An Ascending Multi-Item Auction with Financially Constrained Bidders

A number of heterogeneous items are to be sold to a group of potential bidders. Every bidder knows his own values over the items and his own budget privately. Due to budget constraint, bidders may not be able to pay up to their values. In such a market, a Walrasian equilibrium typically fails to exist and furthermore no existing allocation mechanism can tackle this case. We propose the notion of an `equilibrium under allotment' to such markets and develop an ascending auction mechanism that always finds such an equilibrium assignment and corresponding price system in finitely many rounds. The auction can be viewed as an appropriate and proper generalization of the ascending auction of Demange, Gale and Sotomayor from settings without financial constraints to settings with financial constraints. We examine various properties of the auction and its outcome.Ascending auction, Financial constraint, Equilibrium under allotment.

An Owen-type value for games with two-level communication structures

We introduce an Owen-type value for games with two-level communication structures, being structures where the players are partitioned into a coalition structure such that there exists restricted communication between as well as within the a priori unions of the coalition structure. Both types of communication restrictions are modeled by an undirected communication graph, so there is a communication graph between the unions of the coalition structure as well as a communication graph on the players in every union. We also show that, for particular two-level communication structures, the Owen value and the Aumann-Drèze value for games with coalition structures, the Myerson value for communication graph games and the equal surplus division solution appear as special cases of this new value

The Component Fairness Solution for Cycle- Free Graph Games

In this paper we study cooperative games with limited cooperation possibilities, representedby an undirected cycle-free communication graph. Players in the game can cooperate if andonly if they are connected in the graph, i.e. they can communicate with one another. Weintroduce a new single-valued solution concept, the component fairness solution. Our solution is characterized by component efficiency and component fairness. The interpretationof component fairness is that deleting a link between two players yields for both resultingcomponents the same average change in payoff, where the average is taken over the players in the component. Component fairness replaces the axiom of fairness characterizing the Myerson value, where the players whose link is deleted face the same loss in payoff. Thecomponent fairness solution is always in the core of the restricted game in case the gameis superadditive and can be easily computed as the average of n specific marginal vectors,where n is the number of players. We also show that the component fairness solution canbe generated by a specific distribution of the Harsanyi-dividends.operations research and management science;

A Fixed Point Theorem for Discontinuous Functions

In this paper we prove the following fixed point theorem. Consider a non-empty bounded polyhedron P and a function Æ : P → P such that for every x є P for which Æ (x) ≠ x there exists δ > 0 such that for all y, z є B (x, δ) ∩ P it holds that (Æ(y)-y)2 (Æ(z)-z) ≤ 0, where B (x, δ) is the ball in Rⁿ centered at x with radius δ . Then Æ has a fixed point, i.e., there exists a point x* є P satisfying Æ (x*) = x* . The condition allows for various discontinuities and irregularities of the function. In case f is a continuous function, the condition is automatically satisfied and thus the Brouwer fixed point theorem is implied by the result. We illustrate that a function that satisfies the condition is not necessarily upper or lower semi-continuous. A game-theoretic application is also discussed.mathematical economics and econometrics ;

Generalization of binomial coefficients to numbers on the nodes of graphs

The triangular array of binomial coefficients, or Pascal's triangle, is formed by starting with an apex of 1. Every row of Pascal's triangle can be seen as a line-graph, to each node of which the corresponding binomial coefficient is assigned. We show that the binomial coefficient of a node is equal to the number of ways the line-graph can be constructed when starting with this node and adding subsequently neighboring nodes one by one. Using this interpretation we generalize the sequences of binomial coefficients on each row of Pascal's triangle to so-called Pascal graph numbers assigned to the nodes of an arbitrary (connected) graph. We show that on the class of connected cycle-free graphs the Pascal graph numbers have properties that are very similar to the properties of binomial coefficients. We also show that for a given connected cycle-free graph the Pascal graph numbers, when normalized to sum up to one, are equal to the steady state probabilities of some Markov process on the nodes. Properties of the Pascal graph numbers for arbitrary connected graphs are also discussed. Because the Pascal graph number of a node in a connected graph is defined as the number of ways the graph can be constructed by a sequence of increasing connected subgraphs starting from this node, the Pascal graph numbers can be seen as a measure of centrality in the graph

An efficient and fair solution for communication graph games\ud

We introduce an efficient solution for games with communication graph structures and show that it is characterized by efficiency, fairness and a new axiom called component balancedness. This latter axiom compares for every component in the communication graph the total payo to the players of this component in the game itself to the total payoff of these players when applying the solution to the subgame induced by this component

The Average Tree Solution for Cooperative Games with Communication Structure

We study cooperative games with communication structure, represented by an undirectedgraph. Players in the game are able to cooperate only if they can form a network in the graph. A single-valued solution, the average tree solution, is proposed for this class ofgames. Given the graph structure we define a collection of spanning trees, where eachspanning tree specifies a particular way by which players communicate and determines a payoff vector of marginal contributions of all the players. The average tree solution is defined to be the average of all these payoff vectors. It is shown that if a game has acomplete communication structure, then the proposed solution coincides with the Shapleyvalue, and that if the game has a cycle-free communication structure, it is the solutionproposed by Herings, van der Laan and Talman (2008). We introduce the notion of linkconvexity, under which the game is shown to have a non-empty core and the average tree solution lies in the core. In general, link-convexity is weaker than convexity. For games with a cycle-free communication structure, link-convexity is even weaker than super-additivity.operations research and management science;

The Socially Stable Core in Structured Transferable Utility Games

We consider cooperative games with transferable utility (TU-games), in which we allow for a social structure on the set of players, for instance a hierarchical ordering or a dominance relation. The social structure is utilized to refine the core of the game, being the set of payoffs to the players that cannot be improved upon by any coalition of players. For every coalition the relative strength of a player within that coalition is induced by the social structure and is measured by a power function. We call a payoff vector socially stable if at the collection of coalitions that can attain it, all players have the same power.The socially stable core of the game consists of the core elements that are socially stable. In case the social structure is such that every player in a coalition has the same power, social stability reduces to balancedness and the socially stable core coincides with the core. We show that the socially stable core is non-empty if the game itself is socially stable. In general the socially stable core consists of a finite number of faces of the core and generically consists of a finite number of payoff vectors. Convex TU-games have a non-empty socially stable core, irrespective of the power function. When there is a clear hierarchy of players in terms of power, the socially stable core of a convex TU-game consists of exactly one element, an appropriately defined marginal vector. We demonstrate the usefulness of the concept of the socially stable core by two applications. One application concerns sequencing games and the other one the distribution of water.mathematical economics;

Measuring the Power of Nodes in Digraphs

Many economic and social situations can be represented by a digraph. Both axiomatic and iterative methods to determine the strength or power of all the nodes in a digraph have been proposed in the literature. We propose a new method, where the power of a node is determined by both the number of its successors, as in axiomatic methods, and the powers of its successors, as in iterative methods. Contrary to other iterative methods, we obtain a full ranking of the nodes for any digraph. The new power function, called the positional power function, can either be determined as the unique solution to a system of equations, or as the limit point of an iterative process. The solution is also explicitly characterized. This characterization enables us to derive a number of interesting properties of the positional power function. Next we consider a number of extensions, like the positional weakness function and the position function.mathematical economics and econometrics ;