University of Minnesota Law School

Legal Studies Research Paper Series

Research Paper No. 07-24

Matching Rules

Vincy Fon Francesco Parisi

This paper can be downloaded without charge from the Social Sciences Research Network Electronic Paper Collection at:

http://ssrn.com/abstract=886120

GEORGE MASON UNIVERSITY SCHOOL OF LAW

MATCHING RULES

Vincy Fon Francesco Parisi

06-03

GEORGE MASON UNIVERSITY LAW AND ECONOMICS RESEARCH PAPER SERIES

This paper can be downloaded without charge from the Social Science Research Network at http://ssrn.com/abstract_id=886120

Vincy Fon1 – Francesco Parisi2

Matching Rules3

ABSTRACT: Institutions often utilize matching rules to facilitate the achievement of cooperative outcomes. Yet, in some situations the equilibrium induced by a matching rule may not be socially optimal. After presenting the case in which matching rules yield privately and socially optimal levels of cooperation, this paper identifies the conditions which instead generate inefficient cooperation. Two groups of cases are presented. In one group matching rules undershoot (i.e., the parties cooperate less than is socially optimal). In the other, more puzzling case, matching rules overshoot (i.e., the parties that interact under a matching constraint are induced to cooperate more than is socially optimal). This paper identifies the conditions for such occurrences. The paper then examines the ability of a matching rule to induce a socially optimal level of cooperation, where a social optimum requires equal levels of effort by the two parties, and identifies situations where matching rules fail to induce such an optimum. JEL Codes: K10, D70, C7, Z13 Keywords: Matching rules, Cooperation, Conditions for Social Optimum

In a business context, explicit and implicit matching rules are widespread. Examples

include patent-pooling and information-sharing between competing firms, price-matching

policies adopted by competitors to sustain prices, and other matching agreements among

business partners or in trade associations, such as takeover policies in multinational contexts,

codes of conduct, and most-favored-nation clauses. In a patent-pooling context, a firm that

unilaterally shares intellectual property or discloses information to a competitor faces a private

cost, generating an external benefit to the other firm. If the firms cooperate by sharing their

knowledge or patents, both receive a positive net gain. A sharing arrangement where competing

firms agree to grant licenses to their competitors in exchange for a matching privilege could

improve the position of both firms in the external marketplace. Similarly, in a price-matching

agreement, a seller agrees to match a competitor’s lower price. The maintenance of higher prices

may impose a private cost on the firm (in terms of forgone sale opportunities), generating a

benefit for the competing firm. If both firms refrain from lowering prices, they both obtain a net

1 George Washington University, Department of Economics. 2 University of Minnesota, School of Law & University of Bologna, Department of Economics. We thank Urs Schweizer for his insightful comments and Dan Milkove for his help and encouragement. We are also grateful to the anonymous referees for their thoughtful comments, and we thank Ian Beed for his research assistance. All remaining errors are ours.

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benefit. Price-matching pre-commitment strategies may thus serve a valuable function for the

firms to foster cooperative price-maintenance equilibria. Along similar lines, competing firms

may agree to use matching-expenditure limits on comparative advertising, or to limit the extent

of raiding executive talent from one another through a matching mechanism. Likewise, in a

most-favored-nation clause, one party agrees to govern its business relations with all others

according to the most favorable terms extended to any other business party.4 Although the goals

of these arrangements differ from one another, a common thread of these business settings

involves situations where the well-being of one party depends on the actions undertaken by

another party. Matching rules then operate as constraints, affecting the parties’ choices of action.

In recent law and economics literature, attention has also shifted towards matching rules

operating as an exogenous constraint on human behavior. In many legal situations, matching

rules do not reflect a preference for or spontaneous compliance with social norms of reciprocity.

Rather, the legal system creates an external matching constraint imposed on human behavior. In

particular, the role of legally-created matching rules has been examined in the context of

international and domestic law.5 Interesting applications and extensions are also found in

Hirshleifer (1983) and Arce (2001), showing how matching rules arise in weakest link

environments, and Sandler (1998), applying the weakest link paradigm to the foreign aid

problem.6 In a more theoretical context, Fon and Parisi (2003) considered the effect of

exogenous matching rules - similar in effect to the matching rules examined here - on the parties’

strategies in simultaneous Prisoner’s Dilemma games. In all such settings, matching rules operate

as a binding constraint, facilitating the achievement of cooperative outcomes in many strategic

settings.

In spite of several analytical similarities, our analysis uses the different concept of

“matching rule,” rather than reciprocity, since we treat matching rules as an exogenously

3 An earlier version of this paper was circulated under the title of “The Limits of Reciprocity for Social Cooperation” George Mason Law & Economics Research Paper No. 03-08. Available on-line at SSRN: http://ssrn.com/abstract=384589. 4 Explicit analysis of matching rules in the business context include, among others, Economides et al. (1997) on strategic commitment in interconnection pricing and Becht (2003) on takeovers in multinational settings. 5 Examples in the context of domestic law include Wax (2000) on welfare reform, and Kahan (2002) on community policing. Examples of matching rules and most-favored-nation clauses in public international law are examined by Parisi and Ghei (2003), Parisi and Sevcenko (2003) and Fon and Parisi (2005). 6 An illustration of the “weakest link” problem is provided by Hirshleifer (1983). He considers matching constraints imposed by structural environmental conditions, pointing out that the effectiveness of a dam depends on the weakest (or lowest) section of the protective dam provided by owners of properties adjacent to a river.

2

imposed constraint, rather than an internalized or spontaneously enforced norm of reciprocity or

reciprocal fairness.7 We undertake this approach to separate the issues of emergence and

enforcement of reciprocity from the analytically separate question of the effectiveness of

matching constraints for achieving socially optimal outcomes. In this paper, we thus study the

incentive effects of binding matching rules to facilitate cooperation, and examine the welfare

properties of matching-rule equilibria. This analysis leads to the identification of the merits and

limits of matching rules in inducing socially optimal levels of cooperation.

The paper is structured as follows. Section 1 introduces the concept of matching rules,

comparing the equilibria induced by matching rules to those obtainable in an unconstrained Nash

equilibrium. Section 2 compares the equilibrium induced by matching rules to the social

optimum. First, situations under which the matching-rule equilibrium is identical to the social

optimum are investigated. Next, the possibility that the matching-rule equilibrium induces less

than the optimal level of cooperation effort is studied, and conditions for such an occurrence are

identified. Third, the possibility that excessive levels of cooperation effort will be generated in a

matching-rule equilibrium is discussed. Section 3 defines the notion of social optimum in a

matching-rule regime, requiring the parties to provide the same levels of cooperation. The social

optimum is then compared to the equilibrium induced by matching rules. The analysis reveals

that, even when the social optimum requires the parties to provide equal levels of cooperation,

matching rules may fail to induce efficient levels of cooperation. Section 4 concludes the paper

by outlining cases of convergence and divergence between private and social optima under

matching rules.

1. Matching Rules in a Prisoner’s Dilemma Problem

In a simultaneous Prisoner’s Dilemma with continuous strategies, Fon and Parisi (2003)

examined the role of matching rules in cooperation problems. They considered a matching rule

(in their words, a weak reciprocity constraint), which allows the party who prefers a higher level

of cooperation to revert to the lesser amount of cooperation chosen by the other party. Matching

7 Starting from the seminal contributions of Hamburger (1973), Buchanan (1978) and Sugden (1984), in recent years the notion of reciprocity has gained increasing attention in the social science literature. Experimental and behavioral economists have provided evidence of human attitudes towards reciprocity (see, e.g. Hoffman, McCabe and Smith, 1998). Likewise, sociologists and anthropologists have studied the emergence of reciprocity as part of the social and cultural norms that govern human relationships.

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was treated as an exogenous constraint on the parties’ choice of strategy. The results show that

matching facilitates the achievement of cooperative outcomes in many strategic settings.

Following the framework set forth by Fon and Parisi and assuming the existence of a

matching rule in one-shot Prisoner’s Dilemma games, we examine the effects of matching rules

on parties’ levels of cooperation. Our model treats matching as a rule of the game, which

exogenously constrains the parties’ behavior. This formalization is motivated by a desire to

isolate the effects of matching constraints in cooperation problems from other incentives brought

about by utility profiles, fear of retaliation, and expectations of reciprocation. By treating

matching as a constraint, we consider the best-case scenario in which reciprocal behavior is

always guaranteed by the protocol of the game. Thus, players maximize payoffs and

independently choose their strategies, subject to a binding matching rule.

The motivating framework introduced by Arce and Sandler (2005) provides the context

for our study of matching rules. Unlike prior representations of Prisoner’s Dilemma problems,

Arce and Sandler distinguish between public and private benefits as well as public and private

costs, generating a taxonomy of four categories of Prisoner’s Dilemma games. Despite the

equivalence of their categories with respect to the players’ private incentives, their taxonomy

becomes highly relevant for policy purposes and for unveiling the strategic implications of

alternative Prisoner’s Dilemma problems. The payoff structure of the Prisoner’s Dilemma

problem utilized in this paper falls within their category of “altruism” where cooperative action

provides an external benefit to the other player at a private cost to the cooperative player, and

cooperative outcomes are socially desirable (Arce and Sandler, 2005 pp. 6-7). We extend their

framework to consider asymmetric players with continuous strategies and study the effect of

external matching rules on the players’ strategies and equilibrium outcomes.

For purposes of institutional design, the results of our analysis will help identify the

strengths and weaknesses of contractual or institutional arrangements creating binding matching

rules to foster cooperation within a given group or business organization. Similarly, for policy

purposes, this will help identify desirable legal and social instruments fostering efficient levels of

cooperation.

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1.1 Matching Rules

Building on the above premises, we assume the existence of a matching rule and consider

the effects of this rule on the parties’ levels of cooperation. If the parties’ induced levels of

cooperation under a matching rule yield different levels of cooperation for the two players, the

matching rule allows the lesser of the two amounts of cooperation to become the mutually

binding level of cooperation for both individuals. This corresponds to a weak form of the golden

rule, which binds each player’s strategy to that of his opponent (Parisi, 1998 and 2000).8 Thus,

for example, if one party’s desired level of cooperation under a matching rule equals α , the

other party’s desired level equals β , and βα < , then the matching-rule equilibrium cooperation

is α for both parties. Our matching rule resembles Sugden’s (1984) reciprocity principle.

Sugden’s principal focus is on individual contributions to a public good, where both contributors

and non-contributors benefit from an increase in public goods. Any effort put forth by the party

increases the provision of the public good and the welfare of the voluntary contributor, and a

positive level of supply may occur in spite of the free-riding incentives. Our paper considers s

different cooperation problem in which any effort put forth by a party generates cost but no

benefit to the party itself. Unlike the case of voluntary contributions to a public good, players

derive a benefit only from other players’ effort, not from their own. In our example of patent

pooling, a firm faces a cost but no benefit when allowing the competing firm to use its patent,

and benefits only from the other firm’s sharing of its patents. This exacerbates the Prisoner’s

Dilemma, leading to a zero level of cooperation in equilibrium. We then ask whether any

external matching rules would ameliorate these situations.

We should note that the main virtue of our matching rule is that it encourages the truthful

expression of preferences for both parties. Instruments that induce parties to reveal true

preferences and other private information are extremely valuable in business settings affected by

pervasive agency problems. Under matching rules, parties choose strategies without engaging in

preference falsification, since neither party has an incentive to withhold cooperation below the

privately optimal level of reciprocal cooperation. As a consequence, such matching-rule regimes

trigger a level of cooperation equal to the level desired by the least cooperative player. This

level of cooperation, as shown in Fon and Parisi (2003), always improves upon the Nash level of

8 In the different context of retaliatory norms, Parisi (2001) examined the historical transition from norms of strong retaliation to norms of weak retaliation in biblical times.

5

cooperation. We extend those findings to investigate the relationship between matching-rule

equilibria and socially optimal outcomes.

1.2 Matching Rules: The Model

We consider two parties, each choosing a cooperation effort , where s . When

, there is no cooperation. When

si i ∈[ , ]0 1

si = 0 si = 1, there is full cooperation. Assume that the payoff

functions for the two parties are , where a b and

where . Each party faces a cost to provide cooperation effort, as

P s s as bs1 1 2 12

2( , ) = − + , > 0 P s s cs d s2 1 2 22

1( , ) = − +

c d, > 0 ∂ ∂P si i < 0, and each

party derives a benefit from the cooperation effort of the other party, as ∂ ∂P si j > 0 . Asymmetric

payoffs for the two players are allowed, meaning that a and are not necessarily equal and

similarly for b and d .

c9

Without loss of generality, we assume that one player has a comparative advantage in

cooperation. Specifically, party 1 faces lower net benefits from matching cooperation. That is,

the ratio of benefit to cost at full cooperation for party 1 is lower than that for party 2:

cdab 22 < . In all cases involving asymmetric players we thus maintain the assumption that

.bc a d< 10

1.2.1 We first consider the Nash Equilibrium of this cooperation problem. Since

∂ ∂P s as1 1 12= − < 0 for all , becomes the dominant strategy for party 1. Likewise, the

assumption makes the dominant strategy for party 2. Thus, the Nash equilibrium

strategies are: ( , .

s2 s1 0=

c > 0 s2 0=

) ( , )s sN N1 2 0 0=

To make the problem more interesting, we concentrate on the case in which parties face a

Prisoner’s Dilemma problem. This leads to the introduction of two further assumptions, 0 < <a b

and . These assumptions imply that and . Both

parties would be better off with full cooperation than with no cooperation, as is the case under

the Nash equilibrium. This is to be expected in a classic Prisoner’s Dilemma scenario. In the

0 < <c d P P N1 11 1 0 0( , ) ( , )> P P N

2 21 1 0 0( , ) ( , )>

9 Hence, symmetric parties mean that a and c are equal and b and d are equal. 10 Strictly speaking, bc should have been assumed throughout the paper. However, excluding equality as a possibility often sharpens our understanding of the asymmetric cases examined in this paper. For this reason, we usually assume bc .

ad≤

a d<

6

context of our example, we consider the case in which firms are trapped in a Prisoner’s Dilemma

problem, withholding information and access to patents from one another, in spite of the

potential advantages of cooperation.

1.2.2 The equilibrium obtained in the presence of a binding matching rule can be found

next. In a matching-rule regime, each party knows that the other party will exert a matching

effort level, within the limits of a mutually agreeable level of cooperation. If one party chooses a

smaller level of cooperation, then this level becomes a de facto constraint for the other party,

since any unilateral level of effort which is not matched by the opponent would reduce the

party’s payoff. The matching payoff functions to parties 1 and 2 thus become:11

π1 1 212

1 1

22

2 1

( , )s sas bs s sas bs s s

=− + ≤− + >

⎧⎨⎩

ifif

2

2

1

1

and . π2 1 222

2 2

12

1 2

( , )s scs d s s scs d s s s

=− + ≤− + >

⎧⎨⎩

ifif

Unlike the case without matching constraints, where cooperation efforts increase costs

without generating direct benefits, in a matching-rule regime the parties’ own cooperation efforts

lead to benefits as well as costs. Since ba

as bss2 1

12

1= − +arg max and dc

cs d ss2 2

22

2= − +arg max ,

the maximum levels of agreeable cooperation for party 1 and party 2 are ba2

and dc2

respectively. Note that the maximum agreeable cooperation levels depend on how marginal cost

at full cooperation compares with marginal benefit under the matching rule.

Thus, when 12 2

≤ ≤ba

dc

, given the expectation of matching effort, the two parties’

desired levels of cooperation within the feasible region are ′ =s1 1 and ′ =s2 1 . Here, the parties

are willing to extend full cooperation in a matching-rule regime. Given such willingness, the

strategy chosen by each party matches the expected strategy of the other party. Thus full

cooperation becomes the matching-rule equilibrium strategy for both parties: .1 2( , ) (1,1M Ms s = )

12

11 Note that this matching payoff function

iπ is different from the unconditional payoff function . The matching payoff function incorporates the matching constraint while the payoff function does not.

iP

12 Here there are multiple mutually acceptable equilibria under a matching-rule regime. Our refinement criterion chooses the equilibrium with full cooperation because it is the only strict Nash equilibrium, matching the intuitive criterion of Schelling (1960).

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Next, when ba2

1< , the desired level of cooperation for party 1 is ′ =s ba1 2

. Thus, if party

2 chooses a level of cooperation less than ba2

, party 1 would want to match party 2. On the

other hand, if party 2 chooses a level of cooperation greater than ba2

, party 1 would not match

party 2 and would instead choose ba2

. Therefore, the reaction function of party 1 is given by

ss s

s

ba

ba

ba

12 2 2

2 2 2=

≤>

⎧⎨⎩

ifif

. Likewise, if dc2

1< , the desired level of cooperation for party 2 would be

′ =s dc2 2

, and his reaction function is ss s

s

dc

dc

dc

21 1 2

2 1 2=

≤>

⎧⎨⎩

ifif

.

Given the asymmetry, the desirable levels of cooperation ′s1 and may not coincide for

the two players. If

′s2

12

<a

b , the maximum agreeable cooperation level chosen by party 1 is less

than full. Given that party 1 has a lower benefit-cost ratio, his comparative disadvantage in

cooperation acquires relevance. The lower net benefits from mutual cooperation lead him to

prefer an interior level of cooperation below the level preferred by party 2: 21 22s

cd

abs ′=≤=′ .

In a strict Nash equilibrium the parties will exert effort corresponding to the higher level of

mutually acceptable cooperation.13 Thus, party 1’s maximum agreeable cooperation level

abs

21 =′ becomes the binding strategy for both parties, and the matching-rule equilibrium

strategies are 1 2( , ) ( , )2 2

M M b bs sa a

= .14

13 In fact, the portion between the origin and (b/2a, b/2a) on the 45 degree line in the (s1, s2) space represents all the equilibria. 14 It is easy to show that the equilibrium payoff for party 2 is 2 2 2

(2 )( ) 04

M M b ad bcsa

π−

= > . If d c2 1< , party 2

would have been happier if cds 22 =′ was the matching strategy chosen by both because 2 2 2 2( ) (M M M )s sπ π ′< .

Likewise, if 12 ≥cd , party 2 prefers the matching strategy 1. In spite of these facts, party 2 is still better off

under the matching-rule equilibrium with than under the Nash equilibrium where the payoff would have been 0.

2 2( ) 0M Msπ >

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Summarizing the two cases, we have the following matching-rule equilibria.

1. If a

b2

1 ≤ , then .1 2( , ) (1,1)M Ms s = 15

2. If 12

<a

b , then 1 2( , ) ( , )2 2

M M b bs sa a

= .16

In both cases the matching-rule equilibrium constitutes an improvement over the

alternative Nash equilibrium obtained in the absence of matching rules.17 In our patent-sharing

examples, if firms are subject to a matching constraint – such that their access to the other firm’s

patents is matched by the level of access they grant to the other firm – the equilibrium level of

cooperation and patent sharing between the firms will be higher than that obtainable in an

unconstrained Nash equilibrium.

In the following, we identify the socially optimal levels of cooperation for the parties and

later verify the extent to which the matching-rule equilibrium approaches the social optimum.

1.2.3 We consider the efficiency of the outcome induced by matching rules in light of

the Kaldor-Hicks criterion of welfare. According to this criterion of welfare, socially optimal

outcomes are those that maximize the aggregate payoff for the parties involved. In our 2-party

problem, Kaldor-Hicks efficient strategies are those that maximize the aggregate payoff for the 2

parties:

max

s ,s

S P (s ,s ) P(s ,s ) P (s ,s ) ( as bs ) ( cs ds ) s , s1 2

1 2 1 1 2 2 1 2 12

2 22

1 10 1 0= + = − + + − + ≤ ≤ ≤s.t. 2 1≤

1

.

When we examined individual behavior in a matching-rule regime, party 1 chose its

strategy by balancing its marginal cost MC as1 2= with its private marginal benefit induced by a

matching-rule regime 1MMB = b

.18 Here, find the social optimum by comparing the marginal

15 The equilibrium payoffs for the two parties are: 1 (1) 0M b aπ = − > and 2 (1) 0M d cπ = − > .

16 The equilibrium payoffs for the two parties are: 2

1 ( )2 4

M b ba a

π 0= > and 2 2

( )( ) 02 4

M b b ad bca a

π −= > .

17 From the previous two footnotes, we know that and . 1 1 1 2( , ) 0M N NP s sπ > = 2 2 1 2( , ) 0M N NP s sπ > =18 Note that social marginal cost of cooperation corresponds to the private marginal cost of cooperation. Hence we refer to both simply as marginal cost of cooperation.

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cost of a strategy with its social marginal benefit . In this context, we can

identify four alternative social optima.

MC as1 2= 1 dMB S =1

1. If 12

&12

<<c

ba

d , then 12

,12 21 <=<=

cbs

ads SS .

2. If c

ba

d2

12

≤< , then 1,12 21 =<= SS s

ads .

3. If a

dc

b2

12

≤< , then 12

,1 21 <==c

bss SS .

4. If c

ba

d2

1&2

1 ≤≤ , then . 1,1 21 == SS ss

We see that the socially optimal levels of cooperation depend on relative magnitudes of

social marginal benefit and marginal cost of cooperation, and that the social optimum may

require partial cooperation or full cooperation from either or both parties. In particular, the

social optimum does not necessarily require equal levels of cooperation by the two parties. Since

a matching-rule regime requires equal amounts of cooperation effort from both parties, we do not

expect the matching-rule equilibrium to be the same as the social optimum in general. Hence it is

important to ask whether it is possible to have insufficient cooperation or excessive cooperation

under a matching-rule equilibrium.

Since our main interest is in the extent to which a matching-rule regime can solve the

social cooperation problem, and a social optimum is likely to require unequal levels of

cooperation for asymmetric parties, we further focus on social optima that yield identical

cooperation levels. In order to appraise the efficiency of the matching-rule equilibrium, we

introduce the notion of “matching social optimum.” This represents the case in which the

aggregate payoff for the parties is maximized subject to equal levels of cooperation. After

introducing this concept, we verify the extent to which a matching rule can induce parties to

adopt a level of cooperation equal to the matching social optimum.

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2. Comparing the matching-rule equilibrium and the social optimum

The economic model studied in Fon and Parisi (2003) verified the general intuition that

binding matching rules provide a viable solution to the Prisoner’s Dilemma problem. In that

study, the outcome generated by matching rules was further shown to be both privately and

socially optimal in partial as well as full cooperation cases with symmetric players.19 We now

explore the extent to which this result holds in general for asymmetric players.

In particular, three alternative situations are examined. First, are there situations in which,

in spite of the players’ asymmetries, the matching-rule equilibrium coincides with the social

optimum? Second, is it possible for the matching-rule equilibrium to lead to less than optimal

levels of cooperation? Third, can the matching-rule equilibrium lead to too much cooperation,

where the parties are induced to undertake cooperation efforts in excess of the socially optimal

level?

2.1 When is the matching-rule equilibrium socially optimal?

In this subsection, we find the necessary conditions under which the matching-rule

equilibrium is identical to the social optimum, for the cases of full cooperation and partial

cooperation.

2.1.1 In the case of full cooperation, if the matching-rule equilibrium strategies 1

Ms ,

2Ms , and the social optimum strategies , all equal 1, the parameters of the model must

satisfy the following:

sS1 sS

2

12

≥a

b (as 1 1Ms = ), 12

≥c

d (as 2 1Ms = ), 12

≥a

d (as ), and 11 =Ss 12

≥c

b

(as ). In other words, b and d must hold. These conditions

reflect the fact that in order for the optimal strategies to lead to full cooperation in equilibrium,

both marginal benefits of cooperation efforts from the two parties must be large. More

specifically, for private optimality, marginal benefits of cooperation (given the expectation of

matching cooperation from the other party) should be at least as large as marginal cost at full

cooperation. Likewise, for social optimality, social marginal benefits must be at least as large as

marginal cost at full cooperation. In our example, cooperative equilibria with full sharing of

12 =Ss a≥ max{ , }2 2c ca≥ max{ , }2 2

11

patents between two competing firms will be socially optimal when the social marginal benefit

and the private marginal benefit (under cooperation) of a patent exceed the costs of providing

and sharing a patent license for both firms.

2.1.2 In the case of partial cooperation, two conditions must hold for the matching-rule

equilibrium and the social optimum to coincide. First, party 1’s socially optimal strategy must

equal its matching-rule equilibrium strategy. Since party 1 has a comparative disadvantage in

cooperation, its privately optimal cooperation level becomes the binding strategy under our

matching rule. Hence must hold. Second, the socially optimal strategies chosen by

the two parties must be equal, as the matching rule requires equal levels of cooperation in

equilibrium. Thus must hold. Translating these conditions to the parameters of the

model, we have:

1 1 1S Ms s= <

s sS1 = S

2

1 1 1 12 2

S M d bs s b da a

= < ⇒ = < ⇒ = < 2a and s s da

bc

a cS S1 2 2 2= ⇒ = ⇒ = .

Since and , we see that the conditions for convergence of the matching-

rule equilibrium with the social optimum are fairly restrictive for the case of partial cooperation.

Convergence of private and social incentives can only happen if the two parties are symmetric.

That is, we should expect a partial cooperation outcome to be socially optimal only if the two

parties face symmetric payoff functions. In our example, the matching-rule equilibria with partial

sharing of patents between two competing firms will be socially optimal only when the two firms

are identical.

a c= 2b d a= <

2.1.3 Levels of cooperation induced by matching rules with asymmetric parties are

socially optimal only under restrictive conditions.20 Except when da

bc2 2

= , the social optimum

will be characterized by unequal levels of cooperation between the parties, and thus rendered

unobtainable by a matching rule. Two important conclusions can be drawn from this section.

19 Recall that symmetric parties means that parameters a and c are equal and parameters b and d are equal. 20 Note that the efficiency results induced by matching rules for the case of symmetric parties shown in Fon and Parisi (2003) do not necessarily hold for the asymmetric case.

12

First, the matching-rule equilibrium may be socially optimal when taking place at full

cooperation. The intuition behind this result is that when social marginal benefits of cooperation

exceed marginal costs at full levels of cooperation, the socially optimal levels of cooperation are

also characterized by full cooperation, given the feasibility constraint. Likewise, when private

marginal benefits of cooperation for the parties under a matching-rule regime exceed marginal

costs at full levels of cooperation, the parties would happily extend cooperation beyond full

cooperation, if they had an option to do so. This implies that the differences between the

privately optimal levels of cooperation for the two parties are revealed only in the infeasible

region of more-than-full cooperation, and are thus hidden behind the parties’ visible equilibrium

at full cooperation. Put differently, the parties converge to a full level of cooperation, not because

they have identical preferences, but because such a corner solution gives them the highest

obtainable payoff in the region of feasible cooperation. This privately optimal corner solution

then happens to coincide with the socially optimal level of cooperation in the feasible region. In

the context of our example, full sharing of patents between two firms may be socially optimal

even when firms face asymmetric costs and benefits of patent sharing. Differences may

materialize in the future with respect to potential discoveries and patents, but full pooling is

optimal for both firms with respect to current patents.

Second, partial cooperation outcomes among heterogeneous players acting in a matching-

rule regime will never be efficient. To have an efficient equilibrium strategy under a matching-

rule regime, the privately and socially optimal levels of cooperation for party 1 must coincide,

since party 1’s strategy is the binding strategy in a matching-rule equilibrium. This coincidence

of private and social optima for party 1 requires the marginal benefit in a matching-rule regime b

to equal social marginal benefit d. Further, for a matching-rule equilibrium to be efficient, the

social optimum must require equal levels of cooperation. This implies that social marginal cost

of cooperation for the two parties must be the same since the social marginal benefits b and d are

the same.21 We conclude that in a matching-rule equilibrium, partial cooperation outcomes can

be efficient only if the two parties are homogeneous in that they face the same benefits and costs

of cooperation. In our example, a matching-rule regime will not lead to first-best social optimum

when firms are asymmetric in their costs and benefits of patent sharing.

21 Note that when looking at the private payoff function, the parameter d represents the marginal benefit under a matching rule for party 1. This value also represents the social marginal benefit of party 1’s cooperation.

13

2.2 The case of insufficient cooperation: When does the social optimum require more

cooperation effort than the matching-rule equilibrium?

Having considered the conditions for socially optimal cooperation among parties, we now

investigate the conditions under which a matching-rule regime may induce cooperation efforts

that fall short of the socially optimal levels. As before, we proceed by investigating cases of full

cooperation and partial cooperation in turn.

2.2.1 Consider the case in which the social optimum requires full cooperation from

both parties, so that SS sc

bsa

d21 1

2,1

2=≥=≥ . When social optimum requires full cooperation

from party 1, for example, the social marginal benefit of strategy from party 1, d, must exceed

the marginal cost at full cooperation for party 1, 2a. Thus, both social marginal benefits of effort

must be greater than the corresponding marginal cost at full cooperation in this case.

If the privately optimal strategy for party 1 was characterized by less than full

cooperation, the matching-rule equilibrium would also be characterized by partial cooperation,

since the strategy adopted by party 1 is binding under equilibrium. That is, 1 1 12

M bs sa

′= = < .

Party l prefers less than full cooperation because his marginal benefit in a matching-rule regime

b is less than marginal cost at full cooperation 2a. Collecting all the necessary inequalities, we

have . Equivalently, the following inequalities must hold:

.

d a b c b≥ ≥ <2 2, , a2

c2d a b≥ > ≥2 22 These inequalities further imply the necessary conditions: b and d< c a< .23

When , marginal cost at full cooperation for party 1 is less than the social

marginal benefit from his effort but greater than his private marginal benefit under cooperation.

Private incentive, even with the help of a binding matching rule, and social incentive diverge.

Party 1 could produce some net social surplus by raising his level of cooperation, but he has no

incentive to do so. Note that these conditions imply b

2d a≥ > b

d< , suggesting that social and private

incentives in a matching-rule regime diverge because party 1 does not fully internalize the social

22 Recall that the original assumption of to generate the Prisoner’s Dilemma must also hold. Meanwhile, the maintained assumption that party 1 has the comparative disadvantage in cooperation (bc ) is implied by this inequality.

b a>a d<

14

value of his cooperation (d is the social marginal benefit of effort provided by party 1).

Alternatively, party 1 obtains lower benefits from cooperation (d is the private marginal benefit

through cooperation enjoyed by party 2).

Further, in order for the socially optimal level of cooperation to exceed the level induced

by a matching rule, the ratio of party 2’s social marginal benefit to marginal cost at full

cooperation ( bc2

) must exceed party 1’s ratio of marginal benefit under cooperation to marginal

cost at full cooperation ( ba2

) in a matching-rule equilibrium. This translates to a . That is,

party 1 faces higher costs of cooperation as well.

c>

2.2.2 Consider the case in which the social optimum requires partial cooperation effort

from both parties. That is, 1 12

S dsa

= < and 2 12

S bsc

= < are true. When social optimum requires

less than full cooperation from party 1, for example, the marginal cost at full cooperation for

party 1, 2a, must exceed the social marginal benefit of strategy from party 1, d. Thus, both

social marginal benefits of efforts must be less than the corresponding marginal cost at full

cooperation.

If the matching-rule regime induces insufficient cooperation efforts, the binding strategy

adopted by party 1, 1Ms , must be less than the socially optimal efforts. That is,

1 12 2M Sb ds s

a a= < = and 1 22 2

M Sbs sa c

= < =b . These imply that b d< and c . These are the

familiar conditions found in the previous subsection. As party 1 undertakes a level of

cooperation falling short of the social optimum, its marginal benefit under cooperation must be

lower than the social marginal benefit and

a<

b d< . The condition c a< implies that party 2’s

marginal cost is lower than party 1’s marginal cost, which determines what party 1 does in a

matching-rule equilibrium. Hence party 2 chooses to match the effort of party 1, which falls

short of the social optimum. Both parties thus fail to reach the socially optimal level of

cooperation, in spite of the binding matching rule.

23 For a better understanding of the results, we continue to highlight relative magnitudes between marginal benefit parameters (b and d) and between marginal cost parameters (a and c).

15

2.2.3 In both cases of full and partial cooperation, the conditions b and d< c a<

assure that the matching-rule equilibrium leads to less cooperation effort than the social

optimum. These conditions indicate that the high-cost cooperator, while having a comparative

disadvantage in cooperation, would still produce some net social surplus if engaging in higher

levels of cooperation. In these cases, however, the existence of a matching rule is not sufficient

to induce him to do so. Note that private incentives towards cooperation lead party 1 to compare

the marginal benefit obtainable in a matching-rule regime, 1MMB b= , with marginal cost

. Social optimum instead requires a comparison of social marginal benefit

and marginal cost . Hence, whenever the marginal benefit b is less than the social

marginal benefit d, private and social incentives towards cooperation diverge, in spite of a

binding matching rule, and party 1 chooses a level of cooperation that is less than socially

optimal. In our example, a matching regime may not provide sufficient incentive for the firm

with a comparative disadvantage in cooperation (higher cost-benefit ratio) to pool enough of its

patents with the firm that has a comparative advantage (low cost-benefit ratio), even though it

may be socially optimal to do so. This is true when the firm with a comparative disadvantage

also faces a higher cost in allowing the use of its patents and a lower benefit from accessing the

other firm’s patents. This double disadvantage makes the firm with the comparative disadvantage

unwilling to pool as much intellectual property as is socially desirable.

MC as1 2= 1

1

dMB S =1

MC as1 2=

When the social optimal cooperation effort for party 2, determined by the ratio of social

marginal benefit to marginal cost at full cooperation bc2

, exceeds the matching-rule equilibrium

cooperation level, determined by the ratio of party 1’s marginal benefit under cooperation to his

marginal cost at full cooperation ba2

, the social optimal level of cooperation for party 2 exceeds

the level of cooperation in a matching-rule regime and c a< must hold.24 The efficiency of

party 2’s level of cooperation thus depends on a comparison of the parties’ costs of cooperation a

and c.

The necessary conditions for the undershooting scenarios indicate the existence of

unexploited benefits from cooperation. The asymmetries between the players imply that the

24 Note that in the cases considered in this subsection, given b d< , the comparative disadvantage condition

does not necessarily require . ad bc> a c>

16

benefit (positive externality) for the high-benefit player is greater than the matching benefit

obtainable by the low-benefit player. For a social optimum, the party facing lower costs from

cooperation should increase its level of cooperation to provide a benefit to the other party, but

has no incentive to raise its effort above the level determined by its less-cooperative counterpart.

Under our matching regime, however, outcomes are determined by the party who faces higher

costs and is less willing to cooperate, with no internalization of the forgone benefits of the other

party, and a resulting undersupply of cooperation.

2.3 The case of excessive cooperation: When do matching rules lead to more

cooperation than is socially optimal?

The previous Section considered conditions under which the matching-rule equilibrium

may induce cooperation efforts that fall short of socially optimal levels. We now consider the

more puzzling possibility that matching-rule regimes may induce more cooperation effort than is

socially desirable. As before, we treat cases of full cooperation and partial cooperation

separately.

2.3.1 Consider the case in which the matching-rule equilibrium leads to full

cooperation. That is, we have 112

Mb sa≥ = and 21

2Md s

c≥ = . Naturally, a social optimum could

also require full cooperation from both parties, as was considered in Section 2.1.1. Alternatively,

a social optimum may require less than full cooperation for one or both parties, even though

parties are willing to cooperate at full levels in equilibrium.

Consider first the case in which the socially optimal strategies require partial cooperation

for both parties, but the matching-rule equilibrium dictates full cooperation. We show that this is

impossible through proof by contradiction. For this to happen, we would need: 121> =s d

aS and

122> =s b

cS , which yields:

1 112 2

M Sb ds s ba a≥ = > = ⇒ > d and 2 21

2 2M Sd bs s d

c c≥ = > = ⇒ > b .

17

Clearly this is not possible. Hence, if the matching-rule equilibrium leads to full cooperation, the

social optimal strategies cannot require partial cooperation effort from both parties.

The second possibility is for the social optimum to require partial cooperation effort from

party 1, 121> =s d

aS , but full cooperation effort from party 2, b

csS

21 2≥ = . Combining these

conditions with the fact that matching-rule equilibrium leads to full cooperation, the following

must hold:

1 2 1 21 , 1 , 1 , 1 2 , 2 , 2 , 22 2 2 2

M M S Sb d d bs s s s b a d c a d ba c a c≥ = ≥ = > = ≥ = ⇒ ≥ ≥ > ≥ c

c2

.

This implies that parameters of the model must satisfy b a . In turn, this further

implies that b and . The condition b suggests that party 1 obtains greater benefits

from mutual cooperation, although he faces a comparative disadvantage in cooperation. This

condition further reveals that party 1’s private incentives to cooperate are too strong, since the

private benefit obtained from mutual cooperation exceeds the social benefit of such cooperation.

This leads party 1 to undertake a level of cooperation exceeding the social optimum. The

condition a means that party 2 faces lower cost of cooperation.

d≥ > ≥2

d> a c> d>

c> 25

In the third case, the matching-rule equilibrium yields full cooperation, but the social

optimum requires full cooperation effort from party 1 and partial cooperation effort from party 2.

Combining the conditions, we have:

1 2 1 21 , 1 , 1 , 1 2 , 2 , 2 , 22 2 2 2

M M S Sb d d bs s s s b a d c d aa c a c≥ = ≥ = ≥ = > = ⇒ ≥ ≥ ≥ >c b

a2

.

Thus, the necessary condition d c must hold. These conditions further imply that

and . These conditions are the exact opposite of those found in the previous case.

b≥ > ≥2

b d< a c<

The condition b suggests that party 1 captures lower benefits from cooperation, while

suggests that party 2 faces higher costs in providing cooperation. Here, party 2’s level of

cooperation exceeds the social optimum because party 1 undertakes a choice of cooperation that

does not fully take into account the cost of reciprocal cooperation faced by party 2 in a matching-

rule regime. Party 2 is willing to cooperate at party 1’s chosen level, given his comparative

d<

a c<

25 In this case, the additional condition is necessary in order to preserve the assumption that party 1 has a comparative disadvantage in cooperation.

a c>

18

advantage in cost of cooperation, but does so beyond the socially optimal level, given his higher

costs of cooperation.

To summarize, given a matching-rule equilibrium with full cooperation, no social

optimum can be found which requires strictly less cooperation effort by both parties. It is,

however, possible for the matching-rule equilibrium to “overshoot” in one dimension. Namely, a

social optimum may require less than full cooperation from one party, even when the matching-

rule equilibrium is characterized by full cooperation for both parties.

2.3.2 Consider now the case in which the matching-rule equilibrium yields partial

cooperation: 1 2 12

M M bs sa

= = < . In this case, excessive cooperation implies that matching-rule

regimes induce levels of cooperation exceeding the socially optimal levels for one or both

parties. Consider these possibilities in turn.

In the first case, socially optimal strategies require lower levels of partial cooperation

than those induced by a matching rule for both parties. We prove that this is not possible.

Assume otherwise, so that and hold. These conditions imply the following: 1 1Ss s< M M

2 2Ss s<

1 1 1 12 2S M S Md bs s s s d

a a< ⇔ = < = ⇒ < b and 2 2 2 22 2

S M S Mb bs s s s ac a

c< ⇔ = < = ⇒ < .

But and imply . This contradicts our assumption that party 1 has a

comparative disadvantage in cooperation. Therefore it is not possible for the socially optimal

cooperation efforts of both parties to fall below the levels of partial cooperation induced by

matching-rule regimes.

d b< a c< a d bc<

An almost identical proof would show that, instead, it is possible for socially optimal

cooperation efforts by both parties to be greater than or equal to the cooperation effort induced

by a matching rule in equilibrium: and . The necessary conditions for such

occurrence can easily be shown to be b

1 1Ss s≥ M M

2 2Ss s≥

d≤ and c a≤ . This case is in fact touched upon in

Section 2.2.2. Lastly, it is interesting to point out the possibilities of having

or . The necessary conditions for

1 1 2S M Ms s s s≤ = ≤ 2

S

1S

2S

2 2 1S M Ms s s s≤ = ≤ 1 1 2

S M Ms s s s≤ = ≤ to hold are d b≤ and

, and the necessary conditions for c a≤ 2 2 1S M Ms s s s1

S≤ = ≤ to hold are b d≤ and . a c≤

19

To conclude, when the matching-rule equilibrium leads to partial cooperation efforts, the

resulting level of cooperation will never be higher than both socially optimal levels for the two

parties. A matching rule may lead to too little cooperation by one party and too much

cooperation by the other. This combination of overshooting and undershooting effects may

indeed be expected in cases of asymmetric parties acting under a binding matching rule.

2.3.3 Given a matching-rule equilibrium with full cooperation, no social optimum can

be found which requires strictly less cooperation effort by both parties. Likewise, when the

matching-rule equilibrium leads to partial cooperation efforts, the resulting level of cooperation

is never higher than the socially optimal level for both parties. However, overshooting in one

dimension is possible in the partial cooperation case.

In our patent-sharing context, a matching-rule regime will never induce both firms to

share more than socially optimal. However, the firm who obtains greater benefits from sharing

patents and faces a higher cost in providing them may be induced to share more than is socially

optimal. This is true because differences in parties’ benefits and costs of cooperation efforts

often lead to asymmetric optimal levels of cooperation in a social optimum. When this happens,

the matching-rule equilibrium cannot easily be, and perhaps should not be, compared to the

social optimum, since a comparison involves symmetric versus asymmetric combinations of

strategies. Only under special circumstances would the social optimum lead to identical

cooperation efforts. For this reason, in Section 3 we consider a different concept of social

optimum which focuses on socially optimal levels of cooperation within the subset of equal

levels of cooperation.

3. The matching-rule equilibrium and the matching social optimum

The above analysis revealed the difficulties in evaluating the efficiency of matching-rule

equilibrium where a social optimum leads to unequal levels of cooperation by the parties. We

thus introduce the notion of “matching social optimum.” This concept describes the situation

under which the aggregate payoffs for the parties are maximized, subject to the additional

requirement that the parties undertake the same level of cooperation efforts. In this Section, we

first find the matching social optimum. Next we compare this matching social optimum with the

20

(unconstrained) social optimum discussed in previous sections. Lastly, we compare the

matching-rule equilibrium and the matching social optimum.

3.1 The matching social optimum

The matching social optimum is found by maximizing aggregate payoffs for the parties

subject to the constraint of equal levels of cooperation:

1 2

2 21 2 1 2 2 1 1 2 1 2max ( ) ( ) ( ) s.t. , 0 1 0 1

s ,sP s ,s as bs cs d s s s s , s= − + + − + = ≤ ≤ ≤ ≤ .

This is equivalent to the following optimization problem: . max ( ) ( )

s P(s ) a c s b d s s

11 1

21 10 1= − + + + ≤ ≤s. t.

Possible matching social optima depend on the values of the parameters. 26

If b da c++

≥2

1( )

, then ~ ~s s1 2 1= = .

If b da c++

<2

1( )

, then ~ ~( )

s s b da c1 2 2

1= =++

< .

These two possibilities describe the alternative cases of full and partial cooperation. We now

consider the relation between unconstrained social optimum and matching social optimum.

3.1.1 Whenever the unconstrained social optimum leads to full cooperation,

, the matching social optimum is also characterized by full cooperation, ( , ) ( , )s sS S1 2 1 1=

(~ , ~ ) ( , )s s1 2 1 1= . To see this, note that implies the following: s sS S1 2 1= =

da

s bc

s d a bS S

21

21 21 2≥ = ≥ = ⇒ ≥ ≥and and c2

.

26 We use a ~ above the variables and the functions to denote the matching socially optimal strategies and outcomes.

21

This then implies that b da c

c aa c

++

≥++

=2

2 22

1( ) ( )

. Hence, if the unconstrained social optimum is

characterized by mutual full cooperation, the matching social optimum also requires full

cooperation: ~ ~s s1 2 1= = .

3.1.2 Likewise, whenever the social optimum leads to partial cooperation from both

parties ( ), the matching social optimum also leads to partial cooperation (s sS S1 21< <, 1 ~ ~s s1 2 1= < ).

To see this, consider the case in which unconstrained social optimum is characterized by partial

cooperation efforts for both parties. We have the following:

s da

s bc

d a bS S1 22

12

1 2= < = < ⇒ < <and and c2 .

This implies that b da c

c aa c

++

<++

=2

2 22

1( ) ( )

. Hence ~ ~( )

s s b da c1 2 2

1= =++

< . This indicates that the

matching social optimum also leads to partial cooperation effort.

Further, in this case we show that it is not possible for the matching socially optimal

cooperation levels ~ ~s s1 = 2 to be strictly less than both unconstrained socially optimal levels of

cooperation and . This can be proved by contradiction. Assume the contrary, so that sS1 sS

2

~s sS1 1< and ~s sS

2 < 2 . Then

~( )

s s b da c

da

ab cdS1 1 2 2< ⇒

++

< ⇒ < and ~( )

s s b da c

bc

cd abS2 2 2 2< ⇒

++

< ⇒ <

must both hold. Clearly this is impossible. Likewise, similar logic can show that it is also not

possible that the level of cooperation ~ ~s s1 2= required for a matching social optimum be strictly

greater than both cooperation levels required for an unconstrained social optimum and . sS1 sS

2

This leaves two possibilities. First, if ab cd≠ , then the matching social optimum is

characterized by cooperation efforts ~ ~s s1 2= that lie between the unconstrained socially optimal

cooperation efforts and . Second, whenever sS1 sS

2 ab cd= , all socially optimal cooperation

efforts, constrained or unconstrained, are equal: ~ ~s s s sS1 2 1= = = S

2

.27 This result is quite intuitive

27 Note that what we find here is consistent with our result in subsection 2.2.2. In particular, earlier we show that if

, then must hold. s sS S1 2 1= < ab cd=

22

since in this case the unconstrained social optimum is already characterized by symmetric

strategies. This renders the added constraint immaterial for finding a matching social optimum.

Thus, we can conclude that if the unconstrained social optimum requires partial

cooperation for both parties ( ), the matching social optimum also leads to partial

cooperation (

s sS S1 21< <, 1

~ ~s s1 2 1= < ). Whenever ab cd= , the unconstrained social optimum coincides with

the matching social optimum. On the other hand, if ab cd≠ , the matching social optimum is the

result of a compromise and is characterized by cooperation levels that lie between the two

unconstrained socially optimal strategies for the parties.

3.2 Comparing the matching-rule equilibrium and the matching social optimum

We now compare the equilibrium induced by a matching rule with the matching social

optimum. As before, we start from the matching-rule equilibrium that leads to full cooperation

and then look at the alternative case of partial cooperation.

3.2.1 When a matching-rule equilibrium leads to full cooperation, such equilibrium

always coincides with the matching social optimum. Assume the contrary. Then the following

hold:

1 11 and 1 2 and 2( ) 22 2( )

Mb b ds s b a a c b da a c

c d+≥ = > = ⇒ ≥ + > + ⇒ >

+% .

This implies that 12

>dc

But the assumption that party 1 has the comparative disadvantage

means that dc

ba2 2

> , thus 12

>ba

. This contradicts the assumption that the matching-rule

equilibrium leads to full cooperation in the first place. We conclude that full cooperation under

matching-rule regimes will be observed only if full cooperation is also socially efficient

according to our criterion of optimality. When the matching social optimum requires partial

levels of cooperation, parties never reach full cooperation in equilibrium.

This result is the analogue of the previous result according to which the parties’ levels of

cooperation in a matching-rule regime could never simultaneously exceed the socially optimal

levels. Thus, both parties can never overshoot the social optimum at the same time.

23

3.2.2 Along similar lines, it will be shown that when the matching-rule equilibrium

leads to partial cooperation, such a level of cooperation never exceeds the level required for a

matching social optimum. Assume the contrary so that the level of cooperation induced by a

matching rule is greater than the matching social optimum. That is, assume that and 1 2 1M Ms s= <

1 2 1 2M Ms s s s= < =% % . We know that ~ ~s s1 2= equal either 1 or b d

a c++2( )

. If ~ ~s s1 2 1= = , then the

second assumption 1 2 1 2M Ms s s s= < =% % implies that 11 2

M Ms s< = . This contradicts the assumption

that the matching-rule equilibrium requires partial cooperation in the first place. Consider next

the alternative case in which ~ ~( )

s s b da c1 2 2

= =++

. Since 1 1Ms < , 1 2M bs

a= . We thus have the

following:

1 1 2( ) 2M b d bs s a d b c

a c a+

< ⇒ < ⇒ <+

% .

This last inequality contradicts the assumption bc a d≤ according to which party 1 has a

comparative disadvantage in cooperation. That is, if 1 2 1M Ms s= < then we have either

1 2 1 2M Ms s s s= > =% % or 1 2 1 2

M Ms s s s= = =% % .28 Therefore, whenever matching-rule equilibrium leads

to partial cooperation, either the matching-rule equilibrium is also the matching social optimum

or the matching-rule equilibrium leads to a lower level of cooperation than is required by the

matching social optimum.

4. Conclusion

The conventional wisdom in the social sciences suggests that matching rules facilitate the

achievement of cooperative outcomes. Institutions and legal systems can foster cooperation

creating and enforcing matching-rule regimes. In this paper we considered the limits of matching

rules in fostering cooperative outcomes. Matching rules do not always induce socially optimal

outcomes. In the case of asymmetric players, several conditions need to be satisfied in order for

the matching-rule equilibrium to be efficient.

24

With asymmetric players, the privately optimal levels of cooperation likely differ

between the two parties. Equilibrium level of cooperation under our matching-rule regime is

always dictated by the party with the higher cost-benefit ratio, or relatively less willing

cooperator. Thus, in our example of matching patent licensing between competing firms, the firm

that would obtain lower net benefits from the sharing agreement will share less knowledge with

the competing firm. Further, matching-rule equilibria are always constrained along the principal

diagonal of the game, but social optima may require unequal levels of cooperation for the two

players in response to differences in their benefit-cost ratios. In our example, aggregate payoffs

may be maximized when firms pool unequal number of patents. Asymmetries in the benefits and

costs of cooperation would require asymmetric levels of cooperation and knowledge sharing for

a social optimum, but such a combination of strategies is rendered unachievable by the matching

rule. This leads to a tension between the social and private incentives for cooperation under a

matching-rule regime. Cooperation induced by matching rules would rarely lead to a global

social maximum when applied to heterogeneous players. In this paper, we have shown the

conditions under which a matching-rule regime may lead to too little, or, interestingly, too much

cooperation compared to the social optimum.

In order to facilitate the assessment of the efficiency of matching rules when the

unconstrained social optimum necessitates asymmetric combinations of strategies, we introduced

the concept of matching social optimum. This allowed us to appraise the relative efficiency of

the matching-rule equilibrium in comparison with other reciprocal combinations of strategies.

Here, similar to the case of unconstrained social optimum, the matching-rule equilibrium never

exceeds the matching socially optimal levels of cooperation.

Unlike Sugden (1984), we evaluate the outcome induced by a matching constraint in

terms of an ideal first-best Kaldor-Hicks efficient outcome, rather than a Pareto efficient

outcome. This is a more demanding test, as the set of Kaldor-Hicks efficient outcomes is a subset

of the set of Pareto-efficient outcomes. Our results show that matching rules can lead to

insufficient levels of cooperation, as well as excessive levels of cooperation, when asymmetric

parties are involved. In situations of asymmetry between the parties, matching rules may be

unable to generate efficient outcomes. Whenever the matching-rule equilibrium and the socially

optimal equilibrium do not coincide, the matching-rule equilibrium leaves some unexploited

1 2 1 2

M Ms s s s= > =% % bc a d 1 2 1 2M Ms s s s= = =% % a d= requires bc . requires < while28 It is easy to see that

25

surplus for the parties: a social loss that is likely to increase with an increase in the asymmetries

between the players. In these situations, a move to the socially optimal levels of cooperation

would increase the aggregate payoffs for the parties. The gainers could fully compensate the

losers for the additional cost of cooperation, yet still capture some of the unexploited surplus. In

our example, the matching-rule equilibrium could be improved upon by allowing low-cost firms

to provide a larger number of patent licenses to high-cost firms in exchange for a payment.

Our efficiency metric “matching social optimum” hinges on the fact that a matching rule

requires equal efforts even when asymmetric agents are involved. In real life, these matching

rules are often necessitated by the fact that incorporating attributes of the players to determine

their respective obligations would give parties incentives to conceal or distort relevant

information. Players would want to appear to be high-cost (or low-benefit) cooperators, as a way

to incur lower obligations under matching rules. A firm may be tempted to claim that its patents

are more costly to share than those of its competitor (or that the benefit of using the other firm’s

patents is low) in order to obtain more favorable terms of exchange.

Whenever the matching-rule equilibrium generates aggregate payoffs that are

substantially lower than those obtainable with asymmetric obligations, the parties would have

strong incentives to opt out of the matching-rule regime and enter into contracts with asymmetric

obligations and possible side payments. This obviously poses a critical policy or organizational

dilemma: when the legal system or the relevant institution allows parties to opt out from the

matching-rule regime, the stability of the matching-rule equilibrium may be undermined.

Our results unveil the strengths and limits of matching rules in inducing optimal

cooperation among heterogeneous players. Future applications should investigate the relevance

of these features of matching-rule regimes in specific business contexts, where matching rules

govern relationships among highly heterogeneous parties. Different mechanisms of cooperation,

such as explicit trading and enforceable contracting, could yield better results than binding

matching rules, allowing the parties to undertake asymmetric obligations and converge towards

global maxima. These considerations are also in line with the findings of evolutionary socio-

biology, showing that matching rules and reciprocity norms tend to emerge in close-knit

environments with homogeneous players, but do not thrive in highly heterogeneous groups.

Future extensions should build on these results to examine specific institutional and legal rules

that may facilitate the achievement of optimal levels of cooperation when business entities are

26

known to be heterogeneous. Further, it may be desirable to examine matching-rule regimes

through different mechanism designs to investigate the extent to which matching rules may

induce parties to reveal their true preferences. Consideration could also be given to rules of

asymmetric reciprocation under which heterogeneous parties are subject to scaled matching

rules.

27

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29