Games, Auctions, Learning, and the Price of Anarchy va Tardos - - PowerPoint PPT Presentation

Games, Auctions, Learning, and the Price of Anarchy

Éva Tardos Cornell University

Games and Quality of Solutions

• Rational selfish

action can lead to

• utcome bad for

everyone

Tragedy of the Commons Model:

• Value for each cow

decreasing function of # of cows

• Too many cows: no

value left

Example: Routing Games

• Traffic subject to congestion delays
• cars and packets follow shortest path

Congestion game =cost (delay) depends only on congestion on edges

What is Selfish Outcome?

We will use: Nash equilibrium

– Current strategy “best response” for all players (no incentive to deviate)

Theorem [Nash 1952]:

– Always exists if we allow randomized strategies

Results for non-atomic games

Theorem 3 (Roughgarden-Tardos’02):

• In any network with continuous, nondecreasing

latency functions

cost of Nash with rates ri for all i cost of opt with rates 2ri for all i



Effort Information Advertisement

Today: Online Markets

Online markets use simple auctions for allocations How good is the resulting allocation?

Ideal Auction

Basic Auction: single item Vickrey Auction Player utility 𝑤𝑗 − 𝑞𝑗  item value –price paid Vickrey Auction

– Truthful

(second price)

– Efficient – Simple

$2 $5 $7 $3 $4

Pays $5

Some Simple Auctions

First/Second price GSP, etc All Pay, War of attrition

Effort Advertisement

Most not truthful

Truthful or Simple

Vickrey, Clarke, Groves (VCG) truthful, but not simple

• Assignment efficient

(maximizing social welfare)

• Payment welfare loss to
• thers

Online ads (Display/Search)

customized by information about user: Search term,

History of user, Time of the day, Geographic Data, Cookies, Budget

Millions of ads each minute and all different! Needs a simple and intuitive scheme

Goods Effort Information Advertisement

Should Work in Composition

Multiple Simple Auctions?

Second price auction truthful and simple, but…

Two simultaneous second price auctions? How about sequentially?

repeat offer to same or different seller

Not Not

Goal [Syrkanis-Tardos 2013]: Design mechanisms such that a market composed of such mechanisms is approximately efficient? Local: local mechanism efficient  Global mechanism efficient

Today Mixing Auction Types

First/Second price/All pay/ etc. Each interested in one or more items, but has different values Key idea: Auctions that price

First Price Auction: Good prices

Outcome of first price auction:

Each player i has a bid b’i, such that if current bids are b-i and prices are pj we get 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐𝑗

′, 𝑐−𝑗 ≥ 𝑤𝑗 𝑇𝑗 ∗ −

𝑞𝑦

𝑦∈𝑇𝑗

∗

Players

𝑞1 𝑞2 𝑞3 𝑞1

+

𝑞2

+

𝑇𝑗

∗

𝑐𝑗

′

 Pure Nash sets market clearing prices (Bikhchandani’96)

What are Good Prices?

Market clearing: 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐 ≥ 𝑤𝑗 𝑇𝑗

∗ −

𝑞𝑦

𝑦∈S𝑗

∗

 𝑇𝑗

∗ and  𝑗

Theorem: Market clearing prices guarantee socially

• ptimal outcome: maximizing 𝑤𝑗(𝑇𝑗)

𝑗

• f all

allocations (S1,S2,…) Proof: Each player has value  favorite items (𝑤𝑗 𝑇𝑗 − 𝑞𝑦 ) ≥ ( 𝑤𝑗 𝑇𝑗

∗ − 𝑞𝑦) 𝑦∈𝑇𝑗

∗

𝑦∈𝑇𝑗

𝒋 𝒋

Robust solution concepts

• Pure Nash of Complete Information is very brittle

– Pure Nash might not always exist – Game might be played repeatedly, with players using learning algorithms (correlated behavior) – Players might not know other valuations – Players might have probabilistic beliefs about values of

• pponents

What is Selfish Outcome (2)?

Do players find Nash?

if solution stable that is Nash But….. Finding Nash is hard algorithmic problem (Daskalakis-Goldberg-Papadimitrou’06)

No regret learning: do at least as well as any fixed

strategy with hindsight. If converges: Nash equilibrium…

Learning outcome

time

b11 b21 bn1

… Run Auction on ( b11, b21, …, bn1) Run Auction on ( b1t, b2t, …, bnt)

b12 b22 bn2

…

b13 b23 bn3

…

b1t b2t bnt

…

Maybe here they don’t know how to bid, who are the other players, … By here they have a better idea…

Vanishingly small regret for any fixed strat b’: ∑t utilityi(bit, b-it) ≥ ∑t utilityi(b’, b-it) – o(T) Simple randomized strategies guarantee vanishing regret (regret matching, multiplicative weight)

Bayesian Beliefs

𝑐𝑗(𝑤𝑗) 𝑐1(𝑤1) 𝑐𝑜(𝑤𝑜) 𝑤1 𝑤𝑗 𝑤𝑜 𝐺

1 ∼

𝐺𝑗 ∼ 𝐺

𝑜 ∼

Bayes-Nash Equilibrium: 𝐹𝑤−𝑗 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐 𝑤 ≥ 𝐹𝑤−𝑗 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐𝑗

′, 𝑐−𝑗 𝑤−𝑗

Direct extensions

• What if conclusions drawn for the Pure Nash equilibrium
• f the complete information setting could be directly

extended to these robust notions?

• Possible, but we need to restrict the type of analysis

Market Clearing Prices

Market clearing: Player i has a bid b’i that guarantees 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐𝑗

′, 𝑐−𝑗 ≥ 𝑤𝑗 𝑇𝑗 ∗ −

𝑞𝑦

𝑦∈𝑇𝑗

∗

Robust: bid b’i should not depend on b-i and other players. Do robust bids ever exists?

𝑞1 𝑞2 𝑞3 𝑞1

+

𝑞2

+

𝑇𝑗

∗

𝑐𝑗

′

Example: Approximately market clearing mechanism

Claim: First price auction for a single item is (1

2 , 1) smooth

User of value 𝑤𝑗 bid 𝑐′𝑗 = 1

2 𝑤𝑗, utility

Claim: 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐𝑗

′, 𝑐−𝑗 ≥ 1 2 𝑤𝑗 − 1𝑞𝑗

Proof

• Either wins and has utility 𝑤𝑗 − 𝑞𝑗 =

1 2 𝑤𝑗

• Or looses and hence price was 𝑞𝑗 ≥

1 2 𝑤𝑗

Examples of approximately market clearing auction games

• First price auction (1-1/e,1) approx

– See also Hassidim et al EC’12, Syrkhanis’12

• All pay auction (½,1)-smooth
• First position auction (GFP) is (½,1)-smooth
• Second price auction is (½,0,1)-smooth (no overbidding)
• Generalized second price (GSP) is (½,0,1)-smooth

Other applications include:

• public goods
• bandwidth allocation (Johari-Tsitsiklis),
• etc

Public Projects Bandwidth Allocation

Auctions with OK Prices: Smooth

Approximately market clearing: Player i has a bid b’i, such that if current bids are b-i and item prices are pj we get 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐𝑗

′, 𝑐−𝑗 ≥ 𝜇 𝑤𝑗 𝑇𝑗 ∗ − 𝜈

𝑞𝑦

𝑦∈𝑇𝑗

∗

Or just 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 , 𝑐−𝑗 ≥ 𝜇𝑃𝑄𝑈 − 𝜈 𝑞𝑦

𝑦 𝑗

𝑐𝑗

′, should not depend on b-i

smooth games Roughgarden’09 Theorem Smooth mechanism at Nash has socially approximately optimal outcome (off by at most factor

• f 𝜇/𝑛ax 1, 𝜈 )

Proof: at Nash 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐𝑗

′, 𝑐−𝑗) ≤ 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗(𝑐

Robust Price of Anarchy

Theorem(Syrkganis-T’13) Auction game (,)- smooth, then

• Price of anarchy is at most max(1, )/
• Preserved in Composition (assuming no

complements)

• Also true for mixed equilibria (and even

correlated equilibria = learning outcomes)

• Also true for games with uncertainty, assuming

player types are independent

See in a bit

Robust Price of Anarchy

Theorem (Syrkganis-T’13) Auction game (,)-smooth game, then

• Price of anarchy is at most max(1, )/
• Also true for learning outcomes (= coarse correlated

equilibria) and mixed equilibria Proof: let 𝑐1, 𝑐2, … , 𝑐𝑢, … sequence of bids 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐𝑢 ≥ 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗(𝑐𝑗

′, 𝑐−𝑗 𝑢 ) 𝑢 𝑢

(no regret) 𝑣𝑢𝑗𝑚𝑗𝑢𝑧𝑗 𝑐𝑗

′, 𝑐𝑗 𝑢 ≥ 𝜇𝑈 𝑃𝑞𝑢 − 𝜈 𝑞𝑦 𝑢 𝑦 𝑢 𝑢 𝑗

(smooth)

Robust Price of Anarchy

Theorem (Syrkganis-T’13) Auction game (,)- smooth game, then

• Price of anarchy is at most max(1, )/
• Preserved in Composition (assuming no

complements)

• Also true for mixed equilibria (and even correlated

equilibria = learning outcomes)

• Also true for games with uncertainty, assuming

player types are independent

Simultaneous Composition

Corollary: Simultaneous first price auction has price of anarchy of e/(e-1) if player values have no complements

• Simultaneous all-pay auction: price anarchy 2
• Mix of first price and all pay, price of anarchy ≤ 2

No complements  Submodular

item auctions:

Submodular: Marginal value for any allocation can only decrease, by getting more items: For all sets S  S’, and item x  S’ we have 𝑤 𝑇 + 𝑦 − 𝑤 𝑇 ≥ 𝑤 𝑇′ + 𝑦 − 𝑤(𝑇′)

Across Mechanisms (fractionally subadditive)

𝑵𝟐 𝑵𝟑 … …

e.g., submodular 𝑤𝑦

𝑘 = 𝑤 𝑇𝑦 𝑘 − 𝑤 𝑇−𝑦 𝑘

: marginal value for ordering j, where 𝑇𝑦

𝑘 is prefix till x, and 𝑇−𝑦 𝑘 prefix without x.

Simultaneous Composition

Theorem (Syrkganis-T’13): simultaneous item auctions where each is (,)-smooth and players have fractionally subadditive valuations, then composition is also (,)-smooth Proof: valuation 𝑤 𝑇 is fractionally subadditive  maximum of

linear functions: 𝑤 𝑇 = max

𝑘

𝑤𝑦

𝑘 𝑦∈𝑇

• optimal allocation 𝑇1

∗, 𝑇2 ∗, … values 𝑤𝑗 𝑇𝑗 ∗ =

𝑤𝑦

𝑘𝑗 𝑦∈𝑇𝑗

• User i bids in auction x for value 𝑤𝑦

𝑘𝑗

• Real value 𝑤𝑗 𝑇 ≥

𝑤𝑦

𝑘𝑗 𝑦∈𝑇

Conclusion: Designing for Selfish Users

• Selfish users can ruin social welfare (Tragedy of

the Commons)

• With care, we can design for selfish use and

mitigate the welfare loss

• Guarantees very robust (Nash, learning,

uncertainty)