Colin Rowat

Intermediation by aid agencies

Abstract This paper models aid agencies as financial intermediaries that do not make a financial return to depositors, whose concern is to transfer resources to investor-beneficiaries. This leads to a problem of verifying that the agency is using donations as intended. One solution to this problem is for an agency to employ altruistic workers at below-market wages: altruistic workers, who can monitor the agencys activities, would not work at below-market rates unless they were genuinely transferring resources to beneficiaries. We consider conditions for this solution to be incentive compatible. In a model with pure moral hazard, observability of wages makes incorporation as a not-for-profit firm redundant as a commitment device. In a model with both moral hazard and adverse selection, incorporation as a not-for-profit firm can serve as a costly commitment mechanism reassuring donors against misuse of their funds. Hiring a worker of low ability can also be a valuable commitment device against fraud.

A qualitative comparison of the suitability of four theorem provers for basic auction theory

Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We say: by formal, machine-checked proofs. We investigated the suitability of the Isabelle, Theorema, Mizar, and Hets/CASL/ TPTP theorem provers for reproducing a key result of auction theory: Vickreys 1961 theorem on the properties of second-price auctions. Based on our formalisation experience, taking an auction designers perspective, we give recommendations on what system to use for formalising auctions, and outline further steps towards a complete auction theory toolbox.

Stable Sets in Three Agent Pillage Games

Jordan [2006, “Pillage and property”, JET] characterises stable sets for three special cases of ‘pillage games’. For anonymous, three agent pillage games we show that: when the core is non-empty, it must take one of five forms; all such pillage games with an empty core represent the same dominance relation; when a stable set exists, and the game also satisfies a continuity and a responsiveness assumption, it is unique and contains no more than 15 elements. This result uses a three step procedure: first, if a single agent can defend all of the resources against the other two, these allocations belong to the stable set; dominance is then transitive on the loci of allocations on which the most powerful agent can, with any ally, dominate the third, adding the maximal elements of this set to the stable set; finally, if any allocations remain undominated or not included, the game over the remaining allocations is equivalent to the ‘majority pillage game’, which has a unique stable set [Jordan and Obadia, 2004, “Stable sets in majority pillage games”, mimeo]. Non-existence always reflects conditions on the loci of allocations along which the most powerful agent needs an ally. The analysis unifies the results in Jordan [2006] when n = 3.

A Ramsey Bound on Stable Sets in Jordan Pillage Games

Jordan (J Econ Theory 131(1):26–44, 2006) defined ‘pillage games’, a class of cooperative games whose dominance operator is represented by a ‘power function’ satisfying coalitional and resource monotonicity axioms. In this environment, he proved that stable sets must be finite. We provide a graph theoretical interpretation of the problem which tightens the finite bound to a Ramsey number. We also prove that the Jordan pillage axioms are independent.

Using Theorema in the formalization of theoretical economics

Theoretical economics makes use of strict mathematical methods. For instance, games as introduced by von Neumann and Morgenstern allow for formal mathematical proofs for certain axiomatized economical situations. Such proofs can--at least in principle--also be carried through in formal systems such as Theorema. In this paper we describe experiments carried through using the Theorema system to prove theorems about a particular form of games called pillage games. Each pillage game formalizes a particular understanding of power. Analysis then attempts to derive the properties of solution sets (in particular, the core and stable set), asking about existence, uniqueness and characterization. Concretely we use Theorema to show properties previously proved on paper by two of the co-authors for pillage games with three agents. Of particular interest is some pseudo-code which summarizes the results previously shown. Since the computation involves infinite sets the pseudocode is in several ways non-computational. However, in the presence of appropriate lemmas, the pseudo-code has sufficient computational content that Theorema can compute stable sets (which are always finite). We have concretely demonstrated this for three different important power functions.

An introduction to mechanized reasoning

Mechanized reasoning uses computers to verify proofs and to help discover new theorems. Computer scientists have applied mechanized reasoning to economic problems but -- to date -- this work has not yet been properly presented in economics journals. We introduce mechanized reasoning to economists in three ways. First, we introduce mechanized reasoning in general, describing both the techniques and their successful applications. Second, we explain how mechanized reasoning has been applied to economic problems, concentrating on the two domains that have attracted the most attention: social choice theory and auction theory. Finally, we present a detailed example of mechanized reasoning in practice by means of a proof of Vickreys familiar theorem on second-price auctions.

Sufficient Conditions for Unique Stable Sets in Three Agent Pillage Games

Pillage games (Jordan, 2006a) have two features that make them richer than cooperative games in either characteristic or partition function form: they allow power externalities between coalitions; they allow resources to contribute to coalitions’ power as well as to their utility. Extending von Neumann and Morgenstern’s analysis of three agent games in characteristic function form to anonymous pillage games, we characterise the core for any number of agents; for three agents, all anonymous pillage games with an empty core represent the same dominance relation. When a stable set exists, and the game also satisfies a continuity and a responsiveness axiom, it is unique and contains no more than 15 elements, a tight bound. By contrast, stable sets in three agent games in characteristic or partition function form may not be unique, and may contain continua. Finally, we provide an algorithm for computing the stable set, and can easily decide non-existence. Thus, in addition to offering attractive modelling possibilities, pillage games seem well behaved and analytically tractable, overcoming a difficulty that has long impeded use of cooperative game theory’s flexibility.

Stable Sets in multi-good pillage games are small

It is known that, in one-good pillage games, stable sets are finite. For m goods, it has been conjectured that the stable sets have measure zero. We introduce a class of sets, termed pseudo-indifference sets, which includes level sets of utility functions, quasi-indifference classes associated with a preference relation not given by a utility function, production possibility frontiers, and Pareto efficient sets. We establish the truth of the conjecture by proving that pseudo-indifference sets in Rp have p-dimensional measure zero. This implies that stable sets in n-agent, m-good pillage games have m(n - 1)-dimensional measure zero. We then prove that each pseudo-indifference set in Rp has Hausdorff dimension at most p - 1, a much stronger result than measure zero. Finally, we establish a stronger version of the conjecture: stable sets in n-agent, m-good pillage games have dimension at most m(n-1)-1.

Pillage games with multiple stable sets

We prove that pillage games (Jordan in J Econ Theory 131.1:26–44, 2006, “Pillage and property”, JET) can have multiple stable sets, constructing pillage games with up to