Improved Equilibria via Public Improved Equilibria via Public Service AdvertisingService Advertising

Maria-Florina Balcan

Joint with Avrim Blum and Yishay Mansour

Microsoft Research

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.

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.

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.

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)

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.

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.

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 ?

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.)

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)).

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)

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)

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)

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...

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)

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)

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.

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.

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

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

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)).

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?