Complexity of Determining Nonemptiness of the Core

Vincent Conitzer, Tuomas SandholmComputer Science Department

Carnegie Mellon University

Complexity in cooperative game theory

• Cooperative game theory studies stable coalition formation• Coalitions between agents are useful because

– Coalitions can achieve things an individual cannot– Coalitions can be more efficient than individuals

• However, coalitions must be strategically stable– No subcoalition could do better by breaking off

• We study the computational complexity of finding stable outcomes– To this end, we define a sensible, concise representation of coalitional

situations

Characteristic function games• In characteristic function games, there is a specification of what each

coalition of agents can achieve• In the case of nontransferable utility, each coalition has a set of utility

possibility vectors indicating what utility combinations are possible for that coalition– For instance, (A,B) has vectors (1,2) and (3,1)

• If utility is transferable, each coalition just has a value– For instance, {A,B} has value 4– The implied set of utility possibility vectors is {(a,b):a+b4}

• We assume superadditivity: a coalition can always do at least as well as if the agents had stayed in separate (disjoint) coalitions– If (A) has utility possibility vector (1), and (B) has (2), (A,B) should have (at least) (1,2)– If {A} has value 1, and {B} has 2, {A,B} should have (at least) 3

Characteristic function game: example• Three agents can go out and have fun in Acapulco• Agent 1 has tickets to the cliff diving• Agent 2 has tickets to the flying men• Agent 3 has a car• Every agent values walking to a destination at 1,

driving to a destination at 2• The characteristic function (say utility is transferable)

{}:0 :1 :1 :2

:0 :4 :4 :6

The core• An outcome (value distribution, utility vector) is said to be blocked

by a coalition if everybody in that coalition is better off if the coalition breaks off

• In the case of nontransferable utility, this means that a utility possibility vector dominates the current utility vector for a coalition (all are better off with the first vector)

• In the case of transferable utility, this means that a coalition’s value is larger than the sum of payoffs to that coalition– Equivalently, some implied utility possibility vector dominates the payoff

vector

• We say that an outcome is in the core if it is blocked by no coalition

Core: Examples

A B C

(A):(3) (A,B):(1,7) (B,C):(5,2)

A=4 B=4 C=1 A B C

(A):(3) (A,B):(1,7) (B,C):(5,2)

A=5 B=0 C=4

3<4 1<4 5>4, 2>1 3<5 1<5 2<4

A B C

{A}:3 {A,B}:8 {B,C}:7

A=6 B=3 C=3A B C

A=4 B=8 C=0

3<6 8<6+3 7>3+3 {A}:3 {A,B}:8 {B,C}:7 3<4 8<4+8 7<8+0

Nontransferable utility

Transferable utilityBlocked

Blocked

In the core!

In the core!

Does our example have a nonempty core?• Let us go back to our example:

{}:0 :1 :1 :2

:0 :4 :4 :6

• One solution: 2 2 2

• Another solution: 1 1 4

• Not a solution: 4 1 1

• can break off for 4 > 1+1

Another example• Now the car is necessary to go anywhere, and joint property

– Any 2 agents can claim the car; can go to at most one location

• Agent 1 has tickets to the cliff diving

• Agent 2 has tickets to the flying men

• Agent 3 has passes to go parasailing

• Every agent would prefer going to her own destination (utility 2), would be okay with the next agent’s destination (1), dislikes the previous agent’s destination (0)

• The characteristic function (say utility is transferable)

{}:0 :0 :0 :0

:3 :3 :3 :3

Does this example have a nonempty core?

• Each pair of agents needs to get at least 3 together• So nothing is left for the third agent• Because this holds for all pairs, nobody gets anything

– Contradiction!

• It follows that the core is empty• The same example without transferable utility also has

an empty core

{}:0 :0 :0 :0

:3 :3 :3 :3

Representation of characteristic function games

• Generally, the length of the representation is exponential in the number of agents (2^N values)

• However, usually characteristic games have some special structure allowing for more concise representation– The complexity of finding stable outcomes has already been

studied for certain graph games [Faigle et al. 94, Deng and Papadimitriou 94]

• Our representation captures any superadditive game

How is our representation concise?

A B C

(A):(3) (A,B):(1,7) (B,C):(5,2)

A B C

{A}:3 {A,B}:8 {B,C}:7

Nontransferable utility Transferable utility

(A,B,C):(3,5,2) {A,B,C}:10

• Superadditivity: Coalition can always do at least as well as if the agents had stayed in separate (disjoint) coalitions, so:

Do not explicitly represent utility vectors/values implied by superadditivity

• Lemma: A outcome is in the core if and only if it is in the core with respect to only the explicitly represented utility vectors/values– So, representation allows for easy checking if an outcome is in the core

IMPLIED

Deciding whether the core is nonempty is hard

• Theorem. Given our representation, deciding whether the core is nonempty is NP-complete– Both with and without transferable utility

• Sketch of the reduction (with transferable utility):– Reduction from EXACT-COVER-BY-3-SETS

– (Some) agents correspond to the elements of the 3-sets

– For each 3-set, there is a coalition of the agents in the 3-set with value 3

– The other agents and coalitions involving them are chosen so that:• There is an outcome in the core if and only if the 3-set agents can guarantee

themselves utility 1 each

• But this is possible if and only if there is an exact cover

With grand coalition outcomes, the problem is tractable

• The difficulty in the instance we reduced to was that even collaborative optimization was hard– It was hard just to determine what the grand coalition could do

• What if we know what the grand coalition can do?– I.e. suppose the collaborative optimization has already been done

• The problem becomes tractable!– Without transferable utility: just check for every grand coalition possibility vector if

any coalition blocks it– With transferable utility: use linear programming to divide the value across the agents

in a stable way

• Thus, in these cases, the only hardness came from the collaborative optimization problem

Hybrid games remain hard • However, if the game is hybrid, it turns out that solving the collaborative

optimization problem is not enough– A game is hybrid if only some coalitions can transfer utility

• Consider the case where only the grand coalition can transfer utility– Realistic if the market institution that enforces payments collapses upon deviation by

agents

• In this case, determining nonemptiness of the core is hard even if we are given the value for the grand coalition– Reduction from VERTEX-COVER– Sketch of proof:

• Any two vertices connected by an edge can deviate to get utility 1 each• So, need to give at least one of the vertices on each edge utility 1• Requiring stability elsewhere limits the amount of utility available• But this is just VERTEX-COVER!

Conclusion• Strategically stable coalition formation is a key problem for

multiagent systems with self-interested agents• Determining the existence of stable solutions can be NP-

complete– Oftentimes just the collaborative optimization problem by itself is hard– However, for hybrid games, determining whether there is an outcome in the

core is hard even after the collaborative optimization phase

• Future research includes investigating:– Complexity issues for more restricted classes of games– Designing (worst-case exponential time) algorithms– The effect of complexity issues in determining the synergies between agents

• E.g. when routing problems need to be solved to determine the synergies

Thank you for your attention!