# The Shapley Value and the Core

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1 The Shapley Value and the Core Lecture 23 The Shapley Value and the Core Lecture 23, Slide 1

2 Lecture Overview 1 Recap 2 Analyzing Coalitional Games 3 The Shapley Value 4 The Core The Shapley Value and the Core Lecture 23, Slide 2

3 Bidding Languages OR XOR Composites: OR-of-XOR, XOR-of-OR, etc. Dummy goods The Shapley Value and the Core Lecture 23, Slide 3

4 Definition Transferable utility assumption: the payoffs to a coalition may be freely redistributed among its members. satisfied whenever there is a universal currency that is used for exchange in the system means that each coalition can be assigned a single value as its payoff. Definition (Coalitional game with transferable utility) A coalitional game with transferable utility is a pair (N,v), where N is a finite set of players, indexed by i; and v : 2 N R associates with each coalition S N a real-valued payoff v(s) that the coalition s members can distribute among themselves. We assume that v( ) = 0. The Shapley Value and the Core Lecture 23, Slide 4

5 Using Coalitional Game Theory Questions we use coalitional game theory to answer: 1 Which coalition will form? 2 How should that coalition divide its payoff among its members? The answer to (1) is often the grand coalition the name given to the coalition of all the agents in N though this can depend on having made the right choice about (2). The Shapley Value and the Core Lecture 23, Slide 5

6 Voting Game Our first example considers a social choice setting. Example (Voting game) The parliament of Micronesia is made up of four political parties, A, B, C, and D, which have 45, 25, 15, and 15 representatives, respectively. They are to vote on whether to pass a \$100 million spending bill and how much of this amount should be controlled by each of the parties. A majority vote, that is, a minimum of 51 votes, is required in order to pass any legislation, and if the bill does not pass then every party gets zero to spend. More generally, in a voting game, there is a set of agents N and a set of coalitions W 2 N that are winning coalitions, that is, coalitions that are sufficient for the passage of the bill if all its members choose to do so. To each coalition S W, we assign v(s) = 1, and to the others we assign v(s) = 0. The Shapley Value and the Core Lecture 23, Slide 6

7 Airport Game Our second example concerns sharing the cost of a public good, along the lines of the road-building referendum. Example (Airport game) A number of cities need airport capacity. If a new regional airport is built the cities will have to share its cost, which will depend on the largest aircraft that the runway can accommodate. Otherwise each city will have to build its own airport. This situation can be modeled as a coalitional game (N,v), where N is the set of cities, and v(s) is the sum of the costs of building runways for each city in S minus the cost of the largest runway required by any city in S. The Shapley Value and the Core Lecture 23, Slide 7

8 Minimum Spanning Tree Next, consider a situation in which agents need to get connected to the public good in order to enjoy its benefit. One such setting is the problem of multicast cost sharing. Example (Minimum spanning tree game) A group of customers must be connected to a critical service provided by some central facility, such as a power plant or an emergency switchboard. In order to be served, a customer must either be directly connected to the facility or be connected to some other connected customer. Let us model the customers and the facility as nodes on a graph, and the possible connections as edges with associated costs. This situation can be modeled as a coalitional game (N, v). N is the set of customers, and v(s) is the cost of connecting all customers in S directly to the facility minus the cost of the minimum spanning tree that spans both the customers in S and the facility. The Shapley Value and the Core Lecture 23, Slide 8

9 Auction Finally, consider an efficient auction mechanism. Our previous analysis treated the set of participating agents as given. We might instead want to determine if the seller would prefer to exclude some interested agents to obtain higher payments. To find out, we can model the auction as a coalitional game. Example (Auction game) Let N B be the set of bidders, and let 0 be the seller. The agents in the coalitional game are N = N B {0}. Choosing a coalition means running the auction with the appropriate set of agents. The value of a coalition S is the sum of agents utilities for the efficient allocation when the set of participating agents is restricted to S. A coalition that does not include the seller has value 0, because in this case a trade cannot occur. The Shapley Value and the Core Lecture 23, Slide 9

10 Superadditive games Definition (Superadditive game) A game G = (N,v) is superadditive if for all S,T N, if S T =, then v(s T) v(s) + v(t). Superadditivity is justified when coalitions can always work without interfering with one another the value of two coalitions will be no less than the sum of their individual values. implies that the grand coalition has the highest payoff All our examples are superadditive. The Shapley Value and the Core Lecture 23, Slide 10

11 Convex games An important subclass of superadditive games are the convex games. Definition (Convex game) A game G = (N,v) is convex if for all S,T N, v(s T) v(s) + v(t) v(s T). Convexity is a stronger condition than superadditivity. However, convex games are not too rare in practice. E.g., the airport game is convex. Convex games have a number of useful properties, as we will see later. The Shapley Value and the Core Lecture 23, Slide 11

12 Lecture Overview 1 Recap 2 Analyzing Coalitional Games 3 The Shapley Value 4 The Core The Shapley Value and the Core Lecture 23, Slide 12

13 Analyzing coalitional games 1 Which coalition will form? we ll consider cases where the answer is the grand coalition makes sense for superadditive games, like all our examples 2 How should the coalition divide its payoff? Terminology: in order to be fair in order to be stable ψ : N R 2 N R N : payoff division given a game ψ(n, v) is a vector of payoffs to each agent, explaining how they divide the payoff of the grand coalition ψ i (N, v) is i s payoff shorthand: x R N : payoffs to each agent in N, when the game is implicit. The Shapley Value and the Core Lecture 23, Slide 13

14 Lecture Overview 1 Recap 2 Analyzing Coalitional Games 3 The Shapley Value 4 The Core The Shapley Value and the Core Lecture 23, Slide 14

15 Axiomatizing fairness: Symmetry As we did in social choice, let us describe fairness through axioms. i and j are interchangeable if they always contribute the same amount to every coalition of the other agents. for all S that contains neither i nor j, v(s {i}) = v(s {j}). The symmetry axiom states that such agents should receive the same payments. Axiom (Symmetry) For any v, if i and j are interchangeable then ψ i (N,v) = ψ j (N,v). The Shapley Value and the Core Lecture 23, Slide 15

16 Axiomatizing fairness: Dummy Player i is a dummy player if the amount that i contributes to any coalition is exactly the amount that i is able to achieve alone. for all S such that i S, v(s {i}) v(s) = v({i}). The dummy player axiom states that dummy players should receive a payment equal to exactly the amount that they achieve on their own. Axiom (Dummy player) For any v, if i is a dummy player then ψ i (N,v) = v({i}). The Shapley Value and the Core Lecture 23, Slide 16

17 Axiomatizing fairness: Additivity Consider two different coalitional game theory problems, defined by two different characteristic functions v 1 and v 2, involving the same set of agents. The additivity axiom states that if we re-model the setting as a single game in which each coalition S achieves a payoff of v 1 (S) + v 2 (S), the agents payments in each coalition should be the sum of the payments they would have achieved for that coalition under the two separate games. Axiom (Additivity) For any two v 1 and v 2, we have for any player i that ψ i (N,v 1 + v 2 ) = ψ i (N,v 1 ) + ψ i (N,v 2 ), where the game (N,v 1 + v 2 ) is defined by (v 1 + v 2 )(S) = v 1 (S) + v 2 (S) for every coalition S. The Shapley Value and the Core Lecture 23, Slide 17

18 Shapley Value Theorem Given a coalitional game (N,v), there is a unique payoff division x(v) = φ(n,v) that divides the full payoff of the grand coalition and that satisfies the Symmetry, Dummy player and Additivity axioms. What is this payoff division φ(n,v)? It is called the Shapley value. Definition (Shapley value) Given a coalitional game (N,v), the Shapley value of player i is given by φ i (N,v) = 1 N! S N\{i} [ ] S!( N S 1)! v(s {i}) v(s). The Shapley Value and the Core Lecture 23, Slide 18

19 Understanding the Shapley Value Definition (Shapley value) Given a coalitional game (N, v), the Shapley value of player i is given by φ i (N, v) = 1 N! S N\{i} [ ] S!( N S 1)! v(s {i}) v(s). This captures the average marginal contribution of agent i, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition. imagine that the coalition is assembled by starting with the empty set and adding one agent at a time, with the agent to be added chosen uniformly at random. The Shapley Value and the Core Lecture 23, Slide 19

20 Understanding the Shapley Value Definition (Shapley value) Given a coalitional game (N, v), the Shapley value of player i is given by φ i (N, v) = 1 N! S N\{i} [ ] S!( N S 1)! v(s {i}) v(s). This captures the average marginal contribution of agent i, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition. Within any such sequence of additions, look at agent i s marginal contribution at the time he is added. The Shapley Value and the Core Lecture 23, Slide 19

21 Understanding the Shapley Value Definition (Shapley value) Given a coalitional game (N, v), the Shapley value of player i is given by φ i (N, v) = 1 N! S N\{i} [ ] S!( N S 1)! v(s {i}) v(s). This captures the average marginal contribution of agent i, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition. If he is added to the set S, his contribution is [v(s {i}) v(s)]. The Shapley Value and the Core Lecture 23, Slide 19

22 Understanding the Shapley Value Definition (Shapley value) Given a coalitional game (N, v), the Shapley value of player i is given by φ i (N, v) = 1 N! S N\{i} [ ] S!( N S 1)! v(s {i}) v(s). This captures the average marginal contribution of agent i, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition. Now multiply this quantity by the S! different ways the set S could have been formed prior to agent i s addition The Shapley Value and the Core Lecture 23, Slide 19

23 Understanding the Shapley Value Definition (Shapley value) Given a coalitional game (N, v), the Shapley value of player i is given by φ i (N, v) = 1 N! S N\{i} [ ] S!( N S 1)! v(s {i}) v(s). This captures the average marginal contribution of agent i, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition. Now multiply this quantity by the S! different ways the set S could have been formed prior to agent i s addition and by the ( N S 1)! different ways the remaining agents could be added afterward. The Shapley Value and the Core Lecture 23, Slide 19

24 Understanding the Shapley Value Definition (Shapley value) Given a coalitional game (N, v), the Shapley value of player i is given by φ i (N, v) = 1 N! S N\{i} [ ] S!( N S 1)! v(s {i}) v(s). This captures the average marginal contribution of agent i, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition. Finally, sum over all possible sets S The Shapley Value and the Core Lecture 23, Slide 19

25 Understanding the Shapley Value Definition (Shapley value) Given a coalitional game (N, v), the Shapley value of player i is given by φ i (N, v) = 1 N! S N\{i} [ ] S!( N S 1)! v(s {i}) v(s). This captures the average marginal contribution of agent i, averaging over all the different sequences according to which the grand coalition could be built up from the empty coalition. Finally, sum over all possible sets S and obtain an average by dividing by N!, the number of possible orderings of all the agents. The Shapley Value and the Core Lecture 23, Slide 19

26 Shapley Value Example Consider the Voting game: A, B, C, and D have 45, 25, 15, and 15 votes 51 votes are required to pass the \$100 million bill A is in all winning coalitions, but doesn t win alone B, C, D are interchangeable: they always provide the same marginal benefit to each coalition they add \$100 million to the coalitions {B, C}, {C, D}, {B, D} that do not include them already and to {A} they add \$0 to all other coalitions Grinding through the Shapley value calculation (see the book), we get the payoff division (50,16.66,16.66, 16.66), which adds up to the entire \$100 million. The Shapley Value and the Core Lecture 23, Slide 20

27 Lecture Overview 1 Recap 2 Analyzing Coalitional Games 3 The Shapley Value 4 The Core The Shapley Value and the Core Lecture 23, Slide 21

28 Stable payoff division The Shapley value defined a fair way of dividing the grand coalition s payment among its members. However, this analysis ignored questions of stability. Would the agents be willing to form the grand coalition given the way it will divide payments, or would some of them prefer to form smaller coalitions? Unfortunately, sometimes smaller coalitions can be more attractive for subsets of the agents, even if they lead to lower value overall. Considering the majority voting example, while A does not have a unilateral motivation to vote for a different split, A and B have incentive to defect and divide the \$100 million between them (e.g., (75, 25)). The Shapley Value and the Core Lecture 23, Slide 22

29 The Core Under what payment divisions would the agents want to form the grand coalition? They would want to do so if and only if the payment profile is drawn from a set called the core. Definition (Core) A payoff vector x is in the core of a coalitional game (N,v) if and only if S N, i S x i v(s). The sum of payoffs to the agents in any subcoalition S is at least as large as the amount that these agents could earn by forming a coalition on their own. Analogue to Nash equilibrium, except that it allows deviations by groups of agents. The Shapley Value and the Core Lecture 23, Slide 23

30 Existence and Uniqueness 1 Is the core always nonempty? 2 Is the core always unique? The Shapley Value and the Core Lecture 23, Slide 24

31 Existence and Uniqueness 1 Is the core always nonempty? No. Consider again the voting game. The set of minimal coalitions that meet the required 51 votes is {A, B}, {A, C}, {A, D}, and {B, C, D}. If the sum of the payoffs to parties B, C, and D is less than \$100 million, then this set of agents has incentive to deviate. If B, C, and D get the entire payoff of \$100 million, then A will receive \$0 and will have incentive to form a coalition with whichever of B, C, and D obtained the smallest payoff. Thus, the core is empty for this game. 2 Is the core always unique? The Shapley Value and the Core Lecture 23, Slide 24

32 Existence and Uniqueness 1 Is the core always nonempty? 2 Is the core always unique? No. Consider changing the example so that an 80% majority is required The minimal winning coalitions are now {A, B, C} and {A, B, D}. Any complete distribution of the \$100 million among A and B now belongs to the core all winning coalitions must have both the support of these two parties. The Shapley Value and the Core Lecture 23, Slide 24

33 Positive results We say that a player i is a veto player if v(n \ {i}) = 0. Theorem In a simple game the core is empty iff there is no veto player. If there are veto players, the core consists of all payoff vectors in which the nonveto players get 0. Theorem Every convex game has a nonempty core. Theorem In every convex game, the Shapley value is in the core. The Shapley Value and the Core Lecture 23, Slide 25

34 Auction example In our auction example, we asked whether any coalition (consisting of bidders and the seller) could do better than the payoffs they receive when everyone participates This can be rephrased as asking whether the seller s and the agents payoffs from the auction are in the core. Single-item, second-price auctions: If bidders bid truthfully, the payoffs are always in the core. The seller receives a revenue equal to or greater than the valuations of all the losing bidders Hence she cannot entice any of the losing bidders to pay her more. The Shapley Value and the Core Lecture 23, Slide 26

35 VCG for combinatorial auctions Bidder 1 Bidder 2 Bidder 3 v 1 (x,y) = 90 v 2 (x) = v 2 (x,y) = 100 v 3 (y) = v 3 (x,y) = 100 v 1 (x) = v 1 (y) = 0 v 2 (y) = 0 v 3 (x) = 0 The efficient allocation awards x to bidder 2 and y to bidder 3. Neither bidder is pivotal, so both pay 0. Both bidder 1 and the seller would benefit from forming a coalition in which bidder 1 buys the bundle x,y for any payment 0 < p 1 < 90. Thus in a combinatorial auction the VCG payoffs are not guaranteed to belong to the core. The Shapley Value and the Core Lecture 23, Slide 27

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