William Thomson, A Pioneer of Mechanism Design and Fairness

I was introduced to the "uniform rule"[1] by my Professor William Thomson at Rochester in 1991-92 and was immediately fascinated by Sprumont's (1991)[2] characterization of the uniform rule, which shows that it is the only "non-manipulable" (see the next paragraph) allocation mechanism that satisfies Pareto efficiency and anonymity when individuals' preferences are "single-peaked".

Preferences are usually assumed to be convex in economics and single-peakedness is one such assumption. In mechanism design, it is typical to assume that the exact preference of an individual is private information, i.e. unknown to other individuals, including the mechanism designer. Nevertheless, mechanism designers can overcome such an asymmetric information problem by identifying mechanisms that are non-manipulable, i.e. incentives are provided by structured mechanisms to make truth-telling a dominant strategy for all individuals.

While information asymmetry and incentive compatibility are important aspects of mechanism design, a question dear to the heart of the public is whether "fairness" is served by the mechanisms or not. Mechanism designers should not shy way from this question and William has not. William is a pioneer of and has committed his career to mechanism design and fairness.

William recognizes that there is no unique notion of fairness and contributes to several formulations of it, including but not limited to, consistency (the shares allocated to others should not be affected by an individual leaving the group with his allocated share), resource monotonicity, population monotonicity and replacement principle (solidarity is the common theme of the last three formulations).

When William applied the above formulations of fairness to Sprumont's allocation problem, he obtained a number of alternative characterizations of the uniform rule.[3] The central role of the uniform rule in the literature was established by William and has subsequently been reinforced by numerous researches. I am privileged to take part in the formation of the literature.

The uniform rule is now applied to the real world. The Hong Kong Government started to issue iBond[4] in July 2011 and the inaugural issuance was oversubscribed. The HK Government decided to issue a maximum of HK$10 billion[5] iBond before receiving a total of HK$13.16 billion subscription from 155,835 subscribers. The uniform rule was used to allocate the iBond to the 155,835 subscribers as follows:

"[A]mong all the valid applications, those who have applied for 44 units or less will be allocated the full amount of bond applied for. There are a total of 150,869 such applications. In addition, there are 4,966 valid applications that have applied for more than 44 units. Each of them will be allocated 44 units first, and 1,344 of them will be allocated one additional unit of bond based on ballot results."[6]

In the two subsequent issuances of iBond by the HK Government (May 2012 and May 2013), oversubscriptions were received again. The uniform rule remained to be the allocation mechanism in both cases. The fourth issuance of iBond is on the way, and so will be the uniform rule.

William is a pioneer of mechanism design and fairness. One of his babies, the uniform rule, has made its way to the real world. The overdue of the uniform rule manifests that William is way ahead of many others. The dues of his other babies are uncertain. One thing is sure, though. When more of his babies are born, the world will be very different.

[1] See the seventh paragraph for a real-world example illustrating the uniform rule.
[2] Yves Sprumont, "The Division Problem with Single-Peaked Preferences: A Characterization of the the Uniform Allocation Rule," Econometrica, Vol. 59, No. 2 (March 1991), pp. 509-519.
[3] William Thomson, "Consistent Solutions to the Problem of Fair Division When Preferences Are Single-Peaked," Journal of Economic Theory, Volume 63, Issue 2 (August 1994), pp. 219-245; "Resource-Monotonic Solutions to the Problem of Fair Division When Preferences Are Single-Peaked," Social Choice and Welfare, Volume 11, Issue 3 (July 1994), pp. 205-223; "Population-Monotonic Solutions to the Problem of Fair Division When Preferences Are Single-Peaked," Economic Theory, Volume 5, Issue 2 (1995), pp. 229-246; "The Replacement Principle in Economies with Single-Peaked Preferences," Journal of Economic Theory, Volume 76, Issue 1 (September 1997), pp. 145-168.
[4] iBond is an inflation-linked bond, similar to Treasury Inflation Protected Securities (TIPS) in the US.
[5] The exchange rate between US Dollar and HK Dollar is US$ 1 = HK$ 7.8.
[6] See http://www.hkgb.gov.hk/en/news/press_20110726.html.