Computationally-Efficient Approximation Mechanisms (cont.)

1. Computationally-Efficient Approximation Mechanisms (cont.) Ron Lavi Presented by Yoni Moses

2. Last Week… • Introduction • Combining computational efficiency with game theoretic needs • Monotonicity Conditions • Cyclic Monotonicity • Weak Monotonicity • An Example – Machine Scheduling Problem

3. Today’s Agenda • Approximation for Combinatorial Auctions • Fractional allocation • Integral allocation • Impossibility results

4. Combinatorial Auctions (Review) • m items (Ω) are allocated to n players • i is the value given by player ito a bundle S (a subset of Ω) • Valuations are • Monotone: • Normalized: • Goal: Find allocation such that is maximized.

5. 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!

6. Converting Approximation Algorithms to Truthful Mechanisms • 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

7. The Fractional Domain • 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

8. Formally…

9. Results • 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…

10. Moving from Fractional to Integral • 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

11. Decomposition Lemma • 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)

12. Decomposition-Based Mechanism

13. Results • Individual rationality (non-negative utility) is satisfied, regardless of the randomized choice: • VCG is individually rational: • Thus, by definition: for any l

14. Truthfulness • 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:

15. Truthfulness (cont.) • 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:

16. 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.

17. Remark • 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.

18. Performing The Decomposition • 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. primal dual

19. Performing The Decomposition (cont.) • Constraints 1.11 of P describe the decomposition. • If the optimum satisfies , we’re almost done. • But P has exponentially many variables! • 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.

20. Using The Verifier as 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 :

21. Using The Verifier as Separation Oracle (cont.) • 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)

22. 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.

23. Integrality-Gap Verifier • 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.

24. Integrality-Gap Verifier (cont.) • 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

25. Integrality-Gap Verifier (cont.) • By construction, • To fix the second problem, define : • Clearly, • So now we have , which is feasible for CA-P such that

26. Greedy Approximation Algorithm • 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.

27. What we’ve achieved so far • 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.

28. What’s Next? • 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.

29. Impossibility Results • 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:

30. Impossibility Results (cont.) • 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.

31. 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

32. Positive Association of Differences • 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. f(v’) = x

33. Positive Association of Differences (cont.) • 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.

34. Generalized-WMON • 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 .

35. Generalized-WMON (cont.) • 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.

36. P Construction • 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) =

37. Claim I • 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

38. Claim I • Proof (ii): For any take some and fixsome . • Also, fix some v such thatfor all . • By the above argument, • Since , it follows that f(v)=y. • Thus , as needed.

39. Claim II • 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:

40. Additional Claims • The proof of the thorem follows… • Based on separation lemma