CSC304 Lecture 4 Guest Lecture: Prof. Allan Borodin Game Theory (Cost sharing & congestion games, Potential function, Braess ’ paradox) CSC304 - Nisarg Shah 1
Recap • Finding pure and mixed Nash equilibria ➢ Best response diagrams ➢ Indifference principle • Price of Anarchy (PoA) and Price of Stability (PoS) ➢ How does the Nash equilibrium compare to the social optimum, in the worst case and in the best case? CSC304 - Nisarg Shah 2
Cost Sharing Game • 𝑜 players on directed weighted graph 𝐻 • Player 𝑗 𝑡 1 𝑡 2 1 1 ➢ Wants to go from 𝑡 𝑗 to 𝑢 𝑗 ➢ Strategy set 𝑇 𝑗 = {directed 𝑡 𝑗 → 𝑢 𝑗 paths} ➢ Denote his chosen path by 𝑄 𝑗 ∈ 𝑇 𝑗 10 10 10 • Each edge 𝑓 has cost 𝑑 𝑓 (weight) ➢ Cost is split among all players taking edge 𝑓 1 1 𝑢 1 𝑢 2 ➢ That is, among all players 𝑗 with 𝑓 ∈ 𝑄 𝑗 CSC304 - Nisarg Shah 3
Cost Sharing Game • Given strategy profile 𝑄 , cost 𝑑 𝑗 𝑄 to player 𝑗 is sum of his costs for edges 𝑓 ∈ 𝑄 𝑗 • Social cost 𝐷 𝑄 = σ 𝑗 𝑑 𝑗 𝑄 𝑡 1 𝑡 2 1 1 ➢ Note that 𝐷 𝑄 = σ 𝑓∈𝐹 𝑄 𝑑 𝑓 , where 𝐹(𝑄) ={edges taken in 𝑄 by at least one player} 10 10 10 • In the example on the right: ➢ What if both players take the direct paths? ➢ What if both take the middle paths? 1 1 𝑢 1 𝑢 2 ➢ What if only one player takes the middle path while the other takes the direct path? CSC304 - Nisarg Shah 4
Cost Sharing: Simple Example • Example on the right: 𝑜 players s • Two pure NE ➢ All taking the n-edge: social cost = 𝑜 𝑜 1 ➢ All taking the 1-edge: social cost = 1 o Also the social optimum • In this game, price of anarchy ≥ 𝑜 t • We can show that for all cost sharing games, price of anarchy ≤ 𝑜 CSC304 - Nisarg Shah 5
Cost Sharing: PoA • Theorem: The price of anarchy of a cost sharing game is at most 𝑜 . • Proof: ∗ , 𝑄 2 ∗ , … , 𝑄 ∗ ) , in which ➢ Suppose the social optimum is (𝑄 1 𝑜 ∗ . the cost to player 𝑗 is 𝑑 𝑗 ➢ Take any NE with cost 𝑑 𝑗 to player 𝑗 . ′ be his cost if he switches to 𝑄 𝑗 ∗ . ➢ Let 𝑑 𝑗 ′ ≥ 𝑑 𝑗 ➢ NE ⇒ 𝑑 𝑗 (Why?) ′ ≤ 𝑜 ⋅ 𝑑 𝑗 ∗ (Why?) ➢ But : 𝑑 𝑗 ∗ for each 𝑗 ⇒ no worse than 𝑜 × optimum ➢ 𝑑 𝑗 ≤ 𝑜 ⋅ 𝑑 𝑗 ∎ CSC304 - Nisarg Shah 6
Cost Sharing • Price of anarchy B 10 ➢ All cost-sharing games: PoA ≤ 𝑜 20 A ➢ Example game where PoA = 𝑜 12 C 7 60 • Price of stability? Later… 32 E D • Both examples we saw had 10 players: 𝐹 → 𝐷 pure Nash equilibria 27 players: 𝐶 → 𝐸 ➢ What about more complex 19 players: 𝐷 → 𝐸 games, like the one on the right? CSC304 - Nisarg Shah 7
Good News • Theorem: All cost sharing games have a pure Nash eq. • Proof: ➢ Via “potential function” argument CSC304 - Nisarg Shah 8
Step 1: Define Potential Fn • Potential function: Φ ∶ ς 𝑗 𝑇 𝑗 → ℝ + 𝑜 ∈ ς 𝑗 𝑇 𝑗 , … ➢ For all pure strategy profiles 𝑄 = 𝑄 1 , … , 𝑄 ➢ all players 𝑗 , and … ′ ∈ 𝑇 𝑗 for player 𝑗 … ➢ all alternative strategies 𝑄 𝑗 ′ , 𝑄 −𝑗 − 𝑑 𝑗 𝑄 = Φ 𝑄 𝑗 ′ , 𝑄 −𝑗 − Φ 𝑄 𝑑 𝑗 𝑄 𝑗 • When a single player changes his strategy, the change in his cost is equal to the change in the potential function ➢ Do not care about the changes in the costs to others CSC304 - Nisarg Shah 9
Step 2: Potential F n → pure Nash Eq • All games that admit a potential function have a pure Nash equilibrium. Why? ➢ Think about 𝑄 that minimizes the potential function. ➢ What happens when a player deviates? o If his cost decreases, the potential function value must also decrease. o 𝑄 already minimizes the potential function value. • Pure strategy profile minimizing potential function is a pure Nash equilibrium. CSC304 - Nisarg Shah 10
Step 3: Potential F n for Cost-Sharing • Recall: 𝐹(𝑄) = {edges taken in 𝑄 by at least one player} • Let 𝑜 𝑓 (𝑄) be the number of players taking 𝑓 in 𝑄 𝑜 𝑓 (𝑄) 𝑑 𝑓 Φ 𝑄 = 𝑙 𝑙=1 𝑓∈𝐹(𝑄) • Note: The cost of edge 𝑓 to each player taking 𝑓 is 𝑑 𝑓 /𝑜 𝑓 (𝑄) . But the potential function includes all fractions: 𝑑 𝑓 /1 , 𝑑 𝑓 /2, …, 𝑑 𝑓 /𝑜 𝑓 𝑄 . CSC304 - Nisarg Shah 11
Step 3: Potential F n for Cost-Sharing 𝑜 𝑓 (𝑄) 𝑑 𝑓 Φ 𝑄 = 𝑙 𝑙=1 𝑓∈𝐹(𝑄) • Why is this a potential function? 𝑑 𝑓 ➢ If a player changes path, he pays 𝑜 𝑓 𝑄 +1 for each new 𝑑 𝑔 edge 𝑓 , gets back 𝑜 𝑔 𝑄 for each old edge 𝑔 . ➢ This is precisely the change in the potential function too. ➢ So Δ𝑑 𝑗 = ΔΦ . ∎ CSC304 - Nisarg Shah 12
Potential Minimizing Eq. • There could be multiple pure and multiple mixed Nash equilibria ➢ Pure Nash equilibria are “local minima” of the potential function. ➢ A single player deviating should not decrease the function value. • Minimizing the potential function just gives one of the pure Nash equilibria ➢ Is this equilibrium special? Yes! CSC304 - Nisarg Shah 13
Potential Minimizing Eq. 𝑜 𝑓 (𝑄) 𝑑 𝑓 𝑜 1 𝑑 𝑓 ≤ Φ 𝑄 = ≤ 𝑑 𝑓 ∗ 𝑙 𝑙 𝑙=1 𝑙=1 𝑓∈𝐹(𝑄) 𝑓∈𝐹(𝑄) 𝑓∈𝐹(𝑄) Social cost Harmonic function 𝐼(𝑜) ∀𝑄, 𝐷 𝑄 ≤ Φ 𝑄 ≤ 𝐷 𝑄 ∗ 𝐼 𝑜 𝑜 = σ 𝑙=1 1/𝑜 = 𝑃(log 𝑜) 𝐷 𝑄 ∗ ≤ Φ 𝑄 ∗ ≤ Φ 𝑃𝑄𝑈 ≤ 𝐷 𝑃𝑄𝑈 ∗ 𝐼(𝑜) Potential minimizing eq. Social optimum CSC304 - Nisarg Shah 14
Potential Minimizing Eq. • Potential minimizing equilibrium gives 𝑃(log 𝑜) approximation to the social optimum ➢ Price of stability is 𝑃(log 𝑜) ➢ Compare to the price of anarchy, which can be 𝑜 CSC304 - Nisarg Shah 15
Congestion Games • Generalize cost sharing games • 𝑜 players, 𝑛 resources (e.g., edges) • Each player 𝑗 chooses a set of resources 𝑄 𝑗 (e.g., 𝑡 𝑗 → 𝑢 𝑗 paths) • When 𝑜 𝑘 player use resource 𝑘 , each of them get a cost 𝑔 𝑘 (𝑜 𝑘 ) • Cost to player is the sum of costs of resources used CSC304 - Nisarg Shah 16
Congestion Games • Theorem [Rosenthal 1973]: Every congestion game is a potential game. • Potential function: 𝑜 𝑘 𝑄 Φ 𝑄 = 𝑔 𝑘 𝑙 𝑙=1 𝑘∈𝐹(𝑄) • Theorem [Monderer and Shapley 1996]: Every potential game is equivalent to a congestion game. CSC304 - Nisarg Shah 17
Potential Functions • Potential functions are useful for deriving various results ➢ E.g., used for analyzing amortized complexity of algorithms • Bad news: Finding a potential function that works may be hard. CSC304 - Nisarg Shah 18
The Braess ’ Paradox • In cost sharing, 𝑔 𝑘 is decreasing ➢ The more people use a resource, the less the cost to each. • 𝑔 𝑘 can also be increasing ➢ Road network, each player going from home to work ➢ Uses a sequence of roads ➢ The more people on a road, the greater the congestion, the greater the delay (cost) • Can lead to unintuitive phenomena CSC304 - Nisarg Shah 19
The Braess ’ Paradox • Parkes-Seuken Example: ➢ 2000 players want to go from 1 to 4 ➢ 1 → 2 and 3 → 4 are “congestible” roads ➢ 1 → 3 and 2 → 4 are “constant delay” roads 2 1 4 3 CSC304 - Nisarg Shah 20
The Braess ’ Paradox • Pure Nash equilibrium? ➢ 1000 take 1 → 2 → 4 , 1000 take 1 → 3 → 4 ➢ Each player has cost 10 + 25 = 35 ➢ Anyone switching to the other creates a greater congestion on it, and faces a higher cost 2 1 4 3 CSC304 - Nisarg Shah 21
The Braess ’ Paradox • What if we add a zero-cost connection 2 → 3 ? ➢ Intuitively, adding more roads should only be helpful ➢ In reality, it leads to a greater delay for everyone in the unique equilibrium! 2 𝑑 23 𝑜 23 = 0 1 4 3 CSC304 - Nisarg Shah 22
The Braess ’ Paradox • Nobody chooses 1 → 3 as 1 → 2 → 3 is better irrespective of how many other players take it • Similarly, nobody chooses 2 → 4 • Everyone takes 1 → 2 → 3 → 4 , faces delay = 40 ! 2 𝑑 23 𝑜 23 = 0 1 4 3 CSC304 - Nisarg Shah 23
The Braess ’ Paradox • In fact, what we showed is: ➢ In the new game, 1 → 2 → 3 → 4 is a strictly dominant strategy for each firm! 2 𝑑 23 𝑜 23 = 0 1 4 3 CSC304 - Nisarg Shah 24
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