Robertson–Webb query model


In computer science, the Robertson–Webb query model is a model of computation used by algorithms for the problem of fair cake-cutting. In this problem, there is a resource called a "cake", and several agents with different value measures on the cake. The goal is to divide the cake among the agents such that each agent will consider his/her piece as "fair" by his/her personal value measure. Since the agents' valuations can be very complex, they cannot - in general - be given as inputs to a fair division algorithm. The RW model specifies two kinds of queries that a fair division algorithm may ask the agents: Eval and Cut. Informally, an Eval query asks an agent to specify his/her value to a given piece of the cake, and a Cut query asks an agent to specify a piece of cake with a given value.
Despite the simplicity of the model, many classic cake-cutting algorithms can be described only by these two queries. On the other hand, there are fair cake-cutting problems that provably cannot be solved in the RW model using finitely many queries.
The Eval and Cut queries were first described in the book of Jack M. Robertson and William A. Webb. The name "Robertson–Webb model" was coined and formalized by Woeginger and Sgall.

Definitions

The standard RW model assumes that the cake is an interval, usually the interval . There are n agents, and each agent i has a value measure vi on the cake. The algorithm does not know vi, but can access it using two kinds of queries:
  • An eval query: given two real numbers x and y, Evali asks agent i to report the value of the interval, i.e., vi.
  • A mark query : given two real numbers x and r, Marki asks agent i to report some value y such that vi = r.

Example

The classic Divide and choose algorithm, for cutting a cake between two children, can be done using four queries.
  • Ask Alice an Eval query; let V1 be the answer.
  • Ask Alice a Mark query; let x1 be the answer.
  • Ask George an Eval and an Eval queries.
  • If the former value is larger, give to George and to Alice; else, give to Alice and to George.

Results

Besides divide-and-choose, many cake-cutting algorithms can be performed using RW queries whose number is polynomial in n. For example: Last diminisher can be done by O RW queries and Even–Paz protocol can be done by O RW queries. In parallel, there are many hardness results, proving that certain fair division problems require many RW queries to complete. Some such hardness results are shown below.
Proportional cake-cutting requires Ω RW queries when either
  • the pieces must be connected, or
  • the protocol is deterministic, or
  • the precision of cutting the cake is finite.
  • The only protocol which uses O RW queries is a randomized protocol, which can return disconnected pieces, and the allocation might be only fractionally-proportional.
Proportional cake-cutting with different entitlements requires at least Ω RW queries, where D is the common denominator of the entitlements. There is an algorithm that uses O RW queries for rational entitlements, and a finite algorithm for irrational entitlements.
Envy-free cake-cutting requires
  • Ω RW queries when the pieces may be disconnected, Infintiely many queries when the pieces must be connected and there are at least 3 agents. In other words, there is no algorithm that always finds an envy-free allocation among 3 or more agents using finitely-many RW queries.
  • For any ε > 0, an ε-envy-free connected cake-cutting requires at least Ω queries. For 3 agents, an O protocol exists. For 4 agents, an O protocol exists. For 5 or more agents, the best known protocol requires O, which shows an exponential gap in the query complexity.
Equitable cake-cutting cannot be done using finitely-many RW queries even for 2 agents. Moreover, for any ε > 0:
  • A connected ε-equitable cake-cutting requires at least Ω queries. For 2 agents, an O protocol exists. For 3 or more agents, the best known protocol requires O queries.
  • Even without connectivity, ε-equitable cake-cutting requires at least Ω RW queries.
Exact cake-cutting cannot be done using finitely-many RW queries even for 2 agents. Moreover, for any ε > 0:
  • An ε-perfect cake-cutting with the minimum possible number of cuts requires at least Ω queries. For 2 agents, an O protocol exists. For 3 or more agents, the best known protocol requires O queries.
Maximin share cake-cutting, when the pieces must be separated by a positive distance, cannot be done using finitely-many RW queries. Moreover, even for a single agent, there is no algorithm that computes the agent's maximin-share using finitely-many RW queries. However:
  • For any ε > 0, it is possible to compute a value between the MMS and the MMS-ε using O RW queries.
  • When the cake is circular, it is possible to compute a value between the MMS and the MMS-ε using O RW queries. It is open whether O RW queries suffice.
Average-proportional cake-cutting cannot be computed using finitely-many RW queries, even when there are 2 families with 2 members in each family. The proof is by reduction from equitable cake-cutting.

Variants

Left-mark and right-mark

When the value measure of an agent is not strictly positive, a mark query can, in principle, return infinitely many values. For example, if an agent values at 1 and at 0, then the query Mark can return any value between 0.9 and 1. Some algorithms require a more specific value:
  • The left-mark query, LeftMark, returns the leftmost ''y such that vi = r'';
  • The right-mark query, RightMark, returns the rightmost ''y such that vi = r'';
If only one of these two variants is given, the other variant cannot be computed in finite time.

Two-dimensional cakes

The RW query model has been generalized to two-dimensional cakes and multi-dimensional cakes.

Alternative models

There are many cake-cutting algorithms that do not use the RW model. They usually use one of the following models.

Direct revelation model

Algorithms for restricted classes of valuations, such as piecewise-linear, piecewise-constant or piecewise-uniform, which can be given explicitly as input to the algorithm. Some such algorithms were developed for truthful cake-cutting.

Moving-knife model

In this model, there are knives moving continuously along the cake. This model is related to the RW model as follows: any moving-knife procedure with a fixed number of agents and a fixed number of knives can be simulated using O RW queries.

Simultaneous queries model

In this model, agents simultaneously send discretizations of their preferences. A discretization is a sequence of cut-points, and the values of pieces between these cut-points where the values of and. These reports are used to compute a fair allocation. The complexity of an algorithm in this model is defined as the maximum number of intervals in a required discretization.
One advantage of this model over the RW model is that it enables to elicit preferences in parallel. This allows to compute a proportional cake-cutting in time O by simultaneously asking each agent for a discretization with n intervals. In contrast, in the RW model there is an O lower bound. On the other hand, in the simultaneous model, it is impossible to compute an envy-free cake-cutting using a finite discretization for 3 or more agents; but for every e>0, there exists a simultaneous protocol with complexity O, that attains an e-approximate envy-free division.