Discussion Paper No.432

Reinterpreting the General Rules of Morality and the
Corruption of Moral Sentiments in The Theory of
Moral Sentiments with an Evolutionary Game Model
Takahiko Kan

September 11, 2020TOHOKU ECONOMICS RESEARCH GROUP
Discussion Paper

GRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY
27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPAN1

Reinterpreting the General Rules of Morality and the Corruption of Moral
Sentiments in The Theory of Moral Sentiments with an Evolutionary Game
Model
Takahiko Kan1

1. Introduction
Adam Smith is the author of An Inquiry into the Nature and Causes ofthe Wealth of Nations. He is known mainly as a pioneer of political economy.
However, he was not only an economist but also a moral philosopher. He
published The Theory of Moral Sentiments (TMS) in 1759. In TMS, he
explained an establishment of a social order based on sympathy between
people in a society. Sympathy is sharing of sentiments with others byimaginarily swapping situations with others. People in TMS form theimpartial spectator and regulate their conduct to be sympathized by theimpartial spectator. The impartial spectator is often considered as animportant concept in TMS. However, even if people formed the impartial
spectator, this does not mean that they can always regulate their conduct. To
regulate their conduct absolutely, people need general rules of morality(GRM). People can establish a social order thanks to GRM.
Some preceding studies have reinterpreted TMS with various research
results in contemporary economics. For example, Meardon & Ortmann(1996) reinterprets self-command by using a repeated game theory model.
Ashraf et al. (2005) indicates that Smith foresaw some research findings ofbehavioral economics. Tajima (2007) reinterprets TMS from a perspective of
institutional economics. Bréban (2012) formularizes a behavior of people in
TMS with a utility function, and compares this function with utility
functions in behavioral economics. Khalil (2017) reinterprets TMS from aperspective of rational choice theory. These reinterpretations have shed lighton modern significance of TMS.
However, there is room for reinterpreting important concepts in TMS
with research results in contemporary economics. Following the precedingstudies, this paper reinterprets the GRM formation process and the
Graduate School of Economics and Management, Tohoku University, Sendai, Japan.
Mail: right1414@yahoo.co.jp
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corruption of moral sentiments (CMS) by using a replicator dynamics model,
which is a basic model of evolutionary game theory. GRM are the social
norms in TMS that concern what is fit and proper either to be done or to be
avoided. In TMS, people form GRM through interactions with others. They
continually observe conduct of others, and this can lead them to form certain
GRM. This paper interprets this observation process as a trial-and-error
learning process. To formularize this process, this paper uses a replicator
dynamics model.
The results of the model clarify the character of sympathy in the CRM.
The more sympathetic players exist in a player set, the more corrupted
situation is likely to be realized. This result mathematically supports aninterpretation in preceding studies (Brown 1994, Griswold 1999) that
sympathy involves risk that CMS is progressing.
The paper is organized as follows. In the section 2, we briefly describe
GRM and the CMS. In the section 3, we construct a model of replicator
dynamics. In the section 4, we discuss the results of the model. In the lastsection, we conclude this paper.

2. The Theory of Moral Sentiments
2.1 General Rules of Morality

People in TMS form the impartial spectator and regulate their conduct tobe sympathized by the impartial spectator. The impartial spectator is often
considered as an important concept in TMS. However, even if people formed
the impartial spectator, this does not mean that they can always regulate
their conduct. This is because “so partial are the views of mankind withregard to the propriety of their own conduct, both at the time of action andafter it; and so difficult is it for them to view it in the light in which any
indifferent spectator would consider it” (TMS. III.iv.4). Smith called this kindof problematic situation “self-deceit” (TMS.III.iv.4). According to Smith, tocope with self-deceit, Nature gave people GRM. GRM are the social norms in
TMS concerning what is fit and proper either to be done or to be avoided.
“General rules of conduct, when they have been fixed in our mind by habitualreflection, are of great use in correcting the misrepresentations of self-love3concerning what is fit and proper to be done in our particular situation”(TMS. III.iv.12). GRM are necessary to maintain social orders. “The regard to
those general rules of conduct, is … the only principle by which the bulk of
mankind are capable of directing their actions” (TMS. III. v.1). Smith
referred to cases that people do not have to follow GRM strictly. However, heheld that people should strictly follow GRM of virtue of justice, which isessential for the establishment of the social order in TMS. Virtue of justicerequires not doing “real and positive hurt to some particular persons, frommotives, which are naturally disapproved of” (TMS.II.ii.1.5). “If it [justice] isremoved, the great, the immense fabric of human society… must in a
moment crumble into atoms” (TMS.II.ii.3.4).
GRM about actions to be avoided are formed thorough the following
process. People’s sentiments are shocked from some immoral action of others,with which the impartial spectator has no sympathy. People feel this kind ofaction unseemly. When people hear others express the same feeling as them,“this still further confirms, and even exasperates our [their] natural sense of
their deformity” (TMS.III.4.7). After repeatedly hearing others express thesame feeling as them, people “resolve never to be guilty of the like, nor ever,upon any account, to render ourselves in this manner the objects of universal
disapprobation” (TMS.III.4.7). Then, people form a GRM about an action to
be avoided. GRM about actions to be done are formed through a similar
process as the GRM about actions to be avoided.

2.2 The Corruption of Moral Sentiments

Smith added a chapter about corruption of moral sentiments (CMS) in
the sixth edition of TMS. According to Smith, disposition to “admire, andalmost to worship, the rich and the powerful, and to despise, or, at least, toneglect persons of poor and mean condition, though necessary both toestablish and to maintain the distinction of ranks and the order of society, is,at the same time, the great and most universal cause of the corruption of ourmoral sentiments” (TMS.I.iii.3.1). He distinguished two moral principles: theroad to virtue and the road to fortune. People in the road to virtue try to berespected “by the study of wisdom and the practice of virtue”, whereas people
in the road to fortune do so “by the acquisition of wealth and greatness”. The
road to fortune can involve vices and follies, which contradict the road to4virtue.
In explaining why the CMS occurs, Smith distinguished two different
ranks of life: middling–inferior stations of life and superior stations of life
(Brown 1994, 35). This distinction characterizes the CMS (Hill 2006, 650). In
superior stations, the road to fortune causes the CMS, because, in superior
stations, “success and preferment depend, not upon the esteem of intelligent
and well-informed equals, but upon the fanciful and foolish favour ofignorant, presumptuous, and proud superiors” (TMS.I.iii.3.6). In superior
stations, the impartial spectator is too weak to prevent people from
corrupting (Hill 2006, 650–651).
In middling–inferior stations, the road to fortune basically causes nocorruption, but people in these stations admire and imitate people insuperior stations. This happens due to a character of sympathy with joy(Tajima 2007, 591). People in middling–inferior stations imitate people insuperior stations, and this cause the CMS in middling–inferior stations(TMS.I.iii.3.7). Superior stations popularize not only a fashion but also vices
and follies.
Some authors consider that the CMS was the dominant Smith’s motive
for publishing the sixth edition (Dickey 1986, 608;32 Evensky 1989, 131;
Dwyer 2005, 684; Forman-Barzilai 2010, 97).

3 Formularizing the formation process of GRM
3.1 The model
As we briefly explained above, GRM are “formed, by finding fromexperience, that all actions of a certain kind, or circumstanced in a certain
manner, are approved or disapproved of” (TMS.III.4.8). This formation
process of GRM can be regarded as a process of trial-and-error. Trial-anderror learning is a kind of empirical learning, through which people evaluate
their choice based on its consequence and modify their way of choice. In theformation process, people evaluate justifiability of their feeling based onwhether others express the same feeling as them or not. When they know
others express the same opinion as them, this justifies their opinion. People
evaluate justifiability of their feeling (choice) based on whether others

5express the same opinion (consequence).
To formularize the learning process, this paper uses a replicator
dynamics model with two strategies. Players have their mixed strategies on
the two strategies. The set of players is 𝐼 = {1,2, … , 𝑛}, where 𝑛 = 2𝑚(𝑚 ∈ ℕ).
The set of pure strategies of player 𝑖 is 𝑆𝑖 = {𝐴, 𝐵}. We consider strategy A as
virtue strategy. We consider strategy B as fortune strategy. These two
strategies derive from the road to virtue and the road to fortune in the CMS(TMS. I.iii.3.1).
The mixed strategy of player 𝑖 ∈ 𝐼 is 𝑥𝑖 = (𝑥𝑖𝐴 , 𝑥𝑖𝐵 ) ∈ [0,1]2 . The profile ofmixed strategies of players is 𝑥 = (𝑥1 , 𝑥2 , . . , 𝑥𝑛 ) ∈ [0,1]2×𝑛 .
In the model, choosing a strategy means that the player considers heshould choose this strategy. Justifiability of each strategy is reflected in
mixed strategies. We assume probabilistic choice of players because, before
GRM is formed, people in TMS probabilistically choose what they should do.
Before GRM is formed, people sometimes fail to choose what the impartial
spectator suggests because they suffer from “self-deceit” (TMS. III.iv.4). Once
GRM is formed, they can surely choose what the impartial spectatorsuggests.
Trial-and-error learning processes have been formularized by using someevolutionary game models.2 This paper uses a simple replicator dynamics
model among evolutionary game models, because other models involve
assumptions which cannot be supported by sentences in TMS.
Each player is randomly matched with one player and then observes the
choice of the opponent. This matching with observation is continuously
repeated.
Each 𝑖’s 𝑥𝑖𝐴 changes depending on their payoffs of each game. Purestrategies of player 𝑖 is 𝑠𝑖 ∈ 𝑆𝑖 . The profile of pure strategies is 𝑠 =(𝑠1 , … , 𝑠𝑛 ), and 𝑠 ∈ 𝑆 =×𝑛𝑖=1 𝑆𝑖 . For any 𝑠 ∈ 𝑆, payoff of player 𝑖 is definedwith pure strategy payoff function 𝜋𝑖 : 𝑆 → ℝ.
When mixed strategy profile 𝑥 ∈ [0,1]2×𝑛 is given, 𝑠 = (𝑠1 , … , 𝑠𝑛 ) ∈ 𝑆
has the probability 𝑥(𝑠) = ∏𝑛𝑖=1 𝑥𝑖𝑠𝑖 ∈ [0,1].
Let 𝑢𝑖 : [0,1]2×n → ℝ be𝑢𝑖 (𝑥) = ∑ 𝑥(𝑠)𝜋𝑖 (𝑠).
𝑠∈𝑆Function 𝑢𝑖 is expected payoff of player 𝑖 when 𝑥 ∈ [0,1]2×𝑛 is played.
2See, for example, Roth & Erev(1995) and Erev & Roth(1998).

6𝑠

Let 𝑒𝑖 𝑖 denote the mixed strategy of 𝑖 when 𝑖 chooses pure strategy 𝑠𝑖 .
𝑠𝑠Expected payoff of player 𝑖 when 𝑖 chooses 𝑒𝑖 𝑖 is 𝑢𝑖 (𝑥1 , … , 𝑒𝑖 𝑖 , … 𝑥𝑛 ).
To know the behavior of 𝑥, knowing the behavior of 𝑥A = (𝑥1𝐴 , … , 𝑥𝑛𝐴 ) ∈[0,1]𝑛 is sufficient. Let the dynamics of 𝑥𝑖𝐴 be
𝑥̇ 𝑖𝐴 = [𝑢𝑖 (𝑥1 , … , 𝑒𝑖𝐴 , … 𝑥𝑛 ) − 𝑢𝑖 (𝑥)]𝑥𝑖𝐴 ,where 𝑢𝑖 (𝑥) is the expected payoff of 𝑖 when 𝑥 is played, and 𝑢𝑖 (𝑥) = 𝑥𝑖𝐴 ∙𝑢𝑖 (𝑥1 , … , 𝑒𝑖𝐴 , … 𝑥𝑛 ) + 𝑥𝑖𝐵 ∙ 𝑢𝑖 (𝑥1 , … , 𝑒𝑖𝐵 , … 𝑥𝑛 ) holds. In the dynamics，the higher
expected payoff of a strategy is, the higher the probability of choosing thestrategy becomes.
The payoff matrix of 𝑖 is described as follows.

player 𝑖／ opponent

A

B

A

𝑖
𝑎𝐴𝐴

𝑖
𝑎𝐴𝐵

B

𝑖
𝑎𝐵𝐴

𝑖
𝑎𝐵𝐵

Table

Payoff of player 𝑖𝑖𝑖𝑖𝑖
Let us define 𝛼𝑖 ≡ 𝑎𝐴𝐴− 𝑎𝐵𝐴
and 𝛽𝑖 ≡ 𝑎𝐴𝐵− 𝑎𝐵𝐵. The simultaneous
differential equations of the dynamics are the following. To derive the𝑖𝑖equations, we used 𝑢𝑖 (𝑥1 , … , 𝑒𝑖𝐴 , … 𝑥𝑛 ) = 𝑎𝐴𝐴(∑𝑗≠𝑖 𝑥𝑗𝐴 ⁄𝑛 − 1) + 𝑎𝐴𝐵(1 −𝑖𝑖(∑𝑗≠𝑖 𝑥𝑗𝐴 ⁄𝑛 − 1))，and 𝑢𝑖 (𝑥1 , … , 𝑒𝑖𝐵 , … 𝑥𝑛 ) = 𝑎𝐵𝐴(∑𝑗≠𝑖 𝑥𝑗𝐴 ⁄𝑛 − 1) + 𝑎𝐵𝐵(1 −(∑𝑗≠𝑖 𝑥𝑗𝐴 ⁄𝑛 − 1)).

𝑥̇ 1𝐴 = ((𝛼1 − 𝛽1 )(∑𝑗≠1 𝑥𝑗𝐴⁄𝑛 − 1) + 𝛽1 ) 𝑥1𝐴 (1 − 𝑥1𝐴 )，⋮𝑥̇ 𝑖𝐴 = ((𝛼𝑖 − 𝛽𝑖 )(∑𝑗≠𝑖 𝑥𝑗𝐴⁄𝑛 − 1) + 𝛽𝑖 ) 𝑥𝑖𝐴 (1 − 𝑥𝑖𝐴 )，
⋮

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∑
𝑥
𝑥̇ 𝑛𝐴 = ((𝛼𝑛 − 𝛽𝑛 )( 𝑗≠𝑛 𝑗𝐴⁄𝑛 − 1) + 𝛽𝑛 ) 𝑥𝑛𝐴 (1 − 𝑥𝑛𝐴 )。
From these equations, all edge points are equilibrium points. In thisdynamics, all interior points are not asymptotically stable. We can proof thisresult by using the proposition 6.6 in Weibull(1997). To consider theasymptotically stable points, we only need to check edge points.
This paper assumes that there are three types of players. These types aredefined based on payoff matrices.
𝑖𝑖𝑖𝑖
Player 𝑖 is sympathy type if 𝛼𝑖 > 0，𝛽𝑖 < 0 ， 𝑎𝐴𝐵
< 𝑎𝐴𝐴
， 𝑎𝐵𝐴
< 𝑎𝐵𝐵
.
Sympathy type players get higher payoff when they choose the same strategy
as their opponents.

Player 𝑖 is virtue type if 𝛼𝑖 > 0，𝛽𝑖 > 0. Virtue type players get higher
payoff when they choose strategy A than when they choose strategy B. Thismeans 𝑥̇ 𝑖𝐴 > 0 holds except at equilibrium points.
Player 𝑖 is fortune type if 𝛼𝑖 < 0，𝛽𝑖 < 0 . Fortune type players gethigher payoff when they choose strategy B than when they choose strategy A.
This means 𝑥̇ 𝑖𝐴 < 0 holds except at equilibrium points.

3.2 Seven Cases of Games
There are seven cases of games because we assume three types of players.
Results of the seven cases are shown in the following. This paper provides
supplementary information in the appendix.

Case 1 All players are sympathy type
Both (0,0, … ,0) and (1,1, … ,1) are asymptotically stable. Asymptoticalstability of the other edge points depends on each player’s 𝛼𝑖 and 𝛽𝑖 .

Case 2 All players are fortune type(0,0, … ,0) is the unique asymptotically stable point because 𝑥̇ 𝑖𝐴 < 0holds for any 𝑖 except at equilibrium points.

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Case 3 All players are virtue type
(1,1, … ,1) is the unique asymptotically stable point because 𝑥̇ 𝑖𝐴 > 0
holds for any 𝑖 except at equilibrium points.

Case 4 Sympathy type and fortune type(0,0, … ,0) is asymptotically stable, but (1,1, … ,1) is not asymptoticallystable. Asymptotical stability of other edge points depends on each player’s
𝛼𝑖 and 𝛽𝑖 .

Case 5 Sympathy type and virtue type
(1,1, … ,1) is asymptotically stable, but (0,0, … ,0) is not asymptoticallystable. Asymptotical stability of other edge points depends on each player’s
𝛼𝑖 and 𝛽𝑖 .

Case 6 Virtue type and fortune type
Without loss of generality, assume player 1 to player 𝑝 are fortune type.
The asymptotically stable point is unique, and, at this point, the firstelement to the 𝑝th elements are all 0, and the rest elements are all 1. Thisis because, except at equilibrium points, 𝑥̇ 𝑖𝐴 < 0 holds for any fortune typeplayer, and 𝑥̇ 𝑖𝐴 > 0 holds for any virtue type player.

Case 7 Three types
Asymptotically stable points vary depending on each player’s 𝛼𝑖 and 𝛽𝑖 .

9FigureTwo sympathy type players game4 Discussion
Smith considered the road to fortune causes the CMS. We can consider
that the existence of fortune type characterizes the CMS. That is why thispaper clarifies what the existence of fortune type means. This paper
compares case 3 and 6 first. Then this paper compares case 1 and 4, and case
5 and 7.
We define a degree of corruption as the number of players choosing
fortune strategy. By this definition, the most corrupted point is (0,0, … ,0). In
contrast, the least corrupted point is (1,1, … ,1), where we can consider a
GRM is formed.
Let us assume a change from case 3 to 6. Adding fortune type players
increases the degree of corruption at the asymptotically stable point by the
number of the fortune type. When the number of the fortune type is less, thedegree of corruption is also less.
Next, let us assume a change from case 1 to 4. In case 1, both (0, … ,0)and (1, … ,1) are asymptotically stable regardless of each player’s 𝛼𝑖 and 𝛽𝑖 .
In case 4, (1, … ,1) becomes not asymptotically stable, and (0, … ,0) becomes

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the unique point that is asymptotically stable regardless of each player’s 𝛼𝑖and 𝛽𝑖 . Adding merely one fortune type player makes the most corrupted
point, (0, … ,0), be the unique point.
Last, let us assume a change from case 5 to 7. In case 5, (1,1, … ,1), theleast corrupted point, is asymptotically stable regardless of each player’s 𝛼𝑖and 𝛽𝑖 . This never holds once adding merely one fortune type player.
As shown above, adding fortune type players increases the degree of
corruption of asymptotically stable points. In case 1 and 5, such an addition
can increase the degree of corruption more than by the number of the fortune
type. We can consider such increase of the degree of corruption is more
severe in case 1 (to 4) than in case 5 (to 7). This is because, only in case 4, themost corrupted point (0, … ,0) is the unique asymptotically stable point.
Thus, the more sympathy type players exist, the more corrupted situation
fortune type players cause.

case

Change1 to 4(1,1, … ,1) becomes not asymptotically stable.
(0, … ,0) becomes the unique asymptotically stable point.

3 to 6The degree of corruption of asymptotically stable points increases
by the number of the fortune type players.

5 to 7(1,1, … ,1) becomes not asymptotically stable.
Table

Adding fortune type players

5 Concluding Remarks
This paper provides a formularization of corruption of moral sentiments(CMS), which is consistent with the formation process of general rules ofmorality (GRM). The model reveals that the more sympathy type players
exist, the more corrupted situation fortune type players cause.
The result is consistent with an interpretation in preceding studies(Brown 1994; Griswold 1999) that sympathy involves risk that CMS isprogressing. Sympathy type players are consistent with sympathy as its
name suggests. This paper mathematically supports the interpretation inpreceding studies.

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Appendix
Let us consider the Jacobian matrix of the simultaneous differential
equations of the dynamics. A necessary and sufficient condition forequilibrium points to be asymptotically stable is that the Jacobian is stable.
In the following, we consider only equilibrium edge points. This is because allinterior points are not asymptotically stable in the dynamics. We can proof
this result by using the proposition 6.6 in Weibull(1997).
Diagonal component of the Jacobian at 𝑖th row is𝜕𝑥̇ 𝑖𝐴∑ 𝑥= ((𝛼𝑖 − 𝛽𝑖 )( 𝑗≠𝑖 𝑗𝐴⁄𝑛 − 1) + 𝛽𝑖 ) (1 − 2𝑥𝑖𝐴 ).
𝜕𝑥𝑖𝐴
Non-diagonal component of the Jacobian at 𝑖th row is𝜕𝑥̇ 𝑖𝐴 𝛼𝑖 − 𝛽𝑖=𝑥 (1 − 𝑥𝑖𝐴 ).
𝜕𝑥𝑗𝐴
𝑛 − 1 𝑖𝐴

At equilibrium edge points, non-diagonal components of the Jacobian are
zero. Thus, a necessary and sufficient condition for equilibrium edge points
to be asymptotically stable is that all diagonal components of the Jacobian
are negative.
We can get the results of case 1 to 7 by using the above information. Let
us consider case 1 for instance.

Case 1 All players are sympathy type(0,0, … ,0) and (1,1, … ,1) are asymptotically stable. Asymptoticalstability of the other edge points depends on each player’s 𝛼𝑖 and 𝛽𝑖 .
At (0,0, … ,0) and (1,1, … ,1), all diagonal components of the Jacobian are
negative. Thus, (0,0, … ,0) and (1,1, … ,1) are asymptotically stable. At otheredge points, the sign of diagonal components can be positive depending oneach player’s 𝛼𝑖 and 𝛽𝑖 . For example, at (0,0, … ,0,1) , the diagonalcomponent of the Jacobian at first row is (𝛼1 − 𝛽1 )(1⁄𝑛 − 1) + 𝛽1 , which canbe positive with some 𝛼1 and 𝛽1 because player 1 is sympathy type.

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