1. Non-Cooperative Computation Mark Pearson CS 294 12/01/03

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

3. 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: B n  B Agents all want to compute F, but also have other conflicting interests (e.g., privacy)

4. 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

5. 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, …

6. 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

7. 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

8. 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

9. 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

10. 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 f 0 : B n  B n (usually f 0 : B n  B) And a strategy for a bidder i is a pair (f i,g i ), with – f i : B  Δ(B), the declaration function – g i : B  B  Δ(B), the interpretation function

11. 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)

12. 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

13. Results – 18 orderings left If exclusivity ranked above correctness  not NCC

14. Results – 6 orderings left Correctness not first, but above exclusivity A privacy violation (for i by j) occurs when  v j,x,y,  v -j (F(v j,v -j ) = x)  (v i = y) If privacy ranked above correctness and both are ranked above exclusivity, NCC  not reversible and not dominated and no privacy violations occur

15. 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

16. Summary of Results for 4LEX-NCC Is f(v) reversible or dominated? 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 NCC NCC Y Y Y YY Y N N N N NN

17. Elaborations Partial information gain – Utility for each component is entropy function – Talk about partial privacy violations and expected voyeurism ties – Results basically the same

18. 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

19. 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

20. Bibliography Y. Shoham & M. Tennenholtz Non-Cooperative Computation: Boolean Functions with Correctness and Exclusivity R. McGrew, R. Porter, & Y. Shoham Towards a General Theory of Non-Cooperative Computation Thanks to Y. Shoham for some slides from a talk of his on this topic

21. Questions?