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Price of Anarchy in Games of Incomplete Information

Price of Anarchy in Games of Incomplete Information. Tim Roughgarden. Alon Ardenboim. Full Information Games. The players payoffs are common knowledge.

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Price of Anarchy in Games of Incomplete Information

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  1. Price of Anarchy in Games of Incomplete Information Tim Roughgarden AlonArdenboim

  2. Full Information Games • The players payoffs are common knowledge. • Pure (Mixed) Nash equilibrium – each players maximizes his utility (in expectation) when sticking with his current (probabilistic) strategy.

  3. Price of Anarchy • Choose a goal function (e.g. welfare maximization). • How bad can an equilibrium be w.r.t. the optimal outcome (e.g. maximum welfare)? .

  4. Incomplete Information Games • 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.

  5. Bayes-Nash Equilibrium • 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 ,

  6. Bayes-Nash PoA • 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).

  7. Smooth Full Information Games • Def: A game is -smooth w.r.toutcome 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:

  8. Smooth Incomplete Information Games • 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 iPoAof the game w.r.t. .

  9. Proof of Theorem • 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 .

  10. Proof Cont. • We have: (Payoff dominant) (Lin. of Exp.) (Equilibrium) Bayes-Nash (Def.) (Lin. of Exp.) (Smooth) OPT

  11. Application to GSP • 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 .

  12. GSP (cont.) • 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).

  13. Smoothness of GSP • 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.

  14. Smoothness proof (cont.) • Claim: for every . • : …

  15. Smoothness proof (cont.) • Claim: for every . • : …

  16. Smoothness proof (cont.) • Claim: for every . • : …

  17. Smoothness proof (cont.) • Summing over all players we get:

  18. Directions • Application to other games. • Other smoothness variants. • What to do with correlated type distributions? • Is there a relation between cPoA and sPoA?

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