Initiated by mathematicians in the 1940s, the formal analysis of fair division promptly attracted the interest of economists: what is a just allocation of scarce resources between agents with different needs and tastes?; political scientists: how to distribute fairly the collective decision power between the citizens?; and more recently computer scientists: which tests of fairness are computationally tractable when they involve a very large number of participants, e. g., on the internet?.
A compelling message emerges from eight decades of mathematical and empirical research: mechanisms distributing valuable resources (be they commodities, decision power or computing resources) should always be evaluated from two complementary viewpoints on fairness: Ex Ante and Ex Post.
If I can compare the way the mechanism treats me personally versus other participants, Ex Post fairness makes sure that I have no grounds to complain, no convincing objection questioning the legitimacy of the proposed solution.
But ex ante, before the mechanism is implemented, I may have little or no prior knowledge of the other participants who will compete with me for these resources: what are the preferences guiding their choices, or is their behaviour dictated by other principles, is it strategically sophisticated or crudely myopic, or even adversarial to me? Then I must pay close attention to the worst case scenario where everything about them turns against me. A key parameter is how much protection the mechanism offers me in the gedankenexperiment of the worst case: my guarantee is my worst case welfare, the higher it is, the lower the risk that the other agents manhandle me, the safer it is to participate in - and abide by the outcome selected by - the mechanism.
The seminal discussion of cake-cutting by the mathematicians Hugo Steinhaus and Harold Kuhn is about a canonical guarantee: how to divide a non atomic cake between agents with additive utilities in such a way that each participant can make sure to eat a share worth at least 1/n-th of the total cake to her (n is the number of participants). Later instances of guarantees underpin the fair division problem in the Arrow Debreu microeconomic model, cost and surplus sharing in the cooperative game model, and more recently the allocation of indivisible goods or bads.
We discuss ex ante fairness first in the general collective decision problem variously interpreted as voting, bargaining or social choice. Guarantees there come from the veto rights allocated to individual as well as groups of agents: the proportional veto core gives more voice to minorities than the voting rules a la Condorcet or even Borda, in so doing it eliminates the "tyranny of the majority".
The mathematical model of guarantees is more interesting but much harder to crack when we allow compromises by convex combinations of the final outcomes, interpreted as lotteries, time shares, or the division of a fixed budget. Veto rights still generate guarantees but so does the random dictator rule: everyone gets a 1/n chance of selecting his/her best outcome. Vetoes and random dictatorship can be combined in many ways and are, in a precise sense, dual of one another. The veto guarantee is natural when we choose an expensive and long lasting infrastructure project or a person to hold a position for life; the random dictator approach makes more sense if we are dividing time between substitutable activities, or choosing a pair of roman consuls.
The last part of the talk turns to the fair division of a single homogenous private commodity when preferences are not necessarily monotonic: say when coworkers share a workload or investors distribute shares in a project. In this simple model the versatile concept of ex ante guarantee suggests new ways to organize the division, and also opens challenging new mathematical questions.