Non-Cooperative Computation Mark Pearson CS 294 12/01/03.
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Transcript of Non-Cooperative Computation Mark Pearson CS 294 12/01/03.
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?