Optimal collusion under imperfect monitoring in multimarket contact

Games and Economic Behavior 76 (2012) 636–647

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Optimal collusion under imperfect monitoring in multimarket contact Hajime Kobayashi a,∗ , Katsunori Ohta b a b

Faculty of Economics, Kansai University, 3-3-35 Yamate-cho, Suita, Osaka 564-8680, Japan Faculty of Economics, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan

a r t i c l e

i n f o

Article history: Received 2 December 2010 Available online 20 August 2012 JEL classification: C72 C73 L13

a b s t r a c t We investigate optimal collusion in repeated multimarket contact under imperfect public monitoring, where two firms operate in m markets and in each market, each firm’s decision and public signals are binary. We show that in an optimal pure strategy strongly symmetric perfect public equilibrium, the size of efficiency loss is equal to that in the market with the most tempting deviation under single-market contact. Furthermore, we show a sufficient condition under which the symmetric equilibrium is optimal for joint payoff maximization among any perfect public equilibrium. © 2012 Elsevier Inc. All rights reserved.

Keywords: Multimarket contact Infinitely repeated games Imperfect public monitoring

1. Introduction Modern firms diversify their businesses; even within one business unit, firms may have multiple product lines and geographically separate sales areas. Consequently, firms compete with the same rivals in various markets. Many researchers advocate that collusion may be easier to sustain when firms interact with each other in many markets. For example, Edwards (1955) states that multimarket contact makes collusive agreements easier to sustain because a local deviation from collusion leads to retaliation in all markets.1 Bernheim and Whinston (1990) were the first to provide a theoretical foundation for the relationship between collusion and multimarket contact. They provide the circumstances under which multimarket contact assists in sustaining collusive outcomes, such as market heterogeneity in the number of firms, the growth rates, demand fluctuations, and so on. This heterogeneity makes collusion more stable by transferring enforcement powers across markets. The study of Bernheim and Whinston (1990) relies on the assumption of perfect monitoring.2 Therefore, their main focus is how stable collusion can be. However, a large body of literature on collusion under imperfect monitoring (for example, Green and Porter, 1984 and Abreu et al., 1986) suggests that we cannot always expect firms observe their rivals’ actions. This paper investigates optimal collusion in repeated multimarket contact under imperfect public monitoring, where two firms operate in m markets and in each market, each firm’s decision and the public signals are binary.3 Because our model has only two actions and two signals in each market, the folk theorem by Fudenberg et al. (1994) does not apply. In other

*

Corresponding author. Fax: +81 6 6339 7704. E-mail addresses: [email protected] (H. Kobayashi), [email protected] (K. Ohta). 1 Evans and Kessides (1994) empirically test Edwards’ statement using US airline industry data. 2 Spagnolo (1999) also analyzes the effect of multimarket contact on the stability of collusion under perfect monitoring. He shows that if the payoffs of firms are strictly concave in their profits, then multimarket contact always relaxes the incentive compatibility conditions. 3 For simplicity of exposition, we confine ourselves to the case of two firms. The analysis generalizes straightforwardly to the case of n firms. 0899-8256/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.geb.2012.08.003

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words, our model is an extension of Radner et al. (1986) to multitask partnerships, and diversification of the relationships has the potential to alleviate efficiency loss. We show that in an optimal pure strategy symmetric perfect public equilibrium, the size of efficiency loss is equal to that in the market with the most tempting deviation under single-market contact. That is, firms can collude as if they avoided efficiency loss except in the market with the most tempting deviation. Furthermore, we show a sufficient condition under which the symmetric equilibrium is optimal for joint payoff maximization among any perfect public equilibrium. The key aspects of the optimal equilibrium are that multimarket contact (i) gives firms more information on a rival’s behavior and (ii) makes punishment more severe than single-market competition. The first aspect means that the many signals acquired enable firms to conduct statistical inferences more accurately on the rival’s deviations. The second means that a local deviation leads to global retaliation as in Edwards (1955). In combination with accurate statistical inference, firms hesitate to deviate in the markets with less tempting deviation for fear of global retaliation. Hence, the efficiency loss from those markets disappears. This paper is not the first to examine the effect of multimarket contact on collusion under imperfect monitoring. Matsushima (2001) shows that under a fixed and possibly low discount factor such as that in perfect monitoring, firms achieve the efficient (or first-best) outcome approximately if the number of contacts is sufficiently large. This is because when the number of contacts is sufficiently large, firms can almost certainly detect a rival’s deviations by the law of large numbers. In contrast to Matsushima (2001), we study the optimal (or second-best) outcome of collusion in multimarket contact under imperfect monitoring given an arbitrary number of contacts. By focusing on sufficiently patient firms, we succeed in clarifying the relationship between the degree of temptation to deviate and the value of collusion. In addition, we extend the setting of Matsushima (2001) to the case in which market heterogeneities are allowed. Matsushima (2001) uses market homogeneity to construct an equilibrium. Indeed, Matsushima’s (2001) construction can be extended by allowing a type of asymmetry between markets. Consider the case in which there are finite types of markets and the number of markets is sufficiently large for each type. Then, the law of large numbers would apply. However, our analysis admits market heterogeneities that make the law of large numbers inapplicable. Furthermore, our analysis also admits some firm heterogeneities (Section 4). Another related study is Cai and Obara (2009).4 They examine the effects of horizontal integration on firm reputation in an environment where customers observe only imperfect signals about firms’ effort choices. By horizontal integration, the customers enjoy better monitoring and effective punishment, and the merged firm builds reputation effectively. Specifically, they show that under certain conditions, efficiency loss might be alleviated by merging firms into one large firm, and they clarify that the optimal firm size may naturally arise. Although these results seem similar to ours, their paper and ours have different settings. In particular, they focus on a one-sided moral hazard problem to study firm reputation in the interaction between a long-lived firm and short-lived customers, while we analyze the two-sided moral hazard problem to study the collusion between long-lived firms. As a result, the contrast arises in the analysis of asymmetric strategies, which is absent from their paper. The idea in this paper that accumulating information across markets improves the value of collusion can be related to the literature of repeated games with (in)frequent monitoring. The seminal paper of Abreu, Milgrom and Pearce (1991) (hereafter AMP) showed that by increasing the infrequency of actions while keeping the flow of information constant, efficiency can be improved in a prisoners’ dilemma under strongly symmetric equilibria. Our work is the first study to apply the result of infrequent monitoring to a multimarket contact situation.5 This paper is organized as follows. Section 2 sets up the model. In Section 3, we derive the optimal pure strategy symmetric equilibrium payoffs and extend the analysis by incorporating mixed and asymmetric strategies. Section 4 provides discussions. 2. The model Consider two identical firms (firms 1 and 2) competing with each other in m markets over infinite periods t = 0, 1, 2, . . . . Each firm has binary choices, “to cooperate with the opponent (C)” or “to defect from cooperation (D)”, for each market. m We denote the set of choices for firm i in market k by A ki = {C , D }. Let A i = k=1 A ki , A k = A k1 × A k2 , and A = A 1 × A 2 =

m

Ak . Firms cannot observe their rival’s actions but receive imperfect public signals of the actions. Let Ω k ≡ {G , B } be the set of signals for market k, and let Ω = Πkm=1 Ω k be the product. Here, G represents a “good outcome” whereas B stands for a “bad outcome”. This is a situation in which imperfection of the monitoring technology becomes a serious obstacle for firms to collude in the sense that efficiency loss is inevitable. With this model, we focus on how multimarket contact alleviates the obstacle. k=1

4 Kobayashi and Ohta (2008) also explore the relationship between imperfect information and the effectiveness of multimarket contact in a different setting (continuous-time games). 5 In the stream of literature on frequent monitoring, there are several studies that extend the information structure of AMP and investigate how the type of information can affect the equilibrium values of cooperation. See Sannikov and Skrzypacz (2007, 2010), and Fudenberg and Levine (2009).

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For each k, given an action profile ak = (ak1 , ak2 ) ∈ A k , a signal ωk ∈ Ω k is realized with probability P k (ωk |ak ). In the m 1 2 multimarket contact situation, given an action profile a = (a1 , a2 ) = (a11 , a21 , . . . , am 1 ; a2 , a2 , . . . , a2 ) ∈ A, a signal vector 1 2 m ω = (ω , ω , . . . , ω ) ∈ Ω is realized with probability P (ω|a). We focus on the case in which the public signals of each market are independently distributed and are not influenced by the actions taken in the other markets. Formally, the assumption is as follows. Assumption 1. P (ω|a) =

m

k=1

P k (ωk |ak ).

For notational convenience, let pk2 ≡ P k (G |C , C ), pk1 ≡ P k (G | D , C ) = P k (G |C , D ), and pk0 = P k (G | D , D ). To focus on interesting situations, we make the following standard assumption that a unilateral defection decreases the probability of the good outcome. Assumption 2. 1 > pk2 > pk1 > pk0  0 for all k. Each firm obtains profit r ki = r ki (aki , ωk ) from market k when it takes aki ∈ A ki and

pected payoffs for firm i from market k by uki (aki , akj ) =



ωk is realized. We denote the ex-

k k k k k k ωk P (ω |a )r i (ai , ω ). In total, each firm obtains u i (a i , a j ) = m k k k k=1 u i (a i , a j ) in each stage game. We assume that the stage expected payoff structure in any market k is represented by Table 1.

Table 1 Stage payoffs. C C D

πk, πk π k + g k , −bk

D

−bk , π k + g k 0, 0

The following assumption guarantees that (C , C ) is the efficient action profile for each k, while aki = D is the dominant action for each i, k. That is, the payoff structure has the property of a symmetric prisoners’ dilemma. Assumption 3.

π k > 0, g k > 0, bk > 0, and g k − bk < π k for all k.

In each period, the stage game is played and then the corresponding public signal is revealed. We assume that a public randomization device is available at the end of each period. The randomization device selects a number γ ∈ [0, 1] according to the uniform distribution on [0, 1]. The public history at the beginning of period t is represented by a sequence h(t ) = 1  t −1 {ω(t  ), γ (t  )}tt −  =0 . The private history for firm i at the beginning of period t is h i (t ) = {a i (t )}t  =0 . We denote the set of all ∞ t t public histories and private histories for firm i by H and H i , respectively. Let Hi = t =0 ( H t × H ti ). A strategy of firm i is a map from Hi to the set of (randomized) actions. Given a common discount factor δ ∈ (0, 1) and a sequence of action ∞ t profiles, {a(t )}t∞ , generated by a strategy profile, firm i’s average discounted expected payoff is ( 1 − δ) t =0 δ u i (a(t )). =0 Our solution concept here is perfect public equilibrium (PPE). A strategy of firm i is public if at each time t, it does not depend on the private history. A PPE is a profile of public strategies such that at every date t and for any public history h(t ), the strategies constitute a Nash equilibrium from that date onwards. Note that in a PPE, the players’ beliefs about the opponents’ past play are irrelevant. Therefore, PPE is a special class of sequential equilibria. 3. Analysis We explore the optimal PPE payoffs in the sense of joint payoff maximization for sufficiently patient firms. The standard approach for characterizing the PPE payoff set for a fixed δ is provided by Abreu et al. (1990). Although the set of PPE payoffs is the largest self-generating set, it is not easy to identify the set. Instead, because we restrict our attention to sufficiently patient firms, we can use the linear programming approach by Fudenberg and Levine (1994) and Fudenberg et al. (2007), who characterize the limit set of PPE payoffs as the discount factor goes to one. We utilize this approach to identify a boundary point of the limit set. We divide the computation of the optimal PPE payoffs into two stages. First, we restrict our attention to pure strategy strongly symmetric PPE (SSPPE). Then, we discuss mixed and asymmetric strategies and give a sufficient condition that the sum of the optimal pure strategy SSPPE payoffs is the greatest among any PPE. 3.1. Symmetric equilibrium In this subsection, we derive the optimal pure strategy SSPPE payoffs. A public strategy profile is strongly symmetric if at each time and for any public history, the strategies prescribe the same action for both firms. This restriction to equilibrium strategies makes the continuation payoffs to the one-dimensional set where both firms’ payoffs are identical. Accordingly, we use the algorithm by Fudenberg et al. (2007), who characterize the limit set of PPE payoffs under a restriction to equilibrium strategies such as SSPPE (Section 4.3.1 of Fudenberg et al., 2007). Furthermore, the solution of the linear programming

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problem helps us to identify the form of the SSPPE strategy that exactly achieves the boundary of the limit set of pure strategy SSPPE payoffs as the equilibrium payoff. We show that this strategy profile is the optimal pure strategy SSPPE for sufficiently patient firms. Following Fudenberg et al. (2007), we consider a linear programming problem for identifying the boundary point of the limit set of pure strategy SSPPE payoffs in the direction of the optimal SSPPE payoff. Before analysis, we introduce some  notations. We denote by M = {1, . . . , m} the set of all markets, and for any M  ⊆ M, denote the action C M as taking C ˆ

ˆ ⊆ M arbitrarily, and suppose that C M is an action in the initial period for both firms. in M  and taking D in M \ M  . Take M Let f (ω) ∈ R be a continuation payoff following a signal ω for both firms. Then, consider the following linear programming ˆ

problem for a given C M and δ :



ˆ



V ∗ CM,δ ≡

max

v

v ∈R, f :ω→R

s.t.



ˆ

ˆ

 



v = (1 − δ)u i C M , C M + δ



ˆ



P

ˆ

ˆ



ω|C M , C M f (ω)

ω∈Ω



v  (1 − δ)u i C M , C M + δ

  P

ˆ



for i = 1, 2,

(1)



ω|C M , C M f (ω) for all M  ⊆ M and i = 1, 2,

(2)

ω∈Ω

v  f (ω)

for all ω ∈ Ω.

(3)

Eq. (1) represents the value function, and inequality (2) represents the incentive compatibility constraints. Inequality (3) requires all of the continuation payoffs to be less than the maximized value. Note that the restriction to the continuation ˆ

payoffs ignores the feasibility of the continuation payoffs. This relaxation allows us to compute V ∗ (C M , δ) independently of δ . Given f (ω) that satisfies these constraints and gives the value v under δ , we can construct a continuation payoff f  (ω) that also satisfies these constraints and gives the same value v under δ  = δ (Lemma 3.1 of Fudenberg et al., 2007). ˆ

ˆ

ˆ

Therefore, V ∗ (C M , δ) = V ∗ (C M ), and max Mˆ ⊆ M V ∗ (C M ) is the boundary of the limit set of pure strategy SSPPE payoffs.  To derive the boundary, we first compute V ∗ (C M ). Then, we compute V ∗ (C M ) for all M  ⊂ M and compare them. We set ˆ

C M = C M and define h(ω) = δ( v − f (ω))/(1 − δ). Then the problem can be written as

(LP I)





V∗ CM ≡

max

v ∈R,h:ω→R

v

s.t. v=

m 

πk −

P

P

ω∈Ω

k =1

 

  





ω|C M , C M h(ω), 

(4)



ω|C M , C M − P ω|C M , C M h(ω) 



gk

for all M  ,

(5)

k∈ M \ M 

ω∈Ω

h(ω)  0 for all ω ∈ Ω. 

(6) subject to (5) and (6). Fix an arbitrary M  in constraint (5).

It is clear that firms must minimize ω∈Ω P (ω| C , C )h(ω) Then, the problem can be interpreted as “the expenditure minimization problem by a consumer with a linear utility function”. Let us interpret h(ω) as the level of consumption of the good ω , which must be nonnegative, P (ω|C M , C M ) as the  price of the good ω , and P (ω|C M , C M ) − P (ω|C M , C M ) as utility from the unit consumption of the good ω . Therefore, “goods” are perfect substitutes, and the problem is solved by a “corner solution”. Before presenting the result, let us de note B M as the signal profile {ωk }k∈ M  in which ωk = B for all k ∈ M  . Therefore, B M represents that ω ∈ Ω in which k ω = B for all k ∈ M. Then, we present the result that tells us how to set h(ω) to obtain the boundary. M

M

Lemma 1. Setting h( B M ) > 0 and h(ω) = 0 for ω = B M solves the problem. Proof. This lemma follows from the fact 

B M ∈ arg max

P (ω|C M , C M ) − P (ω|C M , C M ) P (ω|C M , C M )

ω∈Ω

Based on Lemma 1, (LP I) can be reduced to

(LP II)

min h( B

s.t.

M

)>0



 

P B M |C M , C M h B M



for all M  .

2

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H. Kobayashi, K. Ohta / Games and Economic Behavior 76 (2012) 636–647

 







 





P B M |C M , C M − P B M |C M , C M h B M 

gk

for all M  .

(7)

k∈ M \ M 

To solve (LP II), we must find the binding incentive compatibility constraint from (7) and set h( B M ) as small as possible to ensure that the binding constraint holds. For notational convenience, we define the likelihood ratio of the bad signal between collusive behavior and defection as lk ≡ (1 − pk1 )/(1 − pk2 ) for any k ∈ M. Without loss of generality, we permutate the index of the markets as follows. Assumption 4. g 1 /(l1 − 1)  g 2 /(l2 − 1)  · · ·  g m /(lm − 1). Then, we can show that the binding incentive compatibility constraint is deterring the deviation only in market 1 under the constructed continuation payoffs. Lemma 2. Consider the case in which firms collude in all markets. Under h( B M ) > 0 and h(ω) = 0 for ω = B M , if the incentive compatibility constraint is satisfied locally with respect to the deviation in market 1, then all of the constraints are satisfied. Proof. Consider the case in which a firm deviates in markets M \ M  . By Lemma 1, we can verify that, under the construction of the continuation payoffs, the deviation is unprofitable if

 

1 − pk1

 

k∈ M \ M 





1 − pk2 −

k∈ M 



1 − pk2









h BM 

gk .

k∈ M \ M 

k∈ M

Therefore, the optimal deviation is taking D in markets M \ M  to maximize



k∈ M \ M  k k∈ M \ M  l



gk

−1

.

To prove Lemma 2, we use the following two steps to demonstrate that taking D only in market 1 is an optimal deviation. First, we show by induction that deviating only in market 1 dominates deviating in any subset of M that takes the form {1, 2}, {1, 2, 3}, . . . , {1, 2, . . . , m}. Then, based on the inductive argument of the first step, we show that deviating only in market 1 weakly dominates deviating in any subset of M. First, the inequality

g1 l1 − 1

>

g1 + g2

holds because

(8)

⇔

g

⇔

1

1

⇔

(8)

l1l2 − 1

l1 − 1

g1

−



1 l1l2 − 1

l (l − 1) 1 2

> 

g2 l1l2 − 1 g2

>

(l1 − 1)(l1l2 − 1) l1l2 − 1

1  2 g g > 2 l1 × 1 . l −1 l −1

Next, we assume that

g 1 + g 2 + · · · + g k −1 l 1 l 2 · · · l k −1

−1

>

g 1 + g 2 + · · · + gk l 1 l 2 · · · lk − 1

(9)

.

Then, we transform inequality (9) directly:

(9)

⇔ ⇔ ⇔ ⇔



1

2

g + g + ··· + g

k −1



l 1 l 2 · · · lk − 1 l 1 l 2 · · · l k −1 − 1

 − 1 > gk

( g 1 + g 2 + · · · + g k −1 ) gk > l 1 l 2 · · · l k −1 − 1 lk − 1 1 2 k −1 (g + g + · · · + g ) g k l 1 l 2 · · · lk gk g k l 1 l 2 · · · lk l 1 l 2 · · · l k −1 + 12 + 12 > k 1 2 k − 1 k − 1 k l l ···l −1 (l l · · · l − 1)(l − 1) l − 1 (l l · · · lk−1 − 1)(lk − 1) l 1 l 2 · · · l k −1

l 1 l 2 · · · l k −1

( g 1 + g 2 + · · · + gk ) gk > k . 1 2 k l l ···l − 1 l −1

(10)

H. Kobayashi, K. Ohta / Games and Economic Behavior 76 (2012) 636–647

641

From Assumption 4 and lk > 1 for all k, (10) implies that

( g 1 + g 2 + · · · + gk ) g k +1 > l 1 l 2 · · · lk − 1 l k +1 − 1

12  (l l · · · lk+1 − 1) − (l1l2 · · · lk − 1)  1 g k +1 2 k g + g + ··· g > (l1l2 · · · lk − 1)(l1l2 · · · lk+1 − 1) l 1 l 2 · · · l k +1 − 1

l 1 l 2 · · · lk

⇔

g 1 + g 2 + · · · + gk

⇔

l 1 l 2 · · · lk − 1

>

g 1 + g 2 + · · · + g k +1 l 1 l 2 · · · l k +1 − 1

.

Therefore, we prove the first step by the inductive argument. ˜ ⊂ M arbitrarily. Let us denote the minimum number in M ˜ as k˜ and the number of elements of M ˜ as n + 1. We set M ˜ in ascending order and relabel the numbers in M ˜ as {k˜ , k˜ + 1, . . . , k˜ + n}. Then, by the above We arrange the elements in M ˜ of the form {k˜ , k˜ + 1}, {k˜ , k˜ + 1, inductive argument, deviating only in market k˜ dominates deviating in any subset of M k˜ + 2}, . . . , {k˜ , k˜ + 1, . . . , k˜ + n}. Moreover, deviating just in market 1 weakly dominates deviating only in market k˜ by As˜ arbitrarily, deviating only in market 1 weakly dominates deviating in any subset of M. 2 sumption 4. Because we set M By Lemma 2, inequality (7) must be binding at M \ M  = {1}. Then, the inequality can be transformed into



p 12 − p 11

m  

 



1 − pk2 h B M = g 1 .

(11)

k =2

By (4) and (11), the maximized value of the problem is





V∗ CM =

m 

πk −

k =1

g1 l1

−1

(12)

.

The value (12) is a multimarket contact version of the formula introduced by AMP. The formula consists of two terms. The first term represents the benefit of collusion. The second term represents the efficiency loss from the informational imperfection. In the second term, when the deviation gain in market 1 becomes small or when the informativeness of signals in market 1 is improved, the efficiency loss becomes small.  By a similar argument, we can derive V ∗ (C M ) for all M  ⊂ M except M  = ∅ (the case of taking D in all markets).6    Assume that firms take C in the set of M and take D in the remaining markets. Let us denote ω M = {ωk }k∈ M  and ω M \ M = M M M k M \M  M \M   {ω }k∈ M \ M . Then, given C , they achieve the highest value if (i) for any ω , we set h( B , ω ) ≡ h( B ) > 0 and 







h(ω M , ω M \ M ) = 0 for ω M = B M , and (ii) the binding incentive constraint is to deter the deviation in the market, the index of which is the minimum in M  . This construction enables firms to achieve the value



V∗ CM





=



πk −

k∈ M 

gk



k

l −1

,

where k is the minimum number in M  . We are interested in the case in which firms collude in all markets. Then, we investigate the sufficient condition for



max V ∗ C M





M  ⊆M

  = V∗ CM .

If g 1 > π 1 (l1 − 1), there are cases in which firms may attain the highest value by giving up collusion in the markets with more tempting deviations. For example, if π 1 < g 1 /(l1 − 1) − g 2 /(l2 − 1) holds, then m  k =2

πk + π1 −

g1 l1

−1

<

m  k =2

πk −

g2 l2

−1

.

Otherwise, colluding in all markets always leads to the highest value. In sum, we obtain the following result. Lemma 3. Suppose that Assumptions 1, 2, 3, and 4 hold. If g 1  π 1 (l1 − 1), then V ∗ (C M ) is an upper bound of the pure strategy SSPPE payoffs for any δ . Then, by constructing an optimal pure strategy SSPPE that achieves V ∗ (C M ), we show that the boundary is the optimal pure strategy SSPPE payoff. Because our game is a prisoners’ dilemma, a natural candidate for an equilibrium strategy is a trigger strategy. Let us consider a generalized trigger strategy with two phases: phase C and phase D. In phase C , each firm

6

Note that V ∗ (C ∅ ) = 0, which is the worst SSPPE payoff.

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H. Kobayashi, K. Ohta / Games and Economic Behavior 76 (2012) 636–647

plays a “cooperative” action, C M , while each firm plays the dominant action of the stage game (taking D in all markets) in phase D. The phase in period 1 is phase C . If period t − 1 is in phase C and if the signal in period t − 1 is ω , then the phase switches to D with probability ρ (ω) ∈ [0, 1]. With the remaining probability, the phase remains in phase C . If period t − 1 is in phase D, then period t is in phase D with certainty. Let us replace h(ω) in (LP I) and (LP II) with δ( v − f (ω))/(1 − δ) in the original linear programming problem. Then, given the sufficiently high discount factor, the solution of the linear programming problem indicates the following. Lemma 1 shows that to achieve the optimal pure strategy SSPPE payoffs, firms must start punishment only when they observe the bad signals in all markets. That is, ρ ( B M ) > 0 and ρ (ω) = 0 for ω = B M . Moreover, Lemma 2 shows that under the punishment, the incentive constraint for deterring the deviation only in market 1 is binding. Let us denote the value of the optimal generalized trigger strategy as v ∗ (δ) given δ . If the optimal generalized trigger strategy is PPE, the value function and the binding incentive constraint are as follows: ∗

v (δ) = (1 − δ)

m 



k

π +δ 1−

k =1

m  

1−

pk2

 



ρ B

M



v ∗ (δ),

(13)

k =1

m    m      M ∗  k 1 1 k v (δ) = (1 − δ) π + g + δ 1 − 1 − p1 1 − p2 ρ B v (δ). ∗

k =1

(14)

k =2

Now (14) is rewritten as

  m     δ l1 1 − pk2 ρ B M v ∗ (δ) 1−δ k =1 k =1   m m   π k + g 1 + l1 v ∗ (δ) − πk , =

v ∗ (δ) =

m 

π k + g1 −

k =1

(15)

k =1

where the second equality follows from (13). Solving (15), we obtain v ∗ (δ) = V ∗ (C M ). The remaining task is to explore conditions under which the optimal generalized trigger strategy is PPE. Let us take δ that satisfies (13) and (14) evaluated by ρ ( B M ) = 1 and v ∗ (δ) = V ∗ (C M ); that is,

δ=

g1

g 1 + ( p 12 − p 11 )

m

k=2 (1 −

pk2 ) V ∗ (C M )

=

g1 +

m

k=2 (1 −

g1 pk2 )[( p 12 − p 11 )

m

k =1

π k − (1 − p 12 ) g 1 ]

.

(16)

If g 1  π 1 (l1 − 1), there exists δ ∈ (0, 1) given by (16). Eq. (16) and Lemma 2 imply that firms can block any type of deviation in the trigger strategy equilibrium when δ = δ . For any δ > δ , by utilizing the public randomization device, set ρ ( B M ) that satisfies (13) and (14) evaluated by v ∗ (δ) = V ∗ (C M ); that is,





ρ BM =

=

1−δ

δ 1−δ

δ

 

( p 12

−

m

p 11 )

m

k=2 (1 −

If g 1  π 1 (l1 − 1), there exists ρ ( B M ) ∈ (0, 1) given by (17).7

g1

k=2 (1 −

pk2 )[( p 12

−

pk2 ) V ∗ (C M ) g1 p 11 )

m

k =1

π k − (1 − p 12 ) g 1 ]

.

(17)

ρ ( B M ) > 0 given by (17). Then, firms can block all the deviations for any δ > δ by setting

Theorem 1. Suppose that Assumptions 1, 2, 3, and 4 hold. If g 1  π 1 (l1 − 1), then there exists δ ∈ (0, 1) such that for any δ  δ there exists a generalized trigger strategy equilibrium that exactly achieves the optimal pure strategy SSPPE payoff, V ∗ (C M ). The intuitive explanation for the result in Theorem 1 is as follows. Firms statistically infer deviations under imperfect monitoring. Specifically, firms test whether an opponent deviates by using public signals as samples of the opponent’s behavior. Among the many types of statistical tests, the most powerful test in this case is to test the hypothesis that “the opponent deviated” against the hypothesis that “the opponent did not deviate” when observing the bad signals in every market (Lemma 1).8 This is because testing the opponent’s behavior in a market by using the good signal does not increase the likelihood ratio of the hypothesis “the opponent deviated” against the hypothesis “the opponent did not

7

The result relies on public randomizations, but up to integer constraints it is easy to dispense with them by adjusting the length of the punishment. The result that punishment is triggered only after the worst possible signal has appeared in the principal–agent model. See, for example MacLeod (2003). He considers the optimal contract when the principal and agent have private (subjective) measures of performance that are possibly correlated with each other. He shows that when there is no correlation between the principal’s and agent’s beliefs regarding performance, then the optimal contract pays the same bonus to the agent for all but the worst signal of performance. 8

H. Kobayashi, K. Ohta / Games and Economic Behavior 76 (2012) 636–647

643

deviate”. Given the test, when firms are sufficiently patient, firms do not intend to deviate in multiple markets. This is because if a firm deviates in additional markets, the likelihood ratio for the deviation increases exponentially, while the deviation gain increases additively. Therefore, it is sufficient to deter the deviation in the market with the most tempting deviation in which the deviation leads to a larger gain and is less detectable (Lemma 2). As a result, the efficiency loss that appears under single-market contact can be fully alleviated in the other markets. Note that in the generalized trigger strategy, firms should take D in all markets in the punishment phase. This strategy describes the global retaliation proposed by Edwards (1955). Combining the threat of global retaliation with the statistical inference described above, firms are likely to hesitate to deviate. Therefore, the logic of Edwards (1955) proves to be true with imperfect monitoring because the best deviation is to deviate only in one market, whereas punishment occurs in all markets. This is in contrast with what happens with perfect monitoring under which the best deviation would be to deviate in all markets. m Although V ∗ (C M ) can still be enforceable even when g 1 > π 1 (l1 − 1) but g 1 < k=1 π k (l1 − 1), it might not be the best value when the markets are heterogeneous. On the other hand, if the markets are homogeneous (π k = π , g k = g and lk = l for all k) and g < mπ (l − 1), then V ∗ (C M ) is the best value, even when g > π (l − 1). That is, g < mπ (l − 1) is necessary and sufficient for V ∗ (C M ) to be the optimal pure strategy SSPPE payoff. 3.2. Randomization and asymmetric strategies In this subsection, we derive a sufficient condition under which the optimal generalized trigger strategy maximizes the sum of firms’ payoffs among any PPE. Although the analysis in the previous subsection is restricted to symmetric pure strategies, randomization and asymmetric strategies are easily incorporated into the analysis in light of Theorem 1.9 First, let us consider utilizing randomization. Although cooperative payoffs certainly decrease, randomization may improve the value of collusion by alleviating efficiency loss. By Theorem 1, we can verify that efficiency loss is affected by the detectability of the most tempting deviations. If randomization improves the detectability of those deviations, randomization might be effective in increasing the value of collusion. To confirm the argument in the general setting, let us modify the trigger strategy to achieve V ∗ (C M ). For simplicity, consider the situation in which the market with the most tempting deviation is unique (market 1). Consider the trigger strategy in which, in phase C , firms randomize between actions C and D in market 1, and both firms take C in the other markets. Let α 1 = (α11 , α21 ) be a mixed action profile of market 1, where αi1 = ηi1 C + (1 − ηi1 ) D.10 Then, the equilibrium payoff v i satisfies



v i = (1 − δ)

u 1i

C, α

1 j



+

m 



π

k

k =2

 v i  (1 − δ)



u 1i



D, α

1 j



+

m 



+δ 1− P 

π

k

1



m 



k

P ( B |C , C )ρ B M v i ,

k =2

 +δ 1− P

B |C , α

1 j

1



B|D, α

k =2

1 j

m 

 k

P ( B |C , C )ρ B M v i .

k =2

If 0 < ηi1 < 1, the value function and incentive condition must provide an equal payoff so that the incentive constraint is satisfied with equality. Therefore, we can obtain

vi =

m  k =2



π k + η1j π 1 −

g1 η1j (1− p 11 )+(1−η1j )(1− p 10 ) −1 η1j (1− p 12 )+(1−η1j )(1− p 11 )



  − 1 − η1j b1 1 +

1 η1j (1− p 11 )+(1−η1j )(1− p 10 ) −1 η1j (1− p 12 )+(1−η1j )(1− p 11 )

.

If (1 − p 11 )/(1 − p 12 ) < (1 − p 10 )/(1 − p 11 ), taking D with small probability may increase the value of collusion. On the other hand, when (1 − p 11 )/(1 − p 12 )  (1 − p 10 )/(1 − p 11 ), randomization never improves the value of collusion. Therefore, to show that the sum of the values derived in the previous section is the greatest sum of firms’ payoffs among any PPE payoff, we require the following condition. Assumption 5. Let K = {k| g k /(lk − 1) = g 1 /(l1 − 1)} be the set of markets with the most tempting deviations. For all k ∈ K , (1 − pk1 )/(1 − pk2 )  (1 − pk0 )/(1 − pk1 ). Assumption 5 says that a unilateral deviation in the markets with the most tempting deviation is the most detectable when both firms cooperate in the markets. Assumption 5 is derived by the complementarity condition that one firm’s cooperation has greater effects on the marginal probability of receiving a good signal when the opponent cooperates than when he/she defects. That is, if pk2 − pk1  pk1 − pk0 for all k ∈ K , Assumption 5 holds. 9 10

Kandori and Obara (2006) derive the AMP formula in a version of mixed strategy. We employ convex combinations to denote a randomization over two elements.

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Next, to obtain the maximal sum of PPE payoffs, we also need to take into account asymmetric strategies. Suppose a strategy profile in which both firms assign a positive probability to playing C in some markets in the initial period. Let M C be the set of the markets in which both firms assign a positive probability to playing C . Because of the failure of statistical distinguishability of defection in our model, we cannot attribute one bad signal to a particular firm’s defection. To implement the cooperative action profile, mutual punishment in M C is required with some probability when a bad signal is realized in a market of M C . Accordingly, the continuation strategy in those markets becomes symmetric. firms attach zero probability to (C , C ) in M \ M C . Thus, firms at most obtain the total payoffs  On the other k hand, k max { π + g − bk , 0} in those markets. Then, to show that asymmetric strategies cannot achieve optimal collusion C k∈ M \ M maximizing the sum of firms’ payoffs, we introduce the following assumption. Assumption 6. For all k ∈ K ,

2

πk −

gk



lk − 1

  > max π k + g k − bk , 0 .

(18)

Assumption 6 means that the loss term of the optimal SSPPE is less than that of the asymmetric equilibrium in the markets with the most tempting deviation. Then, we show the following theorem that 2V ∗ (C M ) is an upper bound of the sum of PPE payoffs for any δ under Assumptions 5 and 6. Theorem 2. Suppose that Assumptions 1, 2, 3, and 4 are satisfied. Suppose also that Assumptions 5 and 6 hold. If g 1  π 1 (l1 − 1), then for any v = ( v 1 , v 2 ) = ( V ∗ (C M ), V ∗ (C M )) such that v 1 + v 2  2V ∗ (C M ), v is not a PPE payoff pair under any δ ∈ (0, 1). Proof. Let us fix δ arbitrarily. Suppose that there exists an equilibrium payoff vector vˆ = ( vˆ 1 , vˆ 2 ) that satisfies



vˆ 1 + vˆ 2  2V ∗ C M











vˆ = V ∗ C M , V ∗ C M

and

 .

(19)

Without loss of generality, we can assume that vˆ 1 + vˆ 2 is the greatest sum of PPE payoffs over all PPE. Let α = (α 1 , α 2 ) be the mixed action profile played in the initial period under this equilibrium, where α i = (αi1 , αi2 , . . . , αim ), αik = ηki C +

(1 − ηki ) D for k = 1, 2, . . . , m. Because vˆ 1 + vˆ 2 is the greatest sum of PPE payoffs and the continuation payoffs of the PPE are also PPE payoffs, we have:

u 1 (α ) + u 2 (α )  vˆ 1 + vˆ 2 . Let us denote U = {(u 1 (a), u 2 (a))|u 1 (a) + u 2 (a)  also have

(20)

m

k=1 max{

π + g − b , 0} for some a}. By Assumption 6 and (19), we k

k

k

vˆ 1 + vˆ 2 > u 1 (a) + u 2 (a),

(21)

for any (u 1 (a), u 2 (a)) ∈ U . By (20) and (21),

k i

η > 0 for each i in some market k.

Given such α , let us decompose the set of all markets, M, into M C and M \ M C , where M C = {k|ηki > 0 for i = 1, 2}. That is, M C is the set of markets in which both firms assign a positive probability to playing C , and M \ M C is the set of the markets in which either firm plays D with certainty. m Next, let us suppose, without loss of generality, that f i (ω) = k=1 f ik (ω) is the continuation payoff for firm i when a signal profile ω is realized in the initial period. Because vˆ is a PPE payoff vector, we have the following value equations and incentive constraints:

vˆ i = (1 − δ)u i (α i , α j ) + δ



P (ω|α ) f i (ω),

ω

vˆ i  (1 − δ)u i (a i , α j ) + δ



P (ω|a i , α j ) f i (ω)

for all a i ∈ A i ,

ω

for i = 1, 2. If we denote

Wi





ωk |α −k ≡

     P ωκ |α κ f i ωk , ω−k , ω−k κ =k

where α −k

= (α , . . . , αk−1 , αk+1 , . . . , αm ) and ω−k = (ω1 , . . . , ωk−1 , ωk+1 , . . . , ωm ), combined with the fact ηki > 0 for each 1

i and k ∈ M C , then the above equations imply that







vˆ i = (1 − δ) uki C , αkj +







vˆ i  (1 − δ) uki D , αkj +

 κ =k



κ =k

for each i and k ∈ M C .

u κi

   κ κ    αi , α j + δ P k ωk |C , αkj W i ωk |α −k ,

(22)

ωk

u κi

   κ κ    αi , α j + δ P k ωk | D , αkj W i ωk |α −k , ωk

(23)

H. Kobayashi, K. Ohta / Games and Economic Behavior 76 (2012) 636–647

645

Now, let kˆ be the minimum index of the markets in M C . That is, the market kˆ has the most attractive deviation in M C . Inequality (23) for k = kˆ is equivalent to



ˆ  kˆ

Pk

ˆ

ˆ

ˆ 

ˆ



ˆ

ˆ

ω |C , αkj − P k ωk | D , αkj W i ωk |α −k 



1 − δ  kˆ  ˆ ˆ ˆ  u i D , αkj − uki C , αkj

δ

ωkˆ



ˆ



ˆ

W i G |α −k − W i B |α −k 

⇔

1−δ



ˆ

ˆ

δ

ˆ

ˆ

ˆ

ˆ



ˆ

uki ( D , αkj ) − uki (C , αkj )

(24)

.

ˆ

P k (G |C , αkj ) − P k (G | D , αkj )

ˆ we can obtain By rearranging the value equation (22) for k = k,



ˆ

ˆ

vˆ i = (1 − δ) uki C , αkj +



u κi

 κ κ    ˆ ˆ  ˆ ˆ ˆ  αi , α j + δ W i G |α −k − δ P k B |C , αkj W i G |α −k − W i B |α −k .

κ =kˆ

From (24), we have



   κ κ κ  ˆ kˆ kˆ + δ W i G |α −k vˆ i  (1 − δ) u i C , α j + u i αi , α j κ =kˆ kˆ



− (1 − δ) P B |C , α



ˆ

ˆ

ˆ

ˆ

ˆ



ˆ

uki ( D , αkj ) − uki (C , αkj )



kˆ j

ˆ

(25)

.

ˆ

P k (G |C , αkj ) − P k (G | D , αkj )

If we apply the same operations for i = 1, 2, using



ˆ



ˆ

vˆ 1 + vˆ 2  W 1 G |α −k + W 2 G |α −k ,



2

     u κ1 α1κ , α2κ + u κ2 α1κ , α2κ ,

πκ 

κ ∈M C

κ ∈M C

we have

vˆ 1 + vˆ 2 

2

  kˆ

ui C , α

kˆ j



ˆ

−

ˆ

=

η

k j

ˆ

ˆ

P k ( B |C ,αkj )



ˆ

gk

kˆ

π −

kˆ

P ( B | D ,α j )

j =1

ˆ

ˆ

P k ( B |C ,αkj )



+

ˆ

kˆ

P k ( B | D ,α j )

i =1 2  ˆ

ˆ

ˆ

uki ( D , αkj ) − uki (C , αkj )



−1 

kˆ j

−1

 κ  π + g κ − bκ , 0 < 2

πκ +

κ ∈ M C \{kˆ }



− 1−η b

max



+2

kˆ

m 

kˆ

ˆ

P k ( B | D ,α j ) ˆ

k =1

κ ∈M \M C



1 ˆ

P k ( B |C ,αkj )

πk −



g1 l1

max

 κ  π + g κ − bκ , 0

κ ∈M \M C

1+



−1

+2 −1



πκ

κ ∈ M C \{kˆ }

  = 2V ∗ C M .

If kˆ ∈ K , the second inequality holds by Assumptions 3 (π k > g k − bk for all k) and 5. Assumption 5 guarantees that taking D with any probability in market kˆ never improves the likelihood ratio, and thus it never improves the value of collusion. If kˆ ∈ / K , the inequality holds by Assumption 6 because

πk −

2 kˆ

kˆ j

g1 l1



−1



= 2 πk − g

kˆ

π >η π −

ˆ

kˆ

P ( B |C ,α j )

−1



  > max π k + g k − bk , 0 for all k ∈ K ,

−1

  kˆ kˆ − 1 − ηj b 1 +

lk

kˆ

P k ( B | D ,α j ) kˆ

gk



1 ˆ

ˆ

ˆ

ˆ

P k ( B | D ,αkj ) P k ( B |C ,αkj )

. −1

Therefore, vˆ 1 + vˆ 2 < 2V ∗ (C M ) holds. This contradicts inequality (19).

2

The implication of Theorem 2 that we wish to emphasize is that, to improve the value of collusion (per market), the accumulation of information through multimarket contact is effective only when firms adopt trigger strategies. In contrast, by asymmetric strategies that transfer continuation payoffs between firms in some markets, firms can at most achieve the sum of their payoffs, π k + g k − bk , in each market, which can be obtained through single-market contact. That is,

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when firms alternate ( D , C ) and (C , D ) in some markets, accumulating information in those markets does not affect the value of collusion. Therefore, under Assumption 6, firms maximally enjoy the benefit of multimarket contact in the sense of improving the value of collusion when they trigger punishment based on linked information in all markets. Note that the type of private strategy equilibrium studied by Kandori and Obara (2006) does not work under Assumption 5. A private strategy specifies a current action conditional not only on the public history but also on the private history. When firms adopt a mixed private strategy, a combination of one’s own actions in the past and the history of the public signal may contain more information than just a history of the public signals about the opponent’s actions. However, taking noncooperative action with positive probability simply worsens the statistical detectability of deviations because of Assumption 5. Therefore, the combination of the actions and the signals does not contain more information under Assumption 5.11 One thing that is straightforward but may be worth mentioning is that we can just focus on each market with the most tempting deviation to check whether the optimal SSPPE is optimal for joint payoff maximization among any PPE in multimarket contact. That is, if randomization and asymmetric strategies in that market are not useful to increase the greatest sum of PPE payoffs in single market competition, then neither are they useful in multimarket contact. 4. Discussion The analysis in this paper is based on symmetry between firms. This is because the aim of this paper is to characterize optimal SSPPE payoffs cleanly by a version of the formula introduced by AMP. However, it is worthwhile to discuss an extension of the analysis to the case in which some firm heterogeneities are allowed. First, consider the case in which only the information structure and the deviation gain are asymmetric between firms. That is, P k (G | D , C ) = P k (G |C , D ) and uk1 ( D , C ) = uk2 (C , D ) for some k. The difference from the symmetric environment is that the incentive constraints of each firm cannot be identical. Moreover, the market with the most tempting deviation for a firm might be different from that for the opponent. However, firms receive the same payoffs on the equilibrium path in pure strategy SSPPE. Therefore, we can obtain the boundary of the limit set of pure strategy SSPPE payoffs in the same manner as in Section 3.1.12 By the above argument, we must ascertain which firm’s incentive constraint is binding. Now, suppose that firm i’s incentive constraint is binding. Firm i obtains the equilibrium payoff that can be induced by its value function and binding incentive constraint. Therefore, firm i receives the equilibrium payoff represented by the multimarket version of the formula that is also the equilibrium payoff for firm j = i. Then, consider the extension in which asymmetry on cooperative payoffs is incorporated into the above environment. That is, uk1 (C , C ) = uk2 (C , C ) for some k. Under this environment, we can obtain the value that is generated from cooperation in all markets and continuation payoffs on the line between (u 1 (C M , C M ), u 2 (C M , C M )) and (0, 0) as an equilibrium payoff under a sufficiently large discount factor by an optimal trigger strategy. The equilibrium payoff for firm i whose incentive constraint is binding is also represented by the formula. Firm j = i receives the equilibrium payoff that multiplies i’s equilibrium payoff by the ratio of u j (C M , C M ) to u i (C M , C M ). This is because the equilibrium payoff vector is on the line between (u 1 (C M , C M ), u 2 (C M , C M )) and (0, 0). However, many equilibria exist except the optimal trigger strategy equilibrium even if we restrict our attention to pure strategy SSPPE. Note that all of the symmetric action profiles do not necessarily give the payoffs that are located on the line between (u 1 (C M , C M ), u 2 (C M , C M )) and (0, 0) in the stage game. Therefore, in general, the set of continuation payoffs of pure strategy SSPPE might be two dimensional, and we must consider the linear programming problem in Section 3.1 for all action profiles and all directions of the boundary points of the limit set. It is burdensome to compute the optimal SSPPE payoffs in the general settings. Acknowledgments This paper is a substantial revision of Kobayashi and Ohta (2007). We would like to express our gratitude to Tadashi Sekiguchi, whose comments drastically improved this paper. We are also grateful to Michihiro Kandori, Hitoshi Matsushima, Ichiro Obara, Satoru Takahashi, and two anonymous referees and an advisory editor for their valuable comments and encouragement. This research was partly conducted when Ohta was a research fellow under the 21st Century COE Research Program at Kyoto University. Ohta thanks the program for its support. This research is supported in part by JSPS Grants-inAid for Young Scientists 19730145 (Kobayashi) and 19730174 (Ohta), and by the Zengin Foundation for Studies on Economics and Finance. All remaining errors are ours. References Abreu, D., Milgrom, P., Pearce, D., 1991. Information and timing in repeated partnerships. Econometrica 59, 1713–1733. Abreu, D., Pearce, D., Stacchetti, E., 1986. Optimal cartel equilibria with imperfect monitoring. J. Econ. Theory 58, 1041–1063.

11 12

Whether other types of private strategy equilibria improve the value of collusion is an open question. The argument in Section 3.2 would also work by modifying the assumptions in Section 3.2 appropriately for the environment.

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