# The Texas Shoot-Out under Knightian Uncertainty

Gerrit Bauch Center for Mathematical Economics, Bielefeld University.
Financial support through the German Research Foundation Grant Ri-1128-9-1,
the German Academic Exchange Service and the Bielefeld Graduate School of
Economics is gratefully acknowledged. For detailed comments and suggestions on
the working paper version I would like to thank Roland Stauber. I am grateful
for the hospitality of the Economics Department at the University of Arizona
and helpful comments of their members as well as for fruitful discussions with
participants of the Stony Brook conference on Game theory 2022 and the SAET
2022. Corresponding e-mail address<EMAIL_ADDRESS>Frank Riedel
Center for Mathematical Economics, Bielefeld University and School of
Economics, University of Johannesburg. Financial support through the German
Research Foundation Grant Ri-1128-9-1 is gratefully acknowledged.

( August 27, 2024)

###### Abstract

We investigate the allocation of a co-owned company to a single owner using
the Texas Shoot-Out mechanism with private valuations. We identify Knightian
Uncertainty about the peer’s distribution as a reason for its deterrent effect
of a premature dissolving. Modeling uncertainty by a distribution band around
a reference distribution $F$, we derive the optimal price announcement for an
ambiguity averse divider. The divider hedges against uncertainty for
valuations close to the median of $F$, while extracting expected surplus for
high and low valuations. The outcome of the mechanism is efficient for
valuations around the median. A risk neutral co-owner prefers to be the
chooser, even strictly so for any valuation under low levels of uncertainty
and for extreme valuations under high levels of uncertainty. If valuations are
believed to be close, less uncertainty is required for the mechanism to always
be efficient and reduce premature dissolvements.

Key words and phrases: Knightian Uncertainty in Games, Texas Shoot-Out,
Partnership Dissolution
JEL subject classification: C72, D74, D81, D82

##  1 Introduction

Even the friendliest relationships can eventually turn sour. Co-owned
companies are no exception, being at risk of feuding partners. In the event
that co-owners are not able to cooperatively lead the company effectively
anymore, it is reasonable to no longer sustain the partnership. There are two
main kinds of solution concepts available to govern the dissolution of a
partnership. On the one hand, _external solutions_ require a third party or
stock market to serve as an outside option, e.g. a sale of the company to a
prospective buyer or a liquidation in order to distribute asset shares can
terminate a relationship and pay off the co-owners. However, those options are
often not desirable since a liquidation goes along with a loss of jobs, pay-
off of the business’s debts and far reaching tax consequences while a new
owner of the company may not be available on short notice. Furthermore, any
external exit solution suffers from possible loss in market value of the
company due to the publicly observed hostility within. Even more, private
valuations of the co-owners are not taken into account who still might want to
lead the company even without a partner. On the other hand, _internal
solutions_ can mitigate the aforementioned drawbacks, while also taking into
account the co-owners valuation for the company. Writing any such _exit
mechanism_ into the buy-sell contract when founding a business allows an
internal solution to be immediately available independent of whether or not an
external outside option is at hand.

For a two party co-owned company, a frequently used exit mechanism is called
the _Texas Shoot-Out_. Many consulting platforms111See e.g.,
ClaytonCapitalPartners or ExitPlanningSolutions at
https://claytoncapitalpartners.com/navigator/issue66-biz_continuity_part3.html
and https://www.exitplanning.com/blog/texas-shootout. recommend it for its
simplicity and effect as a deterrent to a premature dissolving. The Texas
Shoot-Out works as follows: Any partner has the right to initiate the exit
mechanism at any time she desires, becoming the so-called ”divider”. The
divider commits to a price $p$ for the sole ownership of the company. Her
partner, called the ”chooser” now has exactly two options. Either, she buys
the ownership and interest of the divider at price $p$ or sells hers at price
$p$. In the end, only one partner remains as the sole owner of the business,
having compensated the former co-owner. Although this mechanism is independent
of actual shares of the co-owners, it is typically recommended for equally
sized shareholders.

While simple, the Texas Shoot-Out is notorious for its deterrent effect on
feuding partners: Neither partner knows in which role they will eventually
find themselves, what price they might face in case the co-owner initiated the
exit mechanism or what choice to expect of her if they trigger the mechanism
themselves. The so created uncertainty leads to a very purposeful execution of
the mechanism and thus encourages conciliation in times of minor conflicts
while at the same time offering a promptly available tool in order to solve a
dispute if the partnership reached a dead end.

The focus of this article is to precisely characterize the deterrent component
of the Texas Shoot-Out. Since there is no objective information about a peer’s
valuation, it is natural to assume that co-owners face imprecise probabilistic
information about the valuation of their peers. We model this _Knightian
Uncertainty_ by means of a set of priors about the distribution function of
the co-owner’s valuation. We analyze such a prior set in the form of
_distribution bands_ since they allow for more flexibility than uncertainty
about the valuation alone, being able to incorporate bounds for main
characteristics of the distribution, such as its expectation or variance. More
precisely, a co-owner is willing to entertain the belief that the partner’s
valuation is close to a reference distribution $F$, but prefers a robust
approach to the uncertainty about the exact distribution. The proposed
framework thus models the Texas Shoot-Out in between the Bayesian (single
prior) setting and the full uncertainty (all priors) case explaining what
happens for intermediate levels of Knightian Uncertainty about the co-owner’s
distribution. Requiring mainly strict quasiconcavity of the induced payoff
function, our main finding is the astonishing link that connects those two.
The optimal price announcement of a divider is an increasing function in her
own valuation with two kinks left and right from the median of $F$. In
between, she announces half her valuation, thus hedging herself against
uncertainty. For low (resp. high) valuations she plays as if facing the
Bayesian setting induced by the distribution function being stochastically
dominated by (resp. stochastically dominating) all other distributions in her
prior set. Increasing the level of uncertainty about the co-owner’s
distribution also increases the interval around the median that corresponds to
a full-hedging strategy. This completely pins down the continuous
transformation from the Bayesian case to the one of full uncertainty: Hedging
behavior starts at the median of $F$ and spreads to more and more extreme low
and high valuations continuously with an increase in uncertainty. As long as
uncertainty is not too high, dividers with very low and high valuations make
strategic price announcements. E.g., for low valuations, the divider states a
price exceeding half her valuation, expecting the chooser to still accept the
offer and thus generating a revenue for herself.

Despite allowing for a robust and thus more flexible and realistic approach to
partnership dissolvements, introducing Knightian Uncertainty to the Texas
Shoot-Out elicits not only desirable properties but also explains its
notorious deterrent effect for comparative statics analyses.
Efficiency of the mechanism, in the sense of giving the company to the co-
owner with highest valuation, is increased by a boost to uncertainty: More
valuation profiles will lead to an efficient allocation compared to the
classical Bayesian setting. Especially, a precisely determined range of
valuations of a divider close to the median of $F$ can be guaranteed to
produce efficient allocations whereas the Bayesian setting could only assure
this for a single valuation. This range is the larger the higher the level of
uncertainty about the chooser’s valuation.
Incentives for a profitable selfish exit are diminished the higher the
uncertainty no matter the role a co-owner finds herself in, especially for the
initiator (divider): Fixing a valuation for the company, the interim worst-
case expected utility for a chooser is always higher than for a divider. This
preference is always strict for low levels of uncertainty and stays strict
only for very low and high valuations if uncertainty is increased until it is
too high and both are indifferent. This finding explains why the Texas Shoot-
Out is preventing a premature selfish end of the partnership: Only co-owners
with an extreme low or high valuation expect a revenue exceeding half their
valuation from initiating the exit mechanism. However, they thus become the
divider and suffer from a lower interim expected payoff than the other co-
owner. Hence, the more uncertainty, the fewer types will execute the Texas
Shoot-Out without good reason to do so and expect to get a utility close to
half their valuation. At the same time, the Texas Shoot-Out does offer a fair
way out of a dead end since every co-owner expects to get no less than half
her valuation.
Efficiency and deterrence of the Texas Shoot-Out do not require complete
ignorance of the partner’s valuation. It seems likely that over the course of
the partnership, co-owners share a similar (objective) belief about the
company’s value. In other words, a co-owner suspects the peer’s valuation to
be close or correlated to her own. Surprisingly, this additional intelligence
will decrease a co-owner’s incentive for a selfish exit even for moderate
amounts of uncertainty and would lead again to more efficiency and
conservative price announcements if the Texas Shoot-Out was triggered.

Our contribution contrasts two standpoints on the Texas Shoot-Out in the
literature by connecting them. In the well-known case of a single distribution
$F$ ([mcafee92]) the optimal price announcement describes a strictly
increasing function above the line $\tfrac{x}{2}$ that touches it exactly once
- at the median of $F$. Though the chooser can perfectly identify the
divider’s type in equilibrium, this strategic price announcement can lead to
an inefficient outcome in which the co-owner with lower expectation obtains
the company. In the maxmin equilibrium ([van2020]), a co-owner prepares for
the worst-case the co-owner can inflict on herself by always offering exactly
half her valuation. The allocation will thus always be efficient at the cost
of extreme behavioral assumptions.

The Texas Shoot-Out can be interpreted as variant of a _cake-cutting
mechanism_ (see [moulin2019fair] for a survey). Typically stated for divisible
objects these mechanisms describe discrete or continuous procedures for
proportional or envy-free allocations ([brams1995envy], [brams1997moving]).
Principally, any cake-cutting mechanism can be extended to settings with
indivisible objects by introducing transfer payments, as done in the Texas
Shoot-Out, or having a selling third party with sufficient information
([glazer1989efficient]). The question which agent is cutting or choosing is
significant with regard to an efficient allocation as pointed out by
[de2008efficient]. For the Texas Shoot-Out specifically, [brooks2010trigger]
analyze efficiencies and limitations under the assumption that the roles are
randomly assigned or delegated to the more informed party
([landeo2013shotgun]). For publicly known shares and private valuations of an
indivisible good, [cramton1987dissolving] derive the class of incentive
compatible and individual rational mechanisms that allocate the sole ownership
of an object to a shareholder which reconciles the negative result of
[myerson1983efficient] in which for a sole owner the object cannot be
efficiently allocated.

The remainder of the article is organized as follows. Section 2 gives a formal
introduction to the Texas Shoot-Out and briefly summarizes the results of the
stochastic setting and the one of maxmin strategies, known from the
literature. Section 3 adds Knightian Uncertainty to the setup. The main result
- the optimal price announcement under uncertainty - is stated in Section 3.1
and reveals the link between the for price announcements from the previous
section. In addition to the efficiency of the allocation (Section 3.2),
interim expected utility is derived and compared between the co-owners
(Section 3.3), explaining the deterrent effect of a premature selfish
dissolution of the Texas Shoot-Out. Section 3.4 argues that only small amounts
of uncertainty are necessary for the Texas Shoot-Out to prove its qualities if
correlation of valuations is adopted. How our formal results are to be
interpreted in and apply more general settings is discussed in Section 3.5.
Finally, Section 4 wraps up our findings.

##  2 Bayesian and Maxmin Equilibrium

Consider two222We consider only the simplest two-player case. More general
procedures for proportional or envy-free cake-cutting for $n$ players involve
many more steps or ’trimming techniques’, see [brams1995envy]. We do not know
of any of these $n$-player mechanisms finding actual use in company contracts
for $n\geq 3$. equal owners of a company who have come to the point where they
want to dissolve their partnership. Both owners have a private valuation
$x_{D},x_{C}$ in a compact interval $X:=[{x_{l}},{x_{h}}]\subset\mathbb{R}$
for being the sole owner of the company. In the _Texas Shoot-Out_ , the first
player (”divider”) announces a price333We explicitly allow for negative price
bids, of which we think as compensation, which might occur if a valuation for
the company is negative. That way, the Texas Shoot-Out does not only apply to
allocation decisions of desired objects, but also to undesirable objects, such
as debt. $p\in\mathbb{R}$ that she is willing to offer for obtaining or
forsaking the sole-ownership. In the next step, the second player (”chooser”)
either pays the divider $p$ and becomes the sole owner of the company or sells
her shares at price $p$ for the divider to obtain the company. This procedure
is the simplest cake-cutting mechanism with compensation payments: The Divider
cuts, the chooser selects her piece.

The chooser has all her payoff-relevant information at hand and optimizes by
simply ’picking the larger piece of the cake’ from her perspective: Facing an
offered price $p$ knowing her valuation $x_{C}$, she clearly takes the offer
and sells her shares if the offered price exceeds her private value of the
company minus the required payment, i.e. $p>x_{C}-p$, and buys the divider’s
shares of the company to become the sole owner if $p<x_{C}-p$. In the
following, we stick to this behavior which must be part of any Bayesian
equilibrium on-equilibrium-path444To rule out counter-intuitive off-
equilibrium-path behavior and thus making this behavior unique, a refinement
for extensive form games featuring sequential rationality can be applied,
e.g., weak perfect Bayesian equilibrium or sequential equilibrium
[sequential]. and is free of any assumptions on the distribution of valuations
or the played strategy of the divider: The chooser surely maximizes her
expected payoff by doing so for any of her valuations and any price $p$. In
the case of indifference, i.e. $p=x_{C}/2$, we impose the tie-breaking action
’sell’ as this is more in line with writing out the expected payoff of the
divider by means of (right-continuous) distribution functions. Note that the
so-defined strategy of is always a best reply for the chooser.

The divider’s ideal action is more involved and will take up most of the
upcoming analysis. She faces a trade-off between bidding a price either high
enough to get the company while not going into debt or low enough to get payed
while still getting something out of the deal. Let us assume that the divider
has a belief over the chooser’s valuation given by the distribution function
$F$ on $X$ with a strictly positive and differentiable density function $f$.
Denote by $\mu_{F}$ its (unique) median. Furthermore, we allow the divider to
entertain a utility function which is twice continuously differentiable,
concave and strictly increasing with $u^{\prime}>0$. Taking into account the
aforementioned response of chooser555Recall that, by right-continuity of $F$,
we implicitly assume that the chooser sells her shares if she is indifferent
between accepting and rejecting the offer, i.e. if $2p=x_{C}$. This can be
taken as a convention and is in fact only a null event if $F$ is atom-less.
Allowing for point masses, this tie breaking rule determines whether or not
the supremum of the payoff function can be attained, c.f. Section 3.5, the
divider’s (interim) expected payoff given her valuation $x_{D}$ is

$\pi_{F}(p\mid x_{D})=u(x_{D}-p)\cdot F(2p)+u(p)\cdot(1-F(2p)).$

#### Maxmin and Hedging

Let us first consider the maxmin strategy that was recently discussed by
[van2020]. It supports the idea that the opponent will always choose the
action hurting one the most - not taking into account their own losses. Note
that the divider can remove any uncertainty about the payoff by simply bidding
half of her own valuation, i.e. $\bar{p}=\frac{x_{D}}{2}$, guaranteeing a
payoff of

$\pi_{F}(\bar{p}\mid x_{D})=u\left(\frac{x_{D}}{2}\right)\cdot
F(x_{D})+u\left(\frac{x_{D}}{2}\right)\cdot(1-F(x_{D}))=u\left(\frac{x_{D}}{2}\right).$

The choice $\frac{x_{D}}{2}$ thus hedges the divider’s uncertainty completely
and we call this strategy _full-hedging_.

If the divider offered a price $p\neq\tfrac{x_{D}}{2}$ and thus (exactly) one
of the two possible outcomes $u(x_{D}-p),u(p)$ is strictly smaller than
$u(\tfrac{x_{D}}{2})$, the chooser would hurt the divider the most by choosing
the action leading to the divider getting the lower outcome. Thus, the
following statement holds.

######  Proposition 1 ([van2020]).

In maxmin equilibrium, the divider bids $\bar{p}=\frac{x_{D}}{2}$ and the
chooser sells her shares if $\bar{p}>x_{C}/2$ and buys the divider’ shares if
$\bar{p}<x_{C}/2$.

The outcome of the maxmin equilibrium is efficient in the sense that the
player with the highest valuation obtains the company.

#### Bayesian Equilibrium

The Texas Shoot-Out can be modeled as a Bayesian Game by introducing a (not
necessarily common) joint CDF(s) on the two agent’s valuations. For ease of
exposition, we stick to an i.i.d. distribution of valuations666More generally,
we allow for correlation in Section 3.4. given by a CDF $F$ on $X$ in the
following. Assume that $F$ is atom-less and admits a strictly positive and
differentiable density function $f$. Denote by $\mu_{F}$ its (unique) median.
In Bayes-Nash equilibrium, the divider seeks to maximize her expected payoff
given the aforementioned strategy of the chooser by picking a suitable price
$p$ for each of her valuations $x_{D}$. She faces a trade-off in doing so: The
higher the price, the more likely the chooser will sell her shares and the
divider ends up paying the exorbitant price herself. The lower the price, the
more likely the chooser will take the company and pays her a knocked-down
price. Thus, the optimal price announcement will be a moderate and deliberate
decision. Our first result thus is an immediate boundary for the divider’s
price decision.

######  Lemma 1.

The divider will only announce prices fulfilling $2p\in[{x_{l}},{x_{h}}]$.

###### Proof.

All of the proofs are delegated to the appendix. ∎

For the uniqueness of an optimal price announcement we impose strict
quasiconcavity on the payoff function. For sake of completeness, we include
its definition and relevant properties in Appendix A.

######  Assumption 1.

For a considered valuation $x_{D}$, let the divider’s expected payoff function
$\pi_{F}(p\mid x_{D})$ be strictly quasiconcave in $p$ for
$2p\in[{x_{l}},{x_{h}}]$.

In the literature, it is common to impose monotone hazard rate conditions on
the prior. We show that these conditions are sufficient for strict
quasiconcavity of the resulting payoff function, eliciting quasiconcavity as
the underlying driver of well-known results.

######  Lemma 2.

Assumption 1 is satisfied for all valuations $x_{D}$ if the _standard hazard
rate conditions_ (SHRCs, [mcafee92]) hold, i.e.

$\displaystyle\frac{\partial}{\partial x}\left(x+\frac{F(x)}{f(x)}\right)\geq
0\quad\text{and}\quad\frac{\partial}{\partial
x}\left(x-\frac{1-F(x)}{f(x)}\right)\geq 0.$ (1)

Put simply, a strictly quasiconcave function $f$ has a unique point $m_{f}$
such that $f$ is strictly increasing before and strictly decreasing after
$m_{f}$. It thus should not come as a surprise that it is related to the price
that maximizes expected payoff and can be identified as the solution to the
resp. FOC under smoothness assumptions on $F$. However, in the interplay with
$F$, the optimal price reveals intuitive qualitative properties discovered by
[mcafee92] for a CDF fulfilling the SHRC and now stated under the assumption
of strict quasiconcavity. Especially, there will be strategic choice to
announce a price above what the divider thinks her shares are worth when she
has a low valuation with the goal of getting paid more in the expected case of
the chooser buying ones shares (and vice versa for high valuations).

######  Proposition 2.

The optimal price announcement of the divider is uniquely determined, given by
$m(x_{D}):=m_{\pi_{F}(.\mid x_{D})}$. Furthermore, $m(x_{D})$ is strictly
increasing in $x_{D}$ and fulfills

$\begin{cases}x_{D}&<2m(x_{D})<\mu_{F}\\\ x_{D}&=2m(x_{D})=\mu_{F}\\\
x_{D}&>2m(x_{D})>\mu_{F}\end{cases}\quad\text{ if and only if
}\quad\begin{cases}x_{D}&<\mu_{F}\\\ x_{D}&=\mu_{F}\\\
x_{D}&>\mu_{F}\end{cases},$

where $\mu_{F}$ is the median of $F$. Together with the chooser’s strategy,
this constitutes a Bayesian Nash equilibrium.

Furthermore, for any valuation $x$, interim expected utility is strictly
larger for the chooser.

We now illustrate the implications of Proposition 2 using the uniform
distribution on the unit interval, see Figure 1. This example will be extended
to allow for Knightian Uncertainty in the upcoming section.

######  Example 1.

Let us consider the particularly transparent case of the uniform distribution
$f(x)=1$ for $x\in[0,1]$ and $u=\operatorname{id}$. By Lemma 2, Assumption 1
is satisfied. For $0\leq p\leq 1/2$ we have

$\pi_{F}(p\mid x_{D})=(x_{D}-p)2p+p(1-2p)=2x_{D}\cdot p+p-4p^{2}.$

The optimal bid is given by the first order condition

$\displaystyle 0$ $\displaystyle=2x_{D}+1-8p$ $\displaystyle\iff p$
$\displaystyle=x_{D}/4+1/8.$

$p$$1$$\mu_{F}=\tfrac{1}{2}$$x_{D}$$0$$\tfrac{1}{8}$$\tfrac{x_{D}}{2}$$\tfrac{3}{8}$$m(x_{D})$
Figure 1: Optimal price announcement for
$F\sim\mathcal{U}([0,1]),u=\operatorname{id}$.

Note that the outcome of the Bayes-Nash equilibrium is not always efficient,
in contrast to the maxmin equilibrium outcome. There is an incentive for
dividers with low valuation to bid a relatively high price, because the
chooser is going to accept with a certain probability. It can thus happen that
the divider obtains the company although she has a lower valuation than the
chooser in the case of low valuations and vice versa in case of both players
having a high valuations. Section 3.2 will elaborate more on the issue of
efficiency.

##  3 Equilibrium under Knightian Uncertainty

We now turn towards the case of _Knightian Uncertainty_ , i.e. imprecise
probabilistic information, about the distribution of an agent’s valuation,
where a set of beliefs is deemed possible. More precisely, the divider
considers a whole family of distributions $G\in\mathcal{G}$ for the chooser’s
valuation, bounded by two distribution functions $G_{0},G_{1}$ where $G_{0}$
stochastically dominates777The notion of stochastic dominance is as usual: $G$
stochastically dominates $G^{\prime}$ if $G(x)\leq G^{\prime}(x)$ for all $x$.
Equivalently, $\int u(x)\,G(\mathrm{d}x)\geq\int
u(x)\,G^{\prime}(\mathrm{d}x)$ for all functions $u$ with $u^{\prime}\geq 0$,
see [levy92] or [rothschild1970]. $G_{1}$:

$\mathcal{G}=\left\\{\text{$G$ is a CDF on $X$ satisfying }G_{0}(x)\leq
G(x)\leq G_{1}(x)\,\forall x\right\\}.$

Put differently, the probability that the chooser’s valuation is below a value
$x$ is assumed to be at least $G_{0}(x)$ but no more than $G_{1}(x)$. Such a
set is usually referred to as a _distribution band_ (c.f. [basu1995robust] and
[basu1994variations]) in robust Bayesian analysis. The distributions $G_{0}$
and $G_{1}$ are called _lower_ resp. _upper bound_ of $\mathcal{G}$.

Uncertainty with distribution bands arise naturally when the divider is
willing to entertain a belief $F$ about the chooser’s valuation, yet prefers a
robust approach to account for the uncertainty about the exact distribution.
As a benchmark example, she might consider the family of distributions
$G\in\mathcal{G}$ that are _$\epsilon$ -shifts_ of $F$ for any $\epsilon\geq
0$. Precisely, we define

$\mathcal{G}(F,\epsilon):=\left\\{G\text{ is a CDF on $X$ with
}F(x-\epsilon)\leq G(x)\leq F(x+\epsilon)\,\forall x\right\\}.$

Varying $\epsilon$ allows for changes in the probability weights of each type
as well as shifts of the entire distribution to higher and lower valuations.
The size of $\epsilon$ can be small, even non-existent for $\epsilon=0$, or
large until $\mathcal{G}(F,\epsilon)$ contains all distribution functions on
$X$, representing _full uncertainty_.

The defined distribution band $\mathcal{G}(F,\epsilon)$ has nice topological
properties with respect to the _interval topology_ ( c.f. [nendel2020note]) in
which it is closed and Dedekind super complete. Especially,
$\mathcal{G}(F,\epsilon)$ always admits a stochastically dominant resp.
dominated representative $G_{0}$ resp. $G_{1}$. Furthermore and in contrast to
the Prohorov metric888Also called Lévy-Prohorov metric, c.f.
[billingsley2013convergence], defined by
$d(F,G):=\inf\\{\eta>0:F(x-\eta)-\eta\leq G(x)\leq F(x+\eta)+\eta\mbox{ for
all $x\in[0,1]$}\\}$ the distribution band $\mathcal{G}(F,\epsilon)$ is
compatible with the SHRCs from Lemma 2 on $F$ as we will see in Section 3.1.
We illustrate $\mathcal{G}(F,\epsilon)$ for the case of a uniform distribution
in Figure 2.

Concerning preferences in the presence of Knightian Uncertainty, we assume
both agents to be uncertainty-averse in the sense of Gilboa and Schmeidler
[gilboa1989]. While the chooser can again decide only based on her own interim
valuation and the announced price as before, the divider faces her interim
worst-case expected utility999Considering the worst-case expected utility is a
strong form of ambiguity aversion. One might be intrigued to apply weaker
notions such as the smooth model [smooth]. However, as distribution bands do
not entail any information about the likelihood of one of its CDFs and their
CDFs can be highly discontinuous, there’s no natural parametrization or
computationally accessible choice for a subjective probability over
$\mathcal{G}$. Maximizing the worst-case expected utility on the other hand
can readily be defined without additional parameters. We thus find it
reasonable to assume Gilboa-Schmeidler preferences, especially when
considering $\mathcal{G}(F,\epsilon)$, having an ambiguity averse divider in
mind who chooses a robust approach to uncertainty by not making any
assumptions on distributions functions close to a designated CDF $F$. in the
class $\mathcal{G}$, given by

$\pi(p\mid x_{D})=\min_{G\in\mathcal{G}}\pi_{G}(p\mid x_{D}).$

By strict monotonicity of $u$, we have $u(x_{D}-p)>u(p)$ if and only if
$x_{D}>2p$. Thus, if the divider announces a price $x_{D}>2p$, her worst-case
belief is $G(2p)=G_{0}(2p)$. Vice versa, her worst-case belief for $x_{D}<2p$
is $G(2p)=G_{1}(2p)$. Consequently, only $G_{0}$ and $G_{1}$ are relevant for
evaluating the worst-case for the divider, i.e.

$\displaystyle\pi(p\mid x_{D})$
$\displaystyle=\min_{G\in\\{G_{0},G_{1}\\}}u(x_{D}-p)G(2p)+u(p)(1-G(2p))$
$\displaystyle=\begin{cases}u(x_{D}-p)G_{0}(2p)+u(p)(1-G_{0}(2p))&,2p<x_{D},\\\
u(x_{D}-p)G_{1}(2p)+u(p)(1-G_{1}(2p))&,x_{D}\leq 2p\end{cases}.$

$1$$x$$0$$1$$\mathbb{P}$$\epsilon$$\epsilon$$F$$G_{0}$$G_{1}$$\mathcal{G}$
Figure 2: $\mathcal{G}(F,\epsilon)$ for $F\sim\mathcal{U}([0,1])$ and
$\epsilon=1/5$.

a
In the Texas Shoot-Out, the maxmin setting and the one of _full uncertainty_ ,
i.e. $\mathcal{G}$ contains all CDFs on $X$, are linked and amount to the same
behavior: For any announced price $p$, there exists a valuation $x_{C}^{*}$ of
the chooser that implements the worst-case possible action of the chooser for
the divider. As in the case of full uncertainty, this valuation can always be
assigned probability $1$ by the Dirac measure, the divider’s worst-case belief
will be to always face the deteriorating action of the chooser. For instance,
if the divider announces a low (high) price $p$ with $x_{D}-p>p$ ($x_{D}-p<p$)
then a worst-case distribution is to assign probability $1$ to any valuation
$x_{C}^{*}>2p$ ($x_{C}^{*}<2p$), e.g. $x_{C}^{*}=x_{D}$, resulting in the
chooser buying the divider’s shares (selling her shares), which coincides with
the chooser’s maxmin action. Hence, although different in their
philosophy101010While the maxmin setting considers a mischievous opponent (no
matter their valuation), the full uncertainty setting deals with worst-case
beliefs about the opponent’s valuation., both settings amount to the behavior
of the divider fully-hedging herself.

This observation lets us locate the maxmin and the Bayesian price announcement
on two opposite ends of a setting of Knightian Uncertainty: One of full
uncertainty about the peer’s CDF (maxmin price) and one of no uncertainty,
where only a single CDF is being faced (Bayesian price). The following section
reveals the optimal price announcements for intermediate levels of uncertainty
to be a combination of the former two.

###  3.1 Optimal Price Announcement

In contrast to the chooser, the divider faces a delicate trade-off when
announcing her price if Knightian Uncertainty in the form of a distribution
band $\mathcal{G}$ is dealt with. Nevertheless, the divider’s maximization
problem can be explicitly solved and is our main result, establishing and
revealing a link between the Bayesian setting and the one of full uncertainty
which are recovered as special cases. In fact, the optimal price announcement
will be a combination of the optimal price announcements encountered in
Section 2: An uncertainty averse divider will announce full-hedging prices if
and only if she considers her valuation to be between the smallest and largest
median admissible in $\mathcal{G}$. For a valuation $x_{D}$ outside this
interval, she will announce the respective Bayesian price corresponding to the
stochastically dominant (high $x_{D}$) or stochastically dominated (low
$x_{D}$) representative of $\mathcal{G}$.

We allow for some more generality in the regularity assumptions on the lower
and upper bounds $G_{0},G_{1}$ of the distribution band than in Section 2. Let
$G_{0},G_{1}$ be distribution functions on $X$ that are continuous on
$[{x_{l}},{x_{h}})$111111As usual, distribution functions are assumed to be
right-continuous, thus, continuity from the right in ${x_{l}}$ is immediate.
However, there might still be jumps in ${x_{l}}$ (i.e., $G_{i}({x_{l}})>0$,
putting a point-mass on ${x_{l}}$) and, of course, for $G_{0}$, in ${x_{h}}$.
and piecewise continuously differentiable with positive density and all left
and right limits on $[{x_{l}},{x_{h}}]$. More precisely, there are finitely
many points in $X$ such that on each partition, $G_{i}$ is continuously
differentiable and left and right limits of $G_{i}$ and their derivatives
always exist. We further assume that $G_{1}$ is continuous in ${x_{h}}$ and
thus everywhere. These conditions are automatically fulfilled, if we start
with an atom-less distribution function $F$ with positive density as in
Section 2 and consider $\mathcal{G}(F,\epsilon)$.121212The same is true when
considering a Prohorov-ball around $F$

Note that $\pi$ is continuous, not only in $x_{D}$ but also in $p$ for
$2p\in[{x_{l}},{x_{h}}]$: The possible discontinuity of $\pi_{G_{0}}(.\mid
x_{D})$ in $2p={x_{h}}$ only occurs if $x_{D}<{x_{h}}$. But in that case,
$\pi(.\mid x_{D})$ is equal to the continuous function $\pi_{G_{1}}(.\mid
x_{D})$ around $2p={x_{h}}$.

Thus, the optimal price announcement (correspondence) can be defined

$m(x_{D}):=\arg\max_{p}\pi(p\mid x_{D}).$ (2)

Under Assumption 2, the main driver of our main theorem, it will turn out,
that the maximizer is unique, so the optimal price announcement under
uncertainty is indeed a function rather than a correspondence.

######  Assumption 2.

For a considered valuation $x_{D}\in X$, let both the divider’s payoff
functions $\pi_{G_{0}}(p\mid x_{D})$ and $\pi_{G_{1}}(p\mid x_{D})$ be
strictly quasiconcave in $p$ for $2p\in[{x_{l}},{x_{h}}]$.

The notion of a distribution band $\mathcal{G}(F,\epsilon)$ is compatible with
Assumption 2:

######  Lemma 3.

If $F$ fulfills the SHRC (Lemma 2), $G_{0}$ and $G_{1}$ from
$\mathcal{G}(F,\epsilon)$ induce payoff functions fulfilling Assumption 2.

######  Example 2.

CDFs fulfilling Lemma 3 include triangular distributions, the truncated
standard normal distribution on $[0,1]$ or the class of beta distributions.

However, the sufficient condition is far from being sharp: In the case of risk
neutrality, i.e. $u=\operatorname{id}$, e.g., some beta distributions induce
strictly quasiconcave utility functions although the sufficient condition is
not satisfied, see Figure 3.

Figure 3: Strict quasiconcavity of the payoff functions under the beta
distribution $B(\tfrac{1}{2},1)$ for different values of $x_{D}$ and
$\epsilon$.

We illustrate the Texas Shoot-Out under Knightian Uncertainty by considering a
version of Example 1 that uses a distribution band. It turns out that, on the
one hand, a divider with a rather average valuation will play cautiously and
fully-hedge herself against any losses by playing half her valuation. On the
other hand, a divider with very low (resp. high) valuation will still try to
strategically extract revenue by stating a slightly higher (resp. lower
price), thinking that the chooser will still take the offer (resp. refuse) it.

######  Example 3.

Let $F$ be the uniform distribution on the unit interval $[0,1]$ and let
$u=\operatorname{id}$. For any $\epsilon\leq 1/2$ the distributions defining
the distribution band $\mathcal{G}(F,\epsilon)$ are given by

$\displaystyle G_{0}(x)=\begin{cases}0&,0\leq x\leq\epsilon,\\\
x-\epsilon&,\epsilon<x<1,\\\ 1&,x=1.\end{cases}\quad\text{and}\quad
G_{1}(x)=\begin{cases}x+\epsilon&,0\leq x\leq 1-\epsilon\\\
1&,1-\epsilon<x\leq 1.\end{cases}.$

The optimal price announcement can be explicitly calculated and is given by

$\displaystyle
m(x_{D})=\begin{cases}\frac{x_{D}}{4}-\frac{\epsilon}{4}+\frac{1}{8}&,0\leq
x_{D}<\frac{1}{2}-\epsilon,\\\ \frac{x_{D}}{2}&,\frac{1}{2}-\epsilon\leq
x_{D}\leq\frac{1}{2}+\epsilon,\\\
\frac{x_{D}}{4}+\frac{\epsilon}{4}+\frac{1}{8}&,\frac{1}{2}+\epsilon<x_{D}\leq
1,\end{cases}$

which is depicted in Figure 4.

$p$$1$$x_{D}$$0$$\tfrac{3}{40}$$\tfrac{17}{40}$$m(x_{D})$$\tfrac{x_{D}}{2}$$\mu_{G_{1}}=\tfrac{3}{10}$$=\frac{1}{2}-\epsilon$$\mu_{G_{0}}=\tfrac{7}{10}$$=\frac{1}{2}+\epsilon$$m_{1}(x_{D})$$m_{0}(x_{D})$
Figure 4: Optimal price announcement $m(x_{D})$ for $\mathcal{G}(F,\epsilon)$
where $F\sim\mathcal{U}([0,1]),u=\operatorname{id}$ and
$\epsilon=\tfrac{1}{5}$. The functions $m_{i}(x_{D})$ ($i=0,1$) indicate the
optimal price announcement in the Bayesian setting if, resp., the CDF $G_{i}$
is faced.

For $\epsilon\geq 1/2$ the divider will play the full-hedging strategy
$m(x_{D})=\tfrac{x_{D}}{2}$ for all her valuations.

It is worth mentioning that full uncertainty, i.e. $\mathcal{G}(F,\epsilon)$
containing all CDFs on $X$, is only faced for $\epsilon\geq 1$. Complementary
to [van2020], this indicates that full uncertainty is sufficient, but not
necessary for full-hedging to be optimal for every valuation.

In order to give a precise and complete game theoretic description of the
Texas Shoot-Out under multiple priors, we also allow for the chooser to face
uncertainty about the distribution of the divider’s valuation. Since she has
all her payoff-relevant information at hand when choosing an action, her
belief about the divider’s valuation will only play a role when talking about
interim worst-case expected utility in Section 3.3.

For the following equilibrium concept, assume a symmetric setting in which the
chooser faces the same Knightian Uncertainty about the divider’s valuation
given by the set $\mathcal{G}$ of distribution functions. The game is then
again modeled similar to a Bayesian one with Gilboa-Schmeidler preferences. We
take the set $\mathcal{G}\otimes\mathcal{G}$ to contain independent
draws131313See [muraviev] for a precise description. Also note that we can fit
our game into their framework by adding nature as a player to play an
ambiguous _Ellsberg strategy_ as in [riedel2014ellsberg]. of distribution
functions $G_{D},G_{C}\in\mathcal{G}$, so that, after valuations are privately
observed, both agents face the prior set $\mathcal{G}$ about the peer’s
valuation distribution function. Ex-ante pure141414Picking the favorable pure
strategy of chooser that is always a best reply, our assumptions imply pure
price announcements. Thus, there is no need for considering mixed strategies.
strategies are denoted by $s_{D}\colon X\to\mathbb{R},s_{C}\colon
X\times\mathbb{R}\to\\{\text{buy, sell}\\}$. The expressions
$\widetilde{\pi}_{D},\widetilde{\pi}_{C}$ stand for the realized payoff of the
divider/chooser at the end of the mechanism, given the agents’ valuation
profiles and their strategies. The associated ex-ante worst-case expected
payoff functions can be written as

$\displaystyle U_{D}^{\text{ex}}(s_{D},s_{C})$
$\displaystyle=\min_{(G_{D},G_{C})\in\mathcal{G}\otimes\mathcal{G}}\mathbb{E}_{G_{D}\otimes
G_{C}}\left[\widetilde{\pi}_{D}(s_{D},s_{C};x_{D},x_{C})\right]$
$\displaystyle U_{C}^{\text{ex}}(s_{D},s_{C})$
$\displaystyle=\min_{(G_{D},G_{C})\in\mathcal{G}\otimes\mathcal{G}}\mathbb{E}_{G_{D}\otimes
G_{C}}\left[\widetilde{\pi}_{C}(s_{D},s_{C};x_{D},x_{C})\right].$

Interim worst-case expected utility is given by

$\displaystyle U_{D}^{\text{int}}(p,s_{C}\mid x_{D})$
$\displaystyle=\min_{G_{C}\in\mathcal{G}}\mathbb{E}_{G_{C}}\left[\widetilde{\pi}_{D}(s_{D},s_{C};x_{D},x_{C})\right]$
$\displaystyle U_{C}^{\text{int}}(a,s_{D}\mid x_{C})$
$\displaystyle=\min_{G_{D}\in\mathcal{G}}\mathbb{E}_{G_{D}\mid
s_{D}(x_{D})=p}\left[\widetilde{\pi}_{C}(s_{D},s_{C};x_{D},x_{C})\right]$

From that it is clear that the decision rule ’sell’ if and only if
$x_{C}-p\leq p$ is always a best reply ex-ante and interim for any announced
and observed price $p=s_{D}(x_{D})$ independent of how the divider has come to
set this price, i.e. no matter what updated information the chooser has on
$x_{D}$ or $G_{D}$. Especially, no _dynamic inconsistency_ 151515 _Dynamic
inconsistent_ behavior is characterized by agents having an incentive to
deviate from their ex-ante plan and usually occurs from updating by Bayes rule
in environments with Knightian Uncertainty, see [epstein2003recursive],
[aryal2014note], [pahlke2022dynamic], [hanany2020incomplete] for how this
impacts extensive form games. An established way to avoid this is by
introducing _rectangular beliefs_. arises. If the divider anticipates this
strategy of the chooser, her interim optimal behavior is also ex-ante optimal,
thus it suffices to derive the interim best response for the following
equilibrium concept:

######  Definition 1.

A strategy profile $(s_{D},s_{C})$ is an _ex-ante Knight-Nash equilibrium_ if
it is a Nash equilibrium for the game with ex-ante worst-case expected payoff
functions.
A strategy profile $(s_{D},s_{C})$ is an _interim Knight-Nash equilibrium_ if
it is a Nash equilibrium for every game induced by $x_{D}$ and $(x_{C},p)$
with interim worst-case expected payoff functions.

By continuity of $\pi(.\mid x_{D})$ we can always find an optimizer, thus we
have

######  Proposition 3.

In the Texas Shoot-Out, the following ex-ante and interim Knight-Nash
equilibrium always exists. For any price $p$, a chooser with valuation $x_{C}$
will sell the company if and only if $x_{C}\leq 2p$. A divider with valuation
$x_{D}$ will make an optimal price announcement $p$ belonging to $m(x_{D})$.

Strict quasiconcavity is driving our main result - a full characterization of
the optimal price announcement of the divider. A link between the price
announcements from [mcafee92] and [van2020] for intermediate levels of
uncertainty is established: We explicitly describe the transformation of the
optimal price announcement from the stochastic case (single prior or
$\epsilon=0$) to the setting of full uncertainty (full set of priors or
$\epsilon\gg 0$) by two observations: Firstly, the divider will hedge herself
for valuations between the medians. Secondly, for low resp. high valuations,
the divider will make the optimal price announcement corresponding to facing
the distribution $G_{1}$ resp. $G_{0}$. Note that for increasing levels of
uncertainty (increasing $\epsilon$) the spread of the median increases.

######  Theorem 1.

The optimal price announcement of a divider with valuation $x_{D}$ is given by

$\displaystyle m(x_{D})=\begin{cases}m_{1}(x_{D})&,x_{D}<\mu_{G_{1}}^{-},\\\
\frac{x_{D}}{2}&,\mu_{G_{1}}^{-}\leq x_{D}\leq\mu_{G_{0}}^{+},\\\
m_{0}(x_{D})&,\mu_{G_{0}}^{+}<x_{D},\end{cases}$

where $m_{i}(x_{D}):=m_{\pi_{G_{i}}(.\mid x_{D})}$ denotes the unique point of
the strictly quasiconcave function $\pi_{G_{i}}(.\mid x_{D})$ separating its
strictly decreasing and increasing domains and $\mu_{G_{1}}^{-}$ the smallest
and $\mu_{G_{0}}^{+}$ the largest median of $G_{1}$ resp. $G_{0}$.

Theorem 1 admits the following additional observations encoding a well-behaved
price announcement.

Firstly, the optimal price announcement will be above half of the valuation
while staying below the full-hedging action below the median of $\mu_{G_{1}}$
and vice versa above $\mu_{G_{0}}$, revealing that the divider expects to
extract payoff from her belief that the chooser has a higher resp. lower
valuation than herself. By 1 she will not do so for intermediate valuations,
but rather stick to a safe offer.

######  Corollary 1.

We have $x_{D}<2m(x_{D})<\mu_{G_{1}}^{-}$ if $x_{D}<\mu_{G_{1}}^{-}$ and
$\mu_{G_{0}}^{+}<2m(x_{D})<x_{D}$ if $\mu_{G_{0}}^{+}<x_{D}$.

Secondly, she will announce higher prices for higher valuations.

######  Corollary 2.

The optimal price announcement $m(x_{D})$ is increasing and thus a.s.
continuous and measurable. It is continuous and strictly increasing for
$\mu_{G_{1}}^{-}\leq x_{D}\leq\mu_{G_{0}}^{+}$ and on regions where $G_{0}$
resp. $G_{1}$ are continuously differentiable with positive derivative in
$m_{0}(x_{D})$ resp. $m_{1}(x_{D})$.
Furthermore, $m(x_{D})$ is always continuous in
$x_{D}\in\\{\mu_{G_{1}}^{-},\mu_{G_{0}}^{+}\\}$.

###  3.2 Efficiency

Only one co-owner remains in the company after the Texas Shoot-Out mechanism
has been completed. Desirably, the agent with the highest valuation becomes
the sole owner. In line with [mcafee92], we then call outcome _allocatively
efficient_.

Our previous results determine exactly under which circumstances the Texas
Shoot-Out under Knightian Uncertainty yields an allocatively efficient
outcome: Recall that the chooser sells her shares if and only if $2p\geq
x_{D}$. Now, on the one hand side, for
$x_{D}\in[\mu_{G_{1}}^{-},\mu_{G_{0}}^{+}]$ the divider announces
$2p^{*}=2m(x_{D})=x_{D}$ by Theorem 1 and consequently, the divider obtains
the good if and only if $x_{D}\geq x_{C}$. On the other hand side, for
$x_{D}<\mu_{G_{1}}^{-}$ the divider announces a price fulfilling
$2p^{*}=2m(x_{D})>x_{D}$ by Corollary 1 leading to an inefficient allocation
if and only if $x_{C}\in[x_{D},2m(x_{D})]$, where the divider obtains the
company while having a lower valuation than the chooser. Similarly, for
$x_{D}>\mu_{G_{0}}^{+}$ inefficiencies arise if and only if
$x_{C}\in[2m(x_{D}),x_{D}]$. Figure 5 depicts the profiles $(x_{D},x_{C})$ for
which the Texas Shoot-Out generates inefficient locations in the settings of
Example 1 and Example 3.

$x_{C}$1$\tfrac{1}{2}$$0$$\tfrac{1}{2}$$1$$x_{D}$$\tfrac{1}{4}$$\tfrac{3}{4}$no
uncertainty$x_{C}$1$0$$1$$x_{D}$$\tfrac{3}{20}$$\tfrac{17}{20}$$\tfrac{3}{10}$$\tfrac{7}{10}$uncertainty
Figure 5: Valuation profiles for which the Texas Shoot-Out yields inefficient
allocations in red for the uniform distribution $F$ in the settings of Example
1 (no uncertainty) and Example 3 (uncertainty for
$\mathcal{G}(F,\tfrac{1}{5})$).

Introducing Knightian Uncertainty to the Texas Shoot-Out mechanism increases
allocatively efficiency in comparison to the Bayesian case. More valuation
profiles will lead to an efficient allocation, especially all those for which
the divider’s valuation is close to the believed (uncertain) median of the
chooser’s valuation. But also for extremely low or high valuations, an
increase in Knightian Uncertainty will increase efficiency by weakening the
divider’s bargaining power, leading to her announcing prices closer to half
her valuation.

###  3.3 Interim Utility

Having a valuation $x$ and modeling uncertainty about the peer’s valuation by
a distribution band $\mathcal{G}$, we can ask whether an agent prefers to be
the divider (and initiate the Texas Shoot-Out) or the chooser. Focusing on a
risk neutral agent, we will see that there always is a preference for being
the chooser (rather waiting for the peer to initiate it). This preference is
strict for all valuations if the uncertainty is small, while for high levels
of uncertainty it is strict only for low/high valuations, unless full
uncertainty is faced.

We assume some more regularity on the distribution functions $G_{0},G_{1}$.
Put simply, we require them to be twice continuously differentiable with
positive density where they are not $0$ or $1$. In more detail, in addition to
$G_{1}$ being continuous and $G_{0}$ being continuous on $[{x_{l}},{x_{h}})$,
we assume that there exists values $x^{0},x^{1}\in X$ such that $G_{0}$ is
zero on $[{x_{l}},x^{0}]$, twice continuously differentiable with positive
density on $[x^{0},{x_{h}})$ and $G_{1}$ is twice continuously differentiable
with positive density on $[{x_{l}},x^{1}]$ and equal to $1$ on
$[x^{1},{x_{h}}]$. Especially, they will each have a unique median
$\mu_{G_{0}},\mu_{G_{1}}$. Note that this is automatically fulfilled if we
consider $\mathcal{G}(F,\epsilon)$ around an atom-less CDF $F$ that is twice
continuously differentiable with positive density on $X$.

By Theorem 1 interim worst-case EU of a divider with valuation $x_{D}$ can be
immediately written down to be

$\Phi_{D}(x_{D}):=\pi(m(x_{D})\mid
x_{D})=\begin{cases}\pi_{G_{1}}(m_{1}(x_{D})\mid x_{D})&,x_{D}<\mu_{G_{1}},\\\
\frac{x_{D}}{2}&,\mu_{G_{1}}\leq x_{D}\leq\mu_{G_{0}},\\\
\pi_{G_{0}}(m_{0}(x_{D})\mid x_{D})&,\mu_{G_{0}}<x_{D}.\end{cases}$

So far, we have not investigated the chooser’s interim expected utility, but
will do so now. To this end, assume a symmetric setting where the chooser
faces the same uncertainty about the peer’s valuation as the divider, i.e. she
beliefs the divider’s valuation to be drawn from the set $\mathcal{G}$ while
having own valuation $x_{C}$. Her interim worst-case expected payoff in
equilibrium is given by

$\displaystyle\Phi_{C}(x_{C}):=\min_{G\in\mathcal{G}}\mathbb{E}_{G}\left[\max\\{u(x_{C}-m(z)),u(m(z))\\}\right],$

where the expectation is taken w.r.t. the CDF $G$ over the divider’s
valuations, denoted by the variable $z$, and $m(z)$ is the divider’s optimal
price announcement if her valuation is $z$.

Since $2m(z)$ is increasing with range $[2m({x_{l}}),2m({x_{h}})]$ by
Corollary 2 we can draw the following conclusion about the worst-case beliefs
of the chooser.

######  Lemma 4.

Worst-case beliefs about the divider’s valuation for a chooser with valuation
$x_{C}$ are given by

$\begin{cases}G_{1}&,x_{C}<2m({x_{l}}),\\\ G^{*}_{x_{C}}&,2m({x_{l}})\leq
x_{C}\leq 2m({x_{h}}),\\\ G_{0}&,2m({x_{h}})<x_{C},\end{cases}$

where we define

$G^{*}_{x_{C}}(z)=\begin{cases}G_{0}(z)&,{x_{l}}\leq z<z_{*},\\\
G_{1}(z)&,z_{*}\leq z\leq{x_{h}}\end{cases}$

for the unique $z_{*}$ fulfilling $2m(z_{*})=x_{C}$.

As the chooser can always pick her favorite piece of the cake among
$u(x_{C}-p)$ and $u(p)$, she is worse off, the closer $p$ is to
$\tfrac{x_{C}}{2}$. Thus, if there is a valuation $z_{*}$ of a divider that
induces the price $2m(z_{*})=x_{C}$, the chooser’s worst-case belief is given
by $G^{*}_{x_{C}}$, i.e. the distribution in $\mathcal{G}$ that puts most
weight at and around $z_{*}$.We depicted an example of $G^{*}_{x_{C}}$ in
Figure 6.

$1$$x$$0$$1$$\mathbb{P}$$G_{0}$$G_{1}$$G^{*}_{x_{C}}$$x^{0}=\tfrac{1}{5}$$x^{1}=\tfrac{4}{5}$$z_{*}$
Figure 6: Illustration of the worst-case distribution $G^{*}_{x_{C}}$ of the
chooser in case of $2m(z_{*})=x_{C}$ given
$\mathcal{G}(\mathcal{U}([0,1]),\tfrac{1}{5})$. $G^{*}_{x_{C}}$ puts most
weight on the valuation $z_{*}$ of the divider, that leads to the worst offer
for the chooser with valuation $x_{C}$. Here $x_{C}=\tfrac{1}{2}$ and thus
$z_{*}=\tfrac{1}{2}$ by Example 3

In the following, we will restrict to the case of risk neutral agents only,
i.e. $u=\operatorname{id}$. A useful tool in preparation of the interim EU
comparison and for explicit calculations is the characterization of the
derivatives of $\Phi_{D}$ and $\Phi_{C}$.

######  Lemma 5.

Let $u=\operatorname{id}$. The functions $\Phi_{D}$ and $\Phi_{C}$ are
continuous, increasing and piecewise differentiable with derivatives

$\displaystyle\Phi^{\prime}_{D}(x)$
$\displaystyle=\begin{cases}G_{1}(2m_{1}(x))&,x<\mu_{G_{1}},\\\
\frac{1}{2}&,\mu_{G_{1}}\leq x\leq\mu_{G_{0}},\\\
G_{0}(2m_{0}(x))&,\mu_{G_{0}}<x.\end{cases}$ and
$\displaystyle\Phi^{\prime}_{C}(x)$ $\displaystyle=\begin{cases}0&,{x_{l}}\leq
x<2m({x_{l}}),\\\
\frac{1}{2}\cdot\left(G_{1}(m^{-1}(\tfrac{x}{2}))+G_{0}(m^{-1}(\tfrac{x}{2}))\right)&,2m({x_{l}})\leq
x\leq 2m({x_{h}})\\\ 1&,2m({x_{h}})<x\leq{x_{h}}.\end{cases}$

Thus, the higher the valuation one has, the higher the interim worst-case
expected utility. In fact, from the functional form of the derivatives, the
utility strictly increases with increasing valuation except for a chooser
facing very low valuations.

By means of the above lemma we can show that one is never worse off being the
chooser and can characterize precisely when this preference is strict: One
strictly prefers to be the chooser for all valuations for small levels of
uncertainty. This observation stays true for very low/high valuations if
uncertainty is increased until full uncertainty is faced.

######  Theorem 2.

If $x^{0}<x^{1}$, we have $\Phi_{D}(x)<\Phi_{C}(x)$ for all $x\in X$. If
$x^{1}\leq x^{0}$, $\Phi_{D}(x)=\Phi_{C}(x)=\tfrac{x}{2}$ for
$x\in[x^{1},x^{0}]$ and $\Phi_{D}(x)<\Phi_{C}(x)$ otherwise.

Especially, under full uncertainty both, the divider and the chooser, have an
interim worst-case expected utility equal to half their valuation.

We conclude this section by examining our uniform example for an interim
utility comparison.

######  Example 4.

Consider again $F\sim\mathcal{U}([0,1])$ for risk-neutral agents, i.e.
$u=\operatorname{id}$. Since we have already calculated $m(x)$, we can
calculate $\pi(m(x)\mid x)$ for a divider with valuation $x$, obtaining her
interim worst-case expected utility

$\displaystyle\Phi_{D}(x)=\begin{cases}\frac{1}{4}x^{2}+(\frac{1}{4}+\tfrac{\epsilon}{2})\cdot
x+\frac{\epsilon^{2}}{4}-\frac{\epsilon}{4}+\frac{1}{16}&,0\leq
x<\frac{1}{2}-\epsilon\\\ \frac{x}{2}&,\frac{1}{2}-\epsilon\leq
x\leq\frac{1}{2}+\epsilon,\\\
\frac{1}{4}x^{2}+(\frac{1}{4}-\frac{\epsilon}{2})\cdot
x+\frac{\epsilon^{2}}{4}+\frac{\epsilon}{4}+\frac{1}{16}&,\frac{1}{2}+\epsilon<x\leq
1,\end{cases}$

where for $\epsilon>\tfrac{1}{2}$ the function should be read as being equal
to the function $\tfrac{x}{2}$ everywhere.

In order to calculate $\Phi_{C}$ we need to distinguish three cases
$0\leq\epsilon\leq\tfrac{1}{4}$, $\tfrac{1}{4}<\epsilon\leq\tfrac{1}{2}$ and
$\tfrac{1}{2}<\epsilon\leq 1$ since not only the pasting points for $m$ (or
$m^{-1}$) play a role, but also the kinks of $G_{0},G_{1}$.161616If
$\epsilon\leq\tfrac{1}{4}$, we have
$\epsilon\leq\tfrac{1}{2}-\epsilon\leq\tfrac{1}{2}+\epsilon\leq 1-\epsilon$
while for $\tfrac{1}{4}<\epsilon\leq\tfrac{1}{2}$ we have
$\tfrac{1}{2}-\epsilon<\epsilon\leq 1-\epsilon<\tfrac{1}{2}+\epsilon$ and
$\tfrac{1}{2}<\epsilon$ implies
$\tfrac{1}{2}-\epsilon<1-\epsilon<\epsilon<\tfrac{1}{2}+\epsilon$. Also note
that for $\epsilon\geq 1$ we have full uncertainty and the situation won’t
change anymore. All the explicit formulae are delegated to the appendix.

Figure 7 graphically illustrates $\Phi_{D}$ and $\Phi_{C}$ for
$\epsilon=0.02,0.4,0.6$.

EU$0.5$$0$$0$$1$$x$$\approx
0.25$$\Phi_{C}$$\Phi_{D}$$\frac{x}{2}$$\epsilon=0.02$EU$0.5$$0$$0$$1$$x$$\approx
0.1$$\Phi_{C}$$\Phi_{D}$$\epsilon=0.4$EU$0.5$$0$$0$$1$$x$$0.4$$0.6$$\Phi_{C}$$\Phi_{D}$$\epsilon=0.6$
Figure 7: Interim worst-case EU for risk neutral dividers and choosers with
valuation $x$, facing $\mathcal{G}(\mathcal{U}([0,1]),\epsilon)$ for
$\epsilon\in\\{0.02,0.4,0.6$}.

For $\epsilon<\tfrac{1}{4}$ we see that $\Phi_{C}(x)>\Phi_{D}(x)$ everywhere.
In the case $\epsilon=0.3>\tfrac{1}{4}$ we have $\Phi_{C}(x)=\Phi_{D}(x)$
precisely for $x\in[1-2\cdot 0.3,2\cdot 0.3]=[0.4,0.6]$ as is clear from
Theorem 2. We also note that for increasing $\epsilon$ both functions are
deforming into $x\mapsto\tfrac{x}{2}$, and the curvature of $\Phi_{D}$ is only
visible for very small values of $\epsilon$.

The example combines and stresses two points of our argument for the Texas
Shoot-Out under Knightian Uncertainty to being a deterrent exit mechanism. For
small amounts of uncertainty, i.e. small $\epsilon$, the divider’s interim
worst-case expected payoff is close to the full-hedging payoff given by
$\tfrac{x}{2}$. Thus, the Texas Shoot-Out lowers an agent’s material incentive
to initiate the mechanism and become a divider. We conclude that only co-
owners with relatively high or low valuations might consider the exit
profitable. Furthermore, the higher the uncertainty the more likely the
mechanism is to achieve an efficient allocation. Secondly, even if a co-owner
has a low/high valuation and is thus materially interested in ending the
partnership under the terms of the Texas Shoot-Out, she would rather be in the
position of the chooser and thus not triggering the mechanism in the first
place.

###  3.4 Correlation

So far, we implicitly assumed that $\mathcal{G}$ and $x_{D}$ are independent,
implying that the divider’s (uncertain) belief about the peer’s valuation is
the same for all her valuations. However, in the context of co-ownership one
has good reason to assume that the assessment of the company’s value is
similar or more generally correlated among co-owners. Especially, the agents
know more about the peer’s valuation which could make us believe that the
divider can make use of this additional information to extract profit.
Interestingly however, we will see that suitable correlation will indeed
achieve the exact opposite: The divider will not only play safe for more
valuations but also do so for lower levels of uncertainty. It is thus not the
mere lack of intelligence about the peer’s valuation that fosters the
efficiency of the Texas Shoot-Out in real world applications, but the
interplay between uncertainty and the correlation of valuations.

To start the analysis, we can incorporate the idea of correlation (at an
interim stage) by allowing for the lower and upper bounds $G_{0},G_{1}$ to
depend on $x_{D}$. In that case we write $G_{0}^{x_{D}},G_{1}^{x_{D}}$. As
always, the chooser is not affected by any uncertainty as she has all payoff
relevant information after the price announcement. The divider takes into
account for her price announcement while facing a seemingly more complicated
problem. Fortunately, Theorem 1 is a point-wise statement and thus still
remains valid, provided $G_{i}^{x_{D}}$ ($i=0,1$) fulfill the assumptions from
Section 3.1. We only need to take into account the changes in $G_{i}^{x_{D}}$
and thereby $m_{\pi_{G_{i}^{x_{D}}}}$ with varying $x_{D}$.

For instance, a divider with valuation $x_{D}$ might expect the chooser’s
valuation to be drawn from a distribution $F^{x_{D}}$ with mode $x_{D}$ to
emphasize that she considers $x_{D}$ to be the most likely valuation of the
chooser. As before, she might want to account for a robust approach by means
of a distribution band $\mathcal{G}(F^{x_{D}},\epsilon)$. Note that the
induced upper and lower bounds now depend on both $x_{D}$ and $\epsilon$.

The following example illustrates the impact of correlation by using
triangular distributions.

######  Example 5.

For $c\in[0,1]$ let $\operatorname{Tri}^{c}$ denote the triangular
distribution with mode $c$ on $[0,1]$. Let the divider have valuation $x_{D}$.
She might believe the chooser’s valuation to be close to her own by
considering a triangular distribution $\operatorname{Tri}^{x_{D}}$ with mode
$c=x_{D}$. Accounting for uncertainty, she faces the distribution band
$\mathcal{G}(\operatorname{Tri}^{x_{D}},\epsilon)$. Now, applying Theorem 1
for $\epsilon=\tfrac{1}{5}$ we find that not matter $x_{D}$ the divider will
always fully hedge herself, i.e. announce the price $\tfrac{x_{D}}{2}$. In
comparison, Figure 8 illustrates the optimal price announcement for the case
of a triangular distribution with a fixed and thus uncorrelated mode
$c=\tfrac{1}{2}$.

$p$$1$$x_{D}$$0$$\tfrac{3}{10}$$\tfrac{7}{10}$$\tfrac{x_{D}}{2}$$m_{\text{corr.}}(x_{D})$$m_{\text{uncorr.}}(x_{D})$$\approx
0.08$$\approx 0.42$ Figure 8: Optimal price announcement for
$u=\operatorname{id}$ given the correlated (red) and uncorrelated (blue)
distribution bands $\mathcal{G}(\operatorname{Tri}^{x_{D}},\tfrac{1}{5})$ and
$\mathcal{G}(\operatorname{Tri}^{0.5},\tfrac{1}{5})$ where
$\operatorname{Tri}^{c}$ is the triangular distribution on $[0,1]$ with mode
$c$.

Indeed, we can explicitly characterize the valuations $x_{D}$ for which a
divider will play full-hedging in Example 5.

######  Proposition 4.

Given the distribution band
$\mathcal{G}(\operatorname{Tri}^{x_{D}},\epsilon)$, a divider with valuation
$x_{D}$ will play full-hedging if and only if

$\displaystyle\begin{cases}\sqrt{\frac{x_{D}}{2}}-\epsilon\leq x_{D}&,\text{if
}x_{D}\leq\frac{1}{2}\\\ x_{D}\leq
1+\epsilon-\sqrt{\frac{1-x_{D}}{2}}&,\text{if }\frac{1}{2}\leq
x_{D}\end{cases}.$

Example 5 illustrates that less uncertainty is needed for the divider to play
safely if the chooser’s valuation is believed to be correlated and close to
her own valuation with a high probability. In fact, full-hedging for all
valuations will already be played by the divider for
$\epsilon\geq\tfrac{1}{8}$ in the correlated case whereas $\epsilon$ needs to
exceed $\tfrac{1}{2}$ in the uncorrelated setting to generate this behavior.
Intuitively, this observation arises as the medians of the stochastic
dominated and stochastic dominant distribution of the distribution band lie
around $x_{D}$ for many more valuations in the correlated case than in an
uncorrelated one. In a nutshell, for the Texas Shoot-Out to generate an
efficient outcome or prevent a premature exit less (while still some)
uncertainty is needed if correlation between valuation seems reasonable.

###  3.5 Full Uncertainty on Intervals

So far, we modeled uncertainty as imprecise information about the distribution
from which valuations are drawn. It might seem more natural to instead
directly assume imprecise probabilistic information about the valuations, e.g.
by believing a valuation to lie within an interval
$[a,b]\subset[{x_{l}},{x_{h}}]$. Our framework is flexible enough to
accommodate this setting. By assuming $G_{1}$ to be the (discontinuous)
distribution function that assigns point-mass $1$ to $a$ and $G_{0}$ to do so
at $b$, we have $G(x)=0$ if $x<a$, $G(x)=1$ if $x>b$ and $0\leq G(x)\leq 1$ if
$x\in[a,b]$ for all $G\in\mathcal{G}$. Especially,
$\mu_{G_{1}}=a,\mu_{G_{0}}=b$. Note that $\pi_{G_{0}}(.\mid
x_{D}),\pi_{G_{1}}(.\mid x_{D})$ are both strictly quasiconcave with
$m_{\pi_{G_{0}}}(x_{D})=b,m_{\pi_{G_{1}}}(x_{D})=a$ while generically not
being continuous in that respective point. Even more, $\pi_{G_{0}}(.\mid
x_{D})$ resp. $\pi_{G_{1}}(.\mid x_{D})$ attain their maximum if and only if
$b\leq x_{D}$ resp. $a\leq x_{D}$. However, Theorem 2 still reveals ’the’
optimal price announcement:

The divider will announce a price $\tfrac{x_{D}}{2}$ if $x_{D}\in[a,b]$ and
offer $\tfrac{b}{2}$ if $x_{D}>b$. For $x_{D}<a$, the supremum of the
divider’s payoff function is $\lim_{p\nearrow a/2}\pi(p\mid x_{D})$, which
cannot be attained. Practically however, she can announce a lower price as
close to $\tfrac{a}{2}$ as possible, while still giving the divider an
incentive to sell. Note that this issue is related to assuming the chooser’s
strategy to be ’sell’ if and only if $x_{C}-p\leq p$. If we instead take the
strict inequality, the problem shifts towards valuations above the median.

Intuitively, the divider faces full uncertainty about $x_{C}$ belonging to
$[a,b]$ and thus plays full-hedging if $2p\in[a,b]$. If the divider has a
valuation $x_{D}\geq b$, she is certain to be more interested in buying the
company and will ensure this outcome by offering the lowest price for which
the chooser is selling her shares for sure, which is $b/2$. Vice versa, for a
valuation $x_{D}<a$, she wants to keep the price as high as possible, while
still ensuring the chooser to buy her shares. She can do so by announcing a
price (infinitesimal below) $a/2$. Figure 9 depicts this observation.

${x_{l}}$${x_{h}}$$x_{D}$$a$$b$$\frac{b}{2}$$\frac{a}{2}$$m(x_{D})$ Figure 9:
Optimal price announcement for full uncertainty on $[a,b]$. Between ${x_{l}}$
and $a$ the supremum of the worst-case EU cannot be attained, but approximated
for $p$ close to, but below $\tfrac{a}{2}$.

More generally, we can derive an ’approximate’ optimal price announcement by
invoking the interplay of results from the Appendix (Proposition A1 and Lemma
B2) with limited assumptions on the regularity of the distribution functions
as long as they induce strictly quasiconcave utility functions. The main
analytic issue is just whether the supremum can be attained. By announcing a
price that achieves a worst-case expected payoff arbitrarily close to the
supremum value, Theorem 1 keeps its simple structure and can be applied in
real world scenarios.

##  4 Conclusion

When founding a co-owned company it is preventative and advantageous to agree
in advance on a dissolution mechanism in case things get sour. Any such
dissolution mechanism should be readily available, simple, and it should
discourage a selfish premature termination of the partnership. The so-called
Texas Shoot-Out is a well-known example of such an exit mechanism for two co-
owners. The co-owner initiating the mechanism announces a price for the sole
ownership of the company while the other can choose to sell or buy the company
at that price. While simple and independent of external outside options, it is
notorious for its deterrent effect on a premature dissolving.

This article invokes Knightian Uncertainty as an explanatory source for this
discouragement. Having in mind that co-owners have some idea about the
distribution of their partner, we allow for any degree of confidence in a
reference distribution $F$ by modeling the uncertainty as an distribution band
around $F$ that can range anywhere in between the Bayesian single prior
setting towards one of full uncertainty. If the induced worst-case expected
utility functions are strictly quasiconcave, we derive the co-owners’ optimal
actions and interim expected payoffs. Our main result is the explicit
characterization of the the divider’s price announcement which is a surprising
mixture of the optimal strategies under no and full uncertainty: She will play
cautiously for valuations close to the median valuation of the reference
distribution while still trying to generate a revenue that exceeds half her
valuation for low or high valuations. Hence, only co-owners with low/high
valuations have a material incentive to initiate a Texas Shoot-Out. However,
these co-owners strictly prefer the other co-owner to trigger the exit
mechanism as they otherwise find themselves in an unfavorable position (being
forced to leave the company resp. take over the sole ownership). Welfare is
improved as uncertainty increases efficiency, i.e., for more valuation
profiles the co-owner with higher valuation becomes the sole owner. These
desirable consequences are already visible for small levels of uncertainty and
even enhanced if there is good reason to believe the valuations to be
positively correlated. Knightian Uncertainty can thus explain why
consultancies recommend to include the Texas Shoot-Out in buy-sell agreements.

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##  Appendix A Quasiconcavity

In the following, we give formal definitions of the concept of quasiconcave
functions and some of their properties relevant to our analysis of the Texas
Shoot-Out.

######  Definition A1 (Quasiconcavity).

Let $S\subseteq\mathbb{R}^{L}$ be a convex set. A function $f\colon
S\to\mathbb{R}$ is called _quasiconcave_ if one of the following equivalent
statements holds true:

  1. (i)

For any $x,x^{\prime}\in S$ and any $\lambda\in(0,1)$ we have $f(\lambda
x+(1-\lambda)x^{\prime})\geq\min\\{f(x),f(x^{\prime})\\}$.

  2. (ii)

Every super level set is convex, i.e. for any $\xi\in\mathbb{R}$ the set
$\\{s\in S\,|\,f(s)\geq\xi\\}$ is convex.

If $L=1$ and $S$ is an interval, there is another equivalent statement that
describes the shape of a quasiconcave function. For ease of exposition we
stick to a compact interval $S=[a,b]$ in the following. However, the given
proof works as well for half-open or open ones by just replacing the
respective parentheses accordingly.

  1. (iii)

There exists a number $m\in[a,b]$ such that (at least) one of the following
properties hold:

    1. (a)

$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m]}$ is increasing and
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(m,b]}$ is decreasing.

    2. (b)

$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m)}$ is increasing and
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[m,b]}$ is decreasing.

It is worth noting in the above cases that both, the restriction of $f$ to the
empty set as well as to a single point are considered (strictly) increasing as
well as (strictly) decreasing171717E.g. a function $f\colon X\to Y$ is
increasing by definition if $\forall x,y\in X(x\leq y\Rightarrow f(x)\leq
f(y))$ which is true if $X$ is a singleton or empty. A consequence of this is
that a non-increasing function has at least two elements in its domain..
Especially, if $m=b$ in (iiia) holds, this simply means that $f$ is
increasing. Similarly, $m=a$ in (iiib) corresponds to $f$ being decreasing.

###### Proof.

We first prove the more general equivalence of (i) and (ii) and will then show
(i)$\Leftrightarrow$(iii).

  * •

(i)$\Rightarrow$(ii):
Let $\xi\in\mathbb{R}$ and let $x,x^{\prime}\in\\{s\in S\,|\,f(s)\geq\xi\\}$.
Consider any convex combination $x^{\prime\prime}:=\lambda
x+(1-\lambda)x^{\prime}$. By (i)
$f(x^{\prime\prime})\geq\min\\{f(x),f(x^{\prime})\\}\geq\xi$.

  * •

(ii)$\Rightarrow$(i):
Define $\xi:=\min\\{f(x),f(x^{\prime})\\}$. Then $\\{s\in
S\,|\,f(s)\geq\xi\\}$ is convex and contains $x,x^{\prime}$. Thus, it also
contains $\lambda x+(1-\lambda)x^{\prime}$ for any $\lambda\in(0,1)$ and by
definition $f(\lambda
x+(1-\lambda)x^{\prime})\geq\xi=\min\\{f(x),f(x^{\prime})\\}$.

  * •

(iii)$\Rightarrow$(i):
If $m\in\\{a,b\\}$, $f$ surely fulfills (i). For $m\in(a,b)$ consider now an
$f$ that has property (iiia) or (iiib). Let $x,x^{\prime}\in[a,b]$, wlog
$x<x^{\prime}$ and $\lambda\in(0,1)$. If $\widetilde{x}:=\lambda
x+(1-\lambda)x^{\prime}<m$, we clearly have $f(\widetilde{x})\geq
f(x)\geq\min\\{f(x),f(x^{\prime})\\}$ since $f$ is increasing on
$[a,\widetilde{x}]\subseteq[a,m)$. If $m<\widetilde{x}$, we find
$f(\widetilde{x})\geq f(x^{\prime})\geq\min\\{f(x),f(x^{\prime})\\}$ as $f$ is
decreasing on $[\widetilde{x},b]\subseteq(m,b]$. Finally, if $\widetilde{x}=m$
we find $f(\widetilde{x})\geq f(x)\geq\min\\{f(x),f(x^{\prime})\\}$ if
condition (iiia) holds and $f(\widetilde{x})\geq
f(x^{\prime})\geq\min\\{f(x),f(x^{\prime})\\}$ if condition (iiib) holds.
Thus, (i) is true.

  * •

(i)$\Rightarrow$(iii):
By means of contraposition: Assume the negation of (iii) which implies that
for each $m\in[a,b]$ at a time the restriction
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m)}$ is not increasing or
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(m,b]}$ is not decreasing (or both).

First suppose that there is an $m$ such that both
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m)}$ is not increasing and
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(m,b]}$ is not decreasing. This implies
the existence of $x<x^{\prime}\leq m\leq
x^{\prime\prime}<x^{\prime\prime\prime}$ with $f(x)>f(x^{\prime})$ and
$f(x^{\prime\prime})<f(x^{\prime\prime\prime})$. Especially,
$\min\\{f(x^{\prime}),f(x^{\prime\prime})\\}<\min\\{f(x),f(x^{\prime\prime\prime})\\}$,
showing, that (i) does not hold. A similar argument is valid for the negation
of (iiib).

Second, consider the case where for each $m$ at a time either
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m)}$ is not increasing or
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(m,b]}$ is not decreasing but _never
both_. Equivalently, for each $m$ at a time
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m)}$ is increasing or
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(m,b]}$ is decreasing but never both.
The set
$M_{i}:=\\{m\in[a,b]\,|\,f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m)}\text{ is
increasing}\\}$ is non-empty since formally $a\in M_{i}$. Thus, $m_{i}:=\sup
M_{i}$ is well-defined. Note that $a<m_{i}$ since $a=m_{i}$ implies that $f$
obeys (iiia) which was excluded. Likewise, $m_{i}<b$, since otherwise $f$
would fulfill (iiib). Thus, for the infimum $m_{d}:=\inf M_{d}$ for
$M_{d}:=\\{m\in[a,b]\,|\,f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(m,b]}\text{ is
decreasing}\\}$ is well-defined. In the considered case and by their
definitions we must have $m_{i}\leq m_{d}$ and even $m:=m_{i}=m_{d}\in(a,b)$.
This means that $f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m)}$ is increasing
and $f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(m,b]}$ is decreasing. Since neither
(iiia) nor (iiib) are true by assumption, $\sup_{x<m}f(x)>f(m)$ and
$f(m)<\inf_{m<x}f(x)$. Thus there are $x<m<x^{\prime}$ such that
$f(m)<\min\\{f(x),f(x^{\prime})\\}$, showing that (i) does not hold.

∎

######  Definition A2 (Strict quasiconcavity).

Let $S\subseteq\mathbb{R}^{L}$ be a convex set. A function $f\colon
S\to\mathbb{R}$ is called _strictly quasiconcave_ if

  1. (i’)

For any $x,x^{\prime}\in S,x<x^{\prime}$ and any $\lambda\in(0,1)$ we have
$f(\lambda x+(1-\lambda)x^{\prime})>\min\\{f(x),f(x^{\prime})\\}$.

If $L=1$ and $S$ is an interval, the following assertion (stated for a compact
interval $S=[a,b]$) is an additional characterization of strict
quasiconcavity.

  1. (iii’)

There exists a unique number $m\in[a,b]$ such that (at least) one of the
following properties hold:

    1. (a)

$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m]}$ is strictly increasing and
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(m,b]}$ is strictly decreasing.

    2. (b)

$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[a,m)}$ is strictly increasing and
$f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[m,b]}$ is strictly decreasing.

###### Proof.

The proof is an straightforward adjustment of the one for quasiconcavity. The
uniqueness of $m$ follows as strictly quasiconcave functions can’t have
horizontal sections. ∎

######  Definition A3.

If $f$ is strictly quasiconcave and $S$ an interval, we denote the unique
value that separates the increasing part of the function from the decreasing
one by $m_{f}$.

Note that if $f$ attains a global maximum (e.g. if $f$ is continuous), it must
be $m_{f}$. If $f$ does not have a global maximum, it’s supremum value is
either $\lim_{x\nearrow m_{f}}f(x)$ or $\lim_{x\searrow m_{f}}f(x)$.

Obviously, strictly quasiconcave functions are quasiconcave. Also note that
(strictly) concave functions are (strictly) quasiconcave, but quasiconcave
functions need not be concave nor even continuous.

The following lemma shows some handy properties the minimum of two strictly
quasiconcave functions.

######  Lemma A1.

Let $f,g\colon S\to\mathbb{R}$ be strictly quasiconcave. Then the point-wise
minimum $h:=\min\\{f,g\\}$ is strictly quasiconcave as well. Furthermore, if
$S=[a,b]$ and, e.g., $m_{f}\leq m_{g}$, then $m_{h}=\inf\,\\{x|m_{f}\leq
x<m_{g},f(x)\leq g(x)\\}\cup\\{m_{g}\\}$.

###### Proof.

Let $x<x^{\prime}$ and $\lambda\in(0,1)$. Then

$\displaystyle h(\lambda x+(1-\lambda)x^{\prime})$
$\displaystyle=\min\\{f(\lambda x+(1-\lambda)x^{\prime}),g(\lambda
x+(1-\lambda)x^{\prime})\\}$
$\displaystyle>\min\\{\min\\{f(x),f(x^{\prime})\\},\min\\{g(x),g(x^{\prime})\\}\\}$
$\displaystyle=\min\\{\min\\{f(x),g(x)\\},\min\\{f(x^{\prime}),g(x^{\prime})\\}\\}$
$\displaystyle=\min\\{h(x),h(x^{\prime})\\}.$

We note that, replacing the strict inequality for a weak one, the minimum of
two quasiconcave functions is also quasiconcave.

We now turn towards the points determining increasing and decreasing domains.
By definition, $h$ is strictly increasing on $[a,m_{f})$ and strictly
decreasing on $(m_{g},b]$, thus $m_{h}\in[m_{f},m_{g}]$. If $m_{f}=m_{g}$,
$m_{h}=m_{f}=m_{g}$. We thus assume $m_{f}<m_{g}$ in the following. Consider
the set $M:=\\{x\,|\,m_{f}\leq x<m_{g},f(x)\leq g(x)\\}$. If $M$ is empty, we
have $h\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[m_{f},m_{g})}\equiv
g\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[m_{f},m_{g})}$, which is increasing, and
thus necessarily $m_{h}=m_{g}$. If $M$ is non-empty, let $x_{0}:=\inf M$ and
note $x_{0}<m_{g}$. We have $m_{h}\leq x_{0}$ as
$h\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(x_{0},m_{g})}=f\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{(x_{0},m_{g})}$
keeping in mind that $f$ is strictly decreasing on that domain. By definition,
$m_{f}\leq x_{0}$. If $m_{f}=x_{0}$, we necessarily have $m_{h}=x_{0}=m_{f}$.
If $m_{f}<x_{0}$, the interval $[m_{f},x_{0})$ is non-empty and we find
$h\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[m_{f},x_{0})}=g\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{[m_{f},x_{0})}$,
which is strictly increasing on the considered domain. Thus also $x_{0}\leq
m_{h}$ and hence $x_{0}=m_{h}$. ∎

Note that in a setting with continuous strictly quasiconcave functions, the
proposition simplifies: The point $m_{h}$ is the minimum among $m_{f},m_{g}$
and (if it exists) the unique intersection point between these two.

For our purposes, the following special case is of good use for our main
theorem:

######  Proposition A1.

Let $f,g$ be strictly quasiconcave and assume that $h:=\min\\{f,g\\}$ has the
following form for some $x_{0}$

$\displaystyle h(x)=\begin{cases}g(x)&,x<x_{0},\\\ f(x)=g(x)&,x=x_{0},\\\
f(x)&,x_{0}<x\end{cases}.$

Then,

$\displaystyle m_{h}=\begin{cases}m_{f}&,x_{0}<m_{f},\\\ x_{0}&,m_{f}\leq
x_{0}\leq m_{g},\\\ m_{g}&,m_{g}<x_{0}\end{cases}.$

###### Proof.

We first note that by Lemma A1 $h$ is strictly quasiconcave. Now observe that
the inequality $m_{g}<x_{0}<m_{f}$ is impossible: Otherwise, $h=g$ is strictly
decreasing on $(m_{g},x_{0})$ while $h=f$ is strictly increasing on
$(x_{0},m_{f})$ \- a contradiction to $h$ being strictly quasiconcave.

Hence, it remains to investigate the cases where $x_{0}\leq m_{g}$ or
$m_{f}\leq x_{0}$ (or both) which defines three distinct cases:

  1. (i)

If both are true, i.e. $m_{f}\leq x_{0}\leq m_{g}$, we find that $h=f$ is
strictly increasing on $(a,x_{0})$181818If $a=x_{0}$, the interval $(a,x_{0})$
is empty, but the statement is still true. The same argument applies for
$x_{0}=b$. while $h=g$ is strictly decreasing on $(x_{0},b)$. Thus,
$m_{h}=x_{0}$.

  2. (ii)

If $x_{0}\leq m_{g}$ and $x_{0}<m_{f}$, we find that $h=f$ is strictly
increasing on $(x_{0},m_{f})$ while strictly decreasing on $(m_{f},b)$. Thus,
$m_{h}=m_{f}$.

  3. (iii)

If $m_{f}\leq x_{0}$ and $m_{g}<x_{0}$, we find that $h=g$ is strictly
increasing on $(a,m_{g})$ while strictly decreasing on $(m_{g},x_{0})$. Thus,
$m_{h}=m_{g}$.

∎

##  Appendix B Proofs

###### Proof of Lemma 2.

The proof essentially follows the lines of Lemma 7 in [mcafee92]. It is worth
mentioning that there’s a typo at the second condition in the reference which
is clear from their proof. Also note that the ’price’ in [mcafee92] refers to
the claimed overall price for the indivisible object, while we refer to it as
the value of one’s or the other’s share which is half of that.

We start by calculating

$\displaystyle\frac{\partial}{\partial p}\pi_{F}(p\mid x_{D})$
$\displaystyle=$ $\displaystyle
f(2p)\cdot\left\\{2u(x_{D}-p)-2u(p)+\frac{F(2p)}{f(2p)}\cdot\left(-u^{\prime}(x_{D}-p)-u^{\prime}(p)\right)+\frac{u^{\prime}(p)}{f(2p)}\right\\}.$

Now, the second derivative191919In fact, for the proof it suffices to have
second left and right derivatives of $F$ resp. fulfilling the SHRC as long as
the first derivative is continuous. of $\pi_{F}$ with respect to $p$ and
evaluated in a point $q$ such that $2q\in[{x_{l}},{x_{h}}]$ (if it exists)
with $\tfrac{\partial}{\partial p}\pi_{F}(p\mid x_{D})\mid_{p=q}=0$ is

$\displaystyle\frac{\partial^{2}}{\partial p^{2}}\pi_{F}(p\mid
x_{D})\mid_{p=q}=$ $\displaystyle-f(2q)\frac{\partial}{\partial
p}\left(2p+\frac{F(2p)}{f(2p)}\right)\mid_{p=q}\cdot u^{\prime}(x_{D}-q)$
$\displaystyle-f(2q)\frac{\partial}{\partial
p}\left(2p-\frac{1-F(2p)}{f(2p)}\right)\mid_{p=q}\cdot u^{\prime}(q)$
$\displaystyle+u^{\prime\prime}(x_{D}-q)F(2q)+u^{\prime\prime}(q)(1-F(2q))<0.$

Thus, any $q$ with $\tfrac{\partial}{\partial p}\pi_{F}(p\mid
x_{D})\mid_{p=q}=0$ is an isolated local maximum. There can’t be more than one
such isolated local maximum $q$, since if there were (at least) two, there
must be infinitely many in between them: Let $q_{1}<q_{2}$ with
$\tfrac{\partial}{\partial p}\pi_{F}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=q_{i}}=0$, then by the above
calculation $\tfrac{\partial^{2}}{\partial p^{2}}\pi_{F}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=q_{i}}<0$, thus there exists
$q_{1}<p_{1}<p_{2}<q_{2}$ with $\tfrac{\partial}{\partial p}\pi_{F}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=q_{1}}<0,\tfrac{\partial}{\partial
p}\pi_{F}(p\mid x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=q_{2}}>0$. By
continuity of $\tfrac{\partial}{\partial p}\pi_{F}(p\mid x_{D})$ there must
thus be a point $m_{1}<q_{3}<p_{2}$ with $\tfrac{\partial}{\partial
p}\pi_{F}(p\mid x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=q_{3}}=0$. That
way, we can construct infinitely many isolated maxima in $[q_{1},q_{2}]$. An
accumulation point of these must be a zero of the first derivative of
$\pi_{F}$ itself be continuity of $\tfrac{\partial}{\partial p}\pi_{F}$. But
this can’t be an isolated maximum - a contradiction.

Now, note that if there’s a unique such $q$, then $\tfrac{\partial}{\partial
p}\pi_{F}(p\mid x_{D})$ must be positive before $q$ and negative afterwards by
continuity. Thus, it’s strictly quasiconcave.

Finally, if there’s no such $q$, $\tfrac{\partial}{\partial p}\pi_{F}$ must
either be positive or negative on the whole domain and thus strict
quasiconcavity of $\pi_{F}(p\mid x_{D})$ holds in both cases. ∎

###### Proof of Lemma 1.

Keep in mind that the divider can always ensure a payoff of
$u(\tfrac{x_{D}}{2})$ by the full-hedging strategy $\bar{p}=\tfrac{x_{D}}{2}$.
An optimal price should not be worse than this. Now, firstly consider a price
offer of $2p<{x_{l}}$. As $x_{C}-p>p$, the chooser will take the offer and the
divider faces a payoff of $u(p)<u(\tfrac{x_{D}}{2})$. Secondly, if a price
offer with ${x_{h}}<2p$ is made, the chooser will sell her shares, yielding a
payoff of $u(x_{D}-p)<u(\tfrac{x_{D}}{2})$ for the divider. Consequently, the
divider only considers prices $2p\in[{x_{l}},{x_{h}}]$. ∎

It is worth noting that the proof of Lemma 1 only makes use of $F$ being a
distribution function, i.e. being $0$ resp. $1$ for $x<{x_{l}}$ resp.
${x_{h}}<x$, leading to the functions $u(p)$ resp. $u(x_{D}-p)$.
While we’re at it, the same thought reveals that $\pi_{G}$ being strictly
quasi concave in $p$ for $2p\in X$ (Assumption 1) is equivalent to strict
quasiconcavity everywhere, c.f. the end of the proof of Lemma 3.

###### Proof of Proposition 2.

The proof is adapted from [mcafee92]’s Lemma 7, incorporating the more general
set-up using strict quasi-concavity as the main driver instead of the SHRCs.

The chooser’s best response has already been discussed, thus we focus on the
divider’s best reply.

Note that we can restrict to an interim perspective here as we simply maximize
the integrand of the expectation for each $x_{D}$ at a time and there’s no
correlation between the two valuations. Since $\pi_{F}(p\mid x_{D})$ is
strictly quasiconcave and continuous in $p$, there is a unique price
announcement $m(x_{D})$ for each $x_{D}$ maximizing the function, see Appendix
A.

Firstly, note that $\tfrac{\partial}{\partial p}\pi_{F}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{2p={x_{l}}}=f({x_{l}})\cdot(2\cdot(u(x_{D}-\tfrac{{x_{l}}}{2})-u(\tfrac{{x_{l}}}{2})))+u^{\prime}(\tfrac{{x_{l}}}{2})>0$
since $F({x_{l}})=0$, $x_{D}\geq{x_{l}}$ and $u$ is strictly increasing with
$u^{\prime}>0$. Similarly, we find $\tfrac{\partial}{\partial p}\pi_{F}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{2p={x_{h}}}<0$. Together with
continuously differentiability, this implies that $m(x_{D})$ is an interior
local maximum, i.e. ${x_{l}}<2m(x_{D})<{x_{h}}$ and $\tfrac{\partial}{\partial
p}\pi_{F}(p\mid x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=m(x_{D})}=0$.

Secondly, as a zero of the derivative function, it suffices to show that
$\tfrac{\partial^{2}}{\partial x_{D}\partial p}\pi_{F}(p\mid x_{D})>0$ to
prove that $m(x_{D})$ is strictly increasing. This is immediate from

$\frac{\partial^{2}}{\partial x_{D}\partial p}\pi_{F}(p\mid x_{D})=2f(2p)\cdot
u^{\prime}(x_{D}-p)-F(2p)\cdot u^{\prime\prime}(x-p)>0,$

since $f>0$ and $u$ is strictly increasing and concave with $u^{\prime}>0$.

Recall that $\mu_{F}$ is the median of $F$, i.e. $F(\mu_{F})=\tfrac{1}{2}$.
Calculating $\tfrac{\partial}{\partial p}\pi_{F}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{2p=x_{D}}=u^{\prime}(\tfrac{x_{D}}{2})\cdot(1-2F(x_{D}))$
and checking its sign, we find that $x_{D}\lessgtr 2m(x_{D})$ if
$x_{D}\lessgtr\mu_{F}$. In addition, since $\tfrac{\partial^{2}}{\partial
p^{2}}\pi_{F}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{2p=x_{D}}=-8f(x_{D})u^{\prime}(\tfrac{x_{D}}{2})+u^{\prime\prime}(\tfrac{x_{D}}{2})<0$
the function $\pi_{F}(p\mid x_{D})$ is locally strictly concave in $2p=x_{D}$,
thus, $x_{D}=\mu_{F}$ implies $2m(\mu_{F})=\mu_{F}$.

Finally, we turn to the relation between $2m(x_{D})$ and $\mu_{F}$. Consider,
e.g., $x_{D}<\mu_{F}$. We already know that this implies $x_{D}<2m(x_{D})$. By
the properties of $u$ we thus have $u(x_{D}-m(x_{D}))<u(m(x_{D}))$ and
$u^{\prime}(x_{D}-m(x_{D}))>u^{\prime}(m(x_{D}))$. This implies

$\displaystyle 0$ $\displaystyle=\frac{\partial}{\partial p}\pi_{F}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=m(x_{D})}$
$\displaystyle=-u^{\prime}(x_{D}-m(x_{D}))F(2m(x_{D}))+u^{\prime}(m(x_{D}))(1-F(2m(x_{D}))$
$\displaystyle\phantom{==}+2f(2m(x_{D}))(u(x-m(x_{D}))-u(m(x_{D})))$
$\displaystyle<u^{\prime}(m(x_{D}))\cdot(1-2F(2m(x_{D}))),$

revealing $2m(x_{D})<\mu_{F}$. A similar argument shows that $x_{D}>\mu_{F}$
implies $2m(x_{D})>\mu_{F}$.

Note that the chooser’s decision rule, accepting the offer if and only if
$x_{C}-p\leq p$ for all $p$ and $x_{C}$, is always a best reply. Since the
divider’s price announcement is a unique best reply for each $x_{D}$ they form
an interim Bayesian Nash equilibrium and thus a Bayesian Nash equilibrium in
the ex-ante stage as well.

The statement about interim expected utility is proved exactly as Theorem 9 in
[mcafee92]. We will investigate a generalization of this using Knightian
Uncertainty in Section 3.3. ∎

###### Explicit calculations to Example 3.

It is straightforward to show that the induced respective utility functions
are given by

$\displaystyle\pi_{G_{0}}(x_{D},p)$ $\displaystyle=\begin{cases}p&,0\leq
2p\leq 2\epsilon,\\\ -4p^{2}+4p\epsilon+x_{D}(2p-2\epsilon)+p&,2\epsilon\leq
x<1,\\\ x_{D}-p&,2p=1.\end{cases}$ $\displaystyle\pi_{G_{1}}(x_{D},p)$
$\displaystyle=\begin{cases}-4p^{2}-4p\epsilon+x_{D}(2p+2\epsilon)+p&,0\leq
2p\leq 1-2\epsilon\\\ x_{D}-p&,1-2\epsilon<2p\leq 1.\end{cases}.$

Note that $\pi$ is the function that stays $\pi_{G_{0}}$ until $2p=x_{D}$ and
is $\pi_{G_{1}}$ afterwards. Moreover, on $2p\in[{x_{l}},{x_{h}})$ both,
$\pi_{G_{0}}$ and $\pi_{G_{1}}$ are strictly quasiconcave.

The unique local maxima of the parabolas are located at
$2m_{0}(x_{D})=\tfrac{x_{D}}{2}+\epsilon+\tfrac{1}{4}$ and
$2m_{1}(x_{D})=\tfrac{x_{D}}{2}-\epsilon+\tfrac{1}{4}$, respectively. Note
that $m_{1}(x_{D})<m_{0}(x_{D})$ and moreover $2m_{1}(x_{D})\leq
x_{D}\iff\tfrac{1}{2}-2\epsilon\leq x_{D}$ as well as $x_{D}\leq
2m_{0}(x_{D})\iff x_{D}\leq\tfrac{1}{2}+2\epsilon$. The following conclusions
arise:

Firstly, for $x_{D}\in[\tfrac{1}{2}-2\epsilon,\tfrac{1}{2}+2\epsilon]$ the
highest value of the function $\pi$ is attained at $2p=x_{D}$ since it is
strictly increasing for values below it and strictly decreasing for higher
values of $2p$.

Secondly, for $x_{D}<\tfrac{1}{2}-2\epsilon$ we find $x_{D}<2m_{1}(x_{D})\leq
1$ and the maximum of $\pi$ is attained at $p=m_{1}(x_{D})$ ($\pi_{G_{0}}$ is
increasing, switching into $\pi_{G_{1}}$ and still strictly increasing until
$p=m_{1}(x_{D})$ and strictly decreasing afterwards).

Finally, for $\tfrac{1}{2}+2\epsilon<x_{D}$ we find $0\leq
2m_{0}(x_{D})<x_{D}$ an the maximum of $\pi$ is attained at $p=m_{0}(x_{D})$
($\pi_{G_{0}}$ strictly increases, reaches its top and starts to strictly
decrease and keeps doing to after switching to $\pi_{G_{1}}$). ∎

###### Proof of Lemma 3.

We will indeed show, that the corresponding payoff functions are strictly
quasi-concave not only on $X$ but on $\mathbb{R}$. We first note that the
stochastic dominant/dominated distributions in $\mathcal{G}(F,\epsilon)$ are
given by

$\displaystyle G_{0}(x)$ $\displaystyle=\begin{cases}0&,F(x-\epsilon)<0,\\\
F(x-\epsilon),&0\leq F(x-\epsilon),x<{x_{h}},\\\ 1&,{x_{h}}\leq x\end{cases}$
$\displaystyle G_{1}(x)$ $\displaystyle=\begin{cases}0&,x<{x_{l}},\\\
F(x+\epsilon)&,{x_{l}}\leq x,F(x+\epsilon)<1,\\\ 1&,1\leq
F(x+\epsilon).\end{cases}$

We first look on the resp. domains where the functional form of the
distribution is given by $F(x\pm\epsilon)$. Since $F$ fulfills the SHRC (Lemma
2) so do $G_{0},G_{1}$ as

$\displaystyle\frac{\partial}{\partial
x}\left(x+\frac{F(x\pm\epsilon)}{f(x\pm\epsilon)}\right)=\frac{\partial}{\partial
x}\left(x\pm\epsilon+\frac{F(x\pm\epsilon)}{f(x\pm\epsilon)}\right)\geq 0,$
$\displaystyle\frac{\partial}{\partial
x}\left(x-\frac{1-F(x\pm\epsilon)}{f(x\pm\epsilon)}\right)=\frac{\partial}{\partial
x}\left(x\pm\epsilon-\frac{1-F(x\pm\epsilon)}{f(x\pm\epsilon)}\right)\geq 0,$

by our assumption on $F$. Note that the induced payoff functions on the
domains where a CDF is equal to $0$ resp. $1$ are $u(p)$ resp. $u(x_{D}-p)$
which are strictly increasing resp. decreasing. Thus, to ensure global
strictly quasiconcavity, it we thus only need to look at the (possible)
discontinuities as they could produce jumps that are not in line with
Definition A2. Specifically, it suffices to show
$\lim_{2p\nearrow{x_{h}}}\pi_{G_{0}}(p\mid
x_{D})\geq\pi_{G_{0}}(\tfrac{{x_{h}}}{2}\mid x_{D})$ and
$\lim_{2p\nearrow{x_{l}}}\pi_{G_{1}}(p\mid
x_{D})\leq\pi_{G_{1}}(\tfrac{{x_{l}}}{2}\mid x_{D})$.

Firstly, we have $\lim_{2p\nearrow{x_{h}}}\pi_{G_{0}}(p\mid
x_{D})=u(x_{D}-\frac{{x_{h}}}{2})F({x_{h}}-\epsilon)+u(\frac{{x_{h}}}{2})(1-F({x_{h}}-\epsilon))\geq
u(x_{D}-\frac{{x_{h}}}{2})=\pi_{G_{0}}(\frac{{x_{h}}}{2})$. Thus,
$\pi_{G_{0}}$ is strictly quasiconcave on $\mathbb{R}$.

Secondly, we have $\lim_{2p\nearrow{x_{l}}}\pi_{G_{1}}(p\mid
x_{D})=u(\frac{{x_{l}}}{2})\leq
u(x_{D}-\frac{x}{2})F({x_{l}}+\epsilon)+u(\frac{{x_{l}}}{2})(1-F({x_{l}}+\epsilon))=\pi_{G_{1}}(\frac{{x_{l}}}{2}\mid
x_{D})$. Thus, $\pi_{G_{1}}$ is strictly quasiconcave on $\mathbb{R}$. ∎

###### Calculations for Example 2.

It suffices to check the SHRC on the distribution’s support since all of them
are continuous on $\mathbb{R}$ (compare this to the ’glueing’ argument at the
end of the proof of Lemma 3). a
Triangular distribution:
On an interval $[a,b]$, the triangular distribution with mode $c\in[a,b]$ is
defined by

$F(x)=\operatorname{Tri}^{c}(x)=\begin{cases}0&,x<a,\\\
\frac{(x-a)^{2}}{(b-a)(c-a)}&,a\leq x<c,\\\ \frac{c-a}{b-a}&x=c,\\\
1-\frac{(b-x)^{2}}{(b-a)(b-c)}&,c<x\leq b,\\\ 1&,b<x.\end{cases}$

Note that $x=c$ can be seen as a limit case and that its density $f$ is
continuous on $[a,b]$. The proof of Lemma 2 thus reveals that it suffices to
check the SHRC on the intervals $[a,x)$ and $(x,b]$ with the appropriate left
and right limits/derivatives up to $x=c$. We have

$\displaystyle\frac{\partial}{\partial x}\left(x+\frac{F(x)}{f(x)}\right)$
$\displaystyle=\begin{cases}\frac{3}{2}&,a\leq x<c,\\\
\frac{3}{2}+\frac{(b-a)(b-c)}{2(b-x)^{2}}&,c<x\leq b\end{cases}\quad\geq 0,$
$\displaystyle\frac{\partial}{\partial x}\left(x-\frac{1-F(x)}{f(x)}\right)$
$\displaystyle=\begin{cases}\frac{3}{2}+\frac{(b-a)(c-a)}{2(x-a)^{2}}&,a\leq
x<c,\\\ \frac{3}{2}&,c<x\leq b\end{cases}\quad\geq 0.$

a
Truncated standard normal distribution:
We have
$F(x)=\tfrac{\Phi(x)-\Phi(0)}{\Phi(1)-\Phi(0)},f(x)=\tfrac{\phi(x)}{\Phi(1)-\Phi(0)},f^{\prime}(x)=-xf(x)$.
We find

$\displaystyle\frac{\partial}{\partial x}\left(x+\frac{F(x)}{f(x)}\right)$
$\displaystyle=2+x\cdot\frac{\Phi(x)-\Phi(0)}{\phi(x)}\geq 0,$
$\displaystyle\frac{\partial}{\partial x}\left(x-\frac{1-f(x)}{F(x)}\right)$
$\displaystyle=2-x\cdot\frac{\Phi(1)-\Phi(x)}{\phi(x)}\geq
2-1\cdot\frac{\Phi(1)-\Phi(0)}{\phi(1)}>\frac{1}{2}>0,$

for $x\in[0,1]$. Similar calculations can be made for other truncated normal
distributions.

a
A class of beta distributions:
We only consider the case $\beta=1\leq\alpha$ as the case $\alpha=1\leq\beta$
follows similarly. Note that $F(x)=x^{\alpha},f(x)=\alpha
x^{\alpha-1},f^{\prime}(x)=\alpha(\alpha-1)x^{\alpha-2}$, $x\in[0,1]$. We have

$\displaystyle\frac{\partial}{\partial x}\left(x+\frac{F(x)}{f(x)}\right)$
$\displaystyle=1+\frac{1}{\alpha}\geq 0,$
$\displaystyle\frac{\partial}{\partial x}\left(x-\frac{1-F(x)}{f(x)}\right)$
$\displaystyle=2+\underbrace{\tfrac{\alpha-1}{\alpha}}_{\leq
1}\underbrace{\left(x^{-\alpha}-1\right)}_{\geq-1}\geq 1>0$

for all $x\in(0,1]$.

∎

The proof of Theorem 1 is accompanied by some upfront Lemmata for a piecewise
twice continuously differentiable202020Not necessarily continuous itself.
distribution function $G$ with all left and right limits and strictly
quasiconcave function $\pi_{G}(p\mid x_{D})=u(x_{D}-p)G(2p)+u(p)(1-G(2p))$
with $m_{G}(x_{D}):=m_{\pi_{G}(.\mid x_{D})}$.

######  Lemma B1.

$\pi_{G}(p\mid x_{D})$ is strictly concave in $2p=x_{D}$.

###### Proof.

Let $G=G^{+},G^{-}$ denote the right resp. left limit function and
$g^{\pm}$,$g^{\prime\pm}$ be the left resp. right first resp. second
derivative of $G$. Note that the first left/right derivative is

$\displaystyle\frac{\partial^{\pm}}{\partial p}\pi_{G}(p\mid x_{D})=$
$\displaystyle-u^{\prime}(x_{D}-p)G^{\pm}(2p)+u^{\prime}(p)(1-G^{\pm}(2p))$
$\displaystyle+2g^{\pm}(2p)(u(x_{D}-p)-u(p)).$ (3)

The second left/right derivative is

$\displaystyle\frac{\partial^{2\pm}}{\partial p^{2}}\pi_{G}(p\mid x_{D})=$
$\displaystyle
u^{\prime\prime}(x_{D}-p)G^{\pm}(2p)+u^{\prime\prime}(p)(1-G^{\pm}(2p))$
$\displaystyle-4g^{\pm}(2p)\left(u^{\prime}(x_{D}-p)+u^{\prime}(p)\right)$
$\displaystyle+4g^{\prime\pm}(2p)(u(x_{D}-p)-u(p)).$

Evaluating in $2p=x_{D}$ from left and right yields

$\frac{\partial^{2\pm}}{\partial p^{2}}\pi_{G}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{2p=x_{D}}=u^{\prime\prime}(\tfrac{x_{D}}{2})-8g^{\pm}(x_{D})u^{\prime}(\tfrac{x_{D}}{2})<0,$

proving strict concavity there. ∎

######  Lemma B2.

We have

$\begin{cases}x_{D}<2m_{G}(x_{D})&,G(x_{D})<\tfrac{1}{2},\\\
x_{D}=2m_{G}(x_{D})&,G^{-}(x_{D})\leq\tfrac{1}{2}\leq G(x_{D}),\\\
2m_{G}(x_{D})<x_{D}&,\tfrac{1}{2}<G^{-}(x_{D}),\end{cases}$

where $G^{+}=G$ resp. $G^{-}$ denote the right resp. left limit function of
$G$. Furthermore, if $G(x_{D})<\tfrac{1}{2}$, then
$G^{-}(2m_{G}(x_{D}))<\tfrac{1}{2}$ and $\tfrac{1}{2}<G^{-}(x_{D})$ implies
$\tfrac{1}{2}<G^{-}(2m_{G}(x_{D}))$.

###### Proof.

See Lemma B1 for the notation and recall

$\displaystyle\frac{\partial^{\pm}}{\partial p}\pi_{G}(p\mid x_{D})=$
$\displaystyle-u^{\prime}(x_{D}-p)G^{\pm}(2p)+u^{\prime}(p)(1-G^{\pm}(2p))$
$\displaystyle+2g^{\pm}(2p)(u(x_{D}-p)-u(p)).$

Evaluating this in $2p=x_{D}$ (from left and right) yields

$\frac{\partial^{\pm}}{\partial
p}\pi_{G}\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{2p=x_{D}}=u^{\prime}(\frac{x_{D}}{2})\cdot(1-2G^{\pm}(x_{D})).$
(4)

Now if $G^{-}(x_{D})\leq G(x_{D})<\tfrac{1}{2}$ than (4) is positive and thus
so is the first derivative in an open neighborhood of $2p=x_{D}$ by continuity
of the resp. functions. Thus $x_{D}<2m_{G}(x_{D})$. The additional statement
follows as an adjustment of the end of the proof of Proposition 2: By strict
quasiconcavity, we must have

$\displaystyle 0$ $\displaystyle\leq\frac{\partial^{-}}{\partial
p}\pi_{G}(p\mid x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=m_{G}(x_{D})}$
$\displaystyle=-u^{\prime}(x_{D}-m_{G}(x_{D}))G^{-}(2m_{G}(x_{D}))+u^{\prime}(m_{G}(x_{D}))(1-G^{-}(2m_{G}(x_{D}))$
$\displaystyle\phantom{==}+2g^{-}(2m_{G}(x_{D}))(u(x-m_{G}(x_{D}))-u(m_{G}(x_{D})))$
$\displaystyle<u^{\prime}(m_{G}(x_{D}))\cdot(1-2G^{-}(2m_{G}(x_{D}))),$

revealing $G^{-}(2m_{G}(x_{D}))<\tfrac{1}{2}$.

If $\tfrac{1}{2}<G^{-}(x_{D})\leq G(x_{D})$, a similar argument reveals
$2m_{G}(x_{D})<x_{D}$ and $G^{-}(2m_{G}(x_{D}))<\tfrac{1}{2}$.

Finally, consider $G^{-}(x_{D})\leq\tfrac{1}{2}\leq G(x_{D})$. If both
inequalities are strict, we necessarily have $2m(x_{D})=x_{D}$ by strict
quasiconcavity. If $G^{-}(x_{D})=\tfrac{1}{2}=G(x_{D})$, $2p=x_{D}$ must be an
isolated local maximum by Lemma B1 and thus $x_{D}=2m_{G}(x_{D})$ by strict
quasiconcavity. Assume now exactly one inequality is strict (the other
binding), e.g., $G^{-}(x_{D})=\tfrac{1}{2}<G(x_{D})$ (the other case follows
analogously). Then, $\tfrac{\partial^{+}}{\partial
p}\pi_{G}\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{2p=x_{D}}<0$ by equation (4).
Furthermore, $\tfrac{\partial^{-}}{\partial p}\pi_{G}$ is positive on a
neighborhood $2p\in(x_{D}-\delta,x_{D})$ for some $\delta>0$ by Lemma B1.
Thus, strict quasiconcavity reveals $2m_{G}(x_{D})=x_{D}$. ∎

######  Lemma B3.

The optimal price announcement $m_{G}(x_{D})$ is increasing. If in addition
$G$ is differentiable in $2m_{G}(x_{D})$, $m_{G}(x_{D})$ is strictly
increasing in a neighborhood of $x_{D}$.

###### Proof.

Recall from the proof of Lemma B1 that

$\displaystyle\frac{\partial^{\pm}}{\partial p}\pi_{G}(p\mid x_{D})=$
$\displaystyle-u^{\prime}(x_{D}-p)G^{\pm}(2p)+u^{\prime}(p)(1-G^{\pm}(2p))$
$\displaystyle+2g^{\pm}(2p)(u(x_{D}-p)-u(p)).$

Consequently,

$\displaystyle\frac{\partial}{\partial x_{D}}\frac{\partial^{\pm}}{\partial
p}\pi_{G}(p\mid x_{D})$
$\displaystyle=-u^{\prime\prime}(x_{D}-p)G^{\pm}(2p)+4g^{\pm}(2p)u^{\prime}(x_{D}-p)\geq
0.$ (5)

By strict quasiconcavity we know that

$\displaystyle\frac{\partial^{+}}{\partial p}\pi_{G}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=m_{G}(x_{D})}$ $\displaystyle\leq
0$ $\displaystyle\frac{\partial^{-}}{\partial p}\pi_{G}(p\mid
x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=m_{G}(x_{D})}$ $\displaystyle\geq
0$

with a strict inequality each in a small neighborhood for $m_{G}(x_{D})<p$
resp. $p<m_{G}(x_{D})$. Thus, the inequality (5) reveals that $m(x_{D})$ is
increasing, though not necessarily strictly increasing. But it is strictly
increasing if $g^{-}>0$ (making (5) strict) and $\tfrac{\partial^{-}}{\partial
p}\pi_{G}(p\mid x_{D})\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=m_{G}(x_{D})}=0$
which are implied by, e.g., $G(2p)$ being differentiable in $p=m_{G}(x_{D})$
with positive derivative. If $G$ is even continuously differentiable with
positive derivative, applying the implicit function theorem reveals continuity
of $m_{G}$. ∎

###### Proof of Theorem 1.

Note that $g\hat{=}\pi_{G_{0}}$ and $f\hat{=}\pi_{G_{1}}$ fulfill Proposition
A1 from Appendix A. Now, applying Lemma B2 and realizing $G_{i}=G^{-}_{i}$ by
continuity212121There is no conflict with the possible discontinuity of
$G_{0}$ in ${x_{h}}$ as $2m_{G}(x_{D})={x_{h}}$ is only possible if
$x_{D}={x_{h}}$ and $G^{-}({x_{h}})<1$, where the middle case is active.
yields the functional form of the optimal price announcement. Finally, Lemma
B3 concludes the proof. ∎

###### Proof of Corollaries 1 and 2.

Most statements follow immediately from the proofs of Lemmata B2 and B3 and
since $x_{D}\mapsto\tfrac{x_{D}}{2}$ is continuous and strictly increasing.
For the continuity of $m(x_{D})$ in the ’glueing points’
$x_{D}\in\\{\mu_{G_{1}}^{-},\mu_{G_{0}}^{+}\\}$ for Corollary 2, we can apply
a sandwich argument using Corollary 1 for the limits coming from outside
$[\mu_{G_{1}}^{-},\mu_{G_{0}}^{+}]$. ∎

###### Proof of Lemma 4.

$\Phi_{C}(x_{C})=\min_{G\in\mathcal{G}}\mathbb{E}_{G}\left[\max\\{u(x_{C}-m(z)),u(m(z))\\}\right]$.

If $x_{C}<2m({x_{l}})$, the max-function will always choose $u(m(z))$. Thus,
as $z\mapsto u(m(z))$ is increasing, the worst-case is attained by $G=G_{1}$
which is stochastically dominated by every other distribution in $\mathcal{G}$
and puts most weight on the lowest price announcements. Interim worst-case EU
is thus $\mathbb{E}_{G_{1}}[u(m(z))]$.

If $x_{C}>2m({x_{h}})$, the max-function will always choose $u(x_{C}-m(z))$.
Thus, as $z\mapsto u(x_{C}-m(z))$ is decreasing, the worst-case is attained by
$G=G_{0}$, i.e. the distribution stochastically dominating all other
distributions in $\mathcal{G}$ and puts most weight on the highest price
announcements. Interim worst-case EU is thus
$\mathbb{E}_{G_{0}}[u(x_{C}-m(z))]$.

If $x_{C}\in[2m({x_{l}}),2m({x_{h}})]$, the worst-case is putting as much
weight at (and around) the valuation $z_{*}$ that induces a price announcement
$2m(z_{*})=x_{C}$ as possible. This value is unique since $m$ is strictly
increasing by our assumptions and Corollary 2. More precisely, consider the
partition $X=[{x_{l}},z_{*})\cup[z_{*},{x_{h}}]$. Note that for any
$x\in[{x_{l}},z_{*})$, the max-function will choose $u(x_{C}-m(z))$, thus the
expectation is minimized by $G_{0}$. For $x\in[z_{*},{x_{h}}]$ the max-
function selects $u(m(z))$, thus the expectation is minimized by $G_{1}$.
Piecing both cases together into a distribution function gets us
$G^{*}_{x_{C}}\in\mathcal{G}$ (it must be right-continuous, thus
$G^{*}_{x_{C}}(z_{*})=G_{1}(z_{*})$). Note that interim worst-case EU in this
case can be written as

$\displaystyle\mathbb{E}_{G^{*}_{x_{C}}}[\max\\{u(x_{C}-m(z)),u(m(z))\\}]$
$\displaystyle=$ $\displaystyle\int_{[{x_{l}},z_{*})}u(x_{C}-m(z))\
G_{0}(\mathrm{d}z)+(G_{1}(z_{*})-G_{0}(z_{*}))\cdot u(x_{C}/2)$
$\displaystyle\ +\int_{(z_{*},{x_{h}}]}u(m(z))\ G_{1}(\mathrm{d}z).$

∎

###### Proof of Lemma 5.

Before we get into the calculations, note that by our smoothness assumptions,
$m_{0},m_{1}$ are strictly increasing in $x_{D}$ and continuously
differentiable on the considered resp. intervals: Outside their medians, they
are zeros of a continuously differentiable function $\tfrac{\partial}{\partial
p}\pi_{G_{i}}(p\mid x_{D})$ and thus continuously differentiable themselves by
the implicit function theorem (see Lemma B2). This implies that also $m$ is
continuous, strictly increasing and piecewise continuously differentiable
(Lemma B2 and Theorem 1).

Thus, both $\Phi_{D}$ and $\Phi_{C}$ are continuous (indeed it will turn out
that they are even differentiable except for $\Phi_{C}$ in $x=2m({x_{l}})$),
so it suffices to show differentiability on the corresponding open intervals.
The derivatives of $\Phi_{D}$ for the center interval and of $\Phi_{C}$ for
the outer ones are obvious. We now turn towards the remaining ones.

Firstly, for $G\in\\{G_{0},G_{1}\\}$ consider $\Phi_{D}(x)=\pi_{G}(m(x)\mid
x)$ on the respective (open) interval. A calculation with
$u=\operatorname{id}$ (or using the envelope theorem) reveals

$\displaystyle\frac{\partial}{\partial x}\Phi_{D}(x)=$
$\displaystyle\frac{\partial}{\partial x}\left((x-m(x))\cdot
G(2m(x)+m(x)\cdot(1-G(2m(x)))\right)$ $\displaystyle=$ $\displaystyle
G(2m(x))+\frac{\partial}{\partial
x}m^{\prime}(x)\cdot\underbrace{\left(\frac{\partial}{\partial p}\pi_{G}(p\mid
x)\right)\,\rule[-4.0pt]{0.4pt}{11.0pt}\,{}_{p=m(x)}}_{=0}$ $\displaystyle=$
$\displaystyle G(2m(x)).$

Secondly, consider $\Phi_{C}$ for $x\in(2m({x_{l}}),2m({x_{h}}))$ and recall
$z_{*}=z_{*}(x)=m^{-1}(x/2)$. We now apply a measure theoretic version of the
differentiation of parameter integrals for the following calculation.
Therefore, $C:={x_{h}}+1$ serves as a constant bounding $z_{*}$ from above,
$\mu^{G}$ describes the probability measure associated to the distribution
function $G\in\\{G_{0},G_{1}\\}$ with densities $g_{0},g_{1}$, and we again
use the integration variable $z$ to avoid confusion.

$\displaystyle\frac{\partial}{\partial x}\Phi_{C}(x)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial x}\left(\int_{[{x_{l}},z_{*})}x-m(z)\
G_{0}(\mathrm{d}z)+(G_{1}(z_{*})-G_{0}(z_{*}))\cdot\frac{x}{2}+\int_{(z_{*},{x_{h}}]}u(m(z))\
G_{1}(\mathrm{d}z)\right)$ $\displaystyle=$ $\displaystyle\
\phantom{+}\int_{[{x_{l}},z_{*})}1\,G_{0}(\mathrm{d}z)-(x-m(z_{*}))\cdot\frac{\partial}{\partial
x}\left(\mu^{G_{0}}((z_{*},C))\right)$ $\displaystyle\
+\left(g_{1}(z_{*})\cdot\left(\frac{\partial}{\partial
x}(z_{*})\right)-g_{0}(z_{*})\cdot\left(\frac{\partial}{\partial
x}(z_{*})\right)\right)\cdot\frac{x}{2}+(G_{1}(z_{*})-G_{0}(z_{*}))\cdot\frac{1}{2}$
$\displaystyle\
+\int_{(z_{*},{x_{h}}]}0\,G_{1}(\mathrm{d}z)+m(z_{*})\cdot\frac{\partial}{\partial
x}\left(\mu^{G_{1}}([z_{*},C])\right).$

Realizing $\mu^{G_{0}}((z_{*},C))=1-G_{0}(z_{*})$ we find
$\tfrac{\partial}{\partial
x}\mu^{G_{0}}((z_{*},C])=-g_{0}(z_{*})\cdot\frac{\partial}{\partial x}(z_{*})$
and similarly for $G_{1}$. Since $m(z_{*})=x/2$ we thus obtain

$\displaystyle\frac{\partial}{\partial x}\Phi_{C}(x)$ $\displaystyle=$
$\displaystyle
G_{0}(z_{*})-\frac{x}{2}\cdot(-g_{0}(z_{*})\frac{\partial}{\partial
x}(z_{*}))+\left(g_{1}(z_{*})\cdot\left(\frac{\partial}{\partial
x}(z_{*})\right)-g_{0}(z_{*})\cdot\left(\frac{\partial}{\partial
x}(z_{*})\right)\right)\cdot\frac{x}{2}$ $\displaystyle\
+\frac{1}{2}\cdot(G_{1}(z_{*})-G_{0}(z_{*}))+\frac{x}{2}\cdot(-g_{1}(z_{*})\frac{\partial}{\partial
x}(z_{*}))$ $\displaystyle=$
$\displaystyle\frac{1}{2}\cdot(G_{1}(z_{*})+G_{0}(z_{*})).$

Finally, since the derivatives are non-negative, $\Phi_{D}$ and $\Phi_{C}$ are
increasing. ∎

###### Proof of Theorem 2.

a

Case $x^{0}<x^{1}$:
Under this assumption, we have $G_{1}(z)-G_{0}(z)<1$ for all $z\in X$. In
other words, for all $z\in X$ there is no $G\in\mathcal{G}$ that can assign
point mass $1$ to $z$. Since $m$ is measurable, as it’s strictly increasing,
we find $\mu^{G}(\\{z\,|\,m(z)=\tfrac{x}{2}\\})<1$ for all $G\in\mathcal{G}$.

We now prove that $\Phi_{D}(x)<\Phi_{C}(x)$ for all
$x\in[\mu_{G_{1}},\mu_{G_{0}}]$. On the one hand for any such $x$ we have
$m(x)=\tfrac{x}{2}$ and thus $\Phi_{D}(x)=\tfrac{x}{2}$. On the other hand

$\displaystyle\Phi_{C}(x)=\min_{G\in\mathcal{G}}\mathbb{E}_{G}\left[\underbrace{\max\\{x-m(z),m(z)}_{\geq\tfrac{x}{2}}\\}\right]>\tfrac{x}{2}=\Phi_{D}(x),$

where the strict inequality results from the argument above as the max can
only be $\tfrac{x}{2}$ if $2m(z)=x$.

In the following, we compare the derivatives on the outer intervals. We start
with $x\in[{x_{l}},\mu_{G_{1}})$. We have the following chain of arguments

$\displaystyle x$ $\displaystyle<\mu_{G_{1}}$ $\displaystyle\implies x$
$\displaystyle<2m(2m(x))\qquad(\text{Corollary }\ref{Corollary: B xm and fix
points})$ $\displaystyle\implies m^{-1}(\tfrac{x}{2})$ $\displaystyle<2m(x)$
$\displaystyle\implies G_{1}(m^{-1}(\tfrac{x}{2}))$
$\displaystyle<G_{1}(2m(x))\qquad(\star)$
$\displaystyle\implies\frac{1}{2}\left(G_{1}(m^{-1}(\tfrac{x}{2}))+G_{0}(m^{-1}(\tfrac{x}{2}))\right)$
$\displaystyle<G_{1}(2m(x))\qquad(G_{0}\leq G_{1})$
$\displaystyle\implies\Phi^{\prime}_{C}(x)$
$\displaystyle<\Phi^{\prime}_{D}(x),$

where we note that $\Phi^{\prime}_{C}(x)$ might be zero, but cannot be $1$
since $x<2m_{1}(x)=2m(x)<2m({x_{h}})$ as $2m$ is strictly increasing. The
inequality in $(\star)$ remains a strict one, since (using Corollary 1)
$m^{-1}(\tfrac{x}{2})<x<\mu_{G_{1}}\leq x^{1}$ and thus, by its functional
form, $G_{1}$ is strictly increasing in a neighborhood of
$m^{-1}(\tfrac{x}{2})$.

We now turn towards the case $x\in(\mu_{G_{0}},{x_{h}}]$. In analogy to the
above, the following chain of arguments applies:

$\displaystyle\mu_{G_{0}}$ $\displaystyle<x$ $\displaystyle\implies 2m(2m(x))$
$\displaystyle<x$ $\displaystyle\implies 2m(x)$
$\displaystyle<m^{-1}(\tfrac{x}{2})$ $\displaystyle\implies G_{0}(2m(x))$
$\displaystyle<G_{0}(m^{-1}(\tfrac{x}{2}))$ $\displaystyle\implies
G_{0}(2m(x))$
$\displaystyle<\frac{1}{2}\left(G_{0}(m^{-1}(\tfrac{x}{2})+G_{1}(m^{-1}(\tfrac{x}{2})\right)$
$\displaystyle\implies\Phi^{\prime}_{D}(x)$
$\displaystyle<\Phi^{\prime}_{C}(x),$

where we note that indeed $\Phi^{\prime}_{C}(x)$ could be equal to $1$ and
that applying $G_{0}$ preserves the strict inequality since
$x^{0}\leq\mu_{G_{0}}<x<m^{-1}(\tfrac{x}{2})$.

Together, the above steps imply $\Phi_{D}(x)<\Phi_{C}(x)$ for all $x\in X$.

Case $x^{1}\leq x^{0}$:
For $x\in[x^{1},x^{0}]$ we have by Theorem 1 $z_{*}:=m^{-1}(\tfrac{x}{2})=x$
and there is $G\in\mathcal{G}$, namely $G=G^{*}_{x}$, with
$\mu^{G}(\\{z_{*}|m(z_{*})=\tfrac{x}{2}\\})=1$. Hence, the argument of the
first part of the previous case reveals that
$\Phi_{D}(x)=\Phi_{C}(x)=\frac{x}{2}$ if $x=z_{*}\in[x^{1},x^{0}]$ and
$\Phi_{D}(x)<\Phi_{C}(x)$ otherwise. ∎

###### Explicit formulae for $\Phi_{C}$ in Example 4.

For $\tfrac{1}{2}<\epsilon\leq 1$ we find

$\displaystyle\Phi_{C}(x)=\begin{cases}\frac{1}{4}x^{2}+\frac{1}{2}\epsilon
x+\frac{1}{4}(1-\epsilon)^{2}&,0\leq x<1-\epsilon,\\\
\frac{x}{2}&,1-\epsilon\leq x\leq\epsilon,\\\
\frac{1}{4}x^{2}+\frac{1}{2}(1-\epsilon)x+\frac{1}{4}\epsilon^{2}&,\epsilon<x\leq
1,\end{cases}$

while for $\tfrac{1}{4}<\epsilon\leq\tfrac{1}{2}$ it is given by

$\displaystyle\Phi_{C}(x)=\begin{cases}\frac{1}{8}\epsilon^{2}-\frac{1}{2}\epsilon+\frac{9}{32}&,0\leq
x<2m_{1}(0)=\frac{1}{4}-\frac{1}{2}\epsilon,\\\
\frac{1}{2}x^{2}+(\epsilon-\frac{1}{4})x+\frac{1}{2}\epsilon^{2}-\frac{3}{4}\epsilon+\frac{5}{16}&,\frac{1}{4}-\frac{1}{2}\epsilon\leq
x<\frac{1}{2}-\epsilon\\\ \frac{1}{4}x^{2}+\frac{1}{2}\epsilon
x+\frac{1}{4}\epsilon^{2}-\frac{1}{2}\epsilon+\frac{1}{4}&,\frac{1}{2}-\epsilon\leq
x<\epsilon,\\\
\frac{1}{2}x^{2}+\frac{1}{2}\epsilon^{2}-\frac{1}{2}\epsilon+\frac{1}{4}&,\epsilon\leq
x\leq 1-\epsilon,\\\
\frac{1}{4}x^{2}+\frac{1}{2}(1-\epsilon)x+\frac{1}{4}\epsilon^{2}&,1-\epsilon<x\leq\frac{1}{2}+\epsilon,\\\
\frac{1}{2}x^{2}+(\frac{1}{4}-\epsilon)x+\frac{1}{2}\epsilon^{2}+\frac{1}{4}\epsilon+\frac{1}{16}&,\frac{1}{2}+\epsilon<x\leq\frac{3}{4}+\frac{1}{2}\epsilon,\\\
x+\frac{1}{8}\epsilon^{2}-\frac{1}{2}\epsilon-\frac{7}{32}&,\frac{3}{4}+\frac{1}{2}\epsilon=2m_{0}(1)<x\leq
1,\end{cases}$

and for $0\leq\epsilon\leq\tfrac{1}{4}$ we finally have

$\displaystyle\Phi_{C}(x)=\begin{cases}-\frac{3}{8}\epsilon^{2}-\frac{1}{4}\epsilon+\frac{1}{4}&,0\leq
x<2m_{1}(0)=\frac{1}{4}-\frac{1}{2}\epsilon,\\\
\frac{1}{2}x^{2}+(\epsilon-\frac{1}{4})x-\frac{1}{2}\epsilon+\frac{9}{32}&,\frac{1}{4}-\frac{1}{2}\epsilon\leq
x<\frac{1}{4},\\\
x^{2}+(\epsilon-\frac{1}{2})x-\frac{1}{2}\epsilon+\frac{5}{16}&,\frac{1}{4}\leq
x<\frac{1}{2}-\epsilon,\\\
\frac{1}{2}x^{2}-\frac{1}{2}\epsilon^{2}+\frac{3}{16}&,\frac{1}{2}-\epsilon\leq
x\leq\frac{1}{2}+\epsilon,\\\
x^{2}-(\epsilon+\frac{1}{2})x+\frac{1}{2}\epsilon+\frac{5}{16}&,\frac{1}{2}+\epsilon<x\leq\frac{3}{4}\\\
\frac{1}{2}x^{2}+(\frac{1}{4}-\epsilon)x+\frac{1}{2}\epsilon+\frac{1}{32}&,\frac{3}{4}<x\leq\frac{3}{4}+\frac{1}{2}\epsilon,\\\
x-\frac{3}{8}\epsilon^{2}-\frac{1}{4}\epsilon-\frac{1}{4}&,\frac{3}{4}+\frac{1}{2}\epsilon=2m_{0}(1)<x\leq
1.\end{cases}$

∎

###### Proof of Proposition 4 and calculations for Example 5.

Note that a triangular distribution $\operatorname{Tri}^{c}$ on $[0,1]$ with
mode $c$ has the CDF

$\operatorname{Tri}^{c}(x)=\begin{cases}\frac{x^{2}}{c}&,x<c,\\\ c&,x=c\\\
1-\frac{(1-x)^{2}}{(1-c)}&c<x.\end{cases}$

In $\mathcal{G}(\operatorname{Tri}^{c},\epsilon)$ the medians of
$G_{0}^{c},G_{1}^{c}$ can be derived to be

$\displaystyle\mu_{0}^{c}$
$\displaystyle=\begin{cases}\sqrt{\frac{c}{2}}+\epsilon&,\frac{1}{2}\leq c,\\\
1+\epsilon-\sqrt{\frac{1-c}{2}}&,c\leq\frac{1}{2}\end{cases}$
$\displaystyle\mu_{1}^{c}$
$\displaystyle=\begin{cases}\sqrt{\frac{c}{2}}-\epsilon&,\frac{1}{2}\leq c,\\\
1-\epsilon-\sqrt{\frac{1-c}{2}}&,c\leq\frac{1}{2}\end{cases}$

In the case of correlation we thus find by Theorem 1

$\displaystyle\begin{cases}\sqrt{\frac{x_{D}}{2}}-\epsilon\leq
x_{D}\leq\sqrt{\frac{x_{D}}{2}}+\epsilon&,\text{if }x_{D}=c\leq\frac{1}{2}\\\
1-\epsilon-\sqrt{\frac{1-x_{D}}{2}}\leq x_{D}\leq
1+\epsilon-\sqrt{\frac{1-x_{D}}{2}}&,\text{if }\frac{1}{2}\leq
x_{D}=c\end{cases}.$

Note that for $x_{D}\leq\tfrac{1}{2}$ we always have
$x_{D}\leq\sqrt{\tfrac{x_{D}}{2}}+\epsilon$ and for $\tfrac{1}{2}\leq x_{D}$
the statement $1-\epsilon-\sqrt{\tfrac{1-x_{D}}{2}}\leq x_{D}$ is always true.
In the case $c=\tfrac{1}{2}$ and $\epsilon=\tfrac{1}{5}$ we have
$\mu_{0}=\tfrac{7}{10},\mu_{1}=\tfrac{3}{10}$. The optimal price announcement
outside of $[0.3,0.7]$ can again be derived by the corresponding FOCs and
SOCs. Explicitly, they are given by

$\begin{cases}\frac{x}{6}+\frac{1}{60}\cdot\sqrt{100x^{2}+40x+79}-\frac{1}{15}&,x_{D}<\tfrac{3}{10},\\\
\frac{x}{6}-\frac{1}{60}\cdot\sqrt{100x^{2}-240x+219}+\frac{2}{5}&,\tfrac{7}{10}<x_{D}.\end{cases}$

∎