menu size complexity and revenue continuity of buy many
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MENU-SIZE COMPLEXITY AND REVENUE CONTINUITY OF BUY-MANY MECHANISMS - PowerPoint PPT Presentation

Jan 03, 2024 β€’32 likes β€’72 views

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(

  1. 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)

  2. Buy-one mechanisms and buy-many mechanisms v( )=$1000000 v( )=v( )=$1 1 $5 2 1 $2 3 $999 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. Buy-many mechanisms: buyer interact with the mechanism multiple times. β–ͺ Optimal strategy: repeatedly purchase , then repeatedly purchase , pay $16 in expectation. β–ͺ

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

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

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