How to divide a panettone fairly with mechanism design

"The real test of our success," writes Nobel Prize-winning economist Alvin Roth, "will not only be our understanding of the general principles governing economic interactions, but also our ability to apply this knowledge to practical questions of microeconomic engineering. Let's deal with a practical microeconomic engineering question then. It's Christmas, time for presents and panettone. Those with young children know how much the serenity of the holidays can be undermined by arguments between children, many of which are generated precisely by presents and panettone. Imagine having to divide a panettone cake between your two children, Andrea and Beatrice. The obvious choice is to divide the cake in half, equally between the two. But your concept of fairness does not necessarily coincide with Andrea's or Beatrice's. The eldest may feel, for example, that he is entitled to a bigger slice by virtue of his age. Beatrice, on the other hand, has a bigger sweet tooth than Andrea and might demand a larger portion by virtue of her taste. The reasons why your fair division might not satisfy the two children are endless and you, of course, do not know them all. You would like to achieve a division that satisfies everyone, but you do not have enough information to do so because you do not know the reasons why the children would be satisfied. The question of economic engineering that confronts us, then, is this: is there a safe procedure to achieve a fair division, even if you do not have enough information to say what a fair division is? This, in a nutshell, is the problem underlying what economists call mechanism design, a discipline that today represents one of the most interesting frontiers of research at the intersection of economics, mathematics, law and politics. The answer that the theory of mechanism design provides to the question of panettone is fortunately affirmative and can guarantee, at least on this front, the possibility of preserving the serenity of the Christmas holidays. But on the details of this we will return later.

The so-called 'Hayek Hypothesis' refers to the idea originally put forward by the Austrian economist and philosopher Friedrich Von Hayek that markets function as aggregators of knowledge. The latter, in fact, is scattered and possessed only to a small extent by each of the participants in the market game. This is why, according to Hayek, any form of planned and centralised economy is doomed to failure. Even if the goals and preferences of each consumer and producer were known, a central authority simply would not have the knowledge of the causal links between these goals and the actions necessary to pursue them. It is only from the narrowness of our knowledge that it is possible, according to Hayek, to build institutions capable of fostering the functioning of a well-ordered society. The market is one such institution that, through the price mechanism, in an 'unintentional' manner generates a spontaneous order in which the preferences of individuals can be optimally satisfied. In this sense, the market is a specific example of a broader set of institutions that, a few decades after Hayek's work, would be called 'mechanisms' by Leonid Hurwicz. According to the Polish economist and future Nobel Prize winner, a 'mechanism' is a communication system in which participants exchange messages with each other or with a 'message centre' and in which a predetermined rule assigns a certain outcome for each combination of messages received. A very abstract concept, but for this reason generalisable to many different situations. In a traditional market, for example, messages are conveyed by prices which, together with the rule of equality between supply and demand, determine the allocation of goods and services. But the idea of 'mechanism' can also be applied to the analysis of situations in which the market, so to speak, 'fails'. One of these cases concerns the production of public goods, goods that have the characteristic of being 'non-excludable', such as, in Italy, health, schooling, property protection, but also the air we breathe, parks, roads, and many others. All these goods, precisely because they are 'non-excludable', cannot be produced and allocated efficiently through the market mechanism, as is the case with private goods. Public goods, in fact, do not have a price and therefore it is complicated to determine the optimal level of their production. One could proceed through a production mechanism based on a voluntary basis. In the case of blood for transfusions, one proceeds in this way. The public good 'blood' is produced through a voluntary contribution process. But this mechanism is not generalisable. In fact, it is shown that by its very nature this voluntary process is likely to be inefficient. This is why public goods are generally produced by the state and financed through taxation. But what is the optimal level of these goods? In the case of private goods, it is the interaction between supply and demand that meets this demand. And in the case of public goods? How many do we have to produce to satisfy the needs of citizens? Since they do not have a price, precisely because they are not market goods, answering this question is not easy. Of course, one could ask the citizens themselves how much they would be willing to pay to have an efficient public transport service or a justice system capable of guaranteeing swift and fair trials or, again, a healthcare system capable of promoting and protecting their health in a dignified manner. One could ask them, that is, to reveal the value that these goods have for them. But as Paul Samuelson shows, the answers we would get would all be systematically distorted. "It is, in fact, in every person's self-interest to give false signals, to pretend that they have less interest in a given collective activity than they actually do". This is because if it turned out that people placed a high value on these goods, governments would be induced to raise taxes in order to produce more of them. Citizens who would like to enjoy the goods but not bear the cost of producing them would be induced to send untrue signals. Samuelson came to a pessimistic conclusion. He was convinced that no mechanism existed that could effectively solve this problem.

The path inaugurated by Leonid Hurwicz and subsequently trodden by many others, on the contrary, proved particularly fruitful in addressing and solving this and many other similar problems, giving rise to a whole new field of research that goes by the name of 'mechanism design theory', better known as 'mechanism design'.

Eric Maskin and Alvin Roth have defined 'mechanism design' as the engineering part of economics. Just as engineers use the laws of physics, chemistry, electronics and other disciplines to solve concrete problems, so mechanism theorists use game theory and social choice theory to design institutions to solve specific economic, political and social problems related to the allocation of some kind of resource.

From a logical point of view, 'mechanism design' is a kind of 'reverse engineering' that proceeds in stages: it starts with the definition of an objective to be achieved and the prediction of the behaviour that is expected to be observed according to certain rules; it then proceeds to evaluate the resource allocations - consumption, production, impact on the environment, individual satisfaction, etc. - that result from this behaviour. Finally, we search for the mechanism that, starting from these predictions, best guarantees the achievement of the objective. We would like, for example, customers and taxi drivers to be satisfied, the former with the service and the price paid and the latter with the profits made against the costs incurred. What should be expected? Given the information asymmetry between taxi driver and customer, the former will take advantage of the latter. Anticipating this, the taxi driver will struggle to find customers. Therein lies the market failure. How is it solved? This is where the 'mechanism' comes in: a set of rules capable of aligning the objectives of taxi drivers and customers, making the former act in the interest of the latter because, at this point, the former and the latter have the same interest: a mutually beneficial exchange.