Taxation and Stability in Cooperative Games Yair Zick Maria Polukarov Nick R. Jennings AAMAS 2013

Cooperative TU Games Agents divide into coalitions; generate profit Coalition members can freely divide profits. How should profits be divided? $5 $3 $2

TU Games - Notations Agents: N = {1,…, n } Coalition: S µ N Characteristic function: v : 2 N → R A TU game G = h N, v i is anonymous, if the value of a coalition is only a function of its size. A TU game is monotone, if the value of a coalition can only increase by adding more agents to it.

Payoffs We assume that only the grand coalition N is formed. Agents may freely distribute profits. An imputation is a vector x = ( x 1,…, x n ) such that Σ i 2 N x i = v ( N ). Individual rationality: each agent gets at least what she can make on her own: x i ≥ v ({ i })

The Core The core is the set of all stable outcomes: for all S µ N we have x ( S ) ¸ v ( S ) ; a game with a non-empty core is called stable. May be empty in many games. Example: the 3-majority game. Three players; any set of size two or more has a value of 1; singletons have a value of 0. The total amount to be divided is 1; if (w.l.o.g.) player 1 gets more than 0, then players 2 and 3 get a total payoff of less than 1.

Stabilizing Games The 3-Majority game has an empty core. However, if one reduces the value of all 2- player coalitions by 1/3, the core becomes non-empty (giving 1/3 to each player is a stable outcome). Similarly: reduce the value of {2,3} to 0. Reducing the value of some coalitions can result in a stable game.

Our Work We explore taxation methods (i.e. reductions in coalition value), that ensure stability. What is the minimal amount of tax required in order to stabilize a game? Which taxation schemes are optimal? When are known taxation schemes optimal?

Some Background Taxation is not new " -core: coalition values reduced by ". Least-core: corresponds to the " * -core, where " * is the smallest " for which the " -core is not empty. Reliability extensions: each agent i survives with probability r i ; the value of a coalition is reduced to its expected value. Myerson graphs, etc…

1 23 x 1 = v ({1}) x 2 = v ( N ) − v ({1,3}) x 2 = v ({2}) x 1 = v ( N ) − v ({2,3}) x 3 = v ({3}) x 3 = v ( N ) − v ({1,2}) x 1 + x 2 + x 3 = v ( N )

1 23 x 1 = v ({1}) x 2 = v ( N ) − v ({1,3}) x 2 = v ({2}) x 1 = v ( N ) − v ({2,3}) x 3 = v ({3}) x 3 = v ( N ) − v ({1,2}) x 1 + x 2 + x 3 = v ( N ) " " " " " "

1 23 x 1 = v ({1}) x 2 = v ( N ) − v ({1,3}) x 2 = v ({2}) x 1 = v ( N ) − v ({2,3}) x 3 = v ({3}) x 3 = v ( N ) − v ({1,2}) x 1 + x 2 + x 3 = v ( N ) "

Exploring Taxation Methods Given a game G = h N, v i, we say that G ’ ≤ G if v ( S ) ≥ v ’ ( S ) for all S µ N. A game G ’ is maximal-stable w.r.t. G if G ’ ≤ G It has a non-empty core If G ’’ is stable and G ’ ≤ G ’’ ≤ G, then one of the inequalities holds with equality.

Exploring Taxation Methods Increasing the value of any coalition results in losing either stability or dominance. Observations: Maximal-stable games still distribute v ( N ) to the agents. They are defined by a single vector x = ( x 1,…, x n ); the value of each coalition S is min{ v ( S ), x ( S )}.

Optimal Taxation Schemes The set of dominated games with a non-empty core is a convex polyhedron denoted S ( G ). We are interested in the set of games that minimizes the total tax taken. These games are said to have optimal taxation schemes.

Optimal Taxation Schemes We characterize the optimal taxation scheme for anonymous games. Given an anonymous game, where v ( S ) = f (| S |), the optimal taxation scheme is given by reducing the value of each coalition to min ( f (| S |), f (| N |)/| S |). Good for small coalitions; bad for large coalitions.

Existing Taxation Schemes Suppose that a central authority wants to implement a taxation scheme. What conditions must hold in order for this taxation scheme to be optimal? We find conditions on the underlying cooperative game which ensure this for the " -core and for reliability extensions.

Conclusions & Future Work Given a class of cooperative games, what taxation schemes would be optimal? How much are we “over-taxing” by using a given taxation scheme? Other ways of measuring total taxation. Computational complexity: efficient computation of optimal taxes?

Thank you! Questions? P.S.: I am on the job market!