Post on 15-Apr-2022
EasyChair Preprintโ 3083
Game Theory Analysis of Self-Awareness andPoliteness
Huanhuan Guo and Biao Gao
EasyChair preprints are intended for rapiddissemination of research results and areintegrated with the rest of EasyChair.
March 31, 2020
Game Theory Analysis of Self-awareness and Politeness
Huanhuan Guo1 Biao Gao
2*
1Graduate School of Economics, Kobe University, 2-1, Rokkodai, Nada, Kobe, 657-8501, Japan
2Graduate School of Business Administration, Kobe University, 2-1, Rokkodai, Nada, Kobe, 657-8501, Japan
Abstract
This paper studies human irrational behavior from the perspective of behavioral
economics. Through the establishment of a game model, we can understand why
people do not seek help from others when they are clearly in trouble. By adding
politeness and self-awareness as influencing factors, people's help-seeking and
help-giving behavior was clarified by seeking Bayesian Nash equilibrium. As a result,
we clarified the relationship between politeness, self-awareness and the willingness of
the help seekers as well as the helpers. Specifically, on one hand, from the
perspective of the help seekers, we can distinguish people who are likely to seek help
from who are unlikely to seek help. On the other hand, from the perspective of helpers,
we can distinguish people who are likely to help from who are unlikely to help others.
Keywords: Game Theory; Bayesian Nash equilibrium; Self-awareness; Politeness;
Behavioral economics
*Corresponding author.
E-mail address:Biaogao.edu@outlook.com (Biao Gao)
1
1. Introduction
This study explores the psychological mechanism of human help-seeking and
help-giving behavior based on game theory. Take Japanese society as an example,
living in Japan, the Japanese often refuse to ask for help. One important reason is
politeness. In order not to make trouble for others, Japanese people treat others
politely. However, if help seekers care about making trouble for others too much, it's
hard to seek for help (Bendixsen & Wyller, 2019; Martochhio, 2005). On the other
hand, from the aspect of helpers, helpers will feel sympathy for help seekers when
they are asked for help, so they would decide to help or not immediately (Bohns &
Flynn, 2015). Besides, self-awareness is also an important reason why help seekers
can be embarrassed (Barbee, Rowatt & Cunningham, 1996). In this study,
self-awareness is to speculate about what others think of themselves. In other words,
people with strong self-awareness would care too much about othersโ opinions.
People with low self-awareness would less likely care about what others think. In order
not to be looked down upon by others, help seekers with strong self -awareness are
relatively difficult to send a signal. For the same reason, in order not to be looked
down upon by others, helpers with strong self-awareness may help others forcibly if
they are asked. Hashimoto (2015) stated that there are two reasons to explain: the
first reason why help seekers do not ask is that if their help-seeking is low yield and
high cost to helpers, help seekers would also have negative emotions such as debt
and apology. This politeness as human nature will affect people's help-seeking
behavior. The second reason is that if help seekers accept helper's assistance, they're
afraid they won't be able to repay and being ungrateful, so they have to suppress their
request in advance. From a game theory perspective, this study does not consider the
action of future recompense and only analyzes from the nature of self -awareness. We
assume that people with self-awareness should care about how others think of
themselves, and can thus affect their help-seeking behavior accordingly.
The research objective of this study is as follows: On one hand, from the
perspective of the help seekers, it is important to distinguish people who are likely to
2
seek help from who are unlikely to seek help. On the other hand, from the perspective
of helpers, it is also important to distinguish people who are likely to help from who are
unlikely to help others. This study is thus expected to make important contributions by
analyzing human help-seeking and giving behavior from the aspects of politeness and
self-awareness. We would be able to clarify the relationship between politeness,
self-awareness and the willingness of the help seekers as well as the helpers.
Consequently, to fulfill the objectives of the study, we first set up the game theory
model. After that, we find the Bayesian Nash equilibrium. Finally, the results of
equilibrium are summarized.
2. Model
In this study, we firstly established a game model, which is given by Harsanyi
(1967) and adopted by many researchers, such as Myatt & Wallace (2004); Huang &
Zhu (2020).
The characters on the stage are player A as a help-seeker and player B as a
helper. A can be people with the low ability signed as ๐ฟ or high ability signed as ๐ป .
Player B can also be people with the low ability or high ability. The utility function of
each player is:
๐ ๐ ๐
๐
๐
is the utility of the actual cost paid. ๐
is related to the basic cost c and the
ability cost of that player, thus, ๐ ๐ from player A's point of view, if player B
does not help him, he has to work on it by himself, that is ๐ ๐
๐ด ; If Player B
helps, he can do it without effort, that is ๐๐ด ; from player B's point of view, if he
helps player A, he has to make his own efforts, that is ๐๐ต ๐
๐ต๐; If he does not help,
he does not have to make effort๏ผthat is ๐๐ต . The player with high ability cost less;
on the contrary, the player with low ability cost more.
๐
is the utility of self-awareness, which is related to the degree of
self-awareness๏ผrepresented by ๐ผ, and the speculation about how others think of
3
๐ดโฌ
โฌ
๐ดโฌ
โฌ๐ดโฌ
โฌ
๐ดโฌ
โฌ
(1 ๐๐ด
โฌ)
๐ตโฌ
โฌ
๐ตโฌ
โฌ
๐ป๐ต
โฌ
๐ฟ๐ต
โฌ
๐โฌ
โฌ
๐โฌ
โฌ
๐โฌ
โฌ (((1 ๐
๐ต
โฌ) ๐
๐ต
โฌ
๐๐ต
โฌ (1 ๐
๐ต
โฌ)
๐ตโฌ
โฌ
๐ตโฌ
โฌ
๐ตโฌ
โฌ
๐ตโฌ
โฌ
sos
n n
h
so
sos
so
n
n
h h
n
n
n
n
h ๐ตโฌ
โฌ
๐พโฌ
โฌ
๐ฟโฌ
โฌ
๐โฌ
โฌ
๐โฌ
โฌ
1 ๐โฌ
โฌ
๐โฌ
โฌ
๐โฌ
โฌ
1 ๐โฌ
โฌ ๐
โฌ
โฌ
๐โฌ
โฌ
1 ๐ฟโฌ
โฌ
1 ๐โฌ
โฌ
1 ๐โฌ
โฌ
1 ๐โฌ
โฌ
1-ฮณ
1-ฯ
๐ดโฌ
โฌ
n
n
h
himself, represented by ๐ธ(๐). Thus, ๐ ๐ผ[๐ธ(๐) ๐
]. ๐ธ
๐ต.๐๐ด ๐๐๐๐
/ means player
B speculates on A's ability when player A seeks for help; ๐ธ๐ต.๐๐ด ๐๐๐ก ๐ ๐๐๐
/ means
player B speculates on player A's ability when player A doesn't seek for help;
๐ธ๐ด.๐๐ต ๐ป๐๐๐
/ means player A speculates on player B's ability when player B helps;
๐ธ๐ด.๐๐ต ๐๐๐ก ๐๐๐
/ means player A speculates on player B's ability when player B
doesn't help. ๐
is the effect of politeness, and the degree of politeness ฮฒ (0 < ฮฒ < 1)
is related to the basic cost and his actual ability, thus, ๐ ๐ฝ๐๐.
Player A's utility in seeking help is:
๐๐ด ๐๐๐๐
๐ผ๐ด 0๐ธ๐ต .๐๐ด ๐๐๐๐/ ๐๐ด1 ๐ฝ๐ด๐๐๐ต
Player A's utility when he doesn't seek help is:
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐๐ด๐ ๐ผ
๐ด 0๐ธ๐ต .๐๐ด ๐๐๐ก ๐ ๐๐๐/ ๐๐ด1
The utility of player B's help is:
๐๐ต ๐ป๐๐๐
๐๐ต๐ ๐ผ
๐ต 0๐ธ๐ด .๐๐ต ๐ป๐๐๐/ ๐๐ต1
The utility of player B without help is:
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต 0๐ธ๐ด .๐๐ต ๐๐๐ก โ๐๐๐/ ๐๐ต1 ๐ฝ๐ต๐๐๐ด
When the difference between the ability predicted by others and his actual ability
is positive, it is ๐ผ ; and when it is negative, it becomes ๐ผ
. The relationship between
๐ผ and ๐ผ
is ๐ผ
๐ผ
as follows:
Nature enters twice in this game. The game tree is as follows:
๐ฟ๐ด
โฌ ๐
๐ด
โฌ
๐ป๐ด
โฌ
๐ป๐ต
โฌ
๐ฟ๐ต
โฌ
4
The probability of ๐ฟ๐ด
with high ability is ๐๐ด
; The probability of ๐ป๐ด
with low
ability is ๏ผ๏ผ๏ผ๐๐ด๏ผ; ๐ฟ
๐ดโs probability of seeking help is ๐; its probability of not seeking
help ๏ผ ๐; The probability of ๐ป๐ดโs help-seeking is ๐ ; Its probability of not seeking for
help is ๏ผ ๐ . Let the probability of ๐ฟ๐ด
be ๐๐ด
, the probability of ๐ป๐ด
be ๏ผ๏ผ๏ผ๐๐ด๏ผ.
Let the probability of ๐ฟ๐ต
be ๐๐ต
, the probability of๐ป๐ต
be ๏ผ๏ผ๏ผ๐๐ต๏ผ. The probability of
๐ฟ๐ต
giving help is ๐ , its probability of not help is ๏ผ ๐ . The probability of ๐ป๐ต
giving
help is ๐, its probability of not help is ๏ผ ๐. Thus, the belief of player A and B at
each point is:
๐พ (1๏ผ๐๐ต )๐๐๐ด
๐๐๐ด .1๏ผ๐๐ต / ๐ .1๏ผ๐๐ต /(1๏ผ๐๐ด )
๐๐๐ด
๐๐๐ด ๐ (1๏ผ๐๐ด )ใ 1 ๐พ
๐ (1๏ผ๐๐ด )
๐๐๐ด ๐ (1๏ผ๐๐ด )ใ
๐ ๐๐ต ๐๐๐ด
๐๐๐ด ๐๐ต ๐ ๐๐ต (1๏ผ๐๐ด )
๐๐๐ด
๐๐๐ด ๐ (1๏ผ๐๐ด )ใ 1 ๐
๐ (1๏ผ๐๐ด )
๐๐๐ด ๐ (1๏ผ๐๐ด )ใ
ํ ( ๏ผ๐๐ต )๐๐๐ด ๐
๐๐๐ด ( ๏ผ๐๐ต )๐+๐๐๐ต ๐๐ด ๐ =
( ๏ผ๐๐ต )๐
( ๏ผ๐๐ต )๐+๐๐ต ๐ ใ 1 ํ
๐๐ต ๐
.1๏ผ๐๐ต /๐ ๐๐ต ๐ ใ
๐ (1๏ผ๐๐ต )(1๏ผ๐๐ด )๐ ๐
๐ .1๏ผ๐๐ต /(1๏ผ๐๐ด )๐ ๐ ๐๐ต (1๏ผ๐๐ด )๐
(1๏ผ๐๐ต )๐
.1๏ผ๐๐ต /๐ ๐๐ต ๐ ใ 1 ๐
๐๐ต ๐
( ๏ผ๐๐ต )๐+๐๐ต ๐ ใ
๐ (1๏ผ๐๐ต )๐๐ด ๐(1 ๐)
๐.1๏ผ๐๐ต /๐๐ด (1 ๐) ๐๐๐ต ๐๐ด (1 ๐ )
(1๏ผ๐๐ต )(1 ๐)
.1๏ผ๐๐ต /(1 ๐) ๐๐ต (1 ๐ )ใ 1 ๐
๐๐ต (1 ๐ )
.1๏ผ๐๐ต /(1 ๐) ๐๐ต (1 ๐ )ใ
๐ (1๏ผ๐๐ต )(1 ๐๐ด )๐ (1 ๐)
๐ .1๏ผ๐๐ต /(1 ๐๐ด )(1 ๐) ๐ ๐๐ต (1 ๐๐ด )(1 ๐ )
(1๏ผ๐๐ต )(1 ๐)
.1๏ผ๐๐ต /(1 ๐) ๐๐ต (1 ๐ )ใ1 ๐
๐๐ต (1 ๐ )
.1๏ผ๐๐ต /(1 ๐) ๐๐ต (1 ๐ )ใ
๐ฟ (1 ๐)๐๐ด .1๏ผ๐๐ต /
(1 ๐)๐๐ด .1๏ผ๐๐ต / (1 ๐ ).1๏ผ๐๐ต /(1๏ผ๐๐ด )
(1 ๐)๐๐ด
(1 ๐)๐๐ด (1 ๐ )(1๏ผ๐๐ด )ใ
1 ๐ฟ (1 ๐ ).1๏ผ๐๐ต /(1๏ผ๐๐ด )
(1 ๐)๐๐ด .1๏ผ๐๐ต / (1 ๐ ).1๏ผ๐๐ต /(1๏ผ๐๐ด )=
( ๐ )( ๏ผ๐๐ด )
( ๐)๐๐ด +( ๐ )( ๏ผ๐๐ด )
Therefore, if player A seeks for help, player B is in two situations: help and not
help. Take the utility of player A when player B helps as ๐๐ด๐ ๐๐๐๐
, and the utility of
player A when player B doesn't help A as ๐๐ด๐ ๐๐๐๐
. Thus,
๐๐ด ๐๐๐๐๐
๐ผ๐ด 0๐ธ๐ต .๐๐ด ๐๐๐๐/ ๐๐ด1 ๐ฝ๐ด๐๐๐ต
5
๐๐ด ๐๐๐๐๐
๐๐ด๐ ๐ผ
๐ด 0๐ธ๐ต .๐๐ด ๐๐๐๐/ ๐๐ด1
Player A's utility is:
๐๐ด ๐๐๐๐
๐๐ต 0๐ ๐๐ด ๐๐๐๐๐ (1 ๐ )๐๐ด ๐๐๐๐๐1
(1 ๐๐ต),๐๐
๐ด ๐๐๐๐๐ (1 ๐)๐
๐ด ๐๐๐๐๐-
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐๐ด๐ ๐ผ
๐ด 0๐ธ๐ต .๐๐ด ๐๐๐ก ๐ ๐๐๐/ ๐๐ด1
In the case of ๐ป๐ด high ability:
0๐ธ๐ต.๐๐ด ๐๐๐๐/ ๐ป๐ด 1 (1 ๐
๐ต) [๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐๐ต [๐ฟ๐ด ๐ ๐ป๐ด (1 ๐)] ๐ป๐ด
[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ป๐ด > 0
0๐ธ๐ต.๐๐ด ๐๐๐ก ๐ ๐๐๐/ ๐ป๐ด 1 (1 ๐
๐ต) [๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐๐ต [๐ฟ๐ด ๐ ๐ป๐ด (1 ๐)] ๐ป๐ด
[๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐ป๐ด > 0
Therefore, ๐ผ๐ด
of both parties is ๐ผ๐ด
. Player A's utility in seeking help is:
๐๐ด ๐๐๐๐
๐๐ต 0๐ ๐๐ด ๐๐๐๐๐ (1 ๐ )๐๐ด ๐๐๐๐๐1
(1 ๐๐ต),๐๐
๐ด ๐๐๐๐๐ (1 ๐)๐
๐ด ๐๐๐๐๐-
,๐๐ต๐ (1 ๐
๐ต)๐-๐
๐ด ๐๐๐๐๐ ,๐
๐ต (1 ๐ ) (1 ๐๐ต )(1 ๐)-๐๐ด ๐๐๐๐๐
,๐๐ต๐ .1 ๐
๐ต/ ๐-* ๐ผ
๐ด1[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ป๐ด ] ๐ฝ๐ด๐๐๐ต+ ,๐๐ต (1 ๐ )
.1 ๐๐ต/ (1 ๐)-* ๐๐ป
๐ด ๐ผ
๐ด1[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ป๐ด ]+
Player A's utility when he doesn't seek for help is:
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐๐ด๐ ๐ผ
๐ด 0๐ธ๐ต .๐๐ด ๐๐๐ก ๐ ๐๐๐/ ๐๐ด1
6
๐ป๐ด๐ ๐ผ
๐ด1[[๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐ป๐ด ]
In the case of ๐ฟ๐ด
with low ability:
0๐ธ๐ต.๐๐ด seek
/ ๐ฟ๐ด1 (1 ๐
๐ต) [๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐๐ต [๐ฟ๐ด ๐ ๐ป๐ด (1 ๐)] ๐ฟ๐ด
[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ฟ๐ด 0
0๐ธ๐ต.๐๐ด ๐๐๐ก ๐ ๐๐๐/ ๐ป๐ด 1 (1 ๐
๐ต) [๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐๐ต [๐ฟ๐ด ๐ ๐ป๐ด (1 ๐)] ๐ฟ๐ด
[๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐ฟ๐ด 0
Therefore, ๐ผ๐ด
of both parties are ๐ผ๐ด
. When player A seeks for help, the utility is:
๐๐ด ๐๐๐๐
๐๐ต 0๐ ๐๐ด ๐๐๐๐๐ (1 ๐ )๐๐ด ๐๐๐๐๐1
(1 ๐๐ต),๐๐
๐ด ๐๐๐๐๐ (1 ๐)๐
๐ด ๐๐๐๐๐-
,๐๐ต๐ (1 ๐
๐ต)๐-๐
๐ด ๐๐๐๐๐ ,๐
๐ต (1 ๐ ) (1 ๐๐ต )(1 ๐)-๐๐ด ๐๐๐๐๐
,๐๐ต๐ .1 ๐
๐ต/ ๐-* ๐ผ
๐ด2[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ฟ๐ด ] ๐ฝ๐ด๐๐๐ต+ ,๐๐ต (1 ๐ )
.1 ๐๐ต/ (1 ๐)-* ๐๐ฟ
๐ด ๐ผ
๐ด2[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ฟ๐ด ]+
Player A's utility when he doesn't seek for help is:
๐๐ด ๐๐๐ก ๐๐๐๐
๐ฟ๐ด๐ ๐ผ
๐ด2[[๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐ฟ๐ด ]
In the case of ๐ป๐ต
with high ability:
0๐ธ๐ด .๐๐ต ๐ป๐๐๐/ ๐๐ต 1 ๐๐ด [๐ฟ๐ต (1 ํ) ๐ป๐ต ํ] (1 ๐
๐ด) [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐] ๐ป๐ต
๐ฟ๐ต (1 ํ) ๐ป๐ต ํ ๐ป๐ต > 0
0๐ธ๐ด .๐๐ต ๐๐๐ก โ๐๐๐/ ๐ป๐ต 1 ๐๐ด [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐] (1 ๐
๐ด) [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐] ๐ป๐ต
7
๐ฟ๐ต(1 ๐) ๐ป
๐ต๐ ๐ป
๐ต> 0
Therefore, ๐ผ๐ต
of both parties are ๐ผ๐ต
. At this time, the utility of player B when he
helps is:
๐๐ต ๐ป๐๐๐
๐ป๐ต๐ ๐ผ
๐ต1 0๐ธ๐ด .๐๐ต ๐ป๐๐๐/ ๐ป๐ต 1
๐ป๐ต๐ ๐ผ
๐ต1, ๐ฟ
๐ต(1 ํ) ํ๐ป
๐ต ๐ป
๐ต-
The utility of player B when he doesnโt help is:
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต1 0๐ธ๐ด .๐๐ต ๐๐๐ก โ๐๐๐/ ๐ป๐ต 1 ๐ฝ๐ต๐๐๐ด
๐ผ๐ต1 [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐ ๐ป๐ต ] ๐ฝ๐ต๐๐๐ด
In the case of ๐ฟ๐ต
:
0๐ธ๐ด.๐๐ต help
/ ๐๐ต1 ๐
๐ด0๐ฟ๐ต(1 ํ) ๐ป
๐ตํ1 .1 ๐
๐ด/ 0๐ฟ
๐ต(1 ๐) ๐ป
๐ต๐1 ๐ฟ
๐ต
๐ฟ๐ต (1 ํ) ๐ป๐ต ํ ๐ฟ๐ต 0
0๐ธ๐ด .๐๐ต ๐๐ โ๐๐๐/ ๐๐ต 1 ๐๐ด [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐] (1 ๐
๐ด) [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐] ๐ฟ๐ต
๐ฟ๐ต(1 ๐) ๐ป
๐ต๐ ๐ฟ
๐ต 0
Therefore, ๐ผ๐ต
of both parties are ๐ผ๐ต
. At this time, the utility of player B when he
helps is:
๐๐ต ๐ป๐๐๐
๐ฟ๐ต๐ ๐ผ
๐ต2 0๐ธ๐ด .๐๐ต ๐ป๐๐๐/ ๐ฟ๐ต 1
๐ฟ๐ต๐ ๐ผ
๐ต2, ๐ฟ
๐ต(1 ํ) ํ๐ป
๐ต ๐ฟ
๐ต-
The utility of player B when he doesnโt help is:
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2 0๐ธ๐ด .๐๐ต ๐๐๐ก โ๐๐๐/ ๐ฟ๐ต 1 ๐ฝ๐ต๐๐๐ด
๐ผ๐ต2 [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐ ๐ฟ๐ต ] ๐ฝ๐ต๐๐๐ด
3. Equilibrium Analysis
Now let's analyze the equilibrium. For player A and player B, when they decide
their strategy, because they would select the strategy with larger util ity, thus we should
8
compare the difference of the utility of playersโ different selection.
In the case of ๐ป๐ต :
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ป๐ต๐ ๐ผ
๐ต1 [ ๐ฟ๐ต (1 ํ) ๐ป๐ต ํ ๐ป๐ต ] ๐ผ๐ต1 [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐ ๐ป๐ต ] ๐ฝ๐ต๐๐๐ด
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
In the case of LB :
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ฟ๐ต๐ ๐ผ
๐ต2 0๐ธ๐ด .๐๐ต ๐ป๐๐๐/ ๐ฟ๐ต 1 ๐ผ๐ต2 0๐ธ๐ด .๐๐ต ๐๐๐ก โ๐๐๐/ ๐ฟ๐ต 1 ๐ฝ๐ต๐๐๐ด
๐ฟ๐ต๐ ๐ผ
๐ต2 [๐ฟ๐ต (1 ํ) ๐ป๐ต ํ ๐ฟ๐ต ] ๐ผ๐ต2 [๐ฟ๐ต (1 ๐) ๐ป๐ต ๐ ๐ฟ๐ต ] ๐ฝ๐ต๐๐๐ด
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
In the case of ๐ป :
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐๐๐๐
,๐๐ต๐ .1 ๐
๐ต/ ๐-* ๐ผ
๐ด1[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ป๐ด ] ๐ฝ๐ด๐๐๐ต+ ,๐๐ต (1 ๐ )
.1 ๐๐ต/ (1 ๐)- { ๐๐ป
๐ด ๐ผ
๐ด1[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ป๐ด ]}
๐ป๐ด๐ ๐ผ
๐ด1[[๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐ป๐ด ]
๐ผ๐ด1( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด / (๐ ๐ ๐ ๐ ๐)๐(๐ป๐ด ๐ฝ๐ด๐๐ต)
In the case of ๐ฟ๐ด :
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
,๐๐ต๐ .1 ๐
๐ต/ ๐-* ๐ผ
๐ด2[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ฟ๐ด ] ๐ฝ๐ด๐๐๐ต+ ,๐๐ต (1 ๐ )
.1 ๐๐ต/ (1 ๐)- { ๐๐ฟ
๐ด ๐ผ
๐ด2[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ฟ๐ด ]} ๐ฟ๐ด๐
9
๐ผ๐ด2[[๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐ฟ๐ด ]
๐ผ๐ด2( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด / (๐ ๐ ๐ ๐ ๐)๐(๐ฟ๐ด ๐ฝ๐ด๐๐ต)
Player belief:
ํ .1๏ผ๐
๐ต/ ๐
.1๏ผ๐๐ต/ ๐ ๐
๐ต๐ ใ๐
.1๏ผ๐๐ต/ (1 ๐)
ใ๐ ๐พ ๐๐
๐ด
๐๐๐ด ๐ .1๏ผ๐
๐ด/ใ
๐ฟ (1 ๐)๐
๐ด
(1 ๐)๐๐ด (1 ๐ ) .1๏ผ๐
๐ด/ใ
Furthermore, B with high ability can expect Aโs ability as ๐๐ด ๐พ ๐ฟ
๐ด (1
๐พ) ๐ป๐ด
.The ability of B with low ability can expect Aโs ability as ๐๐ด ๐ ๐ฟ
๐ด (1
๐) ๐ป๐ด
We thus substitute them into utility function. In addition, Player A with either
the high ability or low ability would decide to seek help or not, player B with either
the high ability or low ability would decide to help or not. When player A is helped by
player B with high ability, r = 1; when player B with high ability doesn't help, r = 0;
when player B with low ability helps, ๐ 1; when player B with low ability doesn't
help, ๐ When player A with low ability seek for help, q = 1; when player A with
low ability doesn't seek for help, q = 0; when player A with high ability seeks for help
๐ 1, when player A with high ability doesn't seek for help ๐ In summary,
there are situations including ๐ 1 ๐ ๐ 1 ๐ ๐ 1 ๐ ๐ 1 ๐ . As
a result, there are 16 combinations. According to calculating there are nine
equilibriums. The following section will discuss these equilibriums.
The first equilibrium is: ๐ป๐ต
help; ๐ฟ๐ต
not help; ๐ฟ๐ด
seek; ๐ป๐ด not seek
โ ๐ 1 ๐ 0 ๐ 1 ๐ 1
In the case of ๐ป๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต) ๐(๐ฝ
๐ต , ๐
๐ด๐ฟ๐ด .1 ๐
๐ด/๐ป
๐ด- ๐ป
๐ต)
In order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> , there must be:
10
๐ฝ๐ต> ๐ผ
๐ต1
.๐ฟ๐ต ๐ป
๐ต/
๐, ๐๐ด๐ฟ๐ด .1 ๐
๐ด/๐ป
๐ด-
๐ป๐ต
, ๐๐ด๐ฟ๐ด .1 ๐
๐ด/๐ป
๐ด-
The relationship between ๐ฝ๐ต and ๐ผ
๐ต is shown in the following figure:
In the case of ๐ฟ๐ต
:
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
In order to meet the condition ๐๐ต help
๐๐ต not help
, there must be:
๐ฝ๐ต ๐ผ
๐ต2
.๐ฟ๐ต ๐ป
๐ต/
๐,๐๐ด๐ฟ๐ด .1 ๐
๐ด/๐ป
๐ด-
๐ฟ๐ต
๐๐ด๐ฟ๐ด .1 ๐
๐ด/๐ป
๐ด
The relationship between ๐ฝ๐ต and ๐ผ
๐ต is shown in the following figure:
Next, let ๐ป๐ต
=๐ป๐ด ๐ฟ๐ต
=๐ฟ๐ด ๐๐ต
=๐๐ด ๐ผ๐ต ๐ผ
๐ด =๐ผ
๐ผ๐ต ๐ผ
๐ด =๐ผ
๐ฝ๐ด ๐ฝ
๐ต. In this
case, set the difference between ๐ผ
and ๐ผ
to ฮพ. Thus, ๐ผ ๐ผ
๐ . Because
player B must meet the conditions of ๐ป๐ต
and ๐ฟ๐ต
, we thus find the union set of ๐ป๐ต
and ๐ฟ๐ต
. To put the two figures above together, we must distinguish between
occasions between ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ >๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case 1๐ต and๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ ๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case
๐ต. At this point:
11
Case 1๐ต๏ผ Case ๐ต๏ผโ
In the case of ๐ป๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด1( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด / (๐ ๐ ๐ ๐ ๐)๐(๐ป๐ด ๐ฝ๐ด๐๐ต)
๐ผ๐ด ( ๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (1 ๐
๐ต)๐(๐ป
๐ด ๐ฝ
๐ด๐ป๐ต)๏ผ
Therefore, in order to meet the condition ๐๐ด Seek
๐๐ด Not seek
> , there must be:
๐ฝ๐ด ๐ผ
๐ด
( ๐พ ๐ฟ)(๐ฟ๐ด ๐ป๐ด )
๐๐ป๐ต ( ๐๐ต ) ๐ป๐ด
๐ป๐ต
. Because there are ( ๐พ ๐ฟ) > and ( ๐พ ๐ฟ) , The
relationship between ๐ฝ๐ด and ๐ผ
๐ด is shown in the following figure:
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด / .1 ๐
๐ต/ ๐(๐ฟ
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด2( ๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ .1 ๐
๐ต/ ๐(๐ฟ
๐ด ๐ฝ
๐ด๐ป๐ต-)
Therefore, in order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
> , there must be:
๐ฝ๐ด ๐ผ
๐ด
( ๐พ ๐ฟ)(๐ฟ๐ด ๐ป๐ด )
๐๐ป๐ต .1๏ผ๐๐ต / ๐ฟ๐ด
๐ป๐ต
.
Because there are ( ๐พ ๐ฟ) > and ( ๐พ ๐ฟ) the relationship between
12
๐ฝ๐ด and ๐ผ
๐ด is shown in the following figure:
Because player B must meet the conditions of ๐ป๐ด
and ๐ฟ๐ด
, we thus find the
union set of ๐ป๐ด
and ๐ฟ๐ด
. Owing to ๐ > , ๐๐ฟ๐ด ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
๐ must be less
than ๐๐ป๐ด ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
. Therefore:
Player A and B must satisfy all of the conditions of game equilibrium case1๐ต or
case ๐ต. Thus, the result that satisfies the conditions of player A and B is 0 < ฮพ < C. The
figure is as follows:
โก ๐ 1 ๐ 0 ๐ 1 ๐ 0
In the case of ๐ป๐ต
13
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต) ๐(๐ฝ
๐ต๐ฟ๐ด ๐ป
๐ต)
Therefore, in order to meet the condition ๐๐ต ๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> , there must be:
๐ฝ๐ต> ๐ผ
๐ต1
.๐ฟ๐ต ๐ป
๐ต/
๐๐ฟ๐ด
๐ป๐ต
๐ฟ๐ด
The relationship between ๐ฝ๐ต and ๐ผ
๐ต is shown in the following figure:
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต) ๐(๐ฝ
๐ต๐ฟ๐ด ๐ฟ
๐ต)
Therefore, in order to meet the condition ๐๐ต ๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
, there must be:
๐ฝ๐ต ๐ผ
๐ต2
.๐ฟ๐ต ๐ป
๐ต/
๐๐ฟ๐ด
๐ฟ๐ต
๐ฟ๐ด
The relationship between ๐ฝ๐ต and ๐ผ
๐ต is shown in the following figure:
To put the two figures above together, we must distinguish between occasions
between ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ >๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case 1๐ต and๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ ๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case ๐ต . At this
point:
14
Case 1๐ต๏ผ Case ๐ต๏ผโ
In the case of ๐ป๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด1( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด / (๐ ๐ ๐ ๐ ๐)๐(๐ป๐ด ๐ฝ๐ด๐๐ต)
๐ผ๐ด1.๐ฟ๐ด ๐ป
๐ด/ (1 ๐ )๐(๐ป๐ด ๐ฝ๐ด๐ป๐ต )
Therefore, in order to meet the condition ๐๐ด ๐ ๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
there must be:
๐ฝ๐ด> ๐ผ
๐ด1
.๐ฟ๐ด ๐ป
๐ด/
๐(1 ๐ )๐ป๐ต
1
The relationship between ๐ฝ๐ด and ๐ผ
๐ด is shown in the following figure:
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด / .1 ๐
๐ต/ ๐(๐ฟ
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด2.๐ฟ๐ด ๐ป
๐ด/ .1 ๐
๐ต/ ๐(๐ฟ
๐ด ๐ฝ
๐ด๐ป๐ต)
Therefore, in order to meet the condition ๐๐ด ๐ ๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
> there must be:
๐ฝ๐ด ๐ผ
๐ด2
.๐ฟ๐ด ๐ป
๐ด/
๐๐ป๐ต.1๏ผ๐
๐ต/ ๐ฟ๐ด
๐ป๐ต
The relationship between ๐ฝ๐ด and ๐ผ
๐ด is shown in the following figure:
15
To put the two figures above together, we must distinguish between occasions
between ๐๐ฟ๐ด ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
๐ >๐๐ป๐ด ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
as case 1๐ด and ๐๐ฟ๐ด ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
๐ ๐๐ป๐ด ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
as case
๐ด. At this point:
Case 1๐ด๏ผ Case ๐ด๏ผโ
Player A must satisfy all of the conditions of game equilibrium case1๐ด. Player B must
satisfy all of the conditions of game equilibrium case1๐ต. To satisfy both case1๐ด and
case1๐ต, there must be ๐ .1 ๐๐ต/. The figure is as follows:
โข ๐ 1 ๐ 0 ๐ 0 ๐ 0
In the case of ๐ป๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต) ๐(๐ฝ
๐ต,๐พ ๐ฟ
๐ด (1 ๐พ)๐ป
๐ด- ๐ป
๐ต)
Therefore, in order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> there must be:
๐ฝ๐ต> ๐ผ
๐ต1
.๐ฟ๐ต ๐ป
๐ต/
๐,๐พ ๐ฟ๐ด (1 ๐พ)๐ป
๐ด-
๐ป๐ต
๐พ ๐ฟ๐ด (1 ๐พ) ๐ป
๐ด
The relationship between ๐ฝ๐ต and ๐ผ
๐ต is shown in the following figure:
16
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
๐ผ๐ต (๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต) ๐(๐ฝ
๐ต,๐ ๐ฟ
๐ด (1 ๐)๐ป
๐ด- ๐ฟ
๐ต)
Therefore, In order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
there must be:
๐ฝ๐ต ๐ผ
๐ต2
.๐ฟ๐ต ๐ป
๐ต/
๐,๐ ๐ฟ๐ด (1 ๐)๐ป
๐ด-
๐ฟ๐ต
๐ ๐ฟ๐ด (1 ๐) ๐ป
๐ด
The relationship between๐ฝ๐ต and ๐ผ
๐ต is shown in the following figure:
To put the two figures above together, we must distinguish between occasions
between ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ >๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case 1๐ต and ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ ๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case ๐ต . At this
point:
Case 1๐ต๏ผ Case ๐ต๏ผโ
In the case of ๐ป๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
17
๐ผ๐ด1( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด / (๐ ๐๐ต ๐
โฒ ๐
๐ต๐) ๐ .๐ป
๐ด ๐ฝ
๐ด๐๐ต/
๐ผ๐ด1( ๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ .1 ๐
๐ต/ ๐(๐ป
๐ด ๐ฝ
๐ด๐ป๐ต)
Therefore, In order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
there must be:
๐ฝ๐ด> ๐ผ
๐ด1
( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด /
๐๐ป๐ต.1๏ผ๐
๐ต/
1
When ( ๐พ ๐ฟ) > and ( ๐พ ๐ฟ) , the relationship between ๐ฝ๐ต
and ๐ผ๐ต
is
shown in the following figure:
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด / .1 ๐
๐ต/ ๐ .๐ฟ
๐ด ๐ฝ
๐ด๐๐ต/
๐ผ๐ด2( ๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ .1 ๐
๐ต/ ๐(๐ฟ
๐ด ๐ฝ
๐ด๐ป๐ต)
Therefore, In order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
there must be:
๐ฝ๐ด> ๐ผ
๐ด2
( ๐พ ๐ฟ) .๐ฟ๐ด ๐ป๐ด /
๐๐ป๐ต.1๏ผ๐
๐ต/
๐ฟ๐ด
๐ป๐ต
When ( ๐พ ๐ฟ) > and ( ๐พ ๐ฟ) , the relationship between๐ฝ๐ด and ๐ผ
๐ด is
shown in the following figure:
:
Does not exist
18
Owing to:
๐ผ ๐ผ
๐,
๐๐ฟ๐ด ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
๐ >๐๐ป๐ต ( ๐๐ต )
(๐ฟ๐ด ๐ป๐ด ) as case 1๐ด
๐๐ฟ๐ด ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
๐ ๐๐ป๐ต ( ๐๐ต )
๐ฟ๐ด ๐ป๐ด
as case
๐ด are as follows:
Case 1๐ด๏ผ Case ๐ด๏ผ
Player A must satisfy all of the conditions of game equilibrium case1 A and player
B must satisfy all of the conditions of game equilibrium case1 B and case ๐ต. Because
case ๐ต is an empty set, the union set of case ๐ต and case 1๐ด is also an empty set.
To satisfy both case1๐ต and case1๐ด, there must be ๐ .1 ๐๐ต/ To satisfy both
case1๐ต and case ๐ด, there must be .1 ๐๐ต/ ๐ Both of them are shown in the
following figure:
Does not exist
19
โฃ ๐ 0 ๐ 0 ๐ 1 ๐ 1
In the case of ๐ป๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต),ํ (1 ๐
๐ต)- ๐(๐ฝ
๐ต,๐๐ด๐ฟ๐ด (1 ๐
๐ด)๐ป
๐ด- ๐ป
๐ต)
Therefore, In order to meet the condition ๐๐ด ๐ป๐๐๐
๐๐ด ๐๐๐ก ๐๐๐
there must be:
๐ฝ๐ต ๐ผ
๐ต1
.๐ฟ๐ต ๐ป
๐ต/ ,ํ (1 ๐
๐ต)-
๐,๐๐ด๐ฟ๐ด (1 ๐
๐ด)๐ฟ๐ด-
๐ป๐ต
๐๐ด๐ฟ๐ด (1 ๐
๐ด)๐ฟ๐ด
When (ํ ๐) > (ํ ๐) , the relationship between ๐ฝ๐ต and ๐ผ
๐ต is
shown in the following figure:
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐ ๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
20
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต),ํ (1 ๐
๐ต)- ๐*๐ฝ
๐ต,๐๐ด๐ฟ๐ด .1 ๐
๐ด/๐ป
๐ด- ๐ฟ
๐ต+
Therefore, in order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
there must be:
๐ฝ๐ต ๐ผ
๐ต2
.๐ฟ๐ต ๐ป
๐ต/ ,ํ (1 ๐
๐ต)-
๐,๐๐ด๐ฟ๐ด .1 ๐
๐ด/๐ป
๐ด-
๐ฟ๐ต
,๐๐ด๐ฟ๐ด .1 ๐
๐ด/๐ป
๐ด-
When (ํ ๐) > (ํ ๐) , the relationship between๐ฝ๐ต and ๐ผ
๐ต is
shown in the following figure:
Owing to ๐ > , ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ must be less than ๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
. Therefore, the union
set is:
In the case of ๐ป๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด1(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ป
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด1(๐๐ด ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/
Therefore, In order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
> there must be:
21
(๐๐ด ๐ฟ)
When ( ๐พ ๐ฟ) , the relationship between ๐ฝ๐ด and ๐ผ
๐ด is shown in the
following figure:
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ฟ
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด2(๐๐ด ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/
Therefore, in order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
> there must be:
(๐๐ด ๐ฟ) , When ( ๐พ ๐ฟ) , the relationship between ๐ฝ
๐ด and ๐ผ
๐ด is shown in
the following figure:
The condition of player B and player A is ๐. As shown in the following figure,
โค ๐ 0 ๐ 0 ๐ 0 ๐ 0
In the case of ๐ป๐ต
๐๐ต ๐๐๐๐
๐๐ต ๐๐๐ก ๐ ๐๐๐
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต),ํ (1 ๐
๐ต)- ๐(๐ฝ
๐ต,๐พ๐ฟ
๐ด (1 ๐พ)๐ป
๐ด- ๐ป
๐ต)
22
Therefore, in order to meet the condition ๐๐ต ๐๐๐๐
๐๐ต ๐๐๐ก ๐๐๐๐
there must be:
(๐ฝ๐ต ๐ผ
๐ต
(๐ฟ๐ต ๐ป๐ต ),๐ ( ๐๐ต )-
๐,๐พ๐ฟ๐ด+( ๐พ)๐ป๐ด -
๐ป๐ต
,๐พ๐ฟ๐ด+( ๐พ)๐ป๐ด -, when ( ํ ๐) > and ( ํ ๐) , the
relationship between ๐ฝ๐ต and ๐ผ
๐ต is shown in the following figure:
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต),ํ (1 ๐
๐ต)- ๐*๐ฝ
๐ต,๐๐ฟ
๐ด (1 ฯ)๐ป
๐ด- ๐ฟ
๐ต+
Therefore, in order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
there must be:
๐ฝ๐ต ๐ผ
๐ต2
.๐ฟ๐ต ๐ป
๐ต/ ,ํ (1 ๐
๐ต)-
๐,๐๐ฟ๐ด (1 ฯ)๐ป
๐ด-
๐ฟ๐ต
,๐๐ฟ๐ด (1 ฯ)๐ป
๐ด-
When ( ํ ๐) > and ( ํ ๐) , the relationship between ๐ฝ๐ต and ๐ผ
๐ต is
shown in the upper right of the following figure:
23
Since ๐ > , ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ is always smaller than ๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
. Thus, the union set is
In the case of ๐ป๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด1(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ป
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด1(ฮณ ๐
๐ด) .๐ฟ
๐ด ๐ป
๐ด/
Therefore, in order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
there must
be: (ฮณ ๐๐ด) > , when ( ๐พ ๐ฟ) > , the relationship between ๐ฝ
๐ด and ๐ผ
๐ด is
shown in the upper right of the following figure:
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ฟ
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด2(ฮณ ๐
๐ด) .๐ฟ
๐ด ๐ป
๐ด/
Therefore, in order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
there must
24
be: (ฮณ ๐๐ด) > , when ( ๐พ ๐ฟ) > , the relationship between ๐ฝ
๐ด and ๐ผ
๐ด is
shown in the following figure:
Player B's all conditions and player A's all conditions must be met. Thus, the
result is ๐. The figure is shown as follows:
โฅ ๐ 1 ๐ 1 ๐ 0 ๐ 0
In the case of ๐ป๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต),.1 ๐
๐ต/ ๐- ๐(๐ฝ
๐ต,๐พ๐ฟ
๐ด (1 ๐พ)๐ป
๐ด- ๐ป
๐ต)
Therefore, in order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> there must be:
๐ฝ๐ต ๐ผ
๐ต1
.๐ฟ๐ต ๐ป
๐ต/ ,.1 ๐
๐ต/ ๐-
๐,๐พ๐ฟ๐ด (1 ๐พ)๐ป
๐ด-
๐ป๐ต
,๐พ๐ฟ๐ด (1 ๐พ)๐ป
๐ด-
When ( ํ ๐) > and ( ํ ๐) , the relationship between ใ๐ฝ๐ตand ๐ผ
๐ต is
shown in the following figure:
25
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต),.1 ๐
๐ต/ ๐- ๐*๐ฝ
๐ต,๐๐ฟ
๐ด (1 ฯ)๐ป
๐ด- ๐ฟ
๐ต+
Therefore, in order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> there must be:
๐ฝ๐ต ๐ผ
๐ต2
.๐ฟ๐ต ๐ป
๐ต/ ,.1 ๐
๐ต/ ๐-
๐,๐๐ฟ๐ด (1 ฯ)๐ป
๐ด-
๐ฟ๐ต
,๐๐ฟ๐ด (1 ฯ)๐ป
๐ด-
When ( ํ ๐) > and( ํ ๐) , the relationship between ๐ฝ๐ตand ๐ผ
๐ต is
shown in the following figure:
To combine the two figures above together, we must distinguish between
occasions between ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ >๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case 1๐ต and๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ ๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case ๐ต.
At this time,
26
Case 1๐ต๏ผ Case ๐ต๏ผ
In the case of ๐ป๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด1(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ป
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด1(ฮณ ๐
๐ด) .๐ฟ
๐ด ๐ป
๐ด/ ๐*๐ป
๐ด ๐ฝ
๐ด 0๐๐ต ๐ฟ๐ต .1 ๐๐ต/๐ป
๐ต1+
Therefore, In order to meet the condition ๐๐ด seek
๐๐ด ๐๐๐ก ๐ ๐๐๐
there must be:
๐ฝ๐ด> ๐ผ
๐ด1
(ฮณ ๐๐ด) .๐ฟ
๐ด ๐ป
๐ด/
๐ 0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
๐ป๐ด
,๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต-
When ( ๐พ ๐ฟ) > and ( ๐พ ๐ฟ) , the relationship between ๐ฝ๐ด
and ๐ผ๐ด
is
shown in the following figure:
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ฟ
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด2(ฮณ ๐
๐ด) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐*๐ฟ
๐ด ๐ฝ
๐ด 0๐๐ต ๐ฟ๐ต .1 ๐๐ต/๐ป
๐ต1+
27
Therefore, in order to meet the condition ๐๐ด seek
๐๐ด ๐๐๐ก ๐ ๐๐๐
there must be:
๐ฝ๐ด> ๐ผ
๐ด2
(ฮณ ๐๐ด) .๐ฟ
๐ด ๐ป
๐ด/
๐ 0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
๐ฟ๐ด
0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
When ( ๐พ ๐ฟ) > and ( ๐พ ๐ฟ) , the relationship between ๐ฝ๐ดand ๐ผ
๐ด is
shown in the following figure:
O g o ๐ผ ๐ผ
๐, there is
๐๐ฟ๐ด
๐ฟ๐ด ๐ป๐ด
๐ >๐๐ป๐ด
๐ฟ๐ด ๐ป๐ด
as case 1๐ด ๐๐ฟ๐ด
๐ฟ๐ด ๐ป๐ด
๐ ๐๐ป๐ด
๐ฟ๐ด ๐ป๐ด
as
case ๐ด. At this time,
Case 1๐ด๏ผ Case ๐ด๏ผ
Player A must satisfy all of the conditions of game equilibrium case1๐ด ๐ด. Player
B must satisfy all of the conditions of game equilibrium case1๐ต ๐ต. Thus๏ผthe result
of the union set of case1๐ต and case1๐ด is 0<ฮพ<c. It is shown in the following figure:
28
Since the results of the union set of case 1๐ต and case ๐ด are ๐ , ๐ > , ฮพ
does not exist. Since the results of the union set of case ๐ต and case 1๐ด are ๐ > ใ
๐ ๐ does not exist. The result of the union set of case ๐ต and case ๐ด is ๐ > .
The figure is shown as follows:
โฆ ๐ 1 ๐ 1 ๐ 1 ๐ 1
In the case of ๐ป๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต),.1 ๐
๐ต/ ๐- ๐(๐ฝ
๐ต,๐พ๐ฟ
๐ด (1 ๐พ)๐ป
๐ด- ๐ป
๐ต)
Therefore, in order to meet the condition ๐๐ต ๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> there must be:
๐ฝ๐ต ๐ผ
๐ต1
.๐ฟ๐ต ๐ป
๐ต/ ,.1 ๐
๐ต/ ๐-
๐,๐พ๐ฟ๐ด (1 ๐พ)๐ป
๐ด-
๐ป๐ต
,๐พ๐ฟ๐ด (1 ๐พ)๐ป
๐ด-
When ( ํ ๐) > and ( ํ ๐) , the relationship betweenใ๐ฝ๐ตand๐ผ
๐ต is
shown in the following figure:
29
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต),.1 ๐
๐ต/ ๐- ๐*๐ฝ
๐ต,๐๐ฟ
๐ด (1 ฯ)๐ป
๐ด- ๐ฟ
๐ต+
Therefore, in order to meet the condition ๐๐ต ๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> there must be:
๐ฝ๐ต ๐ผ
๐ต2
.๐ฟ๐ต ๐ป
๐ต/ ,.1 ๐
๐ต/ ๐-
๐,๐๐ฟ๐ด (1 ฯ)๐ป
๐ด-
๐ฟ๐ต
,๐๐ฟ๐ด (1 ฯ)๐ป
๐ด-
When ( ํ ๐) > and ( ํ ๐) , the relationship between ๐ฝ๐ตand ๐ผ
๐ต is
shown in the following figure:
To put the two figures above together, we must distinguish between occasions
between ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ >๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case 1๐ต and ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ ๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case ๐ต. At this
point:
30
Case 1๐ต๏ผ Case ๐ต๏ผ
In the case of ๐ป๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด1(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ป
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด1(๐๐ด ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ ๐*๐ป
๐ด ๐ฝ
๐ด 0๐๐ต ๐ฟ๐ต .1 ๐๐ต/๐ป
๐ต1+
Therefore, in order to meet the condition ๐๐ด ๐ ๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
> there must be:
๐ฝ๐ด> ๐ผ
๐ด1
(๐๐ด ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/
๐ 0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
๐ป๐ด
,๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต-
When ( ๐พ ๐ฟ) > and ( ๐พ ๐ฟ) , the relationship between ๐ฝ๐ดand ๐ผ
๐ด is
shown in the following figure:
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ฟ
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด2(๐๐ด ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ ๐*๐ฟ
๐ด ๐ฝ
๐ด 0๐๐ต ๐ฟ๐ต .1 ๐๐ต/๐ป
๐ต1+
Therefore, in order to meet the condition ๐๐ด ๐ ๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
> there must be:
31
๐ฝ๐ด ๐ผ
๐ด2
(๐๐ด ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/
๐ 0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
๐ฟ๐ด
0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
When ( ๐พ ๐ฟ) > and ( ๐พ ๐ฟ) , the relationship between ๐ฝ๐ดand ๐ผ
๐ด is
shown in the following figure:
Since ๐ > , ๐๐ฟ๐ด
๐ฟ๐ด ๐ป๐ด
๐ is always smaller than ๐๐ป๐ด
๐ฟ๐ด ๐ป๐ด
. Thus, the union set is:
Player B must satisfy all of the conditions of game equilibrium case 1๐ต or
case ๐ต, so does player A. Thus, the result that satisfies the conditions of player A and
B is ๐ . The figure is
32
To satisfy Case ๐ต and player Aโs conditions, the result is 1 > ๐ > . The figure
is as follows:
โง ๐ 1 ๐ 1 ๐ 1 ๐ 0
In the case of ๐ป๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ป
๐ต)
๐ผ๐ต1(๐ฟ๐ต ๐ป
๐ต),.1 ๐
๐ต/ ๐- ๐(๐ฝ
๐ต๐ฟ๐ด ๐ป
๐ต)
Therefore, in order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> there must be:
๐ฝ๐ต> ๐ผ
๐ต1
.๐ฟ๐ต ๐ป
๐ต/ ,.1 ๐
๐ต/ ๐-
๐๐ฟ๐ด
๐ป๐ต
๐ฟ๐ด
The relationship between ๐ฝ๐ตand ๐ผ
๐ต is shown in the following figure:
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต),.1 ๐
๐ต/ ๐- ๐(๐ฝ
๐ต๐ฟ๐ด ๐ฟ
๐ต)
Therefore, in order to meet the condition ๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
> there must be:
33
๐ฝ๐ต> ๐ผ
๐ต2
.๐ฟ๐ต ๐ป
๐ต/ ,.1 ๐
๐ต/ ๐-
๐๐ฟ๐ด
๐ฟ๐ต
๐ฟ๐ด
The relationship between ๐ฝ๐ตand ๐ผ
๐ต is shown in the following figure:
To combine the two figures above together, we must distinguish between
occasions between ๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ >๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case 1๐ต and๐๐ฟ๐ต
๐ฟ๐ต ๐ป๐ต
๐ ๐๐ป๐ต
๐ฟ๐ต ๐ป๐ต
as case ๐ต.
At this point,
Case 1๐ต๏ผ Case ๐ต๏ผ
In the case of ๐ป๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด1(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ป
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด1.๐ฟ๐ด ๐ป
๐ด/ ๐*๐ป
๐ด ๐ฝ
๐ด 0๐๐ต ๐ฟ๐ต .1 ๐๐ต/๐ป
๐ต1+
Therefore, in order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
there must be:
34
๐ฝ๐ด> ๐ผ
๐ด1
.๐ฟ๐ด ๐ป
๐ด/
๐ 0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
๐ป๐ด
,๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต-
The relationship between ๐ฝ๐ดand ๐ผ
๐ด is shown in the following figure:
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐
๐ต๐โฒ ๐
๐ต๐) ๐(๐ฟ
๐ด ๐ฝ
๐ด๐๐ต)
๐ผ๐ด2.๐ฟ๐ด ๐ป
๐ด/ ๐*๐ฟ
๐ด ๐ฝ
๐ด0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1+
Therefore, in order to meet the condition ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
> there must be:
๐ฝ๐ด ๐ผ
๐ด2
.๐ฟ๐ด ๐ป
๐ด/
๐ 0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
๐ฟ๐ด
0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1
The relationship between ๐ฝ๐ดand ๐ผ
๐ด is shown in the following figure:
Since ๐ผ ๐ผ
๐, there are
๐๐ฟ๐ด
๐ฟ๐ด ๐ป๐ด
๐ >๐๐ป๐ด
๐ฟ๐ด ๐ป๐ด
as case 1๐ด and ๐๐ฟ๐ด
๐ฟ๐ด ๐ป๐ด
๐ ๐๐ป๐ด
๐ฟ๐ด ๐ป๐ด
as case ๐ด. At this point,
35
Case 1๐ด๏ผ Case ๐ด๏ผโ
Player B must satisfy all of the conditions of game equilibrium case1๐ต or case ๐ต,
Player A must satisfy all of the conditions of game equilibrium case1๐ด or case ๐ด,
Because case ๐ด is an empty set, the union set of case 1๐ต and case ๐ต is also an
empty set. Thus, the result that satisfies both case1๐ต and case1๐ด is ๐ The
figure is as follows:
Since the result of the union set of case ๐ต and case 1๐ด is ๐ > , ๐ , ฮพ does
not exist.
The above section discusses the 8 equilibriums. The next section will discuss why
the following 8 situations are not equilibrium:
โจ ๐ ๐ ๐ 1 ๐
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
,๐๐ต๐ .1 ๐
๐ต/ ๐-* ๐ผ
๐ด2[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ฟ๐ด ] ๐ฝ๐ด๐๐๐ต+ ,๐๐ต (1 ๐ )
.1 ๐๐ต/ (1 ๐)- { ๐๐ฟ
๐ด ๐ผ
๐ด2[[๐ฟ๐ด ๐พ ๐ป๐ด (1 ๐พ)] ๐ฟ๐ด ]}
๐ฟ๐ด๐ ๐ผ
๐ด2[[๐ฟ๐ด ๐ฟ ๐ป๐ด (1 ๐ฟ)] ๐ฟ๐ด ]
36
๐ผ๐ด2.๐ฟ๐ด ๐ป
๐ด/ (๐ ๐ ๐ ๐ ๐)๐(๐ฟ๐ด ๐ฝ๐ด๐๐ต)
๐ผ๐ด2.๐ฟ๐ด ๐ป
๐ด/
Owing to ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
> must be satisfied, ๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
must
be less than 0. This contradicts ๐ 1.
โฉ ๐ 0 ๐ 0 ๐ 0 ๐ 1
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐ ๐ ๐ ๐)๐(๐ฟ๐ด ๐ฝ๐ด๐๐ต)
๐ผ๐ด2( 1) .๐ฟ
๐ด ๐ป
๐ด/
Owing to ๐ผ๐ด ( 1) .๐ฟ
๐ด ๐ป
๐ด/ is always positive, ๐
๐ด ๐๐๐๐ ๐
๐ด ๐๐๐ก ๐ ๐๐๐ must be
greater than 0. This contradicts ๐ .
โช ๐ 1 ๐ 0 ๐ 0 ๐ 1
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐ ๐ ๐ ๐)๐(๐ฟ๐ด ๐ฝ๐ด๐๐ต)
๐ผ๐ด2( 1) .๐ฟ
๐ด ๐ป
๐ด/ .1 ๐
๐ต/ ๐*๐ฟ
๐ด ๐ฝ
๐ด 0๐๐ต ๐ฟ๐ต .1 ๐๐ต/๐ป
๐ต1+
Owing to ๐ผ๐ด ( 1) .๐ฟ
๐ด ๐ป
๐ด/ and .1 ๐
๐ต/ ๐*๐ฟ
๐ด ๐ฝ
๐ด0๐๐ต๐ฟ๐ต .1
๐๐ต/๐ป
๐ต1+are always positive, ๐
๐ด ๐๐๐๐ ๐
๐ด ๐๐๐ก ๐ ๐๐๐ must be greater than 0. This
contradicts ๐ .
โซ ๐ 0 ๐ 1 ๐ 1 ๐ 0
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)( 1) ๐(๐ฝ๐ต๐ฟ๐ด ๐ฟ๐ต )
Owing to ๐ผ๐ต (๐ฟ๐ต ๐ป
๐ต)( 1) and๐ .๐ฝ
๐ต๐ฟ๐ด ๐ฟ
๐ต/ are always negative, ๐
๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
must be less than 0. This contradicts ๐ 1.
โฌ ๐ 1 ๐ 1 ๐ 0 ๐ 1
In the case of ๐ฟ๐ด
๐๐ด ๐๐๐๐
๐๐ด ๐๐๐ก ๐ ๐๐๐
๐ผ๐ด2(๐พ ๐ฟ) .๐ฟ
๐ด ๐ป
๐ด/ (๐ ๐ ๐ ๐ ๐)๐(๐ฟ๐ด ๐ฝ๐ด๐๐ต)
๐ผ๐ด2( 1) .๐ฟ
๐ด ๐ป
๐ด/ ๐*๐ฟ
๐ด ๐ฝ
๐ด 0๐๐ต ๐ฟ๐ต .1 ๐๐ต/๐ป
๐ต1+
Owing to ๐ผ๐ด ( 1) .๐ฟ
๐ด ๐ป
๐ด/ and ๐*๐ฟ
๐ด ๐ฝ
๐ด0๐๐ต๐ฟ๐ต .1 ๐
๐ต/๐ป
๐ต1+ are
always positive, ๐๐ด Seek
๐๐ด Not seek
must be greater than 0. This contradicts ๐ .
37
โญ ๐ 0 ๐ 1 ๐ 1 ๐ 1
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)( 1) ๐(๐ฝ๐ต๐ฟ๐ด ๐ฟ๐ต )
Owing to ๐ผ๐ต (๐ฟ๐ต ๐ป
๐ต)( 1) and๐(๐ฝ
๐ต๐ฟ๐ด ๐ฟ
๐ต) are always negative, ๐
๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐ ๐
must be less than 0. This contradicts ๐ 1.
โฎ ๐ 0 ๐ 1 ๐ 0 ๐ 0
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)( 1) ๐(๐ฝ๐ต๐ฟ๐ด ๐ฟ๐ต )
Owing to ๐ผ๐ต (๐ฟ๐ต ๐ป
๐ต)( 1) and ๐(๐ฝ
๐ต๐ฟ๐ด ๐ฟ
๐ต) are always negative, ๐
๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
must be less than 0. This contradicts ๐ 1.
โฏ ๐ 0 ๐ 1 ๐ 0 ๐ 1
In the case of ๐ฟ๐ต
๐๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก โ๐๐๐
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)(ํ ๐) ๐(๐ฝ
๐ต๐๐ด ๐ฟ
๐ต)
๐ผ๐ต2(๐ฟ๐ต ๐ป
๐ต)( 1) ๐(๐ฝ๐ต๐ป๐ด ๐ฟ๐ต )
Owing to ๐ผ๐ต (๐ฟ๐ต ๐ป
๐ต)( 1) and๐(๐ฝ
๐ต๐ป๐ด ๐ฟ
๐ต) are always negative, ๐
๐ต ๐ป๐๐๐
๐๐ต ๐๐๐ก ๐๐๐
must be less than 0. This contradicts ๐ 1.
4. Discussion
Based on the above equilibrium analysis results, the difference in self-awareness
ฮพ is greatest when it is larger than c, it becomes equilibriumโฅ and equilibriumโฆ. In
other words, when the degree of loss avoidance of the player is the highest, there are
equilibriumโฅ and equilibriumโฆ. At this time, the helper always helps, regardless of
help-seekersโ ability. When ๐ is small, there are equilibriumโ , equilibriumโก ,
equilibriumโข, equilibriumโฅ, equilibriumโฆ and equilibriumโง. At this time, player B
with high ability would help. ๐ผ
is the largest in equilibrium โฃ , equilibrium โค ,
equilibrium โฅ and equilibrium โฆ . At this time, we concluded that helpers and
38
help-seekers behave similarly, regardless of ability.
With respect to the helper, by equilibrium โฃ and equilibrium โค , if the
self-awareness ฮฑ is large but the politeness ฮฒ is very small, neither player B with high
ability nor with low ability would help. By equilibrium โ , equilibrium โก and
equilibrium โข, if self-awareness ฮฑ and politeness ฮฒ are within a certain range, player
B with high ability would help, player B with low ability would not. In equilibrium โฅ,
equilibrium โฆ and equilibrium โง, when self-awareness ฮฑ and politeness ฮฒ are all
large, either player B with the high ability or low ability would help.
With respect to the help-seeker, in equilibriumโฆ, If politeness ๐ฝ is small when
seeking help, either player A with low ability and with high ability would seek help. In
equilibrium โก and equilibrium โง, if self-awareness ฮฑ and politeness ฮฒ are within a
certain range, player A with high ability would not seek help, however, player A with
low ability would seek help. In equilibrium โข and equilibrium โฅ , when
self-awareness ฮฑ and politeness ฮฒ are both large, either player B with the high ability
or low ability would not seek help.
In situationโซ , situationโญ , situationโฎ and situationโฏ where there is no
equilibrium, it is impossible that player B with high ability does not help but help with
low ability. While the psychological effect of self-awareness ๐ผ and politeness ๐ฝ
affect human behavior, it is impossible to force others to do things beyond your
abilities.
In situationโฉ, situationโช, situationโฌ and situationโฏ, it is unlikely that player
with low abilities would not seek help and player with high abilities would seek help.
5. Implications
This paper studies human irrational behavior from the perspective of behavioral
economics. Through the establishment of a game model, we can understand why
people do not seek help when they are clearly in trouble. Because people have the
nature of politeness, it is difficult to ask for help from others if they are excessively
39
polite, and help others if they are excessively polite. Besides, people with strong
self-awareness are easy to care about what others think of themselves. In order to
build a good impression, they may not ask others for help, but they may help others
although it is not their original intention. Specifically speaking, from the perspective of
the help seeker, it is difficult to make clear what kind of people are likely to seek help
and what kind of people are unlikely to seek help. On the other hand, from the
perspective of helpers, what kind of people are likely to help and what kind of people
are unlikely to help.
Consequently, this study found that when the loss avoidance degree of players is
the largest, the helper will help others. Besides, when the loss avoidance degree of
players is small, people with high ability would help. In addition, people with higher
self-awareness would take the same action. If self-awareness is great, but the degree
of politeness is relatively small, no matter the ability is high or low, they will not help
each other. If the level of politeness is very low, people with the low ability or high
ability would seek help. When politeness and self-awareness are within a certain
range, neither big nor small, people with high ability would help, people with low ability
would not help; people with low ability would seek help, but people with high ability
would not seek help. When politeness and self-awareness are both great, people with
the high ability or low ability would not seek help.
6. Conclusion
This study analyzes whether people would seek help or help others from two
basic human characteristics of self-awareness and politeness. We found that, first, the
more polite the helper is to the help seeker, the easier to help. Second, helper with a
higher level of self-awareness and wish to keep a better impression from the help
seekers would help. However, helper with low ability would not grudgingly help others.
Help seeker with higher the degree of politeness would be less likely to seek help;
Besides, help seeker with a higher degree of self-awareness and care more about
40
other people's impression would not seek help.
7. Limitations and Future Directions
To simplify the model, the study has limitations. First, this study does not consider
the repaying, in other words, whether the help will be repaid in the future. Second, this
study only considered two psychological factors, politeness and self-awareness;
however, there could be other psychological factors which would influence human
behavior in this situation.
Future studies can consider the act of repaying; the game model can be set as a
repeated game. In addition, other psychological factors can be considered in the game
model for a better understanding of human help-seeking and help-giving behavior.
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