Chris Dietz

Union values for games with coalition structure

In a cooperative transferable utility game each decision-making agent is usually represented by one player. We model a situation where a decision-making agent can be represented by more than one player by a game with coalition structure where, besides the game, there is a partition of the player set into several unions. But, whereas usually the decision-making agents are the players in such a game, in this paper the decision-making agents are modeled as the unions in the coalition structure. Consequently, where usually a solution assigns payoffs to the individual players, we introduce the concept of union value being solutions that assign payoffs to the unions in a game with coalition structure. We introduce two such union values, both generalizing the Shapley value for TU-games. The first is the union-Shapley value and considers the unions in the most unified way: when a union enters a coalition then it enters with all its players. The second is the player-Shapley value which takes all players as units, and the payoff of a union is the sum of the payoffs over all its players. We provide axiomatic characterizations of these two union values differing only in a collusion neutrality axiom. After that we apply them to airport games and voting games.

Games with a Local Permission Structure: Separation of Authority and Value Generation

It is known that peer group games are a special class of games with a permission structure. However, peer group games are also a special class of (weighted) digraph games. To be specific, they are digraph games in which the digraph is the transitive closure of a rooted tree. In this paper we first argue that some known results on solutions for peer group games hold more general for digraph games. Second, we generalize both digraph games as well as games with a permission structure into a model called games with a local permission structure, where every player needs permission from its predecessors only to generate worth, but does not need its predecessors to give permission to its own successors. We introduce and axiomatize a Shapley value-type solution for these games, generalizing the conjunctive permission value for games with a permission structure and the

Approximate Results for a Generalized Secretary Problem

\beta

Multi-Player Agents in Cooperative TU-Games

β-measure for weighted digraphs.

Network Structures with Hierarchy and Communication

Games under precedence constraints model situations, where players in a cooperative transferable utility game belong to some hierarchical structure, which is represented by an acyclic digraph (partial order). In this paper, we introduce the class of precedence power solutions for games under precedence constraints. These solutions are obtained by allocating the dividends in the game proportional to some power measure for acyclic digraphs. We show that all these solutions satisfy the desirable axiom of irrelevant player independence, which establishes that the payoffs assigned to relevant players are not affected by the presence of irrelevant players. We axiomatize these precedence power solutions using irrelevant player independence and an axiom that uses a digraph power measure. We give special attention to the hierarchical solution, which applies the hierarchical measure. We argue how this solution is related to the known precedence Shapley value, which does not satisfy irrelevant player independence, and thus is not a precedence power solution. We also axiomatize the hierarchical measure as a digraph power measure.

Comparable Characterizations of Four Solutions for Permission Tree Games

A version of the classical secretary problem is studied, in which one is interested in selecting one of the b best out of a group of n differently ranked persons who are presented one by one in a random order. It is assumed that b is a preassigned (natural) number. It is known, already for a long time, that for the optimal policy one needs to compute b position thresholds, for instance via backwards induction. In this paper we study approximate policies, that use just a single or a double position threshold, albeit in conjunction with a level rank. We give exact and asymptotic (as n tends to infinity) results, which show that the double-level policy is an extremely accurate approximation.

A Generalization of the Classical Secretary Problem

Agents participating in different kind of organizations, usually take different positions in some relational structure. The aim of this paper is to introduce a new framework taking into account both communication and hierachical features derived from this participation. In fact, this new set or network structure unifies and generalizes well-known models from the literature, such as communication networks and hierarchies. We introduce and analyze accessible union stable systems where union stability reflects the communication network and accessibility describes the hierarchy. Particular cases of these new structures are the sets of connected coalitions in a communication graph, antimatroids (and therefore also sets of feasible coalitions in permission structures) and augmenting systems which have numerous applications in the literature. We give special attention to th e class of cycle-free accessible union stable systems. We also consider cooperative games with restricted cooperation where the set of feasible coalitions is an accessible union stable system, and provide an axiomatization of an extension of the Shapley value to this class of games.

Strategic and Co-operative Decision Making

A situation in which a finite set of agents can generate certain payoffs by cooperation can be described by a cooperative game with transferable utility (or simply a TU-game) where each agent is represented by one player in the game. In this paper, we assume that one agent can be represented by more than one player. We introduce two solutions for this multi-player agent game model, both being generalizations of the Shapley value for TU-games. The first is the agent-Shapley value and considers the agents in the most unified way in the sense that when an agent enters a coalition then it enters with all its players. The second is the player-Shapley value which takes all players as units, and the payoff of an agent is the sum of the payoffs over all its players. We provide axiomatic characterizations of these two solutions that differ only in a collusion neutrality axiom. The agent-Shapley value satisfies player collusion neutrality stating that collusion of two players belonging to the same agent does not change the payoff of this agent. On the other hand, the player-Shapley value satisfies agent collusion neutrality stating that after a collusion of two agents, the sum of their payoffs does not change. After axiomatizing the player- and agent-Shapley values we apply them to airport games and voting games.