Regular Equilibria and Negative Welfare Theorems in Delegation Games∗ Tomoya Tajika† November 14, 2017 Preliminary draft‡

Abstract This paper studies a strategic delegation in a strategic form game with complete information and differentiable utility functions. We focus on the equilibrium of the delegation game in which the chosen reaction functions yield regular Nash equilibrium in the sense that the equilibrium action profile is differentiable by parameters. We show a necessary condition that an action profile achieved as a regular equilibrium of the delegation game. In two-player games with misaligned preferences, each efficient action profile violates the condition. We also show that almost action profiles are achieved as an equilibrium of the delegation game by linear reaction functions. This implies that the delegated objective function is written as a quadratic utility function.

Keywords: Strategic delegation; inefficiency; regular Nash equilibrium JEL Classification: C70; L21 ∗ The author is grateful to Tomoya Kazumura, Yuji Muramatsu, Ryusuke Shinohara and Yoshikatsu Tatamitani

for their valuable comments. † Institute of Economic Research, Hitotsubashi University, Naka 2-1, Kunitachi, Tokyo, Japan. 186-8603. E-Mail: [email protected]. ‡ Latest version is available at https://sites.google.com/site/tomoyatajika90/

1

1

Introduction

While players are assumed to behave to maximize their utility in traditional economics, as it has already been emphasized by Schelling (1960), players can benefit from delegating their decision by some other person. Many succeeding researches investigate which type of delegation is chosen in specific strategic situation such as oligopoly competition,1 bargaining,2 tax competition,3 and so on. The aim of this study is to characterize the delegation strategy in a general differentiable strategic form game and to investigate the properties of the equilibrium. We study a strategic form game with continuously differentiable utility functions. Delegation game is a game such that each player can commit a reaction function in advance (delegation stage). The outcome of the game is a Nash equilibrium of the committed reaction functions (execution stage). We focus on the action profiles that are achieved as a subgame perfect Nash equilibrium of the delegation game. We provide a necessary condition that an action profile of the original game is achieved in the delegation stage in terms of the indifference curves and the gradient of the reaction function. Before explaining the condition, we introduce several terms. Note that a Nash equilibrium in the execution stage, which is the realized action profile, is written as an intersection of the reaction functions. A Nash equilibrium is regular if the profile of reaction functions that defines the equilibrium has an invertible Jacobian. The regularity condition guarantees that the Nash equilibrium obtained by the profile of reaction function is locally unique and it is continuously differentiable by parameters. This implies that the equilibrium is continuous under small mistakes of the players. This condition is natural and familiar in the applied theory such as oligopolistic competitions. If an action profile that is achieved as a subgame perfect equilibrium of the delegation game and the Nash equilibrium in the delegation stage is regular, we call that the action profile is regularly implementable. 1Vickers (1985); Fershtman and Judd (1987); Sklivas (1987) and so on. 2Segendorff (1998); Besley and Coate (2003); Gradstein (2004) and so on. 3Ihori and Yang (2009); Pal and Sharma (2013) and so on.

2

We show a necessary condition that an action profile is regularly implementable. The conditions says that if an action profile is regularly implementable, each column of the inverse of the Jacobian matrix is tangent to a player’s indifference curve. In a two-player game, this condition breaks down into the condition that each player’s reaction function is tangent to the other’s indifference curve. The necessary condition also implies that when an action profile is not regularly implementable. We show that if the indifference curves of all players share the same (nonzero) tangential plane at an action profile, the action profile cannot be regularly implemented. In two-player game, this result implies that unless the players’ best action profiles coincide each other, the efficient action profile cannot be regularly implemented. Note that if the players’ best action profiles coincide each other, there is no need of strategic delegation since it is achieved by the reaction function induced by the players’ original utility function. Therefore, the strategic delegation is critical only when the players’ true utilities are misaligned. In that case, each realized action of the delegation game must be inefficient.4 We also provide a sufficient condition for the equilibrium in the delegation stage. We show that if the utility function is quasi-concave at an action profile and with mild conditions, there exists a profile of linear reaction functions that regularly implements the action profile as the unique regular Nash equilibrium. Especially, if the utility function is strictly quasiconcave, almost all action profiles (except for efficient ones in two-player case) are regularly implementable. Constructing an equilibrium by linear reaction functions provides another implication. A linear reaction function can be induced from a quadratic utility function. Our result states that if true utility function is quasi-concave, each player needs only to consider a delegation to an agent who has a quadratic objective function. Our necessary and sufficient conditions may be contrasts of the celebrated welfare the4This result is reminiscent of the results by Dubey (1986) and Bade et al. (2009). They show that the mass of the utility profiles in which the efficient action is achieved is 0. Our result is not a corollary of their results since we consider a different type of games that they consider.

3

orems in the general equilibrium theory, which are briefly stated as the following: Each Walrasian equilibrium allocation is efficient, all efficient allocation is achieved as a Walrasian equilibrium. In contrast with these results, our results roughly state that each regularly implementable action profile is inefficient, and (almost) inefficient action profiles are regularly implementable.5 The latter result suggests the indeterminacy of the equilibrium of the delegation game. Indeed, in a Cournot duopoly, if we allow only both of sales and relative profit bonuses, which are often considered in the literature, indeterminacy emerges: We show that each symmetric positive profit profiles except for efficient one can be achieved as a symmetric subgame perfect equilibrium. Our delegation game has many applications in oligopoly theory, interregional conflicts resolution and so on. In the last of this paper, we provide a few applications of our results.

1.1

Literature review

In the literature, many studies have tackled strategic delegation. In the oligopoly theory, it is shown that firms can gain larger profit by behaving to maximize sales rather than behaving to maximize their profit (Vickers, 1985; Fershtman and Judd, 1987; Sklivas, 1987 so called VFJS model). Recently, many studies demonstrate the effect of other objectives (relative profit for Miller and Pazgal, 2001, 20026, market share for Ritz, 2008, and so on).7 In general, Fershtman et al. (1991) consider a general form of the delegation game with two players and show a partial folk theorem: every payoff profile that exceeds the original Nash equilibrium payoff can be achieved. Polo and Tedeschi (1997, 2000) also consider a general model and show a folk theorem: every individual rational action profile can be achieved. Like as ours, Polo and Tedeschi (1997, 2000) also show the tangency property of Nash equilibria

5Note that although our result shows that efficient allocation cannot be achieved, the second result shows that approximate efficiency can be achieved. 6See also Koçkesen et al. (2000) for a generalization of this type of delegation. 7See Sengul et al. (2012) for a survey.

4

in the delegation stage in two-player cases. Related to this literature, Kalai et al. (2010) consider the space of commitment devices, which enforce the players to take some strategy. Commitment game is the game in which players choose their commitment devices voluntary and simultaneously. The authors show that there exists a commitment device space in which any individual rational payoff is achieved at an equilibrium of the commitment game. Forges (2013) extends Kalai et al. (2010)’s result to the world of incomplete information. Our study is different from these studies in the following points. First, previous studies ignore the regularity of Nash equilibrium in the executing stage. In applications, the types of agents (or applicable objective functions) are limited and thus, selectable reaction functions are also limited. In such case, if the regularity is violated, (locally) uniqueness and differentiability of Nash equilibrium cannot be guaranteed. These properties are natural requirement and convenient for applications like VFJS model. By imposing regularity, we can (slightly) refine the achievable action profile. While many of existing studies complete folk theorem, we show that (specific) efficient allocations cannot be achieved. Relating this point, in most of the existing researches, to find a Nash equilibrium, they construct a mechanism to implement a payoff profile, which may not be differentiable. Therefore, the constructed response function may not be represented as a maximization of continuously differentiable utility function.8 In our study, since the reaction function is linear, the differentiability is satisfied and thus, we can find a quadratic utility function that induces the linear reaction function. Lastly, most of the existing studies focus on the achievable payoff profiles in delegation games. Contrast to their results, we focus on action profiles and show that almost action profile can be achieved. Renou (2009) and Bade et al. (2009) also relate the literature. The authors consider 8Polo and Tedeschi (2000) is an exception. They show a folk theorem using a differentiable contract for two-player case.

5

another type of commitment game in which players can restrict their action spaces in advance and plays a game with the restricted actions. While Renou (2009) considers the case that player can commit to arbitrary subset of action spaces, Bade et al. (2009) restrict the power of commitment so that players can commit only to subinterval on the action set for two-player cases. Delegation game has a closed relation with manipulation game in the mechanism design theory. Related to our study, Reichelstein (1984) show that efficiency cannot be achieved by differentiable mechanisms.

2

Model

Consider a strategic form game consisting of n players. The set of players denoted by N = {1, . . . , n} and let i be a typical element. The action set of player i ∈ N is Ai ⊆ Rmi (mi ∈ N, ∏ ∏ int Ai , œ)9 and his/her utility function is denoted by ui : i∈N Ai → R. Let A = i∈N Ai ∏ and as usual convention, A−i = j∈N\{i} A j . We also denote a−i = (a1, . . . , ai−1, ai+1, . . . , an ) for each a ∈ A. Before the play of the game, player i ∈ N can commit to a reaction function φi : A−i → Ai .10 Formally, each player plays the following game. Stage 1 (Delegation stage). Player i ∈ N chooses a reaction function φi : A−i → Ai . Stage 2 (Executing stage). The game is played by the chosen reaction function. We assume that ui is continuously differentiable. Let

Ci1

{

= φi ∈

AiA−i

: φi is continuously differentiable if (φi (a−i ), a−i ) ∈ int A

}

9For each set S, int S denotes the interior of S. 10Equivalently, each player delegates his/her decision to an agent out of the game whose utility function is vi (a) = −ai2 /2 + ai φi (a−i ) + f (a−i ) for some function f : A−i → R.

6

be the set of all continuously differentiable reaction functions. Let Φi ⊆ Ci1 be the set of ∏ admissible reaction functions of player i. Also as a usual convention, we denote Φ = i∈N Φi . We assume that for each i ∈ N, Φi is nonempty and satisfies the following conditions. Assumption 1 (Piecewise connectedness). For each φ ∈ Φi , if (φ(a−i ), a−i ) ∈ int A, for each k ∈ {1, . . . , mi } there exists a continuously differentiable function g : [0, 1] × A−i → R such that g(0, a−i ) = 0,

∂g(0,a−i ) ∂t

, 0 and for each t ∈ (0, 1), φ(·) + g(t, ·)e k ∈ Φi .11

In many applications, as we show in section 6, this assumption is satisfied. In more familiar expression, if Φi is open and convex, Assumption 1 is satisfied. An action profile a = (a1, . . . , an ) ∈ A is a Nash equilibrium induced by profile of reaction functions (φi )i∈N ∈ Φ if ai = φi (a−i ) for each i ∈ N, which is the realized outcome of the game. The objective of this study is characterization of achievable outcomes in the subgame perfect Nash equilibria of the delegation game. We introduce several definitions. Definition 1. A pair of a profile of reaction functions φ = (φi )i∈N and a fixed point a ∈ A of φ is regular if the following matrix is invertible: ∂φ1 1 − ∂φ © 1m1 ∂a2 (a−1 ) · · · − ∂an (a−1 )ª ­ ® ­ ∂φ2 ® ... ­− ∂a1 (a−2 ) ® ®. J(φ, a) = ­­ ® .. ... ­ ® . ­ ® ­ ® ∂φn − (a ) 1 −n m n « ∂a1 ¬

11ek is the k-th coordinate unit vector.

7

Here, ∂φi1

···

© ∂a j1 ∂φi ­­ .. =­ . ∂a j ­ ­ ∂φim i « ∂a j1

... ···

∂φi1 ∂a jm j ª

® .. ® . ®® . ∂φimi ® ∂aim j ¬

This condition guarantees that the Nash equilibrium of the execution stage is unique in a neighborhood of a and even when a player perturbs his reaction function slightly, the resulted Nash equilibrium of the execution stage remains to exist sufficiently near the original one. To elaborate, note that the Nash equilibrium is a solution to the following system. ©a1 ª © φ1 (a−1 )ª ­ ® ­ ® ­ ® ­ ® ­a2 ® ­ φ2 (a−2 )® ­ ®=­ ® ­ .. ® ­ .. ® ­.® ­ . ® ­ ® ­ ® ­ ® ­ ® a φ (a ) « n ¬ « n −n ¬ The matrix J(φ, a) is the Jacobian matrix of the above system. For each ε ∈ R, φ˜i is a ε-perturbation of φi if φ˜i (a−i ) = φi (a−i ) + ε1m1 . Without loss of generality, we assume that i = 1. By replacing φ1 with φ˜1 , Nash equilibrium a(ε) ˜ satisfies that ©a˜1 (ε)ª ©φ1 (a˜ −1 (ε)) + ε1m1 ª ­ ® ­ ® ­ ® ­ ® ­a˜2 (ε)® ­ φ2 (a˜ −2 (ε)) ® ­ ® ­ ®. ­ .. ® = ­ ® . . ­ . ® ­ ® . ­ ® ­ ® ­ ® ­ ® a ˜ (ε) φ ( a ˜ (ε)) n −n « n ¬ « ¬ To guarantee that a(ε) ˜ exists sufficiently near a for sufficiently small ε and it is differentiable by ε at ε = 0, it is often assumed that J(φ, a) is invertible to use the implicit function theorem. Regularity is a sufficient condition.

8

We also need to consider the first stage behavior of the delegating players. In this stage, each player chooses his/her reaction function simultaneously. Here, subgame perfect Nash equilibrium is the solution concept. In particular, we focus on the action profile achieved by a subgame perfect equilibrium of the delegation game. Since the equilibrium of the execution stage is characterized as an intersection of the chosen reaction functions, subgame perfect equilibrium is defined as follows. Definition 2. A pair of a profile of reaction functions and action profile (φ, a) is a subgame perfect equilibrium of the delegation game if a = φ(a) and for each i ∈ N, and each h ∈ Φi , ui (a) ⩾ ui (b), for each b ∈ A that satisfies bi = h(b−i ) and b j = φ j (b− j ) for each j , i (b is a Nash equilibrium action profile under reaction function profile (h, φ−i )). We focus on action profiles that are achieved as results of subgame perfect equilibrium and regular. Such an action profile is formally defined as follows. Definition 3. An action profile a is regularly implementable via a profile of reaction functions φ if (φ, a) is a subgame perfect equilibrium of the delegation game and regular. An action profile a is regularly implementable if it is regularly implemented via a profile of reaction functions.

3

Illustration in a two-player case

Before we move on to the detail analysis, this section provides an intuition of the condition for regularly implementability by illustration. Sklivas (1987) and Polo and Tedeschi (2000) provide similar illustration for Cournot and Bertrand duopoly, which is referred as tangency condition. Figure 1 shows the reaction functions that is a Nash equilibrium of the delegation 9

a2

2’s indifference curve 1’s reaction function

1’s deviation better for 2

NE

better for 1

1’s indifference curve New NE 2’s reaction function a1 Figure 1: Regularly implementable stage. Observing this figure, we can find that player 1’s reaction function is tangent to player 2’s indifference curve. Similarly, the converse holds. In this case, for any player 1’s deviation, the new NE of executing stage is on 2’s reaction function that is tangent to 1’s indifference curve, and it is out of upper contour set of the original NE of executing stage. If 2’s reaction function fails to be tangent to 1’s indifference curve, it crosses the indifference curve (Figure 2). Then, since it can be that 2’s reaction function crosses the upper contour set of player 1, there can be a profitable deviation of 1. Thus, in a Nash equilibrium of the delegation stage, one’s chosen reaction function needs to be tangent to the other’s indifference curve. This result implies the inefficiency of the equilibrium of the delegation game. If the two players’ indifference curves are tangent to each other at an action profile, the action profile is not regularly implementable (Figure 3). To achieve the action profile, the reaction functions are also tangent to each other. However, with a small perturbation of 1’s reaction function, Nash equilibrium cannot be around of the original one. At any interior Pareto efficient action 10

a2

2’s indifference curve 1’s reaction function

1’s profitable deviation NE New NE

2’s reaction function

1’s indifference curve

a1 Figure 2: Not NE in the delegation stage

a2

2’s indifference curve 1’s reaction function A small perturbation of 1’s reaction function

1’s indifference curve 2’s reaction function a1 Figure 3: Unimplementable action profiles

11

profiles, the indifference curves are tangent to each other and thus, such action profiles are not regularly implementable. The next section formally states and generalizes this intuition.

4

Necessary condition and inefficiency

This section provides a necessary condition such that an action profile is regularly implementable, which takes a form of first-order condition. Proposition 1 (First-order necessary condition). Suppose that Φi satisfies Assumption 1 for ∏ each i ∈ N. Consider a profile of reaction functions φ ∈ i Φi and a Nash equilibrium a∗ ∈ int A that is induced by φ. Then, if a∗ is regularly implementable via φ, for each i ∈ N and each k ∈ {1, . . . , mi }, there exists a normal vector νik of ∇ui (a∗−i ) for each i ∈ N such that (ν11, . . . , νnmn ) = [J(φ, a∗ )]−1 . □

Proof. See Appendix.

This proposition says that each column vector of [J(φ, a∗ )]−1 is tangent to a player’s indifference curve at a∗ . When |N | = 2 and mi = 1 for each i ∈ N, this condition is reduced to the following condition. Corollary 1 (Sklivas, 1987; Polo and Tedeschi, 2000). Suppose that Φi satisfies Assumption 1 for each i ∈ N. Suppose also that |N | = 2 and mi = 1 for each i ∈ N. Consider a profile of ∏ reaction functions φ ∈ i Φi and a Nash equilibrium a∗ ∈ int A that is induced by φ. Then, if a∗ is regularly implementable via φ, for each i ∈ N, one’s reaction function is tangent to

12

the other’s indifference curve at a∗ . That is, (

)© 1 ª ( ­ ® = ∂u2 (a∗ ) ∗ ∗ ® ∂a1 (a ) ∂a2 (a ) ­ ∂φ2 ∂a1 ∗) (a ∂a 1 « ¬

∂u1

∂u1

) © ∂φ1 (a∗ )ª ® = 0. ∗ ­ ∂a2 ® ∂a2 (a ) ­ 1 « ¬

∂u2

Proposition 1 also states that when action profile a∗ is not regularly implementable. To do this, we investigate the condition in terms of action profiles and utility functions. The following proposition shows a sufficient condition that an action profile is not regularly implementable. Proposition 2. Suppose that for each i ∈ N, there exists αi ∈ R \ {0} and a vector µ ∈ R

∑ i

mi

\ {0} such that ∇ui (a−i ) = αi µ. Then, a is not regularly implementable. □

Proof. See Appendix.

To give the above proposition an economic insight, we prepare the following definition. Definition 4. An action profile a ∈ int A is a (local) compromise point between i, j ∈ N if there exists ε ∈ R++ such that for each ε′ < ε and each a′ ∈ Bε ′ (a),12 if u k (a′) > u k (a), ′

′

u k (a′) ⩽ u k (a) for each k ∈ {i, j}, k ′ , k and ∇u k (a−k ) , 0 for each k ∈ {i, j}. Compromise point is similar to efficiency between i and j. However, compromise point excludes the point that maximizes both of player i and j’s utilities. If these players preferences are misaligned, the set of (global) compromise points equals the set of interior efficient points. As usual, compromise point is characterized by the tangency condition. Fact 1. If action profile a ∈ A is a compromise point between {i, j}, ∇ui (a−i ) = λ∇u j (a− j ) for some non-zero real number λ. Compromise point is an arbitration between mutually-opposing two players. We extend the definition to n players. 12Bε (a) = {a′ ∈ A : ∥a − a′ ∥ ⩽ ε}.

13

Definition 5. An action profile a ∈ int A is a compromise point of a binary opposition if there exist N1, N2 ⊆ N such that N1 ∪ N2 = N, N1 ∩ N2 = œ and for each i ∈ N1 and each j ∈ N2 , a is a compromise point between {i, j}. Combining Fact 1 and Proposition 2 implies unimplementability of the compromise points of binary opposition. Theorem 1. Suppose that action profile a ∈ A is a compromise point of a binary opposition. Then, a is not regularly implementable. If there are only two players, it achieves the common local maximum or it is inefficient. When the Nash equilibrium achieves the common maximum, there is no conflicts between the players and thus, it can be achieved without delegation. When such conflicts are inevitable, the delegation game achieves inefficiency at each equilibrium. Remark 1 (Trinary opposition). When |N | ⩾ 3, some efficient action profiles are regularly implementable. This is because, when the number of players is more than 2, at an efficient profile, indifferent curves need not to have a common tangent plane. Consider the following example. Suppose that |N | = 3 and Ai = R for each i ∈ N. Utility functions are defined as follows. u1 (a) = −(a1 + 2)2 − (a2 − 1)2 − (a3 − 1)2, u2 (a) = −(a1 − 1)2 − (a2 + 2)2 − (a3 − 1)2, u3 (a) = −(a1 − 1)2 − (a2 − 1)2 − (a3 + 2)2 . Note that a1 = a2 = a3 = 0 is a Pareto efficient action profile. As a reaction-function profile,

14

consider the following:13 φ1 (a−1 ) = 2a2 − 4a3 φ2 (a−2 ) = −4a1 + 2a3 φ3 (a−3 ) = 2a1 − 4a2 . Clearly, at the intersection, a1 = a2 = a3 = 0. Note that this constructs an equilibrium of the delegation stage. Consider player 3. By fixing φ1 and φ2 , it must hold that a1 = 0 and a2 = 2a3 . Then, for player 3, a3 = 0 is optimal. Therefore, no deviation is profitable. Similar ©−1 2 −4ª ­ ® ­ ® argument stands for player 1 and 2. One can easily check that J(φ, a) = ­−4 −1 2 ® is ­ ® ­ ® 2 −4 −1 « ¬ invertible. Thus, the action profile a1 = a2 = a3 = 1 is regularly implementable.

5

Sufficiency

This section shows a sufficient condition for regular implementability. To implement an action profile, we consider a specific type of reaction functions. Hereafter, to keep the notation simple, we assume that mi = 1 for each i ∈ N. However, this assumption does not lose generality. A reaction function φi is linear if φi (a−i ) =

∑ j∈N\{i}

βi j a j + βi0 . This type of reaction

functions are familiar, which induced by the following quadratic utility function: vi (a) =

13This reaction-function profile is constructed by using Proposition 1. Note that at a = 0, ν1 = (−1/9, 0, −2/9)T is a normal vector of ∇u1 (0) = (−2, 1, 1), ν2 = (−2/9, −1/9, 0)T is that of ∇u2 (0) = (1, −2, 1) and ν3 = (0, −2/9, −1/9)T is that of ∇u3 (0) = (1, 1, −2). Here M T denotes the transpose of matrix M. Then, we have (ν1, ν2, ν3 )−1 = J(φ, a) that we defined above. The reaction-function profile that we show is the system of linear functions that has the matrix as its Jacobian and yields a1 = a2 = a3 = 0 at the intersection.

15

−1 2 2 ai

+ ai

[∑ j∈N\{i}

] βi j a j + βi0 . Then, © 1 −β12 · · · ­ ­ ­−β21 1 −β23 · · · ­ ­ . .. J(φ, a) = ­­ .. . ­ ­ .. ­ . ­ ­ −β ··· « n1

−β1n ª ® ® −β2n ®® ® ® ® ® ® ® ® ® 1 ¬

We now show that a is regularly implementable via linear reaction functions (φi (a−i ))i∈N = ∑ ( j∈N\{i} βi j a j + βi0 )i under reasonable conditions. A utility function ui is quasi-concave at a if the upper contour set U i (a) := {a′ : ui (a′) > ui (a)} is convex. We show that if ui is quasi-concave and {∇ui } is linearly independent at a, a is regularly implementable by linear reaction functions. This construction has some interesting properties. First, since the Nash equilibrium in the executing stage defined by a linear system with invertible Jacobian, the Nash equilibrium in the executing stage is unique and differentiable by parameters. Second, since we assume that {∇ui } is linearly independent at a, a cannot be efficient, which is consistent with Theorem 1. This point is a difference with previous studies that show so-called folk theorems. Before we provide formal proposition, we explain briefly the construction of the equilibrium in the delegation stage. Consider to implement a∗ ∈ A. We first assume that ∂ui+1 ∗ ∂ai (a )

, 0 for each i ∈ N and {∇ui (a∗ )}i∈N is linear independent. Then, we can construct a

profile of reaction function as follows: player 1’s reaction function is a tangent plane of player 2’s indifference curve, player 2’s reaction function is a tangent plane of player 3’s indifference curve, ... and player n’s reaction function is a tangent plane of player 1’s indifference curve. By the assumption, we can create such a cycle. Further, since we assume that {∇ui (a∗ )}i∈N is linear independent, the pair of the profile of reaction function and a∗ is regular. Then, by the construction and the separating hyperplane theorem, no deviation of a given player can 16

achieve any profitable Nash equilibrium. Now we provide the formal statement. Theorem 2. Assume that Ai = R for each i ∈ N. Take a∗ ∈ A. Suppose that {∇ui (a∗ )}i∈N is linearly independent and there exists a permutation π of N such that for each i ∈ {1, . . . , n − 1} and

−1

∂u π (1) ∗ ∂aπ −1 (n) (a )

−1

∂u π (i+1) ∗ ∂aπ −1 (i) (a )

, 0

, 0. Suppose also that ui is quasi-concave at

a∗ for each i ∈ N. Further, we assume that for each i ∈ N, Φi contains all linear reaction functions. Then, a∗ is regularly implementable via a profile of linear reaction functions ∑ (φi (a−i ))i = ( j∈N\{i} βi j a j + βi0 )i . Constructing an equilibrium by linear reaction functions provides another implication. A linear reaction function can be induced from a quadratic utility function. Theorem 2 states that as long as the true utility functions are quasi-concave, each player needs only to consider a delegation to an agent who has a simple quadratic objective function, which are often used in applied theory.

5.1

Restriction on the action space

In many application, the set of action profile is bounded. In Cournot oligopoly, the amount of production is bounded below by zero. Similarly in Bertrand oligopoly, price is also bounded by zero. For the application, we need to deal with the case when the action profile is bounded below. For the case that the action profile is bounded above can be considered similarly.

Lower bound

Assume the all assumptions made in Theorem 2. Suppose that Ai = [ai, ∞)

for each i ∈ N. Let φ˜i be a linear reaction function. A reaction function is the restriction of φ˜i (a−i ) if φi (a−i ) = max{ φ˜i (a−i ), ai }. It is a natural restriction of φ˜i to A and is often used in literature, which induces a corner solution equilibrium. Moreover, it is differentiable at the interior. Note that restricted linear function may not regularly implement every action profiles even the environment satisfies all assumptions made in Theorem 2. For example, see Figure 4. 17

a2

φ˜1 (a2 )

ù

ï

Indifference curve of player 1

(0, 0)

ú

D

E

φ2 (a1 ) a1

Figure 4: Unimplementable action profile In this figure, action profile D cannot implementable by any restricted linear function. This is because, if D is regularly implementable, by Proposition 1, the reaction function must be tangent to the indifference curve of player 1 at D. Then, if a restricted linear reaction function is equilibrium, it is φ2 in the figure. However, in this case, a deviation φ˜1 yields action profile E, which is better than D for player 1. Remark 2. The example illustrated in Figure 4 has another interesting feature. We add the player 2’s indifference curve and the intersection of the players’ indifference curves, namely point F, in Figure 5. Note that both of points D and F offer the same utility for player 1 and 2. While D is not regularly implementable by restricted linear function, as Figure 5 shows, F is regularly implementable. Therefore, regularly implementability does not only depend on the utility profile. Although in a situation, some action profile cannot be implemented, in a specific setting, every action profiles can be regularly implementable. Now we explore the sufficient condition. Let C(φ) = {a ∈ A : φ(a) = a, φ j (a− j ) = a j for some j ∈ N } be the set of allocations that achieved by corner solutions. Take a∗ ∈ int A and assume the all assumptions made in Theorem 2. Further, we assume that for each i ∈ N, Φi contains all restricted linear reaction functions. Let φ˜i be the linear function obtained by Theorem 2 and let φi be the restriction of 18

a2

Indifference curve of player 2 φ1 (a2 ) F φ2 (a1 ) a1

ú

(0, 0)

ù

D

Figure 5: Implementable and unimplementable action profiles φ˜i . Assume U i (a∗ ) ⊆ A \ C(φ) for each i ∈ N. Then, a∗ is regularly implementable via (φi )i . Clearly, int A ⊆ A \ C(φ). Therefore, we have that if U i (a∗ ) ⊆ int A, a∗ is regularly implementable via a profile of restricted linear functions. As an example, any strictly positive price vector in (pure) Bertrand competition satisfies the condition. As another case, consider a Cournot competition. In this case, it might be U i (a∗ )∩Ci (φi ) , œ for any a∗ ∈ A since a−i = 0 is preferred for player i. However, under specific, but natural conditions, we can show that a∗ is regularly implementable by restricted linear reaction functions. To elaborate, let ai = 0 for each i ∈ N and assume that ui (a) = Pi (a)ai − Ci (ai ) for some function Pi, Ci so that ui is concave in ai > 0 and quasi-concave in a ∈ A. Here, Pi is the inverse demand function and Ci is cost function for player i. We assume that Pi (a) ⩾ 0 for each a ∈ A,

∂Pi ∂a j

< 0 for each j , i. This assumption implies that we have that

∂ui ∂a−i

<0

for each i ∈ N. In addition, Ci (0) = 0, which yields that ui (0, a−i ) = 0. In typical Cournot competition model, these assumptions are satisfied. Now we state the following proposition for a generalized case. Proposition 3. Assume that |N | = 2 and Ai = [0, ∞). Suppose that

∂ui ∂a−i

< 0 for each i ∈ N.

Then, for each a∗ ∈ int A such that {∇ui (a∗ )}i∈N is linear independent, ui is quasi-linear at 19

a∗ and ui (a∗ ) > ui (0, 0) for each i ∈ N, it is regularly implementable by restricted linear functions. □

Proof. See Appendix.

By this proposition, in typical Cournot games, if an action profile yields positive profit, that is, ui (a∗ ) > 0 = ui (0, 0), a∗ is regularly implementable by restricted linear reaction functions.

6

Application

Delegation game is applied to many strategic situation, such as oligopoly, tax competition, international conflicts and so on. We apply Propositions 1 and 3 to these specific situations.

6.1

Managerial delegation and indeterminacy of the equilibria

In the literature, separation of ownership and management is a typical example of strategic delegation. In a typical stock company, shareholders have a ownership and pursue the revenue. Shareholders can delegate the management of the company to an agent. The shareholders’ problem is offering an incentive scheme to the agent. To elaborate, consider a Cournot duopoly. Let ai be the production of company i ∈ {1, 2} and P(a1, a2 ) be the inverse demand function. Company i ∈ {1, 2}’s profit is ui (a1, a2 ) = ai [P(a1, a2 ) − ci ]. Then,

∂ui ∂ai

= P(a1, a2 ) − ci + ai Pi (a1, a2 ) and

∂ui ∂a j

1 © 1 − ∂φ ∂a2 ª ­ ® J(φ, a) = ­ ® ∂φ2 − ∂a1 1 « ¬

−1

[J(φ, a)]

=

1 1−

∂φ1 ∂φ2 ∂a2 ∂a1

20

© 1 ­ ­ ∂φ2 « ∂a1

∂φ1 ∂a2 ª ®

1

® ¬

= ai P j (a1, a2 ). Further,

Proposition 1 states that if (φ1, φ2 ) regularly implements an action profile a∗ , P(a1∗, a2∗ ) − c1 + ai Pi (a1∗, a2∗ ) + ai P j (a1∗, a2∗ )

∂φ j = 0 for each i, j , i. ∂ai

When, P(x, y) = −α1 x−α2 y+ β (note that β > max{c1, c2 }), the above condition is rearranged as ) ∂φ2 −1 ( β − 2α1 α1∗ − α2 α2∗ − c1 . = ∗ ∂a1 α2 a1 In the literature, as means of delegation, sales bonus (VFJS model) and relative profit bonus (that is, ui (a) − u j (a), Miller and Pazgal, 2001, 2002) are considered. To generalize their ideas, consider the following objective function. vi (a : λ) =

∑

sj

cj

λi P(a1, a2 )a j − λi a j c j

j sj

cj

sj

cj

The term λi is firm i’s weight on firm j’s sales and λi is that on firm j’s cost. If λi = λi , this is i’s weight on j’s profit. Then, Φi = {φi [λ] : φi [λ](a−i ) = arg maxai vi (a : λ)}. We can easily to check that this specification satisfies Assumption 1. Without loss of generality, we cj

normalize so that λisi = 1. Note also that since when we induce the reaction function, λi , j , i cj

sj

j

brings no effect. Therefore, without loss of generality, we can assume that λi = λi = λi for each i ∈ {1, 2}, j , i, which is the weight on the other’s revenue. Then, the reaction function of vi is j

φi (a j ) = −

(α j + λi αi ) β − λici ci aj + 2αi 2αi

21

and the (interior) equilibrium induced by the reaction function is ai∗ =

where γi =

j

(α j −λi αi ) 2αi

and δi =

β−λici ci 2αi .

δi − γi δ j , 1 − γi γ j

Then, our necessary condition states that

( ) δ j − γ j δi δi − γi δ j 1 − γi γ j β − 2α j − αi − c j , for each i ∈ {1, 2}, j , i. (1) γi = − αi (δ j − γ j δi ) 1 − γi γ j 1 − γi γ j Since each φi is linear, by Proposition 3, when the necessary condition is satisfied, the restriction of this reaction function is an equilibrium. j

Note that there are four free variables ({λi , λici }i∈{1,2}, j,i ) and two equations to satisfy (equation (1)). Therefore, there can be multiple equilibria. To see this property, and to keep the calculation tractable, we consider a symmetric situation, α1 = α2 = α, c1 = c2 = c. We j

cj

also focus on the symmetric equilibrium, that is λi = λij = λ o and λici = λ j = λ c for each i ∈ {1, 2}, j , i. Then, by calculating the condition, there exists (λ o, λ c ) that satisfies equation (1) if λ c , 3(β/c) − 2 and then, 3β − (λ c + 2)c 8α ( ) 3β − (λ c + 2)c β − (2 − λ c )c i ∗ ∗ u (a1, a2 ) = 8α 4 a1∗ = a2∗ =

Here, note that equilibrium λ o is calculated as λo = 8

β − λc c − 3. β − λ c c + 2(β − c)

We can easily check that unless λ c = 2 − β/c or λ c = β/c, it yields the unique equilibrium in the execution stage. As long as β − (2 − λ c )c > 0 and 3β − (λ c + 2)c > 0, that is, λ c ∈ (2 − β/c, β/c) ∪ (β/c, 3(β/c) − 2), the profit is positive and therefore, by Proposition 3, 22

a∗ is regularly implementable. Note that Pareto efficient action profile is achieved if λ c = β/c. This implies that any symmetric positive profit profile except for efficient one can be achieved as the outcome of a subgame perfect equilibrium of the delegation game. Therefore, when each company can offer both of sales and relative profit bonus, indeterminacy of equilibrium profit emerges.

6.2

Delegation via election

Resolving an international conflict is a typical strategic situation. Policy choice in each country is usually delegated to the country’s representative. Then, a strategic delegation problem arises in an election. Via an election, citizens in each country strategically choose their representative and the elected representative chooses the policy that maximizes his/her utility.14 To give a formal discussion, let , T ⊆ R be the set of citizens. Assume that T is an open interval. Let vi (a : t) be the utility function of citizen t ∈ T. Policy choice in the country is denoted by ai ∈ Ai ⊆ R. Assume that vi is a concave function. In country i ∈ N, citizens is distributed by density function fi . The chosen citizen chooses an element a so as to maximize his utility. Let φi (a−i : t) be the solution to ∂vi (φi (a−i : t), a−i : t) = 0. ∂ai Since vi is concave, the function is uniquely determined. Then, the set of admissible reaction functions is Φi = {φi (a−i : t) : t ∈ T }. We assume that ∂vi (a : t) ,0 ∂ai ∂t

14As a research of this type of delegation, for example, Ihori and Yang (2009) study the case for tax competition.

23

for each a ∈ int A and t ∈ T. Since ∂ 2 vi (φi (a−i :t),a−i :t)

∂φi (a−i : t) ∂ai ∂t =− 2 , 0, ∂ vi (φi (a−i :t),a−i :t) ∂t ∂ai2

the set of admissible reaction function satisfies Assumption 1. The election proceeds as follows. There are two parties that aim to their own candidates to be elected. Each party selects a candidate from the set of citizens and the elected candidate decides the policy after the election. Assume that voting is a simple majority voting and voting is costly for each citizen. In a costly voting with two parties, it is known that the elected candidate is chosen to maximize the social welfare.15 Therefore, the objective function in country i ∈ N, denoted by ui is ∫ u (a) =

vi (a : t) fi (t)dt.

i

t∈T

Consider the case that |N | = 2. Then, by Proposition 1, ∫ ∂v (a:t) i − fi (t)dt ∂φ j t∈T ∂ai . (a) = ∫ ∂v (a:t) i ∂ai f (t)dt i t∈T ∂a j Therefore, the elected candidate t ∗j satisfies −

∂ 2 v j (a:t ∗j ) ∂a j ∂ai

∂ 2 v j (a:t ∗j ) ∂a2j

∫ i (a:t) − t∈T ∂v∂a fi (t)dt ∂φ j i ∗ (a : t j ) = ∫ ∂v (a:t) = , i ∂ai fi (t)dt t∈T

φ j (ai : t ∗j ) = a j ,

for each j, i , j,

∂a j

which shows the characteristics of the elected candidate. Example 1 (Greenhouse emission). Let ai ∈ R+ be the amount of production in country i. Cost function is defined as ci ai2 /2, ci ∈ R++ . Production emits greenhouse gas. Let ai + a j 15See for example, Austen-Smith and Banks (2005).

24

be the total amount of gas emission. A continuum of citizens live in each country. They differ in the damage from the greenhouse effect. The damage from greenhouse effect for citizen t ∈ (0, ∞) is defined as t(ai + a j )2 /2. Then, citizen t in country i’s utility is defined as vi (a : t) = ai −

Note that

∂vi ∂ai

= 1 − ci ai − t(ai + a j ),

∂ 2 vi ∂ai ∂t

satisfies our assumption. Note also that

ci 2 t ai − (ai + a j )2 . 2 2

= −(ai + a j ). Therefore, if a j > 0, this specification

∂ 2 vi ∂ai ∂a j

= −t,

∂ 2 vi ∂ai2

= −ci − t and

∂vi ∂a j

= −t(ai + a j ) < 0.

Note that this game satisfies the assumptions of Proposition 3 and thus, there exists an equilibrium by (restricted) liner reaction functions. Since the reaction function of each elected candidate is linear, the necessary condition shown in Proposition 1 is sufficient. Then, the elected candidate t ∗j is characterized by t ∗j c j + t ∗j where t¯ =

∫

∫ =

(1 − ci ai − t(ai + a j )) f (t)dt 1 − ci ai − t¯(ai + a j ) ti∗ − t¯ ∫ = , = t¯(ai + a j ) t¯ t(ai + a j ) f (t)dt

t f (t)dt. The last equation derived by 1 − ci ai − ti∗ (ai + a j ) = 0 since the elected

candidate maximizes his utility against the opponent’s choice. By this equation, we can derive the elected candidates without deriving the optimal action plans. Since t ∗j , c j > 0, t ∗j > t¯. This implies that the elected candidate is more sensitive to the damage from greenhouse effect than the average citizens and therefore, he is more zealous to solve the environmental problem.

7

Conclusion

We study a general delegation game with complete information. This study introduces notion regularly implementability and provide necessary and sufficient conditions by using a method like the first-order condition. Under reasonable conditions, we show that almost action profiles are regularly imple25

mentable, which is reminiscent of the delegation folk theorem known in the literature. However, unlikely in the previous researches, (specific) efficient allocations violate our necessary condition of the regularly implementability. Summarizing these results roughly, regularly implementable action profile is inefficient, and inefficient action profile is regularly implementable. This is a sharp contrast of the welfare theorems in the general equilibrium theory. Furthermore, since the construction of equilibrium is made by linear reaction functions, we need only to consider a quadratic utility function as an objective of delegated agents. Although our study deals with a general model of delegation game, our results are not free from limitations. First, when action space has a bound, our sufficient condition well works only under specific assumptions. Second, we only deal with strategic form with complete information. Extensions to extensive or Bayesian form game should be considered. Future researches would overcome these limitations.

References Austen-Smith, David and Jeffrey S. Banks (2005) Positive Political Theory II, University Michigan Press. Bade, Sophie, Guillaume Haeringer and Ludovic Renou (2009) “Bilateral commitment,” Journal of Economic Theory, 144, 1817–1831. Besley, Timothy and Stephen Coate (2003) “Centralized versus Decentralized Provision of Local Public Goods,” Journal of Public Economics, 87, 2611–2637. Dubey, Pradeep (1986) “Inefficiency of Nash equilibria,” Mathematics of Operations Research, 11, 1–8. Fershtman, Chaim and Kenneth L. Judd (1987) “Equilibrium incentives in oligopoly,” American Economic Review, 927–940. Fershtman, Chaim, Kenneth L. Judd and Ehud Kalai (1991) “Observable contracts: Strategic delegation and cooperation,” International Economic Review 32, 551–559. Forges, Françoise (2013) “A folk theorem for Bayesian games with commitment,” Games and Economic Behavior 78, 64–71. 26

Gradstein, Mark (2004) “Political Bargaining in a Federation: Buchanan meets Coase," European Economic Review, 48, 983–999. Ihori, Toshihiro and C.C. Yang (2009) “Interregional tax competition and intraregional political competition: The optimal provision of public goods under representative democracy,” Journal of Urban Economics, 66, 210–217. Kalai, Adam Tauman, Ehud Kalai, Ehud Lehrer and Dav Samet (2010) “A commitment folk theorem,” Games and Economic Behavior, 69, 127–137. Koçkesen, Levent, Efe Ok and Rejiv Sethi (2000) “The Strategic Advantage of Negatively Interdependent Preferences,” Journal of Economic Theory, 92, 274–299. Miller, Nolan and Amit Pazgal (2001) “The equivalence of price and quantity competition with delegation,” RAND Journal of Economics, 32, 284–301. Miller, Nolan and Amit Pazgal (2002) “Relative performance as a strategic commitment mechanism,” Managerial and Decision Economics, 23, 51–68. Pal, Rupayan and Ajay Sharma (2013) “Endogenizing governments’ objectives in tax competition,” Regional Science and Urban Economics, 43, 570–578. Polo, Mechele and Piero Tedeschi (1997) “Equilibrium and renegotiation in delegation games,” Working paper. Polo, Mechele and Piero Tedeschi (2000) “Delegation games and side-contracting,” Research in Economics, 54, 101–116. Reichelstein, Stefan (1984) “Smooth versus discontinuous mechanisms,” Economic Letters, 16, 239–242. Renou, Ludvic (2009) “Commitment games,” Games and Economic Behavior, 66, 488–505. Ritz, Robert A. (2008) “Strategic incentives for market share,” International Journal of Industrial Organization, 26, 586–597. Schelling, Thomas, C. (1960) The Strategy of Conflict, Oxford University Press. Sengul, Metin, Javier Gimeno and Jay Dial (2012) “Strategic delegation: A review, theoretical integration, and research agenda,” Journal of Management, 38, 375–414. Segendorff, Björn (1998) “Delegation and Threat in Bargaining”, Games and Economic Behavior, 23, 266-283. Sklivas, Steven D. (1987) “Strategic choice of managerial incentives,” RAND Journal of Economics, 18, 452–458. Vickers, John (1985) “Delegation and the theory of the firm,” Economic Journal, 95, 138–147.

27

A

Proofs

Proof of Proposition 1. Suppose that a∗ is regularly implementable by φ. Let k ∈ {1, . . . , mi }. By assumption, we can find h ∈ Φi such that h(t, ·) = φi (·) + g(t, ·)e k and

∂g(0,a−i ) ∂t

= ε for

some ε ∈ R \ {0}. Let at be the Nash equilibrium induced by (h(t, ·), φ−i ). Then, since φi is the optimal reaction function against the others’ reaction function φ j , ui (a∗ ) ⩾ ui (at ) for each at ∈ A t s.t. aiℓ = hiℓ (t, at−i ),

∀ℓ ∈ {1, . . . , mi } \ {k}

atjℓ = φ jℓ (at− j ), ∀ j ∈ N \ {i},

ℓ ∈ {1, . . . , m j }.

Since (φ, a∗ ) is regular, as per the implicit function theorem, at is uniquely determined and differentiable at a neighborhood of t = 0. Then, total differentiation yields that mj t ∑ ∑ ∂φik ∂a j p ∂giℓ (0, a∗−i ) − = =ε ∂t ∂a j p ∂t ∂t p=1

(2)

mj t ∑ ∑ ∂φiℓ ∂a j p − = 0, ∀ℓ ∈ {1, . . . , mi } \ {k} ∂t ∂a j p ∂ε p=1

(3)

mq t ∑ ∑ ∂φ jℓ ∂aqp − = 0, ∀ℓ ∈ {1, . . . , m j }, ∀ j , i ∂t ∂a ∂ε qp p=1

(4)

t ∂aik

j∈N\{i}

t ∂aiℓ

j∈N\{i}

∂atjℓ

q∈N\{ j}

at t = 0. With matrix expression, © ∂a0 © ∂t1 ª ­­ ∂φ ­ ® ­− 2 (a∗ ) ­ .. ® ­ ∂a1 −2 ­ . ®=­ .. ­ ® ­ . ­ ∂a0 ® ­ n ­ ∂φn « ∂t ¬ ∗ «− ∂a1 (a−n ) 1m1

where ei,k is the

∑i−1

j=1 m j

1 ∗ − ∂φ ∂a2 (a−1 ) · · · ...

1 ∗ − ∂φ ∂an (a−1 )ª ® ® ® ® ® ... ® ® ® 1m n ¬

−1

εei,k

+ k-th unit vector and 1m is the m × m unit matrix. 28

We now consider the function ui (at ). If φ is optimal, by the first-order condition, differentiating this function by t at t = 0 equals 0. That is, mj 0 ∑∑ ∂ui ∂a j k =0 ∂a ∂t j k j∈N k=1

Then, νiℓ := ε1 (

∂a0jk ∂t ) j∈N,k∈{1,...,m j }

is a normal vector of ∇ui (a−i ) and thus, νiℓ = J(φ, a∗ )−1 ei,k ,

which implies that (ν11, ν12, . . . , νnmn ) = J(φ, a∗ )−1 .

□

Proof of Proposition 2. Before the proof of the proposition, we note the following fact. Fact 2. Suppose that for each i ∈ N, there exists αi ∈ R \ {0} and vector µ ∈ R

∑ i

mi \ {0}

such

that ∇ui (a−i ) = αi µ. Let νik be a normal vector of ∇ui (a−i ) for each i ∈ N, k ∈ {1, . . . , mi }. Then, {ν11, . . . , νnmn } is linearly dependent. Proof of Fact 2. Let νik be a normal vector of ∇ui (a−i ) for each i ∈ N, k ∈ {1, . . . , mi }. Suppose that {ν11, . . . , νnmn } is linearly independent. Then, since νik is orthogonal to µ, νik · µ = 0. If, {ν11, . . . , νnmn, µ} is linearly dependent, there exists ((λik )i∈N,k=1,...,mi , λµ ) ∈ ∑ ∑ ∑ R i mi +1 \ {0} such that i,k λik νik + λµ µ = 0. Then, i,k λik νik · µ + λµ µ · µ = λµ µ · µ = 0 · µ = 0. Since {ν11, . . . , νnmn } is linearly independent, λ µ , 0. Therefore, µ = 0, a contradiction. Therefore, {ν11, . . . , νnmn, µ} is linearly independent. However, since {ν11, . . . , νnmn, µ} ⊆ R

∑ i

mi

and |{ν11, . . . , νnmn, µ}| =

∑ i

mi + 1, it is a □

contradiction.

If a is regularly implementable, by Proposition 1, there exists a tuple of normal vectors (ν11, . . . , νnmn ) of (∇ui (a−i ))i∈N such that (ν11, . . . , νnmn ) = J(φ, a)−1 . Since (φ, a) is regular,

29

J(φ, a)−1 is invertible. Therefore, (ν11, . . . , νnmn ) is also invertible and thus {ν11, . . . , νnmn } □

is linearly independent. However, it contradicts to Fact 2. Proof of Fact 1. Consider the following maximization problem: max ui (a′) s.t. u j (a′) ⩾ u j (a)

a′ ∈Bε ′ (a)

Since if u j (a′) > u j (a), ui (a) > ui (a′), a′ = a is the unique solution. Then, by the Karush– Kuhn–Tacker condition, we have ∇ui (a−i ) = λ∇u j (a− j ) for some real number λ , 0.

□

Proof of Theorem 2. Since U i (a∗ ) is convex and open, by the separating hyperplane theorem, there exists ξi ∈ Rn, γi ∈ R such that for each a′ ∈ U i (a∗ ), ξi · a′ > γi and ξi · a∗ = γi . Without loss of generality, ξi = ∇ui (a∗ ). By assumption, without loss of generality, si := sn :=

∂u1 ∗ ∂an (a )

∂ui+1 ∗ ∂ai (a )

, 0 for each i ∈ {1, . . . , n−1} and

, 0. Then, there exists (βi j ) j such that Bi = (−βi1, . . . , −βii−1, 1, −βii+1, . . . , −βin ) =

(1/si )∇ui+1 (a∗ ) for each i ∈ {1, . . . , n − 1} and Bn := (−βn1, . . . , −βnn−1, 1) = (1/sn )∇u1 (a∗ ). Let

(

ν1 · · ·

© B1 ª ) ­ ® ­.® νn = ­­ .. ®® ­ ® B « n¬

−1

.

Since {∇ui (a∗ )} is linear independent, (B1, . . . , Bn )t is invertible and thus, (ν1, . . . , νn ) is well defined. Note that νi · Bi = 1 and νi · B j = 0 for each i ∈ N and j ∈ N \ {i}. ∑ Note also that we can find βi0 that satisfies ai∗ = j∈N\{i} βi j a∗j + βi0 . We define φi (a′−i ) = ∑ ′ ′ ∗ j∈N\{i} βi j a j + βi0 for each a−i ∈ A and i ∈ N. Then, clearly, a is the unique intersection

30

of {φi }i∈N and (φ, a∗ ) is regular since © B1 ª ­ ® ­.® ∗ J(φ, a ) = ­ .. ® ­ ® ­ ® B « n¬

−1

.

Now consider a unilateral deviation of player i. Without loss of generality, i = 1. Suppose that player 1 chooses a different function h ∈ Φ1 . If (h, φ−1 ) induces a Nash equilibrium a, ˆ it satisfies aˆ1 = h(aˆ −1 ) aˆ j = φ j (aˆ −1 ) for each j , 1.

By matrix expression, © 1 0 ­ ­ © h(aˆ −1 )ª ­­−β21 1 ­ ® ­ .. ® ­­ .. ­ . ®=­ . ­ ® ­ ­ ® ­ . ­ .. βn0 « ¬ ­ ­ −β ··· « n1

··· −β23 · · · .. .

0 ª ® ® −β2n ®® ©aˆ1 ª ® ­­ . ®® ® ­ .. ® ®­ ® ®­ ® ® ® aˆn ®« ¬ ® 1 ¬

On the other hand, since © 1 −β12 · · · ­ ­ 1 −β23 · · · © β10 ª ­­−β21 ­ ® ­ ­ .. ® ­ .. .. ­ . ®=­ . . ­ ® ­ ­ ® ­ . ­ .. β « n0 ¬ ­ ­ −β ··· « n1 31

−β1n ª ® ® −β2n ®® ©a1∗ ª ® ­­ . ®® ® ­ .. ® ®­ ® ®­ ® ® ∗ ® an ®« ¬ ® 1 ¬

we have © 1 −β12 · · · ­ h( a ˆ ) −1 ª © ­ ­ ® ­−β21 1 −β23 · · · ­ ® ­ ­ 0 ® ­ . .. ­ ® ­ . . ­ .. ® = ­ . ­ . ® ­ ­ ® ­ .. ­ ® ­ . ­ 0 « ¬ ­ −β ··· « n1

−β1n ª ® ® −β2n ®® ©aˆ1 − a1∗ ª ® ­­ . ®® ® ­ .. ® ®­ ® ®­ ® ® ∗ ® aˆn − an ¬ ®« ® 1 ¬

Therefore, ©aˆ1 − a1∗ ª ­ ® ­ .. ® ν1 h(aˆ −1 ) = ­ . ® ⇐⇒ ­ ® ­ ® ∗ aˆn − an « ¬ ©aˆ1 − a1∗ ª ­ ® .. ® 1 ∗ 1 ∗ ­ ∇u (a )ν1 h(aˆ −1 ) = ∇u (a ) ­ . ® ⇐⇒ ­ ® ­ ® ∗ aˆn − an « ¬ ©aˆ1 − a1∗ ª ­ ® .. ® ∗ ­ ∇u1 (a ) ­ . ® = 0. ­ ® ­ ® aˆn − an∗ « ¬ The last equation derived by the fact that ∇u1 (a∗ ) = Bn and Bn · ν1 = 0. Then, ∇u1 (a∗ ) · aˆ = ∇u1 (a∗ ) · a∗ = γ1 and therefore, aˆ < U 1 (a∗ ), which implies that h is □

an unprofitable deviation. ∂ui ∂a−i

< 0 for each i ∈ N. Now as in the proof of ∑ Theorem 2, let Bi = (1/si )∇ui+1 (a∗ ) and φ˜i (a−i ) = j∈N\{i} βi j a j + βi0 for each i ∈ N. Let us Proof of Proposition 3. By the assumption,

check whether the restriction of above reaction functions, denoted by φ is Nash equilibrium. By Theorem 2, we only consider deviations that yields a corner solution. Note that a deviation 32

that yields ai = 0 is not profitable by definition since ui (a∗ ) ⩾ ui (0, 0) ⩾ ui (0, a−i ) for each a−i . Therefore, consider a deviation h such that C(h, φ−i ) , œ. Note also that ui (a∗ ) ⩾ ui (a¯i, 0) since (a¯i, 0) is on the line {(ai, φ˜−i (ai ))}. If C(h, φ−i ) = œ. If

∂ui ∗ ∂ai (a )

∂ui ∗ ∂ai (a )

= 0, then β−i,i = 0. This implies that

> 0, then β−i,i > 0, and therefore C(h, φ−i ) ⊆ [0, a¯i ] × {0} for some

∂ui ∂ai (a¯i, 0)

< 0, there exists ε ∈ R++ such that ui (a¯i − ε, 0) > ui (a¯1, 0). On the

other hand, since

∂ui ∂a j

< 0, ui (a∗ ) < ui (ai∗, 0). Since a¯i < ai∗ , it contradicts quasi-concavity

of ui . Therefore,

∂ui ∂ai (a¯i, 0)

a¯i < ai∗ . If

⩾ 0. Since

∂ 2 ui ∂ai2

⩽ 0,

∂ui ∂ai (a)

> 0 for each a ∈ C(h, φ−i ). Thus,

ui (a¯i, 0) > ui (a) for each a ∈ C(h, φ−i ). Therefore, h is an unprofitable deviation. In the case of

∂ui ∗ ∂ai (a )

< 0 we have that β−i,i < 0. Then, as in the similar way, we can show that no

deviation is profitable, which concludes the proof.

33

□