Obvious Manipulations in Cake-Cutting

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>2 it requires that one agent receives no cake. We show that a weaker version of this property recently suggested by Troyan and Morril, called non-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. Non-obvious manipulability explains why leftmost leaves is manipulated less often in practice than other proportional mechanisms.


Introduction
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. Since its formalization as the cake-cutting problem (Steinhaus, 1948), this research question has inspired a large interdisciplinary literature which has proposed mechanisms that produce fair allocations without giving agents incentives to misrepresent their preferences over the cake. Unfortunately, Brânzei and Miltersen (2015) show that any deterministic strategy-proof mechanism suggests a dictatorial allocation for n = 2, and that one agent must receive no cake at all for n ≥ 3, documenting a strong tension between fairness and incentives properties.
Nevertheless, recent results in applied mechanism design have shown that, even if mechanisms can be manipulated in theory, they are not always manipulated in practice. Some manipulations are more likely to be observed than others, particularly those which are salient or require less computation. Based on this observation, Troyan and Morrill (2019) have proposed a weaker version of strategy-proofness, called non-obvious manipulability (NOM). They define a manipulation as obvious if it yields a higher utility than truth-telling in either the best-or worst-case scenarios. 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 particularly 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.
In this paper, we extend NOM to cake-cutting problems, and show that the stark conflict between fairness and incentives disappears if we weaken strategy-proofness to NOM. In particular, NOM is compatible with the strong fairness property of proportionality, which guarantees each agent 1/n of the cake. Both properties are satisfied by a discrete adaptation of the moving knife mechanism (Dubins and Spanier, 1961), in which all agents cut the cake and the agent with the smallest cut receives all the cake to the left of his cut and leaves. This procedure is also procedurally fair and easy to implement in practice. NOM is violated by most other classical proportional mechanisms, even by the original Dubins-Spanier procedure, which shows that theoretically equivalent mechanisms may have different "obvious" incentive properties for boundedly rational agents, as shown by Li (2017) in the case of the second-price and the ascending-clock auctions. NOM partially explains why leftmost leaves is manipulated less frequently than other cake-cutting mechanisms in practice (Kyropoulou et al., 2019).

Related Literature
The cake-cutting problem has been studied for decades, given its numerous applications to the division of land, inheritances, and cloud computing (Brams and Taylor, 1996;Moulin, 2004). Regarding the compatibility of fairness and strategy-proofness in cake cutting, Mossel and Tamuz (2010) have shown that truth-telling in expectation and approximate proportionality can be obtained with probabilistic mechanisms, which we do not consider here. Chen et al. (2013) provide a complex deterministic mechanism that is both strategy-proof and proportional for a restricted class of utilities called piecewise linear. Their mechanism may waste pieces of cake which remain unassigned, something that never occurs with the proportional and NOM mechanism we identify.
The conflict between strategy-proofness and fairness properties goes beyond cake-cutting, and is ubiquitous in mechanism design. To avoid this conflict, other relaxations of strategy-proofness have been suggested, including: • Mechanisms that are hard to manipulate with a large number of agents (Immorlica and Mahdian, 2005;Azevedo and Budish, 2018).
• Mechanisms that can be manipulated in a small class of preference profiles (Pathak and Sönmez, 2013).
• Mechanisms that are hard to manipulate by boundedly rational agents. In particular, NOM is a stronger version of an existing property in the cake-cutting literature called maximin strategy-proofness, proposed by Brams et al. (2006Brams et al. ( , 2008. We discuss them in detail in the next section.
Here we only focus on this type of weak SP.

Model
In the classical mechanism design problem, there is a set of agents N, a set of outcomes X , and a set of possible types Θ = Π i∈N Θ i . In the cake cutting problem, the interval [0, 1] is the cake, the set of outcomes is the set of partitions of [0, 1] into n disjoint unions of subintervals, and the set of types is the set of possible utilities over the cake. The utility of agent i with type θ i is an additive, non-atomic function u i that maps each subinterval to a nonnegative number. That u i is non-atomic allows us to ignore the boundaries of intervals. The utility of the entire cake is normalized to 1. A standard assumption is that u i is divisible, i.e. that for every subinterval [x, y] and 0 ≤ λ ≤ 1, there exists a point z ∈ [x, y] such that u(x, z) = λ u(x, y). X denotes an arbitrary outcome and X i denotes the outcome for agent i, namely the piece of cake that corresponds to him.
A direct mechanism φ is a function mapping type profiles to outcomes. The notation θ i and θ −i denotes the type of agent i and the type of all agents in N \{i}, respectively. φ i (θ i , θ −i ) denotes agent i's individual allocation. We will only be concerned with proportional mechanisms, which guarantee to a truthful agent a utility of 1/n regardless of what every other agent reports, i.e. those in which The mechanism φ is strategy-proof (SP) if, for each agent, the utility obtained by reporting his type truthfully is weakly larger than the utility derived by reporting a different type, for all possible combinations of other agents' types. This is, if In the cake-cutting literature, agents do not reveal their whole valuation over the cake directly; rather, they are asked to cut the cake or stop a moving knife at some point y such that u i ([0, y]; θ i ) equals some number, or to choose among different parts of the cake available (Robertson and Webb, 1998). We will keep the notation of direct mechanisms to simplify exposition as no confusion will arise, but emphasize that strategy-proofness requires a truthful report every time an agent is asked to perform an action.
A report θ ′ i is a profitable manipulation of mechanism φ for agent i of type If some type of some agent has a profitable manipulation, then mechanism φ is called manipulable.
Troyan and Morrill suggest the following weaker version of SP. A mechanism φ is not obviously manipulable (NOM) if, for any profitable manipulation θ ′ i , the following two conditions hold: 1 They present this definition using maximum and mininum, which may not exists with a continuous cake; instead we consider the supremum and infimum over Θ −i . They also require a third inequality for tie-breaks, namely that there exists some θ −i such that Including this third restriction does not change our results.
If any of the following two conditions do not hold for some manipulation θ ′ i , then θ ′ i is said to be an obvious manipulation for agent i with type θ i ; and the mechanism φ is obviously manipulable.
In Troyan and Morrill, a manipulation is obvious if it either makes the agent better off in the worst-case or if it makes him better off in the best-case. Their definition is a strengthening of the maximin strategy-proofness imposed by Brams, Jones and Klamler, who only impose condition (1). Brams, Jones and Klamler write: "We assume that players try to maximize the minimumvalue pieces (maximin pieces) that they can guarantee for themselves, regardless of what the other players do. In this sense, the players are risk-averse and never strategically announce false measures if it does not guarantee them more-valued pieces".
Both relaxations of strategy-proofness have the advantages that agents do not require beliefs about other agents' actions, and that comparing best-and worst-cases scenarios requires less cognitive effort than comparing expected values using a arbitrary distribution over agents' types. However, maximin strategy-proofness is a mild property that is satisfied by a very large class of mechanisms. 2 On the other hand, NOM is a property that most classical proportional mechanisms in the literature fail, with the leftmost leaves mechanism being a remarkable exception. This is our main result.
The leftmost leaves mechanism works as follows: in period 1 all agents are asked to cut the cake at a point The agent who cuts the cake at the smallest point x 1 leaves with the allocation [0, x 1 ]. In period 2, all remaining agents are asked to cut the cake again at a point ; the agent who submits the smallest point x 2 leaves with the allocation (x 1 , x 2 ]. In period t, all remaining agents cut the cake at . The procedure continues until only one agent remains, who receives (x n−1 , 1]. The leftmost leaves mechanism is not only proportional and NOM, but also anonymous or procedurally fair (Crawford, 1977;Nicolò and Yu, 2008), in that the identities of agents do not affect the allocation produced. It also generates an assignment of a connected piece of cake for each agent, a desirable property for applications such as the division of land and computational resources (Segal-Halevi et al., 2017).
The leftmost leaves procedure is sometimes described in a slightly different way, in which agents at period t are asked to cut the cake at a point Procaccia (2016). This apparently innocuous modification makes leftmost leaves obviously manipulable, as we discuss in the proof of Theorem 1. To clarify the definitions we have introduced, we start with an example.
Example 1. For simplicity, consider a cake-cutting problem with piecewise linear preferences, i.e. agents either like or dislike certain intervals, and each desirable interval of the same length has the same value. The preferences of agent blue appear in Figure 1.
.5 .6 .7 .8 .9 Figure 1: The preferences of a (blue) agent over the cake.
.1 .2 .3 .4 .5 .6 .7 .8 .9 (b) Cut-middle. If the cake is divided using the well-known cut-and-choose mechanism in which the blue agent is the cutter who is asked to cut at a point x such that u i ([0, x]; θ i ) = 1/2, truthful behavior requires him to cut at 0.5 guaranteeing a utility of 0.5. If he chooses a profitable manipulation instead, say to cut the cake at 0.4, the best that can happen to him is that the other agent chooses the left piece of the cake. Thus, he gets a utility of 0.75, and cut-and-choose is therefore obviously manipulable.
We now examine another celebrated mechanism, known as cut-middle. In it, both agents make a cut at a point x i such that u i ([0, x i ]; θ i ) = 1/2, and the cake is divided at x 1 +x 2 2 , with each agent obtaining the part of the cake which contains his cut. If the blue agent is truthful and cuts at 0.5, the best that could happen is that the other agent cuts at ǫ (or 1 − ǫ) and thus the cut point becomes 0.5+ǫ 2 ≈ 0.25, so that the blue agent receives 0.75 utility. Nevertheless, if he would choose a profitable manipulation such as cutting at ǫ, the best that could happen is that the other agent cuts at ǫ 2 , and thus he would receive a utility almost equal to 1. We conclude that cut-middle is also obviously manipulable.
Cut-and-choose and cut-middle are mechanisms to divide cake among two agents. We now turn to mechanisms that can be used to divide cake among two or more agents (we describe the version with two agents to provide the intuition behind the results, and leave the general versions for the proof).
First, we consider the Banach-Knaster last challenger, in which agents are assigned a fixed order. The first agent is asked to cut the cake at the point x 1 for which u i ([0, x 1 ]; θ i ) = 0.5. The second agent is asked whether he wants to challenge this cut (if the value x 2 for which u i ([0, x 1 ]; θ i ) = 0.5 is smaller than x 1 ). If he does, he is asked to cut the cake at x 2 , and he receives [0, x 2 ]; otherwise he receives [x 1 , 1]. This procedure is obviously manipulable. To see this, consider the case in which agent 1 cuts at x 1 > x 2 . Then, the profitable manipulation for agent 2 of announcing x 1 −ǫ yields a higher utility (in the best-and worst-case scenarios, which are the same).
We turn to the Dubins -Spanier moving knife, in which a knife moves from 0 to 1 until the first agent stops it at the point x for which u i ([0, x]; θ i ) = 0.5, receiving himself the interval [0, x]. The best that can happen to a truthful agent is that the other agent will cut the cake at ǫ, so that he receives all the remaining cake which gives him almost 1 utility. However, once the moving knife has reached point x, a truthful agent should stop the knife, implying that he gets 0.5 with certainty, whereas a profitable manipulation would yield a higher utility in the best-case scenario (one example is to stop the knife at 1 − ǫ). Thus, Dubins -Spanier moving knife is also obviously manipulable.
Consider the alternative version of the previous mechanism in which both agents reveal simultaneously the point x i for which u i ([0, x i ]; θ i ) = 0.5, and the one who chooses the smallest x receives the allocation [0, x], whereas the other agent receives [x, 1]. Ties, if any, are broken randomly. Let us call this procedure leftmost leaves. The best that could happen to a truthful agent who reports x i is that the other reports ǫ < x i . He will receive then u i ([x j , 1]; θ i ) = 1 − ǫ, where ǫ can be made arbitrarily small. If the agent was to lie, every other report could guarantee at most a utility of 1, which is achieved by truthful behavior in the best-case scenario. On the other hand, the worst-case scenario of truthfulness gives a utility of 0.5, whereas any manipulation yields a smaller payoff in the worst-case. An extension of this argument to the general case with n agents allows us to obtain the following result.

Theorem 1. The leftmost leaves mechanism is not obviously manipulable and proportional, whereas cut-and-choose, cut-middle, Banach-Knaster last diminisher and Dubins-Spanier moving knife are all obviously manipulable.
Before presenting the proof, we present a few remarks about this result and its possible extensions. First, note that leftmost leaves is theoretically equivalent to Dubins-Spanier moving knife mechanism, in that when applied to truthful agents with the same types, both mechanisms always suggest the same allocation. How can mechanisms that are theoretically equivalent, such as the two we just presented, rank differently in terms of incentives? This idea goes back to Li (2017), who shows that two equivalent mechanisms, such as the ascending auction and the second price auction, in which bidding truthfully is a weakly dominant strategy, are different in terms of incentive properties for boundedly rational agents. The intuition in both results is similar: both in the second price auction and in leftmost leaves, agents have no restriction in the prior about their opponents' types when they reveal their type trough either their bids or their cuts; whereas in both the ascending auction and the moving-knife procedure, the fact that the knife or the clock has reached some point tells the agents' something about their opponents' types, and thus modifies what to expect in the best-and worst-case scenarios.
Second, a follow-up question to Theorem 1 is whether leftmost leaves is the only proportional and NOM mechanism in cake-cutting. The answer is no, as leftmost leaves can be slightly modified in several ways retaining both properties. One of them is to start cutting the cake from the left instead of from the right. Another less trivial one is the modification suggested by Even and Paz (1984), which works exactly the same as leftmost leaves for n = 2 (the proof is an almost verbatim copy of the proof of Theorem 1, so we omit it). 3 Obtaining a characterization of all NOM and proportional mechanisms remains an interesting, albeit challenging, open question. This question is particularly hard to tackle since equivalent mechanisms may be different in terms of NOM.
Third, we may ask whether NOM is compatible with envy-freeness, a stronger fairness property than proportionality (that leftmost leaves fails for n > 2). Envy-freeness requires that no agent prefers the piece of cake received by someone else over his own piece. These properties are incompatible in the canonical envy-free mechanism of Selfridge-Conway (see Brams and Taylor, 1996 for a detailed description of this complex protocol), in which the first agent cuts the cake into three pieces of equal worth. A truthful agent knows that one of those pieces will never belong to him, and thus he can achieve 3 The Even-Paz mechanism works as follows (for clarity let n be a power of 2). Given a cake [y, z], all agents choose cuts x i such that u i ([y, x i ]; θ i ) = u i ([y, z]; θ i )/2. Let x * be the median cut. Then the procedure breaks the cake-cutting problem into two: all agents who choose cuts x i ≤ x * are to divide the cake [y, x * ), whereas all agents who chose cuts above x * are to divide the cake [x * , z]. Each half is divided recursively among the n/2 partners assigned to it. When the procedure is called with a singleton set of agents {i} and an interval I it assigns X i = I. For example, if n = 4, agents cut the cake in two equivalent pieces and the cake is cut at the second smallest cut; then the two agents with the smallest (largest) cuts play leftmost leaves on the left (right) side of the cake. a maximum utility of 0.67. However, a lying agent can cut the cake in one piece of value 1 − ǫ, and two pieces of almost no value at all. In the best case scenario, he will keep the most valued piece entirely, showing that the Selfridge-Conway procedure is obviously manipulable.

Proof of Theorem 1
Proof. We show that leftmost leaves (φ ll ) is not obviously manipulable. Since leftmost leaves is an anonymous procedure in which the identity of the agents does not play a role, it is necessary to check conditions (1) and (2) only for one agent.
First, we prove that in period t, This is, agent i, by being truthful in period t, can expect (in the best-case scenario) to obtain the whole cake available in period t. Let us remember that truthful agent i cuts the cake at a point . In the best-case scenario, the smallest cut at period t is for an arbitrarily small ǫ ′ , i.e. no valuable piece of cake for agent i has been allocated. That we can find an ǫ so small comes directly from the standard assumption that the utilities are divisible. The best-case scenario is that all further smallest cake cuts x t+1 , x t+2 , . . . are epsilon increments of x t , such that in the end agent i receives X i = [x n−1 , 1], which gives him a utility arbitrarily close to u i [x t , 1], i.e. the utility of eating all of the remaining cake available. Since this is the maximum utility attainable, every manipulation must yield a weakly smaller utility. This concludes the proof that there is no profitable manipulation that gives a higher utility to a truthful agent in the best-case scenario. Now we show that any manipulation yields a lower utility in the worstcase scenario. To do so, we start with the following lemma.
Proof. Suppose x t i is not the smallest cut at period t (otherwise the result is immediate), which means A direct implication is that the remainder of the cake [x t , 1] must be worth more than n−t n−t+1 of u i ((t t−1 , 1); θ i ), i.e.
Note that the left-hand side of the previous expression is the utility that the truthful agent i would receive if he cut the cake at the smallest point in period t + 1. If his cut was not the smallest at period t + 1, a recursive formulation shows that he would receive a share of the cake that he values even more in period t + 2, showing that the worst that can happen to a truthful agent in period t is to obtain a utility of . Note that this previous argument shows that leftmost leaves is proportional, because Now we show that any manipulation at period t,x t i = x t i , yields a utility smaller than u i ([x t−1 ,1];θ i ) n−k+1 in the worst-case scenario. Ifx t i < x t i , in the worst-case scenariox t i would be the smallest cut, and agent i would therefore receive the allocation [x t−1 ,x t i ], which by construction yields a weakly lower utility than the allocation [x t−1 , x t i ]. In the other case, ifx t i > x t i , in the worst-case scenario the smallest cut in period t, x t , would be such that x t i < x t <x t i . Thus, the rest of the cake [x t , 1] is worth less than In period t + 1, agent i cannot choose a manipulationx t+1 i ≤ x t+1 i , as per the previous step he would receive a smaller utility, so agent i must choose a manipulationx t+1 i > x t+1 i . But in the worst-case scenario, the smallest cut in period t + 1 is x t+1 such that x t+1 i < x t+1 <x t+1 i , so that the rest of the cake [x t+1 , 1] is worth less than Following this argument, we see that agent i's only alternative is to receive the last piece of cake [x n−1 , 1], which by construction This concludes the proof that any manipulation yields a lower utility than truth-telling in the worst-case scenario. We conclude that no manipulation is better than truth-telling in either the best or the worst-case scenario, thus no manipulation is obvious and leftmost leaves is NOM.
In the main text we have provided examples of obvious manipulations for cut-and-choose, cut-middle, Banack-Knaster last diminisher and Dubins-Spanier moving knife mechanisms, which implies that none of these mechanisms are NOM.
Finally, we conclude by showing that a minor modification of leftmost leaves, sometimes used in the literature, destroys the NOM of the procedure. In this modification, at each period t, agents are asked to cut a cake at a point x t i such that u i ([x t−1 , x t i ]; θ i ) = 1 n . To see that this procedure is obviously manipulable, consider the case of an agent who has uniform preferences over the whole [0, 1] interval and that he has to cut the cake against 4 other agents. His first cut must be x 1 i = 0.2, which guarantees a utility of 0.2. However, suppose the lowest cut (submitted by someone else) was instead x 1 = 0.1. In period t = 2, the remaining cake is [0.1, 1] and if agent i is truthful, he should cut the cake at x 2 i = 0.3 so that u i ([0.1, 0.3]; θ i ) = 1/5 = 0.2. However, there is a manipulation that is guaranteed to yield a higher utility in the worstcase scenario. If he cuts the cake instead atx 2 i = .325 (i.e. the point at which u i ([0.1,x 2 i ]; θ i ) = 1/4), in the worst-case scenario, in which his cut is the lowest, she would guarantee herself a utility of 0.225 > 0.2. If her cut was not the lowest, by continuing to cut the cake at the pointx t i such that x t i = ([x t−1 ,1]) n−t+1 for all subsequent t, he could make sure to receive a utility of at least 0.225 (Lemma 1), which is larger than the worst-case scenario utility received by being truthful, 0.2. We conclude that the modified version of leftmost leaves is obviously manipulable.

Conclusion
Although 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 easily implementable mechanism called leftmost leaves.
Troyan and Morril's notion of NOM not only allows us to escape the tradeoff between fairness and incentives in cake-cutting, but also helps us to better understand real-life behavior when dividing an heterogeneous good. In the first lab experiment comparing cake-cutting mechanisms, Kyropoulou et al. (2019) report truthful behavior in NOM cake-cutting mechanisms in 44% of the cases, whereas the respective number for OM ones is of 31% (the difference is statistically significant with a p-value of 0.0000). In particular, leftmost leaves was significantly less manipulated than Banach-Knaster last diminisher when agents played against 2 opponents (difference of 29 percentage points, p-value of 0.0000) and 3 opponents (difference of 16 percentage points, p-value of 0.0000). The Even-Paz modification of leftmost leaves (which is NOM) was also significantly less manipulated than Banach-Knaster (difference of 35 percentage points, p-value of 0.0000).
Although in general NOM gives us testable predictions that map relatively well to observed behavior, it is intriguing that the Selfridge-Conway procedure also reports high rates of truth-telling, comparable to those of NOM mechanisms. Explaining this puzzling phenomenon remains an open problem which we leave for future research.