CMU 15-251 Game theory Teachers: Anil Ada Ariel Procaccia (this - - PowerPoint PPT Presentation

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CMU 15-251 Game theory Teachers: Anil Ada Ariel Procaccia (this - - PowerPoint PPT Presentation

Oct 07, 2022 278 likes β€’558 views

CMU 15-251 Game theory Teachers: Anil Ada Ariel Procaccia (this time) Normal-Form Game = {1, , } o o : o j ( 1 , ,

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SLIDE 1

CMU 15-251

Game theory

Teachers: Anil Ada Ariel Procaccia (this time)

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Normal-Form Game

  • 𝑂 = {1, … , π‘œ}
  • 𝑇
  • 𝑗 ∈ 𝑂

𝑣𝑗: π‘‡π‘œ β†’ ℝ j ∈ 𝑂 𝑑

π‘˜ ∈ 𝑇

𝑗 𝑣𝑗(𝑑1, … , π‘‘π‘œ)

  • 2
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3

Selling ice cream at the beach. One day your cousin Ted shows up. His ice cream is identical! You split the beach in half; you set up at 1/4. 50% of the customers buy from you. 50% buy from Teddy. One day Teddy sets up at the 1/2 point! Now you serve only 37.5%!

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  • 𝑣𝑗 𝑑𝑗, 𝑑

π‘˜ = 𝑑𝑗+π‘‘π‘˜ 2

, 𝑑𝑗 < 𝑑

π‘˜

1 βˆ’

𝑑𝑗+π‘‘π‘˜ 2

, 𝑑𝑗 > 𝑑

π‘˜ 1 2 ,

𝑑𝑗 = 𝑑

π‘˜

The Ice Cream Wars

  • 𝑂 = 1,2
  • 𝑇 = [0,1]
  • 4
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The prisoner’s dilemma

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The prisoner’s dilemma

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In real life

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On TV

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The professor’s dilemma

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Nash equilibrium

  • 𝒕 = 𝑑1 … , π‘‘π‘œ ∈ π‘‡π‘œ

βˆ€π‘— ∈ 𝑂, βˆ€π‘‘π‘—

β€² ∈ 𝑇, 𝑣𝑗 𝒕 β‰₯ 𝑣𝑗(𝑑𝑗 β€², π’•βˆ’π‘—)

π’•βˆ’π‘— = 𝑑1, … , π‘‘π‘—βˆ’1, 𝑑𝑗+1, … , π‘‘π‘œ

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Nash equilibrium

  • 1.

2.

1

3.

2

4.

3

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Nash equilibrium

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Russel Crowe was wrong

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End of the Ice Cream Wars

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Day 3 of the ice cream wars… Teddy sets up south of you! You go south of Teddy. Eventually…

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15

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Does NE make sense?

  • {2, … , 100}
  • 𝑑

𝑒 𝑑 < 𝑒 𝑑 + 2 𝑑 βˆ’ 2

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Back to prison

  • =
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Anarchy and stability

  • =
  • =

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Example: Cost sharing

  • π‘œ

𝐻

  • 𝑗

𝑑𝑗 𝑒𝑗 𝑑𝑗 β†’ 𝑒𝑗

  • 𝑓

𝑑𝑓

  • 19

𝑑2 𝑑1

𝑒1 𝑒2

10 10 10 1 1 1 1

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Example: Cost sharing

  • π‘œ

π‘œ

  • 1
  • β‡’

β‰₯ π‘œ

  • ≀ π‘œ
  • 20

𝑒 𝑑 π‘œ 1

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Example: Cost sharing

  • 1

2

  • π‘œ
  • 2
  • 21

𝑑1 𝑑2 π‘‘π‘œ 𝑒 … 0 0 1 1 1 2

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Example: Cost sharing

  • = 2
  • β‡’
  • 22

𝑑1 𝑑2 π‘‘π‘œ 𝑒 … 0 0 2

1 1 1 2 1 π‘œ

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Potential games

  • Ξ¦: 𝑗=1

π‘œ

𝑇𝑗 β†’ ℝ 𝑗 ∈ 𝑂 𝒕 ∈ 𝑗=1

π‘œ

𝑇𝑗 𝑑𝑗

β€² ∈ 𝑇𝑗

cost𝑗 𝑑𝑗

β€², π’•βˆ’π‘— βˆ’ cost𝑗 𝒕 = Ξ¦ 𝑑𝑗 β€², π’•βˆ’π‘— βˆ’ Ξ¦(𝒕)

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Potential games *

  • π‘œπ‘“ 𝒕

𝑓 𝒕

  • Ξ¦ 𝒕 =

𝑓 𝑙=1 π‘œπ‘“(𝒕) 𝑑𝑓

𝑙

  • 𝑑𝑓

π‘œπ‘“ 𝒕 +1 𝑑𝑓 π‘œπ‘“ 𝒕

Ξ”cost𝑗 = ΔΦ ∎

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Potential games *

  • 𝑃(log π‘œ)
  • 𝒕 ≀ Ξ¦ 𝒕 ≀ 𝐼 π‘œ β‹… cost(𝒕)
  • π’•βˆ—

Ξ¦

  • π’•βˆ—
  • π’•βˆ— ≀ Ξ¦ π’•βˆ— ≀ Ξ¦ OPT

≀ 𝐼 π‘œ β‹… cost(OPT) ∎

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Cost sharing summary

  • βˆ€

𝒕, cost 𝒕 ≀ π‘œ β‹… cost(OPT)

  • βˆƒ

𝒕 cost 𝒕 ≀ 𝐼 π‘œ β‹… cost(OPT)

  • βˆƒ

𝒕 cost 𝒕 β‰₯ π‘œ β‹… cost(OPT)

  • βˆ€

𝒕, cost 𝒕 β‰₯ 𝐼 π‘œ β‹… cost(OPT)

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What we have learned

  • 

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