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阿部, 宏行

2683

アベ, ヒロユキ

Abe, Hiroyuki

2684

Curriculum guideline (art and handicraft) and formative play (2) : Nishino and Itarashiki's “formative play” and attitude formation

渡辺, 聡

2691

ワタナベ, サトシ

Watanabe, Satoshi

2692

A Study on Risk Management Awareness of Elementary School and Kodomoen Staff

Dobashi, Yoshihito

50374781

English

Short Notes on Labelability of T and Externalization

Japanese

table of contents (Journal of Comparative Cultures)

REVIEW OF THE ECONOMICS AND BUSINESS ADMINISTRATION
SAPPORO UNIVERSITY
Society of Economics and Business Administration, Sapporo University
Nishioka, Sapporo, Japan
 No.148               March                  2023                   Vol.53,   No.1
札 幌 大 学 経 済 ・ 経 営 学 会
ISSN   0389-1119
第 53 巻　第 1号（通巻 148 号）
2023 年 3月
《論　　文》
《研究ノート》
ドイツ基本法「債務ブレーキ」条項の会計上の意義
…………………………………………………権　　大煥　　  1
第一次大戦前フランス農業の組織化 ―信用領域における事例―
 ………………………………………………… 武田　佑太       25
バリュー効果のミスプライシングの要因による検証
 ………………………………………………… 柳樂　明伸       45
労務管理の機能と制度について
………………………………………………… 于 　  暁爽       69
常微分方程式の解の構成と汎弱位相について
…………………………………………………山内　和幸　　89
A note on alternate two-way contracts for
side payments in two-player games
………………………………………………Akira Yamada   　99
中国の経営者報酬と企業統治改革
 ………………………………………………　汪　　志平 　107
関係人口の創出プロセスの研究 (2)
―北海道型ワーケーション施策の考察
 ………………………………………………中山　健一郎  　127
Bedeutung des Grundgesetzes „Schuldenbremse “ im Rechnungswesen
　………………………………………………………Daehwan Kweon 1
Organization of the French agriculture before Wold War I:A case report 
on finance field
　……………………………………………………………Yuta Takeda
25
A Mispricing Explaration on the Value Effect
…………………………………………………………Akinobu Nagira 45
A study on the functions and the systems of human resource 
management
…………………………………………………………Xiaoshuang Yu 69
A note on  the construction of solutions for ODEs  in weak * topology
…………………………………………………Kazuyuki Yamauchi 89
A note on alternate two-way contracts for side payments in 
　two-player games
…………………………………………………………Akira Yamada 99
Top Executive Pay and Corporate Governance Reform in China
…………………………………………………………Wang Zhiping 107
A study of the Process of Creating Related Populations(2)
………………………………………………Kenichiro Nakayama 127
〈Articles〉
〈Note〉


表紙

汪, 志平

2739

オウ, シヘイ

Wang, Zhiping

2740

Top Executive Pay and Corporate Governance Reform in China

文化と言語 
第88号
Culture and Language No.88
札幌大学外国語学系紀要
Sapporo University Journal of 
the Faculty of Foreign Languages
2023.3


中表紙（文化と言語 第88号）

佐藤, 美希

2749

サトウ, ミキ

Sato, Miki

2750

Readers'reviews of Murakami Haruki's retranslations and their translation discourses

山田, 玲良

2763

ヤマダ, アキラ

Yamada, Akira

2764

On the innovator's dilemma without the replacement effect

Akira, Yamada

2737

2738

99A note on alternate two-way contracts for side payments in two-player games
A note on alternate two-way contracts for
side payments in two-player games∗
Akira Yamada†
March 5, 2023
Abstract
We studied side contracting process where two players alternately
offer not only schemes for side payment transfer to the other but
also schemes for side payment receipt from the other before choosing
actions in the underlying game. Then, we found that there may be
a kind of the second mover’s disadvantage in such side contracting
process in contrast to Yamada (2005).
1 Introduction
Yamada (2005b) studied side contracting process where two players alter-
nately offer schemes for side payment transfer to the other before choosing
actions in the underlying game, and found that the second offerer’s equilib-
rium payoff can exceed his threat point level to some extent. Then, we noted
by the result that there is a kind of the second mover’s advantage in such
side contracting process.
∗This note stems from a manuscript, titled ”Alternate two-way contracts for side pay-
ments in two-player games”, which the author presented at the annual meeting of the
Japan Law and Economics Association organized over zoom in November 2021. The au-
thor is grateful to Professor Shinsuke Kambe for his helpful comments on the manuscript
in the conference, in accordance with which the author is still on the way revising the
manuscript.
†This work is part of achievement of 2021 Domestic Research Training Program, Sap-
poro University. The author was engaged in visiting research at the Institute of Innovation
Research, Hitotsubashi University, in this program.
1
経済と経営　53‒1（２０23.3）
〈研究ノート〉
A note on alternate two-way contracts for
side payments in two-player games＊  
Akira Yamada
100 経営と経済　53巻 1号
In line with Jackson andWilkie (2005), a canonical paper in this eld, Ya-
mada (2005b) focused on voluntarily offered side payments and assumed that
such side payments would always be accepted by transferees. This assump-
tion might be thought of as arbitrary, however, since voluntarily offered side
payments could be invalidated by spontaneous rejection to receive them.1
In contrast to Yamada (2005b), we studied side contracting process where
two players alternately offer not only schemes for side payment transfer to the
other but also schemes for side payment receipt from the other before choos-
ing actions in the underlying game. Then, we found that the second offerer’s
equilibrium payoff must be suppressed to his threat point level, and there
may be a kind of the second mover’s disadvantage in such side contracting
process.
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.
Stage 1: Player 1 announces a transfer function (side-payment transfer
scheme) and a receipt function (side-payment acceptance/rejection scheme),
each of which is assumed to be binding.
Stage 2: Player 2 announces a transfer function (side-payment transfer
scheme) and a receipt function (side-payment transfer acceptance/rejection
scheme), each of which is assumed to be binding.
Stage 3: Each player chooses an action.
The players alternately promises two-way schemes (transfer and receipt func-
tions) on side payments in the rst and second stages. Such an alternate con-
tracting process was analyzed by Yamada (2005b), where the players promise
not two-way but one-way (only transfer) schemes. Moreover, the two-way
schemes were called bilateral contracts in Yamada (2003), which studied the
performance of such two-way schemes in not alternate but simultaneous con-
tracting games.
1Yamada (2005a) studied another type of two-player three-stage games which are the
same as Yamada’s (2005b) except that players are allowed to offer negative side payments
as well as nonnegative ones. In Yamada’s (2005a) setting, the second transfer-offerer takes
all the payoff while the rst looses everything.
2
101A note on alternate two-way contracts for side payments in two-player games
The players are given by a set N = {1, 2}. Let i or j denote any given
one of the two players. When a player is denoted by i, let j denote the other
player, and vice versa.
A player i’s nite pure strategy space in the third stage game is denoted
by Xi, with X = ×iXi. Let ∆ (Xi) denote the set of mixed strategies for i,
and let ∆ = ×i∆ (Xi). We denote by xi, x, µi, and µ generic elements of
Xi, X, ∆ (Xi), and ∆ respectively. For simplicity, we sometimes use xi and
x to denote µi and µ respectively that place probability one on xi and x. A
player i’s before-side-payment payoffs in the third stage game are given by a
von Neumann-Morgenstern utility function vi : X → R.
A transfer function announced by i in the rst or second stage is denoted
by ti, where ti : X×Z → R+ with Z = {0, 1} represents i’s promises to j as a
function of actions chosen in the third stage and indicators 0 and 1. Indicator
0 means that according to the transfer and receipt schemes announced in the
rst and second stages, a player rejects transfer from the other. Indicator 1
means that according to the transfer and receipt schemes announced in the
rst and second stages, every player accepts transfer from the other. Let T
be the set of all possible ti. Let t = (t1, t2).
Note that if ti (x, z) = zτi (x) for some τi : X → R+, then the transfer
scheme becomes degenerate, or ti (x, z) = 0 for all x, unless every player ac-
cepts transfer from the other. 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.
A receipt function announced by player i in the rst or second stage
is denoted by ri, where ri : T 2 → {0, 1} represents i’s acceptance (1) or
rejection (0) of transfer from j as a function of proles of transfer functions
announced in the rst and second stages. Let r = (r1, r2).
Given a prole t of transfer functions and a prole r of receipt functions
in the rst and second stages, and a play x in the third stage game, the payoff
Ui to player i becomes
Ui (x, t, r) = vi (x) + (ri (t) tj (x, a (t, r))− rj (t) ti (x, a (t, r)))
where a (t, r) = r1 (t)× r2 (t).
Given a prole t of transfer functions and a prole r of receipt functions
in the rst and second stages, and a play µ in the third stage game, the
expected payoff EUi to player i becomes
3
102 経営と経済　53巻 1号
EUi (µ, t, r) =P
x
×kµk (xk) (vi (x) + (ri (t) tj (x, a (t, r))− rj (t) ti (x, a (t, r))))
where a (t, r) = r1 (t)× r2 (t). Let EUi (µ) =
P
x
×kµk (xk) vi (x).
Let NE (t, r) denote the set of (mixed) Nash equilibria of the third stage
game given (t, r) in the rst and second stages. Let NE represent the set
of (mixed) Nash equilibria of the underlying game (the third stage game
without side contracts). For sake of simplicity and analyzability, we assume
for each i, there exists min
µ∈NE
EUi (µ).
A pure strategy prole x ∈ X of the third stage game is supportable if
there exists a subgame perfect equilibrium of the three stage game where
(i) for t and r announced in the rst and second stages and x played in the
third stage on the equilibrium path, Ui (x, t, r) = ui for some u ∈ R2 such
that
P
i ui =
P
i vi (x), and moreover, (ii) any 1’s deviation in the rst stage
cannot induce a Nash equilibrium of the following (after-deviation) subgame
off the equilibrium path so that 1 may enjoy more payoff than u1. Note (ii)
guarantees a kind of stability of the supportability. This condition requires
that there be no incentive for deviation from the beginning of the game.
3 Analysis
We study the model to obtain the next theorem.
Theorem 1. If x is supportable, then 2’s equilibrium payoff is no more than
min
µ∈NE
EU2 (µ).
Proof of Theorem 1. Since x is supportable, there exists a subgame perfect
equilibrium of the three stage game where (i) for t and r announced in the
rst and second stages, and x played in the third stage on the equilibrium
path, Ui
¡
x, t, r
¢
= ui for some u ∈ R2 such that
P
i ui =
P
i vi (x), and (ii)
any 1’s deviation in the rst stage cannot induce a Nash equilibrium of the
following (after-deviation) subgame off the equilibrium path so that 1 may
enjoy more payoff than u1.
Suppose u2 >min
µ∈NE
EU2 (µ). Note here that there must be iµ for all i such
that iµ ∈ NE and EUi (iµ) ≤ ui. Otherwise some i announces degenerate
ti and ri, or no transfer and no acceptance, and enjoy EUi (µ) > ui for any
4
103A note on alternate two-way contracts for side payments in two-player games
µ ∈ NE. That is, i has an incentive to deviate from the equilibrium path.
Let 2bµ =argmin
µ∈NE
EU2 (µ) in particular.
Consider τi : X → R+ such that τi (x) = 0 for all x 6= x, bui = vi (x) +
τj (x)− τi (x) where bu1 = u1 +
u2− min
µ∈NE
(EU2(µ))
2
and bu2 = u2 −
u2− min
µ∈NE
(EU2(µ))
2
,
and τi (x) > 0 implies τj (x) = 0. Note bu1 > u1 and bu2 >min
µ∈NE
(EU2 (µ))
since u2 >min
µ∈NE
(EU2 (µ)). Let bt and br be as follows.
bti (x, z) =
½
z (τi (x) + max {0, vi (x) + τj (x)− τi (x)− bui}) if x = (xi, xj)
0 otherwise
bri (t) =
½
1 if t = bt
0 otherwise
Suppose 1 deviates to
¡bt1, br1
¢
in the rst stage, considering the following
strategy prole (µ, t2, r2) for the subgame afterwards.
(1) (t2, r2) =
¡bt2, br2
¢
;
(2) if (t2, r2) =
¡bt2, r2
¢
, where
r2
¡bt¢ = 1, then µ = x;
(2-1) if (t2, r2) =
¡bt2, r2
¢
, where r2
¡bt¢ = 0, then 2bµ ∈ NE.
(2-2) if (t2, r2) = (t2, r2), where t2 6= bt2, then 2bµ ∈ NE.
Case (1)-(2). If 2 announces
¡bt2, r2
¢
with r2
¡bt¢ = 1 in the second stage,
x ∈ NE ¡bt, (br1, r2)
¢
and Ui
¡
x,bt, (br1, r2)
¢
= bui for each i where bu1 > u1 and
bu2 >min
µ∈NE
EU2 (µ).
Case (1)-(2-1). Consider 2 announces
¡bt2, r2
¢
with r2
¡bt¢ = 0 in the second
stage.
If µ = (µ1,2 bµ2), then
EU1 (µ, t, r)
=
P
x
×kµk (xk) (v1 (x) + (r1 (t) t2 (x, a (t, r))− r2 (t) t1 (x, a (t, r))))
=
P
x
×kµk (xk) (v1 (x) + (1 · 0− 0 · 0))
=
P
x
×kµk (xk) v1 (x) = EU1 (µ1,2 bµ2) ≤ EU1 (2bµ),
If µ = (2bµ1, µ2), then
EU2 (µ, t, r)
=
P
x
×kµk (xk) (v2 (x) + (r2 (t) t1 (x, a (t, r))− r1 (t) t2 (x, a (t, r))))
5
104 経営と経済　53巻 1号
=
P
x
×kµk (xk) (v2 (x) + (0 · 0− 1 · 0))
=
P
x
×kµk (xk) v2 (x) = EU2 (2bµ1, µ2) ≤ EU2 (2bµ) < bu2.
Case (1)-(2-2). Consider 2 announces (t2, r2) with t2 6= bt2 in the second
stage.
If µ = (µ1,2 bµ2), then
EU1 (µ, t, r)
=
P
x
×kµk (xk) (v1 (x) + (r1 (t) t2 (x, a (t, r))− r2 (t) t1 (x, a (t, r))))
=
P
x
×kµk (xk) (v1 (x) + (0 · t2 (x, a (t, r))− r2 (t) · 0))
=
P
x
×kµk (xk) v1 (x) = EU1 (µ1,2 bµ2) ≤ EU1 (2bµ).
If µ = (2bµ1, µ2), then
EU2 (µ, t, r)
=
P
x
×kµk (xk) (v2 (x) + (r2 (t) t1 (x, a (t, r))− r1 (t) t2 (x, a (t, r))))
=
P
x
×kµk (xk) (v2 (x) + (r2 (t) · 0− 0 · t2 (x, a (t, r))))
=
P
x
×kµk (xk) v2 (x) = EU2 (2bµ1, µ2) ≤ EU2 (2bµ) < bu2.
Thus, (1), (2), (2-1) and (2-2) constitute a Nash equilibrium of the sub-
game of the three-stage game after
¡bt1, br1
¢
is announced in the rst stage.
In its equilibrium path (1)-(2) 1 can enjoy bu1 > u1, which implies 1 has an
incentive to deviate to
¡bt1, br1
¢
in the rst stage. The result contradicts to
x’s supportability in terms of (ii), or its stability condition.
Theorem 1 implies that the second offerer’s equilibrium payoff must be
suppressed to his threat point level. That is, there may be a kind of the
second mover’s disadvantage in such side contracting process.
4 Conclusion
We studied side contracting process where two players alternately offer not
only schemes for side payment transfer to the other but also schemes for side
payment receipt from the other before choosing actions in the underlying
game. Then, we found that there may be a kind of the second mover’s
disadvantage in such side contracting process.
6
105A note on alternate two-way contracts for side payments in two-player games
The result seems in contrast to Yamada (2005b). Yamada (2005b) studied
side contracting process where two players alternately offer only schemes for
side payment transfer to the other before choosing actions in the underlying
game, and found that the second offerer’s equilibrium payoff can exceed his
threat point level to some extent. Then, we noted by the result that there
is a kind of the second mover’s advantage, not disadvantage, in such side
contracting process as in Yamada’s (2005b) framework.
In order to make it clear to what extent there really is a kind of the
second mover’s disadvantage, we must consider in more detail the nature of
equilibrium in our setting. Next, we are to fully characterize equilibrium
outcomes in side contracting process where two players alternately offer not
only schemes for side payment transfer to the other but also schemes for side
payment receipt from the other before choosing actions in the underlying
game.
5 Reference
JACKSON, M. O. and WILKIE, S. (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
Bulletin, 3(6), 1-7.
YAMADA, A. (2005a), ”Surviving Efficient Equilibria in Two-Player Games
with Alternate Unilateral Contracts for Side Payments,” Review of the Eco-
nomics and Business Administration, Sapporo University, 35(2), 81-86.
YAMADA, A. (2005b), ”Alternate Contracts for Side Payments,” Economics
Bulletin, 3(46), 1-5.
7


A note on alternate two-way contracts for side payments in two-player games

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