Computing Shapley values, manipulating value division schemes, and checking core membership in multi-issue domains Vincent Conitzer, Tuomas Sandholm Computer Science Department Carnegie Mellon University

Complexity in cooperative game theory Coalitions between agents are useful because –Coalitions can achieve things an individual cannot –Coalitions can be more efficient than individuals But how should the gains from cooperation be distributed? –Fairness concerns (how is an agent’s contribution best measured?) –Stability concerns (could a coalition be better off by breaking off?) This is what is studied in cooperative game theory –It defines various value distribution schemes with fairness and stability properties (Shapley value, core, …) The computational aspects of such distribution schemes have received relatively little attention –In complex distributed settings, they are nontrivial We define a concise representation of such situations and study the computational implications

Characteristic function games A characteristic function specifies a value for each coalition –Or a vector of utilities if utility is nontransferable (not studied here) This is the value that the coalition can obtain by themselves –Sometimes a coalition is affected by what other agents do Optimistic values: assume that other agents will act to maximize the coalition’s value Pessimistic values: assume that the other agents will act to minimize the coalition’s value Pessimistic values are not necessarily more “conservative” –When we evaluate stability (will agents deviate?), optimistic is actually more conservative

Characteristic function game: example The task is to entertain a visitor for a day Three agents can cooperate to accomplish this Agent 1 has tickets to the theatre Agent 2 has passes to the bowling alley Agent 3 has a car The characteristic function: {}:0 :3 :3 :4 :0 :4 :4 :6

Properties of characteristic function games Some properties that a characteristic function game may or may not have: Increasingness: If S is a subset of T, then value of S is at most value of T Increasingness almost always makes sense –All our results hold with and without increasingness Superadditivity: If S and T are disjoint, and their union is U, then the value of U is at least the sum of the values of S and T The idea: the agents in U can always pretend that they are in separate coalitions S and T Superadditivity does not always make sense –Coordination overhead –The collaborative optimization problem may be much harder Often scales superlinearly in the number of agents –Penalty to collusion (e.g., anti-trust laws) –When the values are optimistic with respect to other agents’ behavior, it may be inconsistent to assume that everyone behaves both in S’s and T’s best interest We will make it clear in which of our results superadditivity plays a role

Concise representation: multiple issues General characteristic function games take 2^N values to represent (where N is the number of agents) There may be structure in the instance that makes it easier to represent We have previously studied a concise representation based on superadditivity [DCR-02; IJCAI-03 (4pm Tuesday)] –Not applicable without superadditivity This paper uses two ideas for concise representation: Decomposability. Sometimes a characteristic function can be written as the sum of other (easier to represent) characteristic functions, each representing an issue –E.g. multiple independent tasks Few relevant agents. It is possible that only a few agents affect the characteristic function of a particular issue –Requires specifying only 2^{|C_i|} << 2^N values Here C_i is the set of agents issue i concerns

Marginal contribution schemes One way to divide the value generated among the agents is by giving each one its marginal contribution The marginal contribution is defined only relative to a particular joining order –May differ across joining orders Back to our example… {}:0 :3 :3 :4 :0 :4 :4 :6 One joining order: gets 0, gets 4, gets 2 Another joining order: gets 3, gets 1, gets 2

The Shapley value The Shapley value [Shapley 1953] resolves the problem of order dependence in marginal contribution schemes as follows: Average the marginal contribution over all possible orders! There are many equivalent characterizations of the Shapley value: –E.g., of the form: it is the only function satisfying … The marginal contribution characterization is nice because of its closed form

Shapley value in our example Let us compute the Shapley value in our example: {}:0 :3 :3 :4 :0 :4 :4 :6 Agent 1 : (1/6)*(2* *2)=5/2 Agent 2 : (1/6)*(2* *2)=5/2 Agent 3 : (1/6)*(2* *2)=1

Computing the Shapley value is easy Theorem. An agent’s Shapley value can be computed in time O(T*2^{max |C_i|}) –T = # of issues –C_i = the set of agents issue i concerns Holds with and without both increasingness and superadditivity

More on marginal contribution schemes Now suppose that we do not want to average over all possible orderings One scheme: choose the order at random –Now, an agent’s expected payoff is its Shapley value! Requires trusted source of randomness, or a distributed cryptographic protocol –E.g., each agent picks a permutation of the agents, encrypts it, and makes the encryption public –Then everybody provides a decryption key –The order is composition of all permutations [Zlotkin & Rosenschein 1994] –Completely random if even one agent gives a random key –But: still have to worry about manipulating the decryption key! We suggest a different approach: –It may be hard to make the order beneficial to yourself even with perfect control!

Hardness of manipulating marginal contribution schemes Definition. MAX-MARGINAL- CONTRIBUTION: given a multi-issue characteristic function, find an order which maximizes a given agent’s marginal contribution. Theorem. MAX-MARGINAL-CONTRIBUTION is NP-complete (even when each issue concerns only 3 agents, and each issue’s characteristic function only takes on values in {0,1,2} and is increasing)

The core An outcome (value distribution, payoff vector) is said to be blocked by a subcoalition if everybody in that subcoalition is better off if the subcoalition breaks off In the case of transferable utility, this means that a subcoalition’s value is larger than the sum of payoffs to that subcoalition We say that an outcome is in the core if it is blocked by no subcoalition

Core: Examples A B C {A}:3 {A,B}:8 {B,C}:7  A =6  B =3  C =3 A B C  A =4  B =8  C = {A}:3 {A,B}:8 {B,C}:7 3<4 8<4+8 7<8+0 Blocked In the core!

Does our example have a nonempty core? Let us go back to our example: {}:0 :3 :3 :4 :0 :4 :4 :6 We need to pay and at least 3 each, or one of them will block Because the total value is only 6, that would mean and would get 3 exactly, and nothing But then, could block for 4 > It follows that the core is empty!

Hardness of determining core membership Theorem. It is NP-complete to determine whether there is some subcoalition that can block a given payoff vector! (Even when each issue concerns only 3 agents, and each issue’s characteristic function is increasing and superadditive, and only takes on values in {0,1}) This not only has computational implications for computing a solution in the core… … but it also suggests that the core may be too conservative a stability concept –If a particular subcoalition can beneficially break off, but nobody is able to detect this, the coalition is stable in a computational sense

Relationship to DCR? In what DCR settings should cooperative game theory notions be applied? –Nodes may have preferences over their settings –How do we fairly decide who is made happy? –How do we make sure nodes do not block the algorithm? What DCR techniques are useful to compute solutions from cooperative game theory? –As we showed, computing these solutions is nontrivial –These settings are inherently distributed

Thank you for your attention!