We characterize efficient equilibrium outcomes of two-player games that remain equilibrium outcomes even when the two players may alternately make binding offers of strategy contingent side payments before the game is played. Our characterization result implies that alternately contracting for side payments has more efficiency of a certain type in equilibria than simultaneously side contracting which is analyzed by Jackson and Wilkie (2005).alternate contracts

Akira Yamada

English

Research Papers in Economics

Alternate contracts for side payments 
Akira Yamada
Sapporo University
Abstract
We characterize efficient equilibrium outcomes of two−player games that remain equilibrium
outcomes even when the two players may alternately make binding offers of strategy
contingent side payments before the game is played. Our characterization result implies that
alternately contracting for side payments has more efficiency of a certain type in equilibria
than simultaneously side contracting which is analyzed by Jackson and Wilkie (2005).
I thank Ryuji Ushizawa, Naoki Yoshihara, and Makoto Okamura for helpful discussions.
Citation: Yamada, Akira, (2005) "Alternate contracts for side payments." Economics Bulletin, Vol. 3, No. 46 pp. 1−5
Submitted: July 28, 2005.  Accepted: October 25, 2005.
URL: http://www.economicsbulletin.com/2005/volume3/EB−05C70029A.pdf1. Introduction
Jackson and Wilkie (2005) explored two-stage games when players may si-
multaneously make binding oﬀers of strategy contingent side payments before
choosing actions. Among their various ﬁndings, there is a fact for two-player
games that given a proﬁle of eﬃcient actions which constitutes an equilib-
r i u mi nt h eu n d e r l y i n gg a m e( t h es e c o n ds t a g eg a m ew i t h o u ts i d ec o n t r a c t s ) ,
the equilibrium payoﬀ distribution under the actions remains an equilibrium
outcome even in the two-stage game if and only if each player’s payoﬀ in the
distribution is no less than what is called his solo payoﬀ.
We study two-player three-stage games where players may alternately
make binding oﬀers of strategy contingent side payments in the ﬁrst and
second stages before choosing actions in the ﬁnal stage. It is shown in our
analysis that given a proﬁle of eﬃcient actions which constitutes an equilib-
rium in the underlying game (the third stage game without side contracts),
the equilibrium payoﬀ distribution under the actions remains an equilibrium
outcome even in the three-stage game if and only if the second transfer-
oﬀerer’s payoﬀ in the distribution is no less than his solo payoﬀ, no matter
how much payoﬀ his counterpart enjoys in the distribution.
Jackson and Wilkie (2005) also discussed about timing problems, and
doubted that players’ alternating in announcing their transfer schemes would
generally improve eﬃciency.1 Indeed their assertion might be true, but at
the same time our characterization result tells that alternately side contract-
ing might have more eﬃcient actions of a certain type in equilibria than
simultaneous side contracting.
In what follows we present the model in Section 2 and the analysis in
Section 3. Our concluding remarks appear in Section 4.
2. Model
We consider two-player three-stage games played as follows.
1Jackson and Wilkie (2005) concluded: “Thus, in order for timing to really be an issue
it must either be that some players are restricted not to be able to respond to the contracts
of others or else there must be some frictions in timing, for instance in the form of time
discounting and some time or eﬀort cost to writing contracts. But note that neither of
these situations should generally improve eﬃciency, and in some cases might harm it” (p.
561).
1Stage 1: Player 1 announces a transfer function (transfer scheme), which is
assumed to be binding.
Stage 2: Player 2 announces a transfer function, which is assumed to be
binding.
Stage 3: Each player chooses an action.
The players alternately make side contracts in the ﬁrst and second stages.
Let i denote any given one of the two players. When a player is denoted
by i,l e tj denote the other player. A player i’s ﬁnite pure strategy space in
t h et h i r ds t a g eg a m ei sd e n o t e db yXi,w i t hX = X1×X2.L e t∆(Xi) denote
the set of mixed strategies for i,a n dl e t∆ = ∆(X1) × ∆(X2).W ed e n o t e
by xi, x, µi,a n dµ g e n e r i ce l e m e n t so fXi, 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.Ap l a y e ri’s payoﬀsi nt h et h i r ds t a g e
game are given by a von Neumann-Morgenstern utility function vi : X → R.
A transfer function announced by player i in the ﬁrst or second stage is
denoted by ti,w h e r eti : X → R+ represents i’s nonnegative promises to j as
a function of actions chosen in the third stage. Let T b et h eu n i v e r s a ls e to f
ti. T contains i’s degenerate transfer function t0
i : X → {0}.L e tt =( t1,t 2).
Given a proﬁle t of transfer functions in the ﬁrst and second stages, and
ap l a yx in the third stage game, the payoﬀ Ui to player i becomes
Ui (x,t)=vi (x)+tj (x) − ti (x).
I ti sa s s u m e dh e r et h a te a c hp l a y e rc o u l dn o tr e j e c tt h eo t h e r ’ so ﬀer of side
payments. Thus, the players’ contracts for side payments are unilateral.2
Given a proﬁle t of transfer functions in the ﬁrst and second stages, and a
play µ in the third stage game, the expected payoﬀ EUi to player i becomes
EUi (µ,t)=
P
x
µ1 (x1) · µ2 (x2) · (vi (x)+tj (x) − ti (x)).
Let NE(t) denote the set of (mixed) Nash equilibria of the third stage
game given t in the ﬁrst and second stages. Let NE represent the set of
(mixed) Nash equilibria of the underlying game (the third stage game without
side contracts).
A pure strategy proﬁle x ∈ X of the third stage game together with
av e c t o ru =( u1,u2) ∈ R2 of payoﬀs such that u1 + u2 = v1 (x)+v2 (x)
2For the case when each player could reject the others’ oﬀers of side payments, see
Yamada (2003).
2is supportable if there exists a subgame perfect equilibrium of the three-
stage game where some t is announced in the ﬁrst and second stages and
x is played in the third stage on the equilibrium path, and Ui (x,t)=ui
for each i. A pure equilibrium strategy proﬁle x ∈ NE of the underlying
game is surviving i nt h et h r e e - s t a g eg a m ei f(x,v (x)) is supportable, where
v(x)=( v1 (x),v 2 (x)).
3. Analysis
Let ui (tj)= s u p
ti∈T
·
min
µ∈NE(tj,ti)
EUi (µ,tj,t i)
¸
. ui
¡
t0
j
¢
is particularly called i’s
solo payoﬀ. We obtain the following two results, which characterize eﬃcient
equilibrium actions of the underlying game that survive in the three-stage
game.
Theorem 1. x ∈ NE is surviving only if v2 (x) ≥ u2 (t0
1).
P r o o fo fT h e o r e m1 .When x is surviving, there exists a subgame perfect
equilibrium of the three-stage game where some t =( t1,t 2) is announced
in the ﬁrst and second stages and x is played in the third stage on the
equilibrium path, and Ui (x,t)=vi (x) for each i. Suppose to the con-
trary that v2 (x) <u 2 (t0
1),n a m e l yv2 (x)=u2 (t0
1) − δ for some δ > 0.
Let t0
2 ∈
(
t2 ∈ T :m i n
µ∈NE(t0
1,t2)
EU2 (µ,t0
1,t 2) >u 2 (t0
1) − δ
)
.L e t b t2 = t1 + t0
2.
Then, since NE
¡
t1,b t2
¢
= NE(t0
1,t 0
2), min
µ∈NE(t1,b t2)
EU2
¡
µ,t1,b t2
¢
=m i n
µ∈NE(t0
1,t0
2)
EU2 (µ,t0
1,t 0
2) >u 2 (t0
1) − δ = v2 (x). That is, Player 2 has an incentive to
deviate from t in the second stage. A contradiction. Thus, v2 (x) ≥ u2 (t0
1).
Theorem 2. Let x ∈ NE such that v1 (x)+v2 (x) ≥ v1 (x0)+v2 (x0) for any
x0 ∈ X.T h e n ,x is surviving if v2 (x) ≥ u2 (t0
1).
P r o o fo fT h e o r e m2 .Given t1,l e tt2 (t1) ∈ arg max
t2∈T
·
max
µ∈NE(t1,t2)
EU2 (µ,t1,t 2)
¸
.
Consider the following strategy proﬁle.
Stage 1: Player 1 announces t1 = t0
1.
Stage 2: If t1 = t0
1,t h e nP l a y e r2 announces t2 = t0
2.I f t1 6= t0
1,t h e n
t2 = t2 (t1).
3Stage 3: If t = t0,t h e nx is chosen. If t1 = t0
1 and t2 6= t0
2,t h e ns o m e
µ ∈ arg min
µ∈NE(t1,t2)
EU2 (µ,t1,t 2) is chosen. If t1 6= t0
1,t h e nµ ∈ arg max
µ∈NE(t1,t2)
EU2 (µ,t1,t 2) is chosen.
What is chosen in the third stage is an element of NE(t).
As for the second stage, if Player 2 announces t2 6= t0
2 when t1 = t0
1,t h e n
his expected payoﬀ is no more than u2 (t0
1) ≤ v2 (x). Hence t2 = t0
2 is the
best response to t1 = t0
1. Clearly, t2 = t2 (t1) is the best reply to t1 6= t0
1 by
the construction of the third stage strategy proﬁle.
As for the ﬁrst stage, if Player 1 announces t1 6= t0
1,t h e nP l a y e r2
chooses t2 = t2 (t1) in the second stage so that Player 2’s expected payoﬀ
becomes max
µ∈NE(t1,t2(t1))
EU2 (µ,t1,t 2 (t1)), which must be no less than v2 (x)
since Player 2 prefers t2 = t2 (t1) w h e r eh ec a nc h o o s et2 = t1 to obtain
max
µ∈NE(t1,t2)
EU2 (µ,t1,t 2)= m a x
µ∈NE(t0
1,t0
2)
EU2 (µ,t0
1,t 0
2) ≥ v2 (x). Then, Player
1’s expected payoﬀ is no more than v1 (x) since v1 (x)+v2 (x) ≥ v1 (x0)+v2 (x0)
for any x0 ∈ X. Therefore, Player 1 has no incentive to choose t1 6= t0
1 in the
ﬁrst stage.
Thus, the strategy proﬁle constitutes a subgame perfect equilibrium of the
three-stage game, where t0 is announced in the ﬁrst and second stages and
x is played in the third stage on the equilibrium path, and Ui (x,t)=vi (x)
for each i.
Corollary 1. Let x ∈ NE such that v1 (x)+v2 (x) ≥ v1 (x0)+v2 (x0) for
any x0 ∈ X.T h e n ,x is surviving if and only if v2 (x) ≥ u2 (t0
1).
Remark 1. Corollary 1 corresponds to Theorem 1 in Jackson and Wilkie
(2005) that is for the case when players may simultaneously make side con-
tracts before choosing actions. In contrast to that theorem, our characteri-
zation does not need any condition for Player 1 like v1 (x) ≥ u1 (t0
2).I no u r
three-stage game, Player 2 decides his transfer scheme after knowing what
transfer Player 1 promised in the previous stage. Player 2 is even able to
propose a transfer scheme which cancels Player 1’s oﬀer. Thus, Player 2 can
arbitrarily aﬀect the payoﬀ structure of the third stage game by his transfer,
no matter what transfer the other promises. Therefore, Player 1’s deviation
from t0
1 in the ﬁrst stage would not reduce Player 2’s payoﬀ. Since the devi-
ation only maintains or destroys eﬃciency, its not reducing Player 2’s payoﬀ
implies that Player 1 cannot enjoy any additional beneﬁt by the deviation.
4This is why the characterization is carried out only by a condition for Player
2.
Remark 2. A c c o r d i n gt oT h e o r e m1 of Jackson and Wilkie (2005) and our
Corollary 1,t h es e to fe ﬃcient equilibrium actions of two-player games that
survive in the three-stage games with alternate contracts for side payments,
includes the set of eﬃcient equilibrium actions that survive in the two-stage
games with simultaneous side contracts. This implies that alternately con-
tracting for side payments has more eﬃcient actions of a certain type in
equilibria than simultaneous side contracting. This induces us to withhold
our full consent to the discussion by Jackson and Wilkie (2005), as mentioned
in Introduction.
4. Conclusion
We characterize eﬃcient equilibrium outcomes of two-player games that
remain equilibrium outcomes even when the two players may alternately
make binding oﬀe r so fs t r a t e g yc o n t i n g e n ts i d ep a y m e n t sb e f o r et h eg a m ei s
played. Our characterization result implies that alternate contracts for side
payments have more eﬃciency of a certain kind in equilibria than simultane-
ously contracting.
To make it clear whether the implication of our result holds more gener-
ally, we would try next to characterize eﬃcient outcomes of two-player games
that may not be equilibrium ones in the underlying game but are realized in
equilibria when such alternate side contracts are allowed.
5. Reference
Jackson, M. O. and S. Wilkie (2005) “Endogenous Games and Mech-
anisms: Side Payments Among Players” Review of Economic Studies 72,
543-566.
Yamada, A. (2003) “Eﬃcient Equilibrium Side Contracts” Economics Bul-
letin 3, No. 6, 1-7.
5

Eﬃcient Equilibrium Side Contracts”

Endogenous Games and Mechanisms: Side Payments Among Players”

Alternate contracts for side payments

http://www.accessecon.com/pubs/EB/2005/Volume3/EB-05C70029A.pdf

Alternate contracts for side payments

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