Computationally-Efficient Approximation Mechanisms

(cont.)Ron Lavi

Presented by Yoni Moses

Last Week… Introduction

◦ Combining computational efficiency with game theoretic needs

Monotonicity Conditions◦ Cyclic Monotonicity◦ Weak Monotonicity

An Example – Machine Scheduling Problem

Today’s Agenda Approximation for Combinatorial Auctions

◦ Fractional allocation◦ Integral allocation

Impossibility results

Combinatorial Auctions (Review)

m items (Ω) are allocated to n players

i is the value given by player i to a bundle S (a subset of Ω)

Valuations are◦ Monotone: ◦ Normalized:

Goal: Find allocation such that is maximized.

Clash between complexity and game theoretic requirements

Problem: a general valuation’s size is exponential is n and m.

Possible representations:◦ Bidding languages model◦ access model

But polynomial algorithms that use these representations only obtain an approximation.VCG requires the exact optimum!

Given: an algorithm for CA that outputs a c-approximation.

Construct: A randomized c-approx. mechanism that is truthful in expectation

Plan:◦ First, solve for the fractional domain◦ Next, move back to the original domain, using

randomization

Converting Approximation Algorithms to Truthful Mechanisms

Solve using Linear Programming Allocation x gives player i a fraction of

subset S. The value is Constraints:

◦ A player receives at most one integral subset◦ An item cannot be over-allocated

Goal:◦ maximize the sum of values

The Fractional Domain

Formally…

The algorithm’s time complexity is polynomial.◦ We can assume the bidding languages model,

where the LP has size polynomial in the size of the bid (for example: k-minded players)

◦ We can assume general valuations with query-access, and the LP is solvable with a poly. num of demand queries

◦ The number of non-zero coordinates is poly. because we obtain x in polynomial-time

Solution is optimal => We can use VCG!◦ but it’s a solution for the fractional domain…

Results

Definition: Algorithm A “verifies a c-integrality-gap” for the LP program CA-P if it receives real numbers and outputs an integral point which is feasible for CA-P and

Moving from Fractional to Integral

Suppose A verifies a c-integrality-gap for CA-P (in poly. time), and x is any feasible point of CA-P.

Then x/c can be decomposed to a convex combination of integral feasible points (in poly. time)

Decomposition Lemma

Decomposition-Based Mechanism

Individual rationality (non-negative utility) is satisfied, regardless of the randomized choice:

VCG is individually rational: Thus, by definition: for any l

Results

Lemma: The decomposition-based mechanism is truthful in expectation, and obtains a c-approx. to the social welfare

Proof: The expected social welfare is . Since x* is the optimal (fractional) allocation, the c-

approx. is obtained. Truthfulness: First, we show that the expected price

equals the fractional price over c:

Truthfulness

Now, fix the other players’ valuation . x* is the fractional optimum obtained when player i

declares . z* is the frac. optimum obtained when i declares . .

Since VCG fractional prices are truthful:

Divide this formula by c. Using the previous formula and by definition of the decomposition, we get:

Truthfulness (cont.)

The left hand side is the expected utility for declaring .

The right hand side is the expected utility for declaring .

Thus, the lemma follows.

Truthfulness (cont.)

This analysis is for one-shot mechanisms, where a player declares his valuation up-front◦ for example: the bidding languages model.

For an iterative mechanism such as the query-access model, the solution is weakened to ex-post Nash◦ If all other players are truthful, player i will

maximize her expected utility by being truthful.

Remark

How de we decompose x/c into ? We use a new LP called P and its dual D. notation: E is the set of nonzero fractions in the

allocation.

Performing The Decomposition

primal dual

Constraints 1.11 of P describe the decomposition. If the optimum satisfies , we’re almost done.

◦ But P has exponentially many variables!

Performing The Decomposition (cont.)

We’ll use the dual D. Its number of variables is poly.◦ Of course, D’s constraints are

analogous to P’s variables => D has exponentially many constraints.

We can still solve D in polynomial time, using the ellipsoid method and our verifier A as a separation oracle.

Claim: If w, z is feasible for D:

If not, A can be used to find a violated constraint in poly. time Proof: Suppose . Let A receive w as input. Its output is an integral allocation . Since A is a c-approx. to the fractional optimum:

Due to the violated inequality of the claim:

Thus constraint 1.12 is violated for :

Using The Verifier as Separation Oracle

Claim: The optimum of D is 1, and the decomposition is polynomial-time computable.

Proof: is feasible, hence the optimum is at least 1.

By the previous claim, it is at most 1.

To solve P, we first solve D with this separation oracle: Given w,z , if , return the separating hyperplane .

Otherwise, find the violated constraint (which implies the separating hyperplane)

Using The Verifier as Separation Oracle (cont.)

Due to the oracle, the ellipsoid method uses a poly. number of constraints

Thus, there is an equivalent program with only these constraints.

Its dual is a program equivalent to P, but with a poly. number of variables.◦ Solving that gives us the decomposition.

Using The Verifier as Separation Oracle (cont.)

We still need an algorithm for verifying a c-integrality-gap…

Claim: We’re given A’, a c-approx. for general CA.◦ The approximation is with respect to the fractional optimum.

Using A’ ,we can obtain A, a c-integrality-gap verifier for CA-P, with a poly. time overhead on top of A’.

Proof: Given (the weights in A’s input), we

need to build from them a valid valuation that can be used as input for A’.◦ We can’t assume that w is non-negative and monotone.

Define for non-negativity Next, Define for monotonicity.

Integrality-Gap Verifier

is valid and can be represented with size |E|. Let

A’ gives c-approx. So such that ◦ Remember that

But in order to construct a verifier, we need this formula to hold for (w instead of ).◦ Now we only consider coordinates in E◦ Some coordinates in w (but not in ) can be negative

To fix the first problem, define : For any (i,S) such that , set:

All other coordinates of are set to 0

Integrality-Gap Verifier (cont.)

'( )A w x

By construction,

To fix the second problem, define :

Clearly,

So now we have , which is feasible for CA-P such that

Integrality-Gap Verifier (cont.)

Now we know how to build a verifier using a c-approx. for CA.

We still have to find an algorithm that approximates the fractional optimum.

The following greedy algorithm will give us a approx. to the fractional optimum (proof is skipped).

Input: Iteration: Let Set . Remove from E all (i’,S’) with i’=i or If E isn’t empty, reiterate.

Greedy Approximation Algorithm

The decomposition-based mechanism with Greedy as the integrality-gap verifier is individually rational and truthful-in-expectation and obtains an approximation of to the social welfare.

What we’ve achieved so far

The notion of truthfulness-in-expectation is inferior to truthfulness◦ It assumes to players are only interested in their expected utility. But

Don’t they care about the variance as well? Stronger notion: universal truthfulness. Players maximize

their utility for every coin toss◦ Still, “deterministic truthfulness” is better. ◦ In classic algorithms, the law of large numbers can be used to

approach the expected performance. But in mechanism design, we cannot repeat the execution because it affects the strategic properties.

Conclusion: deterministic mechanisms are still a better choice.

What’s Next?

Notations: is the domain of values The social choice function is onto A (domain of

alternatives)

Definition: f is an “affine maximizer” if there exist weights such that for all :

Of course, we might prefer other function forms. For example, due to computational complexity, revenue maximization, etc.◦ But what other forms are implementable:

Impossibility Results

Theorem: Suppose and . Then f is dominant-strategy

implementable iff it is an affine maximizer.◦ In other words, if we have unrestricted value domain and nontrivial

alternative domain, we have to use an affine maximizer.◦ Note that any affine maximizer is implementable (can be shown by

generalizing VCG arguments).

We will prove one side of a weaker theorem.

Definition: f is neutral if for all , if an alternative x exists such that for all i and , then f (v)=x◦ In a neutral affine maximizer, all constants will be zero.

Impossibility Results (cont.)

Theorem:

Suppose and . Then if f is dominant-strategy implementable and neutral, it must be an affine maximizer.

The proof will require two monotonicity conditions:◦ Positive Association of Differences (PAD)◦ Generalized-WMON

Impossibility Results (cont.)

Definition: f satisfies PAD if the following holds for any :

f(v)=x. for any and any i,

Claim: Any implementable function f, on any domain, satisfies PAD.

Proof: Let . In other words, players up to i declare according to v’. The rest declare according to v.

Positive Association of Differences

f(v’) = x

Now, suppose that for some and , . For every alternative we have

. In addition:

Reminder: f satisfies W-MON if for every player i, every and every with , .

W-MON implies that . By induction, . Which means f(v’)=x.

Positive Association of Differences (cont.)

In W-MON, we fix a player and fix the other players’ declarations.◦ We can generalize W-MON by dropping this.

Definition: f satisfies Generalized-WMON if for every with f(v)=x and f(v’)=y there exists a player i such that

Another way of looking at it: if f(v)=x and then .

Generalized-WMON

Note: for , , means that ,ni iR i

Claim: If the domain is unrestricted and f is implementable then f satisfies Generalized-WMON

Proof: Fix any v, v’. Suppose that f(v’) = x and

v’(y) – v(y) > v’(x) – v(x). Assume by contradiction that f(v) = y.

Fix a vector such that v’(x) – v’(y) = v(x)- v(y) - . Define v’’:

Using PAD, the transition v->v’’ implies f(v’’)=y and the transition v’->v’’ implies f(v’’)=x. contradiction.

Generalized-WMON (cont.)

Define:

Note that P(x,y) is not empty (assuming that v exists such that f(v) = x)

Also, if then for any , Explanation: take v with f(v)=x and v(x)-v(y)= . Construct v’ by increasing v(x) by and setting the other

coordinates as in v. By PAD, f(v’)=x and v’(x) – v’(y) =

P Construction

, 0nR

( , )P x y

Proof (i): Suppose by contradiction that . There exists

We assumed that , we know that a v’ exists such that v’(x)-v’(y) = and f(v’)=x

Due to our assumption, . This contradicts Generalized-WMON

Claim I

( ) ( ) '( ) '( )v x v y v x v y

Proof (ii): For any take some and fixsome .

Also, fix some v such that for all .

By the above argument, Since , it follows that f(v)=y. Thus , as needed.

Claim I

Proof: For any , fix some . Choose any v such that for all

By Generalized-WMON, f(v)=x. And by adding the 2 equations, we get:

Claim II

Additional Claims

The proof of the thorem follows… Based on separation lemma