1. Computational Mechanism Design
Nathanaël Hyafil
Depth Oral Examination

2. Outline Background I. The three issues and direct ‘solutions’
Mechanism Design
Vickrey Clarke Groves
I. The three issues and direct ‘solutions’
Valuation Complexity
Computation Complexity
Communication Complexity
II. Sequential Mechanisms
Price-based
General query-based
III. Automated Mechanism Design
IV. Future Research

3. Mechanism Design
Choose global outcome according to some objective (Social Welfare, Revenue, ...)
Objective value depends on private information held by self-interested agents  Elicitation + Incentives
Applications:
Economics: Auctions, Public Goods, ...
E-Commerce: multi-attribute Negotiation (Supply chain), ...
Comp.Sc.: Networks, Autonomic computing, ...

4. Vickrey Clarke Groves (VCG)
Objective: Social Welfare
Agents report full valuation function
Outcome: that maximizes SW given reports
Payment: cost of your presence to the others
find outcome that maximizes SW without i, i
truth-telling: dominant strategy
wide application: quasi-linear environments

5. 1. Valuation complexity Valuation: a value for every outcome
Large outcome spaces (e.g. CA):
exponential number of values to compute
1-item: hard to evaluate valuation exactly
e.g., 1 hour of CPU time

6. 1. Valuation complexity
Work focused on analyzing valuation complexity in standard auctions (Parkes-04, Larson&Sandholm-01,03,04)
No direct solution to valuation problem but indirectly through work on communication complexity

7. 2. Computational complexity
Problem: Large Outcome Spaces
e.g., Combinatorial Auctions: number of allocations is K^N
Finding efficient allocation (WD):
NP-complete (RPH-98)
VCG: must do this (n+1) times, for n agents

8. Solutions for Combinatorial Auctions
Faster optimal allocation: (SSGL-01)
faster on ‘common’ instances
Branch & Bound
Heuristic search
Special cases: (RPH-98,Nisan-00,Tenn.-00)
Structure in preferences that make it ‘easy’
Approximate allocation: …

9. Approximate Allocation
Also intractable in worst case, if guarantees on quality of approximation
(Hastad 99)
Lose Incentive Compatibility in VCG
(LOCS 2002; NR 2000)
Settle for ‘hard’ to manipulate
-improvement NP-hard (Sanghvi & Parkes 04)
feasibly-dominant (Nisan&Ronen 00)

10. 3. Communication complexity
Valuation: one real number for every outcome
communication costs:
problem for large outcome spaces, e.g. CA:
worst case exponential (Nisan & Segal 2003)
even 1-item auctions for low value resources (bandwidth, CPU time...)
cost of sending 1 real number is significant
privacy / secrecy issues

11. Communication in 1-Item Auctions
Priority Games:
(Blumrosen & Nisan 02 ; BN & Segal 03)
hard limit on communication: k bits/agent
2k bids possible per agent

12. Priority Games
ex: 2 agents, 1 bit per agent B A

13. Priority Games: allocation
if one sends strictly higher bid: he wins B A

14. Priority Games: allocation
if tie: fixed order; here B > A B A

15. B A Priority Games payment rule:
2k thresholds per agent; Here (0,tA) ; (0,tB) B A

16. B A Priority Games Dominant strategy:
A bids ‘1’ if and only if vA  tA (Threshold strategy) B A

17. Priority Games allocation is not (always) optimal
Any set of thresholds induces a
dominant (threshold) strategy
Given a prior, they show how to
optimize thresholds to limit loss in
Social Welfare

18. Communication in CAs Complete Languages: Restricted Languages:
fully expressive
concise on ‘important’ classes of valuations
easy to interpret
Restricted Languages:
restrict bidding (e.g., to some bundles)
report ‘closest’ allowed valuation

19. Complete Language: LGB (BH 2001)
Ex: Machine m ; Resources r1, r2, r3, …
[ (m  r1;p1)  (m  r2;p2)  … ; p0 ]
Logical combinations of bids and goods
 ,  , 
prices in any sub-formula

20. LGB (BH 2001) more compact (sometimes exponentially)
easily interpretable
easily expressed
IP for Winner Determination with LGB bids:
faster than standard (flat) IP and heuristic search
much larger problems

21. Restricted Languages VCG with restricted revelation not always IC
truth-telling: reporting ‘closest’ valuation
if true valuation not allowed, possibly lie
restrict to classes of valuations (Ronen 01)
feasible-dominance
less successful: unrealistic assumption(s)
restrict to subset of bundles (HKMT-03)
IC: dominant for some choices of subset
analysis of loss in SW ; (not very informative)

22. II. Sequential Mechanisms
Addressing the three issues separately in VCG-based mechanisms:
not successful; everything ‘connected’
e.g., NR-00: less computation, more communication
Move away from VCG to get:
Approximate outcome without losing IC
Same Vickrey outcome without full revelation

23. Price-based: One-Item auctions
ASIA: (Kress&Boutilier 04)
‘sequential partitioning’: 1 price / round
report ‘in’ or ‘out’ ( 1 bit/round/agent)
exact allocation OR no allocation
DS: stay ‘in’ if and only if value  price!
‘Optimal’ price increase rule (MDP solution)
communication vs. efficiency

24. Price-based: CAs iBundle and extensions (Parkes01,Parkes&Ungar-00):
announce prices for each bundle at each round
(truthful) myopic best-response: bid on bundle that maximizes (true) utility given prices
at least ex-post equilibrium
with price adjustments: non-dynamic rev. dominant

25. Price-based: CAs Computation: Communication:
more WD to solve but smaller instances
empirically faster
Communication:
more bids to send but of smaller size
price posting

26. Query-based:Heuristic Search (Conen&Sandholm 01,02) , (Hudson&Sandholm 03)
Centralized elicitation with: value, order, rank queries
Exponential worst case, but in practice...
Information structures: rank lattice, order graph
Incentives only through Vickrey payments:
in some cases ‘for free’
Communication: empirically (on ‘important’ valuations), small fraction of valuation revealed
Computation: exponential complexity to maintain information structures

27. Query-based:Learning Theory (SCS04,ZBS-03,BJSZ04, LP04)
Learn classes of functions while minimizing number of queries to an oracle
Queries: membership=value, equiv.=demand
Important: learn allocation function, not valuations! (Lahaie & Parkes 04)
IC: through Vickrey payments; not free
But: exact learning, even if cost > benefits

28. Query-based: Decision Theory
Before: minimize number of queries to get optimal decision
DT approach: elicit information to improve decision, if it is worth the cost
Value Of Information starting to be used:
Chajewska, Koller, Parr 2000
Boutilier 2002

29. POMDP model of PE (Boutilier-02)
Sequentially optimal elicitation
trades-off quality of decision with cost of
elicitation
arbitrary cost models
computationally very complex: approximation
needed
no IC; Vickrey payments don’t apply

30. III. Automated Mechanism Design
VCG: one mechanism for all priors (SW)
Myerson auction: one algorithm, outputs an auction for each prior (Revenue)
AMD = Myerson for general MD
search for mechanism as an LP
IC and IR imposed as constraints
LP objective = designer ’s expected objective

31. Automated Mechanism Design
Problems:
Continuous outcome space
Continuous type / action space
Partial revelation
Priors hard to quantify

32. Past Research: Strict Uncertainty
Hyafil & Boutilier, UAI 2004:
Games with strict (qualitative) uncertainty
cannot Maximize Expected Utility
should Minimize Max Regret
Minimax-Regret Equilibrium
Minimax-Regret (Partial Revelation) AMD
sequence LPs and MIPs

33. IV. Current / Future Research
Partial Revelation:
partition agent type space
report subset containing type
pick outcome despite remaining
uncertainty (approximately optimal)
Partial Revelation objectives:
ex-post expected full objective
Min ex-post MaxRegret wrt full objective
IC: dominant / BayesNash / MinimaxRegret

34. Current / Future Research
General case: VCG-like algorithm
need ‘ tie-breaking ’ and payment rule
AMD: optimize mechanism + partition
non-linear LP objective
quadratic constraints
for direct mechanisms with dominant strategies, in general setting: ‘impossible’ ?
BayesNash? Minimax Regret?
Sequential mechanisms

35. Current / Future Research
specific application: multi-attribute negotiation
incentives issues
computational issues
MinimaxRegret-based Mechanism Design
use MMR as guide for sequential elicitation
incentives through Vickrey payments
more useful incentives properties??
(MinimaxRegret-Equilibrium applications)