1. Comparing the Fairness of Two Popular Solution Concepts of Coalition Games: Shapley Value and Nucleolus
Jahangir Alam, Ronald I. Frank, and Charles C. Tappert
Seidenberg School of CSIS
Pace University, New York

2. Focus of Study
Develop an analytics model to compare two popular solution concepts of coalition games, and to determine which one is fairer and more appropriate to use in real life applications of coalition games
The algorithm presented sheds light on the effectiveness and fairness of the solution concepts and, in particular, is used to compare the fairness between Shapley Value and Nucleolus solution concepts

3. Background
Game theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers (Wikipedia)
Games are non-cooperative or cooperative
The traditional non-cooperative games include the zero-sum games like chess and checkers
Where a player benefits at the equal expense of others

4. Background (cont.)
Cooperative (coalition) games are games with competition between groups of players (coalitions) that work together in competition with the other coalitions
Example: World War II was a war of two opposing military alliances: the Allies and the Axis
This paper deals with coalition games

5. Background (cont.)
Mathematical models and built-in consistency of game theory make it a suitable framework and basis for modeling and designing of automated bargaining and decision-making software systems in interactive negotiation
These are decision problems with multiple decision makers, whose decisions impact one another and the outcome

6. Background (cont.)
Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when decision-makers interact
The basic assumptions are that decision-makers pursue well-defined rational objectives and reason strategically
The actions of one person have influence on the outcomes of others in the game and vice versa

7. Coalition Games Have Important Applications
Coalition games have found wide application in economics, finance, politics, and computing
For example, a game theory framework can serve as the most efficient bidding rule for Web Services, e-commerce auction website, or tamper-proof automated negotiations for purchasing communication bandwidth
The application of game theory to automated negotiation is still in a nascent stage. The automation of strategic choices enhances the need for these choices to be made efficiently

8. The Problem
In coalition game theory the modeler often must choose one of several substantively different solution methods, or solution concepts, which can lead to different outcomes
The modeler tries to characterize the set of outcomes from a viewpoint of fairness and rationality
In this study, we describe and discuss the main solution concepts, in particular two important solution concepts and their usefulness and limitations in actual applications

9. Example: NYC Public School System Old Method
The NYC public school system had a problem matching incoming freshmen to high schools
The school district students used to mail in a list of their five preferred schools in rank order to the Board of Education which then mailed a photocopy of that list to each of the five schools
The schools could then tell whether or not students had listed them as their first choice, and because many schools only wanted first-choice students this meant that some students really had a choice of only one school rather than five

10. Example: NYC Public School System New Method
Coalition game theory experts (from Duke, M.I.T. and Stanford) designed a new matching method for the NYC school system
In 2003 the new NYC school matching system employed a method based on a coalition game solution concept called the Shapley Value

11. Coalition Games and Solution Concepts
Various coalitions can be formed in a cooperative game
The key assumption in coalition game theory is that the grand coalition (a coalition of all the players) will form because it is guaranteed to yield the highest overall benefit to the players
A solution concept is characterized by the payoff vector specifying the benefit to players
Various solution concepts have been proposed based on different notions of fairness

12. Shapley Value Solution Concept
Wikipedia: The Shapley value is the unique payoff vector that is efficient, symmetric, additive, and assigns zero payoffs to dummy players. It was introduced by Lloyd Shapley (published 1953) 
Detailed example follows

13. Nucleolus Solution Concept
The Nucleolus of a cooperative game is a solution concept that makes the largest unhappiness of the coalitions as small as possible, or, equivalently, minimizes the worst inequity (introduced in 1969 by Schmeidler)
Detailed example follows

14. Measuring Fairness
Fairness means reasonable, equitable, just treatment without favoritism or discrimination
However, to compare coalition game solutions we need a quantitative measure of fairness
Detailed example follows

15. Example: Coalition of Three Neighboring Farms
This is a three-player coalition game involving three neighboring farms connected to each other and to the main highway by a series of trails, see following figure
The farms are planning to build paved roads connecting them to the highway
The farms can build paved roads individually, or jointly by forming coalitions

16. Example: Coalition of Three Neighboring Farms
Paving costs in millions of dollars for road sections for farms F1, F2, and F3

17. Example: Coalition of Three Neighboring Farms
Possible coalitions 2N, here 23 = 8
Opportunity costs
oc(F1)=13, oc(F2)=15, oc(F3)=11, oc(F1,F2)=21, oc(F1,F3)=22, oc(F2,F3)=20, oc(F1,F2,F3)=29
Possible coalitions agreements
V(F1,F2) = 21. If built separately, cost = 28 million
V(F1,F3) = 22. If built separately, cost = 24 million
V(F2,F3) = 20. If built separately, cost = 26 million
Grand coalition V(F1,F2,F3) = 29. If built separately, cost = 39 mil (= )

18. Example: Coalition of Three Neighboring Farms Summary of Grand Coalition vs Individual Costs
Sum of individual farm build costs = 39 million
Grand Coalition build costs = 29 million
Above are two possible savings distributions
Is there a fairer distribution of the savings?
Yes, Solution Concepts discussed below

19. Example: Coalition of Three Neighboring Farms Computing Pro-rata
Characteristic functions - savings
v(F1)=v(F2)=v(F3)=0, v(F1,F2)=7, v(F1,F3)=2, v(F2,F3)=6, v(F1,F2,F3)=10
Pro-rata profits: simple proportional distribution
Pro-rata (profits) imputations: (3.3, 3.9, 2.8)
Used as benchmark to compare various solution concepts
‘Imputation’ is optimal payoff vector for grand coalition

20. Example: Coalition of Three Neighboring Farms Computing Shapley Value
Inductively
c{F1}=v({F1})=0, c{F2}=v({F2})=0, c{F3}=v({F3})=0, c{F1,F2}=v({F1,F2})-c{F1}-c{F2}=7-0-0=7, c{F1,F3}=v({F1,F3})-c{F1}-c{F3}=2-0-0=2, c{F2,F3}=v({F2,F3})-c{F2}-c{F3}=6-0-0=6, c{N}=v({N})-c{F1,F2}-c{F1,F3}}-c{F2,F3}= =-5
Thus, v=7w{1,2}+2w{1,3}+6w{2,3}-5w{1,2,3}
And f1(v) = 7/2 +2/2 - 5/3 = 17/6 = 2.8
f2(v) = 7/2 +6/2 - 5/3 = 29/6 = 4.8
f3(v) = 2/2 +6/2 - 5/3 = 14/6 = 2.3
Shapley imputations: (17/6,29/6,14/6)=(2.8,4.8,2.3)
Numerator sum =60 used in barycentric triangle

21. Example: Coalition of Three Neighboring Farms Computing Nucleolus
Tabular system for calculating Nucleolus
Procedure: start with pro-rata imputations and adjust for excesses (we skip details here)
Nucleolus imputations: (3.5, 3.9, 2.6)

22. Example: Coalition of Three Neighboring Farms Comparing Fairness
Fairness means reasonable, equitable, just treatment without favoritism or discrimination
However, to compare coalition game solutions we need a quantitative measure of fairness
Possible measures
Comparing deviations of imputations from pro-rata
Lexographic ordering
Smaller core of the imputations – we use this measure

23. Shapley Value Barycentric coordinate triangle
Example: Coalition of Three Neighboring Farms Comparing Fairness – Smaller Core
Shapley Value Barycentric coordinate triangle
Plane of plot is
x1+x2+x3=60

24. Example: Coalition of Three Neighboring Farms Comparing Fairness – Smaller Core
Compute area of core = small center D
Outer D: 60 units/side, area =
Area core = area outer D – area of trapezoids Amsq,Boym,Cqzo = sq. units
Nucleolus core area is similarly computed to obtain sq. units
Shapley Value

25. Example: Coalition of Three Neighboring Farms Comparing Fairness – Smaller Core
Comparing the Areas of the Cores
Shapley Value: Area of Core = square units
Nucleolus: Area of Core = square units
A smaller core area is considered better because it gives the players less room (space) to negotiate (less wiggle room), and therefore the grand coalition is less likely to be upset
Based on the area-of-core measure, the Shapely Value solution concept has greater fairness than the Nucleolus solution concept

26. Example: Coalition of Three Neighboring Farms Summary of Grand Coalition vs Individual Costs
Sum of individual farm build costs = 39 million
Grand Coalition build costs = 29 million
Above are two possible savings distributions
Is there a fairer distribution of the savings?

27. Example: Coalition of Three Neighboring Farms Summary of Imputations = Optimal Payoffs
Summary of payoffs in millions of dollars
Nucleolus is similar to Pro Rata basically because it adjusts for excesses from Pro Rata
Shapley Value gives higher payoffs to the players contributing more to the coalition
In this case, Farm 2, the middle farm
Total 10 million
saved
Average savings

28. Example: Coalition of Three Neighboring Farms
Paving costs in millions of dollars for road sections for farms F1, F2, and F3

29. Coalition Games Have Important Applications
Coalition games have found wide application in economics, finance, politics, and computing
For example, a game theory framework can serve as the most efficient bidding rule for Web Services, e-commerce auction website, or tamper-proof automated negotiations for purchasing communication bandwidth
The application of game theory to automated negotiation is still in a nascent stage. The automation of strategic choices enhances the need for these choices to be made efficiently

30. Conclusions
This study investigated the “fairness” of solution concepts in coalition games
In particular, the fairness of the two most popular solution concepts were examined
Computational methods were presented for deriving payoffs and computing fairness

31. References
The bulk of the critical literature is found in either books or in articles by key researchers in the area
Key Books
Thomas Ferguson, Game Theory, lecture notes, UCLA, 2014
Martin Osborne, An introduction to Game Theory, Oxford University Press, 2004
Roger Myerson, Game Theory: Analysis of Conflict, Harvard University Press, 1997
Example Articles
Lloyd Shapley’s “Notes on N-person Game-II, Value of an N-Person Game,” Project RAND, U.S. Air Force, ASTA Doc. No. ATI , 1951