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

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

Normal-Form Game • 𝑂 = {1, … , 𝑜} o 𝑇 o 𝑣 𝑗 : 𝑇 𝑜 → ℝ 𝑗 ∈ 𝑂 o j ∈ 𝑂 𝑡 𝑘 ∈ 𝑇 𝑗 𝑣 𝑗 (𝑡 1 , … , 𝑡 𝑜 ) • 2

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

The Ice Cream Wars • 𝑂 = 1,2 𝑡 𝑗 +𝑡 𝑘 , 𝑡 𝑗 < 𝑡 • 𝑇 = [0,1] 𝑘 2 𝑡 𝑗 +𝑡 𝑘 • 𝑣 𝑗 𝑡 𝑗 , 𝑡 𝑘 = 1 − , 𝑡 𝑗 > 𝑡 𝑘 2 1 2 , 𝑡 𝑗 = 𝑡 𝑘 • 4

The prisoner’s dilemma • • o o • 5

The prisoner’s dilemma 6

In real life • o o • o o • o o 7

On TV 8

The professor’s dilemma 9

Nash equilibrium • • 𝒕 = 𝑡 1 … , 𝑡 𝑜 ∈ 𝑇 𝑜 ′ ∈ 𝑇, 𝑣 𝑗 𝒕 ≥ 𝑣 𝑗 (𝑡 𝑗 ′ , 𝒕 −𝑗 ) ∀𝑗 ∈ 𝑂, ∀𝑡 𝑗 𝒕 −𝑗 = 𝑡 1 , … , 𝑡 𝑗−1 , 𝑡 𝑗+1 , … , 𝑡 𝑜 10

Nash equilibrium • 0 1. 1 2. 2 3. 3 4. 11

Nash equilibrium 12

Russel Crowe was wrong 13

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

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Does NE make sense? {2, … , 100} • • 𝑡 𝑢 𝑡 < 𝑢 • 𝑡 + 2 𝑡 − 2 • 16

Back to prison • • = • 17

Anarchy and stability • • • = o = o 18

Example: Cost sharing • 𝑜 𝐻 𝑡 1 𝑡 2 1 1 𝑗 𝑡 𝑗 𝑢 𝑗 • 𝑡 𝑗 → 𝑢 𝑗 𝑓 𝑑 𝑓 • 10 10 10 • 1 1 • 𝑢 1 𝑢 2 19

Example: Cost sharing 𝑜 • 𝑜 𝑡 1 • • ⇒ ≥ 𝑜 𝑜 1 ≤ 𝑜 • o 𝑢 o 20

Example: Cost sharing 1 • 2 • 0 0 0 𝑜 … 𝑡 1 𝑡 2 𝑡 𝑜 2 • 2 1 1 1 • 𝑢 21

Example: Cost sharing = 2 • • 0 0 0 • ⇒ … 𝑡 1 𝑡 2 𝑡 𝑜 2 • 1 1 1 1 2 𝑜 𝑢 22

Potential games • 𝑜 Φ: 𝑗=1 𝑇 𝑗 → ℝ 𝑜 𝒕 ∈ 𝑗=1 𝑗 ∈ 𝑂 𝑇 𝑗 ′ ∈ 𝑇 𝑗 𝑡 𝑗 ′ , 𝒕 −𝑗 − cost 𝑗 𝒕 = Φ 𝑡 𝑗 ′ , 𝒕 −𝑗 − Φ(𝒕) cost 𝑗 𝑡 𝑗 • 23

Potential games * • • 𝑜 𝑓 𝒕 𝑓 𝒕 o o 𝑜 𝑓 (𝒕) 𝑑 𝑓 Φ 𝒕 = 𝑙 𝑓 𝑙=1 𝑑 𝑓 o 𝑜 𝑓 𝒕 +1 𝑑 𝑓 Δcost 𝑗 = ΔΦ ∎ 𝑜 𝑓 𝒕 24

Potential games * • 𝑃(log 𝑜) • o 𝒕 ≤ Φ 𝒕 ≤ 𝐼 𝑜 ⋅ cost(𝒕) 𝒕 ∗ Φ o 𝒕 ∗ o 𝒕 ∗ ≤ Φ 𝒕 ∗ ≤ Φ OPT o ≤ 𝐼 𝑜 ⋅ cost(OPT) ∎ 25

Cost sharing summary • ∀ 𝒕 , cost 𝒕 ≤ 𝑜 ⋅ cost(OPT) o ∃ 𝒕 cost 𝒕 ≤ 𝐼 𝑜 ⋅ cost(OPT) o • ∃ 𝒕 cost 𝒕 ≥ 𝑜 ⋅ cost(OPT) o ∀ 𝒕 , cost 𝒕 ≥ 𝐼 𝑜 ⋅ cost(OPT) o 26

What we have learned • o o o o o •  o 27