1. Statistical Cost Sharing: Learning Fair Cost Allocations from Samples
Eric Balkanski
Harvard University
Joint work with
Umar Syed and Sergei Vassilvitskii (Google NYC)
Dagstuhl Seminar on
Game Theory in AI, Logic, and Algorithms
\max_S\ & f(S) \\
\notag \text{s.t. } &\sum_{i \in S} p_i \leq B, \\
\notag & p_i \geq c_i \text{ for all } i \in S.
3/16/17
Dagstuhl Seminar

2. Motivating Example: Attributing Battery Consumption
Data: (apps running, units of battery usage)
( , 8)
( , 3)
( , 9)
Goal: fair blaming
: 40%, : 40%, : 20%
3/16/17
Dagstuhl Seminar

3. Cost Sharing
Aims to find an equitable way to split the cost of a service among players N
Cost function Cost allocation
Multiple solutions suggested: Shapley value, core, nucleolus…
But:
Assumes complete knowledge of cost function C
In previous example, only given samples of C
3/16/17
Dagstuhl Seminar

4. Statistical Cost Sharing
Definition: Given samples (S1, C(S1)), … (Sm, C(Sm)) where Si drawn i.i.d. from a distribution D, the Statistical Cost Sharing problem asks to find a cost allocation
[Balcan, Procaccia, and Zick 15]:
“It is the authors’ opinion that the information required in order to compute cooperative solution concepts (much more than the computation complexity) is a major obstacle to their widespread implementation”
3/16/17
Dagstuhl Seminar

5. Outline
The core
Previously and concurrently studied [Balcan et al. 15, 16]
The Shapley value
Never studied before in the statistical cost sharing context
3/16/17
Dagstuhl Seminar

6. The core
3/16/17
Dagstuhl Seminar

7. The Core “no group of players has an incentive to deviate”
Balance: must allocate the total cost:
Core [Gillies 59]: is in the core of C if for all
Definition [Balcan et al. 15]: Given poly(n, 1/δ) samples from D,
is in the probably stable core if
3/16/17
Dagstuhl Seminar

8. Prior Approach C’ (learned) cost function C data S1, C(S1) … Sm, C(Sm)
Traditional
Cost Sharing
Learning (PAC)
data
S1, C(S1) …
Sm, C(Sm)
core
Statistical Cost Sharing
Thm [Balcan et al. 15]: Assume C’ PAC-learns C from samples, then a cost allocation that is in the core of C’ is in the probably stable core of C.
3/16/17
Dagstuhl Seminar

9. Direct Approach cost function C data S1, C(S1) … Sm, C(Sm) core
Thm [B., Syed, Vassilvitskii 16, Balcan et al. 16]: If C has a non empty core, then C has a computable probably stable core .
Traditional
Cost Sharing
Learning (PAC)
data
S1, C(S1) …
Sm, C(Sm)
core
Statistical Cost Sharing
3/16/17
Dagstuhl Seminar

10. High Level Overview of Analysis
Compute any vector that satisfies the core property on samples, i.e., for all samples S.
Bound the generalization error to another set drawn from D using tools from theoretical machine learning.
Using VC dimension: sample complexity linear in n [B., Syed, Vassilvitskii 16].
Using Rademacher complexity: sample complexity logarithmic in n, but dependent on the spread of C and weaker relaxation of the core [B., Syed, Vassilvitskii 16].
3/16/17
Dagstuhl Seminar

11. The Shapley value
3/16/17
Dagstuhl Seminar

12. Shapley Value Unique cost allocation satisfying four natural axioms
Definition: Given poly(n) samples from D, an algorithm α-approximates the Shapley value of C over D if it computes estimates s.t. for all i,
3/16/17
Dagstuhl Seminar

13. Impossibility for Approximating Shapley Value
cost function C
Thm: There is no constant α > 0 such that coverage functions have α-approximable Shapley value, even over the uniform distribution D.
Traditional
Cost Sharing
Learning (PAC)
data
S1, C(S1) …
Sm, C(Sm)
Shapley value
Statistical Cost Sharing
Thm: Functions with curvature κ have √(1 – κ)-approximable Shapley value over the uniform distribution. Moreover, this is tight.
3/16/17
Dagstuhl Seminar

14. Data Dependent Shapley Value
We define four novel natural axioms that are dependent on D
There exists a unique cost allocation satisfying these four axioms
These data dependent Shapley value is (1-ε)-approximable from samples for any distribution and any function.
3/16/17
Dagstuhl Seminar

15. Conclusion
Studied cost sharing problem where the cost function is unknown and only given sampled data from this function
Possible to compute cost allocations from samples that satisfy a simple relaxation of the core for all functions with a non-empty core
Also possible to approximate arbitrarily well data-dependent Shapley value, which is the unique cost allocation satisfying four novel axioms.
Future work: other cost sharing methods, other classes of functions, better sample complexity bounds,…
3/16/17
Dagstuhl Seminar