MENU-SIZE COMPLEXITY AND REVENUE CONTINUITY OF BUY-MANY MECHANISMS - - PowerPoint PPT Presentation

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MENU-SIZE COMPLEXITY AND REVENUE CONTINUITY OF BUY-MANY MECHANISMS - - PowerPoint PPT Presentation

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MENU-SIZE COMPLEXITY AND REVENUE CONTINUITY OF BUY-MANY MECHANISMS Yi Yifen eng Ten eng University of Wisconsin-Madison Joint work with Shuchi Chawla and Christos Tzamos (UW-Madison) Buy-one mechanisms and buy-many mechanisms v(

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MENU-SIZE COMPLEXITY AND REVENUE CONTINUITY OF BUY-MANY MECHANISMS

Yi Yifen eng Ten eng

University of Wisconsin-Madison Joint work with Shuchi Chawla and Christos Tzamos (UW-Madison)

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β–ͺ A seller has π‘œ heterogeneous items to sell to a single buyer. β–ͺ Typical buy-one mechanisms: buyer interact with the seller once. β–ͺ Optimal strategy: purchases the third menu option, pay $999. β–ͺ Buy-many mechanisms: buyer interact with the mechanism multiple times. β–ͺ Optimal strategy: repeatedly purchase , then repeatedly purchase , pay $16 in expectation. v( )=$1000000

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$5

1 3

$2 $999 v( )=v( )=$1

Buy-one mechanisms and buy-many mechanisms

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Menu-size complexity for near-optimal revenue

β–ͺ How many menu options are needed for (1 βˆ’ πœ—)-approx in revenue? β–ͺ Buy-one mechanisms: infinite [Hart Nisan’13]. β–ͺ Buy-many mechanisms: finite. Theorem 1. For any distribution 𝐸 and πœ— ∈ [0,1], exists mechanism 𝑁 with finite menu size 𝑔(π‘œ, πœ—), such that 𝑆𝑓𝑀𝐸 𝑁 β‰₯ 1 βˆ’ πœ— πΆπ‘£π‘§π‘π‘π‘œπ‘§π‘†π‘“π‘€πΈ. Theorem 2. There exists 𝐸 being a distribution over XOS functions, such that for any mechanism 𝑁 with description complexity 22𝑝(π‘œ1/4), πΆπ‘£π‘§π‘π‘π‘œπ‘§π‘†π‘“π‘€πΈ β‰₯ 𝑝 log π‘œ 𝑆𝑓𝑀𝐸 𝑁 . β–ͺ 𝑔 π‘œ, πœ— = 1/πœ— 2𝑃(π‘œ). β–ͺ The doubly-exponential dependency of n is tight.

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Revenue Continuity

β–ͺ When the buyer’s values for the sets of items perturb multiplicatively slightly, how much does the revenue change? β–ͺ Any 𝑀 ∼ 𝐸 is perturbed to 𝑀′ ∼ 𝐸′, such that 𝑀′ 𝑇 ∈ 1 βˆ’ πœ— 𝑀(𝑇), 1 + πœ— 𝑀(𝑇) , βˆ€π‘‡ βŠ† [π‘œ]. β–ͺ Buy-one mechanisms: revenue may change significantly [Psomas et al.’19]. β–ͺ Continuity only holds for weaker additive perturbation [Rubinstein Weinberg’15] [Brustle et al.’20]. β–ͺ Buy-many mechanisms: revenue changes slightly. β–ͺ Note: such dependency on π‘œ is necessary. β–ͺ Full paper: https://arxiv.org/abs/2003.10636 Theorem 3. For any value distribution 𝐸 and any 1 Β± πœ— multiplicative perturbation 𝐸′, πΆπ‘£π‘§π‘π‘π‘œπ‘§π‘†π‘“π‘€πΈβ€² β‰₯ 1 βˆ’ π‘žπ‘π‘šπ‘§ π‘œ, πœ— πΆπ‘£π‘§π‘π‘π‘œπ‘§π‘†π‘“π‘€πΈ. Theorem 4. There exists 𝐸 over unit-demand functions and a 1 Β± πœ— multiplicative perturbation 𝐸′, such that πΆπ‘£π‘§π‘π‘π‘œπ‘§π‘†π‘“π‘€πΈβ€² ≀ 1 πœ—π‘œ πΆπ‘£π‘§π‘π‘π‘œπ‘§π‘†π‘“π‘€πΈ.