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Improved Equilibria via Improved Equilibria via Public Service Public Service Advertising Advertising Maria-Florina Balcan Joint with Avrim Blum and Yishay Mansour Microsoft Research

Improved Equilibria via Public Service Advertising Maria-Florina Balcan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

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Page 1: Improved Equilibria via Public Service Advertising Maria-Florina Balcan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

Improved Equilibria via Public Improved Equilibria via Public Service AdvertisingService Advertising

Maria-Florina Balcan

Joint with Avrim Blum and Yishay Mansour

Microsoft Research

Page 2: Improved Equilibria via Public Service Advertising Maria-Florina Balcan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

Good equilibria, Bad equilibriaGood equilibria, Bad equilibriaMany games have both bad and good equilibria.

• In some places, everyone throws their trash on the street. In some, everyone puts their trash in the trash can.

• In some places, everyone drives their own car. In some, everybody uses and pays for good public transit.

Page 3: Improved Equilibria via Public Service Advertising Maria-Florina Balcan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

Good equilibria, Bad equilibriaGood equilibria, Bad equilibriaMany games have both good and bad equilibria.

s

t

1n-

• Player i wants to get from si to ti. • all players share cost of edges they use with

others.

Fair cost-sharing. • n players in directed graph G, each edge e costs ce.

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Good equilibria, Bad equilibriaGood equilibria, Bad equilibriaMany games have both good and bad equilibria.

s

t

1n-

Good equilibrium: all use edge of cost 1.

• Player i wants to get from si to ti. • all players share cost of edges they use with

others.

Fair cost-sharing.

(paying 1/n each)

• n players in directed graph G, each edge e costs ce.

Page 5: Improved Equilibria via Public Service Advertising Maria-Florina Balcan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

Good equilibria, Bad equilibriaGood equilibria, Bad equilibriaMany games have both good and bad equilibria.

s

t

1n-

Good equilibrium: all use edge of cost 1.

Bad equilibrium: all use edge of cost n-.

• n players in directed graph G, each edge e costs ce. • Player i wants to get from si to ti.

• all players share cost of edges they use with others.

Fair cost-sharing.

(paying 1/n each)

(paying 1- ²/n each)

Page 6: Improved Equilibria via Public Service Advertising Maria-Florina Balcan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

Good equilibria, Bad equilibriaGood equilibria, Bad equilibriaMany games have both good and bad equilibria.

Fair cost-sharing.

• Player i wants to get from si to ti. • all players share cost of edges they use with

others.

…1 1 1 1

s1 sn

t

0 00

k ¿ n

cars

Subway/shared van

Bad eq. result of natural dynamics:

• players entering one at time• minimizing regret

v

• n players in directed graph G, each edge e costs ce.

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Good equilibria, Bad equilibriaGood equilibria, Bad equilibria

Standard motivation for PoS:

Price of Stability (PoS): ratio of best Nash equilibrium to OPT. E.g., for fair cost-sharing, PoS is log(n), whereas PoA is n.

If a central authority could suggest a low-cost Nash (throw away your trash, ride public transit), and everyone followed the suggestion, then this would be stable.

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Good equilibria, Bad equilibriaGood equilibria, Bad equilibria

What if only some fraction will pay attention?

• Can the authority guide behavior to a good state?

• Will it just snap back? How does this depend on ?

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Main ModelMain Model

1. Authority launches advertising, proposing joint action sad.

2. Remaining (non-receptive) players fall to some arbitrary equilibrium for themselves, given play of receptive players.

3. Campaign wears off. All players follow best-response dynamics to an overall Nash equilibrium.

• Only consider potential games.

Each player i follows with probability . Call players that follow receptive players

Notes:

• Focus on social cost

0. n players initially playing some arbitrary equilibrium.

(Except we use makespan for load balancing.)

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Main ResultsMain Results

• If only a constant fraction of the players follow the advice, then we can still get within O(1/) of the PoS.• Extend to cost-sharing + linear delays.

• For any < 1, an fraction is not sufficient. Ratio to OPT can still be unbounded.

(PoS = log(n), PoA = n)

(PoS = 1, PoA = 1)

(PoS = 1, PoA = (n2))

• Threshold behavior: for > ½, can get ratio O(1), but for < ½, ratio stays (n2). (assume degrees (log n)).

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Fair Cost SharingFair Cost Sharing

…1 1 1 1

s1 sn

t

0 00

k

Note: this is best you can hope for. E.g., k =2n.

If only a constant fraction of the players follow the advice, then we get within O(1/) of the PoS.

(PoS = log(n), PoA = n)

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Fair Cost SharingFair Cost Sharing

- Moreover, this option is guaranteed to be at least as good as if other NR players didn’t exist.

If only a constant fraction of the players follow the advice, then we get within O(1/) of the PoS.

(PoS = log(n), PoA = n)

- In any NE a non-receptive player i, can’t improve by switching to his path Pi

OPT in OPT.

- Advertiser proposes OPT (any apx also works)

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Fair Cost SharingFair Cost Sharing

If only a constant fraction of the players follow the advice, then we get within O(1/) of the PoS.

(PoS = log(n), PoA = n)

- In any NE a non-receptive player i, can’t improve by switching to his path Pi

OPT in OPT.

- Advertiser proposes OPT (any apx also works)

Page 14: Improved Equilibria via Public Service Advertising Maria-Florina Balcan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

Fair Cost SharingFair Cost Sharing

If only a constant fraction of the players follow the advice, then we get within O(1/) of the PoS.

(PoS = log(n), PoA = n)

- In any NE a non-receptive player i, can’t improve by switching to his path Pi

OPT in OPT.

- Advertiser proposes OPT (any apx also works)

- Calculate total cost of these guaranteed options.

- Rearrange sum...

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Fair Cost SharingFair Cost Sharing

If only a constant fraction of the players follow the advice, then we get within O(1/) of the PoS.

(PoS = log(n), PoA = n)

- In any NE a non-receptive player i, can’t improve by switching to his path Pi

OPT in OPT.

- Advertiser proposes OPT (any apx also works)

- Calculate total cost of these guaranteed options.

- Take expectation, add back in cost of receptives: get O(OPT/).(End of phase 2)

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Fair Cost SharingFair Cost Sharing

- Finally, in last phase, std potential argument shows behavior cannot get worse by more than an additional log(n) factor.(End of phase 3)

If only a constant fraction of the players follow the advice, then we get within O(1/) of the PoS.

(PoS = log(n), PoA = n)

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Cost Sharing, ExtensionCost Sharing, Extension

- Still get same guarantee, but proof is trickier

+ linear delays:

- Problem: can’t argue as if remaining NR players didn’t exist since they add to delays

- Define shadow game: pure linear latency fns. Offset defined by equilib at end of phase 2.

# users on e at end of phase 2

- Behavior at end of phase 2 is equilib for this game too.

- Show

- This has good PoA.

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Party affiliation gamesParty affiliation games• Given graph G, each edge labeled + or -.• Vertices have two actions: RED or BLUE.

Pay 1 for each + edge with endpoints of different color, and each – edge with endpoints of same color.

• Special cases:

+

+

+

--

• All + edges is consensus game. • All – edges is cut-game.

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Party affiliation gamesParty affiliation games OPT is an equilibrium so PoS = 1.

But even for consensus, PoA = (n2)

Clique with perfect matching removed

all edges labeled plus

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Party affiliation gamesParty affiliation games(PoS = 1, PoA = (n2))

- Threshold behavior: for > ½, can get ratio O(1), but for < ½, ratio stays (n2). (assume degrees (log n)).

- Same example as for consensus PoA, but sparser across cut. Players “locked” into place.

(lower bound)

Degree (1/2 - )n/8 across cut

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Party affiliation gamesParty affiliation games

- Split nodes into those incurring low-cost vs those incurring high-cost under OPT.

(upper bound)

- Advertising strategy = follow OPT.

- Show that low-cost will switch to behavior in OPT. For high-cost, don’t care.

- Cost only improves in final best-response process.

(PoS = 1, PoA = (n2))

- Threshold behavior: for > ½, can get ratio O(1), but for < ½, ratio stays (n2). (assume degrees (log n)).

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Conclusions and Open Conclusions and Open QuestionsQuestions

Analyze ability of a central authority to guide behavior to a good equilibrium even if only ® fraction of players are paying attention.

Main Open Question: Get around problem of natural dynamics

converging to poor equilibrium without central authority by giving players more information about the game?