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Mechanism Design without Money
Lecture 4
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Price of Anarchy simplest example
• G is given• Route 1 unit from A to B, through AB,AXB • OPT– route ½ on AB and ½ on AXB (check!)• NASH – route 1 on AXB• Ratio is 4/3
BA
X0
1
x
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Price of Anarchy: general functions
• G is given• Route 1 unit from A to B, through AB,AXB • OPT – choose y to minimize yC (y) + (1 – y)C (1)• Nash – route 1 on AXB• Ratio is • Also called the Pigou Bound
BA
X0
c(1)
c(x)
𝑐 (1)𝑦 ∙𝑐 (𝑦 )+(1−𝑦 ) ∙𝑐(1)
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Can things be worse?
• No!• For a set of cpu functions C, define
• In our examples before r was 1• Theorem: For any routing game G, if the cost per
unit functions come from C, the Price of Anarchy is at most a(C)
𝑎 (𝐶 )=𝑚𝑎𝑥𝑐𝑝𝑢∈𝐶, 𝑦 ,𝑟
𝑟 ∙𝑐𝑝𝑢(𝑟 )𝑦 ∙𝑐𝑝𝑢 (𝑦 )+(𝑟 − 𝑦 )∙𝑐𝑝𝑢(𝑟 )
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Another property of Nash flows
• Theorem: Let f be a Nash flow. For any other flow f* which routes the same amount, we have
• Note: The cost per unit of every edge is constant, and we just want to route the flow.
∑𝑒
𝑓 𝑒𝑐𝑝𝑢𝑒( 𝑓 𝑒)≤∑𝑒
𝑓 ∗𝑒❑𝑐𝑝𝑢𝑒( 𝑓 𝑒)
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Proof
• Define
Note that
• Therefore, we need to prove that H(f*,f) ≥ H(f,f)
• This follows by using that if a path has any flow in it in a Nash flow, its cost is minimal
𝐻 ( 𝑓 ∗ , 𝑓 )=∑𝑃
𝑓 ∗𝑃𝑐𝑝𝑢𝑃( 𝑓 )
=
∑𝑒
𝑓 𝑒𝑐𝑝𝑢𝑒( 𝑓 𝑒)≤∑𝑒
𝑓 ∗𝑒❑𝑐𝑝𝑢𝑒( 𝑓 𝑒)
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Proof that a(C) is the bound on PoA
Let C be the set of functions on the edges. Let f* be the optimal solution, and f be the Nash.
And in particular setting r=fe and y=f*e
𝑐𝑜𝑠𝑡 𝑜𝑓 𝑓𝑙𝑜𝑤 𝑓 ∗=¿=
≥𝑐𝑜𝑠𝑡𝑜𝑓 𝑓𝑙𝑜𝑤 𝑓
𝑎(𝐶)
𝑎 (𝐶 )=𝑚𝑎𝑥𝑐𝑝𝑢∈𝐶, 𝑦 ,𝑟
𝑟 ∙𝑐𝑝𝑢(𝑟 )𝑦 ∙𝑐𝑝𝑢 (𝑦 )+(𝑟 − 𝑦 )∙𝑐𝑝𝑢(𝑟 )
- )
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How big can a(C) be?
• Theorem: If C is a set of affine functions, a(C) is at most 4/3
• Proof: Do this at home. Hint: compute the derivative. You should get x = r/2
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Fighting selfishness
• Braess paradox shows that selfish agents can improve their situation if an edge is removed from the graph
• Given a graph, which edges should be removed?
• We are looking at
𝐶𝑜𝑠𝑡𝑜𝑓 𝑤𝑜𝑟𝑠𝑡 𝑁𝑎𝑠h 𝑓𝑜𝑟 𝐺𝑐𝑜𝑠𝑡 𝑜𝑓 𝑤𝑜𝑟𝑠𝑡 𝑁𝑎𝑠h 𝑓𝑜𝑟 𝑎𝑠𝑢𝑏𝑔𝑟𝑎𝑝h𝑜𝑓 𝐺
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Fighting selfishness is (computationally) hard
• Problem: Which edges should I cut to improve the worst Nash?
• It is trivial to get a 4/3 approximation – just cut nothing. The worst Nash for a subgraph is always worse than OPT for the original graph, and the PoA with linear cost functions is 4/3
• Thm: It is NP hard to approximate better than 4/3
𝐶𝑜𝑠𝑡𝑜𝑓 𝑤𝑜𝑟𝑠𝑡 𝑁𝑎𝑠h 𝑓𝑜𝑟 𝐺𝑐𝑜𝑠𝑡 𝑜𝑓 𝑤𝑜𝑟𝑠𝑡 𝑁𝑎𝑠h 𝑓𝑜𝑟 𝑎𝑠𝑢𝑏𝑔𝑟𝑎𝑝h𝑜𝑓 𝐺
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Proof• Reduction from 2DPP: Given a graph G, two sources s1,s2
and two targets t1,t2 are there two vertex disjoint paths s1t1 and s2t2
• If there are no two disjoint paths you will always have a path s2t1
s
t2
s1
s2
t1
t
G
x
x1
1
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What happens for non linear cost functions?
• We said price of anarchy can grow, but what about fighting selfishness?
• Thm: There exists a graph with n vertices and non linear cost functions, such that removing edges improves the worst Nash by a factor of n/2
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The Graph
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Bad Nash flow
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After edge removal
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Atomic flows
• Multiple equilibria (remember the examples)• Sometimes there is no pure equilibrium• Weaker bounds, different techniques
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No pure Nash
• P1 routes 1 unit from s to t
• P2 routes 2 units from s to t
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Price of anarchy example
• Not all paths in the equilibrium have the same cost• In the example: PoA of 5/2• This is the worst case for
affine functions if allplayers have the sameamount of flow
• We will prove a weaker boundwhen players control different amounts of flow
U V
W
0
0
x
x xx
s1,s2
t2,t3s4
t1,s3 ,t4
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Atomic flow for affine functions
• Edge e has CPU aex + be
• Player i sends ri
• Let f be Nash, f* be OPT. We have for every player i:
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Summing over the players:
𝑟 𝑖 ∑𝑒∈ 𝑃 𝑖
𝑎𝑒 𝑓 𝑒+𝑏𝑒≤𝑟 𝑖 ∑𝑒∈ 𝑃∗
𝑖❑
𝑎𝑒 ( 𝑓 𝑒+𝑟 𝑖 )+𝑏𝑒
𝑐𝑜𝑠𝑡 ( 𝑓 )≤𝑐𝑜𝑠𝑡 ( 𝑓 ∗)+∑𝑒
𝑎𝑒 𝑓 𝑒 𝑓 𝑒∗
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Manipulations
• You get
• Solving x2-3x+10 gives (3 + 51/2)/2
𝐶 ( 𝑓 )𝐶 ( 𝑓 ∗)
−1≤ √ 𝐶 ( 𝑓 )𝐶 ( 𝑓 ∗)
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Questions?
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Extra Slides
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Chicken
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Road example
A B
1 hour
1 hour
N minutes
N minutes
• 50 people want to get from A to B• There are two roads, each one has two segments. One takes
an hour, and the other one takes the number of people on it
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Nash in road example
• In the Nash equilibrium, 25 people would take each route, for a travel time of 85 minutes
A B
1 hour
1 hour
N minutes
N minutes
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Braess’ paradox
• Now suppose someone adds an extra road which takes no time at all. Travel time goes to 100 minutes
A B
1 hour
1 hour
N minutes
N minutes
Free