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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. - PowerPoint PPT Presentation
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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