We analyze two-stage games where players may make binding offers of schemes for side payment acceptance (or rejection) as well as those for side payments before choosing actions. We find that any set of efficient actions maximizing the total payoff is played on an equilibrium path of the two-stage game when such bilateral contracts on side payments are interdependent.

Akira Yamada

English

Research Papers in Economics

Efficient equilibrium side contracts 
Akira Yamada
Graduate School of Economics, Hitotsubashi University
Abstract
We analyze two−stage games where players may make binding offers of schemes for side
payment acceptance (or rejection) as well as those for side payments before choosing actions.
We find that any set of efficient actions maximizing the total payoff is played on an
equilibrium path of the two−stage game when such bilateral contracts on side payments are
interdependent.
The content of this note was presented at the symposium on mathematical economics sponsored by the Research Institute for
Mathematical Sciences, Kyoto University, which was held from November 29 to December 1, 2002. I extend my cordial thanks
to Takehiko Yamato and Shinsuke Nakamura for their helpful comments and suggestions on my research for the note. All
remaining errors are mine. Moreover, I am thankful to the PINE Foundation for their financial support.
Citation: Yamada, Akira, (2003) "Efficient equilibrium side contracts." Economics Bulletin, Vol. 3, No. 6 pp. 1−7
Submitted: February 28, 2003.  Accepted: April 7, 2003.
URL: http://www.economicsbulletin.com/2003/volume3/EB−03C70004A.pdf1. Introduction
Coase (1960) put forth an idea that if property rights are well-deﬁned, and
bargaining is costless, then rational agents playing a game with externalities
should contract to come to an eﬃcient point. Coase (1960) was not explicit
about the type of agreements between agents that are necessary as a form
of bargaining to reach eﬃciency, but the idea has been widely accepted by
economists.
Contrary to the widespread belief in the idea, Jackson and Wilkie (2000)
pointed out that side contracting does not always lead to eﬃciency even
when there are no transactions costs, complete information, and binding
contracts. They studied games where agents may make binding oﬀers of
strategy-contingent side payments before choosing actions, and found that
if there are only two agents, the agents are not always able to come to an
agreement that supports an eﬃcient strategy proﬁle as an equilibrium point
of the game. What kind of contracts will agents need to reach eﬃciency
generally?
According to Jackson and Wilkie (2000), if there are three or more players,
each eﬃcient strategy proﬁle is played on an equilibrium path in the game
with side payments. However, Jackson and Wilkie (2000) only focused on
voluntarily oﬀered side payments and assumed that such side payments would
always be accepted by transferees. This assumption might be thought of
as arbitrary since voluntarily oﬀered side payments could be invalidated by
spontaneous rejection to receive them. Moreover, the results of Jackson and
Wilkie (2000) depended upon another assumption as well that there is no
budget constraint with players’ transfer. Thus the question proposed above
seems to remain unanswered. What kind of bilateral contracts would lead
to eﬃciency even when agents face budget constraint with their transfer, no
matter what number of players there are? This is the question we address in
this paper.
W ea r et oa n a l y z et w o - s t a g eg a m e sw h e r ep l a y e r sm a ym a k eb i n d i n go ﬀers
of schemes for side payment acceptance (or rejection) as well as those for side
payments before choosing actions. A side payment from a player, say 1,t o
another, say 2, is implemented if and only if 1 oﬀers the payment and 2
accepts it. If 2 rejects, then 1’s oﬀer is not in eﬀect, and the payoﬀ for the
transfer remains with 1. W ew i l ls e et h a te v e r ye ﬃcient strategy proﬁle is
played on an equilibrium path of the two-stage game, no matter what number
of players there are, when the bilateral side contracts (transfer and receipt
1schemes) are somehow interdependent. Moreover, we will reach a similar
result even when equilibrium contracts are required to meet agents’ budget
constraint with their transfer.
In what follows we present the model in Section 2 and our analysis in
Section 3. Our concluding remarks appear in Section 4.
2. The Model
We consider two-stage games played as follows.
Stage 1: Each player announces a transfer function proﬁle (transfer scheme)
and a receipt function proﬁle (transfer acceptance/rejection scheme), each of
which is assumed to be binding.
Stage 2: Each player chooses an action.
2.1 The Underlying Game
T h ep l a y e r sa r eg i v e nb yas e tN = {1,...,n}.A p l a y e r i’s ﬁnite pure
strategy space in the second stage game is denoted by Xi,w i t hX = ×iXi.
Let ∆(Xi) denote the set of mixed strategies for i,a n dl e t∆ = ×i∆(Xi).
We denote by xi, x, µi,a n dµ generic elements of Xi, X, ∆(Xi),a n d∆
respectively. For simplicity, we sometimes use xi and x to denote µi and
µ respectively that place probability one on xi and x.A p l a y e r i’s payoﬀs
in the second stage game are given by a von Neumann-Morgenstern utility
function vi : X → R.
2.2 The Contracts
We are interested in the contracts that are interdependent: each agent’s
transfer scheme (indirectly) depends on the others’ receipt schemes and each
agent’s receipt scheme depends on the others’ transfer schemes.
A transfer function proﬁle announced by player i in the ﬁrst stage is
denoted by ti =
¡
ti1,...,t i(i−1),t i(i+1),...,t in
¢
,w h e r etij : X ×Z → R+ with
Z = {0,1} represents i’s promises to j as a function of actions chosen in the
second stage and indicators 0 and 1.I n d i c a t o r 0 means that according to
the transfer and receipt schemes announced in the ﬁrst stage, a player rejects
transfer from some other. Indicator 1 means that according to the transfer
and receipt schemes announced in the ﬁrst stage, every player accepts transfer
from any other.
2Note that if tij (x,z)=zτij (x) for some τij : X → R+, then the transfer
scheme becomes degenerate, or tij (x,z)=0for all x, unless every player
accepts transfer from the others. That is, when players are expected to
promise acceptance to each other, such transfer function can be sensitive to
a player’s deviation on the receipt scheme.
Let T be the set of all possible tij.L e tt =( t1,...,t n). A transfer function
proﬁle ti =
¡
ti1,...,t i(i−1),t i(i+1),...,t in
¢
announced by player i meets his
budget constraint if
P
j6=itij (x,z) ≤ max{0,v i (x)} for all x and all z.A
proﬁle t =( t1,...,t n) of transfer function proﬁles is called feasible if every ti
meets i’s budget constraint.
A receipt function proﬁle announced by player i in the ﬁrst stage is de-
noted by ri =
¡
ri1,...,r i(i−1),r i(i+1),...,r in
¢
,w h e r erij :( T n−1)
n → {0,1}
represents i’s acceptance (1) or rejection (0)o ft r a n s f e rf r o mj as a func-
tion of proﬁles of transfer function proﬁles announced in the ﬁrst stage. Let
r =( r1,...,r n).
Given a proﬁle t of transfer function proﬁles and a proﬁle r of receipt
function proﬁles in the ﬁrst stage, and a play x in the second stage game,
the payoﬀ Ui to player i becomes
Ui (x,t,r)=vi (x)+
P
j6=i
(rij (t)tji(x,a(t,r)) − rji(t)tij (x,a(t,r)))
where a(t,r)=×i,j,i6=jrij (t).
Given a proﬁle t of transfer function proﬁles and a proﬁle r of receipt
function proﬁles in the ﬁrst stage, and a play µ in the second stage game,
the expected payoﬀ EUi to player i becomes
EUi (µ,t,r)=
P
x
×kµk (xk)
Ã
vi (x)+
P
j6=i
(rij (t)tji(x,a(t,r)) − rji(t)tij (x,a(t,r)))
!
where a(t,r)=×i,j,i6=jrij (t).L e tEUi (µ)=
P
x
×kµk (xk)vi (x).
Let NE(t,r) denote the set of (mixed) Nash equilibria of the second stage
game given (t,r) in the ﬁrst stage. Let NE represent the set of (mixed)
Nash equilibria of the underlying game (the second stage game without side
contracts).
A pure strategy proﬁle x ∈ X of the second stage game together with a
vector u ∈ Rn of payoﬀss u c ht h a t
P
i ui =
P
i vi (x) is supportable if there
3exists a subgame perfect equilibrium of the two stage game where some t and
some r are announced in the ﬁrst stage and x is played in the second stage
on the equilibrium path, and Ui (x,t,r)=ui.
A pure strategy proﬁle x ∈ X of the second stage game together with a
vector u ∈ Rn of payoﬀs such that
P
i ui =
P
ivi(x) is feasibly supportable
if there exists a subgame perfect equilibrium of the two stage game where
some feasible t and some r are announced in the ﬁrst stage and x is played
in the second stage on the equilibrium path, and Ui (x,t,r)=ui.
3. Analysis
The following proposition holds in the model, which implies that any
set of eﬃcient actions maximizing the total payoﬀ is supportable with some
payoﬀ distribution.
Proposition 1. (x,u) such that
P
i ui =
P
ivi(x) is supportable if there
exists iµ for all i such that iµ ∈ NE and EUi (iµ) ≤ ui.
Proof of Proposition 1. Suppose for (x,u) with
P
i ui =
P
i vi (x),t h e r e
exists iµ for all i such that iµ ∈ NE and EUi(iµ) ≤ ui.
Consider τi =
¡
τi1,...,τi(i−1),τi(i+1),...,τin
¢
where τij : X → R+.L e t
τ =( τ1,...,τn) be such that τij (x)=0for all x 6= x, ui = vi (x)+
P
j6=i
(τji(x) − τij (x)),a n dτij (x) > 0 for some j implies τji(x)=0for all j.L e t
t and r be as follows.
tij (x,z)=

   
   
z


τij (x)+
max
(
0,vi(x)+
P
j6=i
(τji(x)−τij(x))−ui
)
n−1


 if x =( x−i,x i)
0 otherwise
rij (t)=
½
1 if t = t
0 otherwise
Consider the following strategy proﬁle (µ,t,r).
(1) (t,r)=
¡
t,r
¢
;
(2) if (t,r)=
¡
t,(r−i,r i)
¢
for some i,w h e r e
rij
¡
t
¢
=1for all j 6= i,t h e nµ = x;
(2-1) if (t,r)=
¡¡
t−i,t i
¢
,(r−i,r i)
¢
for some i,w h e r e
4ti 6= ti or ri 6= ri such that rij
¡
t
¢
=0for some j,t h e nµ =i µ;
(2-2) otherwise µ ∈ NE(t,r).
Note ﬁr s tt h a tf o ra l li, x ∈ NE
¡
t,(r−i,r i)
¢
and Ui
¡
x,t,(r−i,r i)
¢
= ui if
rij
¡
t
¢
=1for all j 6= i.
Suppose (t,r)=
¡¡
t−i,t i
¢
,(r−i,r i)
¢
for some i,w h e r eti 6= ti.I f µ =
(iµ−j,µ j) for some j, then when j 6= i
EUj (µ,t,r)
=
P
x
×kµk (xk)
Ã
vj (x)+
P
k6=j
(rjk(t)tkj (x,a(t,r)) − rkj (t)tjk(x,a(t,r)))
!
=
P
x
×kµk (xk)
Ã
vj (x)+
P
k6=j,k6=i
(rjk(t)tkj (x,a(t,r)) − rkj (t)tjk(x,a(t,r)))
+(rji(t)tij (x,a(t,r)) − rij (t)tji(x,a(t,r)))
!
=
P
x
×kµk (xk)
Ã
vj (x)+
P
k6=j,k6=i
(0 · 0 − 0 · 0) + (0 · tij (x,a(t,r)) − rij (t) · 0)
!
=
P
x
×kµk (xk)vj (x)=EUj (iµ−j,µ j) ≤ EUj (iµ),
and when j = i
EUi (µ,t,r)
=
P
x
×kµk (xk)
Ã
vi (x)+
P
k6=i
(rik (t)tki(x,a(t,r)) − rki(t)tik (x,a(t,r)))
!
=
P
x
×kµk (xk)
Ã
vi (x)+
P
k6=i
(rik (t) · 0 − 0 · tik (x,a(t,r)))
!
=
P
x
×kµk (xk)vi (x)=EUi (iµ−i,µ i) ≤ EUi (iµ) ≤ ui.
Suppose (t,r)=
¡
t,(r−i,r i)
¢
for some i,w h e r eri 6= ri such that rij
¡
t
¢
=0
for some j.I fµ =( iµ−j,µ j) for some j,t h e nw h e nj 6= i
EUj (µ,t,r)
=
P
x
×kµk (xk)
Ã
vj (x)+
P
k6=j
(rjk(t)tkj (x,a(t,r)) − rkj (t)tjk(x,a(t,r)))
!
=
P
x
×kµk (xk)
Ã
vj (x)+
P
k6=j,k6=i
(rjk(t)tkj (x,a(t,r)) − rkj (t)tjk(x,a(t,r)))
+(rji(t)tij (x,a(t,r)) − rij (t)tji(x,a(t,r)))
!
=
P
x
×kµk (xk)
Ã
vj (x)+
P
k6=j,k6=i
(1 · 0 − 1 · 0) + (1 · 0 − rij (t) · 0)
!
5=
P
x
×kµk (xk)vj (x)=EUj (iµ−j,µ j) ≤ EUj (iµ),
and when j = i
EUi (µ,t,r)
=
P
x
×kµk (xk)
Ã
vi (x)+
P
k6=i
(rik (t)tki(x,a(t,r)) − rki(t)tik (x,a(t,r)))
!
=
P
x
×kµk (xk)
Ã
vi (x)+
P
k6=i
(rik (t) · 0 − 1 · 0)
!
=
P
x
×kµk (xk)vi (x)=EUi (iµ−i,µ i) ≤ EUi (iµ) ≤ ui.
Thus, (1)-(2-2) constitutes a subgame perfect equilibrium where
¡
t,r
¢
is
announced in the ﬁrst stage and x is played in the second on the equilibrium
path, and Ui
¡
x,t,r
¢
= ui.
Remark 1. Proposition 1 corresponds to Theorem 6 in Jackson and Wilkie
(2000). In contrast to that theorem, Proposition 1 holds even for two-player
games.
Remark 2. Note that the side contracts
¡
t,r
¢
proposed in the proof of
Proposition 1 is equipped with a kind of invalidation mechanism which makes
it possible to punish any individual deviation. If some i deviate to reject
transfer from some other (z =0 ) when all the players j promise tj,t h e n
t(x,z)=0, or no transfer takes place, and iµ becomes an equilibrium in the
second stage game which punishes i.A t t h e s a m e t i m e , i f s o m e i deviates
from ti (t 6= t) while the others j 6= i promise tj and rj,t h e nr−i (t)=0
(z =0 ) ,w h i c hi nt u r nr e s u l t si nt−i (x,z)=0. Thus again no transfer takes
place, and iµ becomes an equilibrium strategy in the second stage game which
punishes i.
Remark 3. It is a corollary of Proposition 1 that equilibrium strategies and
outcomes of the underlying game are supportable.
Corollary 1. If x ∈ NE,t h e n(x,v (x)) is supportable.
Note that t in the proof of Proposition 1 is sure to be feasible when ui ≥ 0
for all i. That is, even the following proposition holds in the model, which
implies that any set of eﬃcient actions maximizing the total payoﬀ is feasibly
s u p p o r t a b l ew i t hs o m ep a y o ﬀ distribution if there exists an equilibrium of
6the underlying game in which each player enjoys nonnegative payoﬀ without
side payments.
Proposition 2. (x,u) such that
P
i ui =
P
i vi (x) and ui ≥ 0 is feasibly
supportable if there exists iµ for all i such that iµ ∈ NE and EUi(iµ) ≤ ui.
4. Concluding Remarks
We found that there is a class of (feasible) side contracts which may
induce play of eﬃcient actions in equilibria not only in three-or-more-player
games but also in two-player games. What to do next is to see whether the
contracts proposed here are the simplest ones in the class. In fact we already
know that there exist simpler (feasible) side contracts which may lead to
eﬃciency for two-player games (Yamada 2002). We will ﬁnd out whether
three-or-more-player games also have such alternatives.
Reference
Coase, R.H. (1960) “The Problem of Social Cost” Journal of Law and Eco-
nomics 3, 1-44.
Jackson, M.O., and S. Wilkie (2000) “Endogenous Games and Mechanisms:
Side Payments Among Players” mimeo (http://www.hss.caltech.edu/
~jacksonm/Jackson.html).
Yamada, A. (2002) “Eﬃcient Equilibrium Contracts in Two-Player Games”
mimeo.
7

Eﬃcient Equilibrium Contracts

Endogenous Games and Mechanisms: Side Payments Among Players” mimeo (http://www.hss.caltech.edu/ ~jacksonm/Jackson.html).

Efficient equilibrium side contracts

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Efficient equilibrium side contracts

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