Simple near-optimal auctions Posted price auctions Recap Last - PowerPoint PPT Presentation

Algorithmic game theory Ruben Hoeksma December 17, 2018 Simple near-optimal auctions Posted price auctions

Recap Last week: ◮ Optimal revenue single item auction (Myerson’s auction) ◮ Virtual valuations ◮ For any feasible (DSIC, IR) allocation rule a we have     � �  = E v i a ( v ) E π i  i ∈ N i ∈ N Today: ◮ Near-optimal auctions ◮ Posted price auctions ◮ VCG auctions

Optimal auction - realistic? Advantages of the optimal auction: ◮ Provable maximum expected revenue ◮ Computations on a per bidder basis ◮ Nothing structurally changes when adding/removing bidders Disadvantages of the optimal auction: ◮ Necessity to have a good knowledge of bidders valuation distribution ◮ Bidders need to know the whole auction structure ◮ Payments can seem confusing ◮ Highest bid not always wins (counter intuitive)

Posted price auction Simple auction ◮ Set one threshold t at which we sell the item ◮ Sell to anyone with virtual value higher than t Auction is based on The prophet inequality. Game ◮ In n stages, n prizes are offered ◮ In each stage i , you can accept the prize p i or forfeit it forever ◮ n prizes have known independent distributions G 1 , . . . , G n Q: How well can you do compared to a prophet who knows all the realizations?

The prophet inequality Theorem (Prophet inequality) For every sequence of independent distributions G 1 , . . . , G n , there is a threshold t such that accepting any prize p i ≥ t guarantees expected reward 1 2 E p [max i p i ]. Proof. Lower bound on threshold t strategy. ◮ q ( t ) = E p [ p i < t , ∀ i ], probability of not accepting any prize ◮ Simple lower bound: E p [max i p i ] ≥ q ( t ) · 0 + (1 − q ( t )) · t ◮ Improvement: bound the amount that we get extra ◮ If exactly one p i ≥ t then we get an extra p i − t ◮ If more prizes are at least t , we get one of them: bound as if p i = t (1 − q ( t )) · t + � n i =1 P [ p i ≥ t , p j < t , ∀ j � = i ] E [ p i − t | p i ≥ t ] = (1 − q ( t )) · t + � n i =1 P [ p j < t , ∀ j � = i ] P [ p i ≥ t ] E [ p i − t | p i ≥ t ] � �� � � �� � = E [( p i − t ) + ] ≥ q ( t )

The prophet inequality Proof Cont. So, n � E [( p i − t ) + ] E [Rev( t -threshold)] ≥ (1 − q ( t )) · t + q ( t ) i =1 Upper bound on the prophet’s strategy ( p i − t ) + ] E [max p i ] = E [ t + max ( p i − t )] ≤ t + E [max i i i n � E [( p i − t ) + ] ≤ t + i =1 Compare to the lower bound: (1 − q ( t )) · t + q ( t ) � n i =1 E [( p i − t ) + ] Set t such that q ( t ) = 1 2 : n � 1 2 · t + 1 E [( p i − t ) + ] ≥ 1 2 E [max p i ] 2 i i =1

Posted price auction Single item auction: ◮ Single item ◮ n bidders N with distributions ϕ 1 ( · ) , . . . , ϕ n ( · ) ◮ Assume regular distributions Equivalence between prizes p i and virtual valuations v i . E [max i p i ] and E [ � i ∈ N v i a ( v )] = E [max i ( v i ) + ] Posted price auction: ◮ Set t such that P [max i v i ≥ t ] = 1 2 . ◮ For each bidder i find the smallest valuation v i such that v i ≥ t , set r i = v i . ◮ Offer bidder i the item at price r i . If they accept: sell the item to them; otherwise offer the item to bidder i + 1. E [Rev(posted price auction)] ≥ 1 ( v i ) + ] = 1 2 E [max 2 E [Rev(OPT)] i

Posted price auction Advantages of posted price auctions: ◮ Extremely simple for the players ◮ No need to know the whole auction ◮ No strategizing possible ◮ Constant factor of the optimal auction revenue ◮ Posted price mechanisms exist that achieve e − 1 fraction of the e optimal auction revenue Disadvantages: ◮ Computation of virtual valuations still necessary ◮ Reserve price for each bidder individual ◮ Reliant on the correctness of assumed distributions

Simple near-optimal auctions Prior-independent auctions (Bulow-Klemperer Theorem)

Prior independent auctions Prior independent Independent on any extra information that may or may not be know before finding the solution. ◮ Looking for a single item auction ◮ Should not depend on anything except that we want to allocate a single item ◮ Not even on number of players Q: What can we do? Compare different auctions and give an analysis that ensures us that one is the better choice. Assumption: All values are distributed equally (for the analysis)

Bulow-Klemperer Bulow-Klemperer Theorem Let F be a regular distribution and n a positive integer, then E v 1 ,..., v n +1 ∼ F [Rev(SPA( n + 1))] ≥ E v 1 ,..., v n ∼ F [Rev(OPT F ( n ))] , where OPT F ( n ) is the optimal auction for distribution F with n bidders and SPA( n + 1) is the Second price auction / Vickrey auction with n + 1 bidders. Note: Analysis can use the distribution as long as we don’t base our choices on it. Proof. Claim: Vickrey auction maximizes revenue among all auctions that always allocate the item (exercise).

Bulow-Klemperer Proof cont. Consider the following auction with n + 1 bidders, with v 1 , . . . , v n +1 ∼ F : ◮ Optimal auction between first n bidders ◮ If the auction of n bidders does not provide a winner, give away the item to bidder n + 1 for free Properties of this auction: ◮ Revenue is exactly equal to the revenue of OPT F ( n ) ◮ It always allocates the item Thus, SPA( n + 1) has revenue at least as large as this auction (by the Claim) and thus at least as large as OPT F ( n ).

Prior independent auctions Corollary E v 1 ,..., v n ∼ F [Rev(SPA( n ))] ≥ n − 1 · E v 1 ,..., v n ∼ F [Rev(OPT F ( n ))] n Proof. Consider the n − 1 bidder auction that simulates OPT F ( n ) and only considers the n − 1 bidders with largest expected payments. This auction achieves at least n − 1 fraction of OPT F ( n ). n Yet, OPT F ( n − 1) achieves higher expected revenue. Thus, E v 1 ,..., v n ∼ F [Rev(SPA( n ))] ≥ E v 1 ,..., v n − 1 ∼ F [Rev(OPT F ( n − 1))] ≥ n − 1 · E v 1 ,..., v n ∼ F [Rev(OPT F ( n ))] . n Take away: It is better to spend resources on getting more bidders, than on improving the estimation of those bidders’ valuation distributions.