Abstract
In cake-cutting, strategy-proofness is a very costly requirement in terms of fairness: for n = 2 it implies a dictatorial allocation, whereas for n≥3 it requires that one agent receives no cake. We show that a weaker version of this property recently suggested by Troyan and Morril, called not-obvious manipulability, is compatible with the strong fairness property of proportionality, which guarantees that each agent receives 1/n of the cake. Both properties are satisfied by the leftmost leaves mechanism, an adaptation of the Dubins - Spanier moving knife procedure. Most other classical proportional mechanisms in literature are obviously manipulable, including the original moving knife mechanism. Not-obvious manipulability explains why leftmost leaves is manipulated less often in practice than other proportional mechanisms.
Highlights
The division of a single good among several agents who value different parts of it distinctly is one of the oldest fair division problems, going as far back as the division of land between Abram and Lot (Genesis 13)
We provide a natural extension of not-obvious manipulability (NOM) to indirect mechanisms, and show that the stark conflict between fairness and truth-telling in cake-cutting disappears if we weaken strategy-proofness to NOM
It is impossible to cut a cake in a strategy-proof manner that is not completely unfair to some agent, we can divide a cake in a fair, proportional way that cannot be obviously manipulated using an implementable mechanism called leftmost leaves
Summary
The division of a single good among several agents who value different parts of it distinctly is one of the oldest fair division problems, going as far back as the division of land between Abram and Lot (Genesis 13). Some manipulations are more likely to be observed than others, those which are salient or require less computation Based on this observation, Troyan and Morrill (2019) have proposed a weaker version of strategy-proofness for direct mechanisms, called not-obvious manipulability (NOM). A mechanism is NOM if it admits no obvious manipulation Their notion of NOM is a compelling one, since it does not require prior beliefs about other agents’ types, and compares mechanisms only based on two scenarios which are salient and which require less cognitive effort to compute. They show that NOM accurately predicts the level of manipulability that different mechanisms experience in practice in school choice and auctions. NOM partially explains why leftmost leaves is manipulated less frequently than other cake-cutting mechanisms in practice (Kyropoulou et al, 2019)
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