1. Complexity of Determining Nonemptiness of the Core
Vincent Conitzer
Carnegie Mellon University
Computer Science Department
[Paper by Conitzer & Sandholm, 2002]

2. 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

3. Characteristic function games
Each subset of agents has a set of utility possibility vectors, indicating what they can do as a coalition
A utility possibility vector gives a utility for each agent
If utilities are transferable, the set of possibility vectors is given by the maximum attainable total utility (value) for the coalition
Superadditivity means two disjoint coalitions can always do at least as well together as separately
Any concatenation of possibility vectors is a possibility vector
For transferable utility: two coalitions can always get the sum of their values together
A possibility vector for the grand coalition is in the core if
No subset of agents has a utility possibility vector that is strictly better for all of them
For transferable utility: no subset has a value strictly greater than the sum of the utilities given to agents in that subset

4. Representation of characteristic function games
Generally, the length of the representation is exponential in the number of agents
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
We only specify utility possibility vectors that cannot be derived from superadditivity
I.e. we specify where there is synergy
This is effective for checking if an outcome is in the core
Only need to check for deviation by the (specified) synergistic coalitions

5. Deciding whether the core is nonempty is hard
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

6. 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. 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

7. Hybrid games remain hard
However, if the game is hybrid, 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!

8. Conclusion
Strategically stable coalition formation is a key problem for 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
The complexity of other solution concepts
The effect of complexity issues in determining the synergies between agents
E.g. when routing problems need to be solved to determine the synergies