Individual Optimal Choice and Social Consensual Actions∗
-- On Logic Connotation and Practice Characteristics of fair distribution
Guocheng WANG
Kui OUYANG
Abstract: Based on strategic integration among rational behavior of individuals, this paper
proposes a definition of Social Consensual Actions in a sense of game equilibrium. One public
player (with the objective function of social welfare function) is introduced, game theory and
social choice theory are combined together to further propose the concept of socially optimal
contract. As a tool, the paper respectively compares logical connotation and practice
characteristics of several common fairness criteria, further discusses technique realization of the
collective action calculation, and in an attempt to provide a new way of thinking for analyzing
complex behavior relationships between individual and groups and solving real-world equitable
distribution and other relative issues.
Keywords: strategic behaviors of individuals; public player; social choice; fair distribution;
analytical framework of game equilibrium
JEL Classification: C70, D71, D63
Guocheng Wang, Institute of Quantitative & Technical Economics, Chinese Academy of Social Sciences, Beijing
100732 China; [email protected]
Kui Ouyang, School of Economic & Management, Northwest University, Xi’an 710127 China;
[email protected]
How to reasonably organize and effectively achieve individual optimal choice as well as their
coherence with group goals in level of collective action is a major theoretical and practical issue
which has been constantly testing and motivating human wisdom. Currently, contradiction
between diverse interests and global integration, micro-selected personalized (behavior
diversification) and macroeconomic integrity shall be increasingly prominent and fierce,
especially in the face of all kinds of crises, conflicts and other increasingly complex social and
economic activities, the traditional theories appear lag embarrassment when endured repeatedly
relentless shocks and inspection, which is in fact pregnant with a good opportunity and an urgent
practical needs to push contemporary economic theory development to a very important choice
crossing. Principal-agent model and incentive mechanism design have obvious theoretical
tendencies to unilaterally make utility maximization as the goal under asymmetric information
(Rees, 1985; Myerson, 2008); multilateral negotiations and contract theory focuses on joint utility
maximization under the assumption of symmetric behavior (Binmore et.al., 1986; Hart and Moore,
1988); while public choice and explore of collective action logic get more negative conclusion,
although there is improvement of Sen and others, a series of impossibility theorem derived from
rigorous logic and Arrow's work seems construct insurmountable obstacles for meeting maximum
∗
The paper is theoretical support section of the major national research programs of China (973 project, series
number: 2012CB955802).
individual wishes of social choice (Olson, 1965; Arrow, 1951); increasingly hot viewpoints and
methods of using evolutionary theory to study cooperative behavior are also mainly individual
perspective (Nowak, 2006); various efforts and achievements of welfare economics, development
economics and macro-economists in filling artificially macro and micro gully, might be able to
promote us to more focus on research of strategy actions between individual from the
perspective of individual and groups behavior, based on game model and analytical framework so
as to reveal complex intrinsic relationship between individual choice and collective action.
This paper shall broaden horizons of behavior analysis, gradually explore the topic based on
relationship between individual and collective action, and the structure is as follows: section I
defines and interprets social desirable actions on the basis of strategic behaviors of individuals;
section II combines game theory and social choice theory together to construct social selection
function based on game equilibrium; section III focuses on analysis of logical connotation and
their relationships between several common fairness criteria; section IV further analyzes the
basic characteristics and practical significance of socially optimal contract; finally briefly gives
conclusive reviews and prospects.
1. Strategic turning of individual self-interested behaviors and socially
consensual actions
From the perspective of individual self-interest to research individual rational behavior and
third-party-involved nature and from the perspective of behavior relationship to research
collective rationality and social effects shall broaden academic horizons, enrich content of
behavioral research and explore its essential character.
1.1 Strategy of individual behavior
Humans are social animals, material resources available are always limited in interaction with the
natural world, their action space is definitely constrained by mutual influence of different subjects,
and behavior pattern may also be significant differences between a relatively independent decision
of a single individual and a decision under multi-body interaction conditions. Therefore,
individual behavior space is divided into two sections or parts: relatively independent sections I
which is aim to individual utility maximization and section II which shall consider public interest
objectives under influence of strategic behavior (please make reference to deflected straight line in
Figure 1), it is more conducive to make a comprehensive analysis of behavioral properties and
seek its essential character. In a relatively independent of individual decision-making stage,
completely independent choice (just do it within legal framework) shall be made to pursuit utility
maximization; in public interest stage of mutual influence (strategic behavior), common rules shall
be obeyed and individual must not do anything prohibited (any acts must be explicitly authorized
by laws). Actions objectives and methods of the two sections are also significantly different:
individual behavior is pursuit of optimal choice, collective action is avoid of the worst; by
comparison, individual response is flexible and sensitive, collective action mode is slow and lag;
different external conditions of the sections shall induce and stimulate different behaves and
specific decisions behaviors, specific behave under interactions conditions of section II shall be
more affected by strategy selections of the double (multiple) parties; and each individual's turning
point λ0 is closely related to their acts essence, resource endowments, external environment
(including relationships with others) and heterogeneity, strategic and non-linear characteristics of
individual behavior shall be manifested accordingly.
Utility
• • I
• • II
λ0
Behavior space Figure 1: schematic of individual non-linear decision-making behavior
(In the figure, single dotted broken line is individual decision-making section with relative
stability of marginal utility; λ0 is demarcation point or threshold for behavior; double dotted
broken line in section II is strategic behavior stage under interact effects, which may be like
fluctuating shock or piecewise linearity, and shall be shown as solid point; broken line indicates
another possible behavior.)
Behavior assumptions and correlation analysis of homogeneous representative subject shall
be adopted in section I, and interaction behavior analysis of heterogeneity subject shall be adopted
in section II. Specific performance of heterogeneity of individual behavior are as follows: scale
structure (weight distribution) of diverse behavioral properties of each individual endowments,
conditional dependencies and critical point interacting with the outside world and the evolution
process (λ0 in Figure 1) are different, thus transmission mechanism of collective action and the
multiplicity of output results shall be affected. So, in the study of individual decision-making
behavior, we must note that changes may occur in order to achieve public interest under the
interaction of strategic behaviors (thereby increasing their own interests). As long as the individual
selection is non-linear and differentiated, public interest objectives in collective action is bound to
appear inconsistent and uncoordinated. In other words, macro- complex stylized facts shall be
representation of multiplicity of the Nash equilibrium, which shall be radically caused by diversity
of choice of heterogeneity individual in interactive behaviors. By analyzing strategy steering of
individual self-interest behavior to broaden their horizons and expand territory is the inevitable
development of the theory and practical needs.
1.2 Theoretical framework for socially reasonable action
Strategy behavior between group members may lead to co-create surplus (value-added), it also
may reduce individuals' utility caused by uncooperative friction. Homogeneous rational subject
can be directly added to the total generation sum based on cardinal utility, but for heterogeneity
subjects with different dimensions, diverse properties, a variety of subjects shall be not
comparable and additive, and shall not be simple summed and contain sequence factors, different
units of measurement, its ordinary utility should be given more emphasis. This leads to collective
action if and only if a relative social optimum, and basic properties and characteristics of the
collective action at this time shall be desirable; whether optimal contract of social choice (welfare)
function based on desirable actions occurs or not shall depend on choice behavioral attributes of
individual members (consider non-linear decision-making behavior of strategy affect), criteria of
target output (mainstream values), organizational structure evolution, adaptive groups cognition
(cultural identity) as well as response mode to external environmental conditions.
Collective action with identical objectives and absolutely consistent pace (goal or value
function is isomorphic) is usually impossible exist in reality, when individual chooses to extend to
strategic behavior section with mutual influence, interaction between many differentiated
individual will led to what collective action, how to choose scale criteria and descriptions and
measurement methods, what kinds of theoretical framework shall we find and construct? Because
individual is behavior composite body of self-interest in instinctive and is conditionally altruistic
with symbiosis of endowments rationality and learning adaptability, coexist of acting
independently and cooperation tendencies and other diversified properties, but proportion, critical
point changed with the external conditions of each part (component) may be different from each
other, and its inherent essential characteristics shall be more clearly manifested after consideration
of strategic interaction, which shall be specifically showed as crossing bifurcation locus of
selected phases (see Figure 1); due to the non-unity of collective action value function (target),
multi-channel and multi-directional chain between order relations, role of institutional rules and
intervention effect of government behavior in realty, non-linear interaction intrinsic properties of
different individual behaviors in certain circumstances, after their own beliefs and judgments,
mutual trust agitation and fermentation and brewing of organizational structure, shall hide overall
output with a variety of possibilities, evolve various stylized facts in different networks structures,
make relationships between individual choice and collective action be complex and confusing and
lead to a multiplicity of game equilibrium theory and macro-emerge in the practice. It shall be
causes and hypostasis of the socio-economic complexity, but also to some extent indicate
necessity and inevitability of public participant (or intervention of some force) in coordination of
collective action.
The Prisoner's Dilemma1 and other classic stories in game theory reveal complex intrinsic
link between individual optimal choice and collective action results (also known as social choice
paradox), but also construct an exemplary theoretical analysis framework which systemically
show action characters, conduction mechanism and various possible results of individual and
collective subjects, lead to integration and reproduction of organic links among individual choice,
overall output and distribution structure. Under conditions of influence between each other,
effectiveness or output of each individual is not only determined by its own behavior, but also
determined by behaviors of others and changes of external conditions (or even in some cases,
determined by choose of others); at the same time, their behavior also influence others. "Together"
mainly refers to compliance (level), "intention" means intent / desire, can be recognized by most
participant or accepted by wishes of majority participant, relatively shall be understood as a
dynamic boundary asymptotic process until consensus, rather than a point (value) of the state,
which is suitable for analyzed with modern set theory in the game framework. Furthermore, we
shall in the game equilibrium sense define socially reasonable collective action and social optimal
contract, etc., to study their achieve conditions, stability (uniqueness) and so on. Socially
Reasonable Actions defined in such method is impossible to ensure that all participants can
simultaneously maximize their own utility, but it can avoid the worst result of groups: each
participant can’t improve their income relying on a unilateral strategy transformation, but they do
not worry about other participant's behavior change strategy will reduce effectiveness of their own,
and no one shall take initiative to deviate from the common expected outcome of the game
equilibrium, as a result, it shall be relatively stable.
1
Founded by Merrill Flood，Melvin Dresher and Albert W. Tucker，please make reference to: Poundstone, W.
(1992) Prisoner's Dilemma. NY: Doubleday. 2. Constructs social choice function based on game equilibrium
2.1 Game equilibrium and social choice
Nash equilibrium - the core concept of non-cooperative game theory is a self-driven contract1, no
one wants to unilaterally deviate from the mutually agreed contract. However, such contracts are
often multiple and reflect non-unique fair criteria of social selection in the reality. How to select
reasonable, practicable and relatively stable contract from multiple contracts requires impose
certain collective rationality restrictions on the Nash equilibrium caused by individual rationality.
This can be illustrated by extended couple game of the following table with three alternative
events.
Table 1. Extended couple game
1
2
Ballet
Movie
Football
Ballet
1，1
0，0
0，0
Movie
0，0
1，5
0，0
Football
0，0
0，0
4，4
In the above table: (ballet, ballet), (movie, movies) and (football, football) are three pure
strategy Nash equilibrium of the game. However, due to the asymmetry of information, there is no
necessary and internal relations between personal preferences and the largest groups output of
social welfare, then, the actual results of which will happen in the end? While it is intuitively easy
to see (ballet, ballet) mix of strategies are Pareto dominant with the other two Nash equilibrium,
however, intuition is lack of rigorous theoretical support, in reality, there is no absolute certainty to
ensure that other two results shall be more likely to occur. And if participants 1 and 2 can
communicate and coordinate in advance, then (football, soccer) shall be a more credible result.
This result is not only groups optimal for both sides, but is also stable. Thus, communication and
coordination between participants making the game essentially became a kind of social choice.
Specifically, couple game shown in the Table 1 can be transformed into social choice problem in
the following table:
Table 2. Societal choice problem transformed from the extended couple game
Ballet，Ballet
Ballet，Movie
Ballet，Football
Movie，Ballet
Movie，Movie
Movie，Football
Football，Ballet
Football，Movie
Football，Football
At this point, if actions space of participants are regarded as the alternative social state sets,
so that contract selection process during the exchange and coordination of the two participants can
be transformed into a socially optimal choice. If a social welfare function is defined or constructed
based on participants, then a problem of socially optimal choice shall be deemed as a social
welfare maximization problem. In the case of social welfare is regarded as the sum of individual
utility (simplified as algebra sum), the corresponding social welfare levels of social state sets
shown in Table 2 are as follows:
Table 3. Social welfare of extended couple game
2
0
0
0
6
0
0
0
8
1
Brandenberger, A. and E. Dekel. Rationalizability and correlated equilibrium. Econometrica, 1987, 55(6):
1391-1402. Seen from Table 3, (football, football) is a combination action of maximized social welfare.
However, generally speaking, there are still two problems in definition of socially optimal contract
options: first, in theory, it does not guarantee that action combination of social welfare
maximization shall be equivalent to Nash equilibrium1, and participant always hope a good
collective action result is robust, which require socially optimal contract choice must be included
in the Nash equilibrium; second, there is potential of great controversy on social welfare is
expressed as a simple algebra sum of individual utility, how single amount generate a total amount
is still a mystery. The utilitarian ideas and other social welfare functions which comply with
Bergson - Samuelson traditions23 are strongly criticized.
The Arrow impossibility theorem shows that if there are at least three alternative social states,
then in the unrestricted domains, when social preferences satisfy completeness and transitivity,
social welfare function which meets the Pareto criteria and independence of irrelevant choice
definitely meet authoritarian. This is necessary justification to impose restrictive conditions so as
to achieve collective rationality. It is suggested that the choice of social welfare function, the
ordinal principle can’t give a satisfactory answer due to lack of precision. In fact, in many cases,
social choice has to rely on the base properties which can be measured of preference strength. In
interpersonal comparison influenced by strategic behavior, social choice is related to the most
representative social welfare evaluation criteria, including three types: utilitarian4, Rawlsian5 and
Nash bargaining solution6. In a sense, the Nash bargaining solution can be seen as compromise
between utilitarian and Rawlsian. In these three evaluation criteria of social welfare, only the Nash
bargaining solution can meet non- comparability of the base. Of course, this does not necessarily
mean that Nash bargaining solution has more advantages than any other standards. To protect
public interest and achieve collective rationality, the paper's approach is to introduce a common
participant in the game who shall act as a coordinator of all participants in prior exchanges and
consultations in order to construct a social welfare function with richer meaning.
In reality, it is usually to see that preference level of alternatives and value of the utility
function of game participants are heterogeneous, due to self-interest of individual choice,
non-linear characteristics involved, and the resulting complex and changes of community
organizations and pay structure correspondingly, and thus can be inferred that preferences
distribution, behavior attribute and choices of pluralistic individual lead to non-uniqueness of
group decision-making results. As heterogeneity and interactive behaviors and other individual
attributes (diversification behavior) cause a lot of troubles for the analysis of results of collective
action and complexity of macroscopic phenomena, therefore, we must deeply analyze
heterogeneity and interactive behaviors and other individual attributes and regard it as the logical
starting point to reveal the complex mystery. Based on heterogeneous interacting agent to study
complex economic problems7, is not only a major contribution to economic theory, but also
1
Although in this paper, action combination of social welfare maximization is Nash equilibrium, but in a wide
range of cases, combination of social welfare maximization strategy is not stable Nash equilibrium, such as in the
prisoner's dilemma (do not confess, do not confess). 2
Bergson, A. A reformulation of certain aspects of welfare economics, Quarterly Journal of Economics, 1938, 52:
310-334. 3
Samuelson, P. A. Foundations of Economic Analysis, Cambridge: Harvard University Press, 1947. 4
Bentham, J., An Introduction to the Principles of Morals and Legislation, London: Payne, 1789. 5
Rawls, J. A Theory of Justice, Cambridge: Harvard University Press, and Oxford: Clarendon Press, 1971. 6
Nash, J. F., The bargaining problem, Econometrica, 1950, 18, 155-162. 7
Kirman A., Complex Economics: Individual and Collective Rationality, The Graz Schumpeter Lectures, London:
Routledge, 2010. important progress and development in epistemological. In this case, a social welfare function
shall be regarded as a public utility function exogenously, I believe that this attempt is not only
theoretically possible, but also is realistic.
2.2 Game model with public player
Similar to virtual outcry of the general equilibrium price formation mechanism in the perfectly
competitive market, a series of impossibility theorem and related theoretical conclusions of the
study directions, which shall inspire us to construct game equilibrium analysis framework of
public participant. "Existing" form, 1strategy space and responsibilities of public participant is to
promote a common code of conduct, enhance mutual trust and promote exchange of information
on effective communication, its utility (payment) function is social welfare (selection) function
which represents public interest. In order to facilitate comparison, we give strict formal statements
on the basis of interrelated concepts. Set limited participant collection as “I = {1, 2, ..., N}”. For
any participant “i ∈ I”, set its limited action space as “Ai”, and write “A = A1×…×AN”. Utility
function of participant “ i “ is defined as payoff function “ui: A → R “ in “A”. For any “i ∈ I”,
and any combination actions “a=(a1, …, aN)= (ai, a−i)∈A”, “ui(a)∈Ui” presents the payment (or
utility) of the participant “ I” in action combination “a” . Set “ui(A) = Ui” as pay (or utility) space
for the participant i, and write “U= U1×…×UN⊆ N”. We regard “G = (I, A, U)” as a (standard)
game of participant “I”. Set arbitrary probability distribution collection defined in “Ai “as “∆(Ai) “,
shall be called as mixed strategy space of “ i “, with elements “αi∈∆(Ai)” is called as mixed
strategy of participants “I”. Participants mixed strategy space can be written as “∆(A1)×…×∆(AN)
“.
The Nash equilibrium concept2 is simplifies and rigorous expressed as: If there is a mixed
strategy combination “α*= (αi*, α−i*)∈∆(A1)×…×∆(AN)”, set “∀i∈I，∀ai∈Ai，ui(αi*, α−i*) ≥
ui(ai, α−i*)”, and α * is called as Nash equilibrium of G.
Now, we are trying to introduce a public participant in game G. "Public participant " can be
some virtual form, it also can be a real natural or legal person (sometimes even can exist between
each participant); either it can be a physical entity, or an abstract structures ....... Before the game
is carried out, the public participant shall propose strategy portfolio recommendations. If all
participants are willing to follow the public participant’s proposal, you can call the proposal a
contract in this game. If the recommendations made by public participants are Nash equilibrium,
then no one is willing to unilaterally violate the proposal. However, if the Nash equilibrium is
regarded as a contract of the game, then the fact is equivalent to you can assume that behavior
choices of all participants meet the following two criteria:
Criterion 1 (non-cooperative principles): each participant's optimal strategy choice based on
other participant has to follow the recommendation;
Criterion 2 (Nash stable principle): in a case of criterion 1 is implemented, if contrary to the
recommendation are not any good for each participant, then it would follow that recommendation.
The basic features of public participant: the main factors to be considered include overall
image, culture and mainstream values, strategic choice, credibility, etc.; functions performed is to
achieve necessary communication and communication, thereby to guide individual behavior
choices. Evolutionary game3 uses symmetric coordination game as a platform to study conditions
1
A very easy to think of public participant existence in reality is the government or public organizations. Nash, J. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United
States of America, 1950, 36(1): 48-49. 3
Weibull J. W., Evolutionary Game Theory, MA: MIT Press, 1995. 2
occurrence and continuance of cooperative behavior as well as achieve conditions of collective
rationality, individual behavior choice, individual constraints and public morals and so on shall be
specified accordingly. The rules of conduct and personal credit are the most workable variables
and control factors.
Criterion 1 indicates strategy choice independence of participants in the Nash equilibrium,
allowing the Nash equilibrium become a very convincing solution concept under non-cooperative
principle. Criterion2 reflects stability of the Nash equilibrium, namely Nash equilibrium has a
"self-driven" in nature, it can intuitively understood as: due to breach of the contract is no benefit
to any participants involved, but it may bring harm to others, so each participant will comply with
the contract. Therefore, if each participant believes that any other participants will not take
behavior which shall be "no profit to their own while is harmful to others ", then criterion 2 shall
be very reasonable constraints imposed on the solution concept of non-cooperative game.
In this collaboration process, the public participant acts as a coordinator. Naturally, public
participant can also work with other participant involved in the game like everyone else involved.
Therefore, if public participant is repressed as “0”, then participants collection can be set as “I#={0,
1, …, N}”. For public participants 0∈I#”, set its limited action space as “A0”, and written as “A# =
A0×A1×⋯×AN，U# = U0×U1×…×UN ⊆ N+1”. Utility function of public participant is social
welfare function, and for any “a∈A#，u0(a) = W(u1(a), …, uN(a)) “. In particular, we always
assume social welfare function is an increasing function of individual utility function, for any “ i
∈ I”, if “ui ≥ ui “ then “W(ui, u−i）≥ W(ui’, u−i）”. We call “G# = ( I#, A#, U#)” as a public
participant game. If actions space of the public participate are the empty set, then the public
participate game G # shall be degenerate into general game G. Obviously, behavior and strategies
of public participant can easily leave us ample room for imagination. If the public participant have
a dual role of coordinator and participant, it will make the problem even more complicated,
however also more attractive.
Likewise, mixed strategy space of the public participant game shall be denoted by
“∆(A0)×∆(A1)×…×∆(AN)”, therefore, the Nash equilibrium definition can be naturally produced
by public participant game “G#”.
Definition 2.1 If there is a mixed-strategy portfolio “α* = (αi*, α−i*) ∈
∆(A0)×∆(A1)×…×∆(AN)”, set “∀i∈I#，∀ai∈Ai,
ui(αi*, α−i*) ≥ ui(ai, α−i*)”,
“α *” shall be called as Nash equilibrium of the public participant game “G#”.
The introduction of public participant shall promote exchange of information and trust
between participants in the strategic behavior section II, establish common rules of action
(mutually acceptable mainstream values), coordinate collective actions and help to achieve social
optimal contract.
3. Logical connotation and their relationships between several common fairness
criteria
Three types of criteria fairness commonly used in social selection, utilitarianism is simple,
Rawlsian is added with exogenously fairness view, Nash bargaining solution is a bit unpredictable.
The introduction of public participant in the game analysis framework, logical connotation, key
features and their relationships of various fairness standards shall be specified by comparison.
3.1 Collective desirable actions and social optimal contract
In public participant game “G#”, because generally assume that actions combination space of the
participant is finite set, so a mixed strategy space of the participants must be compact set.
Therefore, there exists at least one mixed strategy combination “α0∈∆(A0)×∆(A1)×…×∆(AN)”,
set
“ ∀ a∈A# “，and:
u0(α0) ≥ u0(a）
In other words, “α0” is a mixed strategy combination which can make social welfare
maximization, shall be called as "social optimal strategy mix." Unfortunately, social optimal
strategy mix is not definitely the Nash equilibrium, which makes social optimal strategy mix
difficult to achieve. In the Nash equilibrium, when other participants choose a balanced strategy,
equilibrium strategy of the public participant must be the best response strategy against other
participant's (i.e., equilibrium strategy choice of other participants is given, balanced strategy
selection of the public participants must be socially optimal), but this the optimal reaction is not
necessarily “α00”. Since the multiplicity of Nash equilibrium, in different Nash equilibriums,
public participant would choose different optimal responses. Therefore, if the outcome of the
game can only be Nash equilibrium, then we would naturally expect to get a "best" result - that is
the "best" Nash equilibrium, we call the "best" Nash equilibrium as the socially optimal contract.
That is, when other participants choose balanced strategy, public participant should choose such
actions which shall maximize social welfare. However, due to the multiplicity of game
equilibrium, public participants Nash equilibrium may not be the only behavior combination
maximizing social welfare. Therefore, socially optimal choice and Nash equilibrium are often
inconsistent.
Without loss of generality, we can regard mixed strategy space of the public participate game
"∆(A0)×∆(A1)×…×∆(AN)" as an alternative of social status set. Further, any combination of the
mixed strategy space can be regarded as a probability distribution. Set all corresponding
probability distribution of mixed strategy combination of game "G#" as "M#", and corresponding
probability distribution set of the Nash equilibrium is "E# ⊆ M# ". Now, how make a reasonable
social order of all elements in “M#” has become a matter of social choice. Accordingly, any public
participant involved in the game is accordingly corresponds to a social choice problem.
However, optimal social choice in social choice problem is no necessary connection with the
Nash equilibrium. Although public participant would tend to socially optimal act combination, but
other participants may not agree with this. In the Nash equilibrium, although each participant is
reluctant to unilaterally deviate from equilibrium strategy, but t equilibrium strategy may not
maximize social welfare. In order to take into account social attributes of strategy selection in the
Nash equilibrium, the paper presents the concept of socially optimal contract: that means socially
optimal contract proposed by public participant in various feasible contracts (Nash equilibrium).
Not difficult to find that social preference of public participant only be reflected when the Nash
equilibrium has the multiplicity. The logic here is as follows:
(1) A proposal is made by a public participant and the proposal must be a Nash equilibrium;
(2) When there are multiple Nash equilibriums, the public participant will select an optimal
Nash equilibrium;
(3) Each participant (in t criteria 1 and 2) accept and fulfill their social optimal contract.
However, socially optimal contract may not be optimal social choice, but the highest ranking
Nash equilibrium in multiple Nash equilibriums. Socially optimal contract strictly shall be defined
as follows:
Definition 3.1 In the public participant game “G#”, if there is “ π ∈ E# “ E #, making
π ∈arg maxπ’∈E# ∑a∈A#π’(a)W(u1(a), ⋯, uN(a))，
Then “ π “ shall be regarded as a socially optimal contract.
According to this definition, different assumptions of social welfare functions can get
different social optimal contracts. If the social welfare function “W(·) “ meets the following two
properties:
(1) if for any “i∈I，ui ≥ ui’” , then “W(u)≥W(u’)”;
(2) if for all “i∈I，ui ≥ ui’” , then “W(u)≥W(u’)”;
So social welfare function “W(·)” shall satisfy Pareto nature. If the social welfare function “W (·)”
meets the following two properties:
(1) if for any “i∈I，ui ≥ ui’” , then “W(u)≥W(u’)”;
(2) If there is at least one “ i∈I “, making “ ui > ui’”, then “W(u)>W(u’) “; then social
welfare function “W(·)” shall meet strict Pareto nature.
If the social welfare function meet strict Pareto nature, then social contract ‘ π *’ shall be
called as Pareto optimal contract. In practice, the most commonly used social welfare function
meeting strict Pareto is utilitarian. Utilitarian social welfare function believes that social welfare is
equal to the sum of individual utility (direct algebraic sum).
Definition 3.2 In the public participant game “G#”, if there is “π∈ E# “, making
π ∈ arg maxπ’∈E# ∑a∈A#π’(a)∑i∈I ui(a)，
Then π shall be a utilitarian optimal contract.
Set all the utilitarian optimal contract set of the game “G# “ as “U# “. Utilitarianism has been
widely criticized, with another equally extensive social welfare function with corresponding
application is Rawlsian social welfare function, and it believes that the level of social welfare shall
depend on utility level of the most disadvantaged.
Definition 3.3 In the public participant game “G#”, if there is “π∈E#”, making π ∈ arg
maxπ’∈E# ∑a∈A#π’(a) Min{ui(a)}i∈I，
Then π shall be a Rawlsian optimal contract.
Set all Rawlsian optimal contract set of the game “G# “as “R#”. Please note that although the
Rawlsian social welfare function can meet the Pareto nature, it may not meet the strict Pareto
nature. Thus, Rawlsian Pareto optimal contract may not be the optimal contract. In addition to
utilitarian and Rawlsian addition, the so-called "Nash bargaining solution" made by Nash can also
provide a very important social welfare function. The social welfare function believes that social
welfare should be equal to the product of individual net utility. In particular, in Nash social welfare
function, it is assumed that there is a starting point of negotiations. In public participant game, we
can set this initial point as the corresponding worst results of each participant could get in game
(reservation utility).
Definition 3.4 In the public participant game “G#”, if there is “π∈E#”, making
π ∈ arg maxπ’∈E# ∑a∈A#π’(a) ∏i∈I (ui(a)− Min{ui(a’)}a’∈A)，
Then π shall be a Nash optimal contract.
Set all Nash optimal contract set of the game “G# “as “R#”. It should be noted that a Nash
optimal contract is not necessarily strict Pareto optimal contract. This is because there may be a
participant whose relative utility level is 0. If relative utility levels of all participants are strictly
greater than 0, then the Nash optimal contract must be strictly Pareto optimal.
3.2 Order of socially optimal contract
At this point, based on the three social welfare functions1 to calculate the social optimal contract
of game shown in Table 1. The calculations can be divided into three steps. First, corresponding
social welfare level of the three kinds of social welfare functions on behaviors space are as
follows:
Table 4. Social welfare space of extended couple game
2
0
0
1
0
0
1
0
0
0
6
0
0
1
0
0
5
0
0
0
8
0
0
4
0
0
16
（1）
（2）
（3）
(1), (2) and (3) in Table 4 are respectively from utilitarianism, Rawlsian and Nash social welfare
function. Then, the corresponding distribution action of the game's seven Nash equilibrium
(including mixed strategies) are as follows:
Table 5. Nash equilibrium (pure strategy and mixed strategy) of extended couple
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
(1)
（2）
（3）
5/12
5/12
0
16/25
0
4/25
0
0
0
1/12
1/12
0
0
0
0
0
16/45
4/45
0
0
0
4/25
0
1/25
0
20/45
5/45
（4）
（5）
（6）
80/261
80/261
20/261
16/261
16/261
4/261
20/261
20/261
5/261
（7）
Then, according to Tables 4 and 5, corresponding Nash equilibrium social welfare levels of three
kinds of social welfare functions are as follows:
Table 6. Social welfare evaluation of Nash equilibrium
（1）
（2） （3）
（4）
（5）
（6）
（7）
Utilitarianism
2
6
8
4/3
8/5
136/45
296/261
Rawlsian
1
1
4
1/2
4/5
4/5
116/261
Nash Social
Welfare
1
5
16
5/6
32/25
32/9
240/261
Finally, Table 6 shows that order of seven Nash equilibrium in utilitarian are as follows:
(3)> (2)> (6)> (1)> (5)> (4)> (7),
Therefore, the optimal contract in utilitarianism is (3). According to Rawlsian social welfare
function, we know: (3)> (1) = (2)> (5) = (6)> (4)> (7),
Therefore Rawlsian optimal contract is (3). Likewise, we can see by the Nash social welfare
function (3)> (2)> (6)> (5)> (1)> (7)> (4),
1
Simple statements: utilitarian social welfare is a direct sum of the utility of individual members; Rawlsian social
welfare depends on the utility of the most disadvantaged members of society, reflecting the tendency of a particular
fair, is essentially a weighted sum of outside weights; Nash optimal (negotiated solution) is the product of joint
utility maximization. Sum generation of three kinds of fairness standards of utilitarian, Rawlsian and Nash
bargaining solution are as follows: sum up of algebra / arithmetic (sum and difference relationship), maximum and
minimum methods and product relationship of the joint utility maximization. Therefore, the Nash optimal contract is also (3). All of the above show that achievement of
collective rationality and social optimal contract can be fully realized in individual preferences
identical1 or different and sufficient communicate of information; public participant may propose
jointly conduct recommendations, guide individual to enhance mutual trust and communication,
improve weight distribution of the possible equilibrium outcome and ultimately to achieve socially
reasonable action.
4. Basic characteristics and practical significance of socially optimal contract
In discussion of the above cases, three different social welfare functions have got the same
socially optimal contract. In particular, social sorting results of utilitarian social welfare function
and the Nash social welfare function are highly similar. However, this result does not have
universality. Therefore, it is necessary to discuss different social welfare function and different
meanings of the corresponding allocation principles and strategy choices in realty.
4.1 The linearity of social welfare function
Definition 4.1 In the public participant game “G#”, if there is “π ∈ ∆(A0)×∆(A1)×⋯×∆(AN)”, making
∑a∈A#π(a)W(u1(a), ⋯, uN(a))= W(∑a∈A#π(a)u1(a), ⋯, ∑a∈A#π(a)uN(a))，
Social welfare function of the game is linear.
That is, if the social welfare function is linear, then for any “π∈∆(A0)×∆(A1)×⋯×∆(AN)”, and
E[W(u1(a), ⋯, uN(a))]= W(E[u1(a)], ⋯, E[uN(a)]).
Obviously, the utilitarian social welfare function has the linear nature, it is because
∑a∈A#π(a)W(u1(a), ⋯, uN(a))=∑a∈A#π(a)∑i∈I ui(a)
=∑a∈A#∑i∈I π(a)ui(a)
= ∑i∈I∑a∈A# π(a)ui(a)
= W(∑a∈A#π(a)u1(a), ⋯, ∑a∈A#π(a)uN(a)).
Linear characteristic of social welfare function can be understood as scale invariance in
theory and strategy continuity in practice, utility of each society member decreases in the same
scale, order of the social welfare function shall be unchanged.
But Rawlsian and Nash social welfare function do not have the character. Game shown in
Table 4 for example, the expected utility portfolio of Nash equilibrium is as follows:
Table 7. Expected utility portfolio of Nash equilibrium
（1） （2） （3） （4）
（5）
（6）
（7）
E[u1]
1
1
4
1/2
4/5
4/5
116/261
E[u2]
1
5
4
5/6
4/5
20/9
180/261
E[u1]+ E[u2]
2
6
4
4/3
8/5
136/45
296/261
Min{ E[u1], E[u2]}
1
1
4
1/2
4/5
4/5
116/261
E[u1]× E[u2]
1
5
4
5/12
16/25
80/45
160×180/(2612)
It is not difficult to find that in the seven Nash equilibrium utility combinations, only (2) and (3) is
Pareto optimal contract. It can be seen that utilitarianism satisfies linearity, while Nash social
welfare function does not. Particularly, in this case, Rawlsian can also satisfy linearity, it is
because in this case,
∀ a∈ A，Min{u1(a), u2(a)}= u1(a)，
1
When
u i ( ai )
extends to
u i ( a i , a −i )
for the order-preserving transformation, it is right for any “i ∈ I”, the
two obtain extreme values at the same point. Thus, for any “π∈ ∆(A1)×∆(A2)”, it shows that
∑a∈Aπ(a)W(u1(a), u2(a)) = ∑a∈Aπ(a)Min{ u1(a), u2(a)}
= ∑a∈A π(a)u1(a)
= Min{ ∑a∈A π(a)u1(a), ∑a∈A π(a)u2(a)}
= W(∑a∈A π(a)u1(a), ∑a∈A π(a)u2(a)).
But in the more general case, the utilitarian social welfare function is still linear, and
Rawlsian social welfare function and Nash social welfare function are not linear.
4.2 Invariance of socially optimal contract
Theorem 4.1 Suppose in the game G = (I, A, U), hybrid strategy portfolio α* ∈ ∆(A1)×⋯×∆(AN)
is a Nash equilibrium of the game G. If in the game G '= (I, A, U'), for any i∈I，ui’ = fi(ui) ∈ Ui’,
where “fi “ is any positive linear (or positive affine) transformation, then “α *” is a Nash
equilibrium of the game “G ' “if and only if “α *” is a Nash equilibrium of the game “G “(proof is
omitted).
From theorem 4.1, we can see that in the game G = (I, A, U), the pure strategy Nash
equilibrium is not comparable in ordinal number, mixed strategy Nash equilibrium is not
comparable in cardinality. However, the Arrow impossibility theorem makes this nature in the
public participant game G # no longer valid for the socially optimal contract. In other words, the
socially optimal contract generally does not satisfy ordinal comparability. Moreover, even some
degree of interpersonal comparison is permitted, we still face a challenge in nature, that is
interpersonal comparison shall require extremely rich of utility information. To solve this problem,
we are no longer focus on any positive monotonic transformation of individual utility function, but
rather focus on considering impact of a linear transformation on the social welfare function. The
results have shown that different social welfare functions shall have different invariant features for
different types of linear transformations123.
Theorem 4.2 Suppose in the public participant game G#=(I#, A#, U#) , “π ∈ N “ is a Nash
optimal contract. If in the public participant game G#’=(I#, A#, U#’), for any “i∈I”, ui’=aiui+bi∈
Ui’, where “ai and bi”is arbitrary real number, and “ ai>0”, then π is the Nash optimal contract of
game G#’ if and only if π is Nash optimal contract of the game G#.
Proof is omitted. From theorem 4.2 , it shows that Nash optimal contract does not meet the
base comparability.
Theorem 4.3 Suppose in the public participant game G#=(I#, A#, U#) , “π∈U “ is a Nash
optimal contract. If in the public participant game G#’=(I#, A#, U#’), for any “i∈I”, ui’=aiui+bi∈
Ui’, where “a and bi” is arbitrary real number, and “ a>0”, then π is the utilitarian optimal contract
of game G#’ if and only if π is utilitarian optimal contract of the game G#.
Proof is omitted. From theorem 4.3 , it shows that utilitarian optimal contract meet the base
comparability.
Theorem 4.4 Suppose in the public participant game G#=(I#, A#, U#) , “π∈U “ is a Rawlsian
optimal contract. If in the public participant game G#’=(I#, A#, U#’), for any “i∈I”, ui’=aiui+bi∈
1
Sen, A. K., Collective Choice and Social Welfare, North-Holland, Amsterdam, 1970. Sen, A. K. Informational
bases of alternative welfare approaches: Aggregation and income distribution, Journal of Public Economics, 1974,
3, 387-403. Sen, A. K. On weights and measures: Informational constraints in social welfare analysis,
Econometrica, 1977, 45, 1539-1572. 2
d’Aspremont, C. and L. Gevers. Equity and informational basis of collective choice, Review of Economic Studies,
1977, 44, 199-210. 3
Roberts, K. W. S. Interpersonal comparability and social choice theory, Review of Economic Studies, 1980, 47,
421-439. Ui’, where “a and b” is arbitrary real number, and “ a>0”, then π is the Rawlsian optimal contract
of game G#’ if and only if π is Rawlsian optimal contract of the game G#.
Proof is omitted. From theorem 4.4, it shows that Rawlsian optimal contract meet the base
comparability.
Socially optimal strategy invariance means that total social welfare shall be not affected by
wealth form and distribution method, namely it is independent of the chosen value measurement
and has nothing to do with the measurement tool. If we pay more attention to the linear nature of
social welfare function, utilitarian social welfare function is the best choice undoubtedly. But if we
wish to emphasize the immutability of social welfare function, then the Nash social welfare
function is the best choice. Therefore, the social welfare function can’t have both linear and
invariant features, which will undoubtedly bring difficulties can’t be avoided in solving the game
process for public participant, which is theoretical reason why it is difficult to establish and
implement the principle of equitable distribution in practice.
4.3 Coordination of social optimal contract
Because the concept of Nash equilibrium can be extended to correlated equilibriums, then the
concept of socially optimal contract can also be naturally extended to correlated equilibriums. In
fact, concept of correlated equilibriums shall provide more possibilities for coordination between
participants. We use concepts of traffic game and related equilibriums to explain coordination of
the socially optimal contract.
A classic example of correlated equilibriums is "traffic game": at a crossroads, drivers of
participant 1 and 2 encounter; both parties may choose to give way or not to give way, if both
parties have chosen not to give way, road traffic accidents will occur. The payment structures of
such games are as follows.
Table 8. Traffic game
1
2
Give way
Not to give way
Give way
0，0
0，1
Not to give way
1，0
−10，−10
The game has three Nash equilibriums, namely (give way, not to give way), (not to give way, give
way), as well as10/11 probability of giving way of both parties and 1/11 probability of failing to
give way, their respective actions distribution are as follows:
Table 9. Nash equilibrium of traffic game
0
1
0
0
100/121
10/121
0
0
1
0
10/121
1/121
（1）
（2）
（3）
If both parties can be coordinated in advance, then the result of coordination must be one of three
Nash equilibriums, and so that no one will take the initiative to unilaterally deviate from the result
of coordination. However, in this case, we can further set traffic lights at the crossroads, making
the two parties always see vehicles signals of other direction at a certain probability, and abide by
the road rules of "red light stop and green light start". At this point, "red light stop and green light
start" shall constitute a so-called correlated equilibrium, so as to avoid action combinations of
(give way, not to give way) and (not to give way, give way) with a meaningful of Nash
equilibrium. Under normal circumstances, the occurrence probability of (give way, not to give
way) and (not to give way, give way) is 1/2, then the actions distribution shall make reference to
the following table:
Table 10. Correlated equilibrium of "traffic light"
0
1/2
1/2
0
In this correlated equilibrium, no one dares to take the initiative to unilaterally deviate from
rules of "red light stop and green light start", and this is very consistent with the idea of Nash
equilibrium1. But, in fact, concepts of relate equilibriums make stochastic process of participants
strategy generalized. Therefore, any mixed strategy shall give a probability distribution to space,
namely, an action distribution which can assort with corresponding groups. Any coordinated
action distribution which can be assorted with any probability distribution of space action shall be
called as associated strategy combination, the corresponding game equilibrium is correlated
equilibrium, which reflect that earnings of group is higher due to coordination, which are likely to
increase each participant's gain or utility.
Thus, mixed strategy is related strategy which requires each participant's strategy distribution
is independent from each other. Based on their relevant expected payment, correlated strategy
equilibrium (namely correlated equilibrium) makes each participant is take the initiative to
unilaterally deviate from the equilibrium strategy. The set of traffic lights is essentially exercising
functions of public participant, which coordinate actions of all participants to reach a correlated
equilibrium. After a simple calculation and comparison, social welfare evaluation of related
equilibrium in traffic game is got, and it can be seen that the corresponding social welfare
evaluation of the relevant equilibrium is superior to corresponding social welfare of some of the
Nash equilibrium, as in Rolle fascism and Nash social welfare function, correlated equilibrium of
the "traffic light" is better than all Nash equilibrium2.
Statements of overall coordination of social optimal contract in the sense of the relevant
equilibrium, refer to social welfare maximization possibilities with the sense of Pareto
improvement of corresponding equilibrium in collective action, as previously shown in Table 5
(7 ), the introduction of public participant is contribute to social welfare maximization. The
properties of social optimal contract analyzed and discussed in this section indicate that income
redistribution during the practice of reform must have clear values and corresponding policies and
measures, and seeking a strong theoretical explanations and practical resolve method shall also
have a positive instructive role in regulating income distribution gap based on public interest and
promoting construction of a harmonious society.
5. Conclusions
Based on full respect for individual choice, it is necessary to extract and establish the mainstream
values with a representative of the groups, reasonable fair measure standards and evaluation
criteria and rules of conduct to be abided by together. Although in real life it is absolutely
impossible to reach a consistent collective action based on social welfare maximization, but there
may be desirable actions and driving force based on group cognition and willingness, which is
also need to nurture, support and guide, and this is even the main studies direction of economics
1
The payment structures and probability assumption of the exemplary case is the real alternative to an approximate
traffic conditions, of course, the actual time length of traffic lights shall be determined by traffic flow of
intersection. 2
Rigorous mathematical analysis demonstrations and illustrative examples of this section can make reference to
the academic monographs: "Public participant game and socially optimal contract", China Social Sciences Press,
2013; interested readers can contact the author directly. and humanities social sciences. The paper is based on the introduction of public participant game,
compares social choice and social welfare function with different meanings, study essential,
logical meaning and practice characteristics of the socially optimal contract and makes a
preliminary and positive exploration in-depth analysis of basic framework between individual and
collective actions and selection of a more appropriate model, technical realization approach and so
on.
Essential characteristics and inherent requirements of socially optimal contract in public
participant game are as follows: relatively optimal recommendations of public participant should
be robust. The introduction of public participant makes communication and coordination become
more smooth, equilibrium of non-cooperative game become the result of cooperation with each
other of prior communication and consultation among participants in fact, which will not change
with the participant's will, whether it is social being observable or difficult to observe. When
mankind is facing competition and conflict, they will always try to find the best ways of
cooperation, socially optimal contract in the public participant game shall undoubtedly provide a
new way of thinking to solve this problem. Discovery and respect for individual heterogeneity,
define their own behavior, boundaries and relationships, fully mobilize enthusiasm of various
subjects and coordinate its work consistent with the collective objectives, collect and form
mainstream values to promote social harmony and progress as well as healthy economic
development, and the common action must be the most fundamental and most powerful driving
force.
Under the widespread presence of constraints, the expansion of human desire is faster than
capacity enhancement (wealth supply), due to limited cognitive ability of individual
decision-making and prerequisite dependence on utility optimization, any fair is limited to a
certain value orientation and evaluation criteria, thus, and there is only desirable but not absolutely
identical collective action. One problem of the public participant game is how to determine
appropriate social welfare function. The theory suggests that there is no perfect social welfare
function which can meet the basic good intentions of human; however, once socially reasonable
action is achieved in practice and appropriate social welfare function is constructed accordingly,
social optimal contract shall be perceivable and achievable by use of mathematical methods and
computational solving tools.
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