A Balance for Fairness: Fair Distribution Utilising Physics in Games of Characteristic Function Form
AA Balance for Fairness: Fair Distribution Utilising Physics inGames of Characteristic Function Form
Song-Ju Kim †‡∗ , Taiki Takahashi § , and Kazuo Sano †¶† SOBIN Institute, Kawanishi, Japanhttps://sobin.org ‡ Graduate School of Media and Governance, Keio University, Fujisawa, Japan § Department of Behavioral Science, Research and Education Center for Brain Sciences,Center for Experimental Research in Social Sciences, Hokkaido University, Sapporo, Japan ¶ Department of Economics, Fukui Prefectural University, Fukui, Japan ∗ Email: [email protected] 8, 2021
AbstractIn chaotic modern society, there is an increasing demand for the realizationof true ’fairness’. In Greek mythology, Themis, the ’goddess of justice’, has asword in her right hand to protect society from vices, and a ’balance of judgment’in her left hand that measures good and evil. In this study, we propose a fairdistribution method ’utilising physics’ for the profit in games of characteristicfunction form. Specifically, we show that the linear programming problem forcalculating ’nucleolus’ can be efficiently solved by considering it as a physicalsystem in which gravity works. In addition to being able to significantly re-duce computational complexity thereby, we believe that this system could haveflexibility necessary to respond to real-time changes in the parameter.
Keyword:Natural Intelligence, Natural Computing, Fairness, Cooperative Game, Characteristic Func-tion Form a r X i v : . [ c s . G T ] F e b Introduction
The Buddha taught that the ’goodness’ that embodies ’righteousness’ stands in contrast toduties and to personal ties. In other words, the definition of ’good’ is what will be in theinterest of oneself, in the interest of others, and in the interest of those to be born in thefuture. Humans feel happy when they are needed within a community and when they playa role that benefits others and the whole community. This core aspect of human nature isoften forgotten in actual social activities, seen as a ’beautiful thing’ at a distance from thepractical.In modern society driven by neoliberalism, only the more primitive desires of man arejudged essential. Because neoliberalism assumes that individual negligence are the sourceof poverty, disparity and poverty are the incentives for hard work rather than goodness orspiritual insight [1]. We have come to believe that market-based competition brings efficiencyand an optimization of society. As a result of this, not only the destruction of the naturalenvironment, but also the destruction of the human environment has been promoted as ameans of achieving growth. The result has been a loss of trust and bonds between humansdue to excessive competition.Today, humanity is facing a serious crisis not only because of the COVID-19, but alsobecause of the lack of trust between people. In order to defeat infectious diseases, peopleneed to trust scientists, citizens need to trust public institutions, and countries need to trusteach other. New viruses can occur everywhere, and if they do occur, the important questionwill be whether the government can immediately implement a city blockade. Unfortunately,we must include the economic loss of the blockade as a variable. In such a case, the timingof the blockade may be delayed without international assistance [2].Nevertheless, it has been proven in socialist countries that easy solutions to poverty, suchas equal distribution, do weaken incentives for hard work. Also, the fact that acting alone andselfishly within an altruistic community can produce great benefits for the individual (in theshort term) seems to be an obstacle to formulating a powerful critique of neoliberal ideology.The question that remains is whether it not possible to build a system that mobilizes digitaland new analog technologies so that people can do ’good’ with confidence?The problem of ’uneven distribution of wealth’ [3], has been accelerated in the midst of theCOVID-19 crisis [4, 5]. The ’universal basic income’, which includes wealth redistribution,solving social security problems, and combating poverty, is being actively discussed andexperimented with. This year Spain became the first country in the world to implement a’basic income’ for the needy. In this chaotic ’new turn of capitalism’, there is an urgent needto establish a firm and blameless ’fairness’ in order to avoid inequities and social justice foreveryone. However, there are different kinds of difficulties for the realization as follows.1. Fairness requires proper judgment and the processing of various factors impacting thecurrent situation. The calculation often cannot keep up with the changing dynamicsystem.2. The elements critical to determining fairness that depend on human interpretation aredifficult to formalize.3. As Arrow’s impossibility theorem [6, 7] shows, we do not even know whether there isa ’fair solution’ or not.In this study, we propose a method to realize a ’fair distribution of profit’ in a physicalsense within the restricted domain of games. Previously, Kim proposed a flexible analogcomputation scheme that overcomes the computational difficulties of digital computing bytaking advantage of natural (physical) properties, such as conservation laws, continuity, andfluctuations [8, 9]. In particular, he extracted an efficient decision-making scheme (reinforce-ment learning) called the ’tug-of-war principle’ [10] and created ’self-decision-making phys- cal systems’ by applying it to quantum dots [11], single photons [12], atomic switches [13],semiconductor lasers [14], ionic devices [15], resistive memories [16], and IoTs [17].The physics-based computations are useful in the calculation of ’fairness.’ Themis, theGreek goddess of justice, installed in courthouses and law offices, is considered the ’guardianof justice and order’ (see Figure 1). She wears a blindfold as a sign of her objectivity andselflessness, holding a sword in her right hand to protect society from vice and a ’balanceof justice’ in her left hand to measure right and wrong. In other words, she leaves the im-partial judgment to nature (physical phenomena) rather than perceptions and attachments.Although that balance is symbol, the ’use of nature (physics)’ is thought to have the effectof eliminating artificiality and gaining consensus (concessions) from the people concerningissues.The direct use of physics in computation is not a particularly new concept [18]. Rather,before the advent of the digital computer, everything was analog [19]. For example, supposethat Figure 2 has three villages with 50, 70, and 90 elementary school students living in eachvillage. How can we minimize the total distance travelled for all students’ to school whenwe build an elementary school? As shown in the figure, the center of gravity is the answer(as to where the elementary school should be built) made by hanging weights determinedby the number of students in each village. If we consider only the possible locations, thecalculation would be intractable, however, by ’designing’ the problem in this way, we canleave the calculation (or even part of it) to physics. A cooperative game wherein the players can cooperate with each other can be an effec-tive means of determining how the profit obtained through cooperation can be fairly dis-tributed [21]. Cooperative games have been applied to market analysis, voting analysis, andto cost sharing analysis in economics, as well as to market design, such as auction theoryand matching theory, the theme of the 2020 Nobel Prize in Economics [22].Let us define N = { , , · · · , n } as the set of n players. We call the non-empty subset of N ’coalitions’. If R is a set of real numbers, we call the real value function v : 2 N → R on2 N (the set of the subset of N ) ’a characteristic function’. For each coalition S ( ⊆ N ), v ( S )represents the profit that can be gained by the members of the coalition S by cooperating.More concretely, it is the difference between the cooperative case and the non-cooperativecase that is critical. We call ( N, v ), which is the set of players N and a characteristic function v , a ’game of characteristic function form’. In the following subsections, we introduce twoexamples of the ’taxi problem’ and the ’bankruptcy problem’. Consider a situation in which three persons (1, 2, 3) finish eating at a restaurant and go tohome by taxi. If they return home alone, the charges are person 1: 20 $ , person 2: 21 $ , andperson 3: 25 $ , respectively. In addition, between each of the two houses, the following taxifees are charged: 10 $ between 1 and 2, 14 $ between 1 and 3, and 20 $ between 2 and 3. Inthis case, the shortest route is (2 - 1 - 3), which costs 45 $ .In this example, the characteristic functions are the followings. v (1 , ,
3) = (20 + 21 + 25) −
45 = 21 , (1) v (1 ,
2) = 41 −
30 = 11 , (2) v (1 ,
3) = 45 −
34 = 11 , (3) v (2 ,
3) = 46 −
41 = 5 , (4) v (1) = v (2) = v (3) = 0 . (5) here a characteristic function v ( x, y ) is the profit if x and y are cooperated, that is, thedifference between the total amount of fares for going home alone and the fare for goinghome in a taxi by the shortest route.The question is, ’How should the profit of 21 $ be divided ’fairly’ among the three? Ingeneral, this problem can be solved using the concept of ’nucleolus [23]’ as follows. First,in the games of characteristic function form ( N, v ), we define the ’complaint’ C ( S, x ) of thecoalition S with respect to the allocation x as follows. C ( S, x ) = v ( S ) − (cid:88) i ∈ S x i (6)That is, the profit of the coalition S ’ minus the ’total allocation (distribution) of those whoparticipated in the coalition S ’. The ’nucleolus’ is the allocation that minimize the largestcomplaint and is generally known to be uniquely determined (for simplicity, we avoid anexact mathematical definition of ’nucleolus’ here).In the case of the taxi problem, the complaint against each coalition is as follows C ( { , } , x ) = v (1 , − x − x = 11 − x − x , (7) C ( { , } , x ) = v (1 , − x − x = 11 − x − x , (8) C ( { , } , x ) = v (2 , − x − x = 5 − x − x , (9) C ( { } , x ) = v (1) − x = − x , (10) C ( { } , x ) = v (2) − x = − x , (11) C ( { } , x ) = v (3) − x = − x . (12)Using the total rationality x + x + x = v (1 , , C ( { , } , x ) = x − , (13) C ( { , } , x ) = x − , (14) C ( { , } , x ) = x − . (15)Since the nucleolus is the allocation that minimizes the maximum constraint, we supposeabove six complaints eqs. (10, 11, 12, 13, 14, and 15,) are less than or equal to a certainvalue T , then we minimize T . In other words, consider the following linear programmingproblem. M inimize ( T ) : − T ≤ x ≤ T + 16 , (16) − T ≤ x ≤ T + 10 , (17) − T ≤ x ≤ T + 10 , (18) x + x + x = 21 , (19) x ≥ , (20) x ≥ , (21) x ≥ . (22)In order for there to be x , x , x that satisfies the constraints, the following formulasmust hold from the eqs. (16, 17, 17, and 18) and the individual rationality ( x i ≥ v ( i )). − T ≤ T + 16 ⇐⇒ T ≥ − , (23) − T ≤ T + 10 ⇐⇒ T ≥ − . (24) bove expressions can be transformed to the followings from eqs. (16, 17, and 18) and thetotal rationality eq. (19). − T ≤ ( x + x + x ) = 21 ≤ T + 36 (25) ⇐⇒ T ≥ − and T ≥ − T that satisfies all these conditions is −
5, which, when substituted,yields the following constraint condition, − ≤ x ≤ , (27) − ≤ x ≤ , (28) − ≤ x ≤ , (29) x + x + x = 21 (30) T = − x = 11, x = 5, and x = 5. That is, ’nucleolus’in the taxi problem is (11 , ,
5) and According to the nucleolus’s criterion, person1 wouldreceive 11 $ , and 2 and 3 would both receive the allocation of 5 $ . In general, in games ofcharacteristic function form with n players, we must solve a linear programming problemto minimize the maximum complaint and if there is more than one allocation that achievesthe smallest maximum complaint, the second largest complaint in the allocation must beminimized. To find ’nucleolus’, it is necessary to solve the linear programming problems atmost n − In what can be considered a game of characteristic function form, there exists a famousproblem known since long ago. If a person goes bankrupt leaving a debt of D , D , · · · , D n to each of the n creditors 1 , , · · · , n , How should the bankrupt’s entire estate ( M ) be ’fairly’distributed to each creditor? At first glance, a proportional division according to the size ofthe debt seems to be ’fair’. However, if we consider a case in which the value of the entireestate is smaller than the amount of any debt we encounter a situation in which, becauseall creditors are similarly unable to fully recover their share, it is better to split the debt inequal parts.In fact, this sort of bankruptcy problem is described in the Babylonian Talmud, a Jewishscripture more than 2,400 years ago (Table 1). In Table 1, E represents the total estateand D represents the creditors and the amount of their debts. If the total estate is 300, thedivision is proportional, but if it is 100, it is an equal division. Even though for 200, thecriteria are not as clear at first glance, there exists a clear criterion of ’nucleolus’ which wasfirst formulated in 1969. It is very surprising that the Sages of the Talmud more than 2400years ago had clear ideas and criteria for the concepts of ’justice’ and ’fairness’ in this specificfinancial case. We had to wait until 1985 for a satisfactory explanation of this criterion ofbankruptcy problem [24]. That was accomplished by Aumann and Maschler’s discussion inrelation to game theory [25].Aumann and Maschler went on to read from the following description of other passagesin the Talmud Mishnah (Baba Metzia 2a) Two hold a garment; one claims it all, the other claims half. Two hold a garment;one claims it all, the other claims half. Then the one is awarded 3/4, the other1/4.
This statement says that if there are those who demand the whole thing and those whodemand half of it, it should be divided into 3 / / roportional nor an equal division, but a ’residual equal division’. For some reason the latterallows 1 / / M − D ) and 0, so ( M − D ) + , defined as x + = max ( x, M − D ) + . The shares ofcreditors 1 and 2 X and X are as follows X = ( M − D ) + + { M − ( M − D ) + − ( M − D ) + } , (31) X = ( M − D ) + + { M − ( M − D ) + − ( M − D ) + } . (32)They called the Contested Garment principle the distribution principle in a bankruptcy casewith one bankrupt and two creditors. In the Talmudic bankruptcy problem (three creditors)the CG principle holds for any two persons and is shown to be ’nucleolus’ [24]. In this section, we mainly describe how to solve a game of characteristic function formwith ’gravity’. First, we have two cylinders A (blue) and B (yellow) filled with a liquid(incompressibility) such as Figure 4. The adjusters of A and B are assumed to be interlocked:if one moves, the other moves the same amount in the opposite direction (see Figure 3 ). Bymoving the adjusters in sequence according to the instructions as in Figure 4, each liquidlevel represents the ’amount of complaint’ of the corresponding coalition. The liquid levels E , E and E of the blue liquid correspond to the definitions of complaints C ( { , } , x ), C ( { , } , x ) and C ( { , } , x ) of the previous section, respectively. Finally, when the twocylinders are set up vertically at the same time to exert gravity, we find that gravity setsthe liquid level in motion and that the final equilibrium liquid level will represent the ’fairdistribution’.The procedure for the solution with physics is as follows1. The valve is open so as to allow for the adjustment of the liquid level by moving theadjuster. For a cylinder of blue liquid A, divide equally the profit v (1 , ,
3) that wouldbe obtained if everyone had cooperated. X i = v (1 , , / i , add the profit of two-person coalition other than i (+ v ( j, k )), and thensubtract the profit that would be gained if everyone had cooperated ( − v (1 , , i , we record the current liquid level and then set it to the origin of theupcoming gravitational change in the liquid level y i . That is, each distribution will be X i = y i + v (1 , , /
3, satisfying the requirements of the conservation law y + y + y =0. The current liquid level E i expresses the complaint of the coalition ( j, k ). That is, E i ≡ C ( { j, k } , x ) = v ( j, k ) − ( X j + X k )= X i − v (1 , ,
3) + v ( j, k )4. Then, for each i of yellow liquid cylinder B, add the profit generated by the coalition i (+ v ( i )), and then subtract − v (1 , , /
3. The current liquid level C i will expressesthe complaint of the coalition i . That is, C i ≡ v ( i ) − X i .5. All six complaints will thereby be expressed and the preparation will be complete. Atthis point, the valves are closed and the adjusters of cylinders A and B are interlocked.That is, an increase in cylinder A’s E i results in a corresponding decrease in thecorresponding liquid level C i of the cylinder B.6. Stand the two cylinder systems vertically at the same time so that gravity works (seeFigure 3 ). . Wait until the equilibrium state is attained and then record the final liquid level change y i .8. Note that the liquid level change satisfies the following equation: y i ≥ v ( i ) − v (1 , , / . (33)9. The distribution is determined by the following equation: X i = y i + v (1 , , / . (34) The physical solutions of the taxi problem and the bankruptcy problem (three ways) areshown in Figure 5. We can confirm that we are able to find solutions to all problems.There are various possibilities depending on whether the ’interlocking adjusters’ in the figureare realized with other liquids or implemented mechanically, In any case, at each level ofequilibrium, the tensile force and the pressure inside the liquid (the sum of the statistical andgravitational pressures according to Pascal’s principle) will come to a balance. It is obviousthat the ’equal liquid level’ of every cylinder will be realized in the equilibrium. However,the system cannot reach the ideal equilibrium state due to the constraints of the adjusters inthis system. The highest liquid level (maximum complaint) will gradually decrease, headingtowards the equal liquid level. However, at the same time, other liquid levels will increasedue to interlocking, with the result that the highest liquid level will eventually stop when itcannot drop any further.
In this study, we have shown that the fair distribution of profit in games of characteristicfunction form can be solved utilising physics. More specifically, the linear programmingproblem used to compute ’nucleolus’ can be regarded as a physical system that responds togravity, and the solution can be found efficiently. However, it is only shown to be effectivefor the solving of relatively simple problems at present. It is not yet known whether thismethod for solving by making use of physics can also be effective for more complex problems.Identification of the necessary conditions (physical parameters, constraints, etc.) for thisphysical solution to work is also future works.In order for this approach to become a candidate, a viable method, for overcoming thecomputational difficulties in digital computers, it is necessary to at least address the general N -person problems, and how the solution should be presented. Nevertheless, the employmentof such a physical solution may have the following implications and prospects:1. Able to achieve computational reduction (need to generalize to N -person problem),2. Able to make real-time decisions at the time of information updates, e.g., parameterchanges during calculations (flexibility),3. Able to interact with information between different species,4. Able to interlock the distribution motion directly from physical calculations (waterdistribution, oil distribution), and5. Eliminate artificiality by using natural phenomena, thus making it easier for people tounderstand and to reach consensus (compromise). eal-world problems, such as the distance between two people in a taxi problem, mustbe addressed often without complete information available. We should be flexible and ableto respond flexibly if the information is updated along the way. This system will become acandidate for overcoming the computational difficulties in such problems. Acknowledgements
This work was supported by the research grant SP001 from SOBIN Institute, LLC. We wouldlike to thank Dr. Emanuel Pastreich (The Asia Institute, South Korea) and Dr. AlexanderKrabbe (The Asia Institute, Germany) for fruitful discussions on this work.
Author Information
Correspondence should be addressed to S.-J.K. ([email protected]).
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