ATCHING ULES Vincy Fon Francesco Parisi€¦ · Vincy Fon1 – Francesco Parisi2 Matching Rules3...
Transcript of ATCHING ULES Vincy Fon Francesco Parisi€¦ · Vincy Fon1 – Francesco Parisi2 Matching Rules3...
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.
1
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.
2
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.
3
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.
4
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
10
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<
5
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
6
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.
7
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 > =
8
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.
9
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.
10
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
11
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.
12
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).
13
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
References Arce, D. (2001), “Leadership and the Aggregation of International Collective Action,” Oxford
Economic Papers, 53: 114-37. Becht, M. (2003), “Reciprocity in Takeovers,” ECGI Working Paper N 14/2003. Available at:
http://ssrn.com/abstract=463003. Buchanan, J.M. (1978), “Markets, State, and the Extent of Morals,” American Economic Review
68: 364-368. Economides, N, Lopomo, G. and Woroch, G. A. (2005), "Strategic Commitments and the
Principle of Reciprocity in Interconnection Pricing." Available at: http://ssrn.com/abstract=1736.
Fon, V. and F. Parisi (2003), “Reciprocity-Induced Cooperation,” Journal of Institutional and
Theoretical Economics, 159: 1-17. Fon, V. and F. Parisi (2005), “Revenge and Retaliation” in F. Parisi and V.L. Smith (eds.), The
Law and Economics of Irrational Behavior (Stanford University Press) 141-168. Hamburger, H. (1973), “N-Person Prisoner’s Dilemmas,” Journal of Mathematical Sociology 2:
27-48. Hirshleifer, J. (1983), “From Weakest Link to Best Shot: The Voluntary Provision of Public
Goods,” Public Choice, 43: 371-86.
Hoffman, E., K.A. McCabe, and V.L. Smith (1998), “Behavioral Foundations of Reciprocity: Experimental Economics and Evolutionary Psychology,” Economic Inquiry, 36:3 (July), 335-352.
Kahan, D.M. (2002), “Reciprocity, Collective Action, and Community Policing,” 90 California
Law Review 1513-1539.
Parisi, F. (1998), “Customary Law,” pp. 572-578 in: The New Palgrave Dictionary of Economics and the Law, MacMillan Press: London.
Parisi, F. (2000), “The Cost of the Game: A Taxonomy of Social Interactions,” European
Journal of Law and Economics, 9: 99-114. Parisi, F. (2001), “The Genesis of Liability in Ancient Law,” 3 American Law and Economics
Review, 3: 82. Parisi, F. and N. Ghei (2003), “The Role of Reciprocity in International Law,” Cornell
International Law Journal, 36: 93-123.
25
Parisi, F. and C. Sevcenko (2003), “Treaty Reservations and the Economics of Article 21 of the
Vienna Convention,” Berkeley Journal of International Law 21: 100-126. Sandler, T. (1998), “Global and Regional Public Goods: A Prognosis for Collective Action,”
Fiscal Studies, 19: 221-47.
Schelling, T. (1960), The Strategy of Conflict, Harvard University Press. Sugden, R. (1984), “Reciprocity: the Supply of Public Goods through Voluntary Contributions,”
The Economic Journal, 94: 772-787. Wax, A. (2000), “Rethinking Welfare Rights: Reciprocity Norms, Reactive Attitudes, and the
Political Economy of Welfare Reform,” Law and Contemporary Problems, 63: 257-297.
26