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 under which they would 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 the business context, the use of explicit or implicit matching rules is widespread.

Examples include 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 price-

matching agreement, a seller agrees to match a competitor’s lower price. Firms may agree to use

matching-expenditure limits on comparative advertising, or to limit the extent of raiding of

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.

1 George Washington University, Department of Economics. 2 George Mason University, School of Law. 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. All remaining errors are ours. 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.

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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

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

of the 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

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. 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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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

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.

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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.

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 and the

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

will be α for both parties.9 Our matching rule resembles Sugden’s (1984) reciprocity principle.

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

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

leads to an increase in 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 a more extreme and general 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. This exacerbates the prisoner’s dilemma, leading to 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. Parties choose strategies under matching rules without engaging in

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.

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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, always improves upon the Nash level of

cooperation. In the following, 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 . When

, there is no cooperation. When

si si ∈[ , ]0 1

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

functions for the two parties are , where and

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

P s s as bs1 1 2 12

2( , ) = − + a b, > 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 . Asymm c

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

similarly for b and d .

etri

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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< 11

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=

9 10 Hence, symmetric parties mean that a and c are equal and b and d are equal. 11 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<

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

faced with a Prisoner’s Dilemma problem. This leads to the introduction of two further

assumptions, and 0 < <a b 0 < <c d . 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.

P P N1 11 1 0 0( , ) ( , )>

P P N2 21 1 0 0( , ) ( , )>

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:12

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

12 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

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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 = ) 13

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 is the high-cost cooperator, 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.14 Thus, party 1’s maximum agreeable cooperation level

13 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). 14 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.

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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

= .15

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

1. If a

b2

1 ≤ , then .1 2( , ) (1 ,1M Ms s = ) 16

2. If 12

<a

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

M M b bs sa a

= .17

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

alternative Nash equilibrium obtained in the absence of matching rules.18 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≤

.

15 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( )M Msπ > 0

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

17 The equilibrium payoffs for the two parties are: 2

1 ( ) 02 4

M b ba a

π = > and 2 2

( )( ) 02 4

M b b ad bca a

π−

= > .

18 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 > =

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Earlier when we look at individual behavior in a matching-rule regime, party 1 chooses

its strategy by balancing its marginal cost MC as1 2 1= with its private marginal benefit induced

by a matching-rule regime 1MMB = b

1

.19 Here, to find the social optimum, it is important to

compare the marginal cost of a strategy MC as1 2= with its social marginal benefit .

In this context, we can identify four alternative social optima.

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

From this list, 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, note that 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 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

19 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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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.

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.20 We now

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

In particular, three alternative situations are examined. First, we inquire whether there are

situations in which, in spite of the players’ asymmetries, the matching-rule equilibrium coincides

with the social optimum. Second, we ask whether it is possible for the matching-rule

equilibrium to lead to less than optimal levels of cooperation. Third, we inquire whether the

matching-rule equilibrium can 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 must hold. These conditions

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

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

12 =Ss a≥ max{ , }2 2c c

d a≥ max{ , }2 2

20 Recall that symmetric parties means that parameters a and c are equal and parameters b and d are equal.

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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.

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, since 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 must hold, 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. Indeed, in this case, the 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.

a c b d a= = <, 2

2.1.3 It is interesting to see that the levels of cooperation induced by matching rules

when asymmetric parties are involved will be socially optimal only under fairly restrictive

conditions.21 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.

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

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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.

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

rule regime will never be efficient. This is because in order 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 that marginal

benefit in a matching-rule regime b must equal social marginal benefit d. Further, for a

matching-rule equilibrium to be efficient, the social optimum must fall along the diagonal with

equal levels of cooperation. This implies that social marginal cost of cooperation for the two

parties must be the same since, from above, the social marginal benefits, b and d, are the same.22

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.

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

21 Note that the efficiency results induced by matching rules for the case of symmetric parties shown in Fon and Parisi do not necessarily hold for the asymmetric case. 22 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.

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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=≥=≥ . 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

′= = < . Collecting the necessary inequalities, we have

. Equivalently, the following inequalities must hold: .d a b c b≥ ≥ <2 2, , a2 cd a b≥ > ≥2 2 23

These inequalities further imply the necessary conditions: b d< and c a< .24

If , marginal benefit in a matching-rule regime for party 1 is less than marginal

cost at full cooperation, hence party 1 prefers less than full cooperation. However, if ,

social marginal benefit for party 1 is larger than marginal cost at full cooperation, and party 1

could produce some net social surplus by raising his level of cooperation. When these conditions

occur, private and social incentives towards cooperation may diverge, in spite of a binding

matching rule. Note that the above conditions imply b

b < 2a

d a≥ 2

d< , suggesting that social and private

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

value of his cooperation.

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 implies that a must hold. c>

23 Recall that the original assumption of b to generate the prisoner’s dilemma must also hold. Meanwhile, the maintained assumption that party 1 is the relative high cost-benefit cooperator is implied by this inequality.

a>bc a d<

24 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).

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2.2.2 Consider the case in which the social optimum requires equal levels of partial

cooperation effort for both parties. That is, assume that 12221 <===

cb

adss SS is true. The

above conditions imply that ab must hold. Combining this with the assumption that party

1 has a comparative disadvantage in cooperation, such that bc

cd=

a d< , leads to the following:

bc a d b abd

a d b d b d< ⇒ ⋅ < ⇒ < ⇒ <2 2 .

Meanwhile, and imply that cab cd= b d< a< holds. These are the familiar conditions

found in the previous subsection. The condition b d< means that party 1’s marginal benefit

under cooperation is lower than the social marginal benefit. As a result party 1 undertakes a level

of cooperation falling short of the social optimum. The condition c a< implies that party 2’s

marginal cost is lower than party 1’s marginal cost, which determines what party 2 does in a

matching-rule equilibrium. Hence party 2 chooses a level of cooperation that also 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.

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 comparison between 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.

MC as1 2= 1

1

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

marginal benefit to marginal cost at full cooperation

dMB S =1 MC as1 2=

bc2

, exceeds the matching-rule equilibrium

14

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

marginal cost at full cooperation ba2

, the social optim level of cooperation for party 2 exceeds

the level of cooperation in a matching-rule regime and c a

al

< must hold.25 The efficiency of

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

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

and c.

cooperation than socially optimal?

The previous Section considered conditions atching-rule equilibrium

onsider the

ore pu mes may induce more cooperation effort than is

socially

That is, we have

under which the m

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

zzling possibility that matching-rule regim

desirable. As before, we give separate treatments to cases of full cooperation and partial

cooperation.

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

cooperation. 112

Mb sa≥ = and 21

2Md s

c≥ = . Naturally, a social optimum could

also req

ss than full cooperation for one

parties are willing to cooperate at full level in equilibrium.

. For this to happen, we would need:

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

a social optimum may require le or both parties, even though

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

for both parties, but the matching-rule equilibrium would instead be one of full cooperation. We

show that this is impossible through proof by contradiction

121sda

and > =S 122> =s b

cS . This in turn would yield:

1 11 M Sb ds s b d≥ = > = ⇒ > and 2 21 M Sd bs s d b≥ = > = ⇒ > . 2 2a a 2 2c c

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.

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

15

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

party 1, 12a1> =s dS , but full cooperation effort from party 2, b

c2

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

sS1 2≥ = . Combining these

e following

must hold:

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

s s s b c a d b cc a c≥ = > = ≥ = ⇒ ≥ > ≥ .

This implies that parameters of the model must satisfy b a d c≥ > ≥2 2 . In turn, this further

implies tha

1 , 2 , 2M M S Sb d d bs a da≥ = ≥

t and . The condition 1 has an absolute

dvantage in capturing the benefits of mutual cooperation, although he faces a comparativ

isadvantage in cooperation. This condition further reveals that party 1’s private incentives to

mutual c

b d> a c> b d> suggests that party

a e

d

cooperate are too strong, since the private benefit obtained from ooperation exceeds the

social benefit of such cooperation. This leads party 1 to undertake a level of cooperation

exceeding the social optimum. The condition a c> means that party 2 has an absolute advantage

in the cost of cooperation.26

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.

Putting all the conditions together, we have:

1 2 1 21 ,Mb ds≥ = ≥ =1 , 1 , 1 2 , 2 , 2 , 2M S Sd bs s s b a d c d a c b≥ = > = ⇒ ≥ ≥ ≥ > .

t opposite of those found in the previous case.

The condition suggests that party 1 has an absolute disadvantage in capturing the

benefits of cooperation, while suggests that party 2 has an absolute disadvantage in the

becaus ty 1 st

ion f

2 2 2 2a c a c

Thus, the necessary condition d c b a≥ > ≥2 2 must hold. These conditions further imply that

b d< and a c< . These conditions are the exac

b d<

a c<

cost of providing cooperation. Here, party 2’s level of cooperation exceeds the social optimum

e par undertakes a choice of cooperation that does not fully take into account the co

of reciprocal cooperat aced by party 2 in a matching-rule regime. Party 2 is willing to

cooperate at party 1’s chosen level, given his comparative advantage in cost of cooperation, but

ad bc> does not necessarily require a c> .

26 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>

16

does so beyond the socially optimal level, given his absolute disadvantage in cost 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

ooperation:

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

1 2 12

M M bc s sa

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

e

parties. Consider these possibilities in turn.

Ms hold. These conditions imply the following:

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

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 1 1S Ms s< and 2

Ss < 2

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

a a< ⇔ = < = ⇒ < and 2 2 2 22 2

S M S Mb bs s s s a cc a

< ⇔ = < = ⇒ < .

But d b< and a c< imply bc< . This contradicts our assumption that party 1 has a a d

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

levels of partial cooperation induced y

imes

librium: and . The necessary conditions for such

occurre

cooperation efforts of both parties to fall below the b

matching-rule reg .

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

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

matching rule in equi 1 1S Ms s≥ 2 2

S Ms s≥

nce can easily be shown to be b 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 S M M Ss s s s≤ = ≤

or 2 2 1 1S M M Ss s s s≤ = ≤ . The necessary conditions for 2

M M Ss s

1 1 2 2

Ss s1 1 2≤ = ≤ to hold are d b≤ and

Sc a≤ , and the necessary conditions for 1s s s s2 2 1S M M= ≤ to hold would be and b d≤ a c≤ . ≤

17

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

resulting level of cooperation will never be higher than timal levels for the two

shooting effects m

both socially op

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 under ay

deed

in one

imension is possible in th

combinations of strategies. Only under

special

arties. We

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

ect to the additional

quire

in 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

d e partial cooperation case.

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

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 p

th

under which the aggregate payoffs for the parties are maximized, subj

re ment that the same level of cooperation efforts be undertaken by the parties. In this

Section, we first find the matching social optimum. Next we compare this matching social

optimum with the (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

18

The matching social optimum can be found by maximizing aggregate payoffs for the

parties subject to the constraint of equal levels of cooperation. Consider the following:

2 1 1 2 1 2max ( ) ( ) ( ) s.t. , 0 1 0 1P s ,s as bs cs d s s s s , s= − + + − + = ≤ ≤ ≤ ≤ .

his is

1

Possible matching social optima depend on the values of the parameters. 27

1 21 2 1 2s ,s

2 2

T equivalent to the following optimization problem:

max ( ) ( ) P(s ) a c s b d s s1 12

1 10 1= − + + + ≤ ≤s. t. . s

If b d+≥ 1, then

a c+2( )~ ~s s1 2 1= = .

If b da c+2( )+

< 1, then ~ ~( )a c2 +

s s b d1 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.

.1.1 Whenever the unconstrained social optimum leads to full cooperation,

1 1= , the matching social optimum is also characterized by full cooperation,

3

( , ) ( , )s sS S1 2

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

da

s bc

s d a bS S

21

21 21 2≥ = ≥ = ⇒ ≥ ≥and and . c2

This then implies that b da c

c aa c

++

++2

2 22( ) ( )

by mutual full cooperation, the matching social opti

≥ = 1. Hence, if the unconstrained social optimum is

characterized mum 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= < ).

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

19

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 d bS S1 2= <

as

cd a b c21

21 2 2= < ⇒ < <and and .

his implies that b da c

c aa c

++

<++

=2

2 22

1( ) ( )

. Hence ~ ~( )

s s b da c1 2 2

1= =++

<T . This indicates that the

matching social optimum also leads to partial cooperation effort.

cooperation levels

Further, in this case we show that it is not possible for the matching socially optimal ~ ~s s1 2

cooperation sS1 and sS

2 . This can be proved by contradic

= to be strictly less than both unconstrained socially optimal levels of

tion. Assume the contrary, so that

~s sS1 1< and ~s sS

2 2< . Then

~( )

s s1 1 2< ⇒ and

b da c

da

ab cdS

2++

< ⇒ < ~( )

s s b da c

bS2 2 2< ⇒

+c

cd ab2+

< ⇒

ic can show that it is also not

ossible that the level of cooperation

<

must both hold. Clearly this is impossible. Likewise, similar log

~ ~s s1 2=p required for a matching social optimum be strictly

greater than both cooperation

This leaves two possibilities. First, if

levels required for an unconstrained social optimum sS1 and sS

2 .

ab cd≠ , then the matching social optimum is

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

sS and cooperation efforts . Second, whenever 1 sS2 ab cd= , all socially optimal cooperation

efforts, constrained or unconstrained, are equal: ~ ~s1 2s s sS S1 2= = = .28 This result is quite intuitive

raint

q p l

matchi

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

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

Thus, we can conclude that if the unconstrained social optimum re uires artia

cooperation for both parties ( s sS S1 21 1< <, ), the ng social optimum also leads to partial

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

the matching social o timum n the other hand, if abp . O cd≠ , t

unconstrained socially optimal strategies for the parties.

he matching social optimum is the

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

28 Note that what we find here is consistent with what we found in subsection 2.2.2. In particular, earlier we show that if , then must hold. s sS S

1 2 1= < ab cd=

20

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

rium induced by

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

at the alterna

ume the contrary. Then the following

old:

We now compare the equilib a matching rule with the matching social

and then look tive 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. Ass

h

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

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

+≥ = > = ⇒ ≥ + > + ⇒ >

+% .

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

quilib

ing 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

atchin

levels. Thus, it could never happen that both parties overshoot the social optimum at the same

that and

e rium 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

accord

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

time.

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 1 2

1 2 1 2

1M Ms s= <

b dM Ms s s s= < =% % . We know that ~ ~s s1 2= equal either 1 or a c++2( )

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

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

M Ms s< = . This contradicts the assumption

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

21

the alte ernativ case in which ~ ~( )

s sa c1 2 2

= =+

. Since 1 1Msb d+ < , 1 2Ms

a= . We thus have the

following:

b

1 1 2( ) 2M b d bs s a d bc+

< ⇒ < ⇒ <% . 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 s s1 2M 1M then we have either = <

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

M Ms s s s= = =% % .29

to partial cooperation, either the quilibrium hing social optimum

or the matc

Therefore, whenever matching-rule equilibrium leads

matching-rule e is also the matc

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

atching social optimum.

4. onclusion

ces sug

al systems

creating and enforcing matching-rule r

m to be efficient.

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

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

ays the party with the higher cost-benefit ratio, or relatively less willing

coopera

combination of strategies is rendered unachievable by the matching rule. This leads to a tension

m

C

The conventional wisdom in the social scien gests that matching rules facilitate the

achievement of cooperative outcomes. Institutions and leg can foster cooperation

egimes. 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 equilibriu

b

alw dictated by

tor. 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. Asymmetries in the benefits and costs of

cooperation often require asymmetric levels of cooperation for a social optimum and such a

29 It is easy to see that 1 2 1 2

M Ms s s s= > =% % requires bc a d< while 1 2 1 2M Ms s s s= = =% % requires bc . a d=

22

between the social and private incentives for cooperation under a matching-rule regime.

Cooperation induced by matching rules would thus rarely lead to a global social maximum when

applied

-efficient outcomes. Our results show that matching rules can lead to

insuffic

ally optimal levels of cooperation

would

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

ient 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

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 soci

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.

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 if the attributes of the players were taken into account

to determine their respective obligations, parties would have incentives to conceal, or even

distort information relevant for such assessment. Players would want to appear to be high-cost

(or low-benefit) cooperators, as a way to incur lower obligations under matching rules.

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

23

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 leg

tional and legal rules

that ma

al 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 institu

y facilitate the achievement of optimal levels of cooperation when business entities are

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 preference. Consideration could also be given to rules of

asymmetric reciprocation under which heterogeneous parties are subject to scaled matching

rules.

24

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26