price of anarchy in games of incomplete information
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Price of Anarchy in Games of Incomplete Information. Tim Roughgarden. Alon Ardenboim. Full Information Games. The players payoffs are common knowledge. - PowerPoint PPT PresentationTRANSCRIPT
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Tim Roughgarden
PRICE OF ANARCHY IN GAMES OF INCOMPLETE
INFORMATION
Alon Ardenboim
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The players payoffs are common knowledge.Pure (Mixed) Nash equilibrium – each players
maximizes his utility (in expectation) when sticking with his current (probabilistic) strategy.
FULL INFORMATION GAMES
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Choose a goal function (e.g. welfare maximization).How bad can an equilibrium be w.r.t. the optimal
outcome (e.g. maximum welfare)?.
PRICE OF ANARCHY
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Players are uncertain about each other payoffs.For example, auctions (eBay), VCG mechanisms.Assume players’ private preferences are drawn
independently from prior distributions.Distributions ARE common knowledge.
INCOMPLETE INFORMATION GAMES
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Type space .Action space . sampled from . is common knowledge.A strategy is a function from type space to a
distribution over actions .A strategy profile is a Bayes-Nash equilibrium if for
every , type and action ,
BAYES-NASH EQUILIBRIUM
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The corresponding PoA of such a games measures how bad is the worst Bayes-Nash equilibrium w.r.t the optimal value.
That is,
When is a product dist., this is iPoA (independent).Otherwise, we talk about cPoA (correlated).
BAYES-NASH POA
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Def: A game is -smooth w.r.t outcome and a maximization objective function if for every ,
W is payoff-dominating if it bounds the sum of players’ payoffs from above (non-negative transfers).
Thm: if a game is -smooth w.r.t. an optimal outcome for a payoff-dominating then PoA.
Let be a Nash Eq., we have:
SMOOTH FULL INFORMATION GAMES
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Def: Let be a game structure and a maximization objective function. The structure is -smooth w.r.t. social choice function if for every and feasible to , we have
Thm: If a game structure is -smooth w.r.t. an optimal choice function for a payoff-dominating , then the iPoA of the game w.r.t. .
SMOOTH INCOMPLETE INFORMATION GAMES
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Let be an optimal choice function (that is, if every player plays we get ).
Let be a Bayes-Nash equilibrium. In strategy player samples and plays .
PROOF OF THEOREM
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We have:
PROOF CONT.
(Payoff dominant)
(Lin. of Exp.)
(Equilibrium)
(Def.)
(Smooth)
(Lin. of Exp.)
Bayes-Nash
OPT
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In the Generalized Second Prize (GSP) auction there are ad slots in a web page. Each with an associated click-through rate.
Each bidder has a private information – valuation per click .
No player overbids (feasible space of bids is ).Assume .
APPLICATION TO GSP
𝛼1
𝛼2
𝛼3
…𝛼𝑘
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Assume player gets bids the highest bid.Allocation: assign the slot with CTR .Payment: Charge player the highest bid.Payoff: if .
otherwise ( if bid is feasible).
GSP (CONT.)
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Thm: The GSP is a -smooth game (and therefore the iPoA is ) w.r.t. welfare maximization goal function.
Proof: Consider welfare maximization (payoff dominant). Let’s take the social choice function (). Easy to see it’s optimal. Fix a type vector of players valuations and an outcome
(arbitrary bids). Assume . Let denote the index of the highest bidder.
SMOOTHNESS OF GSP
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Claim:
for every . :
SMOOTHNESS PROOF (CONT.)
𝛼1
…𝛼 𝑗
…𝛼 𝑖…𝛼𝑘
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Claim:
for every . :
SMOOTHNESS PROOF (CONT.)
𝛼1
…𝛼 𝑗
…𝛼 𝑖…𝛼𝑘
𝛼 𝑗≥𝛼𝑖𝑏𝑖𝑑 ( 𝑗+1 )≤𝑏𝑖=𝑣𝑖 /2
𝑢𝑖¿¿𝛼 𝑗 ⋅ (𝑣𝑖−𝑏𝑖𝑑 ( 𝑗+1 ))≥12𝛼𝑖𝑣 𝑖
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Claim:
for every . :
SMOOTHNESS PROOF (CONT.)
𝛼1
…𝛼 𝑗
…𝛼 𝑖…𝛼𝑘
𝑏𝑖𝑑 (𝑖 )≥𝑣𝑖 /2
12𝛼𝑖 𝑣𝑖−𝛼𝑖𝑏𝑖𝑑 (𝑖 )≤ 0≤𝑢𝑖¿
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Summing over all players we get:
SMOOTHNESS PROOF (CONT.)
𝑊 (𝐭 ;𝐜∗ ( 𝐭 ) ) ≤𝑊 (𝐬=𝐯 ′ ;𝐚 ) ∀ 𝐯 ′≥𝐚
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Application to other games.Other smoothness variants.What to do with correlated type distributions? Is there a relation between cPoA and sPoA?
DIRECTIONS