Simultaneous and sequential choice in a symmetric two-player game
with strategy-ranking-dependent payoffs
Chia-Hung Sun†
Abstract
This paper investigates a symmetric two-player game with strategy-ranking-dependent payoffs, in which a player’s payoff function depends on whether he chooses a higher or lower
strategy than the other player, and it is smooth above and below the diagonal, but not differentiable on the diagonal. Under general payoff functions, we discuss situations where two
players move simultaneously or sequentially, or even when the entry sequence is endogenously determined. We demonstrate that there exists a first-mover advantage when two
players move sequentially and a player’s preference to the opponent’s choice is monotonic
and identical between a higher-strategy player and a lower-strategy player, whether the
reaction function of the follower is downward sloping or upward sloping. We also show that
a follower may yield a higher payoff in the sequential game than in the simultaneous game,
resulting in the first-mover advantage outcome in an endogenous timing game.
Keywords: Simultaneous game; Sequential game; Endogenous timing; Strategic complements; Strategic substitutes
JEL classification: D43; D21
†
Department of Economics, Soochow University, No. 56, Kueiyang Street, Section 1, Taipei 100, Taiwan.
Tel.: +886-2-23111531, ext. 3640. E-mail: [email protected].
1. Introduction
The purpose of this paper is to investigate a symmetric two-player game with strategyranking-dependent payoffs, in which a player’s payoff function depends on whether he
chooses a higher or lower strategy than the other player, and it is smooth above and below
the diagonal, but not differentiable on the diagonal. Our assumptions about the two-player
game are designed to be general, while allowing for a regular proof. Apart from symmetry, we assume real intervals as the strategy space, implying that players’ strategies can be
ranked. Moreover, there are concavity and smoothness in the payoff function for a higherstrategy or a lower-strategy player, interiority to the solution, and monotonicity of payoffs
and marginal payoffs in the opponent’s choice. Throughout the paper we use the term
“lower (higher) strategy” for a player’s lower (higher) choice versus his opponent’s choice
in order to make our points as clear as possible as well as to analyze a general framework.1
We discuss situations where two players move simultaneously or sequentially, or even when
the entry sequence is endogenously determined, depending on a player’s upward sloping or
downward sloping reaction curve. We also provide examples of economic models with such
strategy-ranking-dependent payoffs in various areas of industrial organization.
Consider an example characterized by a one-way spillover structure in process research
and development (R&D) based on the model of Amir and Wooders (2000). At the first
stage of their model, two a priori identical firms (players) choose their respective unit cost
reduction (R&D decision), while at the second stage both firms compete in the product
market. The model’s novel assumption is that spillovers flow only from the firm with higher
R&D activity to its rival, but not vice versa. Due to the one-way nature of the spillover
process, a firm’s profit function is piecewise and dependent on its strategy (R&D decision)
ranking, presenting a kink along the diagonal in the R&D strategy space.2
Another line of economic research, in which players are concerned about the relative
positions of strategies, encompasses capacity competition under demand uncertainty. De
Frutos and Fabra (2011) analyze a two-stage game in which firms make capacity decisions
under demand uncertainty and then engage in price competition. In that game, firms’ payoff
1
A player’s lower (higher) strategy in our two-player game may correspond to a player’s (a firm’s) lower
(higher) choice in quality choices, R&D decisions, capacity decisions, etc. in the respective specific economic
model.
2
In this application (and the following) the described non-cooperative game is the resulting one-shot
game where players choose their first-stage actions, conditional on a unique second-stage equilibrium. The
object of our analysis is the first-stage actions of the subgame perfect Nash equilibrium of the two-stage
game.
1
functions are dependent on strategy (capacity) ranking, resulting from large and small firms
that face asymmetric marginal returns to investment.
Some other models in the literature, which deal with quality competition in the standard vertical product differentiation model and spatial competition in the Hotelling (1929)
model, also motivate this study’s objective. Shaked and Sutton (1982) and Choi and Shin
(1992) propose a quality competition model, where two firms are a priori identical, and the
functional forms of the duopoly firms’ payoffs depend on the two firms’ relative positions
of their strategic quality decisions. One key property in their model is that the firm taking
on higher quality and the firm taking on lower quality face asymmetric marginal returns to
action. Similarly, in the Hotelling (1929) model a firm typically serves consumers located
near the left side in a linear city when the firm locates to the left of its opponent.3 If
the same firm chooses to “leapfrog” its rival and locates to the right of its opponent, then
it serves consumers located near the right side in a linear city. The result is that a firm’s
demands are piecewise in strategic location choices, and thus a firm’s overall payoff function
is also piecewise.
In a two-player game with strategy-ranking-dependent payoffs, a player’s overall payoff
function typically has a canyon (valley-like) shape along the diagonal in the strategy space.
As such, a player never responds to an opponent’s action by playing that same action, and
hence a player’s reaction curve has a jump across the diagonal. Somewhat surprisingly,
we show in Section 5 that not only are there myriads of natural economic applications,
in which players’ payoff functions are dependent on strategy ranking, but that the shape
of a player’s reaction curve in such models also varies, which may be (piecewise) upward
sloping, downward sloping, or even non-monotonic between a lower-strategy player and a
higher-strategy player.
Towards the goal of analyzing a game in which players’ payoff functions depend on the
relative positions of their strategies, this study is close in spirit to the research on symmetrybreaking by Amir et al. (2010). In particular, Amir et al. (2010) generalize a number of
games categorized by two key characteristics of two-player symmetric games – actions form
strategic substitutes and each player’s payoff admits a concavity-destroying kink along the
diagonal – that always possess only asymmetric pure strategy Nash equilibria.4 It is worth
3
Strictly speaking, this refers to the revised Hotelling (1929) model with quadratic transportation costs
in distance, as proposed by d’Aspremont et al. (1979).
4
Players’ strategies are called strategic substitutes if the best response to “more aggressive” behavior
is “less aggressive” behavior and called strategic complements if the best response to “more aggressive”
behavior is “more aggressive” behavior.
2
pointing out that Amir et al. (2010) focus on generating endogenous heterogeneity in a
simultaneous game, whereas this present paper investigates players’ competition behaviors
when actions are both strategic substitutes and strategic complements. We further explicitly
compare the follower’s payoff with the leader’s payoff in a sequential game.
This study relates to the literature on the determination of first-mover and secondmover advantages in a symmetric game, commonly characterized by monotonically upward
or monotonically downward sloping reaction functions and monotonic payoffs in an opponent’s choices. Gal-Or (1985) explores the order of moves affecting players’ relative payoffs
between two a priori identical players in a game, depending on the slope of a player’s reaction function. Under the assumption that a player’s payoff function has a general form and
is globally concave in his own strategy, Gal-Or concludes that when the follower’s reaction
function is upward sloping (i.e. actions are strategic complements), like in price setting
games with differentiated substitute products, there is a second-mover advantage. Conversely, when the follower’s reaction function is downward sloping (i.e. actions are strategic
substitutes), like in quantity competition with substitute goods, there is a first-mover advantage.5
The other strand of research concerning the issue of endogenous timing in a game also
corresponds to this present study. Hamilton and Slutsky (1990) carry out a pioneering work
on endogenous Stackelberg leadership in a duopoly game, by embedding the simultaneous
and sequential move of a two-player game into an extended game where players first select the timing of choosing actions in the basic game. Amir (1995) completes the work of
Hamilton and Slutsky (1990) by establishing the necessity of an additional monotonicity
condition. Numerous developments have also been proposed in the literature. For instance,
Sadanand and Sadanand (1996) investigate demand uncertainty, Amir and Grilo (1999) consider multiple Nash equilibria in a simultaneous game and use the theory of supermodular
games, and van Damme and Hurkens (1999) introduce the notion of risk-dominance.
The results in this present study can be summarized as follows. In the simultaneous
game with strategy-ranking-dependent payoffs, if players’ reaction curves are completely
contained in certain compact subsets of the joint strategy space, then a Nash equilibrium
exists, no matter if the reaction function of a player is upward sloping or downward sloping.
Due to the game’s symmetry, the Nash equilibria are asymmetric and occur in pairs.
When two players move sequentially, the analysis can be seen as an extension of the
5
Several developments of this strand of research incorporate private information about the players, in-
cluding Gal-Or (1987), Bagwell (1995), van Damme and Hurkens (1997), and Maggi (1999).
3
investigation of first-mover and second-mover advantages in Gal-Or (1985) to a symmetric
game with strategy-ranking-dependent payoffs.6 First, we find when players’ preferences
to their opponent’s choice are monotonic and identical between a higher-strategy player
and a lower-strategy player, that a leader in equilibrium chooses a higher strategy than the
follower when players prefer an opponent’s lower choice, and that a leader in equilibrium
chooses a lower strategy than the follower when players prefer an opponent’s higher choice.
The player who moves first earns higher payoffs than the player who moves second, whether
the reaction function of the follower is upward or downward sloping. Second, when the
lower-strategy player prefers an opponent’s higher choice while the higher-strategy player
prefers an opponent’s lower choice, a leader may choose a higher or a lower strategy than
the follower in equilibrium. There is still a first-mover advantage when the reaction function
of the follower is upward sloping.
Extending Gal-Or’s (1985) analysis to allow for strategy-ranking-dependent payoff functions in a symmetric game suggests that under almost all kinds of preferences to the opponent’s choice for a player and the shape of a player’s reaction function, a leader can exercise
a preemptive advantage over the follower. In a sense, the strategy-ranking-dependent payoff functions play a crucial role in determining which player can get an upper hand in a
sequential move game.
The intuition behind our results is as follows.
In a game with strategy-ranking-
dependent payoffs, the existence of a jump in the follower’s reaction curve can enlarge
or reverse a leader’s strategic effect on the follower’s choice, which is shown to be beneficial
to the leader. In such a game a player is concerned about the relative positions of players’
strategies and chooses a lower (higher) strategy when seeing an opponent’s higher (lower)
strategy. We note that players’ actions in such a game are, in a global sense, strategic
substitutes in the interval across any jump points of reaction curves. As such, there always
exists a first-mover advantage, and the locally strategic substitutes and locally strategic
complements do not affect players’ relative payoffs.
We are also concerned with endogenizing the timing in the sequential game – that is,
the order of play that determines the roles of leader and follower. It is shown that although
there always exists a first-mover advantage in a sequential game, the follower may still yield
a higher payoff in the sequential game than in the simultaneous game. This implies that our
6
A player’s payoff function in the model of Gal-Or (1985) is globally concave in his own strategy, and
the functional form of a player’s payoff does not depend on the relative positions of players’ strategies (i.e.
strategy-ranking-independent).
4
symmetric two-player game may yield the first-mover advantage outcome in an endogenous
timing game with observable delay, which revises an interesting puzzle proposed by von
Stengel (2010), showing that an endogenous sequential entry with a first-mover advantage
can occur only in an asymmetric game.
The rest of the paper is organized as follows. Section 2 states and discusses our main
assumptions. Section 3 investigates the model’s competition behavior under simultaneous
choice. Section 4 examines the model’s competition behavior under sequential choice. Section 5 provides some examples corresponding to the assumptions and conclusions in the
model. Section 6 offers a conclusion.
2. The model
There are two a priori identical players, denoted by player a and player b, moving simultaneously or sequentially in order during a game. The strategy choice of each player i is denoted
by si ∈ Si , for Si ≡ S = [c, c̄], indicating player i’s strategy space is a real interval, and thus
the two players’ strategies can be ranked in this game. We use the term “lower (higher)
strategy” for a player’s lower (higher) choice versus his opponent’s choice. The payoff of
player i is denoted by π(si , sj ), for i, j ∈ {a, b} and i 6= j. The first argument of a player’s
payoff function corresponds to his own strategy, and the second argument corresponds to
his opponent’s strategy.
The functional form of player i’s payoff is given by:

π l (si , sj ), if si ≤ sj
,
π(si , sj ) = 
π h (si , sj ), if si ≥ sj
where π l (si , sj ) and π h (si , sj ) are defined only for si ≤ sj and si ≥ sj , respectively, and they
are twice continuously differentiable with respect to si and sj , and π l (si , sj ) = π h (si , sj )
when si = sj . Superscript “l” here denotes a lower strategy of player i (i.e. si ≤ sj ), and
superscript “h” denotes a higher strategy of player i (i.e. si ≥ sj ), meaning the payoff
function is dependent on a player’s strategy ranking.7 We set up the game to fulfill the
7
The primary purpose of this paper is to examine a general class of symmetric two-player games with
strategy-ranking-dependent payoffs. The condition of a continuous payoff function is necessary for derivation,
and so the cases of discontinuous payoffs are ignored. We note that some results on the leader-follower
relationship and endogenous timing in two-person games with discontinuous payoff functions have been
investigated in the literature. For instance, Zhu et al. (2014) analyze a generalized framework for endogenous
timing in price-setting games with homogenous goods under a general form of concave industrial revenues
5
following assumptions.8
l (s , s ) < 0 and π h (s , s ) < 0 for all s ≤ s and s ≥ s , respectively.
A1. π11
i
j
i
j
i j
11 i j
A2. π1l (v − , v) < 0 and π1h (v + , v) > 0 for all v > c and v < c̄, respectively.
A3. π1l (c, v) > 0 and π1h (c̄, v) < 0 for all v > c and v < c̄, respectively.
Assumption A1 implies π l (si , sj ) and π h (si , sj ) are strictly concave in si . Assumption
A2 rules out the case that π l (·, v) is maximized at v for all v > c and π h (·, v) is maximized
at v for all v < c̄, implying that there is no symmetric pure strategy Nash equilibrium (i.e.
(si = v, sj = v) for v ∈ [c, c̄]). In this sense, the reaction curves have a jump across the
diagonal and do not intersect the diagonal on the whole strategy space. Assumption A3
rules out the case that π l (·, v) is maximized at c for all v > c and that π h (·, v) is maximized
at c̄ for all v < c̄; when combined with assumptions A1 and A2, it guarantees a player’s
unique interior choice si ∈ (c, sj ) that maximizes π l (si , sj ) as well as a unique interior choice
si ∈ (sj , c̄) that maximizes π h (si , sj ) and holds that each player’s payoff presents a canyon
shape along the diagonal, such as Figure 1 shows.
S ( si , s j )
c
sj
c si
Figure 1: Player i’s payoff function for a given s̄j
under strategy-ranking-dependent payoffs
h (s , s ) and π l (s , s ) have a constant sign for all s ≥ s and s ≤ s , respectively.
A4. π12
i j
i
j
i
j
12 i j
and convex costs, which overcomes the inconsistency across the existing related literature. We thank an
anonymous referee for suggesting this point.
8
The subscripts of payoff functions denote partial derivatives, and the signs “+” and “−” denote one-sided
partial derivatives in a point.
6
t (s , s ) for
The slope (shape) of a player’s reaction function depends on the sign of π12
i j
h (s , s ) <
t ∈ {l, h}, which the next section discusses. We investigate four cases. Case (i): π12
i j
l (s , s ) < 0. Case (ii): π h (s , s ) > 0 and π l (s , s ) > 0. Case (iii): π h (s , s ) > 0
0 and π12
i j
12 i j
12 i j
12 i j
l (s , s ) < 0. Case (iv): π h (s , s ) < 0 and π l (s , s ) > 0.
and π12
i j
12 i j
12 i j
A5. π h (si , sj ) and π l (si , sj ) are strictly monotone in sj ≤ si and sj ≥ si , respectively.
Assumption A5 is made for comparing the leader and follower payoffs in a sequential
game.9 The assumption of π2t (si , sj ) > 0 for t ∈ {l, h} states that a player prefers an
opponent’s higher choice since an increase in the opponent’s choice is beneficial to that
player. The assumption of π2t (si , sj ) < 0 for t ∈ {l, h} conversely means that a player
prefers an opponent’s lower choice since an increase in the opponent’s choice is harmful to
that player.
3. Simultaneous choice
This section investigates the equilibrium outcome when player a and player b move simultaneously in this game. Player i’s payoff function in the two strategies, π t (si , sj ) for t ∈ {l, h},
t (s , s ) < 0 (π t (s , s ) > 0, respecis continuous and strictly concave in his own strategy, π12
i j
12 i j
tively), and both strategy spaces are compact sets. Thus, arg maxsi π t (si , sj ) has a unique
selection (best response, reaction function) g t (sj ), with g l (sj ) ≡ arg maxsi ∈[c,sj ] π l (si , sj )
and g h (sj ) ≡ arg maxsi ∈[sj ,c̄] π h (si , sj ), that is continuous and decreasing (increasing, respectively) in sj .
Lemma 1. In Case (i), g h (sj ) and g l (sj ) are decreasing in sj . In Case (ii), g h (sj ) and
g l (sj ) are increasing in sj . In Case (iii), g h (sj ) is increasing and g l (sj ) is decreasing in sj .
In Case (iv), g h (sj ) is decreasing and g l (sj ) is increasing in sj .
The payoff value function π t (g t (sj ), sj ) for t ∈ {l, h} and the function π h (g h (sj ), sj ) −
π l (g l (sj ), sj ) are then continuous in sj . The best response of player i, g(sj ) ≡ arg maxsi ∈[c,c̄]
π(si , sj ), is determined by comparing π h (g h (sj ), sj ) and π l (g l (sj ), sj ), which is multiplevalued when π h (g h (sj ), sj ) = π l (g l (sj ), sj ).
Amir et al. (2010) show that under Case (i) globally strategic substitutes are inherited
from the strategic substitutes of a player’s two separate strategies (i.e. higher strategy
and lower strategy) in the presence of a concavity-destroying kink along the diagonal in
9
Gal-Or (1985) also adopts a similar assumption for her analysis.
7
the strategy space (implied by assumption A2), which leads to a globally decreasing best
response function. The existence of a Nash equilibrium in the simultaneous game with
monotone reaction functions in compact sets is guaranteed through Tarski’s fixed point
theorem.
Result 1. (Amir et al. (2010)) When the follower’s reaction function is downward sloping,
h < 0 and π l < 0), a player’s reaction
whether playing a higher or a lower strategy (π12
12
function then jumps only once in point d and a Nash equilibrium (ŝa , ŝb ) exists in the set
[c, d] × [d, c̄] (as well as in the set [d, c̄] × [c, d] by symmetry of the game).
Assumption A2 ensures that the best response function must have a downward jump
that crosses the diagonal in all cases. Since monotonicity of the reaction curve is not met in
Cases (ii), (iii), and (iv) (i.e. reaction curves never globally increase or globally decrease),
the corresponding games need not possess pure strategy Nash equilibria.10 Moreover, in
the case of π2l (si , sj ) < 0 and π2h (si , sj ) > 0, the difference π h (g h (sj ), sj ) − π l (g l (sj ), sj ) is
increasing in sj , which contradicts assumption A2, implying that the difference is positive at
sj = c and negative at sj = c̄. We thus discuss only three pairs of monotonic preferences to
the opponent’s choice for a player: (1): (π2l (si , sj ) < 0, π2h (si , sj ) < 0); (2): (π2l (si , sj ) > 0,
π2h (si , sj ) > 0); (3): (π2l (si , sj ) > 0, π2h (si , sj ) < 0).
Lemma 2. If π2l (si , sj ) > 0 and π2h (si , sj ) < 0, then there is a unique jump in a player’s
reaction curve.
Proof: According to the envelope theorem, π l (g l (sj ), sj ) is increasing in sj and π h (g h (sj ), sj )
is decreasing in sj , provided π2l (si , sj ) > 0 and π2h (si , sj ) < 0. The difference π h (g h (sj ), sj )−
π l (g l (sj ), sj ) is positive at sj = c, negative at sj = c̄, and decreasing in sj . Therefore, there
is a single point d solving for π h (g h (sj ), sj ) − π l (g l (sj ), sj ) = 0. Player i’s reaction curve
g(sj ) jumps only once in point d with g(sj ) = g h (sj ) for sj ∈ [c, d] and g(sj ) = g l (sj ) for
sj ∈ [d, c̄]. Q.E.D.
Lemma 2 states that there is a unique jump in a player’s reaction curve when π2l (si , sj ) >
0 and π2h (si , sj ) < 0. In the cases of π2t (si , sj ) < 0 for t ∈ {l, h} and π2t (si , sj ) > 0 for
t ∈ {l, h}, a unique jump in a player’s reaction curve is guaranteed by the single-crossing of
π l (si , sj ) and π h (si , sj ) with respect to the opponent’s action along with his best response:
10
The best response function g is obviously not globally monotone in Cases (iii) and (iv). The existence
of a downward jump in a player’s reaction curve implies that the best response function g is not globally
monotone in Case (ii).
8
π2h (g h (sj ), sj ) < π2l (g l (sj ), sj ) whenever π h (g h (sj ), sj ) = π l (g l (sj ), sj ). We now impose conditions, which secure that the reaction curves are well defined and completely contained
in certain compact subsets of the joint strategy space, given there is a unique jump in a
player’s reaction curve.
Proposition 1. Suppose a player’s reaction curve jumps only once in point d. If π1h (d, c) >
0 and π1l (d, c̄) < 0, then a Nash equilibrium (ŝi , ŝj ), for i, j ∈ {a, b} and i 6= j, exists in the
set [c, d] × [d, c̄] under Cases (ii), (iii), and (iv).
Proof: By the assumption of a unique jump at point d, we have g(si ) = g h (si ) for si ∈ [c, d]
and g(sj ) = g l (sj ) for sj ∈ [d, c̄]. By strict concavity, the assumption of π1h (d, c) > 0
implies that g h (c) > d, and the assumption of π1l (d, c̄) < 0 implies that g l (c̄) < d. From the
definitions of g h and g l , we have g h (d) > d and g l (d) < d. In all three cases, monotonicity
of g h implies that g h (si ) ∈ [d, c̄] for all si ∈ [c, d], and monotonicity of g l implies that
g l (sj ) ∈ [c, d] for all sj ∈ [d, c̄]. The existence of a Nash equilibrium then follows from
Brouwer’s fixed point theorem applied to the mapping (g l (sj ), g h (si )) from [c, d] × [d, c̄] to
itself. Q.E.D.
When there is a unique jump in a player’s reaction curve, Proposition 1 gives the sufficient condition for the reaction curves to be completely contained in certain compact subsets
of the joint strategy space, [c, d]×[d, c̄]. This implies a pure strategy Nash equilibrium exists
in the simultaneous game.
We note that the condition of a unique jump in a player’s reaction curve is satisfied
under various models in the literature of industrial organization (as presented in Section
5), with a long-run investment decision at the first stage and short-run product market
competition at the second stage, although this is not a necessary condition for guaranteeing
a pure strategy Nash equilibrium in the simultaneous game. Claim 1 in Appendix A extends
the condition that guarantees the existence of a pure strategy Nash equilibrium to the case
of multiple jumps in a player’s reaction curve.
We provide explicit explanations in the case of a unique jump in a player’s reaction curve
in order to clarify the ideas of the existence of a pure strategy Nash equilibrium. Figure 3
depicts the possible reaction functions guaranteeing a pure strategy Nash equilibrium under
four cases.
9
sb
sb
c
c
L1
g h ( sa )
g l ( sb )
g l ( sb )L
2
g h ( sa )
L3
d
d
L1
l
g ( sa )
H3
H1
g l ( sa )
h
g ( sb )
c
sa
H2
H1
sb
c sa
d
Case (i)
sb
c
sa
g h ( sb )
sb
c
d
Case (ii)
sa
sb
c
c
g h ( sa )
g h ( sa )
L1
g l ( sb )
L1
g l ( sb )
d
g h ( sb )
d
g l ( sa )
H1
c
sa
g l ( sa )
H1
g h ( sb )
sb
d
Case (iii)
c
c
sa
sa
sb
d
Case (iv)
c sa
Figure 3: The possible reaction functions guaranteeing a pure strategy Nash equilibrium
under four cases
We can partition a player’s strategy space, Si ≡ S = [c, c̄], into two subsets, S L ≡ [c, d]
and S H ≡ [d, c̄]. The concept of reaction curves being completely contained in certain
compact subsets of the joint strategy space requires that: d < g(sj ) ∈ S H for all sj ∈ S L ,
and d > g(sj ) ∈ S L for all sj ∈ S H . This means that player i’s best response to any sj
belonging to the subset S L lies entirely in the subset S H and the best response to any sj
belonging to the subset S H lies entirely in the subset S L . In other words, when the rival has
a relatively low strategic choice, it is optimal for a player to have a relatively high strategic
choice. On the other hand, if a rival has a relatively high strategic choice, it is optimal for
a player to have a relatively low strategic choice.
Due to the game’s symmetry, the multiple equilibria in a simultaneous game are asymmetric (i.e. sa 6= sb ) and occur in pairs. In a sense, Li = (ŝla , ŝhb ) and Hi = (ŝha , ŝlb ), for
i ∈ {1, 2, 3}, in Figure 3, satisfying ŝla < ŝhb (ŝha > ŝlb ), ŝla = ŝlb , and ŝha = ŝhb . We note that in
Cases (i) and (ii) the two reaction curves may intersect several times above and below the
10
diagonal, and thus multiple pairs of pure strategy Nash equilibria may arise. On the other
hand, there is exactly one pair of pure strategy Nash equilibria in Cases (iii) and (iv).11
4. Sequential choice and comparison
This section investigates the equilibrium outcome when player a and player b move sequentially in order during a game and then compares the equilibrium outcomes of simultaneous
and sequential move games. We now consider situations where a player’s reaction curves
have a unique jump and are completely contained in certain compact subsets of the joint
strategy space.
4.1 Sequential choice
The best response of the follower g(sa ) is given by g(sa ) = g h (sa ), or g(sa ) = g l (sa ), or
both when sa is a jump point in the reaction curve. Given the assumption that the best
response of player b has a unique jump at sa = d, the constrained maximization problem
for the leader (player a) holds one of the following two local optima:
sla ∗ ≡ arg max π l (sa , g h (sa )),
sa ∈[c,d]
sha ∗ ≡ arg max π h (sa , g l (sa )).
sa ∈[d,c̄]
(1)
Suppose each of the two local optima is unique, and one of the local maxima, sla ∗ ,
involves player a choosing a lower strategy than player b, while player a chooses a higher
strategy than player b in the other local maximum, sha ∗ . We note that the two local optima
are well-defined, because the payoff function π l (sa , g h (sa )) is continuous in sa ∈ [c, d], and
likewise π h (sa , g l (sa )) is continuous in sa ∈ [d, c̄]. In particular, the functions are continuous
at the only point sa = d by assumption. The equilibrium outcome (s∗a , s∗b ) of the sequential
move game will be (sla ∗ , g h (sla ∗ ) ≡ shb ∗ ) or (sha ∗ , g l (sha ∗ ) ≡ slb ∗ ).12
11
We also note that in the simultaneous game if π2l (si , sj ) < 0 and π2h (si , sj ) < 0 (π2l (si , sj ) > 0 and
π2h (si , sj ) > 0, respectively), then the player choosing a higher strategy (lower strategy) in equilibrium has
an advantage. If π2l (si , sj ) > 0 and π2h (si , sj ) < 0, then the two players’ relative equilibrium payoffs cannot
be determined.
12
We note that player a generally is not indifferent between π l (d, g h (d)) and π h (d, g l (d)). For example, if
π2l > 0 and π2h > 0, then π l (d, g h (d)) > π l (d, d) = π h (d, d) > π h (d, g l (d)). In this case, the only equilibrium
outcome is (d, g h (d)); if player b responds to d by instead playing g l (d), then player a would deviate to an
action sa just below d, and so (d, g l (d)) is not an equilibrium outcome. In particular, player a is indifferent
11
h∗
If an interior maximum exists (i.e. sl∗
a ∈ (c, d) and sa ∈ (d, c̄)), then the first-order
necessary condition for the leader’s (player a’s) payoff maximization holds:
dπ l (s∗a , s∗b )
π l (s∗ , s∗ ) · π h (s∗ , s∗ )
≡ Gl (s∗a , s∗b ) = π1l (s∗a , s∗b ) − 2 a hb ∗ 12∗ b a = 0, if s∗a ≤ s∗b ,
dsa
π11 (sb , sa )
dπ h (s∗a , s∗b )
π h (s∗ , s∗ ) · π l (s∗ , s∗ )
≡ Gh (s∗a , s∗b ) = π1h (s∗a , s∗b ) − 2 a l b ∗ 12∗ b a = 0, if s∗a ≥ s∗b .
dsa
π11 (sb , sa )
(2)
The second-order condition requires:
h (s∗ , s∗ )
d2 π l (s∗a , s∗b )
Gl2 (s∗a , s∗b ) · π12
l ∗ ∗
b a
=
G
(s
,
s
)
−
< 0, if s∗a ≤ s∗b ,
1
a
b
h
∗
∗
ds2a
π11 (sb , sa )
l (s∗ , s∗ )
d2 π h (s∗a , s∗b )
Gh2 (s∗a , s∗b ) · π12
h ∗ ∗
b a
=
G
(s
,
s
)
−
< 0, if s∗a ≥ s∗b .
1
a
b
2
l
∗
∗
dsa
π11 (sb , sa )
(3)
If the second-order condition holds globally, then the maximum is unique. As with many
sequential games with general payoff functions, the sign of inequality (3) depends on the
third-order derivative of a player’s payoff function. In terms of sufficient conditions on
primitives, it can be shown that condition (i), whereby the third-order derivatives of the
payoff function vanish, is a sufficient condition for the second-order condition in Cases (iii)
and (iv). Condition (i), when combined with condition (ii) in which own effects dominate
¯ t ¯
¯ ¯
¯ > ¯π t ¯, for t ∈ {l, h}), becomes the sufficient condition for the
cross effects (i.e. ¯π11
12
second-order condition in Cases (i) and (ii).
We note that an interior equilibrium exists in many other cases (as presented in Section
5), which is in fact sufficient for the results presented in this paper. In this case, we simply
assume the existence of a unique interior maximum, following the related literature, such
as Gal-Or (1985) who also adopts a similar assumption in her study.
A6. The two local optima in the sequential game are interior and unique.
For analysis of a general framework, assumption A6 simply assumes a unique interior
maximum and thus the first-order necessary condition in equation (2) holds. We note that
h∗
our analysis is still valid when a maximum occurs at sl∗
a = c, d or sa = c̄, d (i.e. a boundary
solution), and the first-order condition in equation (2) nevertheless still holds.
When players’ preferences to their opponent’s choice are monotonic and identical between a higher-strategy player and a lower-strategy player, the following Lemmas 3 and 4
provide a useful criterion about the equilibrium outcome.
between π l (d, g h (d)) and π h (d, g l (d)) only if π2l > 0 and π2h < 0. We thank the Editor for suggesting this
point.
12
Lemma 3. Suppose π2l (si , sj ) < 0 and π2h (si , sj ) < 0.
h < 0 and π l < 0, then ŝl < sl ∗ < d < sh ∗ < ŝh and sl ∗ < ŝl < d < ŝh < sh ∗ .
(i) If π12
a
a
a
a
12
b
b
b
b
h > 0 and π l > 0, then sl ∗ < ŝl < d < sh ∗ < ŝh and sl ∗ < ŝl < d < sh ∗ < ŝh .
(ii) If π12
a
a
a
a
12
b
b
b
b
h > 0 and π l < 0, then sl ∗ < ŝl < d < sh ∗ < ŝh and sl ∗ < ŝl < d < ŝh < sh ∗ .
(iii) If π12
a
a
a
a
12
b
b
b
b
h < 0 and π l > 0, then ŝl < sl ∗ < d < sh ∗ < ŝh and sl ∗ < ŝl < d < sh ∗ < ŝh .
(iv) If π12
a
a
a
a
12
b
b
b
b
Proof: See Appendix B.
Lemma 4. Suppose π2l (si , sj ) > 0 and π2h (si , sj ) > 0.
h < 0 and π l < 0, then sl ∗ < ŝl < d < ŝh < sh ∗ and ŝl < sl ∗ < d < sh ∗ < ŝh .
(i) If π12
a
a
a
a
12
b
b
b
b
h > 0 and π l > 0, then ŝl < sl ∗ < d < ŝh < sh ∗ and ŝl < sl ∗ < d < ŝh < sh ∗ .
(ii) If π12
a
a
a
a
12
b
b
b
b
h > 0 and π l < 0, then ŝl < sl ∗ < d < ŝh < sh ∗ and ŝl < sl ∗ < d < sh ∗ < ŝh .
(iii) If π12
a
a
a
a
12
b
b
b
b
h < 0 and π l > 0, then sl ∗ < ŝl < d < ŝh < sh ∗ and ŝl < sl ∗ < d < ŝh < sh ∗ .
(iv) If π12
a
a
a
a
12
b
b
b
b
Proof: See Appendix B.
The notation that has “hats” refers to the equilibrium strategies in simultaneous games.
We note that although there may be multiple pairs of pure strategy Nash equilibria in Cases
(i) and (ii), they are payoff-ranked under assumption A5. Player a can always obtain the
most preferred Nash equilibrium payoff by choosing the corresponding equilibrium strategy,
which means that only this equilibrium is relevant in the sequential move game.13 Lemmas 3
and 4 and the following Lemma 5 take into account only the most preferred Nash equilibrium
for player a.
For a given player a who chooses a lower (or higher) strategy than player b, Lemmas
3 and 4 first compare player a’s equilibrium strategy in a sequential game versus that in a
simultaneous game. The relationship between player b’s equilibrium strategy in a sequential
game and that in a simultaneous game then follows from the strategic substitutes or strategic
complements in the respective case. Proposition 2 follows from Lemmas 3 and 4.
Proposition 2. Suppose a player’s preference to the opponent’s choice is monotonic and
identical between a higher-strategy player and a lower-strategy player (i.e. π2t (si , sj ) < 0
13
For example, suppose π2h < 0. If there are two Nash equilibria (ŝha , ŝlb ) and (s̃ha , s̃lb ) in Case (i) or Case
(ii), then ŝlb < s̃lb implies π h (ŝha , ŝlb ) > π h (s̃ha , ŝlb ) > π h (s̃ha , s̃lb ). This means the Nash equilibrium with the
smallest strategy for player b is the most preferred for player a regardless of whether player b’s best response
is upward or downward sloping. Player a can obtain an even higher payoff by marginally increasing ŝha in
Case (i) or by marginally decreasing ŝha in Case (ii).
13
for t ∈ {l, h} or π2t (si , sj ) > 0 for t ∈ {l, h}). There then exists a first-mover advantage in
all the four cases.
Proof:
(1) Suppose players’ payoff functions are monotonically decreasing in their opponent’s
choice, whether choosing a higher or a lower strategy (i.e. π2l (si , sj ) < 0 and π2h (si , sj ) <
0). Under Cases (i) and (iv), we claim that π h (sha ∗ , slb ∗ ) ≥ π h (ŝha , ŝlb ) = π h (ŝhb , ŝla ) >
π h (sla ∗ , ŝla ) > π h (sla ∗ , sla ∗ ) = π l (sla ∗ , sla ∗ ) > π l (sla ∗ , shb ∗ ). The first inequality follows since the
first-mover always has the option of enforcing the Nash equilibrium payoff in the sequential
game; the second equality follows from the symmetry of the two pure strategy Nash equilibria; the third inequality follows since ŝhb is the best response for a given ŝla and sla ∗ > ŝla
shown in Lemma 3; the fourth inequality follows from π2h (si , sj ) < 0 and sla ∗ > ŝla shown
in Lemma 3; the fifth equality follows from the assumption of the model; and the sixth
inequality follows from π2l (si , sj ) < 0 and shb ∗ > sla ∗ shown in Lemma 3.
We now claim that π h (sha ∗ , slb ∗ ) ≥ π h (ŝha , ŝlb ) = π h (ŝhb , ŝla ) > π l (sla ∗ , ŝla ) > π l (sla ∗ , shb ∗ )
under Cases (ii) and (iii). The first inequality follows since the first-mover always has the
option of enforcing the Nash equilibrium payoff in the sequential game; the second equality
follows from the symmetry of the two pure strategy Nash equilibria; the third inequality
follows since ŝhb is the best response for a given ŝla and sla ∗ < ŝla shown in Lemma 3; and
the fourth inequality follows from π2l (si , sj ) < 0 and shb ∗ > ŝla shown in Lemma 3.
Therefore, the first-mover chooses a higher strategy s∗a = sha ∗ in equilibrium in all the
four cases, meaning π(s∗a , s∗b ) = π h (sha ∗ , slb ∗ ) and π(s∗b , s∗a ) = π l (slb ∗ , sha ∗ ). We next show that
π h (sha ∗ , slb ∗ ) ≥ π h (ŝha , ŝlb ) > π l (slb ∗ , ŝlb ) > π l (slb ∗ , sha ∗ ). The first inequality follows since the
first-mover always has the option of enforcing the Nash equilibrium payoff in the sequential
game; the second inequality follows since ŝha is the best response for a given ŝlb and slb ∗ < ŝlb
shown in Lemma 3; and the third inequality follows from π2l (si , sj ) < 0 and sha ∗ > ŝlb shown
in Lemma 3. We conclude that the first-mover chooses a higher strategy s∗a = sha ∗ , and that
there is a first-mover advantage in equilibrium.
(2) Suppose players’ payoff functions are monotonically increasing in their opponent’s choice,
whether choosing a higher or a lower strategy (i.e. π2l (si , sj ) > 0 and π2h (si , sj ) > 0). Under
Cases (i) and (iii), we claim that π l (sla ∗ , shb ∗ ) ≥ π l (ŝla , ŝhb ) = π l (ŝlb , ŝha ) > π l (sha ∗ , ŝha ) >
π l (sha ∗ , sha ∗ ) = π h (sha ∗ , sha ∗ ) > π h (sha ∗ , slb ∗ ). The first inequality follows since the first-mover
always has the option of enforcing the Nash equilibrium payoff in the sequential game; the
second equality follows from the symmetry of the two pure strategy Nash equilibria; the
14
third inequality follows since ŝlb is the best response for a given ŝha and sha ∗ < ŝha shown in
Lemma 4; the fourth inequality follows from π2l (si , sj ) > 0 and sha ∗ < ŝha shown in Lemma 4;
the fifth equality follows from the assumption of the model; and the sixth inequality follows
from π2h (si , sj ) > 0 and sha ∗ > slb ∗ shown in Lemma 4.
We next claim that π l (sla ∗ , shb ∗ ) ≥ π l (ŝla , ŝhb ) = π l (ŝlb , ŝha ) > π h (sha ∗ , ŝha ) > π h (sha ∗ , slb ∗ )
under Cases (ii) and (iv). The first inequality follows from the first-mover always having the
option of enforcing the Nash equilibrium payoff in the sequential game; the second equality
follows from the symmetry of the two pure strategy Nash equilibria; the third inequality
follows since ŝlb is the best response for a given ŝha and sha ∗ > ŝha shown in Lemma 4; and
the fourth inequality follows from π2h (si , sj ) > 0 and ŝha > slb ∗ shown in Lemma 4.
Therefore, the first-mover chooses a lower strategy s∗a = sla ∗ in equilibrium in all the
four cases. In this sense, π(s∗a , s∗b ) = π l (sla ∗ , shb ∗ ) and π(s∗b , s∗a ) = π h (shb ∗ , sla ∗ ). We next
show that π l (sla ∗ , shb ∗ ) ≥ π l (ŝla , ŝhb ) > π h (shb ∗ , ŝhb ) > π h (shb ∗ , sla ∗ ). The first inequality follows
since the first-mover always has the option of enforcing the Nash equilibrium payoff in the
sequential game; the second inequality follows since ŝla is the best response for a given ŝhb
and shb ∗ > ŝhb shown in Lemma 4; and the third inequality follows from π2h (si , sj ) > 0 and
ŝhb > sla ∗ shown in Lemma 4. We conclude that the first-mover chooses a lower strategy
s∗a = sla ∗ , and that there is a first-mover advantage in equilibrium. Q.E.D.
We first discuss the cases where the slope of the follower’s reaction function is negative
in Case (i) and positive in Case (ii), whether he plays a higher or a lower strategy, in order
to compare with the analysis of Gal-Or (1985). When the follower’s reaction function is
downward sloping, whether choosing a higher or a lower strategy, Proposition 2 shows that
there is a first-mover advantage, which is consistent with Gal-Or’s (1985) findings. When
the follower’s reaction function is upward sloping, whether choosing a higher or a lower
strategy, Proposition 2 shows that strategy-ranking-dependent payoff functions lead to the
first-mover getting an upper hand in the game, which runs opposite to Gal-Or (1985) in
which the second-mover has the advantage. We use the following Figure 4 to illustrate the
intuition behind the results.
15
sb
sb
c
c
g h ( sa )
g h ( sa )
r ( sa )
d
d
r ( sa )
g l ( sa )
g l ( sa )
c
sa
sa
sb
d
Case (i)
c
c sa
sb
d
Case (ii)
c sa
Figure 4: The reaction functions under Cases (i) and (ii)
The continuous downward (upward) sloping curve r(sa ) in Figure 4 is the follower’s
reaction curve in a game with strategy-ranking-independent payoffs. On the other hand,
the follower’s reaction curve in a game with strategy-ranking-dependent payoffs is given by
g(sa ) = g h (sa ) when sa ≤ d and is given by g(sa ) = g l (sa ) when sa ≥ d.
We first note that when a player prefers an opponent’s lower choice, π2l (si , sj ) < 0
and π2h (si , sj ) < 0, there is a higher-strategy advantage (i.e. the player choosing a higher
strategy in equilibrium has an advantage), whether or not a player’s payoff function in a
game depends on the relative positions of the players’ strategies.14
Suppose a player’s payoff function is strategy-ranking-independent and the Nash equilibrium in a simultaneous game is unique and symmetric (ŝa = ŝb ). If π2 (si , sj ) < 0, then
the leader can choose a relatively higher choice in order to induce the follower to choose a
relatively lower choice under strategic substitutes and then s∗a > s∗b . In this sense, a leader
14
Suppose that the payoff functions are strategy-ranking-independent, with player i’s payoff given by
π(si , sj ) and player j’s payoff given by π(sj , si ), and that the Nash equilibrium in a simultaneous game is
unique and symmetric (ŝa = ŝb ). If π2 (si , sj ) < 0 and π12 (si , sj ) > 0, then Lemma 1 and Proposition 1 in
Gal-Or (1985) show that s∗a < s∗b under strategic complements and the second-mover earns higher payoffs
than the first-mover, which means there is a higher-strategy advantage. If π2 (si , sj ) < 0 and π12 (si , sj ) < 0,
then π1 (sa , r(sa )) + π2 (sa , sb ) · r0 (sa ) > π1 (sa , r(sa )) and the leader chooses a higher choice in a sequential
game versus that in a simultaneous game, s∗a > ŝa . Therefore, s∗a > ŝa = ŝb > s∗b under strategic substitutes
and the first-mover earns higher payoffs than the second-mover, meaning there is also a higher-strategy
advantage.
16
can exercise a preemptive advantage over the follower. By contrast, the leader must choose
a relatively lower choice to induce the follower’s lower choice under strategic complements
and then s∗a < s∗b . In this case, the leader’s lower choice matches the follower’s preference. This is in reference to markets in which a follower copies the leader and then has an
advantage.
We now suppose a player’s payoff function is strategy-ranking-dependent. If π2l (si , sj ) <
0 and π2h (si , sj ) < 0, then the leader can choose a relatively higher choice to induce the
follower’s lower choice under strategic substitutes. In particular, the jump in the follower’s
reaction curve enlarges the follower’s lower choice, which implies the first-mover advantage
is enhanced in Case (i) and then s∗a > sb∗ , such as Case (i) in Figure 4 shows. Although the
follower’s reaction function is increasing in the leader’s choice for sa ∈ [c, d] and sa ∈ [d, c̄],
respectively, the downward jump in the follower’s reaction curve allows the leader to choose
a relatively higher choice to induce the follower’s lower choice under strategic complements.
In particular, the jump in the follower’s reaction curve reverses a leader’s strategic effect
on the follower’s choice, which implies the second-mover advantage is reversed in Case (ii)
and then s∗a > s∗b , such as Case (ii) in Figure 4 shows. It follows that there is a first-mover
advantage.15
In a two-player game with strategy-ranking-dependent payoffs, a player’s overall payoff
function has a canyon (valley-like) shape along the diagonal in the strategy space, resulting
in a player never responding to an opponent’s action by playing that same action, and then
a player’s reaction curve has a jump across the diagonal. The existence of a jump in the
follower’s reaction curve can enlarge or reverse a leader’s strategic effect on the follower’s
choice, which is always beneficial to the leader.
h > 0, π l < 0) and (π h < 0, π l > 0), respectively,
In Cases (iii) and (iv), where (π12
12
12
12
Proposition 2 shows that a first-mover advantage still exists, provided a player’s preference
to the opponent’s choice is monotonic and identical between a higher-strategy player and a
lower-strategy player. During such a game, a player is concerned about the relative positions
of players’ strategies and chooses a lower (higher) strategy when seeing an opponent’s higher
(lower) strategy. It can be seen from Figure 3 that players’ actions in this game are, in
a global sense, strategic substitutes in the interval across any jump points of the reaction
curves. It follows that a leader can always make a preemptive move and get an upper
15
We discuss just the case of π2h , π2l < 0. In Case (ii), the logic for why there is a second-mover advan-
tage with strategy-ranking-independent payoffs but a first-mover advantage with strategy-ranking-dependent
payoffs does not depend on whether player a always wants player b to decrease or increase his strategy, i.e.,
whether π2h , π2l < 0 or π2h , π2l > 0.
17
hand under strategy-ranking-dependent payoff functions, no matter whether the follower’s
reaction curve is downward sloping or upward sloping.
We finally discuss the situations where the player choosing a lower strategy prefers an
opponent’s higher choice (π2l (si , sj ) > 0), while the player choosing a higher strategy prefers
an opponent’s lower choice (π2h (si , sj ) < 0). Lemma 5 in Appendix B gives a useful criterion
about the equilibrium outcome. Proposition 3 then follows from Lemma 5.
Proposition 3. Suppose a player choosing a lower strategy prefers an opponent’s higher
choice (π2l (si , sj ) > 0), while a player choosing a higher strategy prefers an opponent’s lower
choice (π2h (si , sj ) < 0). A first-mover advantage then exists in Case (ii), in Case (iii) when
s∗a < s∗b in equilibrium, and in Case (iv) when s∗a > s∗b in equilibrium.
Proof: We first show that a first-mover advantage exists in Case (ii). If s∗a ≤ s∗b , then
π(s∗a , s∗b ) = π l (sla ∗ , shb ∗ ) and π(s∗b , s∗a ) = π h (shb ∗ , sla ∗ ). We claim that π l (sla ∗ , shb ∗ ) ≥ π h (ŝhb , ŝla ) >
π h (shb ∗ , ŝla ) > π h (shb ∗ , sla ∗ ). The first inequality follows since the first-mover always has the
option of enforcing the Nash equilibrium payoff in the sequential game; the second inequality follows since ŝhb is the best response for a given ŝla ; and the third inequality follows from
π2h (si , sj ) < 0 and sla ∗ > ŝla shown in Lemma 5. If s∗a ≥ s∗b , then π(s∗a , s∗b ) = π h (sha ∗ , slb ∗ ) and
π(s∗b , s∗a ) = π l (slb ∗ , sha ∗ ). We claim that π h (sha ∗ , slb ∗ ) ≥ π l (ŝlb , ŝha ) > π l (slb ∗ , ŝha ) > π l (slb ∗ , sha ∗ ).
The first inequality follows since the first-mover always has the option of enforcing the Nash
equilibrium payoff in the sequential game; the second inequality follows since ŝlb is the best
response for a given ŝha ; and the third inequality follows from π2l (si , sj ) > 0 and ŝha > sha ∗
shown in Lemma 5. We conclude that in either case there always exists a first-mover
advantage.
By a similar argument it can be shown that π l (sla ∗ , shb ∗ ) ≥ π h (ŝhb , ŝla ) > π h (shb ∗ , ŝla ) >
π h (shb ∗ , sla ∗ ) in Case (iii) and that π h (sha ∗ , slb ∗ ) ≥ π l (ŝlb , ŝha ) > π l (slb ∗ , ŝha ) > π l (slb ∗ , sha ∗ ) in
Case (iv). It follows that there exists a first-mover advantage in Case (iii) when s∗a < s∗b in
equilibrium and in Case (iv) when s∗a > s∗b in equilibrium.
For other situations it is not possible to guarantee whether there is a higher-strategy
advantage or a lower-strategy advantage. Thus, the two players’ relative equilibrium payoffs
cannot be further determined without incorporating some more restrictions on players’
payoff functions. Q.E.D.
h >0
We take a closer look at the strategic interaction among players in Case (ii) (i.e. π12
18
l > 0), illustrated in the following Figure 5.
and π12
l
h*
l*
sˆal g ( sb ) sa d
sbl * sˆbl
sˆbh sbh*
l*
h* h
h
d sa g ( sb ) sˆa
Figure 5: The strategic interaction among players
when π2l (si , sj ) > 0 and π2h (si , sj ) < 0
Suppose the leader chooses a lower strategy than the follower. Figure 5 shows that
π(s∗a , s∗b ) = π l (sla ∗ , shb ∗ ) = π l (sla ∗ , g h (sla ∗ )) ≥ π h (shb ∗ , g l (shb ∗ )) > π h (shb ∗ , sla ∗ ) = π(s∗b , s∗a ). In
this sense, the leader can hence make a preemptive move by choosing a relatively higher
choice in order to induce his opponent to choose a relatively higher choice under strategic
complements. We note that the size of the difference between the two players’ equilibrium
strategies, shb ∗ and sla ∗ , is smaller than that between shb ∗ and g l (shb ∗ ), which results in a
first-mover advantage. In this sense, from the follower’s viewpoint, the leader’s preemptive
move is too close to his best response (i.e. sla ∗ > g l (shb ∗ )).
Suppose the leader chooses a higher strategy than the follower. Figure 5 presents that
π(s∗a , s∗b ) = π h (sha ∗ , slb ∗ ) = π h (sha ∗ , g l (sha ∗ )) ≥ π l (slb ∗ , g h (slb ∗ )) > π l (slb ∗ , sha ∗ ) = π(s∗b , s∗a ).
Under this situation, the leader can make a preemptive move by choosing a relatively lower
choice to induce the follower to choose a relatively lower choice. Similarly, the size of the
difference between the two players’ equilibrium strategies, slb ∗ and sha ∗ , is smaller than that
between slb ∗ and g h (slb ∗ ), which results in a first-mover advantage. Therefore, from the
follower’s viewpoint, the leader’s preemptive move is also too close to his best response (i.e.
sha ∗ < g h (slb ∗ )). It follows that a leader can always exercise a preemptive advantage over
the follower.
Extending Gal-Or’s (1985) analysis to allow for strategy-ranking-dependent payoff functions in a symmetric game suggests that for a player under almost all kinds of preferences
to the opponent’s choice and the shape of a player’s reaction function, a leader can exercise a preemptive advantage over the follower. In a sense, the strategy-ranking-dependent
payoff functions play a crucial role in determining which player can get an upper hand in a
sequential move game.
19
4.2 Comparison
We now compare the leader and follower payoffs in the game, as they arise in sequential play,
with the Nash payoff under simultaneous play. We only discuss situations where a player’s
preference to the opponent’s choice is monotonic and identical between a higher-strategy
player and a lower-strategy player.
Recall that due to the game’s symmetry, the multiple equilibria are asymmetric and
occur in pairs. For analysis, we assume player i chooses a higher (or lower) strategy both
l∗
h l
in a simultaneous game and in a sequential game – that is, πa∗ = π(sh∗
a , sb ), π̂a = π(ŝa , ŝb ),
h∗
l h
∗
l∗ h∗
l
h
∗
h∗ l∗
πb∗ = π(sl∗
b , sa ), and π̂b = π(ŝb , ŝa ), or πa = π(sa , sb ), π̂a = π(ŝa , ŝb ), πb = π(sb , sa ),
and π̂b = π(ŝhb , ŝla ). Proposition 2 shows that a first-mover advantage always exists in the
sequential game. Proposition 4 below states that the follower may still yield a higher payoff
in the sequential game than that in the simultaneous game.16
Proposition 4. Player b’s equilibrium payoff is higher in a sequential game than in a
h > 0,
simultaneous game (i.e. πb∗ > π̂b ) under any of the following four situations: (i) (π12
l > 0) and (π l (s , s ) < 0, π h (s , s ) < 0); (ii) (π h > 0, π l > 0) and (π l (s , s ) > 0,
π12
2 i j
2 i j
12
12
2 i j
h > 0, π l < 0) and (π l (s , s ) > 0, π h (s , s ) > 0); (iv) (π h < 0,
π2h (si , sj ) > 0); (iii) (π12
12
2 i j
2 i j
12
l > 0) and (π l (s , s ) < 0, π h (s , s ) < 0); otherwise, player b’s equilibrium payoff is lower
π12
2 i j
2 i j
in a sequential game than in a simultaneous game (i.e. πb∗ < π̂b ).
We next discuss an extended endogenous timing game with observable delay, which
Hamilton and Slutsky (1990) first propose, by adding a pre-play stage whereby players
simultaneously decide whether to select actions in the basic game (discussed above) in the
first period or to wait until observing the opponent’s first-period actions and selecting one’s
own actions in the second period. More specifically, the basic game is played according to
these timing decisions: If one player moves in the first period and the other player moves in
the second period, then they become leader and follower, respectively. If they move in the
same period, their payoffs are the same as that under simultaneous play. In the subgame
perfect Nash equilibrium (SPNE) of the extended game, the equilibrium outcome for the
basic game endogenously emerges.
Comparisons of payoffs under simultaneous play and sequential play in the basic game
determine what order of play occurs. It is clear that πa∗ ≥ π̂a and πa∗ > πb∗ . The leaderfollower (sequential entry) is an equilibrium outcome in the endogenous timing game if the
16
The proof of Proposition 4 is simply through monotonicity of a player’s preference to the opponent’s
choice.
20
follower’s payoff is not smaller than his Nash payoff in the simultaneous game. Proposition
5 follows directly from Proposition 4.
Proposition 5. Under the four situations stated in Proposition 4, in which player b’s
equilibrium payoff is higher in a sequential game than in a simultaneous game, sequential
entry is the equilibrium outcome in the endogenous timing game with observable delay and
there is a first-mover advantage;17 otherwise, simultaneous entry is the equilibrium outcome.
Von Stengel (2010) demonstrates an interesting puzzle in an endogenous timing game
by showing that if a game is symmetric and certain standard assumptions hold, then the
follower gets either less than that in the simultaneous game, or more than the leader.18
It follows in the seemingly natural case that both players profit from sequential play as
compared to simultaneous play, and the leader gets more than if he was a follower, which
never occurs in a symmetric game. We use Gal-Or’s (1985) example of a horizontal product
differentiation model with a representative consumer and substitute products, in which
players’ payoffs are strategy-ranking-independent, to illustrate von Stengel’s (2010) findings.
It can be shown that with price competition, the follower’s equilibrium payoff is higher
than the leader. In quantity competition, the follower’s equilibrium payoff is lower in a
sequential game than in a simultaneous game. It follows that in the endogenous timing
game with observable delay, both firms play a sequential game under price competition
with a second-mover advantage and play a simultaneous game under quantity competition.
It is worth noting that Proposition 5 indicates that our symmetric two-player game with
strategy-ranking-dependent payoffs may revise such a puzzle by showing the existence of an
endogenous first-mover advantage outcome. The reason is due to the symmetry-breaking
result during such a strategy-ranking-dependent payoff game.19
17
By symmetry of the game, both sequential entry subgames – with player a as the leader and player b
as the follower, and player b as the leader and player a as the follower – are outcomes of the equilibrium in
the endogenous timing game.
18
Von Stengel (2010) assumes a symmetric two-player game with intervals as strategy spaces, a unique
best response, a unique Nash equilibrium in the simultaneous game, and monotonicity of payoffs in the other
player’s strategy along one’s own best response.
19
Due to the game’s symmetry, the Nash equilibria in the simultaneous game occur in pairs. The assumption that a player chooses a higher (or lower) strategy both in a simultaneous game and in a sequential
game is meant to simplify the analysis and to directly show this study’s main results, but is not a very
restrictive one. In particular, if we relax such an assumption, then there are multiple equilibria in the
subgame with a simultaneous move, which results in multiple SPNE outcomes of the entire game existing
in some cases. We note that the sequential entry with a first-mover advantage stated in Proposition 5 is
21
5. Examples
A symmetric game with strategy-ranking-independent payoff functions can be seen in
Gal-Or (1985). To illustrate the generality and wide scope of application in our study,
we present examples of economic models with strategy-ranking-dependent payoff functions
in a game – corresponding to the R&D model with one-way spillovers, the horizontal product differentiation model, and the vertical product differentiation model in the literature –
that constitute special cases of the four cases of the general framework developed above.
Going over these examples allows us to provide some contextual interpretations of a symmetric game with strategy-ranking-dependent payoff functions and to illustrate that the
slope (shape) of a player’s reaction curve and a player’s preference to the opponent’s choice
both vary in the respective specific economic model.
Each of the following examples is constituted by a two-stage game, and only the equilibrium payoffs of the second stage are relevant for the characterization of a subgame perfect
Nash equilibrium. We restrict the consideration to a subgame perfect Nash equilibrium and
analyze the first-stage strategy. A player’s payoff function in each of the following examples
is strategy-ranking-dependent, continuous everywhere, and has a kink along the diagonal.
There exists a unique pair of interior asymmetric Nash equilibrium in the simultaneous game
(assumption A2 and the conditions in Proposition 1 are satisfied). In particular, Examples
2 and 3 originally investigate both simultaneous and sequential games, while Examples 1
and 4 originally investigate only a simultaneous game in the literature. We focus on the
sequential game and on the shape (slope) of reaction curves and a player’s preference to
the opponent’s choice. We apply the theory developed in this paper to show that there is
a first-mover advantage in all the examples. In particular, the general results in this paper
lead to new findings for Examples 1 and 4.
Example 1 below refers to Case (i), for which π2l (si , sj ) < 0 and π2h (si , sj ) < 0, while
Example 2 below refers to Case (ii), for which π2l (si , sj ) > 0 and π2h (si , sj ) < 0, in our
model. Example 3 below refers to Case (iii), for which π2l (si , sj ) < 0 and π2h (si , sj ) < 0,
and Example 4 below refers to Case (iv), for which π2l (si , sj ) > 0 and π2h (si , sj ) > 0, in our
model.
still the SPNE outcome in the endogenous timing game. The endogenous heterogeneity generated from
the strategy-ranking-dependent payoff game indeed plays a crucial role in determining such an endogenous
first-mover advantage outcome.
22
Example 1: Process R&D model with one-way spillovers (Amir and Wooders (2000))20
Consider an industry composed of two identical firms, each with initial unit cost c > 0,
that engage in a two-stage game. At the first stage, firms sequentially choose their respective
unit cost reduction si ∈ [0, c] for i ∈ {a, b}, on the basis of R&D cost s2i . At the second
stage, the firms simultaneously compete in the product market by choosing their respective
outputs qi , given a linear market demand P (Q) = α − Q where P denotes the market
price, Q = qa + qb denotes industry output, and α > c is a positive constant. Suppose the
cost reductions by firms a and b are sa and sb , respectively, say, with sa ≥ sb ; then the
effective cost reduction for firm a is sa , and the effective cost reduction for firm b is sa with
probability β and sb with probability 1 − β, for 0 < β < 1.21
At the first stage of the game, the expected payoff to firm i is given by:

π l (si , sj ) = β · π(c − sj , c − sj ) + (1 − β) · π(c − si , c − sj ) − s2i , if si ≤ sj
π(si , sj ) = 
.
π h (si , sj ) = β · π(c − si , c − si ) + (1 − β) · π(c − si , c − sj ) − s2i , if si ≥ sj
±
Here, π(ki , kj ) = (α − 2ki + kj )2 9 is denoted as a firm’s equilibrium payoff at the second
stage of the game when its own unit cost is ki and its opponent’s unit cost is kj . Note that
π l (si , sj ) and π h (si , sj ) are strictly concave in si :
l
π11
(si , sj ) =
−10 8β
−10 2β
h
−
< 0, π11
(si , sj ) =
−
< 0.
9
9
9
3
The cross partial derivative of a firm’s payoff is negative, implying the reaction function of
firm b is downward sloping, whether it plays a higher or a lower strategy:
l
h
π12
(si , sj ) = π12
(si , sj ) =
4(β − 1)
< 0.
9
Note that π h (si , sj ) is decreasing in sj and there is a parameter restriction
c < α(2β − 1)/[2(β − 1)], implying β < 0.5, and π l (si , sj ) is also decreasing in sj :
2(β − 1)(α − c + 2si − sj )
< 0,
9
2(2αβ − α − 2βc + c + 2βsi − 2si + sj )
π2l (si , sj ) =
< 0.
9
π2h (si , sj ) =
Solving for the equilibrium outcome yields results showing that when the condition
c < α(2β − 1)/[2(β − 1)] is satisfied, meaning π2l (si , sj ) < 0 and π2h (si , sj ) < 0, then firm
20
Amir and Wooders (2000) originally investigate a simultaneous game and obtain a symmetry-breaking
result.
21
We note in this example that if β = 0, then the payoff function turns out to be strategy-rankingindependent and there is a first-mover advantage with a downward sloping reaction function, such as what
Gal-Or (1985) shows.
23
a chooses an interior higher strategy in equilibrium, s∗a = sha ∗ and s∗b = g l (sha ∗ ) = slb ∗ , and
there is a first-mover advantage in this game:22
π(s∗a , s∗b ) − π(s∗b , s∗a ) =
9(α − c)2 (2 + β + 6β 2 )(18 + 52β − 77β 2 − 74β 3 )
>0
(81 + 335β + 212β 2 + 20β 3 )2
if β < 0.642.
Example 2: Horizontal product differentiation model with (mill) price competition (Tabuchi
and Thisse (1995) and Lambertini (1997))
Consider a situation in which consumers are uniformly distributed over the unit interval
[0, 1], and their total density is one. We assume each consumer at most buys one unit of
a good and that a consumer located at s ∈ [0, 1], who decides to buy one unit from firm i
located at si , derives a surplus U = ū − pi − t(s − si )2 , where ū is the maximum a consumer
is willing to pay for a product (i.e. reservation price), pi is the mill price of firm i’s product,
and t > 0 denotes the transportation rate. Let the reservation price ū of consumers be large
enough for total demand to be always equal to one (i.e. all consumers will definitely buy
a product from the industry). At the first stage of the game, the firms sequentially choose
their respective location, si ∈ [−c, c] for i ∈ {a, b}, and the limit c is a positive constant.23
Both firms are engaged in simultaneous price competition at the second stage of the game.
Consumers finally enter the market and make their purchase decisions. There are no fixed
costs, and variable costs are equal among the firms and normalized to be zero.
When firms are located at the same point (i.e. si = sj ), payoffs are nil in the price
subgame as in any Bertrand equilibrium with homogeneous goods. When firms are not
located at the same point, we assume si < sj , without loss of generality. Solving the
marginal consumer, ŝ, for pi + t(ŝ − si )2 = pj + t(ŝ − sj )2 , the demands of the duopoly firms,
qi and qj = 1 − qi , are given by:

si +sj
p −p
s +s
+ 2t(sj j −si i ) , if 0 < i 2 j +
 2

p −p
j
qi =  1, if si +s
+ 2t(sj j −si i ) ≥ 1
2

s +s
p −p
0, if i 2 j + 2t(sj j −si i ) ≤ 0
22
Note
that
pj −pi
2t(sj −si )
<1
.
one
needs
to
have
the
parameter
restriction,
c
>
2
h∗
α(36 + 79β − 14β − 20β ) 9(13 + 46β + 22β ) , in order that c − sa > 0, and then a firm’s
2
3
payoff decreases with its own cost.
23
The unconstrained Hotelling line in Tabuchi and Thisse (1995) and Lambertini (1997) can smoothly be
revised as a compact set, [−c, c], without changing any qualitative results.
24
Each firm i’s payoff is given by: pi · qi for i ∈ {a, b}. When 0 < qi < 1, solving for the
first-order conditions of the two firms’ payoff maximization yields the equilibrium payoff of
firm i in the price subgame:

±
π l (si , sj ) = t(sj − si )(2 + sj + si )2 18, if si ≤ sj
π(si , sj ) = 
.
±
π h (si , sj ) = t(si − sj )(si + sj − 4)2 18, if si ≥ sj
Turning to the first stage, it can be checked that π l (si , sj ) is strictly concave in si and
l (s , s ) < 0 and π l (s , s ) > 0), and π h (s , s ) is strictly concave in s
increasing in sj (π11
i j
i j
i
2 i j
h (s , s ) < 0 and π h (s , s ) < 0) for −1 < s , s < 2, such that each
and decreasing in sj (π11
i j
i j
2 i j
firm’s equilibrium demand in the price subgame is positive.
The cross partial derivative of a firm’s payoff is positive, whether it plays a higher or
a lower strategy, implying the follower’s reaction function is upward sloping:
l
h
π12
(si , sj ) = t(sj − si )/9 > 0, π12
(si , sj ) = t(si − sj )/9 > 0.
Solving for the equilibrium outcome with a sequential move yields results showing that
firm a may choose an interior higher or lower strategy than firm b will choose, but there
always exists a first-mover advantage in this game:24
s∗a = 1/2, s∗b = −1/2 or s∗b = 3/2.
π(s∗a , s∗b ) = 8t/9 > 2t/9 = π(s∗b , s∗a ).
Example 3: Vertical product differentiation model with quantity competition (Aoki (2003))
Consider a situation in which consumers are uniformly distributed over the unit interval
[0, 1], and each consumer at most buys one unit of a good. A consumer in a taste parameter
t ∈ [0, 1], who decides to buy one unit from firm i providing product quality si at price
pi , derives a surplus U = tsi − pi . At the first stage of the game, the firms sequentially
choose their respective product quality, si ∈ [0, c] for i ∈ {a, b}, and the limit c > 0 is
a constant. Both firms are engaged in simultaneous quantity competition at the second
stage of the game. Consumers finally make their purchase decisions. The costs of quality
are independent of output and are convex in quality: C(qi ) = k · s2i , where k is a positive
constant. Variable costs are equal among the firms and assumed to be zero.
24
A similar result showing a first-mover advantage that exists under strategic complements can also be
obtained in a horizontal product differentiation model with delivered price competition (Hamilton et al.
(1989)) and in a vertical product differentiation model with price competition (Aoki and Prusa (1997), Aoki
(1998), and Lehmann-Grube (1997)).
25
Routine manipulation yields the equilibrium payoffs of firm i in the quantity subgame:

π l (si , sj ) = si s2j /(4sj − si )2 − k · s2i , if si ≤ sj
π(si , sj ) = 
.
±
π h (si , sj ) = si (2si − sj )2 (4si − sj )2 − k · s2i , if si ≥ sj
Note that π l (si , sj ) and π h (si , sj ) are both decreasing in sj :
±
±
π2l (si , sj ) = −2s2i sj (4sj − si )3 < 0, π2h (si , sj ) = −4s2i (2si − sj ) (4si − sj )3 < 0.
The second-order derivatives of π l (si , sj ) and π h (si , sj ) with respect to si can be calculated as:
l
π11
(si , sj ) =
2s2j (si + 8sj )
8s2j (si − sj )
h
−
2k,
π
(s
,
s
)
=
−
− 2k < 0.
i
j
11
(si − 4sj )4
(4si − sj )4
Here, π h (si , sj ) is obviously globally concave in si . If sj is not too low, then π l (si , sj ) is
also globally concave in si . For a sufficiently low sj , the optimal choice of a lower-strategy
firm i is given by g l (sj ) = sj , and π h (g h (sj ), sj ) > π l (g l (sj ), sj ), implying our derivations
are qualitatively valid.
The following equations indicate that the follower’s reaction function when choosing a
higher strategy is upward sloping, and that the follower’s reaction function when choosing
a lower strategy is downward sloping:
±
h
π12
(si , sj ) = 8si sj (si − sj ) (4si − sj )4 > 0,
±
l
π12
(si , sj ) = −2si sj (si + 8sj ) (4sj − si )4 < 0.
The equilibrium outcomes showing firm a choosing an interior higher strategy and there
being a first-mover advantage in this game are given by: s∗a = 0.1363/k > 0.0432/k = s∗b ,
and π(s∗a , s∗b ) = π h (s∗a , s∗b ) = 0.0099/k > 0.0013/k = π l (s∗b , s∗a ) = π 2 (s∗b , s∗a ).25
Example 4: Vertical product differentiation model with bad characteristics and quantity
competition (Garella and Lambertini (1999) and Bontems and Requillant (2001))26
The model in Example 4 is the same as that in the above Example 3, except that
the specification of quality si , which is now defined as a bad characteristic for a consumer,
means the surplus of a consumer purchasing a product from firm i is decreasing in si . We
suppose consumers are uniformly distributed over the unit interval [−1, 0], and a consumer
25
26
See Aoki (2003) for a complete proof in this example.
Garella and Lambertini (1999) and Bontems and Requillant (2001) originally investigate a vertical
product differentiation model with bad characteristics and price competition.
26
in taste parameter t ∈ [−1, 0], who decides to buy one unit from firm i providing product
characteristic si ∈ [−c, 0] at price pi , derives a surplus U = tsi − pi .27
The equilibrium payoffs of firm i in the quantity subgame are given by:

±
π l (si , sj ) = −si (2si − sj )2 (4si − sj )2 − k · s2i , if si ≤ sj
π(si , sj ) = 
.
π h (si , sj ) = −si s2j /(4sj − si )2 − k · s2i , if si ≥ sj
Note that π l (si , sj ) and π h (si , sj ) are both increasing in sj :
±
±
π2l (si , sj ) = 4s2i (2si − sj ) (4si − sj )3 > 0, π2h (si , sj ) = 2s2i sj (4sj − si )3 > 0.
The follower’s reaction function when choosing a higher strategy is downward sloping, and
the follower’s reaction function when choosing a lower strategy is upward sloping:
±
h
π12
(si , sj ) = 2si sj (si + 8sj ) (4sj − si )4 < 0,
±
l
π12
(si , sj ) = −8si sj (si − sj ) (4si − sj )4 > 0.
The equilibrium outcomes showing firm a choosing an interior lower strategy and there being
a first-mover advantage in this game are given by: s∗a = −0.1363/k < −0.0432/k = s∗b , and
π(s∗a , s∗b ) = π l (s∗a , s∗b ) = 0.0099/k > 0.0013/k = π h (s∗b , s∗a ) = π(s∗b , s∗a ).
6. Conclusion
Under a general payoff function, this paper investigates a symmetric two-player game with
strategy-ranking-dependent payoffs. We first treat the timing of the move, whether two
players move simultaneously or whether they move sequentially, as exogenously given and
the timing of the move is then endogenously determined. We conclude that under almost all
kinds of a player’s preferences to the opponent’s choice and the shape of a player’s reaction
function, a leader in a sequential game can exercise a preemptive advantage and get a higher
payoff over the follower. The findings present that the jump in the follower’s reaction curve
during a game with strategy-ranking-dependent payoffs can enlarge or reverse a leader’s
strategic effect on the follower’s choice, which is beneficial to the leader. We also show
that the subgame perfect Nash equilibrium of an endogenous timing game may induce an
endogenous sequence of moves that exhibit a first-mover advantage.
This paper’s analysis can be applied to myriads of natural economic applications in
which players’ payoff functions are dependent on strategy ranking, including Hotelling-type
27
This example is just the reverse of the order on a player’s strategy space, Si , in Example 3.
27
models in location competition, vertical product differentiation models in quality competition, capacity expansion models under demand uncertainty, and R&D models with one-way
spillovers.
Appendix A. The existence of a pure strategy Nash equilibrium with multiple jumps of reaction curves.
The difference π h (g h (sj ), sj ) − π l (g l (sj ), sj ) is positive at sj = c and negative at sj =
c̄. We suppose the jumps across the diagonal of a player’s reaction curve g(sj ), with
g(c) = g h (c) and g(c̄) = g l (c̄), to be finite and also denote dn , for n ∈ {1, 2, ..., N }, as the
n-th jump point, where c < d1 < d2 < · · · < dN < c̄.28 The jump point is found by solving π h (g h (sj ), sj ) − π l (g l (sj ), sj ) = 0. Claim 1 extends the condition that guarantees the
existence of a pure strategy Nash equilibrium in Case (ii) with multiple jumps in a player’s
reaction curve.
Claim 1. Denote d0 ≡ c and dN +1 ≡ c̄. If l, m ∈ {0, 1, 2, ..., N }, with l as an even
number involving zero, m is an odd number, and l < m, such that (1): (π1h (dm , dl ) > 0,
π1h (dm+1 , dl+1 ) < 0) and (2): (π1l (dl , dm ) > 0, π1l (dl+1 , dm+1 ) < 0), then a Nash equilibrium
(ŝi , ŝj ), for i, j ∈ {a, b} and i 6= j, exists in the set [dl , dl+1 ] × [dm , dm+1 ] under Case (ii).
Proof: We take the following Figure 2 for example, in which there are three jump points in
a player’s reaction curve (N = 3). By the assumption of l being an even number involving
zero, m is an odd number, and l < m, we have g(si ) = g h (si ) for si ∈ [dl , dl+1 ] and
g(sj ) = g l (sj ) for sj ∈ [dm , dm+1 ]. By strict concavity, the assumption of (π1h (dm , dl ) >
0, π1h (dm+1 , dl+1 ) < 0) implies g h (dl ) > dm and g h (dl+1 ) < dm+1 , and the assumption
of (π1l (dl , dm ) > 0, π1l (dl+1 , dm+1 ) < 0) implies that g l (dm ) > dl and g l (dm+1 ) < dl+1 .
Monotonicity of g h implies that g h (si ) ∈ [dm , dm+1 ] for all si ∈ [dl , dl+1 ], and monotonicity
of g l implies that g l (sj ) ∈ [dl , dl+1 ] for all sj ∈ [dm , dm+1 ]. The two reaction curves, g l (sj )
and g h (si ), must intersect in the set [dl , dl+1 ] × [dm , dm+1 ]. Q.E.D.
28
We cannot exclude the possibility that a player’s reaction curve may have countably infinite jumps. We
thank an anonymous referee for suggesting this point.
28
sb
c
g l ( sb )
g h ( sa )
d3
d2
L2
H2
g l ( sa )
g h ( sa )
g h ( sb )
L1
l
d1
g ( sb )
H1
g l ( sa )
c
sa
g h ( sb )
sb
d1
d2
d3
c sa
Figure 2: The existence of a pure strategy Nash equilibrium
with multiple jumps of reaction curves under Case (ii)
Claim 1 states that if the reaction curves are restricted to certain compact subsets of
the joint strategy space, [dl , dl+1 ] × [dm , dm+1 ], then the existence of a Nash equilibrium
in Case (ii) is guaranteed. If condition (2) in Claim 1 is replaced by (π1l (dl+1 , dm ) < 0,
π1l (dl , dm+1 ) > 0), then a Nash equilibrium (ŝi , ŝj ) exists in the set [dl , dl+1 ] × [dm , dm+1 ]
under Case (iii). Similarly, if condition (1) in Claim 1 is replaced by (π1h (dm+1 , dl ) < 0,
π1h (dm , dl+1 ) > 0), then a Nash equilibrium (ŝi , ŝj ) exists in the set [dl , dl+1 ] × [dm , dm+1 ]
under Case (iv).
Appendix B.
Lemma 3. Suppose π2l (si , sj ) < 0 and π2h (si , sj ) < 0.
h < 0 and π l < 0, then ŝl < sl ∗ < d < sh ∗ < ŝh and sl ∗ < ŝl < d < ŝh < sh ∗ .
(i) If π12
a
a
a
a
12
b
b
b
b
h > 0 and π l > 0, then sl ∗ < ŝl < d < sh ∗ < ŝh and sl ∗ < ŝl < d < sh ∗ < ŝh .
(ii) If π12
a
a
a
a
12
b
b
b
b
h > 0 and π l < 0, then sl ∗ < ŝl < d < sh ∗ < ŝh and sl ∗ < ŝl < d < ŝh < sh ∗ .
(iii) If π12
a
a
a
a
12
b
b
b
b
h < 0 and π l > 0, then ŝl < sl ∗ < d < sh ∗ < ŝh and sl ∗ < ŝl < d < sh ∗ < ŝh .
(iv) If π12
a
a
a
a
12
b
b
b
b
Proof: Suppose s∗a ≤ s∗b . In Cases (i) and (iv) since π1l (sa , g h (sa )) + π2l (sa , sb ) · g 0 (sa ) >
29
π1l (sa , g h (sa )) for all sa ≤ sb , the leader chooses a higher choice in a sequential game
versus that in a simultaneous game, sla ∗ > ŝla , which follows that shb ∗ < ŝhb under strategic
substitutes. In Cases (ii) and (iii) since π1l (sa , g h (sa )) + π2l (sa , sb ) · g 0 (sa ) < π1l (sa , g h (sa ))
for all sa ≤ sb , the leader chooses a lower choice in a sequential game versus that in a
simultaneous game, sla ∗ < ŝla , which follows that shb ∗ < ŝhb under strategic complements.
Suppose s∗a ≥ s∗b . In Cases (i) and (iii) since π1h (sa , g l (sa )) + π2h (sa , sb ) · g 0 (sa ) >
π1h (sa , g l (sa )) for all sa ≥ sb , the leader chooses a higher choice in a sequential game versus
that in a simultaneous game, sha ∗ > ŝha , which follows that slb ∗ < ŝlb under strategic substitutes. In Cases (ii) and (iv) since π1h (sa , g l (sa )) + π2h (sa , sb ) · g 0 (sa ) < π1h (sa , g l (sa )) for all
sa ≥ sb , the leader chooses a lower choice in a sequential game versus that in a simultaneous
game, sha ∗ < ŝha , which follows that slb ∗ < ŝlb under strategic complements. Q.E.D.
Lemma 4. Suppose π2l (si , sj ) > 0 and π2h (si , sj ) > 0.
h < 0 and π l < 0, then sl ∗ < ŝl < d < ŝh < sh ∗ and ŝl < sl ∗ < d < sh ∗ < ŝh .
(i) If π12
a
a
a
a
12
b
b
b
b
h > 0 and π l > 0, then ŝl < sl ∗ < d < ŝh < sh ∗ and ŝl < sl ∗ < d < ŝh < sh ∗ .
(ii) If π12
a
a
a
a
12
b
b
b
b
h > 0 and π l < 0, then ŝl < sl ∗ < d < ŝh < sh ∗ and ŝl < sl ∗ < d < sh ∗ < ŝh .
(iii) If π12
a
a
a
a
12
b
b
b
b
h < 0 and π l > 0, then sl ∗ < ŝl < d < ŝh < sh ∗ and ŝl < sl ∗ < d < ŝh < sh ∗ .
(iv) If π12
a
a
a
a
12
b
b
b
b
Proof: Suppose s∗a ≤ s∗b . In Cases (i) and (iv) since π1l (sa , g h (sa )) + π2l (sa , sb ) · g 0 (sa ) <
π1l (sa , g h (sa )) for all sa ≤ sb , the leader chooses a lower choice in a sequential game versus that in a simultaneous game, sla ∗ < ŝla , which follows that shb ∗ > ŝhb under strategic
substitutes. In Cases (ii) and (iii) since π1l (sa , g h (sa )) + π2l (sa , sb ) · g 0 (sa ) > π1l (sa , g h (sa ))
for all sa ≤ sb , the leader chooses a higher choice in a sequential game versus that in a
simultaneous game, sla ∗ > ŝla , which follows that shb ∗ > ŝhb under strategic complements.
Suppose s∗a ≥ s∗b . In Cases (i) and (iii) since π1h (sa , g l (sa )) + π2h (sa , sb ) · g 0 (sa ) <
π1h (sa , g l (sa )) for all sa ≥ sb , the leader chooses a lower choice in a sequential game versus that in a simultaneous game, sha ∗ < ŝha , which follows that slb ∗ > ŝlb under strategic
substitutes. In Cases (ii) and (iv) since π1h (sa , g l (sa )) + π2h (sa , sb ) · g 0 (sa ) > π1h (sa , g l (sa ))
for all sa ≥ sb , the leader chooses a higher choice in a sequential game versus that in a
simultaneous game, sha ∗ > ŝha , which follows that slb ∗ > ŝlb under strategic complements.
Q.E.D.
Lemma 5. Suppose π2l (si , sj ) > 0 and π2h (si , sj ) < 0.
h < 0 and π l < 0, then sl ∗ < g l (sh ∗ ) < ŝl < d < ŝh < sh ∗ and sl ∗ < ŝl < d < ŝh <
(i) If π12
a
a
a
12
b
b
b
b
b
g h (slb ∗ ) < sha ∗ .
30
h > 0 and π l > 0, then ŝl < g l (sh ∗ ) < sl ∗ < d < ŝh < sh ∗ and sl ∗ < ŝl < d <
(ii) If π12
a
a
12
b
b
b
b
b
sha ∗ < g h (slb ∗ ) < ŝha .
h > 0 and π l < 0, then g l (sh ∗ ) < ŝl < sl ∗ < d < ŝh < sh ∗ and sl ∗ < ŝl < d <
(iii) If π12
a
a
12
b
b
b
b
b
g h (slb ∗ ) < ŝha < sha ∗ .
h < 0 and π l > 0, then sl ∗ < ŝl < g l (sh ∗ ) < d < ŝh < sh ∗ and sl ∗ < ŝl < d <
(iv) If π12
a
a
12
b
b
b
b
b
sha ∗ < ŝha < g h (slb ∗ ).
Proof:
(i) As with Lemmas 3 and 4, if s∗a ≤ s∗b , then we can show the leader chooses a lower choice in
a sequential game versus that in a simultaneous game, sla ∗ < ŝla , and shb ∗ > ŝhb and g l (shb ∗ ) <
ŝla under strategic substitutes. We then claim that g l (shb ∗ ) > sla ∗ . Assume for a moment
that this is not true – that is, g l (shb ∗ ) ≤ sla ∗ . Thus, π1l (sla ∗ , shb ∗ ) ≤ π1l (g l (shb ∗ ), shb ∗ ) = 0. The
l < 0, and the second equality follows from the definition of
first inequality follows since π11
h (sh ∗ , sl ∗ ) < 0, π l (sl ∗ , sh ∗ ) ≤ 0 from equation (2), which
g l . Since π1l (sla ∗ , shb ∗ ) ≤ 0 and π12
a
2 a
b
b
is a contradiction.
The leader conversely chooses a higher choice in a sequential game versus that in a
simultaneous game, sha ∗ > ŝha , if s∗a ≥ s∗b . It follows that slb ∗ < ŝlb and g h (slb ∗ ) > ŝha under
strategic substitutes. We then claim that g h (slb ∗ ) < sha ∗ . Assume that this is not true
– that is, g h (slb ∗ ) ≥ sha ∗ . Thus, π1h (sha ∗ , slb ∗ ) ≥ π1h (g h (slb ∗ ), slb ∗ ) = 0. The first inequality
h < 0, and the second equality follows from the definition of g h . Since
follows since π11
l (sl ∗ , sh ∗ ) < 0, π h (sh ∗ , sl ∗ ) ≥ 0 from equation (2), which is a
π1h (sha ∗ , slb ∗ ) ≥ 0 and π12
a
2 a
b
b
contradiction.
(ii) A similar argument applies when showing that sla ∗ > ŝla . It follows that shb ∗ > ŝhb and
g l (shb ∗ ) > ŝla under strategic complements. We can thus show that g l (shb ∗ ) < sla ∗ from the
definition of g l and equation (2). On the other hand, a similar argument applies when
showing that sha ∗ < ŝha , slb ∗ < ŝlb , and g h (slb ∗ ) < ŝha . We now claim that g h (slb ∗ ) > sha ∗ from
the definition of g h and equation (2).
The proofs of Lemma 5 (iii) and Lemma 5 (iv) are similar to those of Lemma 5 (i) and
Lemma 5 (ii), which are excluded in the paper to save space. Q.E.D.
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