CCST9017. Hidden Order in Daily Life:

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1 CCST9017 Hidden Order in Daily Life: A Mathematical Perspective Lecture 4 Shapley Value and Power Indices I Prof. Patrick,Tuen Wai Ng Department of Mathematics, HKU

2 Example 1: An advertising agent approaches three movie stars, 1, 2, and 3, and asks them to sign an advertisement contract to promote certain soft drink. The agent says that he is interested in obtaining at least two signatures. If 1 and 2 sign, the agent will pay them a total of $100k(=$100,000). If 1 and 3 sign, the agent will pay them a total of $100k. On the other hand, if 2 and 3 sign, the agent will only pay them a total of $50k.

3 If all three agree to sign, the agent will pay them a total of $120k. In this example, the formation of a coalition means the agreement of its members to sign an advertisement: The mathematical model is: N= {1, 2, 3} (set of all players) Possible coalitions (subsets) and the value they generated: Ø (the empty set), {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} (the grand coalition) v(ø ) = v(1)=v(2)=v(3)=0; v(1,2)=v(1,3)=100; v(2,3)=50; v(1,2,3)=120 Question: Which coalition will be formed?

4 In discussing coalition games, we can focus on how players bargain with each other over how to divide the payoff that they can achieve in a game by banding together and joining forces in a coalition. We shall now present a possible example of negotiations between players in the game described above: Player 2 proposes an equal division to Player 1: (50,50, 0) Player 3, who might get nothing, proposes to 1: (60,0,40)

5 Player 2 lowers his demands so he won't be left out: (70, 30, 0) Player 3 is prepared to settle for: (80, 0, 20) Player 2 turns to player 3 and proposes: (0, 25, 25) Player 1 is forced to compete and proposes: (70, 0,30) Player 2 suggests forming a coalition of all participants with a payoff division of: (70, 20, 30) If all players reach agreement at this stage, the game terminates, and it is said that the outcome of the game is (70,20,30) and that the coalition {1, 2, 3} was formed.

6 There is a rich theory that attempts to describe what outcomes are likely to form, but it is beyond the scope of this course. L.C.Thomas, Games, Theory and Applications Philip D. Straffin, Game Theory and Strategy. MATH3911 Game Theory and Strategy Instead, we address the question of what a judge is likely to decide if the three players decide to form a coalition and bring the game before him and ask him to propose a fair'' division of the total $120k.

7 Example 2: 24 Hours McDelivery (2014) After the lecture, Feel hungry

8 $20.3 Amy Clara $12.5 Betty $13.4

9 Order a Big Mac Meal $37.6

10 Use a Diagram/ Function to Model It Set of all players N = {1, 2, 3} where 1= Amy, 2= Betty, 3= Clara Can form different coalitions (subsets): Ø (the empty set), {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} (the grand coalition) Payment for different coalitions: v(ø)=0; v(1)=20.3; v(2)=13.4; v(3)=12.5; v(1, 2)=33.7; v(1, 3)=32.8; v(2, 3)=25.9; v(1, 2, 3)=37.6 How to split the bill if they agree to order the Big Mac Meal?

11 Cooperative Games Games in coalitional form A solution concept for cooperative games----- Shapley value Shapley-Shubik power index for weighted voting power (will cover in the next lecture) Lloyd Shapley ( )

12 Lloyd S. Shapley Born in June 2, Graduated from Harvard with an A.B. in Mathematics in Awarded a PhD (Math) from Princeton in 1953 under the direction of Alan W. Tucker (supervisor of Johan Nash and introduced the game Prisoner s Dilemma). A Professor Emeritus at UCLA, affiliated with Departments of Mathematics and Economics. I m a mathematician, I m not an economist. Lloyd Shapley ( )

13 Nobel Prize in Economics (2012) Shapley was awarded the John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences in With Alvin E. Roth, Shapley won the 2012 Nobel Memorial Prize in Economic Sciences "for the theory of stable allocations and the practice of market design." Lloyd Shapley is the most important researcher in the filed of cooperative game theory. from dvanced-economicsciences2012.pdf

14 Nobel Prize in Economics (2012) The Shapley value has played a major role in the development of cooperative game theory, with a large variety of applications. The most important single-valued solution concept in cooperative game theory is the Shapley value. anced-economicsciences2012.pdf

15 Cooperative Games We now turn to the theory of cooperative games, where the focus of interest is the way in which the players bargain together over the division of the available payoff, rather than the way this payoff can be attained by the use of certain strategies. A large part of the theory of cooperative games deals with games in coalitional form or coalition function games.

16 Coalitions Let n 2 be the number of players in the game, numbered from 1 to n, and let N = {1, 2,..., n} be the set of players. A subset S of N (S N) is called a coalition and the set of all coalitions is denoted by 2 N. For example, if there are 3 players, i.e., N={1,2,3}, then we have 8 coalitions and 2 N ={Ø, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3},N}. The empty set, Ø is called the empty coalition while N is called the grand coalition.

17 Coalitional Form Definition. A game in coalitional form (N;v) consists of 1) a set N (the set of all players), and 2) a real-valued function v defined on 2 N ={S:S N} such that v(ø ) = 0. The function v is called the coalition function/characteristic function. The number v(s) is called the value or worth of the coalition S. We may think it this way, whenever a coalition S is formed, an amount of (transferable) utility or money or power or cost v(s) is generated.

18 Example 3: One Seller and Two Buyers Game Player 1 has a house which she values at $100,000 and offers it for sale. There are two potential buyers, Player 2 and 3, who each has $200,000 in cash, and value the house at $200,000. We can then model this situation as a game in coalitional form or coalitional game. Here the values are measured in the units of $100,000. First of all, let N={1,2,3}. The characteristic function v is given by: v(ø )=0 [by definition], v(1)=1,v(2)=v(3)=2 [initial conditions].

19 One Seller and Two Buyers Game v(1,2)=[0+ p]+[(2-p)+2]=4. It is because if player 1 and player 2 form a coalition and player 1 sells the house to 2 for some price p (1<p 2). Player 1 will then have p units in cash and player 2 will have 2-p units plus a house he values at 2 units. Hence the total wealth of the coalition is 4 units. v(1,3)=4, [for the same reasons]. v(2,3)=2+2=4. What should be v(1,2,3)?

20 One Seller and Two Players Game The formation of coalition {1,2,3} denotes a procedure in which all players take part. Such a procedure may be carried out as follows. Player 1 can sell the house to either player 2 or player 3. Suppose player 1 is thinking to sell the house to player 2 for some price p>1. Then we must have p=2 because if p < 2, then player 3 can offer q (>p) to player 1. So eventually we have v(1,2,3)=[0+2]+[(2-2)+2]+[2]=6.

21 Example 4: Weighted Voting Games In a voting body like a parliament, the players are parties and every party has a certain number of representatives. For example, a certain country with three parties in its parliament obtained the following election results: Party 1: 5 representatives; Party 2: 3 representatives; Party 3: 7 representatives. The number of representatives of party i is called the weight'' of party i and will be denoted by w i. In our example, w 1 = 5, w 2 = 3, w 3 = 7.

22 Example 4: Weighted Voting Games To pass a bill, we need the votes of at least half of members. Can model it as the following cooperative game (N; v) : N={1,2,3}, v is defined by v(s) = 1 if Σ i S w i 8 v(s) = 0 if Σ i S w i < 8 Remark: Σ i S means summing over i inside S. Party 2 has only 3 members but quite often both Party 1 and Party 3 needs the support from Party 2 in order to pass a motion. Question: How to measure the political power of each party?

23 Main Question: Given that the grand coalition N is formed, how should the players share the coalition worth among themselves? Each theory that attempts to answer this question is called a solution concept. We shall study one particular solution concept, namely, the Shapley value, which can be regarded as a division of money that an unbiased judge is likely to recommend. One possible way to obtain a solution concept is to lay down rules (axioms) that will enable an unbiased judge to suggest an allocation of v(n) among the players.

24 Axiomatic Approach We shall present a simple system of four conditions or axioms and show that this system of axioms offers the unbiased judge an opportunity to decide how to divide v(n) fairly among the players in any given game. The axioms presented later were first formulated in 1953 by Lloyd Shapley, who showed that indeed they dictate to the judge how to decide in every case. The division of payoffs according to this decision is called the Shapley value.

25 Value Functions A value function,, is function that assigns to each possible coalition function v of an n-person game (N;v), an n-tuple or vector, (v) = ( 1 (v), 2 (v),...., n (v)) of real numbers. i (v) (or simply i ) represents the worth or value of player i in the game with coalition function v. We shall impose four conditions/axioms of fairness on the vector (v) = ( 1 (v), 2 (v),...., n (v)).

26 Symmetric Players Definition: Players i and j of the game (N;v) are substitutes or symmetric players in the game (N;v) if for all subset S containing neither i nor j, v(s {i}) = v(s {j}) In particular, if S = Ø, we have v(i) = v(j). Two players are symmetric in the game if they can replace each other in every coalition that contains one of them; that is, if one player replaces the other the coalition worth does not change.

27 Null Players Definition: A player i of (N;v) is called a null player if v(s {i}) = v(s) for all S N. In particular, take S=Ø, we have v(i)=0. This simply means a null player does not contribute anything through his/her participation in any coalition.

28 Sum of Games Definition: The game (N;v) is called the sum of two games (N;u) and (N;w) if for every coalition S N, v(s)=u(s)+w(s) This definition is also correct in the other direction; that is, given any game (N;v), it is possible to split it into two games whose sum is the original game, for example

29 Shapley Value A Shapley value or value on N is a value function satisfying the following four conditions: 1. Symmetry condition: if i and j are symmetric players in (N;v), then i (v)= j (v) 2. Null player condition: if i is a null player, then i (v)= 0 3. Efficiency condition: Σ i N i (v) = v(n) 4. Additivity condition: i (v + w) = i (v) + i (w), for all i in N.

30 Remark The efficiency condition is reasonable, because the players want to divide among themselves everything they can achieve in the game when they unite, namely, exactly v(n). The additivity condition says that if we split the original game into a sum of individual games, the division of payoffs among the players in the original game should be the sum of divisions obtained in the individual games. The additivity condition is kind of artificial, but it provides a simple way to do the calculation.

31 Example (Ein-Yu Gura and Michael B. Maschler): Consider N ={1,2,3}, v(ø )=0, v(1) = 6, v(2) = 12, v(3) = 18, v(1,2) = 30, v(1,3) = 60, v(2,3) = 90, v(1,2,3) = 120 Classwork 1. Draw a figure to represent this game. 2. Is there any symmetric player or null player in this game? Answer:

32 Find the Shapley Value of the Game This game can be split into a sum of very simple games so that there will be null players and/or symmetric players in every game. We can then calculate the Shapley value of these games by the axioms above, and then, by the additivity axiom, we can calculate the Shapley value of the original game.

33 Find the Shapley Value of the Game We now present one way of splitting the original game into games in which there are mainly null players and/or symmetric players.

34 How is This Split Achieved? First, we split the original game into two games. We then split the right-hand game into two games.

35 We can again split the right-hand game into two games. And so on. Explanation: The split presented above is not random. At every stage we split the game into two games, one of which has a special property; namely, it contains a coalition T such that v(t) = v(s) whenever S contains T and v(s) = 0 for every other coalition. Such a game is called a carrier game and the coalition T is called its carrier.

36 Carrier Game Definition: A carrier game (N; v) is a game in which there is a coalition T, called the carrier of the game, such that v(s) = v(t), whenever T S; v(s) = 0, otherwise. In the first three games the carriers are {1}, {2}, and {3}. The fourth game is obtained by splitting the following game: The carrier coalition in the fourth game is {1,2}.

37 Why Did We Split the Game in This Way? We now present one way of splitting the original game into games in which there are mainly null players and/or symmetric players.

38 Find the Shapley Value of the Game If we examine the diagram of the split, we can calculate the following values: (6,0,0) (0,12,0) (0,0,18) (6,6,0) (18,0,18) (0,30,30) + (-8, -8, -8) (22,40,58) Ultimately, by the additivity axiom, the value obtained is the Shapley value of the original game. In general, Shapley established a unique value for every game.

39 Theorem 1 (Shapley, 1953): There exists a unique Shapley = ( 1,..., n ). Theorem 2 (Shapley, 1953): The Shapley value is given by = ( 1,..., n ), where for i = 1,..., n, i =1/n!Σ all permutations [v(n i {i}) v(n i )]. where N i represents the set of players preceding i in N corresponding to the permutation on N. Here, n!=1x2x3 xn is the total number of permutations of the set N and means union (e.g. {1,2} {4}={1,2,4}). Also, Σ means summation.

40 Example: The permutation = [3,2,5,4,1] is a mapping sending 1 to 3, 2 to 2, 3 to 5, 4 to 4 and 5 to 1. Example: For the permutation = [3,2,5,4,1], we have Classwork: N 4 = {3,2,5}. 3. For the permutation = [2,3,4,5,1], find N 5 and N 5 {5}? Answer:

41 Shapley s Formula Recall that the Shapley value should measure the overall significance of a player in the game. For example, when player i enters a coalition S, he receives the marginal contributions by which his entry increases the value of the coalition S he enters, i.e. v(s {i}) v(s). Of course, this measure of the marginal contributions/significance of player i depends on the coalition S he enters and we need to take into account the contributions of all possible coalition S.

42 Shapley s Formula So we take all possible orders of the players and average the corresponding marginal contributions. This is the reason why we would like to consider the difference [v(n i {i}) v(n i )] which measures the contribution of i when he joins N i. This explains the formula of the Shapley value stated in the above theorem.

43 Example 1 Recall that N ={1,2,3}, v(ø ) = v(1)=v(2)=v(3)=0; v(1,2)=v(1,3)=100; v(2,3)=50; v(1,2,3)=

44 i =1/n!Σ all permutations [v(n i {i}) v(n i )]. In order to compute the Shapley value for this game, we first notice that there are 3! =3x2x1=6 orderings/permutations of the 3 players: [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2] and [3,2,1]. marginal contributions: [v(n i {i}) v(n i Permutation )] i 340 / / /

45 Therefore, (v) = (170/3,95/3, 95/3) The Shapley value shows that Player 1 should gets more than the other two players and this reflects the fact that Player 1 is actually more valuable. Checking Since player 2 and 3 are symmetric players, 2 (v) = 3 (v). The efficiency condition should give Σ i N i (v) = 170/3+95/3+95/3=v(N)=v(1,2,3)=120

46 Assignment Four Due date: Oct 2 (before 1:00pm) Please put your assignment into the assignment box at the second floor, Hui Oi Chow Building. Please write your tutorial group number on the right hand corner of your assignment Question One Amy, Betty and Clara are planning to run a new hair salon. They have complementary skills and can offer three different services: Amy can do the shampoo blow dry; Betty can do the haircut; while Clara knows how to do the hot oil treatment.

47 Question One They agree to have the following menu for their hair salon: A) Shampoo Blow Dry: $125 B) Haircut: $130 C) Hot Oil Treatment: $200 D) Shampoo Blow Dry + Haircut + Hot Oil Treatment: $350 The price of any combination of the services not mentioned above would simply be the sum of the price of the individual services. a) Model the situation as a suitable cooperative game (N; v). b) Use a diagram to represent the game defined in part a).

48 Question One c) Since there is a big discount if one chooses "Shampoo Blow Dry + Haircut + Hot Oil Treatment", most customers would choose this service. In this case, Amy, Betty and Clara agree that they will split the income $350 among themselves according to the Shapley value of the game defined in part (a). Find the Shapley value of the game by using two different methods. Hence, find out the amount Amy, Betty and Clara should earn from a customer who has chosen "Shampoo Blow Dry + Haircut + Hot Oil Treatment.

49 Question Two (Joseph Malkevitch s example in Theoretical Mathematics Finds Use in Economics-- A Tribute to Lloyd Shapley ) Suppose three towns A, B, and C are under court orders to clean up their sewage, currently being sent to the ocean but with more bacteria than meets the law. Because of the different fixed costs involved, if the towns teamed up, they would lower their costs of meeting the state standards. To minimize the cost, the towns are willing the form coalitions and share the costs.

50 The different costs of cleaning up the sewage building the power station are summarized below: A s cost of cleaning up the sewage alone is $12 million B s cost of cleaning up the sewage alone is $6 million C s cost of cleaning up the sewage alone is $8 million A and B together would cost $15 million A and C together would cost $17 million B and C together would cost $11 million A, B and C together would cost $20 million

51 Question Two a) Model this situation as a game in coalitional form (N;v), where N={1,2,3} and v is the cost function. b) Assume the grand coalition N is formed. Find the Shapley value of the game by splitting the game into some suitable games. c) Suppose A s cost is decreased from $12 million to $11 million and all the other costs are unchanged. Find the new Shapley value of the game. Reading assignment Please read the article Theoretical Mathematics Finds Use in Economics--A Tribute to Lloyd Shapley (from Feature Column of AMS, September 2016)

52 Possible Project Topic: You may do a group project on labor share problem: Let there be one owner of capital and a large number of workers, what should be the reasonable share of the profit for the workers? References J Weinstein, Fairness and Tax Policy: A Response to Mankiw's Proposed Just Deserts, Eastern Economic Journal,

53 Possible Project Topic: You may do a group project on Shapley s Nobel winning work on stable allocations. References Gale, D.; Shapley, L.,College admissions and the stability of marriage. Am. Math. Mon. 1962, 69, opular-economicsciences2012.pdf dvanced-economicsciences2012.pdf

54 Reference: Ein-Ya Gura and Michael B. Maschler, Insights into Game Theory, Cambridge University Press, Philip D. Straffin, Game Theory and Strategy, MAA,1993. We will consider application of Shapley value in political science in the next lecture. Thank You!