## Refereed publications

**Axioms for measuring utility on partial mixture sets,**

This paper extends Herstein and Milnor's famous representation of preferences on a mixture set to settings where some pairs of prospects are not connected via a path of mixtures. This is useful when we wish to model a decision maker using cardinal, nonlinear utility. Eg when beliefs are ambiguous, paths between prospects that are not comonotonic are excluded.

*Journal of Mathematical Economics*(2018)*Published version***Axioms for parametric continuity when the topology is coarse**

*,**Journal of Mathematical Economics*(2017), vol. 72, p. 88-94 Published version

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Sub-models for interactive unawareness, written in collaboration with S. Grant, J. J. Kline and J. Quiggin.

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*Theory and Decision*(2015), p. 1-13 Published versionPDF file

## Papers under review

## Current working papers

*Optimal taxation in networks with informal production*

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**Abstract**: Informal production is a key part of any developing economy. We extend Diamond--Mirrlees to a game-theoretic setting where informal firms (and consumers) are too-small-to-catch. That is, the government's cost of observing a transaction (and taxing it) is infinite if both parties to the transaction are small and zero otherwise. Thus, our starting point is that a transaction is only formal if at least one party is formal (faces initially increasing returns to scale). We assume a fixed production network (without cycles) of markets (with free entry) and postulate a zero-tax equilibrium. A withholding tax on formal inputs increases the local capacity for taxation. The constrained optimal policy is a correspondence on transaction bundles. The formation of conglomerates minimises price distortions and reconfigures the network. Production efficiency is restored if, and only if, (i) every supply chain contains a formal transaction and (ii) the level of development is enough to mitigate diseconomies of scope due to bundling across goods that are unrelated in production. The latter condition suggests a role for government investment in key sectors (such as infrastructure, IT and finance).

## Further projects

*Eliciting beliefs about stock prices*

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**Abstract**: Stock prices evolve in continuous time generating a state space that is infinite dimensional. In such spaces, even the simplest horse-lotteries (binary options) on the most basic events (cylinder sets) are unwieldy and difficult to evaluate. Yet events are the basic building blocks of the standard Bayesian model of Savage or Anscombe and Aumann. This supports the view (of Savage) that Bayesian beliefs are better suited to small worlds, where events are easy to describe. We propose an alternative foundation for eliciting subjective beliefs about the law of a given stock price from an agent. We consider the daily trajectory of a given stock price. Each trajectory is the image of a random path in the set of prospects. Together, these paths form a partial mixture set which we use to elicit subjective statistical moments about the stock prices. Using the generality of (partial) mixture sets we translate axioms on moments into standard axioms for a subjective expected utility representation. Since a full elicitation of beliefs is unfeasible when the state space is infinite-dimensional, we formalise a procedure for partially eliciting beliefs via a finite number of moments and propose a maximum entropy distribution to characterise beliefs.

*From rough sets to crisp partitions: learning in sequential games with large state spaces*, written in collaboration with J. Quiggin.

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*Unawareness and partially observable expected utility maximisation in the context of a game*

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