Multi-item auctions with identical items limited supply: M items (M smaller than number of bidders, n). Three possible bidder types: –Unit-demand bidders –Decreasing marginal values –General valuations –We only consider private values We will see both strategic considerations and computational considerations.

Unit demand bidders Each bidder desires one item. Two popular “sealed-bid” auction formats: –Uniform-price auctions: The M highest bidders win, each pays the M+1 highest bid. –Discriminatory auctions: The M highest bidders win, each pays her bid. Two equivalent “open-cry” auctions: –Ascending price (English): The price ascends until M bidders remain. –Descending price (Dutch): The price descends until M bidders accept. Similarly to the single-item case, uniform-price is equivalent to English, and Discriminatory price is equivalent to Dutch.

Efficiency and Revenue in Unit-Demand Uniform-price is in fact a VCG mechanism (check at home). Therefore: –Truth-telling is a dominant strategy –The resulting allocation is efficient Discriminatory-price with symmetric bidders has a symmetric efficient equilibrium (similar analysis to the first-price auction). The revenue equivalence theorem and the optimal auction analysis can be extended to unit-demand bidders: –Any two auctions with the same outcome in equilibrium raise the same revenue (e.g. unifrom-price and discriminator-price). –The optimal auction is to sell the M items to the M bidders with the highest marginal valuations.

Example “Real” Applications Government securities were sold by the US government using discriminatory auctions, until From 1992, some securities (e.g. 2-years and 5-years) are being sold using a uniform-price auction. In the UK, electricity generators bid to sell their output on a daily basis. Until 2000 the auctions were uniform-price, and after that they switched to discriminatory price

Decreasing Marginal Valuations Each player has a marginal valuation function v i : {1,…,M}-> R –The value of receiving q items is v i (1)+…+v i (q) Marginal decreasing means: v i (q+1) < v i (q) for any 1<q<M Implication: Every bidder submits many bids Example for the uniform-price auction with two items: Red is player 117 Black is player Result: the red player wins two items7 and pays 2·14=28 => utility=46 Observation: if the red player only bids 17 then he will win one item and will pay price=6, increasing his utility!

Conclusions and remarks It is no longer true that the dominant strategy of a player in the uniform-price auction is to bid truthfully. –Therefore uniform-price in this case is different than VCG. As we saw, it is beneficial for the players to decrease their stated values for the items. This phenomena is termed “demand reduction”. There are no dominant strategies. However, the uniform-price auction is known to have a pure strategy equilibrium, in which: –“demand reduction” occurs. –the result is inefficient. It is also possible to show that every equilibrium of the discriminatory auction is inefficient.

VCG VCG continues of-course to have dominant strategies and an efficient outcome. The VCG price for this case: suppose player i won q items, and let x 1,…,x q be the q highest non-winning bids of the other players. Then player i pays x 1 +…+x q. –Recall that in general the price is -Σ j≠i v i (a) + Σ j≠i v i (b), where a is the allocation chosen when i participates and b is the allocation chosen when i does not participate. In the previous example (2-item auction),17 15 Result: the red player wins two items 14 and pays 14+6=20 => utility=12 7 6

The residual supply d i (p) = max {q | v i (q) > p } s -i (p) = M - Σ j≠i d j (p) bids s -i (p) d i (p) VCG price bids quantity s -i (p) d i (p) uniform price quantity

The Ausubel auction An ascending auction that is equivalent to VCG: –We start with a very low price (at this point s -i (p)=0). –The price is raised until, for at least one player i, s -i (p)>0. –Every player i gets s -i (p) units, for a price-per-unit p. –Continue in the same manner. The residual supply of i increases exactly at the marginal value of some other player, i’, for one of his units (say q). This means that this other player i’ will win at most q-1 units. Therefore, player’s price exactly equals the marginal value of the others for the units he got. As a result, truthfulness is an ex-post equilibrium in this auction.

Example 1’s bid: 17, 15, 7. 2’s bid: 14, 6. While price < 6, the demand of both players is at least two, so the residual supply < 0. At a price=6, 2’s demand decreases to 1. Therefore 1’s residual supply is 1, so he gets one unit. At price=14, 2’s demand decreases to 0, and 1 gets another unit.

General valuations (with complementarities) In general, marginal valuations may increase. For example v(1)=0, v(2)=100 represents a situation where the player must get two units in order to obtain any value from the items. –Let V i (q) denote the total value of player i for q items. We still assume that V i (q) < V i (q+1) (“free disposal”). In this case, the discriminatory-price and the uniform-price have no real meaning. VCG, again, has dominant strategies, and reaches the efficient outcome. However, no “natural” way of representing VCG or its price is known. We simply use the general mechanism.

Computational issues How do we compute the optimal allocation? Solve with dynamic programming: –O(i,q) = the optimal welfare for players 1,…,i, obtained with q items. –O(i,q)=max q’<q { O(i-1,q’) + V i (q-q’) } We need at most (an order of) M operations to compute every O(i,q), and so in total we need an order of n·M 2 operations to compute O(n,M), which is what we need. Sometimes M is extremely large (tens of thousands of items), and we want faster algorithms. It is known that it is impossible to achieve the exact optimum in a faster way, but can we design truthful approximation auctions?

Truthful approximations To achieve truthfulness with the VCG method, we must choose the alternative with the maximal welfare. But we can restrict the alternatives as we wish! A faster VCG mechanism: –Bundle the items in n 2 bundles of size b =  M/n 2 , and one “remainder” bundle of size r such that M=b·n 2 + r. –Allocate items only in bundles, i.e. each player can receive a multiple of b items, or the entire set of items. –Ask the players for their values of all these possible n 2 +1 bundles, and find the allocation with maximal welfare among all these allocations.

Example n=2, M=100. The values are: –Player 1: v 1 (12)=10, v 1 (64)=23, v 1 (80)=30, and all the rest are the minimal possible. –Player 2: v 2 (26)=15, v 2 (36)=20, v 2 (40)=22, and all the rest are the minimal possible. players can get only multiples of M/n 2 =25. Therefore we ask each player for his value for 25 items, 50 items, 75 items, and 100 items. Some of the alternatives we can choose: –0 for player 1 and 100 for player 2 (total value of 0+30=30) –25 for player 1 and 75 for player 2 (total value of 10+22=32) –75 for player 1 and 25 for player 2 (total value of 23+0=23) We cannot choose 64 for player 1 and 36 for player 2 (total value of 23+20=43). We choose 25 for player 1 and 75 for player 2 (total value 32).

Properties Claim: We can find the optimal allocation (among all possible allocations) with order of n 5 operations. Proof: with dynamic programming, very similar to before.

Properties Claim: We can find the optimal allocation (among all possible allocations) with order of n 5 operations. Proof: with dynamic programming, very similar to before. Claim: The obtained allocation, a, has total welfare at least half of the optimal welfare, o (i.e. Σ j v j (o) < 2 Σ j v j (a) ).

Properties Claim: We can find the optimal allocation (among all possible allocations) with order of n 5 operations. Proof: with dynamic programming, very similar to before. Claim: The obtained allocation, a, has total welfare at least half of the optimal welfare, o (i.e. Σ j v j (o) < 2 Σ j v j (a) ). Proof: Let i be a player with o i > M/n. Case 1: v i (o i ) > Σ j≠i v j (o j ). Then Σ j v j (o) < 2 v i (o) < 2 v i (M) < 2 Σ j v j (a).

Properties Claim: We can find the optimal allocation (among all possible allocations) with order of n 5 operations. Proof: with dynamic programming, very similar to before. Claim: The obtained allocation, a, has total welfare at least half of the optimal welfare, o (i.e. Σ j v j (o) < 2 Σ j v j (a) ). Proof: Let i be a player with o i > M/n. Case 1: v i (o i ) > Σ j≠i v j (o j ). Then Σ j v j (o) < 2 v i (o) < 2 v i (M) < 2 Σ j v j (a). Case 2: v i (o i ) < Σ j≠i v j (o j ).Consider the allocation d in which player i gets nothing and every j ≠ i gets o j rounded up to the next multiple of b. We added at most nb items and removed at least M/n items, and so, since nb < M/n, the allocation d is valid. We get: Σ j v j (o) < 2 Σ j≠i v j (o j ) < 2 Σ j v j (d) < 2 Σ j v j (a).

Conclusion There exists a computationally-efficient truthful multi-unit auction that always obtains at least half of the optimal welfare. Main open question: what about the revenue? –(Nothing is currently known about the revenue!)