Equitable cake-cutting
Equitable cake-cutting
- Equitable (EQ) cake-cutting** is a kind of a [[fair-cake-cutting]] problem, in which the fairness criterion is [[equitability]]. It is a cake-allocation in which the subjective value of all partners is the same, i.e., each partner is equally happy with his/her share. Mathematically, that means that for all partners and :
:<math>V_i(X_i) = V_j(X_j)</math>
Where: <math>X_i</math> is the piece of cake allocated to partner ; <math>V_i</math> is the value measure of partner . It is a real-valued function that, for every piece of cake, returns a number that represents the utility of partner from that piece. Usually these functions are normalized such that <math>V_i(\emptyset)=0</math> and <math>V_i(EntireCake)=1</math> for every . See the page on equitability for examples and comparison to other fairness criteria.
Finding an equitable cake-cutting for two partners ### One cut - full revelation When there are 2 partners, it is possible to get an EQ division with a single cut, but it requires full knowledge of the partners' valuations. Assume that the cake is the interval [0,1]. For each <math>x\in[0,1]</math>, calculate <math>u_1([0,x])</math> and <math>u_2([x,1])</math> and plot them on the same graph. Note that the first graph is increasing from 0 to 1 and the second graph is decreasing from 1 to 0, so they have an intersection point. Cutting the cake at that point yields an equitable division. This division has several additional properties: * It is EF, since each partner receives a value of at least 1/2. * It is not EX, since the value per partner may be more than 1/2. It is Pareto efficient (PE) among all divisions that use a single cut. However, there may be more efficient divisions that use two or more cuts. which satisfies a weaker kind of equitability and a stronger kind of truthfulness. The procedure first finds the median points of each partner. Suppose the median point of partner A is <math>a</math> and of partner B is <math>b</math>, with <math>0<a<b<1</math>. Then, A receives <math>[0,a]</math> and B receives <math>[b,1]</math>. Now there is a surplus - <math>[a,b]</math>. The surplus is divided between the partners in equal proportions*. So, for example, if A values the surplus as 0.4 and B values the surplus as 0.2, then A will receive twice more value from <math>[a,b]</math> than B. So this protocol is not equitable, but it is still EF. It is weakly-truthful in the following sense: a risk-averse player has an incentive to report his true valuation, because reporting an untrue valuation might leave him with a smaller value.
Two cuts - moving knife Austin moving-knife procedure gives each of the two partners a piece with a subjective value of exactly 1/2. Thus the division is EQ, EX and EF. It requires 2 cuts, and gives one of the partners two disconnected pieces.
Many cuts - full revelation When more than two cuts are allowed, it is possible to achieve a division which is not only EQ but also EF and PE. Some authors call such a division "perfect".
The minimum number of cuts required for a PE-EF-EQ division depends on the valuations of the partners. In most practical cases (including all cases when the valuations are piecewise-linear) the number of required cuts is finite. In these cases, it is possible to both find the optimal number of cuts and their exact locations. The algorithm requires full knowledge of the partners' valuations.
Finding an equitable division for three or more partners ### Moving knife procedures Austin's procedure can be extended to n partners. It gives each partner a piece with a subjective value of exactly <math>1/n</math>. This division is EQ, but not necessarily EX or EF or PE (since some partners may value the share given to other partners as more than <math>1/n</math>).
There is another procedure, using n-1 moving knives, that can be used to find a connected equitable allocation for any ordering of the agents. They do this by describing 3 specific value measures on a 1-dimensional cake, with the following properties: With 2 cuts, every EQ allocation is not EF nor PE (but there are allocations which are EF and 2-PE, or EQ and 2-PE). With 3 cuts, every EQ allocation is not PE (but there is an EQ+EF allocation). * With 4 cuts, every EQ allocation is not EF (but there is an EQ+PE allocation).
Pie cutting A pie is a cake in the shape of a 1-dimensional circle (see fair-pie-cutting).
Barbanel, Brams and Stromquist study the existence of divisions of a pie, which are both EQ and EF. The following existence results are proved without providing a specific division algorithm:
- For 2 partners, there always exists a partition of a pie which is both envy-free and equitable. When the value measures of the partners are absolutely continuous with respect to each other (i.e. every piece which has a positive value for one partner also has a positive value for the other partner), then there exists a partition which is envy-free, equitable and undominated.
- For 3 or more partners, it may be impossible to find an allocation that is both envy-free and equitable. But there always exists a division that is both equitable and undominated.
Divisible goods The adjusted-winner-procedure calculates an equitable, envy-free and efficient division of a set of divisible goods between two partners.
Query complexity An equitable cake allocation cannot be found using a finite protocol in the robertson–webb-query-model, even for 2 agents. Moreover, for any ε > 0:
- A connected ε-equitable cake-cutting requires at least Ω(log ε−1) queries. For 2 agents, an O(log ε−1) protocol exists.
- Even without connectivity, ε-equitable cake-cutting requires at least Ω(log ε−1 / log log ε−1 ) queries.