Design and analysis of algorithms for non-cooperative environments - PowerPoint PPT Presentation

Worst-case characterization Mechanism 𝑵 with allocation function 𝑕 and payment function 𝑞 • For every 𝒕 , the worst case game where 𝒕 is an equilibrium has a • very special structure 𝑤 𝑗 𝑦 = 𝑑 𝑗 + 𝜇 𝑗 𝒕 𝑦 𝑑 1 = +∞ 𝑤 1 𝑦 = 𝜇 1 𝒕 𝑦 𝑑 𝑗 = 𝑞 𝑗 (𝒕) 0 1 0 1

Worst-case characterization Mechanism 𝑵 with allocation function 𝑕 and payment function 𝑞 • For every 𝒕 , the worst case game where 𝒕 is an equilibrium has a • very special structure 𝑤 𝑗 𝑦 = 𝑑 𝑗 + 𝜇 𝑗 𝒕 𝑦 𝑑 1 = +∞ 𝑤 1 𝑦 = 𝜇 1 𝒕 𝑦 𝑑 𝑗 + 𝜇 𝑗 𝒕 𝑕 𝑗 (𝒕) 𝜇 1 𝒕 𝑕 1 (𝒕) 𝑑 𝑗 = 𝑞 𝑗 (𝒕) 𝑕 1 (𝒕) 𝑕 𝑗 (𝒕) 0 1 0 1 equilibrium LW 𝑕(𝒕) = σ 𝑗≥2 𝑞 𝑗 (𝑡) + 𝜇 1 𝒕 𝑕 1 (𝒕)

Worst-case characterization Mechanism 𝑵 with allocation function 𝑕 and payment function 𝑞 • For every 𝒕 , the worst case game where 𝒕 is an equilibrium has a • very special structure 𝑤 𝑗 𝑦 = 𝑑 𝑗 + 𝜇 𝑗 𝒕 𝑦 𝑑 1 = +∞ 𝑤 1 𝑦 = 𝜇 1 𝒕 𝑦 𝜇 1 𝒕 𝑑 𝑗 = 𝑞 𝑗 (𝒕) 𝑦 1 (𝒕) = 1 0 = 𝑦 𝑗 (𝒕) 0 1 optimal allocation LW 𝑦(𝒕) = σ 𝑗≥2 𝑞 𝑗 (𝑡) + 𝜇 1 𝒕

Worst-case characterization Mechanism 𝑵 with allocation function 𝑕 and payment function 𝑞 • For every 𝒕 , the worst case game where 𝒕 is an equilibrium has a • very special structure LPoA 𝒕 −game = LW 𝑦(𝒕) σ 𝑗≥2 𝑞 𝑗 𝒕 + 𝜇 1 (𝒕) LW 𝑕(𝒕) = σ 𝑗≥2 𝑞 𝑗 𝒕 + 𝜇 1 𝒕 𝑕 1 (𝒕)

Worst-case characterization Mechanism 𝑵 with allocation function 𝑕 and payment function 𝑞 • For every 𝒕 , the worst case game where 𝒕 is an equilibrium has a • very special structure LPoA 𝒕 −game = LW 𝑦(𝒕) σ 𝑗≥2 𝑞 𝑗 𝒕 + 𝜇 1 (𝒕) LW 𝑕(𝒕) = σ 𝑗≥2 𝑞 𝑗 𝒕 + 𝜇 1 𝒕 𝑕 1 (𝒕) Theorem The liquid price of anarchy of mechanism 𝑵 is σ 𝑗≥2 𝑞 𝑗 𝒕 + 𝜇 1 (𝒕) LPoA 𝑵 = sup σ 𝑗≥2 𝑞 𝑗 𝒕 + 𝜇 1 𝒕 𝑕 1 (𝒕) 𝒕 where: −1 𝜖𝑕 1 (𝑧, 𝑡 −1 ) ∙ 𝜖𝑞 1 (𝑧, 𝑡 −1 ) 𝜇 1 𝒕 = ቚ ቚ 𝑒𝑧 𝑒𝑧 𝑧=𝑡 1 𝑧=𝑡 1

Tight bound for the Kelly mechanism Theorem The liquid price of anarchy of the Kelly mechanism is exactly 2

Tight bound for the Kelly mechanism Theorem The liquid price of anarchy of the Kelly mechanism is exactly 2 Every player pays her signal: σ 𝑗≥2 𝑞 𝑗 𝒕 = σ 𝑗≥2 𝑡 𝑗 = 𝐷 •

Tight bound for the Kelly mechanism Theorem The liquid price of anarchy of the Kelly mechanism is exactly 2 Every player pays her signal: σ 𝑗≥2 𝑞 𝑗 𝒕 = σ 𝑗≥2 𝑡 𝑗 = 𝐷 • 𝑡 1 For player 1 : 𝑕 1 𝒕 = • 𝑡 1 +𝐷

Tight bound for the Kelly mechanism Theorem The liquid price of anarchy of the Kelly mechanism is exactly 2 Every player pays her signal: σ 𝑗≥2 𝑞 𝑗 𝒕 = σ 𝑗≥2 𝑡 𝑗 = 𝐷 • 𝑡 1 For player 1 : 𝑕 1 𝒕 = • 𝑡 1 +𝐷 𝑧+𝐷 ⇒ 𝜖𝑕 1 (𝑧,𝒕 −1 ) 𝑧 𝐷 ȁ 𝑧=𝑡 1 = 𝑕 1 𝑧, 𝒕 −1 = 𝜇 1 𝒕 = (𝑡 1 + 𝐷) 2 (𝑡 1 +𝐷) 2 𝑒𝑧 𝐷 𝑞 1 𝑧, 𝒕 −1 = 𝑧 ⇒ 𝜖𝑞 1 (𝑧,𝒕 −1 ) ȁ 𝑧=𝑡 1 = 1 𝑒𝑧

Tight bound for the Kelly mechanism Theorem The liquid price of anarchy of the Kelly mechanism is exactly 2 Every player pays her signal: σ 𝑗≥2 𝑞 𝑗 𝒕 = σ 𝑗≥2 𝑡 𝑗 = 𝐷 • 𝑡 1 For player 1 : 𝑕 1 𝒕 = • 𝑡 1 +𝐷 𝑧+𝐷 ⇒ 𝜖𝑕 1 (𝑧,𝒕 −1 ) 𝑧 𝐷 ȁ 𝑧=𝑡 1 = 𝑕 1 𝑧, 𝒕 −1 = 𝜇 1 𝒕 = (𝑡 1 + 𝐷) 2 (𝑡 1 +𝐷) 2 𝑒𝑧 𝐷 𝑞 1 𝑧, 𝒕 −1 = 𝑧 ⇒ 𝜖𝑞 1 (𝑧,𝒕 −1 ) ȁ 𝑧=𝑡 1 = 1 𝑒𝑧 𝐷 + (𝑡 1 + 𝐷) 2 /𝐷 LPoA Kelly = sup 𝐷 + 𝑡 1 (𝑡 1 + 𝐷)/𝐷 = 2 □ 𝑡 1 ,𝐷

Overview of results mechanism LPoA ≥ 2-1/ 𝒐 all Kelly 2 SH 3 E 2 -PYS 1.79 E 2 -SR 1.53

Overview of results mechanism LPoA ≥ 2-1/ 𝒐 all no mechanism can achieve full efficiency Kelly 2 SH 3 E 2 -PYS 1.79 E 2 -SR 1.53

Overview of results mechanism LPoA ≥ 2-1/ 𝒐 all Kelly 2 almost best possible among all mechanisms with many players SH 3 E 2 -PYS 1.79 E 2 -SR 1.53

Overview of results mechanism LPoA ≥ 2-1/ 𝒐 all Kelly 2 SH 3 different picture than the no-budget setting E 2 -PYS 1.79 E 2 -SR 1.53

Overview of results mechanism LPoA ≥ 2-1/ 𝒐 all Kelly 2 SH 3 E 2 -PYS 1.79 E 2 -SR 1.53 The allocation functions are solutions of simple linear differential equations , which are defined by properly setting the payment function (PYS/SR) and using the worst-case characterization theorem

Overview of results mechanism LPoA ≥ 2-1/ 𝒐 all Kelly 2 SH 3 E 2 -PYS 1.79 best possible PYS mechanism for two players E 2 -SR 1.53

Overview of results mechanism LPoA ≥ 2-1/ 𝒐 all Kelly 2 SH 3 E 2 -PYS 1.79 almost best possible mechanism E 2 -SR 1.53 for two players

Opinion formation games

A simple model • There is a set of individuals, and each of them has a (numerical) personal belief 𝑡 𝑗 However, she might express a possibly different opinion 𝑨 𝑗 • • Averaging process: all individuals simultaneously update their opinions according to the rule 𝑡 𝑗 + σ 𝑘∈𝑂 𝑗 𝑨 𝑘 𝑨 𝑗 = 1 + ȁ𝑂 𝑗 ȁ 𝑂 𝑗 indicates the social circle of individual 𝑗 • – Friedkin & Johnsen ( 1990 )

Game-theoretic interpretation • The limit of the averaging process is the unique equilibrium of an opinion formation game that is defined by the personal beliefs of the individuals • The opinions of the individuals (players) can be thought of as their strategies • Each player has a cost that depends on her belief and the opinions that are expressed by other players in her social circle 2 cost 𝑗 𝒕, 𝒜 = 𝑨 𝑗 − 𝑡 𝑗 2 + ෍ 𝑨 𝑗 − 𝑨 𝑘 𝑘∈𝑂 𝑗 • The players act as cost-minimizers – Bindel, Kleinberg, & Oren ( 2015 )

Co-evolutionary games • The social circle of an individual changes as the opinions change 𝒍 -NN games (Nearest Neighbors) • • There is no underlying social network The social circle 𝑂 𝑗 𝒕, 𝒜 consists of the 𝑙 players with opinions • closest to the belief of player 𝑗 • Same cost function 2 cost 𝑗 𝒕, 𝒜 = 𝑨 𝑗 − 𝑡 𝑗 2 + ෍ 𝑨 𝑗 − 𝑨 𝑘 𝑘∈𝑂 𝑗 𝐭,𝐴 – Bhawalkar, Gollapudi, & Munagala ( 2013 )

Compromising opinion formation games 𝒍 -COF games • • There is no underlying social network • The social circle 𝑂 𝑗 𝒕, 𝒜 consists of the 𝑙 players with opinions closest to the belief of player 𝑗 • Different cost function definition cost 𝑗 𝒕, 𝒜 = max 𝑨 𝑗 − 𝑡 𝑗 , ȁ𝑨 𝑗 − 𝑨 𝑘 ȁ 𝑘∈𝑂 𝑗 𝐭,𝐴

Compromising opinion formation games 𝒍 -COF games • • There is no underlying social network • The social circle 𝑂 𝑗 𝒕, 𝒜 consists of the 𝑙 players with opinions closest to the belief of player 𝑗 • Different cost function definition cost 𝑗 𝒕, 𝒜 = max 𝑨 𝑗 − 𝑡 𝑗 , ȁ𝑨 𝑗 − 𝑨 𝑘 ȁ 𝑘∈𝑂 𝑗 𝐭,𝐴 – Do pure equilibria always exist? – Can we efficiently compute them when they do exist? – How efficient are equilibria (price of anarchy and stability)?

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for any 𝑙

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for 𝑙 = 1 0 1 2 [ 1 ] [ 1 ] [ 1 ]

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for 𝑙 = 1 𝒚 < 1 0 1 2 [ 1 ] [ 1 ] [ 1 ]

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for 𝑙 = 1 𝒚/ 2 𝒚 < 1 0 1 2 [ 1 ] [ 1 ] [ 1 ]

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for 𝑙 = 1 𝒚/ 2 𝒚 < 1 1 + 𝒚/ 2 0 1 2 [ 1 ] [ 1 ] [ 1 ]

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for 𝑙 = 1 𝒚/ 2 1 −𝒚/ 2 𝒚/ 2 𝒚 < 1 1 + 𝒚/ 2 0 1 2 [ 1 ] [ 1 ] [ 1 ]

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for 𝑙 = 1 1 + 𝒚/ 4 𝒚/ 2 𝒚 < 1 1 + 𝒚/ 2 0 1 2 [ 1 ] [ 1 ] [ 1 ]

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for 𝑙 = 1 1 / 2 𝒚 = 1 3 / 2 0 2 [ 1 ] [ 1 ] [ 1 ]

Existence of equilibria Theorem There exists a 𝑙 -COF game with no pure equilibria, for 𝑙 = 1 3 / 4 1 / 2 𝒚 = 1 3 / 2 0 2 [ 1 ] [ 1 ] [ 1 ] □

A lower bound on the price of anarchy Theorem For 𝑙 = 1 , the price of anarchy is at least 3

A lower bound on the price of anarchy Theorem For 𝑙 = 1 , the price of anarchy is at least 3 - 15 - 3 15 3 [ 2 ] [ 1 ] [ 1 ] [ 2 ]

A lower bound on the price of anarchy Theorem For 𝑙 = 1 , the price of anarchy is at least 3 6 6 - 15 -9 - 3 9 15 3 [ 2 ] [ 1 ] [ 1 ] [ 2 ] SC 𝒕, 𝒜 = 12

A lower bound on the price of anarchy Theorem For 𝑙 = 1 , the price of anarchy is at least 3 2 2 - 15 -9 - 3 9 15 -1 1 3 [ 2 ] [ 1 ] [ 1 ] [ 2 ] SC 𝒕, 𝒜′ = 4

A lower bound on the price of anarchy Theorem For 𝑙 = 1 , the price of anarchy is at least 3 - 15 -9 - 3 9 15 -1 1 3 [ 2 ] [ 1 ] [ 1 ] [ 2 ] PoA ≥ SC (𝒕, 𝒜) SC 𝒕, 𝒜′ = 12 4 = 3 □

Overview of results Pure equilibria may not exist , for any 𝑙 ≥ 1 • For 𝑙 = 1 , we can efficiently compute the best and the worst • equilibrium – Shortest and longest paths in DAGs The price of anarchy and stability depend linearly on 𝑙 • – Proofs based on LP duality and case analysis – Tight bound of 3 on the price of anarchy for 𝑙 = 1 – Lower bounds on the mixed price of anarchy

Ownership transfer

Ownership transfer • Privatization of government assets – Public electricity or water companies, airports, buildings , … • Sports tournaments organization – World cup, Olympics, Formula 1 , …

Ownership transfer • Privatization of government assets – Public electricity or water companies, airports, buildings , … • Sports tournaments organization – World cup, Olympics, Formula 1 , … • How should we decide who the new owner is going to be? – Use of historical data related to the possible owners – Run an auction among the possible buyers

Ownership transfer • Privatization of government assets – Public electricity or water companies, airports, buildings , … • Sports tournaments organization – World cup, Olympics, Formula 1 , … • How should we decide who the new owner is going to be? – Use of historical data related to the possible owners – Run an auction among the possible buyers • The new owner wants to maximize her own profit – Her decisions as the owner might critically affect the welfare of the society (company’s employees and consumers, or the citizens)

Ownership transfer • The goal is to make a decision that will sufficiently satisfy both the society and the new owner (if one exists)

Ownership transfer • The goal is to make a decision that will sufficiently satisfy both the society and the new owner (if one exists) • Auction + expert advice – The auction guarantees that the selling price is the best possible – The expert guarantees the well-being of the society

A simple model • One item for sale Two possible buyers 𝑩 and 𝑪 • – Each buyer 𝑗 has a monetary valuation 𝑥 𝑗 for the item • One expert – The expert has von Neumann-Morgenstern valuations 𝑤(∙) for the three options: (1) sell the item to buyer 𝛣 (2) sell the item to buyer 𝛤 (3) Do not sell the item ( ⊘ ) – vNM valuations: [ 1 , 𝑦 , 0 ]

A simple model • Design mechanisms that – incentivize the buyers and the expert to truthfully report their preferences, and – decide the option 𝑗 ∊ {𝐵, 𝐶, ⊘ } that maximizes the social welfare 𝑥 𝑗 𝑤 𝑗 + max(𝑥 𝐵 , 𝑥 𝐶 ) , 𝑗 ∊ {𝐵, 𝐶} SW 𝑗 = ൞ 𝑤 ⊘ , otherwise

A simple model • Design mechanisms that – incentivize the buyers and the expert to truthfully report their preferences, and – decide the option 𝑗 ∊ {𝐵, 𝐶, ⊘ } that maximizes the social welfare 𝑥 𝑗 𝑤 𝑗 + max(𝑥 𝐵 , 𝑥 𝐶 ) , 𝑗 ∊ {𝐵, 𝐶} SW 𝑗 = ൞ 𝑤 ⊘ , otherwise • Combination of approximate mechanism design – with money for the buyers (Nisan & Ronen, 2001 ) – without money for the expert (Procaccia & Tennenholtz, 2013 )

Problem difficulty • Mechanism: given input by the buyers and the expert, choose the option that maximizes the social welfare – Can this mechanism incentivize the participants to truthfully report their valuations?

Problem difficulty • Mechanism: given input by the buyers and the expert, choose the option that maximizes the social welfare – Can this mechanism incentivize the participants to truthfully report their valuations? 0.1 1 0.5 1 0

Problem difficulty • Mechanism: given input by the buyers and the expert, choose the option that maximizes the social welfare – Can this mechanism incentivize the participants to truthfully report their valuations? 0.1 1 0.5 1 0

Problem difficulty • Mechanism: given input by the buyers and the expert, choose the option that maximizes the social welfare – Can this mechanism incentivize the participants to truthfully report their valuations? 0.1 1 0 1 0

Examples of truthful mechanisms 1 .99 0 0.7 1

Examples of truthful mechanisms • Mechanism: choose the favorite option of the expert 1 .99 0 0.7 1

Examples of truthful mechanisms • Mechanism: choose the favorite option of the expert • SW(mechanism) = SW(no-sale) = 1 vs. SW(green) ≈ 2 – approximation ratio = 2 1 .99 0 0.7 1

Examples of truthful mechanisms • Mechanism: with probability 2/3 choose the expert’s favorite option, and with probability 1/3 choose the expert’s second favorite option SW(mechanism) = SW(no-sale) · 2/3 + SW(green) · 1/3 ≈ 4/3 • – 3/2 -approximate 1 .99 0 0.7 1

Overview of results class of mechanisms approx 1.5 ordinal bid-independent 1.377 expert-independent 1.343 randomized template 1.25 deterministic template 1.618 ≥ 1.618 deterministic ≥ 1.14 all

Overview of results class of mechanisms approx Mechanisms that base their 1.5 ordinal decision only on the relative order of the values reported by bid-independent 1.377 the expert or the buyers expert-independent 1.343 randomized template 1.25 deterministic template 1.618 ≥ 1.618 deterministic ≥ 1.14 all

Overview of results class of mechanisms approx 1.5 ordinal Mechanisms that base their bid-independent 1.377 decision solely on the values expert-independent 1.343 reported by the expert randomized template 1.25 deterministic template 1.618 ≥ 1.618 deterministic ≥ 1.14 all

Overview of results class of mechanisms approx 1.5 ordinal bid-independent 1.377 Mechanisms that base their expert-independent 1.343 decision solely on the values randomized template 1.25 reported by the buyers deterministic template 1.618 ≥ 1.618 deterministic ≥ 1.14 all

Overview of results class of mechanisms approx 1.5 ordinal bid-independent 1.377 expert-independent 1.343 randomized template 1.25 Mechanisms that base their deterministic template 1.618 decision on the values reported by the expert and the buyers ≥ 1.618 deterministic ≥ 1.14 all

Overview of results class of mechanisms approx 1.5 ordinal bid-independent 1.377 expert-independent 1.343 randomized template 1.25 deterministic template 1.618 ≥ 1.618 deterministic Unconditional lower bounds for ≥ 1.14 all all mechanisms

Revenue maximization in combinatorial sales