PRICE OF ANARCHY IN
GAMES OF INCOMPLETE
INFORMATION
Alon Ardenboim
Tim
Roughgarden
FULL INFORMATION GAMES
 The players payof fs are common knowledge.
 Pure (Mixed) Nash equilibrium – each players maximizes his
utility (in expectation) when sticking with his current
(probabilistic) strategy.
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)?
𝑊(𝑆)
.
𝑊(𝑂𝑃𝑇)
𝑆∈𝑁𝐸
𝑃𝑜𝐴 = min
INCOMPLETE INFORMATION GAMES
 Players are uncertain about each other payof fs.
 For example, auctions (eBay), VCG mechanisms.
 Assume players’ private preferences are drawn independently
from prior distributions.
 Distributions ARE common knowledge.
BAYES-NASH EQUILIBRIUM
Type space 𝑇 = 𝑇1 × ⋯ × 𝑇𝑛 .
Action space 𝐴 = 𝐴 1 × ⋯ × 𝐴 𝑛 .
𝐭 ∈ 𝑇 sampled from 𝐹 . 𝐹 is common knowledge.
A strategy 𝜎𝑖 is a function from type space 𝑇𝑖 to a distribution
over actions 𝐴 𝑖 .
 A strategy profile 𝜎 is a Bayes-Nash equilibrium if for every 𝑖 ,
type 𝑡 𝑖 ∈ 𝑇𝑖 and action 𝑎 𝑖′ ∈ 𝐴 𝑖 ,




𝐄𝐭
𝑡𝑖
−𝑖 ~𝐹−𝑖
[𝐄 𝐚~𝜎
𝐭
[𝑢 𝑖 𝑡 𝑖 ; 𝐚 ] ≥ 𝐄 𝐭
𝑡𝑖
−𝑖 ~𝐹−𝑖
[𝐄 𝐚 −𝑖 ~𝜎 −𝑖
𝐭 −𝑖
[𝑢 𝑖 𝑡 𝑖 ; (𝑎 𝑖′ , 𝐚 −𝑖 ]]
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,
𝐄 𝐭~𝐹 [𝐄 𝐚~𝜎 𝑡 [𝑊(𝐭; 𝐚)]]
𝐄 𝐭~𝐹 [𝑊(𝑂𝑃𝑇(𝐭))]
 When 𝐹 is a product dist., this is iPoA (independent).
 Otherwise, we talk about cPoA (correlated).
SMOOTH FULL INFORMATION GAMES
 Def: A game 𝐴, 𝑢 is (𝜆, 𝜇)-smooth w.r.t outcome 𝐚 ∗ and a
maximization objective function 𝑊 if for every 𝐚 ∈ 𝐴,
𝑛
𝑢 𝑖 𝑎 𝑖∗ , 𝐚 −𝐢 ≥ 𝜆 ⋅ 𝑊 𝐚 ∗ − 𝜇 ⋅ 𝑊(𝐚)
𝑖=1
 W is payof f-dominating if it bounds the sum of players’
payof fs from above (non-negative transfers).
 Thm: if a game is (𝜆, 𝜇)-smooth w.r.t. an optimal outcome 𝐚 ∗
𝜆
for a payof f-dominating 𝑊 then PoA≥
.
1+𝜇
 Let 𝑎 be a Nash Eq., we have:
𝑊 𝑎 ≥
𝑢 𝑖 𝑎 𝑖∗ , 𝑎 −1 ≥ 𝜆 ⋅ 𝑊 𝐚 ∗ − 𝜇 ⋅ 𝑊(𝐚)
𝑢𝑖 𝑎 ≥
𝑖
𝑖
SMOOTH INCOMPLETE INFORMATION GAMES
 Def: Let Γ = (𝑇, 𝐴, 𝑢) be a game structure and 𝑊 : 𝑇 × 𝐴 → ℝ + a
maximization objective function. The structure Γ is 𝜆, 𝜇 smooth w.r.t. social choice function 𝐜 ∗ if for every 𝐭, 𝐬 ∈ 𝑇 and
𝐚 ∈ 𝐴 feasible to 𝐬, we have
𝑢 𝑖 (𝑡 𝑖 ; (𝑐𝑖∗ 𝐭 , 𝐚 −𝐢 )) ≥ 𝜆 ⋅ 𝑊 𝐭; 𝐜 ∗ 𝐭
− 𝜇 ⋅ 𝑊(𝐬; 𝐚)
𝑖
 Thm: If a game structure Γ is 𝜆, 𝜇 -smooth w.r.t. an optimal
choice function for a payof f-dominating 𝑊 , then the iPoA of
𝜆
the game w.r.t. 𝑊 ≥
.
1+𝜇
PROOF OF THEOREM
 Let 𝑐 ∗ be an optimal choice function (that is, if every player
plays 𝑐 𝑖∗ (𝑡) we get 𝑂𝑃𝑇(𝑡)).
 Let 𝜎 be a Bayes-Nash equilibrium.
 In strategy 𝜎𝑖′ player 𝑖 samples 𝑠 −𝑖 ~𝐹 and plays 𝑐 𝑖∗ 𝑡 𝑖 , 𝑠 −𝑖 .
PROOF CONT.
 We have:
(Payoff dominant)
𝐄 𝐭~𝐹 [𝐄 𝐚~𝜎
𝐭
[𝑊 𝐭; 𝐚 ]] ≥ 𝐄 𝐭~𝐹 [𝐄 𝐚~𝜎
𝐭
[
𝑢 𝑖 𝑡 𝑖 ; 𝐚 ]]
𝑖
(Lin. of Exp.) =
𝐄 𝐭~𝐹 [𝐄 𝐚~𝜎
𝐭
[𝑢 𝑖 (𝑡 𝑖 ; 𝐚)]]
𝑖
(Equilibrium) ≥
𝐄 𝐭~𝐹 [𝐄 𝑎 ′ ~𝜎 ′
𝑖
𝑖 𝑡 𝑖 ,𝐚~𝜎 𝐭
[𝑢 𝑖 𝑡 𝑖 ; 𝑎 𝑖′ ; 𝐚 −𝐢 ]]
𝑖
(Def.)
=
𝐄 𝐭~𝐹 [𝐄 𝐬 −𝐢 ~𝐹 −𝑖 ,𝐚~𝜎
𝐭
[𝑢 𝑖 𝑡 𝑖 ; 𝑐𝑖∗ (𝑡 𝑖 , 𝐬 −𝐢 ); 𝐚 −𝐢 ]]
𝑖
𝑢 𝑖 (𝑡 𝑖 ; 𝑐𝑖∗ 𝐭 ; 𝐚 −𝐢 )]]
(Lin. of Exp.) = 𝐄 𝐭,𝐬~𝐹 [𝐄 𝐚~𝜎(𝐬) [
(Smooth)
≥ 𝜆 ⋅ 𝐄 𝐭~𝐹 [𝑊(𝐭;
OPT
𝐜∗
𝑖
𝐭 )] − 𝜇 ⋅ 𝐄 𝐬~𝐹 [𝐄 𝐚~𝜎
𝐭
[𝑊 𝐬; 𝐚 ]]
Bayes-Nash
APPLICATION TO GSP
 In the Generalized Second Prize (GSP) auction there are 𝑘 ad
slots in a web page. Each with an associated click -through
rate.
 Each bidder has a private information – valuation per click 𝑣 𝑖 .
 No player overbids (feasible space of bids is [0, 𝑣 𝑖 ]).
𝛼1
 Assume 𝛼 1 ≥ 𝛼 2 ≥ ⋯ ≥ 𝛼 𝑛 .
𝛼2
𝛼3
…
𝛼𝑘
GSP (CONT.)
 Assume player 𝑖 gets bids the 𝑗 𝑡ℎ highest bid.
 Allocation: assign 𝑖 the slot with CTR 𝛼𝑗 .
 Payment: Charge player 𝑖 the 𝑗 + 1 highest bid.
 Payoff: 𝛼𝑗 × (𝑣 𝑖 − 𝑏𝑗 +1 ) if 𝑗 ≤ 𝑘 .
0 otherwise (𝑢 𝑖 ≥ 0 if bid is feasible).
SMOOTHNESS OF GSP
1
 Thm: The GSP is a (1, )-smooth game (and therefore the iPoA
2
is ≥ 1/4) w.r.t. welfare maximization goal function.
 Proof:
Consider welfare maximization (payoff dominant).
Let’s take the social choice function 𝐜 ∗ = 𝐯/2 (𝑐𝑖∗ = 𝑣 𝑖 /2).
Easy to see it’s optimal.
Fix a type vector 𝐭 = 𝐯 of players valuations and an outcome 𝐚
= (𝑏1 , … , 𝑏𝑛 ) (arbitrary bids). Assume 𝑣1 ≤ 𝑣 2 ≤ ⋯ ≤ 𝑣𝑛 .
 Let 𝑖𝑑(𝑖) denote the index of the 𝑖 𝑡ℎ highest bidder.




SMOOTHNESS PROOF (CONT.)
 Claim:
𝑢 𝑖 (𝑣 𝑖 ;
for every 𝑖 .
 𝑗 ≤ 𝑖:
𝑐 𝑖∗
𝐯 , 𝑎 −𝑖
1
) ≥ 𝛼 𝑖 𝑣 𝑖 − 𝛼 𝑖 𝑏 𝑖𝑑
2
𝑖
𝛼1
…
𝛼𝑗
…
𝛼𝑖
…
𝛼𝑘
SMOOTHNESS PROOF (CONT.)
 Claim:
𝑢 𝑖 (𝑣 𝑖 ;
for every 𝑖 .
 𝑗 ≤ 𝑖:
𝑐 𝑖∗
𝐯 , 𝑎 −𝑖
1
) ≥ 𝛼 𝑖 𝑣 𝑖 − 𝛼 𝑖 𝑏 𝑖𝑑
2
𝑖
𝛼1
…
𝛼𝑗 ≥ 𝛼𝑖
𝑏𝑖𝑑
𝑗+1
≤ 𝑏𝑖 = 𝑣𝑖 /2
𝛼𝑗
…
𝛼𝑖
𝑢𝑖 (𝑣𝑖 ;
𝑐𝑖∗
𝐯 , 𝑎−𝑖 ) = 𝛼𝑗 ⋅ 𝑣𝑖 − 𝑏𝑖𝑑
𝑗+1
1
≥ 𝛼𝑖 𝑣𝑖
2
…
𝛼𝑘
SMOOTHNESS PROOF (CONT.)
 Claim:
𝑢 𝑖 (𝑣 𝑖 ;
𝑐 𝑖∗
𝐯 , 𝑎 −𝑖
for every 𝑖 .
 𝑗 > 𝑖:
1
) ≥ 𝛼 𝑖 𝑣 𝑖 − 𝛼 𝑖 𝑏 𝑖𝑑
2
𝑖
𝛼1
…
𝛼𝑗
…
𝑏𝑖𝑑
1
𝛼 𝑣 − 𝛼𝑖 𝑏𝑖𝑑
2 𝑖 𝑖
𝑖
≥ 𝑣𝑖 /2
𝛼𝑖
…
𝑖
≤ 0 ≤ 𝑢𝑖 (𝑣𝑖 ; 𝑐𝑖∗ 𝐯 , 𝑎−𝑖 )
𝛼𝑘
SMOOTHNESS PROOF (CONT.)
 Summing over all players we get:
1
𝑢 𝑖 (𝑣 𝑖 ; 𝑐 𝑖∗ 𝐯 , 𝑎 −𝑖 ) ≥
2
𝑖
𝛼𝑖 𝑣𝑖 −
𝑖
𝑊 𝐭; 𝐜 ∗ 𝐭
𝛼 𝑖 𝑏 𝑖𝑑
𝑖
𝑖
≤ 𝑊 𝐬 = 𝐯 ′ ; 𝐚 ∀𝐯 ′ ≥ 𝐚
DIRECTIONS




Application to other games.
Other smoothness variants.
What to do with correlated type distributions?
Is there a relation between cPoA and sPoA?