Online Ascending Auctions for Gradually Expiring Items Ron Lavi and Noam Nisan SISL/IST, Caltech Hebrew University

The Model (I) M identical items that “expire” at different times. Players arrive over time, and desire one item between their arrival time and their deadline Items: Expiration times: Player 1 arrival time deadline Player 2 Player 3 Time 1 Time 2

Player i has value v i for receiving a desired item. Players are selfish: –All information (arrival time, deadline, value) is private, known only to the player. –Each player acts in order to maximize his own utility: value - price. Our goal is to maximize the sum of (true) values of players that receive an item (the “social welfare”). Applications: –In economic settings e.g. transportation tickets –In computational settings e.g. bandwidth allocation The Model (II)

Algorithmic Status Well studied - equivalent to scheduling of unit jobs. Offline optimal allocation is poly-time computable (has a matroid structure). Lower bound of for online approximation. [Hajek] Online greedy is a 2 - approximation: greedy: at time t, allocate item t to the player with highest value. –This assumes obedient players that simply reveal their private information.

Truthfulness and its difficulties A popular approach: truthful auctions. –Motivating the player to reveal his true parameters. – Strong argument of dominant strategy: no matter what others do, the truth will maximize “my” utility. –Many recent positive examples for truthful auctions. Unfortunately, we show that: Theorem: Any deterministic truthful auction for our allocation problem cannot obtain an approximation ratio better than M. –A simple truthful M - approximation exists.

How to approach this difficulty? Relax the equilibrium notion to Bayesian - Nash: –Not a worst-case analysis. Requires strong distributional assumptions. Add assumptions about player types. E.g. assume values in [v min, v max ]. Then a randomized truthful 2 log(v max - v min ) approximation exists (a special case of [BSZ] ). –vs. a deterministic 2 - approximation without any assumptions when truthfulness is dropped. Our approach: –New, relaxed, notion of equilibrium. –Worst - case analysis. No distributional assumptions. –No additional assumptions about player types.

Outline for rest of the talk Describe two ascending auctions: –Their algorithmic properties –Intuition to an equilibrium notion that fits well Describe a new notion of equilibrium –discuss its properties Main theorem –Intuition for the proof

The Online Iterative Auction Maintain temporary prices and owners for each item (initialized to 0). At each time unit t=1,2… : Repeat: Some player that doesn’t currently own an item temporarily takes an item, and increases the price by . Until no losing player wishes to make a new bid. Allocate item t to its current owner for the listed price - . Keep prices and temporary owners for next time unit. This is an adaptation of the Iterative Auction of [DGS].

Example 12 Item Temp. winner Temp. price Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1  =1

Example 12 Item Temp. winner Temp. price I-- 10 Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1 (phase 1)  =1

Example 12 Item Temp. winner Temp. price III 11 Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1 (phase 2)  =1

Example 12 Item Temp. winner Temp. price IIIII 21 Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1 (phase 3)  =1

Example 12 Item Temp. winner Temp. price III 31 Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1 (phase 4)  =1

Example 12 Item Temp. winner Temp. price III 31 Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1 (phase 4) Player I did not bid for the item with lowest price.  =1

Example 12 Item Temp. winner Temp. price III 31 Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1 Result: Player I wins item 1 and pays 2. If no new player will arrive, player II will win item 2 for a price of 0. But, player II might not win at all if a new high valued player will now arrive.  =1

Players’ behaviors (the offline case) DFN( [DGS] ): A player is myopic if he always bids on the item with lowest price among those he desires. THM( [DGS],[GS] ): Assume all players arrive at time 1: When all players are myopic then the online iterative auction finds the optimal allocation *. When all other players are myopic, player i will maximize * his utility by behaving myopically. * up to a difference of about .

A tight block B  S: |B|=d and  j  B d(j) < d. Tight blocks must be prefixes of S, thus contained one in the other. Special focus in the minimal tight block f. Basic structure of allocations d12...M S = the optimal allocation j’’ j j’

A tight block B  S: |B|=d and  j  B d(j) < d. Tight blocks must be prefixes of S, thus contained one in the other. Special focus in the minimal tight block f. Every j in f can be located first. Therefore, its “social cost” is the value of the highest unallocated player. Basic structure of allocations d f Highest un-allocated player determines VCG price of all players in f 12...M S = the optimal allocation j’’ j j’ i*

The offline iterative auction with myopic players finds the optimal allocation All prices in f are equal (because of the structure of swaps): –p(j’) < p(j’’) since j’ is myopic –p(j’’) < p(j’) since j’’ is myopic and has far-enough deadline. Prices will continue to go up exactly until v(i*). d f Highest un-allocated player determines VCG price of all players in f 12...M S = the optimal allocation j’’ j j’ i*

In the online case, non myopic behaviors might perform better. E.g. bidding more aggressively for the current item makes sense if one anticipates that many competitive players will arrive later on. DFN : A player is semi - myopic if he bids on some item with price lower than his value. THM : If all players are semi - myopic then the online iterative auction obtains a 3 - approximation. Players’ behaviors (the online case)

The Sequential Japanese Auction Item t is sold at time t using a classic Japanese auction: –The auctioneer starts raising a price. –Each player decides whether to drop or to stay as the price ascends. –We allow to observe how many players remain at each moment. –The price halts when only one player remains. This player wins and pays the price that was reached (up to some tie breaking adjustment rule).

Example Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1 What if players I and II decide not to participate at all in the auction for item 1? Player III will win item 1. Player I will certainly not win anything. Player II might win item 2, but for a price of 3.

Example continued Player I: v=3, d=2 Player II: v=5, d=2 Player III: v=2, d=1 Suppose players I and II decide to stay until the price reaches their value, or until there remain two players in the auction (including themselves): 2 At price=2, player III will drop. Immediately afterwards, both I and II drops. So either I or II wins and pays 2. Price

Players’ behaviors (the offline case) Surprisingly, a notion of myopic behavior leads to the optimal allocation here as well: DFN: A player is myopic if, at any time t, he drops exactly: –when the price reaches his value, or –when d - t other players remain (where d is his deadline). THM: If all players arrive at time 1, and are all myopic, then the Sequential Japanese Auction finds the optimal allocation.

Proof p*=value of highest unallocated player i*, |f|=d Price < p* implies that no one from f drops: –At least d+1 players still remain (all f + i*) –Price is still low. At price = p* all remaining unallocated players drop, and after them all remaining players of S. Players of f start to drop only after all others have dropped.  winner of item 1 = optimal item 1 winner. Continue inductively. p* Price

In the online case, again, bidding more aggressively for the current item makes sense if one anticipates that many competitive players will arrive later on. DFN: A player is semi - myopic if, at any time t, he drops: –not earlier than d-t other players remain, and –not later than when the price reaches his value. THM: If all players are semi - myopic then the Sequential Japanese Auction obtains a 3 - approximation. Players’ behaviors (the online case)

Summary of auctions Online Iterative Sequential Japanese Myopic behaviorSemi-myopic behavior bid for the item with the lowest price bid for some item with price < value Drop when (i) price reaches value or (ii) Exactly d-t other players remain Drop in between (i) price reaches value and (ii) d-t other players remain

Proving the approximation Lemma: Any semi - myopic algorithm obtains a 3 - approximation. Lemma: When players are semi - myopic then both our auctions are semi - myopic algorithms. MyopicGreedy Allocate to bidder with highest value Allocate according to current best allocation Semi - myopic Allocate to someone with value > value of the winner of item t in a current best allocation ( = an optimal allocation of items t,…,M among the active players at time t ). Semi - Myopic Algorithms

Set - Nash Equilibrium The above intuition implies that we do not expect a player to follow a specific strategy. Instead, we define a set of “recommended strategies” R i for player i. DFN: The strategy sets R 1 … R n are in Set – Nash equilibrium if a best response to every s -i  R -i exists in R i Comment 1: If | R i |=1, then equivalent to regular Nash. Comment 2: Best response to mixed strategies might be outside R i – stronger definitions can require that too. Comment 3: Only interesting if you can say something about the outcome when everyone plays in R i Comment 4: Naturally generalizes to games with incomplete information without a Bayesian prior: R i (t i )

Stronger set notions >Set domination: Player i’s strategies Strategies of other players RiRi R-i (coordinate-wise)

Stronger set notions >Set mixed Nash: Player i’s strategies Strategies of other players RiRi R-i E π ( )

Stronger set notions  Ri Ri Set-Nash: Player i’s strategies Strategies of other players RiRi R-i MAX( ) MAX

Implementation in Set-Nash equilibrium Definition: Given a set of outcomes A and a social choice rule F:T 1 x T 2 x … x T n  2 A, an implementation in Set-Nash equilibrium is a game form with strategy sets S 1 … S n and an outcome function g(s 1 …s n )  A, such that there exists a Set- Nash equilibrium { R i (t i ) } with g(s 1 …s n )  F(t 1 … t n ) for every s i  R i (t i ). Definition: In an extended revelation mechanism, players say (t i, l i ), where l i encapsulates the degree of freedom in R i. Revelation principle: Any Set-Nash equilibrium can be converted into an extended-truthful form: R i (t i ) = (t i,*).

Main Theorem: The Online Iterative Auction and the Sequential Japanese Auction Set - Nash implement a 3 - approximation of the welfare. I.e., both auctions have Set - Nash equilibrium that are all semi - myopic, hence guarantee a 3 - approximation. All the recommended strategies are not dominated. The recommended strategies contain best responses even if the strategies of the others are from a much larger set. The recommended strategies do not necessarily contain b.r. to mixed recommended strategies -- We think this is an interesting open problem.

Proof structure Basic building block: Semi Myopic Mechanism Recommended Strategies that are in Set - Nash Sequential Japanese: Semi Myopic Mechanism “Ignorable extension” Online Iterative: “Ignorable extension” Semi Myopic Mechanism

Reminder: with myopic players, the ascending auctions compute f t and VCG prices Reminder: basic structure of allocations d ftft Highest un-allocated player determines VCG price of all players in f t tt+1...M S t = the optimal allocation j’ j i*

Semi Myopic Mechanisms Strategy space. Extended direct revelation: { arrival time, value, “false” deadline, “true” deadline } (Similar in spirit to “2nd chance mechanisms” [NR]) Allocation rule. Compute S t according to “pretend deadlines”: –Allocate item t to some player in f t. Payment Rule. –For any player i, let c t (i) be his VCG price for entering S t. –Set temporary prices –The winner i pays max t’<t p t’ (i) “false” deadline If this has not passed. “true” deadline Otherwise. “pretend deadline” = c t (i) If i  f t.  [0, c t (i) ]If i  S t - f t. 0Otherwise p t (i) =

Set - Nash in Semi Myopic Mechanisms Recommended strategies: declare true arrival time, value, and true “true deadline”, and any “false deadline” < true deadline. Lemma 1: When all players play recommended strategies then the allocation rule of a semi myopic mechanism is a semi myopic algorithm. Lemma 2: These recommended strategies form a Set - Nash Equilibrium.

The structure of offline allocations Several possibilities to allocate the items to the players of St. DFN : P( S t,d) = { j |  ordering of S t in which player j receives one of the items t..d } Properties (1) f t = P( S t,t) (2) It is a prefix of any ordering of S t. (3) Suppose we have S t and now add a new player i with deadline d i. Then i’s VCG price will be min P( S t, d i ). (4) min P( S t,d) < min P( S t+1,d) t t M d P( S t,d) d’ P( S t,d’) S t =

The structure of best replies Suppose others play s -i, and player i does not show up at all: v* t = min f t -i ; t* = argmin t v* t Lemma 1: Player i cannot win and pay less than v* t* Proof: Fix any strategy. Suppose t is the first time in which f t has changed. Then some player from f t -i is now out of S t, so i’s price is at least his value. Lemma 2: The recommended strategy with a “false deadline” equals to t* will cause i to win item t* for a price of v* t* Proof: For any t<t* : v( j) = min P(S t -i,t*) < v* t* < min f t -i  f t -i = f t  The winner will remain the same one. Corollary: There exists a recommended strategy that is a best response to s -i. f t -i P(S t -i,t*) j i

Semi Myopic Mechanism  Ascending Auctions Recommended strategies for the Online Iterative Auction: play myopically with a fake deadline until it has passed, and myopically with the real deadline afterwards. Lemma: The Semi Myopic Mechanism is embedded in the Online Iterative Auction. Proof sketch: need to show that the requirements of the semi- myopic mechanism hold: –winners belong to f t –prices are VCG Already know these from the offline analysis

Summary We study an online setting with “gradually expiring items”. We first saw that truthful auctions cannot perform well. We then explored a new approach to this difficulty. –Worst case, no additional assumptions on players. Analyzed two adaptations to classical ascending auctions. –Both obtain a 3 - approximation under a large family of selfish behaviors. Introduced the notion of “Set - Nash equilibrium”. –Both our auctions have Set - Nash equilibrium that guarantees a 3 - approximation of the social welfare.