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

Oct 07, 2022 β€’278 likes β€’555 views

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

  1. CMU 15-251 Game theory Teachers: Anil Ada Ariel Procaccia (this time)

  2. Normal-Form Game β€’ 𝑂 = {1, … , π‘œ} o 𝑇 o 𝑣 𝑗 : 𝑇 π‘œ β†’ ℝ 𝑗 ∈ 𝑂 o j ∈ 𝑂 𝑑 π‘˜ ∈ 𝑇 𝑗 𝑣 𝑗 (𝑑 1 , … , 𝑑 π‘œ ) β€’ 2

  3. One day your cousin Ted shows up. His ice cream is identical! Selling ice cream at the beach. 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%! 3

  4. The Ice Cream Wars β€’ 𝑂 = 1,2 𝑑 𝑗 +𝑑 π‘˜ , 𝑑 𝑗 < 𝑑 β€’ 𝑇 = [0,1] π‘˜ 2 𝑑 𝑗 +𝑑 π‘˜ β€’ 𝑣 𝑗 𝑑 𝑗 , 𝑑 π‘˜ = 1 βˆ’ , 𝑑 𝑗 > 𝑑 π‘˜ 2 1 2 , 𝑑 𝑗 = 𝑑 π‘˜ β€’ 4

  5. The prisoner’s dilemma β€’ β€’ o o β€’ 5

  6. The prisoner’s dilemma 6

  7. In real life β€’ o o β€’ o o β€’ o o 7

  8. On TV 8

  9. The professor’s dilemma 9

  10. Nash equilibrium β€’ β€’ 𝒕 = 𝑑 1 … , 𝑑 π‘œ ∈ 𝑇 π‘œ β€² ∈ 𝑇, 𝑣 𝑗 𝒕 β‰₯ 𝑣 𝑗 (𝑑 𝑗 β€² , 𝒕 βˆ’π‘— ) βˆ€π‘— ∈ 𝑂, βˆ€π‘‘ 𝑗 𝒕 βˆ’π‘— = 𝑑 1 , … , 𝑑 π‘—βˆ’1 , 𝑑 𝑗+1 , … , 𝑑 π‘œ 10

  11. Nash equilibrium β€’ 0 1. 1 2. 2 3. 3 4. 11

  12. Nash equilibrium 12

  13. Russel Crowe was wrong 13

  14. End of the Ice Cream Wars Day 3 of the ice cream wars… Teddy sets up south of you! You go south of Teddy. Eventually… 14

  15. 15

  16. Does NE make sense? {2, … , 100} β€’ β€’ 𝑑 𝑒 𝑑 < 𝑒 β€’ 𝑑 + 2 𝑑 βˆ’ 2 β€’ 16

  17. Back to prison β€’ β€’ = β€’ 17

  18. Anarchy and stability β€’ β€’ β€’ = o = o 18

  19. Example: Cost sharing β€’ π‘œ 𝐻 𝑑 1 𝑑 2 1 1 𝑗 𝑑 𝑗 𝑒 𝑗 β€’ 𝑑 𝑗 β†’ 𝑒 𝑗 𝑓 𝑑 𝑓 β€’ 10 10 10 β€’ 1 1 β€’ 𝑒 1 𝑒 2 19

  20. Example: Cost sharing π‘œ β€’ π‘œ 𝑑 1 β€’ β€’ β‡’ β‰₯ π‘œ π‘œ 1 ≀ π‘œ β€’ o 𝑒 o 20

  21. Example: Cost sharing 1 β€’ 2 β€’ 0 0 0 π‘œ … 𝑑 1 𝑑 2 𝑑 π‘œ 2 β€’ 2 1 1 1 β€’ 𝑒 21

  22. Example: Cost sharing = 2 β€’ β€’ 0 0 0 β€’ β‡’ … 𝑑 1 𝑑 2 𝑑 π‘œ 2 β€’ 1 1 1 1 2 π‘œ 𝑒 22

  23. Potential games β€’ π‘œ Ξ¦: 𝑗=1 𝑇 𝑗 β†’ ℝ π‘œ 𝒕 ∈ 𝑗=1 𝑗 ∈ 𝑂 𝑇 𝑗 β€² ∈ 𝑇 𝑗 𝑑 𝑗 β€² , 𝒕 βˆ’π‘— βˆ’ cost 𝑗 𝒕 = Ξ¦ 𝑑 𝑗 β€² , 𝒕 βˆ’π‘— βˆ’ Ξ¦(𝒕) cost 𝑗 𝑑 𝑗 β€’ 23

  24. Potential games * β€’ β€’ π‘œ 𝑓 𝒕 𝑓 𝒕 o o π‘œ 𝑓 (𝒕) 𝑑 𝑓 Ξ¦ 𝒕 = 𝑙 𝑓 𝑙=1 𝑑 𝑓 o π‘œ 𝑓 𝒕 +1 𝑑 𝑓 Ξ”cost 𝑗 = ΔΦ ∎ π‘œ 𝑓 𝒕 24

  25. Potential games * β€’ 𝑃(log π‘œ) β€’ o 𝒕 ≀ Ξ¦ 𝒕 ≀ 𝐼 π‘œ β‹… cost(𝒕) 𝒕 βˆ— Ξ¦ o 𝒕 βˆ— o 𝒕 βˆ— ≀ Ξ¦ 𝒕 βˆ— ≀ Ξ¦ OPT o ≀ 𝐼 π‘œ β‹… cost(OPT) ∎ 25

  26. Cost sharing summary β€’ βˆ€ 𝒕 , cost 𝒕 ≀ π‘œ β‹… cost(OPT) o βˆƒ 𝒕 cost 𝒕 ≀ 𝐼 π‘œ β‹… cost(OPT) o β€’ βˆƒ 𝒕 cost 𝒕 β‰₯ π‘œ β‹… cost(OPT) o βˆ€ 𝒕 , cost 𝒕 β‰₯ 𝐼 π‘œ β‹… cost(OPT) o 26

  27. What we have learned β€’ o o o o o β€’  o 27

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