi_{I})\mu)}{2}\geq(1-\pi_{I})\kappa$. For the
parameters used in Theorem 4.1, this results in $A\rightarrow 1,B\rightarrow
0$. The expected utility of the primary firm is thus

$\begin{split}&Pr[HypothesisI]\cdot\left(Pr[A\land
1|HypothesisI]+\frac{1}{2}Pr[B\land 0]\right)\\\
&+Pr[HypothesisII]\bigg{(}Pr[s_{1}=A\rightarrow 0,B\rightarrow
0|HypothesisII]\cdot\frac{1}{2}Pr[0|HypothesisII]\\\ &+Pr[s_{1}=A\rightarrow
1,B\rightarrow 0|HypothesisII]\cdot\left(\frac{1}{2}Pr[B\land
0|HypothesisII]\right)\bigg{)}\\\
&=\pi_{I}\cdot(\kappa+\frac{(1-\kappa)(1-\lambda)}{2})+(1-\pi_{I})((\kappa+(1-\kappa)\mu)\cdot\frac{1-\kappa}{2}+(1-\kappa)(1-\mu)\cdot\frac{(1-\kappa)(1-\lambda-\mu)}{2}),\end{split}$

and the utility of the secondary firm is

$\begin{split}&Pr[HypothesisI]\cdot\left(Pr[A\land
0|HypothesisI]+\frac{1}{2}Pr[B\land 0]\right)\\\
&+Pr[HypothesisII]\bigg{(}Pr[s_{1}=A\rightarrow 0,B\rightarrow
0|HypothesisII]\cdot\frac{1}{2}Pr[0|HypothesisII]\\\ &+Pr[s_{1}=A\rightarrow
1,B\rightarrow 0|HypothesisII]\cdot\left(Pr[A\land
0|HypothesisII]\cdot\frac{1}{2}Pr[B\land 0|HypothesisII]\right)\bigg{)}\\\
&=\pi_{I}\cdot((1-\kappa)\lambda+\frac{(1-\kappa)(1-\lambda)}{2})+(1-\pi_{I})((\kappa+(1-\kappa)\mu)\cdot\frac{1-\kappa}{2}\\\
&+(1-\kappa)(1-\mu)\cdot\frac{(1-\kappa)(\lambda+\mu)+(1-\kappa)(1-\lambda-\mu)}{2}).\end{split}$

(Under no-sharing)

The secondary firm always predicts $0$. Thus, the strategy of the primary firm
follows a direct application of Equation 5 where we have the strategy
$s_{2}=0$ of firm 2.

The primary firm knows Hypothesis I is correct with certainty when it sees
$(A,1)$ (which happens w.p. $\pi_{I}\kappa$), and then predicts $A\rightarrow
1,B\rightarrow 0$. It knows Hypothesis II is correct with certainty when it
sees $(B,1)$ (which happens w.p. $(1-\pi_{I})\kappa$) and then always predicts
$0$. For $(A,0)$, we have
$Pr[(A,0)|HypothesisI]=(1-\kappa)\lambda,Pr[(A,0)|HypothesisII]=(1-\kappa)(\lambda+\mu)$,
and so
$w^{\prime}=Pr[HypothesisI|(A,0)]=Pr[(A,0)|HypothesisI]\frac{Pr[HypothesisI]}{Pr[(A,0)]}=(1-\kappa)\lambda\cdot\frac{\pi_{I}}{\pi_{I}(1-\kappa)\lambda+(1-\pi_{I})(1-\kappa)(\lambda+\mu)}$.
Similarly, for $(B,0)$, we have
$w^{\prime\prime}=Pr[HypothesisI|(B,0)]=\frac{(1-\kappa)(1-\lambda)\pi_{I}}{(1-\kappa)(1-\lambda)\pi_{I}+(1-\pi_{I})(1-\kappa)(1-\lambda-\mu)}$.
We conclude (similarly to as we did in the train-sharing case for the original
prior) that if the primary firm sees $(A,0)$ it always predicts $0$, and if
the primary firm sees $(B,0)$, it predicts $A\rightarrow 1,B\rightarrow 0$.
The primary firm’s expected utility is then

$\begin{split}&Pr[HypothesisI]\cdot\bigg{(}Pr[(B,0)\lor(A,1)|HypothesisI]\cdot\bigg{(}Pr[A\land
1|HypothesisI]+\frac{1}{2}Pr[B\land 0]\bigg{)}\\\
&+Pr[(A,0)|HypothesisI]\cdot\left(\frac{1}{2}Pr[0|HypothesisI]\right)\bigg{)}\\\
&+Pr[HypothesisII]\cdot\bigg{(}Pr[(B,1)\lor(A,0)|HypothesisII]\cdot\frac{1}{2}Pr[0|HypothesisII]\\\
&+Pr[(B,0)|HypothesisII]\cdot\left(\frac{1}{2}Pr[B\land
0|HypothesisII]\right)\bigg{)}\\\
&=\pi_{I}\cdot\bigg{(}(1-(1-\kappa)\lambda)(\kappa+\frac{(1-\kappa)(1-\lambda)}{2})+(1-\kappa)\lambda\frac{1-\kappa}{2}\bigg{)}\\\
&+(1-\pi_{I})\cdot\bigg{(}(1-(1-\kappa)(1-\lambda-\mu))\frac{1-\kappa}{2}\\\
&+(1-\kappa)(1-\lambda-\mu)\left(\frac{(1-\kappa)(1-\lambda-\mu)}{2}\right),\end{split}$

and firm2’s expected utility is

$\begin{split}&Pr[HypothesisI]\cdot\bigg{(}Pr[(B,0)\lor(A,1)|HypothesisI]\cdot\bigg{(}Pr[A\land
0|HypothesisI]+\frac{1}{2}Pr[B\land 0]\bigg{)}\\\
&+Pr[(A,0)|HypothesisI]\cdot\left(\frac{1}{2}Pr[0|HypothesisI]\right)\bigg{)}\\\
&+Pr[HypothesisII]\cdot\bigg{(}Pr[(B,1)\lor(A,0)|HypothesisII]\cdot\frac{1}{2}Pr[0|HypothesisII]\\\
&+Pr[(B,0)|HypothesisII]\cdot\bigg{(}Pr[A\land
0|HypothesisII]+\frac{1}{2}Pr[B\land 0|HypothesisII]\bigg{)}\bigg{)}\\\
&=\pi_{I}\cdot\bigg{(}(1-(1-\kappa)\lambda)((1-\kappa)\lambda+\frac{(1-\kappa)(1-\lambda)}{2})+(1-\kappa)\lambda\frac{1-\kappa}{2}\bigg{)}\\\
&+(1-\pi_{I})\cdot\bigg{(}(1-(1-\kappa)(1-\lambda-\mu))\frac{1-\kappa}{2}\\\
&+(1-\kappa)(1-\lambda-\mu)\left((1-\kappa)(\lambda+\mu)+\frac{(1-\kappa)(1-\lambda-\mu)}{2}\right).\end{split}$

(Under infer-sharing)

The Bayesian updating phase based on the historical sample is the same as in
the no-sharing case, and so the primary firm attains the various posteriors
under the same probabilities. The secondary firm, regardless on the sample,
has the same posterior as the prior, as the two hypotheses look the same for
its signal structure. For the pair of signals $Ab$ and $Bb$, both firms always
predict $0$. For the signal $Ba$, since
$\frac{\pi_{I}(1-\kappa)\mu}{2}>(1-\pi_{I})\kappa$, it is dominant for the
secondary firm to predict $0$ regardless of how the primary firm predicts, and
similarly for $Aa$, since $\frac{(1-\pi_{I})(1-\kappa)\mu}{2}>\pi_{I}\kappa$,
it is also dominant for the secondary firm to predict $0$.

For the signal $Ba$, since the secondary firm always predicts $0$, the primary
firm predicts $1$ for it when its posterior is $w^{\prime}$, or when it knows
with certainty that Hypothesis II is correct. It predicts $0$ for it when its
posterior is $w^{\prime\prime}$, since

$w^{\prime\prime}\frac{(1-\kappa)\mu}{2}>(1-w^{\prime\prime})\kappa,$

and also when it knows with certainty that Hypothesis I is correct.

For the signal $Aa$, since the secondary firm always predicts $0$, the primary
firm predicts $0$ for it when its posterior is $w^{\prime}$, or when it knows
with certainty that Hypothesis II is correct. It predicts $1$ for it when its
posterior is $w^{\prime\prime}$, since

$w^{\prime\prime}\kappa>(1-w^{\prime\prime})\frac{(1-\kappa)\mu}{2},$

and also when it knows with certainty that Hypothesis I is correct.

The primary firm’s utility is then:

$\begin{split}&Pr[HypothesisI]\bigg{(}Pr[(B,0)\lor(A,1)|HypothesisI]\cdot(Pr[1|HypothesisI]+\frac{1}{2}Pr[0|HypothesisI])\\\
&+Pr[(A,0)|HypothesisI](\frac{1}{2}Pr[0\land b|HypothesisI])\bigg{)}\\\
&+Pr[HypothesisII]\bigg{(}Pr[(B,0)|HypothesisII]\cdot(\frac{1}{2}Pr[0\land
b|HypothesisII])\\\
&+Pr[(A,0)\lor(B,1)|HypothesisII](\frac{1}{2}Pr[0|HypothesisII]+Pr[1|HypothesisII])\bigg{)}\\\
&=\pi_{I}\bigg{(}(1-(1-\kappa)\lambda)(\kappa+\frac{1-\kappa}{2})+(1-\kappa)\lambda\frac{(1-\kappa)(1-\mu)}{2}\bigg{)}\\\
&+(1-\pi_{I})\bigg{(}(1-\kappa)(1-\lambda-\mu)\frac{(1-\kappa)(1-\mu)}{2}\\\
&+(1-(1-\kappa)(1-\lambda-\mu))(\frac{1-\kappa}{2}+\kappa)\bigg{)}.\end{split}$

The utility of the secondary firm is:

$\begin{split}&Pr[HypothesisI]\bigg{(}Pr[(B,0)\lor(A,1)|HypothesisI]\cdot(\frac{1}{2}Pr[0|HypothesisI])\\\
&+Pr[(A,0)|HypothesisI](\frac{1}{2}Pr[0\land b|HypothesisI]+Pr[0\land
a|HypothesisI])\bigg{)}\\\
&+Pr[HypothesisII]\bigg{(}Pr[(B,0)|HypothesisII]\cdot(\frac{1}{2}Pr[0\land
b|HypothesisII]\\\ &+Pr[0\land
a|HypothesisII])+Pr[(A,0)\lor(B,1)|HypothesisII]\cdot(\frac{1}{2}Pr[0|HypothesisII])\bigg{)}\\\
&=\pi_{I}\bigg{(}(1-(1-\kappa)\lambda)(\frac{1-\kappa}{2})+(1-\kappa)\lambda(\frac{(1-\kappa)(1-\mu)}{2}+(1-\kappa)\mu)\bigg{)}\\\
&+(1-\pi_{I})\bigg{(}(1-\kappa)(1-\lambda-\mu)(\frac{(1-\kappa)(1-\mu)}{2}+(1-\kappa)\mu)\\\
&+(1-(1-\kappa)(1-\lambda-\mu))(\frac{1-\kappa}{2})\bigg{)}.\end{split}$

(Under full-sharing) The Bayesian updating phase based on the historical
sample is the same as in the train-sharing case, and so both firms (that see
the full single sample together) either know the hypothesis with certainty, or
maintain the prior $\pi_{I}$.

Given that the firms maintain the prior $\pi_{I}$, upon seeing the signals
$Ab$ or $Bb$ they both predict $0$. If they see the signal $Ba$, since

$\pi_{I}\frac{(1-\kappa)\mu}{2}>(1-\pi_{I})\kappa,$

both firms predict $0$. If they see the signal $Aa$, since

$(1-\pi_{I})\frac{(1-\kappa)\mu}{2}>\pi_{I}\kappa,$

both firms predict $0$.

We thus have that the utility of firm 2 is, if we denote

$p^{*}=Pr[(Aa,1)\lor(Ba,1)\lor(Aa,0)\lor(Ba,0)]=\kappa+(1-\kappa)\mu,$

is

$\frac{1}{2}(p^{*}+(1-p^{*})(1-\kappa))=\frac{1}{2}(\kappa(\kappa+(1-\kappa)\mu-1)+1).$

Finally, we note that the parameter choice
$\kappa=\frac{5}{32},\lambda=\frac{1}{8},\mu=\frac{1}{2},\pi_{I}=\frac{1}{2}$
satisfies both the conditions detailed above and the conditions detailed in
the proof of Theorem 4.1.

##  Appendix 0.D Robustness of Our Results in Section 5

We wish to deepen our empirical results of Section 5. In particular, we wish
to add more variability into the feature selection process. For this purpose,
we introduce the following sampling method. First, for every
$\epsilon\in\\{1,0.85,0.7,0.55,0.4,0.25\\}$, we randomly sample $\epsilon$ of
the data-set features. Denote the resulting partial set of features $X$. Next,
we randomly sample $\frac{0.25}{\epsilon}$ of the features in $X$ to decide
the features for Firm1. Similarly, we sample $\frac{0.1}{\epsilon}$ of the
features in $X$ to decide the features for Firm2. The end result is, as in
Section 5, that Firm1 sees $25\%$ random features out of the original, and
Firm2 sees $10\%$. However, one could hope that the features (and so, the
resulting models after training) will be more correlated as $\epsilon$ is
smaller, since an overlap of the features the firms see is more likely. For
each value of $\epsilon$, we run $32$ random experiments, where we repeat the
above process, followed by the analysis described in Section 5. Overall this
results in $32\cdot 7=224$ experiments where we train a random model for each
of the firms.

We then quantify the emergence of optimal contracts in the following way. For
each cost in the range $0.5$ to $1.5$ in $0.05$ steps, we find what are the
optimal contracts (in the IRPO sense). We break equivalencies in favor of the
more “natural” contracts, i.e., if any contract has exactly the same expected
utilities as no-sharing, we would not consider it optimal, and if train-
sharing or infer-sharing are equivalent to full-sharing, we would similarly
not consider them optimal. If by the end of this process we have more than one
optimal contract for a specific cost, we divide the ‘benefit’ between all the
optimal contracts. Overall, for each experiment, we get a score for each
contract of the frequency it is optimal. Since we are interested in seeing the
possible influence of correlation on optimality, we directly calculate the
correlation of Firm1’s model and Firm2’s model predictions over the validation
set, using the Matthews correlation for a confusion matrix. In this way, we
can directly compare the two variables we are interested in, instead of using
the indirect $\epsilon$ parameter we use as part of our process, which we only
expect to have a probabilistic negative connection with the correlation (i.e.,
as $\epsilon$ higher, we could expect lower correlation).

We present our results in Figure 9. We also provide a smoothed presentation of
our results in Figure 10, where for every $0.1$ range of the correlation that
appeared in the experiments (i.e., $-0.1$ to $0$, $0$ to $0.1$, and so on, up
to $0.8$ to $0.9$), we average the correlation that appear in the range, and
the frequencies associated with them.

Figure 9: Contract Optimality vs. Firm Models’ Correlation  Figure 10:
Smoothed Contract Optimality vs. Firm Models’ Correlation

Overall, the results reiterate the findings we detail in Section 5. A few
surprising aspects to notice are the following:

  * –

Infer-sharing appears as a unique IRPO contract, contrary to our prediction.
We believe that this is likely due to the full-sharing empirical decision that
is based on the test data, where infer-sharing turns out to be better on the
validation data.

  * –

Train-sharing, which our example shows to emerge with high correlation rates,
appears mostly with lower (or negative) correlations. It is reasonable,
however, to expect that with negative correlation rates there could be
examples of it as well. This is interesting as an indication of where train-
sharing might be most relevant.

  * –

In our implementation, the decision rules for no-sharing and infer-sharing are
determined based on an assumption of independence (i.e. no correlation). We
would thus expect the contracts to be more frequently optimal when this
assumption is justified (low correlation rates). However, no-sharing is found
to be somewhat more frequently optimal when the correlation is high.