Programme Director, MSc Mathematical Finance

Department of Economics

University of Birmingham

Edgbaston B15 2TT, UK

[o] Room 2127, University House

[t] +44 121 414 3754

[f] +44 121 414 7377

[e] c.rowat@bham.ac.uk

Spring term office hours: Mondays 15:00 - 17:00 and Tuesdays 16:00 - 17:00

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- An introduction to mechanized reasoning, with M. Kerber and C. Lange

*Journal of Mathematical Economics*, vol 66, pp. 26 - 39, October 2016. - Pillage games with multiple stable sets, with S. MacKenzie and M. Kerber

*International Journal of Game Theory*, vol 44(4), pp. 993 - 1013, November 2015. - Sound auction specification and implementation, with M. B. Caminati, M. Kerber and C. Lange

*Proceedings of the Sixteenth ACM Conference on Economics and Computation*, pp. 547 - 564, 2015. - 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)* - Efficient sets are small with A. Beardon.

*Journal of Mathematical Economics*, vol 49(5), pp. 367 - 374, October 2013. - 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. - 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. - 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'. - 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'. - Non-linear strategies in a linear quadratic differential game.

*Journal of Economic Dynamics and Control*, vol 31(10), pp. 3179 - 3202, October 2007. - The commons with capital markets, with J. Dutta.

*Economic Theory*, vol 31(2), pp. 225-254, May 2007. - Intermediation by aid agencies, with P. Seabright.

*Journal of Development Economics*, vol 79(2), pp. 469 - 491, April 2006.

Asymmetric Shapley values: incorporating causal knowledge into model-agnostic explainability,
with C. Frye and I. Feige.

October 2019, arXiv 1910.06358.

Explaining AI systems is fundamental both to the development of high performing models and to the trust placed in them by their users. A general framework for explaining any AI model is provided by the Shapley values that attribute the prediction output to the various model inputs ("features") in a principled and model-agnostic way. The outstanding strength of Shapley values is their combined generality and rigorous foundation: they can be used to explain any AI system, and one always understands their values as the unique attribution method satisfying a set of mathematical axioms. However, as a framework, Shapley values are too restrictive in one significant regard: they ignore all causal structure in the data. We introduce a less-restrictive framework for model-agnostic explainability: "Asymmetric" Shapley values. Asymmetric Shapley values (ASVs) are rigorously founded on a set of axioms, applicable to any AI system, and can flexibly incorporate any causal knowledge known a-priori to be respected by the data. We show through explicit, realistic examples that the ASV framework can be used to (i) improve model explanations by incorporating causal information, (ii) provide an unambiguous test for unfair discrimination based on simple policy articulations, (iii) enable sequentially incremental explanations in time-series models, and (iv) support feature-selection studies without the need for model retraining.

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. vf.data is a sample data file.

"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)

- ForMaRE (Formal Mathematical Reasoning in Economics) project page

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)

- 11 January 2019: the syllabus is here
- here are the (evolving) lecture slides
- 11 Feb 2019: the list of groups' presentation dates is here
- some past examples of student work:
- initial presentations, 2009: Borel card games, Eastern Europe, foreign aid, prisoner's dilemma, prospect theory, stag hunt, addiction
- final submissions: 2018, pyramid scheme; 2011, war of attrition
- final presentation, 2013: tax havens

- 29 September 2019: the syllabus is here
- 1 October 2019: the (evolving) lecture slides are here
- 12 November 2019: the sample class test is here
- 10 December 2019: the class test is here

- 9 January 2012: the syllabus is here
- 9 January: the list matching students to firms is here
- 9 January: the (preliminary) lecture slides are here
- 11 January: the problem set for class 1 is here
- 24 January: the problem set for class 2 is here
- 9 February: the problem set for class 3 is here
- 22 February: the problem set for class 4 is here
- 7 March: the problem set for class 5 is here

- 10 October 2019: the syllabus is here
- the (evolving) lecture slides are here
- the Meucci problem sets are here
- supplemental notes on some of the Meucci questions are here

- ESRC Festival of Social Science
- 2019: Shapley's value: from game theory to explainable AI: how did a 1953 tool from an obscure branch of game theory end up in explainable AI?
- 2018: Fully automated luxury communism: will big data and state power help China 'win' the AI revolution?
- 2017: gigaGIGO big data increases the need for careful thought about causality and significance
- 2016: Good data gone bad: playing fast and loose with data
- articles for The Conversation
- July 2015: Five misconceptions about the Greek debt crisis
- March 2014: Explainer: do sanctions work?
- May 2014: Applying artificial intelligence to auctions, a TEDx talk; the slides are here
- June 2013: Electricity auctions in Iraq slides for CWC's Iraq Power 2013 conference
- 21 October 2011: Restless intellect of a deserving Nobel winner, letter to the FT on Sargent's award
- January 2011: presentation to the Humanitarian Dialogue Foundation on the legacy of sanctions on Iraq (video: part 1, part 2, part 3, part 4)
- 23 April 2010: Let's be realistic about sophistication FT letter on Goldman, Abacus, financial sophistication and computational complexity
- May 2009: I advised on the Birmingham Rep's production of Caryl Churchill's Serious Money
- May 2007: written evidence on the effectiveness of economic sanctions submitted to the House of Lords' Select Committee on Economic Affairs
- October 2006: some back-of-the-envelope calculations of the economic cost to Iraq of the 2003 invasion, New York Times