Truthful cake-cutting
Truthful cake-cutting
- Truthful cake-cutting** is the study of algorithms for [[fair-cake-cutting]] that are also truthful mechanisms, i.e., they incentivize the participants to reveal their true valuations to the various parts of the cake.
The classic divide-and-choose procedure for cake-cutting is not truthful: if the cutter knows the chooser's preferences, they can get much more than 1/2 by acting strategically. For example, suppose the cutter values a piece by its size while the chooser values a piece by the amount of chocolate in it. So the cutter can cut the cake into two pieces with almost the same amount of chocolate, such that the smaller piece has slightly more chocolate. Then, the chooser will take the smaller piece and the cutter will win the larger piece, which may be worth much more than 1/2 (depending on how the chocolate is distributed).
Randomized mechanisms There is a trivial randomized truthful mechanism for fair-cake-cutting: select a single agent uniformly at random, and give him/her the entire cake. This mechanism is trivially truthful because it asks no questions. Moreover, it is fair in expectation: the expected value of each partner is exactly 1/n. However, the resulting allocation is not fair. The challenge is to develop truthful mechanisms that are fair ex-post and not just ex-ante. Several such mechanisms have been developed.
Exact division mechanism An exact division (aka consensus division) is a partition of the cake into n pieces such that each agent values each piece at exactly 1/n. The existence of such a division is a corollary of the Dubins–Spanier convexity theorem. Moreover, there exists such a division with at most <math>n(n-1)^2</math> cuts; this is a corollary of the stromquist–woodall-theorem and the necklace splitting theorem.
In general, an exact division cannot be found by a finite algorithm. However, it can be found in some special cases, for example when all agents have piecewise-linear valuations. Suppose we have a non-truthful algorithm (or oracle) for finding an exact division. It can be used to construct a randomized mechanism that is truthful in expectation.***' The randomized mechanism is a direct-revelation mechanism - it starts by asking all agents to reveal their entire value-measures:
Ask the agents to report their value measures. # Use the existing algorithm/oracle to generate an exact division. # Perform a random permutation on the consensus partition and give each partner one of the pieces.
Here, the expected value of each agent is always 1/n regardless of the reported value function. Hence, the mechanism is truthful – no agent can gain anything from lying. Moreover, a truthful partner is guaranteed a value of exactly 1/n with probability 1 (not only in expectation). Hence the partners have an incentive to reveal their true value functions.
Super-proportional mechanism A super-proportional division is a cake-division in which each agent receives strictly more than 1/n by their own value measures. Such a division is known to exist if and only if there are at least two agents that have different valuations to at least one piece of the cake. Any deterministic mechanism that always returns a proportional division, and always returns a super-proportional division when it exists, cannot be truthful.
- Mossel and Tamuz present a super-proportional *randomized* mechanism that is truthful in expectation:
Deterministic mechanisms: piecewise-constant valuations For deterministic mechanisms, the results are mostly negative, even when all agents have piecewise-constant valuations.
- Kurokawa, Lai and Procaccia** prove that there is no deterministic, truthful and envy-free mechanism that requires a bounded number of Robertson-Webb queries.
- Aziz and Ye prove that there is no deterministic truthful mechanism that satisfies either one of the following properties:*'
- ε-truthful, approximately-proportional and non-wasteful (for approximation constants at most 1/n);
- Truthful, approximately-proportional and connected (for approximation constant at most 1/n).
They present a minor modification to the even–paz-protocol and prove that it is ε-truthful with ε = 1 - 3/(2n) when n is even, and ε = 1 - 3/(2n) + 1/n2 when n is odd.
- Bei, Chen, Huzhang, Tao and Wu** prove that there is no deterministic, truthful and envy-free mechanism, even in the direct-revelation model, that satisfies either one of the following additional properties:
- Connected pieces;
- Non-wasteful;
- Position oblivious - the allocation of a cake-part is based only on the agents' valuations of that part, and not on its relative position on the cake.
Note that these impossibility results hold with or without free disposal.
On the positive side, in a replicate economy, where each agent is replicated k times, there are envy-free mechanisms in which truth-telling is a nash-equilibrium:
There are only two agents; Agents are hungry: each agent's valuation is positive (i.e., cannot be 0); * The mechanism is allowed to leave some part of the cake unallocated. It is open whether this impossibility result extends to three or more agents.
On the positive side, Tao presents two algorithms that attain a weaker notion called "proportional risk-averse truthfulness" (PRAT). It means that, in any profitable deviation for agent i, there exist valuations of the other agents, for which i gets less than his proportional share. This property is stronger than "risk-averse truthfulness", which means that, in any profitable deviation for i, there exist valuations of the other agents, for which i gets less than his value in a truthful reporting. He presents an algorithm that is PRAT and envy-free, and an algorithm that is PRAT, proportional and connected.
Piecewise-uniform valuations Suppose all agents have piecewise-uniform valuations. This means that, for each agent, there is a subset of the cake that is desirable for the agent, and the agent's value for each piece is just the amount of desirable cake that it contains. For example, suppose some parts of the cake are covered by a uniform layer of chocolate, while other parts are not. An agent who values each piece only by the amount of chocolate it contains has a piecewise-uniform valuation. This is a special case of piecewise-constant valuations. Several truthful algorithms have been developed for this special case.
- Chen, Lai, Parkes and Procaccia present a direct-revelation mechanism that is *deterministic*, proportional, [[envy-freeness|envy-free]], Pareto-optimal, and polynomial-time. Consider the special case of two agents with piecewise-uniform valuations, where the cake is [0,1], Alice wants only the subinterval [0,*a*] for some a<1, and Bob desires only the subinterval [1−*b*,1] for some b<1. Consider only non-wasteful mechanisms - mechanisms that allocate each piece desired by at least one player to a player who wants it. Each such mechanism must give Alice a subset [0,*c*] for some c<1 and Bob a subset [1−*d*,1] for some d<1. In this model:
- A non-wasteful determininstic mechanism is truthful iff, for some parameter t in [0,1], it gives Alice the interval [0, min(*a*, max(1−*b*,*t*))] and Bob the interval [1−min(*b*,max(1−*a*,1−*t*)),1]
- Such mechanism is envy-free iff t=1/2; in this case it is equivalent to the CLPP mechanism
They also show that, even for 2 agents, any truthful mechanism achieves at most 0.93 of the optimal social welfare.
- Li, Zhang and Zhang** show that the CLPP mechanism works well even when there are [[externality|externalities]] (i.e., some agents derive some benefit from the value given to others), as long as the externalities are sufficiently small. On the other hand, if the externalities (either positive or negative) are large, no truthful non-wasteful and position independent mechanism exists.
- Alijani, Farhadi, Ghodsi, Seddighin and Tajik** present several mechanisms for special cases of piecewise-uniform valuations:
- The expansion process handles piecewise-uniform valuations where each agent has a single desired interval, and moreover, the agents' desired intervals satisfy an ordering property. It is polynomial-time, truthful, envy-free, and guarantees connected pieces.
- The expansion process with unlocking handles piecewise-uniform valuations where each agent has a single desired interval, but without the ordering requirement. It is polynomial-time, truthful, envy-free, and not necessarily connected, but it makes at most 2n−2 cuts.
- Bei, Huzhang and Suksompong** present a mechanism for two agents with piecewise-uniform valuations, that has the same properties of CLPP (truthful, deterministic, proportional, envy-free, Pareto-optimal and runs in polynomial time), but guarantees that the *entire* cake is allocated:
Find the smallest x in [0,1] such that Alice's desired length in [0,*x*] equals Bob's desired length in [*x*,1]. # Give Alice the intervals in [0,*x*] valued by Alice and the intervals in [*x*,1] not valued by Bob; give the remainder to Bob.
The BHS mechanism works both for cake-cutting and for chore-division (where the agents' valuations are negative). Note that BHS does not satisfy some natural desirable properties:
- It does not guarantee connected pieces, for example when Alice wants [0,1] and Bob wants [0,0.5], then x=0.25, Alice gets [0,0.25] and [0.5,1], and Bob gets [0.25,0.5].
- It is not anonymous (see [[symmetric-fair-cake-cutting]]): if Alice wants [0,1] and Bob wants [0,0.5], then Alice gets a desired length of 0.75 and Bob gets 0.25, but if the valuations are switched (Alice wants [0,0.5] and Bob wants [0,1]), then x=0.5 and both agents get desired length 0.5.
- It is not position oblivious: if Alice wants [0,0.5] and Bob wants [0,1] then both agents get value 0.5, but if Alice's desired interval moves to [0.5,1] then x=0.75 and Alice gets 0.25 and Bob gets 0.75.
This is not a problem with the specific mechanism: it is provably impossible to have a truthful and envy-free mechanism that allocates the entire cake and guarantees any of these three properties, even for two agents with piecewise-uniform valuations. proves that no truthful mechanism can attain a utilitarian-optimal cake-cutting, even when all agents have piecewise-uniform valuations. Moreover, no truthful mechanism can attain an allocation with utilitarian welfare at least as large as any other mechanism. However, there is a simple truthful mechanism (denoted Lex Order) that is non-wasteful: give to agent 1 all pieces that he likes; then, give to agent 2 all pieces that he likes and were not yet given to agent 1; etc. A variant of this mechanism is the Length Game, in which the agents are renamed by the total length of their desired intervals, such that the agent with the shortest interval is called 1, the agent with the next-shortest interval is called 2, etc. This is not a truthful mechanism, however:
- If all agents are truthful, then it produces a utilitarian-optimal allocation.
- If the agents are strategic, then all its well-behaved Nash equilibria are Pareto-efficient and envy-free, and yield the same payoffs as the CLPP mechanism.