Asynchronous Revision Games with Deadline: Unique
Equilibrium in Coordination Games
Yuichiro Kamada and Takuo Sugaya
June 7, 2010
Abstract
Players prepare their actions before they play a normal-form game at a predetermined deadline. In
the preparation stage, each player stochstically obtains opportunities to revise their actions, and …nallyrevised action is played at the deadline. We show that, (i) a strictly Pareto-dominant Nash equilibrium,
if there exists, is the only equilibrium in the dynamic game; and (ii) in “battle of the sexes” games,
(ii-a) the equilibrium payo¤ set is a full-dimensional subset of the feasible payo¤ set under symmetric
structure, but (ii-b) a slight asymmetry can select a unique equilibrium.
Keywords: Revision games, …nite horizon, equilibrium selection, asynchronous moves
JEL codes: C72, C73
1
Introduction
Strategic interactions often require the preparation of the strategies in advance and even though players
interact once and for all, there can be a long period for the preparation. For example, consider two …rms
which have an investment opportunity. The date to make the actual investment is …xed. However, before the
actual investment, two …rms need to make the preparation: negotiating with banks to have enough liquidity,
allocating agents working for the project, and so forth.
It is natural to think that whether or not the investment is pro…table to one …rm depends on whether or
not the other …rm does the investment. Moreover, while the …rms make the preparation for the investments,
it is often the case that the …rms cannot always change their past preparation. That is, they have several
constraints for changing preparations, such as the administrative procedure to go through for changing
the past plan, obligations due to another project, and so forth. Rather, opportunities for revision of the
preparation may arrive stochastically over time.
This stochastic opportunity for the revisions allows players to credibly commit to the strategy. In the
above example, whether or not one …rm should invest depends on whether or not the opponent will invest.
If one …rm observes the opponent is making preparation for the investment, there is a probability that the
opponent will not have an opportunity to revise its decision and will invest, which makes the investment
more pro…table for the observant.
The present paper models such situations as follows: An n-player normal-form game is played once and
for all at a predetermined deadline, before which players have opportunities to revise their actions. These
opportunities arrive stochastically and independently among players. To model the preparation of actions,
we use the framework of a …nite horizon version of “revision games,” proposed by Kamada and Kandori
(2009). In revision games, players prepare their actions at opportunities in continuous time that arrive
Kamada: Department of Economics, Harvard University, Cambridge, MA, 02138, e-mail: [email protected]; Sugaya: Department of Economics, Princeton University, Princeton, NJ, 08544, e-mail: [email protected]; We thank Dilip
Abreu, Gabriel Carrol, Sylvain Chassang, Drew Fudenberg, Michihiro Kandori, Fuhito Kojima, Barton Lipman, Stephen Morris,
Satoru Takahashi, and seminar participants at Research in Behavior in Games seminar at Harvard for helpful comments.
1
with a Poisson process until a predetermined deadline. The actions that are prepared most recently at the
deadline are played once and for all. They consider the limit that the length of the preparation stage goes
to in…nity, so that there is enough number of opportunities by the deadline with a high probability. This is
equivalent to saying that they consider the limit that the arrival rate becomes very high for a …xed length of
preparation stage, that is, they consider the limit that players can revise their actions very frequently. This
assumption is also maintained in the present paper. We also assume that the Poisson arrival of the revision
opportunity is independent among players.
Kamada and Kandori (2009) show that, with certain conditions such as continuous strategy space, nonNash “cooperative”action pro…les can be played at the deadline. As they have shown, this result would not
be possible with a …nite game with a unique pure Nash equilibrium.1 Hence their focus is on expanding the
set of equilibria when the static Nash equilibrium is ine¢ cient relative to non-Nash pro…les. We ask a very
di¤erent question in this paper: we consider games with multiple e¢ cient static Nash equilibria, and ask
which of these equilibria is selected.2
We show that, even though the revision opportunity arrives with a very high probability, the small
probability of not being able to change the prepared action has surprisingly strong e¤ects on the equilibrium
selection: (i) if there exists a Nash equilibrium that strictly Pareto-dominates all the other action pro…les,
then it is the only equilibrium of the corresponding revision game; and (ii) in “battle of the sexes”games in
which Nash equilibria are not Pareto ranked, (ii-a) while with perfectly symmetric structure, the equilibrium
payo¤ set is a full-dimensional subset of the feasible payo¤ set, (ii-b) a slight asymmetry is enough to select
a unique equilibrium, which corresponds to the Nash equilibrium in the static game that gives the highest
payo¤ to the “strong”player. Actually, for (ii), it will be turned out that our result is valid for all the 2 x 2
games with two pure strategy Nash equilibrium. However, for the simple explanation, let us concentrate on
the battle of the sexes in the Introduction.
The rough intuition for our results is as follows: First, consider a game with strictly Pareto-dominant
Nash equilibrium. Firstly, we show that once all the players prepare the strictly Pareto-dominant Nash
equilibrium strategy, they will not escape from that state. Expecting that, if a unilateral change of the
preparation by player i can induce the strictly Pareto-dominant Nash equilibrium, she will do so. In turn,
player j whose unilateral change in the preparation can induce the situation where player i’s unilateral
change can induce the strictly Pareto-dominant Nash equilibrium, she will also do so if the deadline is far
expecting that player i will go to strictly Pareto dominant Nash equilibrium.3
Second, consider the “battle of the sexes”game with symmetric payo¤s and homogeneous arrival rate of
the revision opportunities. We prove that there are, in particular, three types of equilibria: The …rst one is
such that the process sticks to the Nash pro…le that player 1 prefers, from the beginning until the end of the
game. The second one is such that the process sticks to the Nash pro…le that player 2 prefers, again from
the beginning until the end of the game. The most important is the third one, in which the process starts at
an ine¢ cient non-Nash pro…le that would give a player the best payo¤ if the opponent switches to another
action. In this equilibrium, there is a “cuto¤” time such that if a player obtains an opportunity after this
cuto¤ while the opponent still sticks to the ine¢ cient action, the player switches her action. Once the action
pro…le reaches a Nash equilibrium, players will stick to that pro…le. Notice that, because of the symmetry,
two players’cuto¤s must be the same, and the indi¤erence condition at the cuto¤ implies that each player
expects their respective “worse” Nash equilibrium payo¤ in this equilibrium.4 We can then show that all
the equilibrium payo¤ set is such that at each point in the set, at least one player gets her “worse” payo¤.
1 The possibility of cooperation in …nite horizon in Kamada and Kandori (2009) is closely related to that of …nitely repeated
games with multiple Nash equilibria (Benoit and Krishna, 1985).
2 See also Ambrus and Lu (2009) for a variant of revision games model of bargainig in which the game ends when an o¤er is
accepted.
3 Similar results are obtained in the literature. See Farrell and Saloner (1985) and Dutta (2003) for early works on this topic.
Takahashi (2005) proves these results in a very general context.
4 The intution is similar to the one of the “war of attrition." The war of attrition is analyzed in Abreu and Gul (2000) and
Abreu and Pearce (2000). They consider the war of attrition problem in the context of bargaining. They show that if there
is an “irrational type" with a positive probability, then the agreement delays in equilibrium because rational players try to
imitate the irrational types. Players give in at the point where imitation is no longer pro…table. Although the war of attrition
is a common feature, the focus of their work and ours is quite di¤erent.
2
Hence, the equilibrium payo¤ set is a full-dimensional subset of the feasible payo¤ set.
Third, consider the “battle of the sexes”game with asymmetric payo¤s or asymmetric arrival rates of the
revision opportunities. We prove that the only equilibrium payo¤ is the one that corresponds to the Nash
equilibrium that the “strong” player (player 1) prefers, where by the “strong” player we mean the one who
expects more in his preferred Nash equilibrium than the opponent (player 2) does in the other pure Nash
equilibrium or the player with the lower arrival rate for the revision, i.e., the one who can more credibly
commit to the currently prepared action. The reason is simple. Consider the “chicken race equilibrium,”
the third type of the equilibrium discussed in the case of the symmetric “battle of the sexes”game. Because
we have asymmetry now, the cuto¤s of two players must di¤er. In particular, the strong player has to
have a cuto¤ closer to the deadline than that of the weak player. This implies that, in the chicken race
equilibrium, if it exists, the strong player expects strictly more than his “worse” Nash equilibrium payo¤,
while the weak player expects strictly less than her “worse” Nash equilibrium payo¤. Hence, the strong
player would not want to stick to the “worse”Nash equilibrium, which rules out the possibility of the second
type of equilibrium in the symmetric case. Also, the weak player would not want to stick to the chicken
race equilibrium, which rules out the third type of equilibrium in the symmetric case. Therefore the only
possibility is the …rst type of equilibrium. Thus a unique pro…le is selected.
We provide a detailed explanation of the intuition in Section 4.2 in a simpli…ed setting where the two Nash
equilibria are absorbing states. Note that, although the intuition looks simple, proving that the equilibrium
is unique is far less simple. This is because, no Nash action pro…le being a priori absorbing, hence when the
deadline if far away, players might have incentives to move away from the Nash pro…le. The above explanation
captures a part of this e¤ect because player 1 does not want to stick to his “worse” Nash equilibrium. We
will show that even player 2 does not want to stick to this pro…le (which he likes the best) when the deadline
if far away.
Finally, we note that the selection of equilibrium depends also on the heterogeneity of two players’arrival
rates. Naturally, a player with small arrival rate, i.e. the player with infrequent revision opportunities,
generally expects more from the game, with everything else symmetric. The reason is again simple. When
the arrival rate is small, the player can e¤ectively “commit” to his action for a while, so he can behave as a
“Stackelberg leader.”
Our results crucially hinges on asynchronicity of the revision process. If the revision process were synchronous, the very same indeterminancy among multiple strict Nash equilibria would be present. The result
that asynchronous moves select an equilibrium is not new, but the cases in which the asynchronous moves
result in equilibrium selection is very limited. Laguno¤ and Matsui (1997) show that in pure coordination
games (games in which all players’ payo¤ functions are the same) the Pareto e¢ cient outcome is chosen,
while Yoon (2001) shows that this result is nongeneric.5 Our results, although in a bit di¤erent context in
which the game is played only once at a deadline, show that the payo¤ structure can be anything in order
to obtain the best outcome as a unique equilibrium.6 ;7
Let us compare our work with the large literature on equilibrium selection. The key di¤erence between
our work and the past works in the literature is that, we consider a di¤erent class of situations than the ones
considered in the literature. Speci…cally, four features distinguish our model from the existing literature on
equilibrium selection. That is, we assume that (i) players are rational, (ii) they are nonanonymous, (iii) the
structure of the game is common knowledge, and (iv) the game is played once and for all. Let us overview
the related literature below. All of these four features seem to be present in the example mentioned in the
very …rst paragraph of this Introduction.
Kandori, Mailath, and Rob (1993) and Young (1993) consider an evolutionary learning model in which
5 Laguno¤ and Matsui (2001) argue that this nongenericity result curucially depends on the order of the limits with respect
to the discount factor and the purity of coordination.
6 For asynchronicity with respect to the timing of signals in imperfect monitoring setting, see Fudenberg and Olszewski
(2009).
7 Caruana and Einav (2008) consider a similar setting as ours, with …nite horizon and asynchronous moves, and show that
the equilibrium in generic 2 x 2 game there is a unique equilibrium, irrespective of the order and timing of moves and the
speci…cation of the protocol. They assume that players incur a cost whenever they revise their actions, which substantially
simplify their problem. In particular, on the equilibrium path, each player revises their actions at most once.
3
players interact repeatedly, and each player’s action at each period is stochastically perturbed. The key
di¤erence between their assumptions and ours is that in their model players are assumed to be boundedly
rational –players are assumed to play myopically in repeated interactions, following a rule-of-thumbs type
of decision rule, while we assume completely rational players. In addition, the game is repeated in…nitely in
their models, while the game is played once and for all in our model.
An evolutionary model called “perfect foresight dynamics” drops the bounded rationality assumption.
Matsui and Matsuyama (1994) and Oyama, Takahashi, and Hofbauer (2008) consider a model in which
patient players take into account the future path of the population dynamics. The key assumption to select
an equilibrium is that there is a population of agents who are randomly and anonymously matched over
time. In our model, on the other hand, …xed …nite players play a …nite horizon revision game.
Global games are another successful way to select an equilibrium. Rubinstein (1981), Carlsson and van
Damme (1993), Morris and Shin (1998), Sugaya and Takahashi (2009) show that non-existence of almost
common knowledge due to the incomplete information can select an equilibrium. Without almost common
knowledge, it is possible that a player thinks that his opponent thinks that the player thinks ... that the
game is very di¤erent from the one that the player believes, which is the key to rule out risk-dominated
equilibria. Thus, in particular, it is impossible to have a common knowledge that the game is in a small
open neighborhood of some a priori speci…ed one.
All of these three lines of the literature result in a selection of risk-dominant equilibrium of Harsanyi and
Selten (1988) in 2 x 2 games.8 In our model, however, a di¤erent answer is obtained: a Pareto-dominant
strict Nash equilibrium is played even if it is risk-dominated. Roughly speaking, since we assume complete
information with nonanonymous players, there is no “risk” of mis-coordination, which ensures that rational
players select a Pareto-dominant equilibrium.
Another line of the literature analyzes the e¤ect of the switching costs on the equilibrium selection.
Lipman and Wang (2000) consider a …nite horizon repeated games with small switching costs, in which the
per-period length (hence the per-period payo¤) is very small relative to the costs. They too use backward
induction argument to select a unique equilibrium in games that we consider in the present paper. However,
there are three important di¤erences. First, as the frequency of revision increases, it is potentially possible
in our model that players change their actions very often, while in Lipman and Wang (2000), the switching
cost essentially precludes this possibility. Second, the game is repeatedly played in their model, while it
is played only once at the deadline in our model. Hence, in particular, their model cannot have “chicken
race equilibrium” type of strategies in equilibrium. Third, in their model, the prediction of the game is not
robust to a¢ ne transformation of payo¤s.9 Hence, in some payo¤ speci…cations, their selection matches ours,
while in other cases it does not. The reason for this scale-dependence is that the switching cost is directly
payo¤-relevant. In our case, detailed speci…cations of the preparation stage (such as the arrival rate) is not
directly payo¤-relevant, so our result is robust to the a¢ ne transformation of the payo¤s.
Caruana and Einav (2008) consider a game where players play the actual game once and for all at the
…xed date and before playing the case, players commit to the actions. Though players can change their
commitment before the deadline, they have to pay the switching cost. Our paper is di¤erent from theirs in
two aspects. First, in our model, players can revise the preparation without paying the cost when they get
the opportunity. Second, Caruana and Einav consider the perfect information game, that is, at a …xed date,
only one player can change the past commitment. In their model, the order of the movement at each date
is predetermined. In our model, the opportunity of the movement is stochastic. With the symmetric arrival
rate, we can consider the situation that at each date, each player can move equally likely.
In the context of revision games with …nite strategy sets, Kamada and Sugaya (2010a) consider a model
8 As an exception, Young (1998) shows that in the context of contracting, his evolutionary model does not necessarily lead
to risk-dominant equilibrium (p-dominant equilibrium in Morris, Rob and Shin (1995)). But he considers a large anonymous
population of players and repeated interaction, so the context he focuses on is very di¤erent from the one we focus on.
9 This nonrobustness is …ne for their purpose since they focus on pointing out that the set of outcomes discoutinuously
changes by the introdction of a small switching cost. However, it would not be very ideal in our context because we focus on
identifying which outcome is selected by the introduction of the preparation stage. For that purpose, we want to minimize the
information necessary for the selection: the preference of each player is su¢ cient information to identify the outcome and the
information to compare di¤erent players’payo¤s is not necessary.
4
of election campaign with three possible actions (Left, Right, and Ambiguous). The main di¤erence from
their work is that they assume that once a candidate decides which of Left and Right to take, she cannot
move away from that action. Thus the characterization of the equilibrium is substantially more di¢ cult in
the model of the present paper, because in our model an action a player has escaped from can be taken again
by that player in the future.
2
Model
We consider a normal-form
game (I; (Xi )i ; ( i )i ) where I is the set of players, Xi is the …nite set of player
Q
i’s actions, and i : j Xj ! R is player i’s utility function. Before players actually take actions, they need
to “prepare” their actions. We model this situation as in Kamada and Kandori (2009): time is continuous,
t 2 [ T; 0] with T = 1, and the normal form game (referred to as a “component game”) is played once and
for all at time 0. The game proceeds as follows. First, at time 1, players simultaneously choose actions.
Between time 1 and 0, each player independently obtains opportunities to revise their prepared action
according to a Poisson process with arrival rate ( i )i . At t = 0, the action pro…le that has been prepared
most recently by each player is actually taken and each player receives the payo¤ that corresponds to the
payo¤ speci…cation of the component game. Each player has perfect information about past events at any
moment of time. No discounting is assumed, although this assumption does not change any of our results.10
We consider the limit of the set of subgame-perfect equilibrium payo¤s of this game as the arrival rate
goes to in…nity.
Let ( i )i ( ) be the set of subgame-perfect equilibrium payo¤s given arrival rate ( i )i .
De…nition 1 A payo¤ set S
lim i !1 for al l i ( i )i ( ). If
i = n =ri
R is a revision equilibrium payo¤ set of
if S = (ri )i ( ) :=
( ) is a singleton, we say its element is a revision equilibrium payo¤ .
(ri )i
for al l i
The set of action pro…les that correspond to the revision equilibrium payo¤ set is a revision equilibrium
set. If the revision equilibrium set is a singleton, we say its element is a revision equilibrium.
That is, a revision equilibrium payo¤ set is the set of payo¤s achievable by the revision game de…ned in
this section. Note that ri represents the relative frequency of the arrivals of the revision opportunities for
player i compared to player n. It will turn out in the sequel that this set is often a singleton. The term
“revision equilibrium payo¤” is used for a convenient abbreviation that represents the element of such a
singleton set. “Revision equilibrium set” and “revision equilibrium” are analogously de…ned.
Note well that whenever we refer to some action pro…le (resp. a payo¤ pro…le) as a revision equilibrium
(resp. a revision equilibrium payo¤), we implicitly mean that it is the unique element of the revision
equilibrium set (resp. revision equilibrium payo¤ set).
3
Pareto-Dominant Strict Nash Equilibrium in a Component Game
In this section we consider a component game with a Nash equilibrium that Pareto-dominates all other
action pro…les. Note that this condition is stronger than Pareto-ranked Nash equilibria. We will show that
this strategy pro…le is selected in our model.
De…nition 2 A strategy pro…le x is
Q said to be strongly Pareto-dominant strategy pro…le if there exists > 0
such that for all i and all x 2 X
i Xi with x 6= x , i (xi ) > i (x) + .
Proposition 3 Suppose that x is strongly Pareto-dominant strategy pro…le and inf
i. Then, x is a unique revision equilibrium.
1 0 Discounting
only scales down the payo¤ at time 0.
5
i
=:
i
>
1 for all
Intuitive explanation goes as follows. Firstly, we show that once all the players prepare the strongly
Pareto-dominant Nash equilibrium strategy, they will not escape from that state. Expecting that, if a
unilateral change of the preparation by player i can induce the strongly Pareto-dominant Nash equilibrium,
she will do so. In turn, player j whose unilateral change in the preparation can induce the situation where
player i’s unilateral change can induce the strictly Pareto-dominant Nash equilibrium, she will also do so
if the deadline is far expecting that player i will go to strongly Pareto dominant Nash equilibrium. By
induction, we can show that the strongly Pareto-dominant action will be selected. The formal proof requires
more solid arguments since player i takes into account the possibility that a player m taking xm now will
prepare some other action when she moves.
Notice that the above equilibrium selection always selects the strong Pareto-dominant equilibrium in
…nite component games. In particular, even if the better Nash equilibrium is risk-dominated by a worse
Nash equilibrium, the asynchronous revision process leads to the better equilibrium. The key is that, if the
remaining time is su¢ ciently long, since it is almost common knowledge that the opponent will move to the
Pareto-dominant equilibrium afterwards, the risk of mis-coordination can be arbitrarily small.
Notice also that we do not require that the game is of “pure-coordination,” in which all players’payo¤
functions are the same. This result is in a stark di¤erence from Laguno¤ and Matsui (1997), in which they
need to require that the game is of pure coordination, as otherwise their result would not hold (Yoon, 2001).
4
Pareto-Unranked Nash Equilibria in a Component Game: Battle of the Sexes
The result in the previous section suggests that the Pareto-dominant equilibrium is selected in games in
which such a pro…le exists. But what happens if there are multiple Pareto-optimal Nash equilibria? An
example of this class of games is the game of “battle of the sexes,” as follows:
g( ) = U
D
L
R
a + ; 1 0; 0 ;
0; 0
1; a
with a > 1 and
0. Observe that there are two pure strategy Nash equilibrium, (U; L) and (D; R), where
player 1 prefers the former equilibrium while player 2 prefers the latter.
If 1 = 2 = and = 0, then two players are perfectly symmetric. Then it is obvious that the set of
equilibrium payo¤ is symmetric. In Subsection 4.1, we show that this set is a full-dimensional subset of the
feasible payo¤ set. This set for the case with a = 2 is depicted in Figure 1.
However, if is strictly positive or 1 6= 2 , then what would happen is no longer obvious. We show in
Subsection 4.3 that for 1 = 2 and > 0, (U; L), which corresponds to the payo¤ (a + ; 1), is a revision
equilibrium. The revision equilibrium payo¤ in this case with a = 2 is depicted in Figure 2. Similarly, if
1 < 2 and = 0, (a; 1) is a revision equilibrium.
4.1
Anti-Equilibrium Selection in Symmetric Battle of the Sexes Game
In this subsection, we consider a 2 x 2 game g with
L
U
D
R
1 (U; L);
2 (U; L)
1 (U; R);
2 (U; R)
1 (U; D);
2 (U; D)
1 (D; R);
2 (D; R)
with two pure strategy equilibrium (U; L) and (D; R) and
1 (U; L)
>
(U;
L)
<
2
6
1 (D; R)
2 (D; R)
(D; R)
(U; L)
2 (U; L)
2 (D; R)
(U; R)
+
(D; R)
2 (U; R)
+
2 (U; L)
1
1
1
1
=
2
2
1
1
(U; L)
(U; L)
2 (D; R)
2 (D; R)
(U; R)
(D; R)
1 (U; R)
:
2 (U; L)
1
1
1
1
Since there are only two players, we de…ne r
r1 for simplicity. The battle of the sexes with r = 1 and
= 0 is a special case.
The purpose of this section is to highlight the di¤erence between two cases: A nongeneric case in which
a very special joint condition on the arrival rates and payo¤ structure is satis…ed (as in the case with r = 1
and = 0) and a generic case that encompasses all the other cases.
Proposition 4 Suppose that
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
(U; R)
+
1 (D; R)
2 (U; R)
+
2 (U; L)
1
1
=
2
(U; L)
1 (U; L)
2 (D; R)
2 (D; R)
1
1
2
1
(U; R)
1 (D; R)
1 (U; R)
2 (U; L)
1
in the game g. Then, the revision equilibrium payo¤ set satis…es
r
(g(0))
f(y1 ; y2 ) 2 co f(
1
(D; R) ;
2
(U; L)) ; (
1
(U; L) ;
2
(U; L)) ; (
1
(U; L) ;
2
The basic logic can be explained by our illustrative example of the battle of sexes with
7
(U; L))gg:
1
=
2
=
and
= 0. Figure 1 below depicts the set
1
(g(0)), for the case of a = 2.
Feasible payo¤ set and revision equilibrium payo¤ set in symmetric “battle of the
sexes” game with a = 2.
Let us …rst discuss how to obtain the extreme points of the sets, assuming a = 2 for the sake of concreteness. Point (2; 1) can be obtained by just sticking to (U; L) on the path of the play. In a perfectly symmetric
way, (1; 2) can be obtained by just sticking to (D; R) on the path of the play. To obtain the payo¤ pro…le
(1; 1), we construct an equilibrium as follows: When t < t for some appropriately chosen t , players 1
and 2 stick to U and R, respectively. For time t
t , if a player gets an opportunity, he gives up and
change his action (to D if player 1 gives up and to R if player 2 gives up). From now on we refer to this
equilibrium by “chicken race equilibrium,” as the equilibrium has a favor of the “chicken race” game,” in
which two drivers drive their cars towards each other until one of them gives in, while if both do not give in
then the cars crash and the drivers die.
The key feature of the game to sustain this “chicken race equilibrium” is that, since two players are
symmetric, their “cuto¤s,” t , must be the same. Note that the structure of the chicken race equilibrium is
close to war of attrition. Having the same t implies that both of the players are equally patient to give in.
8
Actually, the condition
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
1
1
=
2
(U; R)
+
(D;
R)
1
2 (U; R)
+
2 (U; L)
1
(U; L)
(U;
L)
1
2 (D; R)
2 (D; R)
1
2
1
(U; R)
(D;
R)
1
1 (U; R)
2 (U; L)
1
is the generalized condition so that two players have the same “giving-up” cuto¤. The natural conjecture
from the war of attrition is that, if we break the symmetry, we can destroy the chicken race equilibrium.
This conjecture will be shown formally in the next section. That is, with
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
1
1
6=
2
(U; R)
+
1 (D; R)
2 (U; R)
+
2 (U; L)
(U; L)
1 (U; L)
2 (D; R)
2 (D; R)
1
1
2
1
(U; R)
1 (D; R)
1 (U; R)
2 (U; L)
1
we show that the equilibrium is essentially unique.
Let us …nish this section by explaining how to obtain the payo¤s other than the extreme points of the
equilibrium payo¤ set de…ned in Proposition 4. Because we are not assuming any public randomization
device, it is not immediately obvious that these payo¤s can be achieved. However, we can use the Poisson
arrivals when t is large as if it were a public randomization device: players start their play at (U; R), and
“count” the numbers of Poisson arrivals to each player until t < t and then decide which of three
equilibria to play after t. Since the length of the time interval during which players count the arrivals can
be arbitrary rational and irrational numbers, we can sustain all the payo¤s in the equilibrium payo¤ set
de…ned in Proposition 4.
4.2
Illustration: Three State Example
In the next subsection, we show that in generic coordination games, a unique equilibrium can be selected.
To gain some intuition for why we have equilibrium selection in generic coordination games, this subsection
considers a simpli…ed setting of a revision game-like model. The example also reveals the di¢ culty of the
analysis of our general setting.
Suppose that there exist three states, s0 , s1 and s2 , in which payo¤s are, (0; 0), (2 + ; 1) and (1; 2),
respectively. Time is t 2 [ T; 0] as before, and the initial state (state at t = T ) is predetermined and
s0 . The Poisson arrival rate is symmetric and 1 = 2 = > 0. Each player i = 1; 2 has two actions, Ii
(Insist) and Yi (Yield). Starting from state s0 , action Ii does not change the state, while action Yi changes
the state from s0 to s i . Thus, i’s action Yi changes the state that the opponent i favors the most. We
suppose (only in this subsection) states s1 and s2 are “absorbing states.”That is, once the state si , i = 1; 2,
is reached, the state cannot change to any other states (See Figure 2).
The transition of 3-state example
As in our general model, we suppose that each player has a chance to revise their actions according to a
Poisson process with arrival rate , and the …nal state will be realized. As in the analysis of the general
9
model, we consider the limit case of T ! 1, which is equivalent to the case when the frequency of revision
( ) goes to in…nity with …xed T > 0.
The following proposition holds:
Proposition 5 There is a unique subgame perfect equilibrium. In this equilibrium, the action pro…le at time
t = T is (I1 ; Y2 ). Hence state s1 is reached immediately.
Since this proposition is a straightforward corollary of our general result, we do not provide a proof.
Rather, we explain the intuition of this result. First, it is intuitive that players use cuto¤ strategies: there
exists ti for each i such that if the state is still in s0 at time t, for t < t1 , player i takes Ii , for t = ti ,
is indi¤erent between Ii and Yi and for t > ti , takes action Yi .
Suppose …rst that two players have exactly the same cuto¤ t1 = t2 . Then, at this cuto¤, if player i is
given a chance to revise her action, her expected payo¤ would be exactly equal to 1, as she must be indi¤erent
between Ii and Yi . But this is impossible, because the payo¤ structure is asymmetric. More precisely, if
player 1 is indi¤erent, player 2 must strictly prefer taking Y2 . On the other hand, if player 2 is indi¤erent,
player 1 must strictly prefer taking I1 . Notice that this argument suggests that t2 < t1 . That is, player
2’s cuto¤ must be further away from the deadline than player 1’s cuto¤. This is intuitive: player 1 expects
more from insisting (since 2 + > 2), so he has a larger incentive to wait.
Given t2 < t1 , we argue that 1 < t1 < 0. This is straightforward: note that player 2 will take Y2
as soon as possible after t1 . We have 1 < t1 : suppose not. By instead waiting until a …nite but large
(in absolute value) T , he can guarantee approximately 2 + as by player 2 will take Y2 as soon as possible.
Hence, t1 = 1 is not optimal. We also have t1 < 0 because the payo¤ from taking I1 at time
will
converge to 0 as goes to 0, while the payo¤ from taking Y1 is always 1.
Finally, we argue that for any t2 2 [ T; t1 ], the payo¤ from taking I2 (assuming that she takes Y2
whenever she gets a chance to revise at later periods; the same quali…cation applies whenever we consider
the expected payo¤ of taking Y2 in this paragraph) is strictly less than 1, hence taking Y2 is a unique best
response at time T from the following reasons. First, we show that player 2’s payo¤ from taking action I2
at time t1 is strictly less than 1. Since player 1 is indi¤erent between I1 and Y1 at t1 , player 1’s expected
payo¤ from I1 at t1 is exactly equal to 1. But the strategies after time t1 is perfectly symmetric, while
the only di¤erence between players 1 and 2 is that the payo¤ that they obtain when they yield is higher
for player 1. This implies that player 2 expects strictly less than 1 at time t1 from I2 . Let this payo¤ be
v2 < 1. Now, consider any time t2 < t1 . At this time, the expected payo¤ from playing I2 is a convex
combination of (i) 1 with a weight that is equal to the probability that she obtains a chance to revise her
action between time t2 and t1 , and (ii) v with a weight that is equal to 1 minus the probability speci…ed
in case (i). Since the weight on (ii) is strictly positive, the overall payo¤ is strictly less than 1 for any t2 ,
which implies t2 = T as desired.
Therefore, we conclude that the action pro…le at time t = T is (I1 ; Y2 ). We note that the analysis
of the general setting is substantially more di¢ cult than the case we discussed here. The reason is that
not both s1 and s2 are a priori absorbing states since players can revise their actions in s1 and s2 . Indeed,
it will turn out that the action pro…le that corresponds to state s2 is not absorbing in the general model.
Nonetheless, the example in this section captures an essential argument for our equilibrium selection: a
slight payo¤ asymmetry leads to a slight di¤erence of the incentives, which causes a huge di¤erence in the
equilibrium behavior in the dynamic setting.
4.3
Equilibrium Selection in Generic Coordination Games
In this subsection we consider generic coordination games with payo¤ matrix as follows:
L
U
D
R
1 (U; L);
2 (U; L)
1 (U; R);
2 (U; R)
1 (U; D);
2 (U; D)
1 (D; R);
2 (D; R)
10
with two pure strategy equilibrium (U; L) and (D; R) and
player 1 prefers (U; L) to (D; R) :
player 1 prefers (D; R) to (U; L) :
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
1
1
6=
2
(U; R)
+
1 (D; R)
2 (U; R)
+
2 (U; L)
1
For example, the “ Battle of the Sexes” game with
1
=
1 (U; L)
>
(U;
L)
<
2
(U; L)
1 (U; L)
2 (D; R)
2 (D; R)
1
2
1
2
1 (D; R)
2 (D; R)
(U; R)
1 (D; R)
1 (U; R)
:
2 (U; L)
1
and " > 0 is a generic game.
Proposition 6 Suppose that
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
1
1
<
2
(U; R)
+
(D;
R)
1
2 (U; R)
+
2 (U; L)
(U; L)
(U;
L)
1
2 (D; R)
2 (D; R)
1
1
2
1
(U; R)
(D;
R)
1
1 (U; R)
:
2 (U; L)
1
Then, (U; L) is a unique revision equilibrium. On the other hand, Suppose that
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
1
1
>
2
(U; R)
+
(D;
R)
1
2 (U; R)
+
2 (U; L)
(U; L)
(U;
L)
1
2 (D; R)
2 (D; R)
1
1
2
1
(U; R)
(D;
R)
1
1 (U; R)
:
2 (U; L)
1
Then, (D; R) is a unique revision equilibrium.
The proposition says that the revision equilibrium payo¤ set in 2 x 2 games is generically a singleton.
Let us consider the chicken race equilibrium, that is, consider what players take if they have a chance to
revise the action when the current preparation is (U; R). For simple explanation, we assume (only for this
intuitive explanation) that (D; R) and (U; L) are absorbing. Without loss of generality, we assume
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
1
1
<
2
(U; R)
+
(D;
R)
1
2 (U; R)
+
2 (U; L)
1
(U; L)
(U;
L)
1
2 (D; R)
2 (D; R)
1
2
1
(U; R)
(D;
R)
1
1 (U; R)
:
2 (U; L)
1
since the other case is completely symmetric. Consider player 1’s incentive to give in at time
both players will give in after t. If player 1 skips the opportunity, player 1 gets
(D; R)
1 (U; L)
1
t given that
(U; R) if player 1 moves …rst,
1 (U; R) if player 2 moves …rst.
1
Note that (i) if 1 (U; L)
1 (U; R) becomes larger, then the gain of waiting goes up and that (ii) 2 is larger,
then player 2 is more likely to move …rst and player 1 will get higher gain 1 (U; L)
1 (U; R) and hence the
gain of waiting goes up. Therefore, as we explained in the previous section, in the chicken race equilibrium,
player 1 becomes more tough to give in, which makes player 2 gives in earlier. The above inequality shows
that in these two cases, it is more likely that the payo¤ of the equilibrium (U; L) preferable to player 1 will
be selected.
11
Actual equilibrium path is illustrated by the following …gures. If
prepares the static best response to the other:
t is su¢ ciently close to 0, every player
Then, suppose (U; R) is prepared at time t. As we explained in Section 4.2, the “strong”player 1 sticks
to (U; R) if t is su¢ ciently far away from 0, knowing that the “weak” player 1 gives in and goes to (U; L)
if player 2 can move afterwards:
There is more complication that arises when t is further way from the deadline 0. Suppose (D; R) is
prepared. As we mentioned in Section 4.2, (D; R) is not an absorbing state, that is, player 2 does not want
to stick (D; R) when t is su¢ ciently far away from the deadline. The reason is that (D; R) is a “risky”
choice. If player 1 can move, player 1 will prepare (U; R). Then, if player 2 moves …rst, they go to (U; L).
If player 1 moves …rst after player 1 starts to give in, they go to (D; R). Otherwise, they stay at (U; R).
Since player 1 needs to move …rst to go to (U; R), the time they spend after going to (U; R) depends on how
fast player 1 will have an opportunity to move. Hence, if player 1 has an opportunity late, then there is a
probability that they will end up with (U; R), a bad outcome. On the other hand, if player 2 prepares (D; L)
today, then only if player 1 gets one opportunity to move, they will end up with (U; L).
Let us go back to the illustrative example of the battle of the sexes. Let us consider the case with the
same arrival rate 1 = 2 = but a slight asymmetry in the payo¤ matrix:
L
R
a + ; 1 0; 0
0; 0
1; a
U
D
The proposition argues that even a very small
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
1
<
is su¢ cient to choose a unique equilibrium and, if
(U; R)
+
(D;
R)
1
2 (U; R)
+
2 (U; L)
1
12
(U; L)
(U;
L)
1
2 (D; R)
2 (D; R)
1
(U; R)
(D;
R)
1
1 (U; R)
:
2 (U; L)
1
> 0,
and (U; L) will be selected. Figure 3 depicts the revision equilibrium payo¤, for the case of a = 2. Therefore,
the slight asymmetry is enough to choose a unique equilibrium.
Similarly, if 1 < 2 with = 0, we also have
(D; R)
1 (U; L)
2 (U; L)
2 (D; R)
1
1
<
2
(U; R)
+
1 (D; R)
2 (U; R)
+
2 (U; L)
1
(U; L)
1 (U; L)
2 (D; R)
2 (D; R)
1
2
1
(U; R)
1 (D; R)
1 (U; R)
:
2 (U; L)
1
since player 1 has more commitment power. Hence, (U; L) is selected.
Feasible payo¤ set and revision equilibrium payo¤ in asymmetric “battle of the
sexes” game with a = 2.
5
Concluding Remarks
We analyzed the situation in which two players prepare their actions before they play a normal-form coordination game at a predetermined date. In the preparation stage, each player stochastically obtains
opportunities to revise their actions, and …nally-revised action is played at the deadline. We showed that,
(i) If there exists a Nash equilibrium that strictly Pareto-dominates all the other action pro…les, then it is
13
the only equilibrium; and (ii) in “battle of the sexes”games in which Nash equilibria are not Pareto ranked,
(ii-a) while with perfectly symmetric payo¤ structure, the equilibrium set is a full-dimensional subset of the
feasible payo¤ set, (ii-b) a slight asymmetry is enough to select a unique equilibrium, which corresponds to
the Nash equilibrium in the static game that gives the highest payo¤ to the “strong” player.
Let us mention possible directions of future research. First, our analysis has been restricted to 2 x 2
games when we have dealt with the case with multiple Pareto optimal equilibria, but the basic intuition
seems to extend to more general cases. Second, even in a 2 x 2 games, we have not yet considered all the
cases. For example, in the game as follows, what happens in revision games seems not straightforward:
U
D
L
10; 11
11; 1
R
1; 10
0; 0
In this game, D is a dominant action for player 1, while player 2’s best response depends on player 1’s action.
Although (D; R) is the unique Nash equilibrium, its payo¤ is Pareto-dominated by that of (U; L). Hence, it
is not obvious whether players want to switch to the Nash action as soon as possible.
Third, it would be interesting to consider the case in which there exists only one mixed equilibrium. For
example, in a symmetric “matching pennies” game, it is obvious that the probability distribution over the
outcome is the same as in the mixed strategy equilibrium of the component game. A question is whether
this is true for asymmetric case.
Fourth, it would be interesting to see the hybrid version of synchronized and asynchronized revision
games.
Fifth, in Kamada and Sugaya (2010b), we are working on a model in which players can choose the
frequency of revision (the arrival rate) at the beggining of the game. The result of this paper seems to
suggest that players would be better o¤ by choosing small arrival rates. However, this is not an entirely
correct intuition, because lowering one player’s arrival rates have two e¤ects: On the one hand, it makes
the opponent more willing to lower her arrival rate, to maintain her “strength” as a Stackelberg leader. On
the other hand, it makes her less willing to lower it because she does not want to end up in sticking at the
nonequilibrium state that gives her a bad payo¤. It turns out that the strength of these two e¤ects depend
in a nontrivial way on the payo¤ structure.
These possibilities are out of the scope of this paper, but we believe that the present paper laid out
motivations that are enough to pursue these generalizations.
References
[1] Abreu, Dilip and Faruk Gul “Bargaining and Reputation” Econometrica, 2000, 68, 85-117.
[2] Abreu, Dilip and David Pearce “Bargaining, Reputation and Equilibrium Selection in Repeated
Games” Princeton Economic Theory Papers, 2000.
[3] Ambrus, Attila and Shih-En Lu, “A continuous-time model of multilateral bargaining,” 2009,
mimeo.
[4] Benoit, Jean-Pierre and Vijay Krishna “Finitely Repeated Games,”Econometrica, 1985, 53, 905922.
[5] Carlsson, Hans and Eric van Damme “Global Games and Equilibrium Selection,” Econometrica,
1993, 61, 989-1018.
[6] Caruana, Guillermo and Liran Einav “A Theory of Endogenous Commitment,” Review of Economic Studies, 2008, 75, 99-116.
[7] Dutta, Prajit K. “Coordination Need Not Be A Problem,” 2003, mimeo.
14
[8] Farrell, J., and Saloner, G., “Standardization, Compatibility, and Innovation,” RAND Journal of
Economics, 1985, 16, 70-83.
[9] Fudenberg, Drew and Wojciech Olszewski “Repeated Games with Asynchronous Monitoring”
2009, mimeo.
[10] Harsanyi, John, C. and Reinhard Selten “A general Theory of Equilibrium Seletion in Games,”
MIT press, Cambridge, MA. 1988.
[11] Kamada, Yuichiro and Michihiro Kandori, “Revision Games,” 2009, mimeo.
[12] — and Takuo Sugaya, “Policy Announcement Game: Valence Candidates and Ambiguous Policies,”
2010a, mimeo.
[13] — and Takuo Sugaya, “Who Wants to Tie the Hands?,” 2010b, mimeo.
[14] Kandori, Michihiro, George J. Mailath, and Rafael Rob “Learning, Mutation, and Long Run
Equilibria in Games,” Econometrica, 1993, 61, 29-56.
[15] Lipman, Barton, L. and Ruqu Wang “Switching Costs in Frequently Repeated Games,” Journal
of Economic Theory, 2000, 93, 149-190.
[16] Matsui, Akihiko and Kiminori Matsuyama “An Approach to Equilibrium Selection,” Journal of
Economic Theory, 1995, 65, 415-434.
[17] Morris, Stephen, Rafael Rob, and Hyun Song Shin “p-Dominance and Belief Potential,”Econometrica, 1995, 63, 145-167.
[18] Morris, Stephen and Hyun Song Shin “Unique Equilibrium in a Model of Self-Ful…lling Currency
Attacks” American Economic Review, 1998, 88, 587-597.
[19] Oyama, Daisuke, Satoru Takahashi, and Josef Hofbauer “Monotone Methods for Equilibrium
Selection under Perfect Foresight Dynamics,” Theoretical Economics, 2008, 3, 155-192.
[20] Rubinstein, Ariel “The Electronic Mail Game: A Game with Almost Common Knowledge,”American
Economic Review, 1989, 79, 389-391.
[21] Sugaya, Takuo and Satoru Takahashi “Coordination Failure in Repeated Games with Private
Monitoring,” 2009, mimeo
[22] Satoru Takahashi “In…nite Horizon Common Interest Games with Perfect Information,” Games and
Economic Behavior, 2005, 53, 231-247.
[23] Young, H. Peyton “The Evolution of Conventions,” Econometrica, 1993, 61, 57-84.
[24] Young, H. Peyton “Conventional Contracts,” Review of Economic Studies, 1998, 65, 773-792.
A
Appendix: Proof of Propositions
Since …xing T and letting converge to in…nity is equivalent to …xing and letting T converge to in…nity,
we consider the latter formulation for the sake of “backward induction.” The following notations are useful:
let be a strategy in the revision game. As we will see, only depends on the calendar time t and the action
pro…le most recently prepared by the player. Therefore, letting xi be the action most recently prepared by
player i at t, we can write ti (x) 2 Ai .11 Finally, let BRi (x i ) be a static best response to x i by player i
in the component game.
1 1 We
can concentrate on the pure strategies without loss of generality.
15
A.1
Proof of Proposition 3
Firstly, we prove the following two lemmas:
Lemma 7 Suppose that x is strongly strictly Pareto-dominant strategy pro…le and inf i =: i > 1.
Then if a subgame starts with a pro…le x , there is a unique subgame perfect equilibrium in that subgame and
this equilibrium designates actions xi for each i on the equilibrium path of the play.
Proof. Let be a game with strongly strictly Pareto e¢ cient strategies. Let the Pareto-dominant action
pro…le in the component game be x . Take an arbitrary > 0 such that for all i and all x 2 X with x 6= x ,
i (xi ) > i (x) + .
The lower bound of the payo¤ from taking action xi at time > 0 given the opponent’s current action
x i is
P
P
exp
exp
i (x ) + 1
i:
j j
j j
The upper bound of the payo¤ from taking action x
^i 6= xi at time
x i is
P
exp
xi ; x i ) + 1 exp
i (^
j j
Hence taking xi is strictly better at time
P
exp
P
> exp
P
() exp
i (x
j
j
i (x
0
(= > @
)+ 1
xi ; x i )
i (^
j
j
xi ; x i )
i (^
exp
1
1
P
j
j
j
<P
ln
j
j
P
j
j
exp
> 1
i
P
P
exp
1
(x )
(x )
(x )
j
j
j
1A [ (x )
j
Notice that the right hand side is strictly positive.
Let
(
1
= min P
ln
i2I
(x ):
j
j
exp
+ 1
)
()
P
conditional on any history if
j
j
> 0 given the opponent’s current action
[ (x )
j
i] :
i] :
+1 :
i
+1
i
)
:
Notice that
is strictly positive. Also notice that for all t 2 [0; ), each player i chooses xi at t conditional
on the opponent’s current action x i .
Now we make a backward induction argument. Suppose that for all time t 2 [0; k ), each player i chooses
xi at t conditional on the opponent’s current action x i . We show that for all time t0 2 [k ; (k + 1) ),
each player i chooses xi at t0 conditional on the opponent’s current action x i .
The lower bound of the payo¤ from taking action xi at time t with t 2 [k ; k + 0 ) with 0 > 0 is
P
0
exp
j
j
i (x
)+ 1
exp
0
exp
0
P
j
j
xi ; x i )
i (^
+ 1
16
exp
j
0
The upper bound of the payo¤ from taking action x
^i 6= xi at time
0
P
j
i:
> 0 is
P
j
j
(x ):
0
Hence taking xi is strictly better at time
P
0
exp
P
0
> exp
i (x
j
j
j
j
(=
0
conditional on the opponent’s current action x
)+ 1
xi ; x i )
i (^
<P
1
+ 1
ln
j
j
if
i
P
0
exp
(x )
j
j
+1 :
(x )
j
j
P
0
exp
i
i
Thus each player i strictly prefers playing xi at all time t0 < k
n
+ mini2I
P1
j
j
ln
(x )
i
+1
o
=
(k + 1) conditional on the opponent’s current action x i . Since we have already proven that each player
i plays xi at all time t 2 [0; ) conditional on the opponent’s current action x i , the backward induction
argument is complete. This completes the proof.
Let V i ( jht ) be player i’s value at history ht . The following Lemma shows that, if there is a player
whose value is close to i (x ), then for all the other players, the value is close to i (x ).
Lemma 8 (Takahashi (2005)) Suppose that x is strongly strictly Pareto-dominant strategy pro…le. Then,
there exists K < 1 such that for any > 0, for any a and t, if there exists i with V i ( jht )
,
i (x )
then
V j ( jht )
K for all j:
j (x )
Now we prove the proposition by mathematical induction. Let V im (t) be the in…mum of player i’s payo¤
such that at least m players take xj at t. We will show, by mathematical induction with respect to m,
that V im (t) is su¢ ciently close to i (x ) when t is su¢ ciently large.
By Lemma 7, when m = n, V im (t) = i (x ) for all i; m.
Suppose limt!1 V im (m + 1; t) = i (x ). Take > 0 arbitrarily. There exists T0 ( ) such that
V im+1 (t)
i
(x )
for all t T0 ( ). Consider the situation where m players take xj at t with t = T0 ( ) + . Then, if player
j that is not taking xj at time t will move next by T0 ( ), then player j will yield at least j (x )
by
taking xj . Hence, Lemma 8 implies each player i will yield i (x ) K in this case. Therefore,
min
P
V im (t)
j
j
j
+ 1
n
1
min
P
j
j
j
min
P
j
+ 1
j
j
P
exp
n
1
j
n
1
P
j
j
j
j
n
1
min
P
j
1
j
j
min
P
j
n
1
j
j
exp
n
1
exp
P
j
j
P
j
o
j
(
i
(x )
j
o
K )
!
o
j
j
exp
for all i. Note that there exists T ( ) such that for all
o
P
exp
exp
min
P
j
i
(x )
i
P
!
o
j
j
K
i
T ( ),
i
(x )
!
o
17
min
P
j
i
1
j
i
(x )
j
min
P
j
j
j
!
2
i
2
;
which implies
V
i
m
min
P
(t)
j
i
(x ) +
min
P
1
j
j
j
j
j
!
i
(K + 1)
for all i. Consider V im (t) with t = T0 ( ) + T ( ) + . Then, if player j that is not taking xj at time t will
move next by T0 ( ) + T ( ), then she will yields at least j (x )
by taking xj . Again, Lemma 8 implies
each player i will yield i (x ) K in this case. Otherwise, from above, player i’s value at T0 ( ) + T ( ) is
min
min
(K + 1) . Therefore,
at least P jj i (x ) + 1 P jj
i
j
j
min
P
V im (t)
j
j
j
+ 1
n
1
min
P
j
j
j
min
P
j
j
j
n
1
j
j
j
(K + 1)
Then, for
n
1
o
j
j
i
o
j
j
i
K )
!
o
min
P
j
(x )
P
exp
(x )
j
j
P
n
1
(
P
exp
exp
min
P
+ 1
P
exp
!
o
j
j
min
P
j
j
i
(x ) +
min
P
1
j
j
j
i
(x ) +
min
P
1
j
j
j
j
j
!
!
(K + 1)
i
i
!
!
T ( ),
V
i
m
(t)
(
min
P
j
j
+
j
1
min
P
j
j
j
!
min
P
j
j
j
)
i
(x ) +
1
min
P
j
j
j
!2
Recursively, we can show that for any , for any l, for all i and t T0 + lT ( ),
(l 1
!m
)
!l
X
min j
min j
min j
i
P
P
1 P
V m (t)
i (x ) + 1
m=0
j
j
j
j
j
Therefore, for any " > 0, there exists l such that for any l l,
(l 1
!m
)
X
min j
min j
P
1 P
i (x )
m=0
j
j
j
1
"
,
3(K+l)
we have, for any i and t
i
min
P
j
j
!l
i
j
(x )
j
j
Taking
j
i
i
(K + 2) :
(K + l) :
"
:
3
"
:
3
T0 + lT ( ),
V im (t)
i
(x )
"
as desired.
A.2
Proof of Proposition 4
Firstly, we prove the existence of an equilibrium with expected payo¤ ( 1 (D; R) ; 1 (U; L)) by construction:
let t be
1
1 1 (D; R)
1 (U; R)
1 (U; L)
1 (U; R)
log
+
(1)
t =
+
(U;
L)
(D;
R)
(U;
L)
1
2
2 1
1
1
1 (D; R)
1
2 2 (U; L)
2 (U; R)
2 (D; R)
1 (U; R)
=
log
+
:
+
(D;
R)
(U;
L)
(D;
R)
1
2
1 2
2
2
2 (U; L)
18
Note that at t , staying at (U; R) gives ( 1 (D; R) ; 1 (U; L)) to both players, given that for
t
i (x) = BRi (x). We verify that the following strategy pro…le constitutes an equilibrium:
Player 1 takes U at
T.
Player 2 takes R at
T.
t 2 ( t ; 0],
Player 1 takes the following Markov strategy:
– For
– For
t 2 [ T; ; t ),
t 2 [ t ; 0],
t
t
1 (U; R) = 1 (D; R)
t
1 (x) = BR1 (x).
= U and
t
1 (U; L)
=
t
1 (D; L)
= U.
Player 2 takes the following Markov strategy:
– For
– For
t
t
2 (U; L) = 2 (U; R)
t
2 (x) = BR2 (x).
t 2 [ T; t ),
t 2 [ t ; 0],
= R and
t
2 (D; L)
=
t
2 (D; R)
= R.
Then, (1) gives the condition that, given that players follow the above strategy after t , player 1 is
indi¤erent between U and D at x = (U; R) and player 2 is indi¤erent between R and L at x = (U; R). Given
this, at (U; R) and (D; R) at t with t
t , (1) ensures that the expected payo¤ for player 1 is 1 (D; R).
Similarly, at (U; L) and (U; R) at t with t t , the expected payo¤ for player 2 is 1 (U; L). Therefore,
At
T , given player 2 takes R, both U and D are optimal.
At
T , given player 1 takes U , both L and R are indi¤erent.
For t 2 (t ; T ), at (U; R) and (D; R), both U and D are optimal. At (U; L) and (D; L), U is strictly
optimal.
For t 2 (t ; T ), at (U; L) and (U; R), both L and R are optimal. At (D; L) and (D; R), R is strictly
optimal.
For t 2 [t ; 0],
t
i (x)
= BRi (x) is optimal.
Secondly, we prove the existence of an equilibrium with expected payo¤ close to ( 1 (U; L) ; 2 (U; L)) by
construction. Let T > 0 be a large number. We verify that the following strategy pro…le constitutes an
equilibrium:
Player 1 takes U at
T.
Player 2 takes R at
T.
Player 1 takes the following Markov strategy:
– for
– for
t 2 [ T; t ),
t 2 [ t ; 0],
t
t
1 (U; R) = 1 (D; R)
t
1 (x) = BR1 (x).
= U and
t
1 (U; L)
=
t
1 (D; L)
= U.
Player 2 takes the following Markov strategy:
– for
– for
t 2 [ T; T ),
t 2 [ T ; 0],
t
2 (U; L)
t
2 (x)
=
t
2 (U; R)
= R and
t
2 (D; L)
=
t
2 (D; R)
= R.
= BR2 (x).
Note that for su¢ ciently large T , the expected payo¤ is su¢ ciently close to ( 1 (U; L) ; 2 (U; L)). We
verify that the above strategy pro…le constitutes an equilibrium: Note that at (U; R) and (D; R) at t with
t
t , T < t ensures that the expected payo¤ for player 1 is higher at (U; R). At (U; L) and (U; R)
at t with t
t , the expected payo¤ for player 2 is 2 (U; L). Therefore,
19
At
T , given player 2 takes R, U is optimal.
At
T , given player 1 takes U , both L and R are indi¤erent.
For
t 2 [ T; t ], at all x, U is optimal.
For
t 2 [ T; t ], at (U; L) and (U; R), both L and R are optimal. At (D; L) and (D; R), R is optimal.
For
t 2 [ t ; 0], each player i takes
t
i (x)
= BRi (x).
Symmetrically, the following equilibrium approximates (
Player 1 takes U at
T.
Player 2 takes R at
T.
1
(D; R) ;
2
(D; R)).
Player 1 takes the following Markov strategy:
– For
– For
t 2 [ T; T ),
t 2 [ T ; 0],
t
1 (U; R)
t
1 (x)
=
t
1 (D; R)
= U and
t
1 (U; L)
=
t
1 (D; L)
= U.
= R and
t
1 (D; L)
=
t
1 (D; R)
= R.
= BR1 (x).
Player 2 takes the following Markov strategy:
– For
– For
t 2 [ T; t ),
t 2 [ t ; 0],
t
2 (U; L)
t
2 (x)
=
t
1 (U; R)
= BR2 (x).
Therefore, we construct equilibria approximating ( 1 (D; R) ; 2 (U; L)), ( 1 (U; L) ; 2 (U; L)), and ( 1 (U; L) ;
Note that the three equilibrium, nobody moves until T and the following is also an equilibrium for any
N; M : with T > T ,
Player 1 takes U at
T.
Player 2 takes R at
T.
Player 1 takes the following Markov strategy:
– For
t 2 [ T; T ),
t
1 (U; R)
=
t
1 (D; R)
= U and
t
1 (U; L)
t
1 (D; L)
=
= U.
– If the number of chances where player 1 can move for [ T; T ) is less than N , going to the …rst
equilibrium, that is,
For
For
t 2 [ T ; t ), t1 (U; R) = t1 (D; R) = U and
t 2 [ t ; 0], t1 (x) = BR1 (x).
t
1 (U; L)
=
t
1 (D; L)
= U.
– If the number of chances where player 1 can move for [ T; T ] is no less than N and no more
than M , going to the second equilibrium, that is,
For
For
t 2 [ T ; t ), t1 (U; R) =
t 2 [t ; 0], t1 (x) = BR1 (x).
t
1 (D; R)
= U and
t
1 (U; L)
=
t
1 (D; L)
= U.
– If the number of chances where player 1 can move for [ T; T ] is no more than M , going to the
third equilibrium, that is,
For
For
t 2 [ T ; T ), t1 (U; R) = t1 (D; R) = U and
t 2 [ T ; 0], t1 (x) = BR1 (x).
t
1 (U; L)
=
t
1 (D; L)
= U.
Player 2 takes the following Markov strategy:
– For
t 2 [ T; T ),
t
2 (U; L)
=
t
2 (U; R)
= R and
20
t
2 (D; L)
=
t
2 (D; R)
= R.
2
(U; L)).
– If the number of chances where player 1 can move for [ T; T ] is less than N , going to the …rst
equilibrium, that is,
For
For
t 2 [ T ; t ), t2 (U; L) = t2 (U; R) = R and
t 2 [ t ; 0], t2 (x) = BR2 (x).
t
2 (D; L)
t
2 (D; R)
=
= R.
– If the number of chances where player 1 can move for [ T; T ] is no less than N and no more
than M , going to the second equilibrium, that is,
For
For
t 2 [ T ; T ), t2 (U; L) = t2 (U; R) = R and
t 2 [ T ; 0], t2 (x) = BR2 (x)
t
2 (D; L)
t
2 (D; R)
=
= R.
– If the number of chances where player 1 can move for [ T; T ] is no more than M , going to the
third equilibrium, that is,
For t 2 [ T ; t ), t2 (U; L) = t2 (U; R) = R and
for t 2 [ t ; 0], t2 (x) = BR2 (x).
t
2 (D; L)
=
t
2 (D; R)
= R.
It is straightforward to show that this is an equilibrium and with appropriate choices of T , N , M , and
su¢ ciently large T
T , we can attain any payo¤ pro…le that can be expressed by a convex combination
of ( 1 (D; R) ; 2 (U; L)), ( 1 (U; L) ; 2 (U; L)), and ( 1 (U; L) ; 2 (U; L)).
A.3
Proof of Proposition 6
We assume that
(U; R) 1
>
(D;
R) 1
1
(D; R)
1 (U; L)
1
(U; L)
(D;
R)
2
(D; L) 1
:
2 (U; L) 2
2
1
2
The other case is completely symmetric.
We use backward induction to derive the (essentially) unique subgame perfect equilibrium. Let xti be the
prepared action by player i at t. In addition, let Vxi (t) be player i’s expected payo¤ when players prepare
an action pro…le x at t and no player has a chance to move at t. Firstly, note that there exists
>0
such that, for each t 2 (
; 0], given player i’s prepared action xt i , player i prepares the best response
to xt i , that is,
t
t
t (x i ) = BR(x i ):
Suppose the players “know”the game proceeds as explained above after
action pro…le, each player’s value is given as follows:
t. Then, at
t, at each prepared
At (U; L),
VU L (t) = (
1
At (U; R), if player 2 moves …rst, then 1 (U; L) ;
and otherwise, 1 (U; R) ; 2 (U; R). Therefore,
VU1R (t)
2
=
|
2
(1
1+
(U; L) ;
2
2
(U; L)) :
(U; L), if player 1 moves …rst, then
exp ( (
{z
1
+
2 ) t)) 1
player 2 m oves …rst
1
+
+
|1
(1
2
exp ( (
{z
1
+
player 1 m oves …rst
+exp ( ( 1 +
|
{z
2 ) t) 1
nob o dy m oves
21
}
(U; R)
}
(U; L)
2 ) t)) 1
}
(D; R)
1
(D; R) ;
2
(D; R),
and
VU2R (t)
=
2
2
(U; L)
+
1
+
2
(1
1
(D; R)
exp ( (
1
+
exp ( (
1
2 ) t))
2
+ 2
+ 2 (U; R) exp ( (
(1
+
2 ) t))
1
At (D; L), if player 1 moves …rst, then 1 (U; L) ;
and otherwise, 1 (D; L) ; 2 (D; L). Therefore,
1
VDL
(t)
1
=
1
+
(1
+
2 ) t) :
(U; L), if player 2 moves …rst, then
exp ( (
1
+
2 ) t))
1
1
(D; R) ;
(U; L)
2
2
+
2
1
1+ 2
+ exp ( (
(1
1
exp ( (
+
2 ) t)
+
1
2 ) t))
1
(D; L)
1
+
1
(D; R)
and
2
VDL
(t)
1
=
1
+
(1
2
+
exp ( (
2 ) t))
2
(U; L)
2
1+ 2
+ exp ( (
(1
1
exp ( (
+
2) t 2
+
1
2 ) t))
2
(D; R)
(D; L)) :
At (D; R),
VDR (t) = (
Let t1 be the solution for
1
(D; R) ;
1
(D; R) = VU1R (t)
2
(U; L) = VU2R (t) :
and t2 be that for
2
(D; R))
That is,
t1 =
1
+
1
t2 =
1
+
1
and
log
2
(D; R)
1 (U; L)
1
1
2
log
2
By assumption, t1 < t2 .
Then, for t 2 ( t1 ; 0], noting that
(U; L)
(D;
R)
2
2
2
1
2
(U; R)
+
(D;
R)
1
(U; L)
(U;
L)
1
1
(U; R)
+
2 (U; L)
2
1
(D; R)
2 (D; R)
2
(U; R)
(D;
R)
1
1
(U; R)
2 (U; L)
1
(U; L) > VU2R (t1 ) since t2 > t1 , we have the following:
If xt2 = L, player 1 will take U since
1
1
(U; L) > VDL
(t) :
1
(D; R) > VU1R (t)
2
(U; L) > VU2R (t)
If xt2 = R, player 1 will take D since
If xt1 = U , player 2 will take L since
22
2
(D; R),
If xt1 = D, player 2 will take R since
2
2
(D; R) > VDL
(t) :
At t = t1 , player 1 is indi¤erent between U and D when player 2 takes R.
For slightly before t1 , since every inequality except for 1 (D; R) = VU1R (t1 ) is strict at
t1 ,
If xt2 = L, player 1 will take U .
If xt2 = R, player 1 will take U .
If xt1 = U , player 2 will take L.
If xt1 = D, player 2 will take R.
While players follow the above strategy, for
t<
t1 ,
If xt2 = L, player 1 will take U .
If xt2 = R, player 1 will take U .
If xt1 = U , player 2 will take L. Suppose player 2 is indi¤erent between L and R at t. If player 2
takes L, his expected payo¤ is de…nitely 2 (U; L). If player 2 takes R, it is a convex combination of
2 (U; R) and 2 (U; L), with a strictly positive weight on the former. Since 2 (U; R) < 2 (U; L) by
assumption, there does not exist such t.
If xt1 = D, there exists T such that player 2 will become indi¤erent between L and R.
We need to verify the fourth argument. Assuming that player 2 will take R after t, the
L is
8
(1 exp ( 2 (t t1 s))) 2 (U; L)
>
>
{z
}
> |
Z t t1
<
player 2 secondly m oves by t1
2
VDR (t) =
1 exp(
1 s)
+ exp ( 2 (t t1 s)) VU2R (t1 )
>
{z
}
|
0
>
>
|
{z
}
player 1 …rstly m oves by t :
1
+
exp (
|
1 (t
{z
t1 ))
}
player 2 do es not m ove by t1
2 (D; R)
player 1 does not m ove by t1
and the payo¤ after taking to L is
1
|
+
1
Z
+
(1
exp ( (
{z
2
+
2 ) (t
t1 )))
}
2
(U; L)
player 1 …rstly m oves by t1
T
t1
exp( 1 s)
| {z }
0
|
1
|
2
exp(
{z
2 s)
}
player 1 do es not m ove by splayer 2 m oves at s
{z
player 2 …rstly m oves by t1
+exp( (
|
1
+ 2 ) (t
{z
nob ody m oves by t1
2
t1 ))VDL
(t1 ) :
}
23
2
VDR
(t
s)ds
}
value of taking
9
>
>
>
=
>
>
>
;
ds
Note that
2
VDR
(t)
=
Z
t t1
1
exp(
1 s) 2
(U; L) ds
0
Z
t t1
exp (
1
1s
2
(t
t1
s))
2
(U; L)
VU2R (t1 ) ds
0
+ exp ( 1 (t t1 )) 2 (D; R)
= (1 exp( 1 (T t1 ))) 2 (U; L)
exp ( 1 (T t1 )) exp ( 2 (T
t1 ))
1
2
+ exp (
(T
1
t1 ))
2
2
VU2R (t1 )
(U; L)
1
(D; R) :
and the payo¤ after taking L is
1
+
Z T
1
+
(1
exp ( (
1
+
2 ) (T
2
t1
2
exp ( (
1
+
2 ) s)
0
+ exp ( (
1
=
1
1
(1
+
+
2 ) (T
1
+
+
exp (
1
(T
exp( (
1
+
2 ) (T
exp(
exp( (
1 2
exp( (
1
1 + 2 )(T
1+
(U; L)
2 (U; L)
t1 )) 2 (D; R)
1
1
(T
s
t1 )))
2
2
+
VU2R (t1 ) A ds
(U; L)
(U; L)
1
2
1 (T
2
t1 )) exp(
2 (T
1
2
(U; L)
!
t1 ))
2
t1 ))
(U; L)
VU2R (t1 )
(exp ( ( 1 + 2 ) (T t1 )) exp ( 1 (T t1 ))) 2 (D; R)
2
+ exp ( ( 1 + 2 ) (T t1 )) VDL
(t1 )
= (1 exp ( 1 (T t1 ))) 2 (U; L)
t1 ))
1 exp (
1 (T
2 exp (
exp ( ( 1 + 2 ) (T t1 )) +
2
(exp ( ( 1 + 2 ) (T t1 )) exp (
2
+ exp ( ( 1 + 2 ) (T t1 )) VDL
(t1 )
1
2
(t1 )
t1 )) exp(
2 )(T
s t1 )))
1 (T
s t1 )) exp(
2 (T s t1 ))
2
2
1
1 (T
exp(
+ exp (
t1 ))
t1 ))
(U; L)
1
2 ) (T
2
2
(1
@
2
t1 )) VDL
exp ( (
2
1
2
0
2
+
+
t1 )))
1
(T
t1 )))
2
2
(T
t1 ))
2
(U; L)
1
VU2R (t1 )
(D; R)
The di¤erence is given by
2
(U; L)
VU2R (t1 ) +
2
VDL
(t1 )
2 (D; R)
2
(U;
L)
V
(t
2
UR 1)
exp (
1
(t
t1 ))
exp (
2
(t
t1 ))
2
Therefore, since 2 (D; R) VDL
(t1 ) > 0, there exists unique T such that the di¤erence is positive
for t 2 [ T ; t1 ] and negative for t < T .
Therefore, for
t 2 ( T ; t1 ), we have
If xt2 = L, player 1 will take U .
If xt2 = R, player 1 will take U .
24
If xt1 = U , player 2 will take L.
If xt1 = D, player 2 will take R.
Since every incentive except for player 2 with x1 = D is strict, for a while, the equilibrium strategy is
If xt2 = L, player 1 will take U .
If xt2 = R, player 1 will take U .
If xt1 = U , player 2 will take L.
If xt1 = D, player 2 will take L.
As long as players follow the above strategy,
If xt2 = L, player 1 will take U .
If xt2 = R, player 1 will take U
If xt1 = U , player 2 will take L since, assuming player 2 will take L after that,
VU2L (t)
=
2 (U; L)
> (1 exp ( 2 (t
{z
|
t1 )))
}
2
(U; L) + exp (
t1 )) VU2R (t1 )
(t
player 2 can m ove by t1
=
VU2R
(t) :
since
2
(U; L) > VU2R (t1 ) :
If x1 = D, player 2 will take L from above.
We need to verify the second argument: at
payo¤ of taking U is given by
VU1R (t) = (1
|
exp (
{z
2
(t
t, if players follow the strategy above afterwards, the
t1 )))
}
1
player 2 can m ove by t1
25
(U; L) + exp (
2
(t
t1 )) VU1R (t1 ) :
On the other hand, the payo¤ of taking D is given by
Z
t T
exp ( ( 1 +
{z
|
0
+
player 1 m oves …rst by T
(U;R)
Z
t T
exp ( ( 1 +
|
{z
0
1
(t
1V
} UR
2 ) s)
s) ds
2
player 2 m oves …rst by T
(D;L)
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
2 ) s)
(1
|
}
exp (
(t
{z
T
1
s)))
}
player 1 m oves by T
(U;L)
+ exp (
|
1
(t T
{z
1
s))
}
3
(U; L)
player 1 do es not m ove by T
(D;L)
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
1
|1
0
+
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
B
B
B
B
B
B
t1 B
B
B
B
B
B
B
@
RT
0
+exp ( (
|
1
1+
2 ) (T
player 1 m oves …rst by t1
(U;L)
exp ( ( 1 +
{z
|
2)
)
t1 )))
}
1 (U; L)
2
}
player 2 m oves …rst by t1
(D;R)
exp (
|
1
(T
{z
t1
))
}
player 1 do es not m ove until t1
(D;R)
t1
exp ( 1 ) 1 VU1R
|
{z
}
1
1
(D; R)
(T
C
C
C
C
C
C
C
A
)d
player 1 m oves by t1
(D;R)
+exp ( (
|
The …rst term is
Z
2
B
B
B
B R
B
B + 0T
B
@
+ 2 ) (t
{z
(U; L)
+
exp ( (
{z
0
nob ody m oves by T
(D;R)
1
(1
1
+ 2 ) (T
{z
nob o dy m oves by t1
(D;L)
8
>
>
>
>
>
>
>
<
exp (
|
(T
{z
1
(t1 )
t1 ))VDL
}
t1 ))
}
1
1
player 1 do es not m ove by t1
(D;R)
R T t1
+ 0
exp ( 1 ) 1 VU1R
T ))
}>
>
>
>
>
>
>
:
|
{z
}
9
>
>
>
>
>
>
>
=
(D; R)
(T
>
>
>
>
>
>
>
;
)d
player 1 m oves by t1
(U;R)
1
C
C
C
C
C
C
C
Cd
C
C
C
C
C
A
:
t T
1
exp ( (
0
1 (U; L)
VU1R (t1 )
Z
1
+
2 ) s)
t T
1
exp ( (
1
+
2 ) s) exp (
2
(t
t1
s)) ds
(t
T )))
0
1
=
2
+
1
(1
exp ( (
1
+
2 ) (t
T )))
1
(U; L)
1
(U; L)
VU1R (t1 ) (exp (
2
(t
26
t1 ))
exp (
2
(t
t1 )
1
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7 ds
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
The second term is
1
(U; L)
2
+
1 (t
1
+ (exp (
(
(1
exp ( (
+
1
2 ) (t
T ))
exp ( (
exp ( ( 1 +
+ ( 1 (U; L)
1 (D; R))
1
+
2 ) (t
2 ) (T
exp(
2
T
t1
exp (
1
)
1
1 VU R
exp (
1
)
1 1
(T
t1 ))
1 (T
T )))
1 (U; L)
t1 ))+exp(
2
The calculation goes as follows: the integral with respect to
Z
T )))
2
2 (T
t1 ))
1
)
yields
)d
0
=
Z
T
t1
0
1
(U; L)
VU1R (t1 )
Z
(U; L)
T
t1
1
exp (
1
) exp (
2
(T
t1 )) d
0
=
1
(U; L) (1
1
(U; L)
exp (
1
(T
VU1R (t1 )
t1
)))
1
2
(exp (
1
27
1
(T
t1
))
exp (
2
(T
t1
))) :
Substituting this to the integral with respect to yields
0
1 (U; L)
Z T t1
B
( 1 (U; L)
t1
))
1 (D; R)) exp (
1 (T
B
exp ( ( 1 + 2 ) ) 2 @
exp ( 1 (T
t1
))
1
1
0
VU R (t1 ) 2 1
1 (U; L)
exp ( 2 (T
t1
))
=
1
(U; L)
+(
1
1
exp ( (
+ 2 ) (T
t1 ))
2+ 1
1 (D; R)) (exp ( ( 1 + 2 ) (T
2
(U; L)
1
exp( (
+
=
1
2
(U; L)
2
=
+(
1
(U; L)
+
1
(U; L)
1
2
+
+
(1
1
1
exp ( (
1
1
(1
+
exp (
(U; L)
1
+
exp(
1
1
(D; R) (exp ( (
1
(T
1 (T
+
1
1
2
(U; L)
1
1
=
1
+
B
(D; R) B
@
++(
VU1R
1
(1
exp ( (
1
(T
+
exp(
2
2
exp(
2 (T
1
+
2 ) (T
1 (T
+
(T
1
(1
1
1
t1 ))
+
1
(D; R))
2
exp(
2
(T
2 (T
1
t1 ))
t1 ))
(T
!
t1 )))
+
2 (T
t1 ))
t1 ))
2 ) (T
1
t1 )))
t1 ))
1
2 ) (T
exp (
2
exp (
(T
!
1
1
C
C
A
t1 )))
1
1
t1 ))
t1 )))
(D; R)
exp ( (
2 (T
t1 ))
t1 ))
2 ) (T
t1 ))+
2
exp ( (
2
(U; L)
2 ) (T
exp (
2
(t1 ) =
2+
+
t1 ))
2
1
exp(
1
exp ( ( 1 + 2 ) (T
t1 ))
exp ( 1 (T
t1 ))
1
2
t1 )) exp(
t1 )))
!
t1 ))
t1 )))
t1 )) + exp (
2
0
(U; L)
t1 ))+
1
(T
1 (T
1
2 ) (T
1 (T
2
2
1
1
exp (
(U; L)
since
exp(
+
exp ( (
2
+
1
t1 ))
t1 ))+
2 ) (T
2
=
2 ) (T
t1 )) + exp (
2
1
t1 )))
1
1
VU1R (t1 )
1 + 2 )(T
2 ) (T
exp ( (
exp ( (
exp (
t1 )) exp(
1
(D; R)) (exp ( (
VU1R (t1 )
+
exp( (
2
1
2
(U; L)
2 1
VU1R (t1 )
(U; L)
1
t1 ))
1 + 2 )(T
(T
t1 )) + exp (
2
28
1
2
(T
t1 ))
:
1
C
Cd
A
Substituting this into the integral with respect to s yields
Z
t T
exp ( (
+
1
2 ) s)
2
0
2
6
6
6
6
6 + exp (
4
=
1 (U; L)
(
(t
(t T
s))) 1 (U; L)
1
(1
exp
(
( 1 + 2 ) (T
t1 ))) 1 (U; L)
1+ 2
t1 ))) 1 (U; L)
+ 2 +2 1 (1 exp ( ( 1 + 2 ) (T
s))
exp(
t1 ))+exp(
1 (T
2 (T
>
+ ( 1 (U; L)
1 (D; R)) 2
>
2
1
>
:
1
+ exp ( ( 1 + 2 ) (T
t1 )) VDL (t1 )
T
t T
exp ( (
1
+
exp (
2 ) s)
+ 2 ) (T
1 (D; R))
t1 ))
2 ds
t1 )) (
1
3
9
7
>
>
7
>
= 7
7 ds
7
>
5
>
>
;
1
0
exp ( (
+ ( 1 (U; L)
+
Z
Z
1
8 (1
>
>
>
<
1 (U; L)
1 (D; R))
t1 ))+exp(
t1 ))
1 (T
2 (T
exp(
2
2
1
)
t T
exp ( (
+
1
2 ) s)
2
exp (
(t
1
T
s))
0
=
1
2
(U; L)
(1
+
1 (t
1
+ (exp (
(
exp ( (
+
1
2 ) (t
T )))
2
T ))
exp ( (
+ ( 1 (U; L)
exp ( (
+ 2 ) (T
1 (D; R))
+
1
2 ) (t
1
VDL
1 (U; L)
t1 ))+exp(
1 (T
t1 ))
1
T )))
exp(
2
2
(t1 )
2 (T
t1 ))
1
)
The third term is
exp ( (
=
=
exp ( (
1
+
1
+
+ exp ( (
1
exp ( (
1
+
+ exp ( (
1
2 ) (t
T ))
2 ) (t
+
T )) exp (
2 ) (t
2 ) (t
+
T ))
T ))
2 ) (t
since VU1R (t1 ) =
+
1
RT
exp ( 1 (T
t1
exp (
0
1
(T
t1 ))
+ VU1R (t1 )
1
T ))
(U; L)
2 exp (
1
1
t1 )) 1 (D; R)
)d
) 1 VU1R (T
(D; R)
exp (
1 (U; L) (1
1
(exp
(
(U;
L)
1
2
1
1
(T
t1 ))
1
2
1
exp (
2
t1 )))
1 (T
(T
t
))
exp (
1
1
(T
t1 ))
(
1
(D; R)
(D; R)
since
Z
T
t1
exp (
0
=
1
(U; L)
Z
T
1
)
1
1 VU R
)d
t1
exp (
0
+ VU1R (t1 )
(T
1 (U; L)
)
1
Z
1d
T
t1
exp (
)
1
1
exp (
2
(T
t1 )) d
0
=
1
(U; L) (1
+ VU1R (t1 )
exp (
1
1
(T
t1 )))
1
(U; L)
2
(exp (
1
29
1
(T
t1 ))
exp (
2
(T
t1 ))) :
2
(T
1
t1 )))
(U; L))
Therefore, in total, the value of taking D is
1
(1
2+
1
+
1
exp ( (
1
+
2 ) (t
T )))
1
(U; L)
1
VU1R (t1 ) (exp (
(U; L)
(U; L)
2
(1
+ 2
T ))
1 (t
2
(t
t1 ))
exp (
exp ( (
1
+
2 ) (t
T )))
exp ( (
1
+
2 ) (t
T )))
2
(t
t1 )
(t
T )))
(T
t1 ))
1
1
+ (exp (
(
exp ( (
+ ( 1 (U; L)
=
1
+ 2 ) (T
1 (D; R))
+ exp ( (
1
+
2 ) (t
T ))
exp ( (
1
+
2 ) (t
T ))
t1 ))
exp(
2
1
1
VDL
(t1 )
1 (U; L)
(T
t
))
exp(
t1 ))
2
1 (T
1
2
(U; L)
2 exp (
1
(T
1
t1 ))
1
2
1
)
exp (
2
(U; L)
VU1R (t1 ) (exp ( 2 (t t1 )) exp ( 2 (t t1 )
1 (U; L)
1 (t
+ (exp ( 1 (t T )) exp ( ( 1 + 2 ) (t T )))
)
(
1
(t1 )
exp ( ( 1 + 2 ) (T
t1 )) 1 (U; L) VDL
exp(
t1 )) exp(
t1 ))
2 (T
1 (T
+ ( 1 (U; L)
1 (D; R)) 2
2
1
(
1
(U; L)
1
(D; R))
(
1
(U; L)
1
(D; R))
1
exp ( (
1
+
2 ) (t
2
T ))
exp (
1
(T
t1 ))
1
2
1
exp (
2
(T
T )))
t1 ))
Subtracting VU1R (t) yields
(
1
(U; L)
+ (exp (
(
+
1
1
(t
(D; R))
exp(
exp ( (
T ))
exp ( (
2
exp (
1
2 (T
exp ( (
+
2 ) (T
t1 )) exp(
2
1
+
1
+
2 ) (t
2 ) (t
2
(t
T ))
t1 )
2
1 (t
1 (T
exp(
T )))
1
(U; L) VDL
(t1 )
t1 ))
( 1 (U; L)
1 (D; R))
t1 ))
1 (T
1
1
)
T ))
t1 ))
1
2
1
exp(
2 (T
t1 ))
:
Since
(exp ( 1 (t
exp ( ( 1 +
exp (
T ))
2 ) (T
2 (T
exp ( ( 1 + 2 ) (t T ))) > 0
1
t1 )) 1 (U; L) VDL
(t1 ) < 0
t1 )) exp ( 1 (T
t1 ))
< 0;
2
it su¢ ces to show that F (t)
F (t)
:
= exp (
1
0 with
2
exp ( (
(t
1
+
t1 )
1
(t
2 ) (t
T ))
T ))
2 exp (
1
At the limit where t goes to in…nity, we have
lim F (t) = 0.
t!1
Hence, it su¢ ces to show that
F 0 (t) > 0.
30
(T
t1 ))
1
2
1
exp (
2
(T
t1 ))
!
Since
F 0 (t)
1+ 2
=
exp (
2
+ exp ( (
=
+
if
2
>
(
1,
1
+
1
t1 )
+
exp ( 2 (t
2 exp (
1
2 ) (t
(t
2
T ))
t1 )
1 (t
T )
2 (t
F 0 (t)
=
2
exp (
2
(t
2
+
=
as desired. Symmetrically, if
exp (
1
(T
T ))
t1 ))
1 (t
1
2
1
exp (
2 exp (
(t
2 (t
2
2
<
2
1,
t1 )
1
T )
t1 )
(t t1 ))
T )) +
1 (t
t1 ))
1
2
1
exp (
1
(t
exp (
T )
2
2
(T
(t
t1 ))
t1 ))
;
(t
T )) +
1
1
exp (
exp (
2 exp (
1
2
(t
(t
1 (t
1
t1 )
1
(t
T ))
T )
t1 )
(t t1 ))
T )) > 0
2 (t
2
we have
(
1)
2
1
to
T ))
we have
1)
2
(t
+
F 0 (t) < 0:
2
When 2 = 1 , our model allows the continuity of the cases between
1 . Therefore, the same result holds.
31
2
=
1
and limit where
2
converves