STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
SVETLANA BOYARCHENKO∗ AND SERGEI LEVENDORSKIǏ†
∗
Department of Economics, The University of Texas at Austin,
1 University Station C3100, Austin, TX 78712, U.S.A.
e-mail: [email protected]
†
Department of Mathematics, University of Leicester,
University Road, Leicester, LE1 7RH, U.K.
e-mail [email protected]
First Draft
Abstract. In a stochastic version of Fudenberg and Tirole’s [22] preemption game,
we analyze how the drift ambiguity in the underlying demand uncertainty affects
equilibrium strategies. Two firms contemplate entering a new market where the
demand follows a geometric Brownian motion with a known variance and unknown drift distributed over the ignorance interval [µ, µ] according to a set of priors
P = {Qµ | µ ≤ µ ≤ µ}. Firms differ is the sunk costs of entry. In the initial state,
entry is optimal to none of the firms. The standard results on entry under ambiguity without strategic interactions predict that the left boundary of the ignorance
interval is the the worst case prior, and only µ matters for entry decisions. Our
model demonstrates that the worst case prior of the low cost firm depends on the
state variable in a non-trivial way. Moreover, if the cost disadvantage between the
firms is sufficiently small so that the preemption zone is non-empty for every drift
in the ignorance interval, the preemption zone in the stopping time game under
ambiguity may disappear if if better priors are added (µ increases). In this case,
the value of the low cost firm increases by a non-zero margin, so ambiguity can be
good for the low cost firm.
Keywords: stopping time games, preemption, ambiguity
JEL: C73, C61, D81
The first author is thankful for discussions to Drew Fudenberg and Max Stinchcombe. The
authors are grateful to the participants of conferences SAET 2011, Faro, June 26-July 1, and especially to Frank Riedel and Jacco Thjissen for useful comments and suggestions. The usual disclaimer
applies.
1
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STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
1. Introduction
Stopping time games have important applications in economics and finance, such
as, for example, investment timing, product innovations, patenting, mergers and acquisitions, asset sales, pricing of convertible bonds (see, for instance, [1, 11, 12, 42,
45, 46], a survey [43], and a collection of papers [30]). Stopping time games are a
special case of stochastic games, where each instant each player has only two available
strategies: “wait” and “stop”. The latter strategy is irreversible. If one of the players
plays “stop”, the game is either terminated or the players’ payoffs become predetermined. If both players wait, the game environment evolves according to an exogenous
stochastic process, {Xt }t≥0 . If player i stops earlier than player j, then player i is
the leader, and player j is the follower. The stopping time game for two players was
initially formulated by Dynkin [13], as a generalization of optimal stopping problems
for the case of a zero sum game in discrete time, and later generalized by various
authors: see, for example, [29, 43, 21, 20] and the references therein. When players
are restricted to stopping times, the value of the game does not necessarily exist, and
the main part of the research on stopping games is focused on existence of the value
of the game and optimal stopping strategies. In more simple cases such as entry and
exit problem in duopoly and oligopoly, explicit solutions are available.
In the present paper, we consider two firms choosing optimal entry into a new
market under demand uncertainty. Demand shocks follow the geometric Brownian
motion (GBM) with a known variance and unknown drift distributed over the ignorance interval [µ, µ] according to a set of priors P = {Qµ | µ ≤ µ ≤ µ}. Notice that
the standard setting in the duopoly entry problems and related preemption game,
the future is either deterministic as, for example, in Fudenberg and Tirole [22], or
players know the objective probability law of the underlying stochastic process and
their beliefs are identical to this probability law (see, for example, Dixit and Pindyck
[9], Dutta and Rustichini [11], Grenadier [24, 25], Pawlina and Kort [38] Thijssen et
al. [46, 45], Weeds [48] and references therein). The latter assumption is quite strong,
especially if one is thinking about modeling entry into new markets or adopting a
completely new technology. In many real life situations, information is too imprecise
to be described adequately by a single prior. According to Knight [31], economic
agents face both measurable uncertainty (or risk) and unmeasurable (Knightian) uncertainty (or ambiguity). Furthermore, Knight [31] argues that ambiguity is quite
common in decision-making settings, therefore studies focusing on risk only, overlook an important factor governing decision making. Ellsberg Paradox demonstrates
that distinction between risk and ambiguity is important in decision making because
agents prefer to act on the basis of known probabilities rather than on the basis of
ambiguous probabilities.
Entry problem for a monopolist under Knightian uncertainty was considered in
Miao and Wang [34], Nishimura and Ozaki [37]. Riedel [41] and Cheng and Riedel
[7] developed a general theory of optimal stopping under ambiguity in discrete and
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
3
continuous time, respectively, and rigorously justified the results in [37]. However,
there are no general results for the cases of strategic (multi-player) environments.
For a monopolist and the standard model with the monotone payoff stream, effects
of ambiguity are straightforward: in the case of GBM with uncertain drift µ ∈ [µ, µ],
the worst case prior is the lowes possible value of the drift, so that the investment
threshold and value of the firm are as in the case µ = µ, and the better priors make no
impact on investment. Riedel (2009) and Cheng and Riedel (2011) have interesting
examples with non-monotone payoff functions, when not only one prior matters. We
show that strategic interactions may bring more than one prior into play even when
the payoff stream is monotone.
Fudenberg and Tirole [22] developed equilibrium concepts for a continuous time
preemption game for the case of symmetric players (firms) in a deterministic environment. Preemption games are a subset of stopping time games. In preemption games,
the leader’s payoff exceeds the follower’s payoff in some region of the state space. This
region is called the preemption zone. In the preemption zone, it may happen that
it is optimal to move for one firm, but not for both, hence there is a coordination
problem: who should move first. If players are symmetric, then there is a unique
(mixed strategies) symmetric equilibrium in the preemption zone: each of the players
becomes the leader with probability one half and the follower with probability one
half; and the probability of simultaneous action (coordination failure) is zero. The
game ends as soon as the left boundary of the preemption zone is reached. At this
point, the value functions of the leader and the follower are the same, therefore the
players are indifferent between becoming the first or the second mover. This is called
rent equalization.
The Fudenberg and Tirole [22] model was later extended to a stochastic environment (see, for example, Dixit and Pindyck [9], Dutta and Rustichini [11], Grenadier
[24, 25], Thijssen et al. [46, 45], Weeds [48] and references therein). In a stochastic
setting, if the initial shock is low, then none of the players has an incentive to act
(become the leader). If the underlying uncertainty is Gaussian or there are no upward
jumps in the stochastic process, then eventually, the left boundary of the preemption
zone is reached, and the outcome is qualitatively the same as in the deterministic
case.
Pawlina and Kort [38] consider the preemption game with asymmetric players under Gaussian uncertainty. They analyze the situation where two firms contemplating
investment into a profit enhancing project differ in the sunk cost of investment. Cost
asymmetry can be motivated by various factors such as regulations, liquidity constraints, credit history etc (see [38] for other factors). The cost asymmetry in [38]
uniquely defines the role of the firms provided that the initial shock is low so that
immediate investment is not optimal. The low cost firm is the leader and the high
cost firm is the follower. In [38], the game ends no later than the left boundary of
the preemption zone is reached. If the low cost firm had not invested earlier, then at
the boundary of the preemption zone, this firm strictly prefers to be the leader, and
4
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
the high cost firm is indifferent between being the leader and the follower, therefore
the low cost firm moves first, and coordination failure does not happen.
When the demand shocks follow the GBM (or, more generally, a Lévy process without positive jumps), and there is no ambiguity, the main results for the preemption
game with asymmetric firms can be summarized as follows. Two types of equilibria
are possible: (1) sequential one, when the low cost firm enters at the level optimal for
a monopolist, and the high cost firm enters at a higher level optimal for the follower,
and (2) preemptive one, when the low cost firm is forced to enter earlier, at the lower
bound of the preemption zone.
As in [38], we consider firms that differ in the sunk cost of investment and assume
that the initial realization of the demand shock is low enough so that the immediate
investment is optimal to none of the firms. Since the underlying process has continuous trajectories, then the roles of the firms are predetermined as in the case without
ambiguity: the low cost firm is the leader and the high cost firm is the follower. We
demonstrate that unlike in the case of the monopolist, µ is not always the worst-case
prior, and the worst-case prior non-trivially depends on the state variable.
There are at least two reasons for this non-trivial worst-case prior. The first one is
that the value of the leader at entry is a non-monotone function of the state variable.
Indeed, if the demand shock is much smaller than the entry barrier of the follower, the
leader enjoys the monopolists’s profits and the value function increases in the state
variable. As the shock moves closer to the level that triggers the entry of the follower
the leader’s value starts to decrease in the state variable: the expected time until
the entry of the second firm occurs and the profit flow drops to the duopolist’s profit
becomes smaller, and therefore the first mover’s advantage also becomes less valuable.
The leader’s value function reaches its local minimum at the entry threshold of the
follower and then becomes an increasing function, because the duopolist’s profit flow
increases in the value of the shock.
One might argue that with such a pattern of behavior of the leader’s value function,
one should expect that only µ and µ should matter as the worst-case priors (µ – for
the intervals where the value function is increasing, and µ – for the interval where
the value function is decreasing). The lower panel of Figure 1 clearly demonstrates
that this is not the case. The second reason for such non-trivial behavior of the
worst-case prior is that the leader’s value depends on the follower’s threshold. The
value of the leader is increasing in the follower’s threshold (the higher the follower’s
threshold, the longer will the leader enjoy the monopolist’s profits); and the follower’s
trigger is a decreasing function of µ (see Nishimura and Ozaki [37], Riedel [41] and
Cheng and Riedel [7]). If we forget for a moment about the demand uncertainty, then
from the point of view of the leader, the worst case is when the follower enters too
early. So had the leader make her decisions based only on the entry threshold of the
follower, she would have used µ to evaluate that threshold. The interplay of these
two considerations creates a complicated dependence of the worst-case prior on the
state variable.
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
5
If the demand shocks do not admit positive jumps, then the behavior of the worstcase priors discussed above is irrelevant, because the game stops either at the threshold chosen by the monopolist or at the left boundary of the preemption zone. In
a right neighborhood of either of this points, µ is the worst-case prior. However, if
there are positive jumps in the stochastic demand, then the demand shock may enter
the preemption zone, and there non-trivial dependence of the worst-case prior on the
state variable will matter.
Suppose that in the case of the Gaussian ambiguity the cost disadvantage between
firms (measured by k > 1 – the ratio of sunk costs of the firms) is sufficiently small so
that the preemption zone exists for each µ in the ignorance interval [µ, µ]. Calculations as in Boyarchenko and Levendorskiǐ [5] show that for a given µ the preemption
zone is an interval (SL (µ), SH (µ)), such that for S < SL (µ), and S > SH (µ), the
value of the follower for the high cost firm is greater than the value of the leader.
If S = SL (µ) or S = SH (µ) then the high cost firm is indifferent between being the
leader or the follower, but the low cost firm gets a positive benefit from the leadership. Our preliminary results show that for small k (the case we are interested in),
both SL and SH are decreasing in µ.
We assume that the high cost firm will not become the first mover, unless the gain
from the leadership is positive under every prior, and therefore the preemption zone
under ambiguity is the intersection of the intervals (SL (µ), SH (µ)) for all µ ∈ [µ, µ].
If such an intersection is non-empty, then the preemption zone under ambiguity is
∗
∗
an interval (SL∗ , SH
= SH (µ). If this is the case, then
), where SL∗ = SL (µ), and SH
the low cost firm enters either at the threshold Hl (µ), which is the threshold of
the monopolist in the model with drift ambiguity, provided Hl (µ) ≤ SL∗ , and then
sequential equilibrium happens; or at SL∗ , if SL∗ < Hl (µ), then preemptive equilibrium
happens. Notice that the value of the low cost firm in the sequential equilibrium is
higher than in the preemptive equilibrium.
The interesting case, which shows how ambiguity and strategic interactions distort
standard results, is when SH (µ) < SL (µ). Then the preemption zone under ambiguity
is empty even though it is non-empty for every µ ∈ [µ, µ], and the low cost firm
always enters at Hl (µ). Thus, ambiguity may destroy preemptive equilibria. In
particular, the preemption zone may disappear if better priors are added (µ increases).
If SL∗ < Hl (µ), the value of the low cost firm increases by a non-zero margin if the
preemption zone disappears, because the low cost firm enters at the threshold optimal
for the monopolist: ambiguity can be good for the low cost firm.
From the technical point of view, the optimal stopping problems arising in the
preemption model are relatively simple, and, as such, do not require the heavy machinery of the backward stochastic differential equations used in a much more general
context in Chen and Epstein [6] and Cheng and Riedel [7]. After the natural guess
is made, we use (the ambiguity version of) Dynkin’s formula and directly verify that
(1) in the action region, the value of the firm is a local supermartingale under at least
one of the priors, hence, it is not optimal to wait; (2) in the inaction region, the value
6
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
of the firm is a local martingale under the worst prior and local sub-martingale under
other priors. Hence, it is not optimal to stop.
The rest of the paper is organized as follows. In Section 2, we describe the economic
environment in more detail, recall the standard results in the no-ambiguity case, and
conduct the comparative statics of the investment thresholds and value functions as
functions of drift. This analysis is crucial for the study of the same problem with
the uncertain drift in Section 3. Section 4 concludes. Proofs of technical results are
relegated to the Appendix.
2. The model and study for a known drift
2.1. The model. Each of two firms has a single irreversible investment opportunity
in a new market. The firms differ in the sunk cost of investment. Assume that firm
1’s sunk cost is I1 = I > 0, and firm 2’s sunk cost is I2 = kI, where k > 1. Hence,
firm 1 is the low cost firm, and firm 2 is the high cost firm. After the investment is
made, each firm can produce one unit of output. For simplicity, assume that there
are no variable costs of production. Since there are only two firms in the industry, the
market supply is Q ∈ {0, 1, 2}. The investment is risky because the market demand
is stochastic. If only one firm is on the market, the revenue flow is D(1)St , and if
both firms are, then D(2)St . In this paper, we concentrate of the study of effects
of ambiguity in the simplest case when the underlying stochastic factor follows a
geometric Brownian motion St = eXt , where Xt is a Brownian motion with volatility
σ > 0 and unknown drift µ; the probability measure corresponding to µ is denoted by
Qµ , and P = {Qµ | µ ≤ µ ≤ µ} denotes the set of priors. Parameters I1 , I2 , D(1), D(2)
and σ are presumed known to both firms but the same firms know only interval [µ, µ]
for drift µ. Both firms discount the future at the same rate q > 0. We assume that,
for each µ ∈ [µ, µ], the no-bubble condition holds
q − Ψµ (1) > 0,
(2.1)
σ2
where Ψµ (β) = 2 β 2 + µβ; this condition ensures that the net present value of the
project (NPV) is finite under each prior Qµ . In the initial state, entry is optimal
to none of the firms. The ambiguity will be resolved when at least one of the firms
enters the market.
In the remaining part of this section, µ is presumed known to both firms.
2.2. Explicit analytical expressions for the value functions and investment
thresholds. Under the no-bubble condition (2.1), the characteristic equation (“fundamental quadratic” in the terminology of Dixit and Pindyck [9]) q − Ψ(γ) = 0 has
two roots βµ− < 0 < 1 < βµ+ .
Lemma 2.1. a) Ψµ (1) is an increasing function of µ.
b) βµ± are decreasing functions of µ.
Proof. a) Evident. b) The derivative dβµ± /dµ = (−1 ± √ 2 µ 2 )/σ 2 < 0, since
µ +2qσ
√
2
|µ| < µ + 2q for q > 0.
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
7
[ ]
For BM Xt , E eXt = eΨµ (1)t , therefore, under the non-bubble condition (2.1), the
expected present value of the stream St is
[∫ ∞
]
S
−qt Xt
X0
e e dt | e = S =
.
(2.2)
E
q − Ψµ (1)
0
Thus, the value of firm i in the case of simultaneous entry is
(2.3)
Pi (µ, S) =
D(2)S
− Ii ,
q − Ψµ (1)
where S is the spot value of the underlying stochastic factor at the moment of entry.
If firm i is the follower, its value at entry is given by Vfi (µ, S) = Pi (µ, S). Therefore,
the optimal entry threshold of a follower is determined from the well-known correction
to the NPV entry threshold (see Dixit and Pindyck [9])
D(2)Hfi (µ)
βµ+
= +
Ii ,
q − Ψµ (1)
βµ − 1
(2.4)
and the value function of the follower in the inaction region S < Hfi (µ) is of the form
+
αS βµ . Using the continuous pasting condition Vfi (µ, S) = Pi (µ, S) at S = Hfi (µ), we
obtain
+
+
Ii
(2.5)
Vfi (µ, S) = Pi (µ, Hfi (µ))(S/Hfi (µ))βµ = +
(S/Hfi (µ))βµ .
βµ − 1
Denote by τ (Hfj (µ)) is the first entrance time into [Hfj (µ), +∞). If firm i becomes
the leader at level S0 = S < Hfj (µ), where j ̸= i, then its value at the moment of
entry is
[∫
]
τ (Hfj (µ))
1
Vlead
(µ, S) = E
e−qt D(1)eXt dt | eX0 = S
0
[∫
]
+∞
+E
[∫
τ (Hfj (µ))
+∞
= E
−qt
e
[∫
e−qt D(2)eXt dt | eX0 = S − Ii
]
D(1)e dt | e
0
+∞
+E
τ (Hfj (µ))
Xt
X0
=S
]
e−qt (D(2) − D(1))eXt dt | eX0 = S − Ii .
The first term is proportional to the RHS of (2.2). On S < Hfj (µ), the second term
+
is of the form αS βµ (see Dixit and Pindyck [9]). Constant α can be found from the
1
value matching condition Vlead
(µ, S) = Pi (µ, S) at S = Hfj (µ), and the final result is
(2.6)
i
(µ, S)
Vlead
(D(2) − D(1))Hfj (µ)
+
D(1)S
=
− Ii +
(S/Hfj (µ))βµ .
q − Ψµ (1)
q − Ψµ (1)
8
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
Lemma 2.2. Assume that the high cost firm finds it optimal to be the follower or
commits itself to be the follower.
Then the optimal entry threshold for the low cost firm is defined by
βµ+
D(1)Hlp (µ)
=I +
,
(2.7)
q − Ψµ (1)
βµ − 1
and, for S < Hlp (µ), the value of the low cost firm is given by
+
1
Vlp (µ, S) = Vlead
(µ, Hlp (µ))(S/Hlp (µ))βµ .
(2.8)
Proof. First, note that (2.7) defines the optimal entry threshold of the monopolist (see
Dixit and Pindyck [9]); and that Hlp < Hf2 (µ) because D(2) < D(1) and I < I2 = kI.
The low cost firm faces the optimal stopping problem with the instantaneous pay1
off function Vlead
(µ, S), equivalently, the optimal stopping problem with the payoff
stream
f (µ, St ) = D(1)St 1St <Hf2 (µ) − qI + ((D(2) − D(1))St 1St ≥Hf2 (µ) .
As it was proved in Boyarchenko and Levendorskiǐ [2, 3, 4], the optimal entry threshold is determined by the stream f (µ, S t ), where S t = inf 0≤s≤t Ss is the infimum
process of St . If the initial point S < Hf2 (µ), all trajectories of the infimum process
do not leave (0, Hf2 (µ)). But, on this set, f (µ, St ) = D(1)St − qI is the payoff stream
for the monopolist. Hence, firm 1 finds it optimal to enter at the level optimal for
the monopolist. This proves (2.7).
+
In the inaction region, the value of the firm is of the form αS βµ . Applying the
value matching condition at Hlp (µ), we obtain (2.8).
It follows from (2.4), (2.7) and inequalities I2 = kI1 > I1 , D(1) > D(2) that
(2.9)
Hf2 (k, µ) = kHf1 (µ) > Hf1 (µ) = AHlp (µ) > Hlp (µ),
where A := D(1)/D(2) > 1.
Proposition 2.3. a) Investment thresholds Hfi (µ) and Hlp (µ) are decreasing in µ.
b) Value functions Pi (µ, S) and Vfi (µ, S) are increasing in µ.
Proof. a) Since
(2.10)
βµ−
βµ+
q
,
= +
q − Ψµ (1)
βµ − 1 βµ− − 1
equations for the investment thresholds Hfi (µ) and Hlp (µ) can be rewritten as
)
(
qIi 1 − βµ−
qIi
1
i
(2.11)
=
Hf (µ) =
1+
D(2) −βµ−
D(2)
−βµ−
)
(
qIi 1 − βµ−
qI
1
(2.12)
Hlp (µ) =
=
1+
D(1) −βµ−
D(1)
−βµ−
By Lemma 2.1 b), 1/(−βµ− ) is a decreasing function of µ, and a) follows.
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
9
b) Pi (µ, S) is decreasing on the strength of (2.3) and Lemma 2.1 a), and, for
S ≥ Hfi (µ), the same holds for Vfi (µ, S) = Pi (µ, S). In the region {(µ, S) | S <
Hfi (µ)}, (2.5) holds. If µ1 < µ2 , then βµ+1 > βµ+2 , and, therefore, for S close to 0,
Vfi (µ1 , S) < Vfi (µ2 , S). Since the curves S 7→ Vfi (µℓ , S), ℓ = 1, 2, intersect at one
point only, it remains to prove that Vfi (µ1 , Hfi (µ1 )) < Vfi (µ2 , Hfi (µ2 )). From (2.5) and
Lemma 2.1 b), Vfi (µ, Hfi (µ)) = Ii /(βµ+ − 1) is increasing in µ, which finishes the proof
of b).
Remark 2.4. The thresholds and value functions depend not only on µ but on the
other parameters of the model as well. We stress the dependence on µ because this
dependence is crucial for the study of ambiguity. However, in the next subsection,
where we study the preemption zone, whose very existence strongly depends on k,
we will indicate the dependence on some of the other parameters explicitly.
i
(k, A, µ, S).
2.3. Preemption zone. Denote DVfi (k, A, µ, S) = Vfi (k, A, µ, S) − Vlead
i
If, at some S, DVf (k, A, µ, S) < 0, i = 1, 2, then each firm prefers to be the leader
rather than the follower. The union of such S is called the preemption zone, where
the game is played. In the appendix, we prove
Lemma 2.5. 1. For each 0 < S < Hf1 (A, µ), DVf2 (k, A, µ, S) is an increasing
function of k on [1, +∞).
2. For each k > 0, DVf2 (k, A, µ, S) is a convex function of S on (0, Hf1 (A, µ)).
3. There exists k ∗ = k ∗ (µ, A) ≥ 1 such that
(a) if k > k ∗ , then DVf2 (k, A, µ, S) > 0 for any S > 0;
(b) if k = k ∗ , there exists a unique S ∗ = S ∗ (k, A, µ) such that DVf2 (k, A, µ, S ∗ ) = 0
but DVf2 (k, A, µ, S) > 0 for all S > 0, S ̸= S ∗ ;
(c) if 1 < k < k ∗ , there exist 0 < SL (k, A, µ) < SH (k, A, µ) < Hf1 (A, µ) such that
• DVf2 (k, A, µ, S) < 0 if and only if SL (k, A, µ) < S < SH (k, A, µ);
• DVf2 (k, A, µ, S) = 0 if and only if SL (k, A, µ) = S or S = SH (k, A, µ);
(d) numbers YL := SL (k, A, µ)/Hf1 (A, µ) and YH := SH (k, A, µ)/Hf1 (A, µ) are solutions of equation
(2.13)
+
+
[k 1−βµ + (A − 1)βµ+ ]Y βµ − Aβµ+ Y + k(βµ+ − 1) = 0.
Consider the interval S < Hf2 (µ). In Case (i), it is not optimal for the high cost
firm to be the leader at any S < Hf2 (µ), in Case (ii), the high cost firm is indifferent
between being the leader or follower at S = S ∗ (k, A, µ), and prefers to be the follower
at any other level S < Hf2 (µ). Finally, in Case (iii), the high cost firm prefers to be
the leader at S ∈ (SL (k, A, µ), SL (k, A, µ)), is indifferent between being the leader
or follower at S = SL (k, A, µ), SH (k, A, µ), and prefers to remain the follower at any
other S < Hf2 (µ).
it follows from the first statement of Lemma 2.5 that if firm 2 prefers to be the leader
rather than the follower, then firm 1 prefers to be the leader as well. We conclude
that the preemption zone exists if and only if 1 < k < k ∗ (µ), and then it is of the
10
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
100
0
−100
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.01
0.012
0.014
0.016
0.018
0.01
0.012
0.014
0.016
0.018
S
100
0
−100
0
0.002
0.004
0.006
0.008
S
100
0
−100
0
0.002
0.004
0.006
0.008
S
2
Figure 1. Value functions of firm 2: dots-dashes: Vlead
(µ, S) (leader);
2
dots: Vf (µ, S) (follower); dashes: P2 (µ, S) (in the case of simultaneous
investment). Upper panel: large k; lower panel: k close to 1, the
preemption zone exists; middle panel: the case k = k ∗ .
form (SL (k, A, µ), SH (k, A, µ)), where 0 < SL (k, A, µ) < SH (k, A, µ) < Hf1 (A, µ). At
the boundary of the preemption zone, the low cost firm enters, and the high cost
cannot preempt because it does not gain even if it becomes the leader, and loses in
the case of the simultaneous entry.
In figures 1 and 2 we plot the value functions of firm 2 under different scenarios
for k. The following lemma is important for our study of the effects of ambiguity.
Lemma 2.6. Let k < k ∗ (A, µ) and
(2.14)
then
(
)−1
A − 1 > e(β + µ − 1)
,
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
11
a)
∂DVf2 (k, A, µ, S) < 0,
S=SL (k,A,µ)
∂µ
and SL (k, A, µ) is a decreasing function of µ;
b) for k in a right neighborhood of one and k in a left neighborhood of k ∗ (A, µ),
∂DVf2 (k, A, µ, S) < 0,
(2.16)
S=SH (k,A,µ)
∂µ
and SH (k, A, µ) is a decreasing function of µ.
(2.15)
Proof in the appendix.
Remark 2.7. Note that condition (A.5) is sufficient but by no means necessary for
(2.15) to hold and SL (k, A, µ) to be a decreasing function of µ. At the same time,
if we assume that A := D(1)/D(2) ≥ 2 and βµ+ is not too close to 1 (βµ+ close to 1
corresponds to an extremely fast growing stream of profits), then (A.5) holds. Lemma
2.6 establishes that (2.16) holds if k is sufficiently close to one. It also holds if k is
sufficiently close to k ∗ . We were unable to find numerical examples where the LHS
of (2.16) is positive for any reasonable parameter values.
Fig.2 illustrates the typical behavior of DVf2 (µ), the other parameters being fixed.
2.4. Entry threshold and value for the low cost firm. If Hlp (µ) < SL (µ), then
the low cost firm enters at Hlp (µ), and, in the inaction region S < Hlp (µ), its value
coincides with the value of the monopolist given by (2.8); if Hlp (µ) ≥ SL (µ), the low
cost firm is forced to enter at the left boundary SL (µ) of the preemption zone, and,
in the inaction region S < SL (µ), its value if given by
(2.17)
+
1
VSL (µ, S) = Vlead
(µ, SL (µ))(S/SL (µ))βµ .
In Section A.3, we prove
Lemma 2.8. For any µ ∈ (µ, µ] and S ≤ Hlp (µ),
(2.18)
1
1
Vlead
(µ, S) > Vlead
(µ, S).
3. Strategic investment under ambiguity about the drift
3.1. Main results for asymmetric firms. Assuming that high cost form does not
preempt, low cost firm faces the optimal stopping problem with the instantaneous
payoff function
1
1
(µ, S)
Vlead;amb
(S) = min Vlead
µ≤µ≤µ
1
Vlead;amb
(S)
The properties of
are non-trivial (kinks, non-monotonicity and nonconcavity); see Fig. 3, upper panel, and the minimizing prior depends on S (lower
panel). However, if the high cost firm does not preempt, then the investment threshold and value function of the low cost firm in the inaction region is determined by
12
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
100
µmin
(µmax+µmin)/2
µmax
DVf
2
50
0
−50
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
S
0.02
Figure 2. Graph of DVf2 (µ) for different µ. Points of intersection of
the graphs with S-axis are SL (µ) and SH (µ).
the worst prior only as it is the case in the standard entry problem under ambiguity
but without strategic interactions.
Theorem 3.1. If the high cost firm does not preempt, then Hlp (µ) is the optimal entry
threshold for the low cost firm, and the value of the low cost firm in the inaction region
S < Hlp (µ) is given by
(3.1)
+
1
Vlp;amb (S) = Vlead
(µ, Hlp (µ))(S/Hlp (µ))βµ .
Proof in the appendix.
Remark 3.2. In a natural extension of the model of the present paper, when posi1
tive jumps are added, the qualitative behavior of Vlead
(µ, S) remains the same (this
is shown in Boyarchenko and Levendorskiǐ (2011)), therefore, one may expect that,
typically (for the case of a small jump component, at least), the behavior of Vlp;amb (S)
and minimizing prior will remain as shown in Fig.3. Possibility of a jump into the
region, where priors different from Qµ determine the value of Vlp;amb (S), will make the
low cost firm enter earlier than the monopolist would, even if there is no preemption
zone (in Boyarchenko and Levendorskiǐ (2011), it was shown that, if the preemption
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
13
60
40
20
0
−20
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.014
0.016
0.018
0.02
0.022
S
0.015
µ
0.01
0.005
0
0.004
0.006
0.008
0.01
0.012
S
1
Figure 3. Upper panel: solid line: graphs of Vlead;amb
(S) where it is
1
1
different from Vlead (µ, S); dot-dashes: Vlead (µ, S); dots: value Vlp;amb (S)
of the low cost firm in the inaction region. Lower panel: dependence
of minimizing µ on S.
zone is non-empty, then, in equilibria of certain types, possibility of jumps up make
the low cost firm enter earlier than the monopolist would), and the investment threshold and value function in the inaction region will depend on more than one prior.
Moreover, addition of better priors makes the low cost firm worse off. We leave a
rigorous study of these and other effects of interactions of jumps and ambiguity for
the future.
In Lemma 2.6, we proved that, under condition (A.5), which is satisfied for reasonable parameter values, the low boundary SL (µ) of the preemption zone is a (strictly)
decreasing function of µ. Moreover, condition (A.5) is not necessary, and we were
unable to find sets of parameters for which SL (µ) is not strictly decreasing in µ. The
situation is somewhat more complicated for SH (µ) because, for relatively reasonable
parameter values, SH (µ) can be non-monotone. Therefore, below, we assume that
SL (µ) is monotone on [µ, µ], and one of the following 3 conditions is satisfied:
(i) SH (µ) is decreasing on [µ, µ] (the case typically observed);
14
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
(ii) SH (µ) is increasing on [µ, µ];
(iii) the minimum of SH (µ) on [µ, µ] is attained at a unique point µ∗ ∈ (µ, µ).
(Generically, one of these 3 conditions holds.) Then the intersection of the closures
∗
], where
of the preemption zones [SL (µ), SH (µ)], µ ≤ µ ≤ µ, is of the form [SL∗ , SH
∗
∗
SL∗ = SL (µ), and, in Case (i), SH
= SH (µ), in Case (ii), SH
= SH (µ), and, in Case
∗
= SH (µ∗ ) < SH (µ). We assume that the preemption zone is non-empty in
(iii), SH
∗
] ̸= ∅,
the case of the worst prior: (SL (µ), SH (µ)) ̸= ∅. Then, in Case (ii), [SL∗ , SH
∗
] ̸= ∅ if the ignorance interval is narrow and µ is
and, in Cases (i) and (iii), [SL∗ , SH
close to µ.
∗
Theorem 3.3. a) The preemption zone under ambiguity is [SL∗ , SH
].
∗
∗
b) In particular, if SL > SH , then the preemption zone under ambiguity is empty.
∗
, the preemption zone under ambiguity is a point {SL∗ }.
c) If SL∗ = SH
Proof. The high cost firm finds it non-optimal to exchange its “follower state” for the
“leader state” if the gain is negative under at least one probability measure. More
specifically, high cost firm does not invest at S if, at least for one µ, the difference
2
Vlead
(µ, S) − Vf2 (µ, S) is negative. Therefore, the preemption zone under ambiguity
∗
contains the intersection (SL∗ , SH
) = ∩µ≤µ≤µ (SL (µ), SH (µ)) of the preemption zones
2
in the models with known µ ∈ [µ, µ]. We have Vlead
(µ, SL∗ ) − Vf2 (µ, SL∗ ) = 0 but
2
Vlead
(µ, SL∗ ) − Vf2 (µ, SL∗ ) > 0 for µ > µ. Since the agents consider not only Qµ but
priors different from Qµ as well, it is natural to presume that the agents assign a
non-zero (albeit unknown) probability to the event µ > µ. Therefore, contrary to
the standard argument about the boundary of the preemptive equilibrium in the
case without ambiguity, the ambiguity makes it optimal for the high cost firm to
∗
try to enter at the left boundary of the preemption zone. The argument for SH
is
similar.
∗
Corollary 3.4. a) If Hlp
≥ SL∗ , then the low cost firm is forced to enter (arbitrarily)
earlier than the process St reaches SL∗ , and the sequential ϵ-equilibrium results.
b) Assuming that the low cost firm enters at SL∗ − ϵ, where ϵ > 0, the value function
of the low cost firm in the inaction region S ≤ SL∗ − ϵ is given by
(
)βµ+
S
∗
1
.
(3.2)
Veps (S) = Vlead (µ, SL − ϵ)
SL∗ − ϵ
Proof. a) is immediate from Theorem 3.3. Part b) is proved in Appendix.
Corollary 3.5. a) In the case of ϵ-equilibria, the ambiguity (marginally) decreases
both the value of the low cost firm and investment threshold.
b) The preemption zone may be empty even if it is non-empty in the model without
ambiguity, for any µ ∈ [µ, µ]. Thus, ambiguity may destroy the preemptive equilibria,
and this may happen for an arbitrary small asymmetry.
c) In the most typical Case (i), the preemption zone shrinks as the ignorance interval increases.
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
15
8
V
6
4
2
0
3.95
4
4.05
4.1
4.15
4.2
4.25
4.3
4.35
4.4
4.45
4.25
4.3
4.35
4.4
4.45
S
4
ln V
2
0
−2
3.95
4
4.05
4.1
4.15
4.2
S
Figure 4. Effect of increase of the upper bound of the ignorance interval. Upper and low panel: graphs of the value function and log-value
function of the low cost firm. Dot-dashes: after the preemption zone
disappears (µ > −0.005, approx.) Dots: when the preemption zone
is non-empty (µ ≤ −0.005, approx.) Other parameters: µ = −0.069,
q = 0.03, σ = 0.08, I = 100, D2 = 1, D1 = 2.3, k = 1.01
d) It is possible that as better priors are added, the propensity of the high cost firm
to preempt and the preemption zone vanish. In the result, the low cost firm can enter
at the level optimal for a monopolist (under the worst prior assumption), and the
value of the low cost firm increases: ambiguity can be good for the low cost firm. This
may happen for an arbitrarily small asymmetry.
An example in Fig. 4 shows that the value function of the low cost firm below the
ambiguity zone for the worst prior can increase significantly (by a factor of 2.13) as
better priors are added and the preemption zone vanishes; the investment threshold
increases by 3.72% only. It remains to consider the high cost firm. After the low
cost firm enters and the ambiguity about µ is resolved, the entry threshold and value
function of the high cost firm will be the same as in the case without ambiguity.
However, before the low cost firm enters, the high cost firm evaluates its own value
under ambiguity.
Let H ∗ be the entry threshold of the low cost firm. At S = H ∗ (but before the low
cost firm enters), the ex-ante value of the high cost firm equals
2
Vf,amb
(H ∗ ) = inf Vf2 (µ, H ∗ ) = Vf2 (µ, H ∗ ).
µ≤µ≤µ
In the appendix, we prove
16
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
Theorem 3.6. Let H ∗ be the entry threshold of the low cost firm. Then, in the region
S < H ∗ , the value function of the high cost firm is given by
+
2
Vamb
(S) = Vf2 (µ, H ∗ )(S/H ∗ )βµ .
(3.3)
4. Conclusion
Appendix A. Technical proofs
A.1. Proof of Lemma 2.5. Introduce Y = S/Hf1 (A, µ), set β = βµ+ , and write
F (k, A, β, Y ) := I −1 DV 2 (k, A, µ, S) in the form
[ 1−β
]
k
β
Aβ
F (k, A, β, Y ) =
(A.1)
+ (A − 1)
Yβ +k−
Y.
β−1
β−1
β−1
1. Calculate
∂F (k, A, β, Y )
=1−
∂k
( )β
Y
> 0,
k
because Y < 1, and k ≥ 1. Hence F (k, A, β, Y ) is increasing in k for k ≥ 1 and
Y < 1.
2. Calculate
)
]
∂F (k, A, β, Y )
β [( 1−β
(A.2)
=
k
+ β(A − 1) Y β−1 − A ;
∂Y
β−1
2
( 1−β
) β−2
∂ F (k, A, β, Y )
(A.3)
=
β
k
+
β(A
−
1)
Y
> 0.
∂Y 2
Hence, F (k, A, β, Y ) is a convex function of Y .
As Y ↓ 0, F (k, A, β, Y ) → k > 0, hence, F (k, A, β, Y ) > 0 for Y in a neighborhood
of 0. To prove that F (k, A, β, Y ) > 0 for Y close to 1, note that, as Y ↑ 1,
(A.4) F (k, A, β, Y ) →
k 1−β
β
Aβ
k 1−β − β
+ (A − 1)
+k−
=
+k
β−1
β−1
β−1
β−1
The derivative of the RHS in (A.4), w.r.t. k, equals −k −β + 1, which is positive
because k > 1 and β > 0, and the limit of the RHS in (A.4) as k ↓ 1 equals 0. Hence,
F (k, A, β, 1) > 0 for k > 1, and F (k, A, β, Y ) > 0 for k > 1 and Y < 1 sufficiently
close to 1. We conclude that equation F (k, A, β, Y ) = 0 has either 0 solutions or
1 or 2, all of which belong to (0, 1). If F (k, A, β, Y ) > 0 for all k > 1 (this is
possible if β is sufficiently close to 1), then k ∗ = 1, and only case (i) is possible.
If, for k = 1, there exists Y ∗ such that F (1, A, β, Y ∗ ) < 0, then, for k sufficiently
close to 1, equation F (k, A, β, Y ) = 0 has two solutions, and case (iii) is observed.
Since Fk (k, A, β, Y ) = −k −β + 1 > 0, F (k, A, β, Y ) increases in k, and, evidently,
F (k, A, β, Y ) → +∞ as k → +∞. Therefore, as k increases, we will arrive ar k ∗ such
that (ii) holds, and, for k > k∗, case (i) will be observed.
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
17
A.2. Proof of Lemma 2.6. Function F (k, A, µ, Y ) decreases in Y near YL (k, A, µ) =
SL (k, A, µ)/Hf1 (A, µ) and increases in Y near YH (k, A, µ) = SH (k, A, µ)/Hf1 (A, µ).
Since βµ+ is decreasing in µ, F (k, A, µ, Y ) is decreasing (increasing) in µ if and only
if it is increasing (resp., decreasing) in βµ+ .
in a subregion, where
+
ln k · k 1−βµ < A − 1,
(A.5)
we have It follows from (A.2), that the derivative ∂F/∂Y is negative if and only if
+
+
(k 1−βµ + (A − 1)βµ+ )Y βµ − AY < 0. Using (2.13), we obtain
Aβµ+ Y − k(βµ+ − 1) − AY < 0,
Y = YL (k, A, µ),
which is equivalent to AYL (k, A, µ) < k. To finish the proof of a), we need to demonstrate that if (A.5) holds at Y = YL (k, A, µ), then ∂F/∂β + > 0 at Y = YL (k, A, µ).
(To simplify formulas below, we suppress the dependence of β + on µ, which is natural
because we regard β + as an independent variable). Direct calculations give
∂F
+
+
= −(β + − 1)−2 [k 1−β + β + (A − 1)]Y β + (β + − 1)−2 AY
+
∂β
+(β + − 1)−1 [− ln k · k 1−β + (A − 1)]Y β
] ∂Y
1 [ 1−β +
+
+ +
(k
+ (A − 1)β + ) ln Y Y β − β + A
β −1
∂β +
+
+
To calculate the derivative ∂Y /∂β + , note first, that
Y =
S
SD(2) −
SD(2) −β −
=
·
κ
(1)
=
·
.
q
Hf1
qI
qI
1 − β−
Introducing ρ = 2q/σ 2 , we can write −β − = ρ/β + , κ− = ρ/(ρ + β + ), and
(A.6)
∂Y
SD(2)
ρ
SD(2) −
1
Y
=−
·
=−
· κq (1) ·
=−
.
+
+
2
+
∂β
qI
(ρ + β )
qI
ρ+β
ρ + β+
Using (A.6), we write
[
(
) +]
∂F
+
−1
1−β +
+
(β − 1) + = (β − 1)
AY − k
+ β (A − 1) Y β
∂β
(
) +
1−β +
+ A − 1 − ln k · k
Yβ
[
]
Y
+
1−β +
+
β+
Aβ
−
(k
+
(A
−
1)β
)
ln
Y
Y
,
+
ρ + β+
+
For Y = YL and Y = YH , using (2.13), we simplify further
(
) +
∂F +
= k − AY + A − 1 − ln k · k 1−β Y β
(β + − 1) + ∂β Y =YL,H
[
]
Y
+
1−β +
+
β+
+
(A.7)
Aβ
−
(k
+
(A
−
1)β
)
ln
Y
Y
.
ρ + β+
18
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
At Y = YL , k > AY , and the first term on the RHS of (A.7) is positive. Let
+
ϕ(k) = ln k · k 1−β .
Observe that
)
dϕ
+ (
= k −β 1 − ln k(β + − 1) ,
dk
hence ϕ(k) is maximized at k = exp(1/(β + − 1), and
(
)
1
+
ϕ e1/(β −1) =
.
+
e(β − 1)
Therefore, condition (A.5) ensures that the second term on the RHS of (A.7) is
positive as well for all k. The last term is positive because ln Y < 0. This finishes
the proof of a).
b) We use (A.7) with Y = YH . In this case, k < AY , and the first term is negative,
but the second term is positive. To prove that the sum is negative if k is sufficiently
close to 1, note first that YH (k) ↑ 1 as k ↓ 1 (to see this, it suffices to note that the
LHS in (2.13) equals zero at (k, Y ) = (1, 1)). But at (k, Y ) = (1, 1), the RHS in (A.7)
is
Aβ +
Aβ +
1 − A + (A − 1) +
=
> 0.
ρ + β+
ρ + β+
By continuity, it is positive at (k, YH (k)), for k in a right neighborhood of 1.
Also, if k ↑ k ∗ , then YH (k) ↓ k ∗ /A, so that the first term on the RHS of (A.7)
vanishes, and the other two terms are positive, hence the RHS in (A.7) is positive.
By continuity, it is positive at (k, YH (k)), for k in a left neighborhood of k ∗ .
A.3. Proof of Lemma 2.8. In a moment, we will prove that, for any Y ∈ (0, A],
1
(A.8) the curve {(Y Hlp (µ), Vlead
(µ, Y Hlp (µ))) | µ ≤ µ ≤ µ} is downward sloping.
(It will be seen from the proof that, in fact, even larger Y are admissible). Since
1
Hlp (µ) is a decreasing function of µ, we conclude that, for S ≤ AHlp (µ), Vlead
(µ, S) >
1
Vlead (µ, S) if µ > µ. Under an assumption AHlp (µ) ≥ Hlp (µ), which is equivalent to
(A.9)
1 − βµ−
1 − βµ−
A
≥
−βµ−
−βµ−
(see (2.12)), Lemma is proved. Note that under a natural assumption about the demand, A = D(1)/D(2) > 2, and then (A.9) is satisfied even for fairly wide ignorance
intervals [µ, µ].
It remains to prove (A.8). Since Hlp (µ) and z := βµ+ − 1 are decreasing in µ, (A.8)
1
is equivalent to the statement that Vlead
(µ, Y Hlp (µ)) is a decreasing function of z.
Using (2.6) and equalities
βµ+
D(1)Hlp (µ)
= I +
q − Ψµ (1)
βµ − 1
Hlp (µ)/Hf2 (µ) = 1/(kA)
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
19
we obtain
(
)βµ+
D(1)Y
H
(µ)
D(2)
−
D(1)
Y
H
(µ)
lp
lp
1
+
Hf2 (µ)
I −1 Vlead
(µ, Y Hlp (µ)) =
−1
2
I(q − Ψµ (1))
I(q − Ψµ (1))
Hf (µ)
)
z+1(
=
(A.10)
Y − (1 − A−1 )(kA/Y )−z − 1.
z
We need to prove that the derivative of the RHS of (A.10) w.r.t. z
[(
)
]
(A.11) −z −2 Y − (1 − A−1 )(kA/Y )−z + (1 − A−1 )(kA/Y )−z ln(kA/Y )z(z + 1)
is non-positive for A > 1, k > 1, z > 0 and Y ≤ A.
Consider first the case Y = 1. Set ρ = ln(kA) and write expression (A.11) in the
form −e−ρz f (A, ρ, z)/z 2 . Then
f (A, ρ, z) = eρz − (1 − A−1 )(1 + ρz(z + 1)),
and we need to prove that f (A, ρ, z) ≥ 0 for z > 0, A > 1, ρ > ln A. A couple of
sufficient conditions are f (A, ln A, z) ≥ 0 and fρ (A, ρ, z) ≥ 0. Explicitly, the first
condition is
(A.12)
f (A, ln A, z) = Az − (1 − A−1 )(1 + ln Az(z + 1)) ≥ 0.
Using the Taylor expansion at z = 0, we conclude that, for fixed A > 1, (A.12)
holds in a small vicinity of z = 0. It is evident that, for fixed A > 1, (A.12) holds
if z is sufficiently large, and, for fixed z > 0, (A.12) holds if A is sufficiently large.
Therefore, there exist 0 < z0 and A1 > A0 > 1 such that f (A, ln A, z) > 0 outside
[1, z0 ] × [A0 , A1 ]. We have found such z0 , A0 , A1 , and verified numerically that (A.12)
holds for (z, A) ∈ [z0 , z1 ] × [A0 , A1 ].
Next, we prove the second condition fρ (A, ρ, z) ≥ 0. Set a = ln A. Condition
fρ (A, ρ, z) ≥ 0 is equivalent to eρz − (1 − A−1 )(z + 1) > 0. Since ρ > ln A, it suffices
to prove that for a := ln A ≥ 0 and z ≥ 0,
(A.13)
g(a, z) := eaz − (1 − e−a )(z + 1) > 0.
We have g(a, 0) = e−a > 0, therefore, it suffices to prove that the derivative gz (a, z) =
aeaz − 1 + e−a is positive for z ≥ 0. We have gz (a, 0) = a − 1 + e−a > 0 for a > 0
(indeed, gz (0, 0) = 0 and gza (a, 0) = 1 − e−a > 0 for a > 0), and gz (a, z) is increasing
in z; hence, gz (a, z) > 0, and g(a, z) > 0 as well.
+
For Y ∈ (0, 1), we represent expression (A.11) in the form −z −2 [Y − bY βµ ]. The
+
proof for the case Y = 1 implies that b < 1. Hence, −z −2 [Y − bY βµ ] < 0 for all
Y ∈ (0, 1].
Finally, if Y > 1, we set ρ = ln(kA/Y ), and represent expression (A.11) in the
form
−z −2 Y (kA/Y )−z [eρz − Y −1 (1 − A−1 )(1 + ρz(z + 1))]
< −z −2 Y (kA/Y )−z [eρz − (1 − A−1 )(1 + ρz(z + 1))].
20
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
The proof for the case Y = 1 gives that eρz − (1 − A−1 )(1 + ρz(z + 1)) > 0 if A > 1.
This finishes the proof of Lemma.
Finally, note that although we proved Lemma under certain weak conditions on
the parameters of the model, in numerous numerical examples that we considered,
the statement of Lemma holds.
A.4. Proof of Theorem 3.1. For any µ > µ, βµ+ > βµ+ > 1. From these inequalities
1
(µ, S), S ≤ Hf2 (µ), it follows that, for µ ≥ µ, function
and expression (2.6) for Vlead
+
+
1
(µ, Y 1/βµ ) is concave on 0 < Y ≤ (Hf2 (µ))βµ (strictly concave if
W (µ, Y ) = Vlead
+
µ > µ; function W (µ, Y ) is affine on (0, (Hlp (µ))βµ ). Since the infimum of a family
of the concave functions is a concave function, we infer from Lemma 2.8, that there
+
exists a unique α > 0 such that the graph of S 7→ αS βµ has a unique common point
1
1
with the graph of Vlead;amb
(S), and this point is (Hlp (µ), Vlead
(µ, Hlp (µ))). Define
+
1
1
1
(S) for S ≥ Hlp (µ).
Vlp;amb
(S) = αS βµ for S ≤ Hlp (µ) and Vlp;amb
(S) = Vlead;amb
Since βµ+ is a solution of the fundamental quadratic q − Ψµ (β) = 0,
1
(q − Lµ )Vlp;amb
(S) = 0, S < Hlp (µ).
+
1
Since αS βµ is an increasing function, we have (q−Lµ )Vlp;amb
(S) < 0 for any µ > µ and
∗
S < Hlp (µ). Denote by τ the first entrance time into [Hlp (µ), +∞). Let S0 < Hlp (µ)
and let a stopping time τ ≤ τ ∗ . Then, applying Dynkin’s formula
[∫ τ
]
[
]
1
Qµ
−qt
1
1
Vlp;amb (S0 ) = E
e (q − Lµ )Vlp;amb (St )dt + EQµ e−qτ Vlp;amb
(Sτ ) ,
0
[ −qτ 1
]
[
]
1
1
1
we derive Vlp;amb
(S0 ) = E
e Vlp;amb (Sτ ) , and Vlp;amb
(S0 ) < EQµ e−qτ Vlp;amb
(Sτ )
1
(St ) is a local martingale w.r.t. Qµ ,
for µ > µ. This means that on t < τ ∗ , Vlp;amb
and a local submartingale w.r.t. Qµ for all µ > µ. Hence, when the process remains
below Hlp , the agent chooses the “worst” measure Qµ for valuation.
To prove that it is not optimal not to invest in the region S > Hlp (µ), it suffices
to prove that, for each point S0 of this region, there exists a measure Qµ , µ ∈ [µ, µ],
1
and stopping time τ > 0 such that Vlp;amb
(St ) started at S0 satisfies
[
]
1
1
(A.14)
Vlp;amb
(S0 ) > EQµ e−qτ Vlp;amb
(Sτ )
Qµ
1
1
1
Take µ ∈ [µ, µ] such that (Vlp;amb
(S0 ) =)Vlead;amb
(S0 ) = Vlead
(µ, S0 ). We know that
1
(q − Lµ )Vlead (µ, S) > 0 for all S > Hlp (µ) (the non-strict inequality follows from the
fact that Hlp (µ) is the optimal entry threshold; it can be verified that the inequality
is, in fact, strict), therefore, if S0 > Hlp (µ), and τ does not exceed the first entry
time into (−∞, Hlp (µ)], then Dynkin’s formula gives
[
]
1
1
1
(A.15)
(Vlead;amb
(S0 ) =)Vlead
(µ, S0 ) > EQµ e−qτ Vlead
(µ, Sτ ) .
1
1
1
Since Vlead
(µ, Sτ ) ≥ Vlead;amb
(Sτ ), we can replace Vlead
(µ, Sτ ) on the RHS of (A.15)
1
1
with Vlead;amb (Sτ ) = Vlp;amb (Sτ ), and obtain (A.14).
STOPPING TIME GAMES UNDER KNIGHTIAN UNCERTAINTY
21
By Blumental 0-1 law, the process started at Hlp (µ) instantly enters (Hlp (µ), +∞),
1
hence, we can define Vlp;amb
(Sτ ) by continuity.
A.5. Proof of (3.2). Since no optimizing decision is involved, it is sufficient to verify
that Veps (St ) is a local martingale on t < τ (SL∗ − ϵ) under the worst prior Qµ and a
local submartingale under Qµ , µ > µ. The verification is the same as in the proof of
Theorem 3.1.
A.6. Proof of (3.3). The same as the proof of (3.2).
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