Tim Roughgarden

PRICE OF ANARCHY IN GAMES OF INCOMPLETE

INFORMATION

Alon Ardenboim

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

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

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

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

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

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

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

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

We have:

PROOF CONT.

(Payoff dominant)

(Lin. of Exp.)

(Equilibrium)

(Def.)

(Smooth)

(Lin. of Exp.)

Bayes-Nash

OPT

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

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.)

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

Claim:

for every . :

SMOOTHNESS PROOF (CONT.)

𝛼1

…𝛼 𝑗

…𝛼 𝑖…𝛼𝑘

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

Claim:

for every . :

SMOOTHNESS PROOF (CONT.)

𝛼1

…𝛼 𝑗

…𝛼 𝑖…𝛼𝑘

𝑏𝑖𝑑 (𝑖 )≥𝑣𝑖 /2

12𝛼𝑖 𝑣𝑖−𝛼𝑖𝑏𝑖𝑑 (𝑖 )≤ 0≤𝑢𝑖¿

Summing over all players we get:

SMOOTHNESS PROOF (CONT.)

𝑊 (𝐭 ;𝐜∗ ( 𝐭 ) ) ≤𝑊 (𝐬=𝐯 ′ ;𝐚 ) ∀ 𝐯 ′≥𝐚

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

DIRECTIONS