An Algorithm for Automatically Designing Deterministic Mechanisms

without Payments

Vincent Conitzer and Tuomas SandholmComputer Science Department

Carnegie Mellon University

Mechanism design• One outcome must be selected

• Select based on agents’ reported preferences (types)

mechanism outcometypes

• Agents report types strategically• Mechanism designer designs mechanism to achieve good outcomes in spite of strategic behavior

– “Good” measured relative to true types– Designer only knows prior distribution over types

Mechanism design constraints

• Incentive compatibility: Agents should always be best off reporting truthfully– Revelation principle justifies restriction to truthful

mechanisms

• Participation: Agents should not be made worse off by participating in mechanism– Otherwise will not participate

Classical mechanism design

• Classical mechanism design has created a number of canonical mechanisms– Vickrey, Clarke, Groves mechanisms; Myerson auction; …

– These obtain a particular goal over a range of settings

• It has also created impossibility results– Gibbard-Satterthwaite; Myerson-Satterthwaite; …

– Show that no mechanism obtains a goal over a range of settings

Automated mechanism design

Solve mechanism design as an optimization problem automatically, only for setting at hand

• Advantages: Can be used when…– Canonical mechanisms are impossible for general class

that setting lies in– Canonical mechanisms are not known

• Unusual objective; payments not possible; …

– Canonical mechanism is known, but can do better• Higher objective, stronger nonmanipulability/participation…

Special case studied in this paper

General case This paper

k agents

payments may/may not be possible

randomization may/may not be possible

1 (type-reporting) agent

payments are not possible

deterministic mechanisms only

Computational problem• Input: instance is given by

– O: set of possible outcomes : set of possible types for the agent– p: [0, 1]: probability distribution over types– u: O : agent’s utility function – g: O : designer’s objective function

• Output: mechanism o: O which– is nonmanipulable: , ’ , u(, o()) u(, o(’))– satisfies participation constraint: , u(, o()) 0– under these constraints, maximize expected objective:

p()g(o())• NP-complete [C.& S. ICEC-03]

Application: one-on-one bartering

• Two agents each have initial endowment of goods• Each agent has value for every subset of all goods• Can trade goods, but payments are not possible• Agent 1 gets to design rules of the game

– “Mechanism designer”– Will maximize own expected utility with mechanism– Must honor agent 2’s participation constraint

• Agent 2 reports type– “Agent” participating in mechanism– Mechanism specifies outcome for each type

Naïve search vs. more sophisticated

• Naïve: Search over functions o: O• Naïve search space has size |O|||

• We will show how to search over subsets of outcomes instead

• This search space has size 2|O|

– Typically || >> |O|• If can independently have k different values for each outcome, || = k|O|

– Otherwise, can restrict search to subsets of outcomes of at most size || - still smaller

The mechanism oX for X O • With every subset X of outcomes, associate mechanism oX over these outcomes:

– Every type should get outcome that maximizes utility among outcomes in X– Given this, objective maximizing outcome in X should be chosen– Further ties broken arbitrarily

• Thrm. Any oX is truthful.• Thrm. For some X, oX is an optimal truthful mechanism.• Example: X={o1, o3} in following setting:

Type u(o1) u(o2) u(o3) g(o1) g(o2) g(o3) outcome

1 1 2 3 3 2 12 1 2 1 1 3 23 3 2 1 1 2 3

Expected objective value:

P(1)·1+P(2)·2+P(3)·1

o3

o3

o1

Searching over subsets of outcomes • Now we can search over subsets of outcomes

– Let X be those that are definitely in, Y those that are definitely out

X={}, Y={}

X={o1}, Y={} X={}, Y={o1}

X={o1}, Y={o2}

X={o1, o3}, Y={o2}

…

…

… …

Q: How do we evaluate nonleaf nodes?

A: Would like admissible heuristic

O1 in

O2 out

O3 in

O1 out

O2 outO2 inO2 in

O3 out

3211233

2311212

1233211

outcomeg(o3)g(o2)g(o1)u(o3)u(o2)u(o1)Type

3211233

2311212

1233211

outcomeg(o3)g(o2)g(o1)u(o3)u(o2)u(o1)Type

Expected objective value:

P(1)·1+P(2)·2+P(3)·1

o3

o3

o1

Mechanism oX,Y for X,Y O • Suppose we have decided for only some outcomes whether they are in or out

– X are definitely in, Y are definitely out.

• What is an optimistic estimate of the best mechanism from here?– Every type should get at least as much utility as the utility-maximizing outcome among those that are

definitely in. (X)– Given this, every type should get an objective-maximizing outcome among those that may still be in. (O-

Y)

• Thrm. The expected value of the objective for oX,Y is at least that of any truthful (leaf) mechanism down the tree.

– oX,Y not necessarily truthful; objective computed as if it were

• Example: X={o1}, Y={o2}

Expected objective value:

P(1)·3+P(2)·2+P(3)·1

Type u(o1) u(o2) u(o3) g(o1) g(o2) g(o3) outcome

1 1 2 3 3 2 12 1 2 1 1 3 23 3 2 1 1 2 3

o1

o3

o1

Using oX,Y as the heuristic• Can use the expected value of the objective for oX,Y as the heuristic

– Admissible because it always has at least the value of any descendant

X={}, Y={}

X={o1}, Y={} X={}, Y={o1}

X={o1}, Y={o2}

X={o1, o3}, Y={o2}

…

…

… …

O1 in

O2 out

O3 in

O1 out

O2 outO2 inO2 in

O3 out

3211233

2311212

1233211

outcomeg(o3)g(o2)g(o1)u(o3)u(o2)u(o1)Type

3211233

2311212

1233211

outcomeg(o3)g(o2)g(o1)u(o3)u(o2)u(o1)Type

Expected objective value:

P(1)·1+P(2)·2+P(3)·1

o3

o3

o1

Expected objective value:

P(1)·3+P(2)·2+P(3)·1

3211233

2311212

1233211

outcomeg(o3)g(o2)g(o1)u(o3)u(o2)u(o1)Type

3211233

2311212

1233211

outcomeg(o3)g(o2)g(o1)u(o3)u(o2)u(o1)Type

o1

o3

o1

Participation constraint

• Thrm. oX satisfies participation constraint iff for every type , there is some o X such that u(, o) 0

• So, during search, simply do not expand nodes where this is no longer true

Different search methods• A* runs out of memory too quickly; not used• Branch-and-Bound DFS

– DFS except only go to child node if its heuristic exceeds current best mechanism found

• IDA*– Same, but have an initial cutoff objective value– If no mechanism reaching that objective value is found, start again with cutoff

reduced by a fixed percentage• Or cutoff set to highest heuristic value not explored – whichever is lower

• In both cases, we also use some special tricks and data structures for computing node’s heuristic– Use information computed at parent node

Special-purpose algorithms vs. CPLEX

log(seconds)

Uniform utility

functions,no

participation

log(seconds)

log(seconds)

Uniform utility

functions,with

participation

Bartering a number of items

0

0

0

0

0 0

3

2

0.5

90

90

50

0.5

1.5

2.5

30

30

35

#types

#types

#types

#outcomes

#outcomes

#outcomes

CPLEXCPLEX

IDA*

CPLEX CPLEX

CPLEX CPLEX

B&BB&B

IDA*

B&B

IDA*

B&B

IDA*

B&BIDA*

IDA*B&B

Future research

• Extend algorithm to more general cases…

• …or at least find algorithms for other special cases

• Can extra structure on the instances be used?– E.g. CPLEX seems able to exploit bartering structure– Optimal combinatorial auctions?

Thank you for your attention!