Minimally Invasive Mechanism Design Distributed Covering with - - PowerPoint PPT Presentation

Games Dynamics models Results

Minimally Invasive Mechanism Design

Distributed Covering with Carefully Chosen Advice Nina Balcan, Sara Krehbiel, Georgios Piliouras, Jinwoo Shin

Georgia Institute of Technology

December 11, 2012

Games Dynamics models Results

Games Dynamics models Results

General Game G = N, S, (costi)i∈N N - set of agents S = (Si)i∈N - strategy sets for each agent costi : S → R

Games Dynamics models Results

Covering Game Agents - elements [n] in a hypergraph Strategies - {on, off } Incentives - i pays ci if on; wσ for each adjacent uncovered hyperedge σ if off

Games Dynamics models Results

Covering Game Agents - elements [n] in a hypergraph Strategies - {on, off } Incentives - i pays ci if on; wσ for each adjacent uncovered hyperedge σ if off Best response dynamics: BRi(s−i) = arg mina∈Si costi(a, s−i)

Games Dynamics models Results

Covering Game Agents - elements [n] in a hypergraph Strategies - {on, off } Incentives - i pays ci if on; wσ for each adjacent uncovered hyperedge σ if off Best response dynamics: BRi(s−i) = arg mina∈Si costi(a, s−i)

Games Dynamics models Results

Covering Game Agents - elements [n] in a hypergraph Strategies - {on, off } Incentives - i pays ci if on; wσ for each adjacent uncovered hyperedge σ if off Best response dynamics: BRi(s−i) = arg mina∈Si costi(a, s−i)

Games Dynamics models Results

Covering Game Agents - elements [n] in a hypergraph Strategies - {on, off } Incentives - i pays ci if on; wσ for each adjacent uncovered hyperedge σ if off Best response dynamics: BRi(s−i) = arg mina∈Si costi(a, s−i)

Games Dynamics models Results

Covering Game Agents - elements [n] in a hypergraph Strategies - {on, off } Incentives - i pays ci if on; wσ for each adjacent uncovered hyperedge σ if off Best response dynamics: BRi(s−i) = arg mina∈Si costi(a, s−i)

Games Dynamics models Results

Covering Game Agents - elements [n] in a hypergraph Strategies - {on, off } Incentives - i pays ci if on; wσ for each adjacent uncovered hyperedge σ if off Best response dynamics: BRi(s−i) = arg mina∈Si costi(a, s−i) Nash equilibrium: s st si = BRi(s−i) for all i ∈ N

Games Dynamics models Results

costi(s) = ci if on, or wσ for uncovered adjacent σ if off Good news: The game is a potential game (∆Φ = ∆costi) Φ(s) =

• i on

ci +

• uncovered σ

wσ Best response dynamics always converge to a pure Nash equilibrium in a potential game.

Games Dynamics models Results

costi(s) = ci if on, or wσ for uncovered adjacent σ if off Bad news: Nash equilibria can vary dramatically in social cost cost(s) =

• i on

ci +

• uncovered σ

|σ| · wσ PoA/PoS = Θ(n)

Games Dynamics models Results

System designer has a range of options: Have faith in best response (eg [CamposNa´

• ez-Garcia-Li-08])

Enforce outcome of central optimization

Games Dynamics models Results

System designer has a range of options: Have faith in best response (eg [CamposNa´

• ez-Garcia-Li-08])

Enforce outcome of central optimization Both! Self-interest + globally-aware advice Result: low-cost Nash equilibria

Games Dynamics models Results

Public Service Advertising [Balcan-Blum-Mansour-09]

1 With independent constant probability α, each agent plays

advertised strategy for Phase 1; others play BR.

2 Everyone plays BR.

Games Dynamics models Results

Public Service Advertising [Balcan-Blum-Mansour-09]

1 With independent constant probability α, each agent plays

advertised strategy for Phase 1; others play BR.

2 Everyone plays BR.

Learn-Then-Decide [Balcon-Blum-Mansour-10]

1 T ∗ rounds of random update: i plays sad

i

with probability pi ≥ β and BR otherwise.

2 Agents commit arbitrarily to sad or BR; best responders

converge to partial Nash equilibrium.

Games Dynamics models Results

Our Results (Informal)

1 Arbitrary sad =

⇒ expected cost after PSA or LTD is O(cost(sad)2) in general, O(cost(sad)) for non-hypergraphs.

2 Particular sad =

⇒ cost after PSA is O(log n) · OPT w.h.p.

Games Dynamics models Results

Our Results (Informal)

1 Arbitrary sad =

⇒ expected cost after PSA or LTD is O(cost(sad)2) in general, O(cost(sad)) for non-hypergraphs.

2 Particular sad =

⇒ cost after PSA is O(log n) · OPT w.h.p. Compare: Best response may converge to cost Ω(n) · OPT Centralized set cover can’t guarantee better than ln n · OPT

Games Dynamics models Results

PSA Model:

1 Some play advertised strategy; others play BR. 2 Everyone plays BR.

Games Dynamics models Results

PSA Model:

1 Some play advertised strategy; others play BR. 2 Everyone plays BR.

Theorem (Arbitrary advertising in PSA) For any advertising strategy sad, the expected cost at the end of Phase 2 of PSA is O(cost(sad)2), assuming ci, wσ, Fmax, ∆2 ∈ Θ(1). Proof overview: Enough to bound cost at end of Phase 1 End of Ph1 cost ≤ cost(sad) +

bad sets wσ + bad vertices ci

Games Dynamics models Results

PSA Model:

1 Some play advertised strategy; others play BR. 2 Everyone plays BR.

Theorem (Carefully chosen advertising in PSA) For an advertising strategy sad of a particular efficient form, the cost at the end of Phase 2 of PSA is O(cost(sad)) w.h.p.,

assuming ci, wσ, Fmax, ∆2 ∈ Θ(1).

Proof idea: Each on agent uniquely covers many sets in sad W.h.p. all these agents must turn on in Phase 1

Games Dynamics models Results

PSA Model:

1 Some play advertised strategy; others play BR. 2 Everyone plays BR.

Theorem (Carefully chosen advertising in PSA) For an advertising strategy sad of a particular efficient form, the cost at the end of Phase 2 of PSA is O(cost(sad)) w.h.p.,

assuming ci, wσ, Fmax, ∆2 ∈ Θ(1).

Proof idea: Each on agent uniquely covers many sets in sad W.h.p. all these agents must turn on in Phase 1 Corollary: There exists a PSA advertising strategy such that expected cost at the end of Phase 2 is O(log n) · OPT.

Games Dynamics models Results

Two contributions:

1 Extend [BBM09, BBM10] to natural set cover game 2 Show how to improve results with carefully chosen advice

Future work: Give results for broader classes of games (eg all potential games) in PSA and LTD models Enhance models to allow for heterogeneity of strategies for different agents

Games Dynamics models Results

Thank you!