Jahangir Alam, Ronald I. Frank, and Charles C. Tappert Seidenberg School of CSIS

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 SystemOld 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 (= 13 + 15 + 11)

18. Example:Coalition of Three Neighboring FarmsSummary 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 FarmsComputing 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 FarmsComputing 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}=10-7-2-6=-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 17+29+14=60 used in barycentric triangle

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

24. Example:Coalition of Three Neighboring FarmsComparing Fairness – Smaller Core Compute area of core = small center D Outer D: 60 units/side, area = 1558.845 Area core = area outer D – area of trapezoids Amsq,Boym,Cqzo = 0.104 sq. units Nucleolus core area is similarly computed to obtain 0.517 sq. units Shapley Value

25. Example:Coalition of Three Neighboring FarmsComparing Fairness – Smaller Core • Comparing the Areas of the Cores • Shapley Value: Area of Core = 0.104 square units • Nucleolus: Area of Core = 0.517 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 FarmsSummary 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 FarmsSummary 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 3.33 3.33 3.33

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 210720, 1951