Non-Cooperative Computation

Mark Pearson

CS 294

12/01/03

Map of the Talk

Introduction and motivation (cryptography) Game theoretic problem formulation Results (with some proofs) Other elaborations of setting, results.

Introduction and Motivation

n agents, each with a private value b B– b chosen by nature from commonly-known, joint

distribution– for definitions, B is unimportant; results assume

B={0,1}

A function F: Bn B Agents all want to compute F, but also have

other conflicting interests (e.g., privacy)

Motivation

Cryptography – esp. secure function evaluation– agents are honest, “curious” or “malicious”– typically, you look for a distributed solution– typically, you worry about collusion (n/2 and n/3 results)

Game Theory (NCC)– agents have specific utility functions– typically, you look for a centralized solution– typically, you analyze only the (Bayes-)Nash eqm

Setting – Utility Functions

Correctness – wishes to compute function correctly Exclusivity – wishes others do not compute function

correctly Privacy – wishes others do not discover my private

input Voyeurism – wishes to discover the private inputs of

others They consider all lexicographic orderings Motivation: joint venture, scientific research, …

Setting – Mechanism Design

Mechanism designer wants agents to know F What functions F have a mechanism?

– i.e., in induced game, it is Bayes-Nash equilibrium to tell the truth

Setting – Mechanism Design

Informational mechanism design (IMD) Agents’ utilities are also purely a function of

who knows what NCC is special case of IMD

– Mechanism designer’s objective is agents know true value of given function

– Agents have specific utility functions

Setting – Minor Details (relax later)

Assume joint distribution over types factors– Each agent’s type is independent

“Whole” information gain setting– Partial knowledge has zero utility

Results – Revelation Principle

Reminder of the standard setting:– Direct mechanisms– Truthfulness– Revelation Principle

The standard simulation argument underlying the Revelation Principle assumes that the mechanism can effect any outcome

Here the mechanism cannot effect arbitrary knowledge conditions

But the argument goes through anyway

Setting – Strategy Space

Informally:– Each agent declares its value (truthfully or not)– Based on all messages the center computes a signal for each agent

(usually the same signal for all agents) in a commonly-known way– Based on its original value and the signal from the center, each agent

calculates the value of the function

Thus a mechanism is f0: Bn Bn (usually f0: Bn B)

And a strategy for a bidder i is a pair (fi,gi), with – fi: B Δ(B), the declaration function– gi: BB Δ(B), the interpretation function

Results – Need Definitions First

A function is locally dominated (by i) if for some value of his, i uniquely determines the value of the function (e.g. OR function)

A function is reversible (by i) if by flipping its value i changes the value of the function, for any values of the other agents (e.g., parity)

Results – 24 orderings to consider

Dominated not NCC Reversible not NCC If correctness ranked first, and F

not dominated and not reversible NCC

Thus, have necessary and sufficient conditions

Results – 18 orderings left

If exclusivity ranked above correctness not NCC

Results – 6 orderings left

Correctness not first, but above exclusivity A privacy violation (for i by j) occurs when

vj,x,y,v-j(F(vj,v-j) = x) (vi = y)

If privacy ranked above correctness and both are ranked above exclusivity, NCC not reversible and not dominated and no privacy violations occur

Results – remaining 2 orderings

Voyeurism first, correctness second Voyeurism tie: learn same amount of other

agents’ private inputs regardless of what you say

In these two orderings, NCC non-reversible, non-dominated, and a voyeurism tie holds for every agent

Example: unanimity function

Summary of Results for 4LEX-NCC

Is f(v) reversible or dominated?

Not NCC

Not NCC

NCC

Exclusivity preferred over correctness?

Correctness ranked first?

Privacy ranked over correctness?

Is there a (partial) privacy violation? Is there a (partial) voyeurism tie?

Not NCCNot NCC NCCNCC

Y

Y

Y

Y Y

Y

N

N

N

N

N N

Elaborations

Partial information gain– Utility for each component is entropy function– Talk about partial privacy violations and expected

voyeurism ties– Results basically the same

Elaborations

Common prior does not factor– Types are correlated

Probabilistic mechanisms Quasi-linear environments

– Mechanism can pay agents– Some things can be overcome, like domination

Elaborations

Non-boolean– Order statistics: min, max generally are not NCC,

other order statistics generally are

Non-lexicographic order Different utility ordering for each agent To do: apply to SFE and other cryptographic

settings

Bibliography

Y. Shoham & M. TennenholtzNon-Cooperative Computation: Boolean Functions

with Correctness and Exclusivity

R. McGrew, R. Porter, & Y. ShohamTowards a General Theory of Non-Cooperative

Computation

Thanks to Y. Shoham for some slides from a talk of his on this topic

Questions?