Colin Rowat: research and teaching

Programme Director, MSc Mathematical Finance
Department of Economics
University of Birmingham
Edgbaston B15 2TT, UK
[o] Room 220, J. G. Smith building
[t] +44 121 414 3754
[f] +44 121 414 7377
Summer term office hours: 13:00 - 16:00 on 29th April, 6th May, 3rd June and 9th June; by appointment in all other weeks

Research: publications, discussion papers, book chapters and grants


  1. Pillage games with multiple stable sets, with S. MacKenzie and M. Kerber
    International Journal of Game Theory, forthcoming.
  2. Sufficient Conditions for Unique Stable Sets in Three Agent Pillage Games, with M. Kerber.
    Mathematical Social Sciences, vol 69, pp. 69 - 80, May 2014.
    Updated version of Discussion Papers 12-11 (Nov 2012) and 09-07 (Jun 2009)
  3. Efficient sets are small with A. Beardon.
    Journal of Mathematical Economics, vol 49(5), pp. 367 - 374, October 2013.
  4. A Qualitative Comparison of the Suitability of Four Theorem Provers for Basic Auction Theory with C. Lange, M. B. Caminati, M. Kerber, T. Mossakowski, M. Wenzel, and W. Windsteiger.
    Lecture Notes in Computer Science, vol 7961, pp. 200 - 215, 2013.
    The accompanying code is available in the auction theory toolbox here.
  5. Using Theorema in the Formalization of Theoretical Economics with M. Kerber and W. Windsteiger.
    Lecture Notes in Computer Science, vol 6824, pp. 58 - 73, 2011.
    The accompanying Theorema code is available here.
  6. A Ramsey bound on stable sets in Jordan pillage games with M. Kerber.
    International Journal of Game Theory, vol 40(3), pp. 461 - 466, August 2011.
    The accompanying LISP code is available here. Previously circulated as `A Ramsey bound on Jordan stable sets'.
  7. Optimal voting rules for two member tenure committees with I. Ayres and N. Zakariya.
    Social Choice and Welfare, vol 36(2), pp. 323 - 354, February 2011.
    The accompanying code can be found here. Previously circulated as `Optimal two stage committee voting rules'.
  8. Non-linear strategies in a linear quadratic differential game.
    Journal of Economic Dynamics and Control, vol 31(10), pp. 3179 - 3202, October 2007.
  9. The commons with capital markets, with J. Dutta.
    Economic Theory, vol 31(2), pp. 225-254, May 2007.
  10. Intermediation by aid agencies, with P. Seabright.
    Journal of Development Economics, vol 79(2), pp. 469 - 491, April 2006.

Discussion papers

Sound Auction Specification and Implementation, with M. Caminati and M. Kerber and C. Lange.
March 2015, Department of Economics Discussion Paper, University of Birmingham, 15-08.
We introduce `formal methods' of mechanized reasoning from computer science to address two problems in auction design and practice: is a given auction design soundly specified, possessing its intended properties; and, is the design faithfully implemented when actually run? Failure on either front can be hugely costly in large auctions. In the familiar setting of the combinatorial Vickrey auction, we use a mechanized reasoner, Isabelle, to first ensure that the auction has a set of desired properties (e.g. allocating all items at non-negative prices), and to then generate verified executable code directly from the specified design. Having established the expected results in a known context, we intend next to use formal methods to verify new auction designs.

A formal proof of Vickrey's theorem by blast, simp, and rule, with M. Kerber and C. Lange.
January 2014, Department of Economics Discussion Paper, University of Birmingham, 14-01.
Formal methods use computers to verify proofs or even discover new theorems. Interest in applying formal methods to problems in economics has increased in the past decade, but - to date - none of this work has been published in economics journals. This paper applies formal methods to a familiar environment - Vickrey's theorem on second-price auctions - and provides, as background, an introduction to formal methods.

Pillage Games with Multiple Stable Sets, with S. MacKenzie and M. Kerber.
February 2013, Department of Economics Discussion Paper, University of Birmingham, 13-07.
We prove that pillage games [Jordan, 2006, “Pillage and property”, JET] can have multiple stable sets, constructing pillage games with up to 2^{(n-1)/3} stable sets, when the number of agents, n, exceeds four. We do so by violating the anonymity axiom common to the existing literature, instead endowing some agents to overpower all but a small number of opposing configurations of agents. Thus, when the core is non-empty, it dominates all but finitely many allocations. As the core must belong to any stable set, derivation of stable sets then requires considering dominance relations among these finite sets of allocations – reminiscent of stable sets’ derivation in classical cooperative game theory. While our constructions are most easily illustrated for non-empty core, we also present a pillage game with multiple stable sets but an empty core. Finally, we construct a multi-good pillage game with only three agents that also has two stable sets.

Stable Sets in Three Agent Pillage Games , with M. Kerber.
June 2009, Department of Economics Discussion Paper, University of Birmingham, 09-07.
Jordan (2006, 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 when n = 3.

The road to extinction: commons with capital markets, with J. Dutta.
January 2007, Department of Economics Discussion Paper, University of Birmingham, 04-11RR.
We study extinction in a commons problem in which agents have access to capital markets. When the commons grows more quickly than the interest rate, multiple equilibria are found for intermediate commons endowments. In one of these, welfare decreases as the resource becomes more abundant, a `resource curse'. As marginal extraction costs become constant, market access instantly depletes the commons. Without markets - the classic environment - equilibria are unique; extinction dates and welfare increase with the endowment. When the endowment is either very abundant or very scarce, market access improves welfare. As marginal costs of extraction from the commons become constant, market access can reduce welfare if the subjective discount rate exceeds the interest rate.
Revises Discussion Paper 04-11 (May 2004)

Functional Nash equilibria in commons games
August 2002, Department of Economics Discussion Paper, University of Birmingham, 02-13.
This paper explores functional Nash equilibria in three static commons problems. The first yields a non-existence result. Two linear equilibrium strategies are found in the second. Unlike the result in Klemperer and Meyer (1989, Eca), this is unaffected by the domain of the stochastic variable. The third model finds two FNE when the second's strategy space is expanded to allow transfers. While these equilibria are improvements over their equivalents in the second model, the models cannot be Pareto ranked.

Asymmetric play in a linear quadratic differential game with bounded controls
August 2002, Department of Economics Discussion Paper, University of Birmingham, 02-12.
This paper uses computational techniques to identify the Markov perfect equilibria in a two agent linear quadratic differential game with bounded controls. No evidence is found of asymmetric equilibria when the agents are symmetric or of non-linear equilibria when the agents are asymmetric. This suggests that the standard continuum result for identical agents is not robust and that non-linear strategies are not of general interest in the analysis of linear quadratic differential games. The techniques presented here are applicable to a broader class of differential games as well.

Additive Externality Games, November 2000, PhD thesis, University of Cambridge (revised November 2001). This thesis attempts to address the question of how national greenhouse gas emissions might be set in the absence of an international enforceable treaty. It explores differential game models to first ask how emissions might be set when nations interact exclusively through their emissions. It then explores a one-shot game in which nations set transfer functions as well as emissions functions.

v2a.c is the C code used to assess asymmetric play in the linear quadratic differential games examined above. is a sample data file.

Book chapters

"Technological and regulatory developments in broadcasting: an overview", in The Economic Regulation of Broadcasting Markets (eds. Paul Seabright and Jürgen von Hagen), Cambridge University Press, 2007.


EPSRC grant EP/J007498/1: Formal representation and proof for cooperative games, with M. Kerber, January 2012 - December 2014 (£389,557)

ESRC grant RES-156-25-0022: Weak Property Rights: Financial Markets and Development (World Economy and Finance Programme), with J. Dutta, April 2005 - March 2009 (£57,002)


112: Mathematical Modelling for Economists

208: The Economics of Corporate Finance

G53: Risk Analytics